{"Dojo":{"knowledgePointLabels":{"grad_mathphys":{"probability":{"conditional_probability":"Conditional probability and sequential sampling","random_variables_distributions":"Random variables and probability distributions","expectation_covariance":"Expectation, variance, and covariance","truncated_normal_observations_and_standardization":"Truncated-normal observations and standardization"},"statistics":{"estimation":"Parameter and unbiased estimation","hypothesis_testing":"Hypothesis testing, significance, and power","confidence_intervals":"Confidence intervals and sample size","regression_design_survival":"Regression, experimental design, and survival analysis"},"linear_algebra":{"maps_rank_kernel":"Linear maps, rank, image, and kernel","eigen_diagonalization":"Eigenvalues, eigenspaces, and diagonalization","jordan_minimal_polynomial":"Minimal polynomials and Jordan structure","operators_algebras":"Matrix operators and generated algebras","inner_products_projections":"Inner products, projections, and quadratic forms","singular_values_least_squares":"Singular values, polar decomposition, and least squares","matrix_functions_quadratic_forms":"Matrix functions, anticommutation, and inertia indices","parameterized_inverses":"Invertibility and inverses of parameterized matrices","finite_field_subspaces":"Subspace counting over finite fields","jordan_similarity_and_intertwining_matrices":"Jordan similarity and intertwining matrices","schur_complements_and_block_matrices":"Schur complements and block matrices","matrix_inversion_and_verification":"Matrix inversion and verification","polynomial_spaces_eigenvalues_and_bifurcation":"Polynomial spaces, eigenvalues and parameter bifurcation","jordan_forms_and_generalized_eigenvectors":"Jordan forms and generalized eigenvectors","linear_maps_and_canonical_forms":"Linear maps and canonical forms"},"algebra":{"groups_actions":"Groups, group actions, and Sylow theory","representations":"Representations, irreducible decomposition, and invariants","rings_ideals":"Rings, ideals, and quotient rings","fields_galois":"Field extensions and Galois theory","modules_lattices":"Modules, lattices, and invariant factors","finite_fields_polynomials_forms":"Finite fields, polynomials, and quadratic forms","chinese_remainder_theorem_and_tripotents":"Chinese remainder theorem and tripotents","commutative_rings_cyclic_vectors_and_invertibility":"Commutative rings, cyclic vectors and invertibility"},"analysis":{"sequences_series":"Sequences, series, and convergence","uniform_convergence":"Uniform convergence and function sequences","measure_integration":"Measure, improper integration, and integrability","asymptotics":"Asymptotics, error bounds, and leading terms","differentiation":"Differentiation, convexity, and implicit functions","critical_points_global_minima":"Critical-point classification and global minima","vector_calculus_integral_theorems":"Vector calculus, circulation, and integral theorems"},"complex":{"holomorphic":"Holomorphic and harmonic functions","residues_contours":"Residues, contours, and Fourier integrals","singularities":"Complex singularities and branches"},"functional":{"banach_hilbert":"Banach/Hilbert spaces and weak convergence","compact_operators":"Compact operators and operator convergence","integral_pde":"Integral operators, variational methods, and stability"},"geometry":{"coverings_degree":"Covering spaces, degree, and lifting","topology_fundamental_group":"Fundamental groups and quotient spaces","manifolds_curvature":"Manifolds, curvature, and differential geometry","homology_cohomology":"Homology, cohomology, and Euler characteristic","metric_compactness":"Metric spaces, compactness, and continuity","surface_bundles_surfaces":"Surfaces, circle bundles, and bundle sections","degree_maps":"Mapping degree and boundary extension","algebraic_geometry":"Projective and rational maps in algebraic geometry","topological_groups":"Topological groups and group extensions"},"differential_equations":{"ode_boundary":"Ordinary differential equations and boundary-value problems","pde_fourier":"PDEs, Fourier methods, and separation of variables","heat_poisson":"Heat kernels and Poisson equations","initial_value_dynamical_systems":"Initial-value problems, dynamical systems, and resonance","complex_line_integrals_and_resonant_forcing":"Complex line integrals and resonant-forcing differential equations"},"optimization":{"constrained_convex":"Constrained optimization and strong convexity","integral_geometry":"Multiple integrals and surface area"},"physics":{"classical_orbital":"Classical mechanics, central forces, and orbits","rigid_body_vibration":"Rigid bodies, vibration, and resonance","quantum_operators":"Quantum mechanics, operators, and perturbation","quantum_states":"Quantum states, ground states, and excitations","thermodynamics_statistical":"Thermodynamics, statistical mechanics, and partition functions","electromagnetism_waves":"Electromagnetic fields, waves, and boundary conditions","electrostatics":"Electrostatics, spherical potentials, and Poisson equations","relativity_particle_kinematics":"Relativistic particle kinematics and invariant mass","optics_interference_absorption":"Optics, interference, absorption, and lenses","spring_vibration":"Spring systems and vibrational response","classical_statics_constraints":"Constraints and statics in classical mechanics","lagrangian_dynamics":"Lagrangian dynamics and geometric constraints","particle_interactions":"Particle interactions and event classification","quantum_foundations":"Quantum foundations, mixed states, and local realism","lattice_condensed":"Lattices, phonons, and condensed-matter models","measurement_inverse":"Physical measurement and inverse problems","variable_mass_dynamics":"Variable-mass dynamics and momentum conservation","precession_angular_momentum":"Precession, angular momentum, and damping torque","electromagnetic_circuits_induction":"Magnetostatics, circuits, and electromagnetic induction","charged_particles_plasma":"Charged-particle motion and plasma drifts","quantum_scattering_boundaries":"Quantum scattering, tunneling, and boundary conditions","quantum_geometric_phase":"Quantum geometric phase and adiabatic parameter evolution","quantum_many_body_statistics":"Quantum gases, degeneracy pressure, and quantum statistics","thermodynamic_processes_cycles":"Thermodynamic processes, cycles, and phase equilibria","transport_diffusion_entropy":"Diffusive transport and mixing entropy","fluid_dynamics_surface_waves":"Fluid dynamics, settling, and surface waves","radiative_transfer":"Radiative transfer and optical depth","optical_spectroscopy_detection":"Spectroscopic measurement and photon detection","angular_momentum_tides":"Angular momentum, tides, and rotational dynamics","nuclear_decay_chain":"Nuclear decay, decay chains, and particle balance"},"chemistry":{"atomic_inorganic_structure":"Atomic electronic structure and inorganic chemistry","acid_base_redox_equilibria":"Acid-base, redox, and chemical equilibria","thermochemical_phase_equilibrium":"Chemical thermodynamics and multiphase equilibrium","kinetics_reaction_networks":"Chemical kinetics and reaction networks","spectroscopy_quantitative_analysis":"Absorption spectroscopy and quantitative analysis","physical_kinetics_interfaces":"Physical chemistry, kinetics, and interfaces","molecular_spectroscopy":"Molecular rovibrational spectroscopy and predissociation"},"materials":{"crystallography_mineral_structures":"Crystal structures, mineral polymorphs, and diffraction"},"earth":{"geophysics_seismology_rheology":"Geophysics, elastic waves, and rheology","plate_tectonics_geodesy":"Plate tectonics, kinematics, and geodesy","rock_mechanics_effective_stress":"Rock mechanics, effective stress, and Mohr circles","petrology_phase_equilibria":"Rock identification, mineral equilibria, and phase diagrams","structural_sedimentary_geology":"Structural geology, sedimentary processes, and stratigraphy","isotopes_geochronology_biostratigraphy":"Isotopes, geochronology, and biostratigraphy","planetary_atmosphere_climate":"Planetary atmospheres, climate, and ocean environments","planetary_geology_interior":"Planet formation, geology, and interior structure","planetary_formation_magnetospheres":"Planet formation and magnetospheric science","geology_geochronology_interior":"Geology, geochronology, and planetary interiors","biogeochemical_climate_feedback":"Carbon cycling, weathering, and climate feedback"},"engineering":{"signal_processing_filters":"Discrete signals, spectra, and filter design"},"real_analysis":{"continuous_extension_and_uniform_convergence":"Continuous extension and uniform convergence","improper_integrals_and_differentiation_under_the_integral_sign":"Improper integrals and differentiation under the integral sign","uniform_continuity_series_and_multiple_integrals":"Uniform continuity, function series and multiple integrals"},"differential_geometry":{"critical_points_and_rank_of_parametrized_maps":"Critical points and rank of parametrized maps"},"quantum_mechanics":{"canonical_transforms_ladder_operators_and_uncertainty":"Canonical transforms, ladder operators and uncertainty","two_particle_exchange_symmetry":"Two-particle exchange symmetry","spectral_problems_boundary_conditions_and_position_representation":"Spectral problems, boundary conditions and position representation"},"statistical_mechanics":{"finite_level_partition_functions_magnetization_and_heat_capacity":"Partition functions, magnetization and heat capacity of finite-level systems","coupled_oscillators_classical_quantum_statistics":"Classical and quantum statistics of coupled oscillators","phase_stability_and_orbital_entropy":"Phase stability and orbital-state entropy"},"complex_analysis":{"contour_integrals_residues_and_laurent_expansions":"Contour integrals, residues and Laurent expansions"},"calculus":{"variational_curves_and_stationarity":"Variational calculus and stationarity of curves"},"topology":{"metric_spaces_closure_and_connectedness":"Metric spaces, closure and connectedness"},"electromagnetism":{"maxwell_integral_laws_and_conductor_boundary_conditions":"Maxwell integral laws and conductor boundary conditions"},"crystallography":{"bragg_condition_uniaxial_strain_and_ewald_sphere":"Bragg condition, uniaxial strain and the Ewald sphere"},"recovered":{"p_a2662ba733e2f90e":"Probability and sample means from bounded scale families","p_907f180332ddb14f":"Probability, maximum order statistics and scale transformations","p_25c2cb6d7f78ece3":"Mathematical statistics, upper quantiles and one-sided confidence bounds","p_40e1ff432c7a3f1e":"Parameter-dependent support and uniformly most powerful tests","p_885e5914b0f3d716":"Power functions and required sample sizes","p_05fc55b0fc7f2beb":"Hypothesis testing and two-stage decisions","p_5d56b5897b1807c8":"Differences in proportions and confidence intervals for two independent samples","p_c273c36c43dcccd1":"Contingency tables, event coding and log odds ratios","p_f17b386b87a7c444":"Correlation, stratification and influential points","p_ade53592c1a41f9a":"Multiple regression, partial coefficient of determination and test for additional variables","p_186793e8eea964c9":"Change of variables and moments of sums of independent variables","p_be2a9a63f0643b37":"Convolution and cumulative distribution function","p_f5f5737e1fba45b9":"Lower quantiles and uniqueness of the critical value","p_8c067bc3e0c14051":"Likelihood ratio and uniformly most powerful test","p_a0aae8e6328e1e52":"Power function and boundary values","p_39ac798808c7d89f":"Binary classification, sensitivity, specificity and conditional dependence","p_60bfc83e891e20f8":"Combined judgment, errors in series and parallel","p_2fa8b1ad5e242842":"Bayes' theorem and posterior probability","p_65700c9836ebac64":"Posterior probability curve, monotonicity and curvature","p_7a15349fe066b127":"Posterior probability and compatibility of operating conditions","p_f069b1992811d90c":"Cumulative distribution function","p_9c90c4e05874d064":"Existence conditions for the mean and the median","p_634de53760be1160":"Normalization of probability density","p_6ecdad23a3bf6d26":"Probability of the relative magnitudes of independent variables and parameter dependence","p_55de58e54d653d3e":"Nonadjacent selection and endpoint constraints","p_96da20408e5cdc2a":"Common-mode noise removal and covariance transformation","p_0554e6006079b19e":"Imputation of missing values and transformation of descriptive statistics","p_22105bb5ee3422a4":"Within-group variation, between-group variation and coefficient of variation","p_8dec87de7c656728":"Standard error of the mean of repeated samples","p_72f4723757861769":"Conditional variance after selective recording","p_ef38fba021baf0f4":"Comparison of incidence rates per observation time and cumulative risk","p_2a4c088a307dc7d6":"Maximum likelihood estimation of an exponential lifetime model with corrected acceleration conditions","p_0174f112277ebdb8":"Observed information and variance estimation using the delta method","p_c36557210734398c":"Joint attention and attentional synchrony","p_157e49f640a2d16e":"Action selection and bimanual coordination","p_5ad5c0b50b0069f4":"Condition dependence of social facilitation","p_8ebd14855f33c315":"Continuity of research consent","p_10cdd93930875876":"Spatiotemporal conditions for apparent motion","p_9bbeaed83d4d1ee4":"Identification of conditioned stimuli and control conditions","p_75488b1991f505f1":"Zone of proximal development and evaluation of support","p_ec5e7fc7fdd51ef0":"Operationalization of self-attention and self-evaluation","p_a890c241643bef1f":"Distinguishing meaning from causality in cross-cultural comparisons","p_deb606dc7b9a8a5c":"Interpretation of sequential requests and compliance","p_969a5a7f6da321a9":"Self-presentation and private attitudes","p_f73eae14f869d99b":"Sense of body ownership and sense of agency","p_291309b432143777":"Decision criteria in signal detection theory","p_bfb4ea17bf089981":"Temporal binding window and simultaneity judgment","p_dce24eff2932aac4":"Sensation and context in sound evaluation","p_be2855939c704595":"Choice overload and conditional dependence","p_eb21f9f14799c14f":"Attraction effect and compromise effect","p_86b897b10b9cd289":"Understanding false belief and self-knowledge","p_3e40089baab7777f":"Latent inhibition and blocking","p_2598667f4579b6f6":"Convergent validity and common method","p_2f5aab526335513e":"Generalized trust and predictions about specific individuals","p_c8d09d46ca975fd8":"Competing explanations of group polarization","p_f0657f4480e0fe17":"Cognitive style and confidence calibration","p_06104a4613026c7d":"Visual field manipulation and functional asymmetry","p_d58043b2a6d8463a":"Confounding, mediation and total effect","p_1971b51192c2897c":"Time measurement in online experiments","p_2081e94dc43e54af":"Infant anticipatory looking and transfer","p_8c9b4f479125b9f3":"Rolling motion of an eccentric rigid body","p_8c437c5cb5707c83":"Composite dielectrics and variable capacitance","p_b3e85a055b0b5d5b":"Response function of a nonideal gas","p_7c5a67f0b30f5355":"Functional discrimination of learning principles","p_dced7ee47751771d":"Learning theory and family systems","p_accee87e7f8134d4":"Ethics of psychological support","p_6f1f21fb37131d21":"Integration of developmental assessment","p_d01e857778713ac2":"Biopsychosocial support plan","p_194690ad32ea258e":"Rating scale method","p_0bf6011f0f5e8b84":"Work-sample test","p_65211e50a73fcd0a":"Interpretation of neuropsychological tests","p_827e4351e9656611":"Experiential process and focusing","p_885e3b5c6f4b9bb1":"The six movements of focusing","p_b15d4aeb58e99990":"Expectation effects and causal inference","p_9fb9c9107f68642b":"Research ethics and authorship","p_de412ccdf20da0fb":"Multiple pathways to equivalent outcomes model","p_dcba1635c2087e47":"Response coefficients of a nonideal fluid","p_5565ca58c7822ea3":"Identities for heat capacity differences and stability boundaries","p_4006251a5f561b14":"Centroid and moment of inertia of a rectangular frame","p_ec2efe51c7c05d5b":"Equations of motion for translation and rotation","p_10c150f3fda982b6":"Elimination of constraints and angular motion","p_75aa2159eb7b3572":"Rolling of a disk and the velocity of a point","p_f29be1e451e0852b":"Maintaining contact and static friction conditions","p_115674ac2b39e3c9":"Electric fields and free charges in a nonuniform dielectric","p_7ee7b588c9330ad3":"Potential difference and stored energy","p_599ddbf2e7137f33":"Position-dependent capacitance","p_a7394d483277658c":"Potential curve of an isolated electrode","p_bf75831c17eff0e4":"Coaxial dielectrics, fixed free charge and grounding current","p_fc6a7c2aecb86c29":"One-dimensional lattice, next-nearest-neighbor coupling and dispersion relation","p_f25c8dfd7221146e":"Non-ideal gases, response coefficients and heat capacity difference","p_33b2d825497bd36e":"Ideal gas, entropy and maximum work design","p_8f42d4d53006a16d":"Overexpansion cycle, two-stage heat rejection and efficiency comparison","p_5c3fd4639f3ba753":"Equation of motion for a lattice with next-nearest-neighbor interactions","p_05b87c77f1df34be":"Nonmonotonic dispersion relation and band edges","p_4b6cbe1b803c7cc8":"Lattice equations of motion with internal degrees of freedom","p_ceb4906aa2f6b9d4":"Bloch amplitude matrix and reduced mass","p_dde20bc60cb28b89":"Two-branch dispersion and forbidden frequencies of a locally resonant lattice","p_fcfca1a899975450":"Energy of a biased harmonic oscillator","p_c300d1346ff6e52d":"Time evolution of a two-level superposition and its expectation value","p_d89cb0ddcd1b2875":"Temperature-dependent heat capacity and entropy increase","p_29010e907dbe5c3e":"The second law by component and the work limit of a refrigerator","p_4fc896451e64ce87":"Momentum balance with simultaneous inflow and outflow","p_8be08e8a1e792b03":"Integration and limits of variable-mass motion with accretion","p_3968598e050490a7":"Motion of an open system under gravity and acceleration conditions","p_1a48544b680176de":"Variable-mass motion with time-dependent thrust and linear drag","p_8630de8f60b0399e":"Sign convention for couples and rotational equations of motion","p_23c0a1a669b07fe3":"Unstable modes and initial conditions in uncontrolled systems","p_7b00502c1944a3e4":"Asymptotic stability of proportional control systems with damping","p_3d3051943965eaaa":"Stability limits of proportional-derivative control with finite response time","p_08283846957e3145":"Momentum balance in variable-mass systems","p_2b42b9fd053f8cad":"Integration and arrival conditions for a variable-mass system","p_d00ac16958923d93":"Exponential mass loss and velocity extrema","p_9a1b81c71cd6a99b":"Matter waves and electrostatic deceleration","p_2a86828073e1fe7b":"Inverse estimation of accelerating voltage from time of flight","p_b55f5fa534308d15":"Invariance of the speed of light and Lorentz transformations","p_7ca150e68164aa56":"Inverse problem of relativistic matter waves","p_ece158eba0da2a98":"Capacitance of a spherically symmetric dielectric","p_1d3226a9ae3da71d":"Field energy at constant voltage","p_c273be1269a9f545":"Interface displacement and charge response","p_0a512acc6b36cf0a":"Virtual work and interface forces with a power supply","p_fa41cd144a488e43":"Discharge transient response and electric force","p_303987a0c09eab44":"Control of the average interfacial force by alternating-current resonance","p_be2cef63e68b4001":"Voltage phase and energy exchange at resonance","p_0f7ca2d73f1d908e":"Temperature-dependent heat capacity and maximum work","p_f0da3f19778b2cb9":"Maximum efficiency of a heat engine with heat leakage","p_13eb29550710eaea":"Volume-dependent pressure and definition of constant-pressure heat capacity","p_ab29f3d127c52a52":"Maxwell relations and the volume derivative of internal energy","p_4a06c56c1754a607":"Heat capacity of photons and conservative elastic mechanisms","p_781c30caeb356b19":"Inverse Carnot cycle including conservative forces","p_b0af9fe49571b60e":"Clausius equality and coefficient of performance of a refrigerator","p_1e49349b57fde886":"Thermodynamics","p_167b63f404d6d940":"Wave optics","p_db089df1e56f968d":"Matter waves","p_2204b91e61ac36f5":"Special relativity","p_18e027a0430960e1":"Rigid-body dynamics","p_14970aa317b8e96d":"Moment of inertia of a nonuniform disk","p_ff7e36a960d9a1e9":"Coupled rotational motion with a torsion spring","p_9e52815973ca16f9":"Moment of inertia of a compound roller","p_93aaff288c02cb6f":"Sliding motion on a moving belt","p_e3e6f43b80086e18":"Transition from slip to pure rolling","p_def4334f45d664a0":"Frictional heating and work of the drive mechanism","p_0ec092c8f9dd504b":"Velocity excitation by out-of-phase support points","p_1cb5fef0817306b1":"Amplitude curve and full width at half maximum","p_7c81e4981a6efdfa":"Displacement reconstruction and phase error at high frequencies","p_06ae2a46e191429c":"Velocity measurement and calibration at low frequencies","p_486b00b4a6ed5cac":"Potential and electric field on the axis of two coaxial rings","p_0185756d7b40aaa7":"Far-field expansion and elimination of the quadrupole term","p_8734f7538d276e35":"Geometric limits of charge distributions","p_cf1f662ff7ba20ed":"Design of a test charge with a specified reflection position","p_df2ac328a1a0eea4":"Axial small oscillations and three-dimensional stability","p_02650c52f4cc5b51":"Bound currents of nonuniform axial magnetization","p_4742d42421168dba":"Distinction between H and B in cyclic magnetization","p_6d9576f29d78852f":"Toroidal coil with radially varying permeability","p_4b23e21df404826b":"Equivalent permeability and error in the mean-radius approximation","p_351f64e78d219b15":"Helmholtz free energy and conjugate variables","p_80fde45c415e1ebf":"Internal energy and heat of isothermal deformation","p_52e4815387a7131d":"Finite isothermal extension specified by final tension","p_4a6541e71cdd189b":"Thermodynamics: temperature-dependent specific heat and adiabatic process","p_e6d0bc5946448fe2":"Thermodynamics: uniqueness of adiabatic compression and work","p_3b37753ac6c730a4":"Thermodynamics / heat absorption in a regenerative heat engine","p_b9f5105d83694760":"Thermodynamics: external dissipation and heat rejection balance","p_d14b43d4c762ed1d":"Thermodynamics/useful work and upper bound on efficiency","p_3774cf076c54b70b":"Thermodynamics: generalized work and thermodynamic potential","p_212dd36e3507403e":"Thermodynamics / internal energy and reversible heat in thermoelasticity","p_c367472f960fbb6e":"Thermodynamics: isothermal energy balance of a thermoelastic element","p_317a75ad66b0675e":"Mechanics: translational coordinate systems and coupled equations of motion","p_a75e65b316ac1ef5":"Mechanics: common-mode and differential-mode separation from the input","p_92ce930ce38571f4":"Mechanics / base excitation and absolute displacement transmissibility","p_8edc2b74ed3f8f20":"Mechanics: plotting oscillation transmissibility and comparing damping","p_72bff2d446711019":"Mechanics: high-frequency vibration isolation","p_2cb60d523e8e9dc5":"Mechanics: amplitude and phase errors of a low-frequency accelerometer","p_750e21b16b9a5084":"Thermodynamics: free energy and thermoelasticity","p_343b8a98bd36fd1a":"Thermodynamics: state dependence of internal energy","p_ae9b71825843b83d":"Thermodynamics: work and heat in an isothermal process","p_4b769fa1faa4287d":"Mechanics: moment of inertia and the parallel axis theorem","p_063b10bc53a2dd42":"Mechanics: constrained rotational and translational motion","p_0c797f55b86984dc":"Mechanics: internal torsional coupling and relative oscillation","p_5e6ed63e72fb6cd8":"Mechanics: constraints and force balance at a moving contact surface","p_3331d37d333c6bfa":"Mechanics: internal torque and effective moment of inertia","p_8ba3046b778f33a5":"Mechanics: friction-coupled synchronous motion","p_015f2dae42cf8e58":"Mechanics: dissipation and work done by a moving boundary","p_d2544b7340bf468c":"Electromagnetism: steady current and heat generation in a nonuniform conductor","p_41278a6fd4448ddd":"Electromagnetism: Ampere's law and extrema of the magnetic field","p_2c58ec1222969f21":"Electromagnetism: one-dimensional equation of an alternating field","p_ee20dfaa96627de9":"Electromagnetism: good conductor approximation and penetration depth","p_2cdeaf7256c2d7c8":"Electromagnetism: asymmetric boundary conditions and net current","p_015eb52466aaa254":"Electromagnetism: low-frequency asymptotic form of AC losses","p_f17f35aa36aa8f60":"Electromagnetism: high-frequency asymptotic form of AC losses","p_804ae1622cad060b":"Thermodynamics: consistency of the equation of state and heat capacity","p_03a54ea374c2cbb7":"Thermodynamics: identification of the equation of state from reversible adiabatic measurements","p_f448c961945d70ce":"Thermodynamics: isothermal expansion by electrical work and entropy production","p_a85988de15267dca":"Thermodynamics: conserved quantities and initial pressure of an isolated system","p_766f7d862032a37b":"Thermodynamics: final state of identical gases and total entropy difference","p_f2eb0b6df7dbe7c2":"Thermodynamics: distinction between isobaric conditions and reversibility","p_5cd61d4337e5722b":"Thermodynamics, equations of state and specific heat","p_c28bc9330c237e2f":"Mechanics, angular momentum and torsional oscillation","p_fba907a76ea82c47":"Mechanics: collisions and free oscillations at different frequencies","p_080d10f68275caea":"Mechanics: inverse problems of inelastic collisions and damped oscillations","p_e895dd1f5fe1bf89":"Mechanics: collision with a moving boundary and conditions for meeting again","p_50cd98c54b5a2686":"Collisions, angular momentum and torsional vibration","p_1e060567541e4e8d":"Simple harmonic oscillation with phase and first return","p_c78c47e33c1f47a6":"Two-body collision: momentum and coefficient of restitution","p_31e6969d5dc38a8c":"Two harmonic oscillations with different equilibrium positions","p_a7d1818ea9f8fd5b":"Velocity immediately before a subsequent collision, including the coefficient of restitution","p_dc539e355ec58d0b":"Identification of the coefficient of restitution from the time of the next collision","p_3f287415ecd03064":"A compressed beam on an elastic foundation and its equation of motion","p_cec359e107b0b4c9":"Sine modes and the method of separation of variables","p_d97fd63d3d8fb986":"Critical load of a compressed beam and minimization of discrete modes","p_14783ff86a6b4440":"Double zero modes and nonuniqueness of neutral shapes","p_7d14f0f7bdfa4b8a":"Rotational electromotive force in a nonuniform magnetic field","p_e59bae917102428e":"Generator braking torque and steady-state angular velocity","p_3348ac0f3836a8d3":"Electromechanical energy balance and variable-load optimization","p_01107d23a879cce2":"Coupling of two conductors, a circuit and motion in a reversed magnetic field","p_9c9bd4bafd4d6194":"Conserved linear combinations and reduction to a first-order equation","p_21d2d14f71d88d19":"Current relaxation, residual motion and total heat generation","p_8577d61c35f9f1c8":"Collisions, conservation of angular momentum and torsional vibration","p_862e1ede7c37de4b":"Initial phase, first return and collision loss","p_f12a2e3ef1f8a811":"Physical pendulum and parallel-axis theorem","p_e88f4f375a7ce3ca":"Finite-amplitude pendulum and constrained optimization","p_9f4a111d7ae8eb7c":"Motional electromotive force and rotating conductor","p_abfa5ac68debd6ee":"Electromagnetic braking, steady rotation and stability","p_5711119d46c7391d":"Energy balance and load power maximization","p_a55b12e1f78686ce":"Circuit equations and electromotive force due to relative motion","p_9761fe0da8f9170c":"Electromagnetic coupling, reduced mass and Conserved quantities","p_8e2d91d40d5700bf":"Coupled transient response, terminal velocity and dissipated heat","p_320de2c87658d1c2":"Composite rigid bodies, center of mass and moment of inertia","p_ea2c8322d6379c04":"Equations of motion of a physical pendulum in an accelerating frame","p_281710bffd188175":"Effective potential and angular velocity","p_999c57e805bc60f8":"Support reaction and tangential acceleration","p_05b596b3fca5c668":"Support reactions, centripetal acceleration and dynamic loads","p_f266bf514fb60b07":"Motional electromotive force and electromagnetic torque","p_35178f08bd1878dc":"Transient response of rotation under electromagnetic braking","p_6a5454a01c9b2241":"Power balance and load optimization","p_1586a8d49c7a0298":"Center of mass and moment of inertia of composite rigid bodies","p_731a4e37dfb9f319":"Coupling of gravity and elastic torque","p_783b980b32241b80":"Energy integral of a rigid pendulum","p_466bb0907e6ba298":"Conversion from angular acceleration to support reactions","p_26302d3840a305c0":"Constraint force at the lowest point","p_77e043d23bcbf92b":"Natural period of an adjustable rigid pendulum","p_e3ac3181ea3d025f":"Optimization of arrival time with finite amplitude","p_1e9362679080d231":"Repulsive collision and center of percussion","p_1fcab9c238c740ec":"Energy thresholds for sticking collisions and complete rotation","p_0fcf5e8a5e2a5085":"Coupling of two moving conductors and a capacitive circuit","p_1c36d4e0482c607e":"Reduction of coupled equations using conserved quantities","p_ec96ebfd237e6bfa":"Final state and heat balance of an electromechanical system","p_950f2878934dcda4":"Electromagnetic induction and motion","p_f143666fa391577d":"Mechanics and rigid-body motion","p_7202448c207c74ff":"Electromagnetism: fields and charges","p_b6b83661dfa67ee4":"Electric and magnetic fields of multilayer conductors","p_49c58b823560ba9e":"Charge relaxation in a nonuniform medium","p_c9f8191e2db58701":"Charge, energy and stability at a dielectric interface","p_130ee7c5f4bfdd78":"Rotational motion of a right-angle rigid frame","p_3baf3ab1e3545b76":"Pendulum with an adjustable pivot and an annular plate","p_adaf6f9b33853202":"Rotational motion of a rigid body and small oscillations","p_68dce19e1b9263b3":"Finite-duration external force and residual oscillation","p_3f5cbb88b55ac336":"Cylindrically symmetric static magnetic fields and current distributions","p_d2dae0e10950533d":"Electrostatic shielding, connection of a floating conductor and field energy","p_0b718e809448bc05":"Resistance and capacitance of a nonuniform medium","p_23f9ebb2a7cdd456":"Charge relaxation and matched current pulses","p_a21a37a502a5f8bc":"Steady current and free interfacial charge in composite media","p_012cecac7a0e88a5":"Electrostatic energy and work done by the power source at fixed voltage","p_2d8ca67c76eb072e":"Electric interfacial force and stability of equilibrium","p_9c4e737075aea2a5":"Vertical force balance of coupled pendulums","p_3ee9d11a22374b58":"Spring elongation and linear restoring force","p_0cd5e8d307144197":"Coupled equations and normal oscillations","p_2cd75729035bac5e":"Superposition of normal oscillations and initial value problem","p_f0fd0f86bd7c25ae":"electromagnetism","p_c6bfd46fda7ab7f5":"Mechanics & Vibrations","p_f93808d21ee0bb7e":"Coupled oscillations of a pendulum with elastic support","p_9344ebe953a214c1":"Equations of motion for coupled systems","p_cf9b8e3af603bf2e":"Natural frequency and mode shape","p_86d09c220fb011b5":"Modal superposition from initial displacement","p_95e5041b85587f70":"Finite-duration base excitation and residual oscillation","p_86dafe1f520478d5":"Physical pendulum of an eccentric disk","p_5f12ced3d61ba0d8":"Angular momentum and impact response of a rigid-body pendulum","p_8fd824fa27dd22ef":"Spherically symmetric nonuniform charge distribution","p_67f70e5ccef3ee2f":"Induced charge on a neutral conducting shell","p_aa01cb31a5b0976b":"Grounding and potential boundary conditions","p_dedc916c9e5e3841":"Comparison of potential integrals and conductor boundary conditions","p_5933f3a78badd652":"Maxwell equations and standing waves","p_b7a2af2ff43af486":"Energy exchange between conductor boundaries and standing waves","p_2d6ef5c7325a2061":"Electromagnetism: standing waves in a perfect conductor resonator","p_2ae529f5c79b2cc1":"Electromagnetism: resonance conditions and electromagnetic energy","p_e4ae2ee4255f9d8f":"Electromagnetism: medium identification by propagation measurements","p_47b0ae7e62bd8498":"Electromagnetism: boundary conditions including surface currents","p_9fd8a09ac28df2a8":"Electromagnetism: flux and absorptivity of a resistive film","p_da93565ae138abc9":"Electromagnetism/optimization of absorptivity","p_0c62fe3fe0b914d4":"Electromagnetism: nonuniform spherically symmetric charge and Gauss's law","p_eb3ab2bdcc5d3a47":"Electromagnetism: grounded conducting spherical shell","p_ef22b7a35ace371e":"Electromagnetism: charge conservation after disconnection from ground","p_d89618e2a2f4d1bd":"Electromagnetism: potential response to a nonuniform distribution","p_31b2089a134a7fbf":"Mechanics: constraint force of a pendulum with a moving pivot","p_a6957fc10fa4b018":"Mechanics: equations of motion for coupled oscillations","p_264055bbdc7f5f0e":"Mechanics: eigenmodes of a support with finite mass","p_dd9c4a53e7b9f997":"Mechanics: initial-value response of coupled systems","p_c5561fc45e4b66e3":"Mechanics: relative equilibrium on a rotating constrained surface","p_e93cc9292a01ac26":"Mechanics/Small oscillations about relative equilibrium","p_8529351876a755a8":"Mechanics: small oscillations with two degrees of freedom","p_0bebf0e59c0897fb":"Mechanics: forced oscillation over a finite time","p_933b43740bbc09bb":"Electromagnetism, superposition and energy transport","p_dc8def5280e04100":"Electromagnetism: electrostatic field with planar symmetry","p_92bec1af05a58469":"Electromagnetism: wave response of conducting thin films","p_0e98cc003a5f1674":"Electromagnetic waves and Maxwell equations","p_81db0337d114947e":"Energy conservation in electromagnetic fields","p_4950b4bb551123a2":"Material constants and phase matching of electromagnetic waves","p_d8fe0ca4cdc9f8f9":"Boundary conditions with surface currents","p_de8d090dab6a4a8e":"Poynting flux and Joule loss","p_904192450dc4c0de":"Global maximization of absorptivity","p_7e88346e7383ecd9":"Rigid-body pendulum and non-inertial frame","p_add7ec9a0957351d":"Linearization about an equilibrium point and angular impulse","p_55dce9386b65079c":"Kinetic energy of a rigid body and quadratic approximation","p_a5823a5d9007fa98":"Coupled equations of motion","p_8b3eecdb5f597804":"Separation of normal modes","p_23dbb33eed981678":"Initial value response and irrational frequency ratios","p_c969e51c188645e5":"Forced oscillation by piecewise external forces and input design","p_8e83fbb2cf550d9f":"Coupled oscillations of a movable platform and a pendulum","p_bcfafd7ae6f7ecf7":"Potential-controlled coaxial conductors and nonuniform charge","p_128d2e3b9cc7b875":"Rolling oscillations of a wheel with a mass distribution","p_7159a235f38c8965":"Pendulum and slackening string after an inelastic collision","p_a9bd19522cfe7e6f":"Scattering and absorption of plane waves by a resistive sheet","p_46f63c409bb8f378":"Compensating charge and spherically symmetric potential","p_de45b93e24783124":"Compensation layer and planar-symmetric potential","p_6ceca36a969e2b70":"Central force and elliptical orbit","p_a3f9446cf3b75470":"Piecewise potential and oscillation period","p_052836c8657e78a2":"Moment of inertia of a ring-shaped rigid body","p_f86dea5995b26ad4":"Rolling constraints and static balance","p_f7a6745dfea7cf23":"Equations of motion of a constrained system","p_c31bcb16eb4f1042":"Rolling oscillation with an initial velocity","p_91c9208a1d60db69":"Limiting static friction and tension conditions","p_0623943b6aeb5c85":"Rigid-body impact and reachability condition","p_2b89a4cc87158810":"Dielectric interfaces and the method of images","p_6dcb6590f48ad49b":"Boundary conditions for potential","p_0c77721ae4c1e52d":"Continuity of electric flux density and image coefficient","p_73def60b0d328c2a":"Interfacial force between neutral charge pairs","p_4ee7648aa2031294":"Energy of induced polarization","p_b030d9b995a12a16":"Composite bound charge density at an interface","p_3200e7c00abf9847":"Full-plane and half-plane integrals of polarization charge","p_f72e544281c56d17":"Maxwell equations and elliptical polarization","p_3f29baaf4061c86e":"Energy transport velocity of electromagnetic waves","p_e214540be97ef62b":"Cylinders, layered charges and motion","p_d5c1abc71857ca56":"A nonuniform disk driven by a couple","p_97c34108e1352118":"Impulse on a counter-rotating hollow sphere","p_831981f43c4562b4":"Neutral charge pair and dielectric interface","p_0bada197654a68a2":"Design and forces of a two-layer coaxial capacitor","p_36e6a93281459c42":"Electromagnetism: nonuniform charge distribution and weakly nonlinear oscillation","p_27ccab9c0f4bf0a9":"Electromagnetism: spherical boundary, image method and induced charge","p_47432007ea7f84ca":"Electromagnetism: two-layer coaxial line and distributed parameters","p_8809f08576a3b455":"Mechanics: rotating bodies on a moving belt, slip and rolling","p_e7c6687e7be99cc4":"Mechanics: changing the pivot, angular momentum and reachability conditions","p_8f085506d3f72da7":"Electrostatics, dielectric boundaries and capacitance","p_5dda34874298f06d":"Magnetic field energy and distributed-parameter lines","p_872a6c4092b6f292":"Magnetic material boundaries, image currents, force and flux","p_5cbc78224bb44b38":"Moment of inertia of a rigid body and frictional impulse","p_bd07b0086d4d4573":"Constraints on a rigid body, inelastic capture and gravitational motion","p_38b4354095da919d":"Mechanics: rigid-body motion","p_f95effd2f055e3bd":"Electromagnetism and electrostatics","p_977b00e7f2cd3118":"Electromagnetism: transmission lines","p_c4b9fff81e87066f":"Electromagnetism: method of images and magnetic energy","p_37b902124a252267":"Moment of inertia of a rigid body","p_8d2c46cc70d95f25":"Rolling constraint at a moving boundary","p_a9a77e673396a1d8":"Inverse design of impulse and kinetic energy","p_cf7b721a2bf3fb60":"Conserved energy of a rolling body","p_3fdf9ae09a38e648":"Constraint switching and conservation laws for collisions","p_3a773883907e07ce":"Angular momentum about a pivot and collision loss","p_26eea288ef30d943":"Barrier-crossing condition after collision","p_b1a7dca128656339":"Cylindrically symmetric electric flux density and boundary conditions","p_ac13abda5b4de0ee":"Discontinuous permittivity and electric field","p_e68d9fa1b157136f":"Potential integral of a piecewise electric field","p_dece6b17fbf90c6e":"Equalization of capacitance and stored energy","p_42ba7ba65bdd857a":"Virtual work of the electric force under constant-voltage conditions","p_c8ed8f99d74d05c5":"Magnetic energy in a spatially nonuniform medium","p_609c9e55fe2301a5":"Capacitance, characteristic impedance and phase velocity","p_785c95ce990a125d":"Image currents and magnetic forces at a right-angle boundary","p_3223839478232cb0":"Comparative proof of inductance using the method of images","p_af9320c5f85fe542":"Impulse of a rigid body and transition from slipping to rolling","p_db610be4cd866bd1":"Rigid-body collision, angular momentum about a pivot and reachability condition","p_ac4f4264aab36b75":"Nonuniform dielectrics, capacitance and electromechanical force","p_398ee35930e6ae59":"Transmission lines in composite media and impedance matching","p_afb57c261914cc10":"Magnetic boundary, image current and inductance","p_c0c1d75447cd8157":"Moment of inertia of a rigid body, impulse and rolling","p_0b50711ac0017225":"Constraint forces, motion of the center of mass and free rotation","p_896f7b52f9fcfade":"Electrostatics, capacitance and the energy method","p_c42fc97f6ed051fd":"Multipole expansion, conductor boundary and vector potential","p_7a7b2aabc7b5de9f":"Magnetic boundary, image dipole and inductance","p_c1fd41c659e80417":"Mechanics","p_5d1988ec4eedcd9b":"Linear algebra","p_58e49c7c5a1d3388":"Calculus and mathematical analysis","p_94c4f08767b03dc2":"Discrete mathematics and periodic points","p_8ac59b9f126c3d7e":"Rotation about the center of mass and work","p_02c9f418d425430e":"Velocity decomposition of a rigid body","p_491561fc2df49289":"Projectile motion and conservation of angular momentum","p_316741be59d3d808":"Far-field expansion of an electric quadrupole","p_e0f390d34b426894":"Boundary value problem for a dielectric sphere","p_cc4f591fc032a9c7":"Magnetic dipole expansion of a current loop","p_9090f93d79fce0e0":"Inverse functions and higher derivatives","p_9ff2187e435ec9f7":"Nonlinear separable differential equation","p_2516440b4f756f36":"Matrix exponential and singular parameter limit","p_1048d6befbc5fcd0":"Eigenvalues of symmetric matrices and orthogonal diagonalization","p_8a831b91d0d83710":"Differentiability and analyticity at the origin","p_94bfad37e10fc6a8":"Periodic points of affine torus maps","p_6e0aa1a249db7715":"Uniform convergence of weighted sequences of functions","p_69d9469fd325fc99":"Reflections for a weighted inner product and generated group","p_92d391e03447510d":"Upper triangular matrix ring and two-sided ideals","p_66453f7bf29e7c32":"Biquadratic fields and minimal polynomials","p_01c08a6f36ca1fc2":"Fundamental forms of surfaces of revolution and minimal surfaces","p_99b2a66bdc2bc3fc":"Covers and homology of a torus","p_6d0560702987d6db":"Constrained critical points on regular hypersurfaces","p_6238de9a8f497e26":"Convergence in measure and completeness of Lp spaces","p_7fb06cba2945c808":"Residue integrals and differentiation with respect to a parameter","p_f00ddceb77ea4b76":"Solutions of the Riccati equation and maximal interval of existence","p_169939aad9d7cc34":"Maximum likelihood estimation, unbiasedness and squared error","p_d20ff771678ecc9f":"Integer programming and feasibility of linear inequalities","p_4a48dc0750d87c8d":"Symmetric matrices and positive semidefinite conditions","p_61ef1f20b8e7da9c":"Differentiability of logarithmic functions","p_66a3dd11816fc0ae":"Damped equation and resonance","p_eabf6ae880cca333":"Convolution iteration and Laplace transform","p_6b83164b8acdeca2":"Continuous distributions and expectation of the number of intervals","p_ce9131252d555510":"Double poles and sine transforms","p_a163cef7ff6ca1e8":"Eigenspace of the conjugate action","p_7b9f6ec15617b1ae":"Oscillatory integrals with cancellation","p_ff224cd049eaa08a":"Change of variables in Euler-type equations","p_bfdfb35223b6cb97":"Parameter integrals and Laplace transforms","p_85962bb90e4f7d45":"Density on a triangle and least-squares prediction","p_49c1f9d257ae0a8f":"Coalescence of distinct poles and real integrals","p_913768ccd30187b0":"Group of positive affine transformations","p_c310c61d34e71b33":"Prime ideals and factorization over a finite field","p_19204c3db9baa2d6":"Real subfield of a cyclotomic field and quadratic intermediate fields","p_420ddf6da0c3c3bf":"Radially symmetric graph and height in the tangent plane direction","p_8be72e867c8001a1":"Homology of a space glued at two points","p_93d126b7455ba35b":"Critical points on a superellipse","p_0278a4a3e3c554b2":"Triangular kernel and second-order approximation error","p_424120fa9356756f":"Two poles of a strip domain and Fourier integrals","p_09ff83765ef75fe9":"Radially symmetric divergence-form equation","p_5308f9311563d0af":"Efficient estimation by logarithmic transformation","p_a168d15b099a6046":"Repeated interpolation conditions and uniform error","p_f6399e8c0237d77e":"Self-adjoint operators involving transpose and trace","p_5823a52891c6e49f":"Oscillatory improper integral with a square phase","p_6f12cd6807aaf9aa":"Spectral radius and nonnormality","p_27568b85e046f2bb":"Monotonicity and asymptotic rate of normalized integrals","p_b9efdcf9b0c6eb64":"Permutation group on four letters and its action on partitions","p_a0bfb190020e994b":"Field decomposition of a cubic quotient ring","p_a53c9f5722de90a5":"Cubic splitting fields in different characteristics","p_078ce47ad14e9e18":"From a surface of revolution to a closed ellipsoid","p_9f644d8a762e6a03":"Homology of a torus with an attached disk","p_efbcd7dd566af5d1":"Covering map of the real projective line","p_694730c702e5f2e9":"Eigenfunctions of an ergodic transformation","p_7bc681c0ede3f35c":"Entire functions constructed from the remainder of the exponential function","p_ab22e387abd1845c":"Mixed boundary conditions and eigenvalue locations","p_27d462a912ef738f":"Estimation of absolute-value statistics and square scales","p_d5ef1669768ab9e1":"Termination and maximum number of carry updates","p_fd3e8d1353fddd87":"Projection decomposition and singular systems of equations","p_1c0022a9b1c5ef33":"Extrema of a function of two variables with degenerate points","p_aab8c2c885e3942d":"Change of variables and resonance at a repeated root","p_ae8b6f03369e5397":"Convolution of different time scales","p_4b98e2cf8f0490ef":"Logarithms of ratios and continuous distributions","p_2e107cd4936ba157":"Double pole and trigonometric integral","p_7462507856ea32f6":"Interpolation conditions and degenerate observations","p_e19de8e78ae542de":"Curvilinear coordinates and bifurcations of stationary points","p_5b82d33e3333aa68":"Invariant subspace decomposition from polynomial identities","p_31cc2fd533640035":"Improper integrals over triangular regions and orders of vanishing","p_84721e9d787afba6":"Phase plane of a competitive system and invasion criterion","p_e9f7468d5e33a322":"Alternating ratios of increase and decrease and hitting probability","p_340ddca033ee85a5":"Transient response and contact conditions","p_0fb0843d62d9bfae":"Affine geometry with a specified inner product","p_a342620f04eddaeb":"Implicitly defined inverse functions and cubic limits","p_b2de97dc5696b90a":"Nonlinear equation with the inverse function as the time variable","p_d45efde8128291fa":"Matrix exponential of a rotation and nilpotent coupling","p_49b6a84115e2107b":"Differentiation of quadratic forms","p_ffd2289f1633e778":"Equality-constrained minimization","p_d295786dc9903b57":"Time transformation and first-order equation","p_5c207a3a1fa36613":"Harmonic input in logarithmic time","p_e04710de7b93c0c7":"Amplitude, phase and time scales","p_a9cdc63f6331704e":"Endpoint conditions and rank of polynomials","p_4920cb7ef325b457":"Oblique coordinates and generalized double integrals","p_7881cea3d7f9c505":"Kernels of matrix evaluations and quotient rings","p_ee848c1e01ac0887":"Closed relations in Hausdorff spaces","p_98674f16fcc1dd26":"Global bifurcation through density and connectedness","p_9ca9c502337dc233":"Growth conditions at both ends and holomorphic functions","p_2d16b21de9099709":"Kernel and solvability of a cyclic matrix","p_9fc95764cbe21868":"Gaussian-type double integrals with cancellation","p_b465cc9c3edd811f":"Nonmonomial ideals and nilpotent elements","p_1056638d8c9123e1":"Simultaneous flattening of two surfaces","p_6e72565bac04d243":"Double poles and Fourier-type integrals","p_db68d767fb94af94":"Decomposability of a bivector","p_55d621fe83035803":"Factorial expansions and the boundary of rationality","p_3043a862fa79c4f5":"Cyclic direct summands of finite abelian groups","p_8a95dcff0c79af89":"Immersion of a torus","p_ee585fa3d63b2e03":"Finite covers and embeddings of quotient spaces","p_dcab69299a303235":"Reciprocal transformation and parameter differentiation","p_e1d492bb673a79ae":"Rank of powers and Jordan canonical form","p_a8dce01f21c4ea15":"Weighted differences and convergence boundary","p_6b1621114e27448b":"Units of finite rings with nilpotent ideals","p_5f68ad5facaf2f14":"Local degree cancellation for a surjective map","p_bf945c9e0b2ce564":"Logarithmic moments and complex integration","p_6c84ad69711040cf":"Complex analysis","p_7daf955db536bf3e":"Algebra","p_a674f42d9295a435":"Geometry and topology","p_45c52975b2080861":"Functional analysis","p_08e55c22118b9757":"Partial differential equations","p_aad37b3fec84fa85":"Fluid mechanics","p_eefd71289768ce37":"Computation theory and formal languages","p_fb4b487f148eac7e":"Discrete mathematics and graph theory","p_7b27f11ee3671115":"Quantum mechanics","p_bd73964c656b5edb":"Change of variables and improper double integrals","p_69d3fd489efc61cb":"Range of a rational function and attainment of endpoints","p_9a1fdcefca99a19f":"Eigenspaces of polynomial operators","p_8d55892161374700":"Similarity conditions and Jordan chains","p_44f659ea8e78285b":"Rank classification of two-column augmented matrices","p_b6b85ef126ed7a6f":"Error estimates for period averages with nonlinear phase","p_83c4499de3700867":"Weighted inner product and resolvent","p_e1ae2fb0842ba6ac":"Inverse correspondence of a weighted Cayley transform","p_154ec42463d4f878":"Parameter differentiation and residue integration","p_aee828331d036d38":"Prime power congruences and the Chinese remainder theorem","p_0829f28ea56abcd6":"Critical points and degenerate bifurcations on a sphere","p_4b51e0bece5f1ce8":"Euler-type inhomogeneous equation with repeated roots","p_4624500c4b94a2f8":"Finite trace tests and nilpotency","p_d2fda61b861b0017":"Normalizer of a diagonal subgroup","p_0c745d467df0fba5":"Normal subgroups and characteristic subgroups of a direct product","p_381759d593760fdd":"Representation of the cusp ring and finite extensions","p_b90eb4ab8d6b7ef7":"Computation of the conductor ideal","p_fd8b31d0b1927266":"Integral closure and nonnormality of local rings","p_d6b5de43e7c5deec":"Contraction of maximal ideals","p_37ec1e16e5a97fa8":"Two-step extension of a rational function field","p_3e93332754b87cad":"Galois groups and quadratic intermediate fields","p_09390d521cda870b":"Closed 1-forms and periods on a torus","p_f1a4f54733146899":"Exactness of 2-forms on the torus","p_bb084b7e4271c89a":"Integer homology of a mapping torus","p_a4e533e613d67227":"Orientation-reversing map and torsion component","p_9248da6ecad44f8b":"Weak convergence in infinite measure spaces","p_ca669a79521c8b6f":"Tail conditions and limitations of strong convergence","p_997f8efa4af14ac9":"Norm of convolution operators and noncompactness","p_d153136886fe1bb8":"Restriction to a finite interval and Hilbert–Schmidt norm","p_d1e1d713d8076109":"Quantitative decay of critical reaction-diffusion equations","p_7308f39dd0eb501b":"Motion of a floating body with changing waterplane area","p_a7e36cf01edf4010":"Period correction for asymmetric oscillations","p_37139d1b3ee14b02":"Energy loss due to weak resistance","p_b3f901750aafb5c0":"Recursive procedures and iteration of partial functions","p_169a7eb6bb7b1b38":"Globality and closed form of the recursive hierarchy","p_8fcb606dd4f50346":"Maximality test using vertex cuts and residual paths","p_7f32d86b36c2abd9":"Uncertainty relation involving covariance","p_6c9f092e5486686a":"Variance and commutator estimates for projective measurements","p_d4666ba06a772224":"Time-dependent Hamiltonian and transition rate","p_8f859a67257d5aba":"Attaining the speed limit in a two-level system","p_11761c0698efd63f":"Case analysis of Jordan types of nilpotent matrices","p_f060f24256e8fa6a":"Absolute value integral over a hyperbolic strip","p_8cf179a3697c85c9":"Homomorphisms to the ring of dual numbers and tangent directions","p_f43e1d2a4c96207d":"Compactness of a family of accumulating line segments and connected components","p_ee0bd5a11d274f6f":"Fourier integral of hyperbolic functions over a rectangular contour","p_69ea189eb9e527c2":"Image of a differential map with boundary conditions","p_f2160cfe48fec628":"Eigenvectors of parameterized matrices","p_dcaa81a9400d3a5e":"Diagonalizability and positive-definite inner products","p_ba3a730dd003cbe5":"Mean limit of correlated fractional parts","p_2571a373d62c9db0":"Noncompact domains and unattained infima","p_dca29173d94f8ed0":"Fluid mechanics (physics)","p_2f46a35aee439eab":"Double integrals with nonsmooth weights","p_5800ee6ec4688b46":"Kernel, compatibility conditions and minimum norm","p_05a2f1e2e8cfac0b":"Spectrum of a block operator","p_2fab7a65c8539dfc":"Asymptotic periodicity and uniform convergence","p_b13bd0ec861b6223":"Stability of second-order equations and integrable external forces","p_adda399cb44aa796":"Compact parameter and minimum","p_4b73c8de2028c5bb":"Quantitative separation by distance functions","p_6bfa638adbbe4882":"Automorphisms of finite abelian groups","p_0afdf72bdc22e413":"Invariant ring of a cyclic action","p_71fb6c4a78cc8ab9":"Normal subgroup and quintic equation","p_19e24ebcc3ec5660":"Periods and differential forms on a punctured surface","p_d8e7cf22c831d37a":"Homology of an attached cell and a punctured torus","p_4fe7f204aae39e8b":"Supremum of a directed family and square integrability","p_aa5c71cf35c1cb1c":"Laplace transform and completeness","p_61e5b713e7e3be29":"Higher-order diffusion with neutral modes","p_91644949fce71874":"Boundary layers in magnetic fields and pressure-driven flow","p_3b399c65c15cb021":"Construction and nonexistence of linear forms","p_cda4003f27ad7fad":"Interval uncertainty and a unique minimum spanning tree","p_c94ff3a983bf6903":"Displacement operator and geometric phase","p_2214d30d53c269f2":"Linear maps using moments","p_785839687209b730":"Abel averages with periodic weights","p_f346e152e1a00e90":"Finite group decomposition from relations","p_966022de692f0ee4":"Dual representation of Hausdorff distance","p_04591bc906a3d2b4":"Double pole and Fourier-type integrals","p_8524eb35fac2154c":"Intersection of subspaces and exceptional parameters","p_2d4278d2bf4fbaa1":"Telescoping series and an obstruction to uniform convergence","p_f3f002a135b654e9":"Unit groups and a specified power","p_635f7f223ff56c46":"Perturbation stability of nondegenerate critical points","p_dfd0091f6f7b1dd0":"Boundary modulus and Blaschke factorization","p_64f065b970cfe54a":"Multiple integrals over unbounded domains and convergence boundary","p_740210d6a00b7de8":"Case split of the rank of symmetric blocks","p_a4b5dde40ec42615":"Coalescence of poles and parameter integrals","p_e2642aa091941dca":"Squeezing of distribution functions and uniform convergence","p_a672fd7164277b07":"Semidirect product and one-dimensional character","p_57ccd396213fcd4a":"Critical points on an orthogonal frame","p_b677634f77f72c8f":"Stability kernel and Fitting decomposition","p_5273fc54a4450f8e":"Local confluence and termination","p_e7dea20851884281":"Zero set defined by a determinant","p_2ea7a8a7be0678e3":"Nontrivial action on a cyclic group","p_ce0e13aba40aabff":"Counting splitting fields and intermediate fields","p_82a1416e5c81cd0c":"Product manifolds and the degree of a map","p_e160c73e1c13e3c9":"Complements of subspaces and duality","p_6ee8db41431ec133":"Identification of measures using weighted moments","p_df420cc4665efd88":"Two-sided compression and compactness","p_d9b614bf9ac15e6f":"Backward heat equation and reachability","p_fa01ddfd0d1a02c5":"Nonlinear flux and viscous traveling waves","p_2115c2c10b070055":"Stopping time of synchronous label propagation","p_50699c94be024e7b":"From matchings to edge coloring","p_5be28e98fce7b503":"Square-number spectrum and quantum recurrence","p_52492faa5b87806c":"Observation map of polynomials","p_037aae270de37d84":"Uniform Cauchy condition for a series of functions","p_a64ec06bd8a6d7d2":"Cyclic subgroups of the unit group","p_85c89e083e968b42":"Perturbation stability of critical points","p_e02f9561e8da8087":"Boundary values on an annulus and integer conditions","p_7a5dd6c0163f79f4":"Intersections of graph subspaces","p_03c37ba22bef90eb":"Improper integrals by differentiation with respect to a parameter","p_087ebafb60c7fb99":"Counting elements in a nonreduced quotient ring","p_ee6d33fb679d667b":"Continuity of a unique optimal point","p_422cb692e0a20ca5":"Rational level sets and limits at infinity","p_7828f9c41826e57e":"Second moment over a sector","p_45fe8536cd5defb6":"Sylvester operator on a matrix space","p_e4b3e9ecd80418a5":"Periodic orbits with position-dependent angular velocity","p_61de73ebdcd4cbda":"Boundary exponent for integrals near periodic zeros","p_3a6abbc9efd3810b":"Diagonalization and kernel stabilization","p_56baa3da3659c194":"Characteristic maps of spaces with two ends","p_4f12dc346c37f348":"Distance matrix of a weighted star tree","p_92f22b72514ef8e4":"Normalization and conductor of a curve ring","p_011508e1a8352816":"Number of solutions of power equations and cyclicity","p_9e7e57fbf4deb646":"Real subfield of reciprocal polynomials","p_48a157db2373d9dc":"Manifold of rank 1 matrices","p_5c90892b983128dd":"Mapping torus of a self-map of the torus","p_89ebae7b43629d03":"Limit from finite products to integrals","p_70dde44031b08ed8":"Periodic antiderivatives with zero mean","p_928775b10f8a800d":"Estimates for the heat equation with an absorption term","p_71e0db43ba48c10e":"Periodic Neumann response in an exterior domain","p_6539ed9774916501":"Displacement representation of a driven quantum oscillator","p_5eb81b9701292271":"Integer load transfer and guaranteed termination","p_37de12559cfb650a":"Edge data and optimal vertex potentials","p_971b4ecceed26f2d":"Interpolation conditions and the kernel of a linear map","p_601a427d0e84c713":"Pole of order two and parameter integral","p_2b1c9b41af094210":"Chinese remainder theorem and idempotents","p_99a09217c8d875a1":"Power equations in non-reduced finite rings","p_f6b775474433125f":"Optimal value on a moving constraint set","p_e5f759b950429dc2":"Finite level sets and extension at endpoints","p_5e98d57569712aa7":"Dimensions of sums and intersections using quotient spaces","p_f8851573618fd9fe":"Telescoping series and uniform convergence on an interval","p_c455b55e9009ebbf":"Uniform vanishing of terms and nonuniform convergence of series","p_7387c0b5bf37eacd":"Order of a power of an element and the unit group","p_12cf0c078151e7a5":"Finite fibers near a regular value","p_6209609e195e7c63":"Prescribed zeros and boundary modulus","p_1a8c6af4da8ae57d":"Coalescence of eigenvalues of a parameter-dependent matrix","p_97430d994f508902":"Eigenspace and minimal polynomial","p_97091a81bf586cf5":"Coalescing poles and a holomorphic branch of the logarithm","p_9e07ba7700f3eeb3":"Quotient of an integer lattice and the order of a specified element","p_f5672fe1aaca1557":"Accumulation set of a sequence of points with small increments","p_38ddea4c01bc2a53":"Endpoint derivatives on a logarithmic scale","p_737cf410f97821c9":"Multiple integrals and symmetry in an ellipsoid","p_ab3f3c1c56cadfae":"Rank-one perturbation of an inverse operator","p_b22c69e2868802f9":"Triple direct sum via nested projections","p_d5745d6d05a3f575":"Integrability and dyadic partitions","p_6f09acde90918466":"Decay of integrable monotone functions and the converse","p_632e29a112baa8ff":"Parameter differentiation of integrals and removable singularities","p_c38cc9cda6fc2c88":"Regular values and tangent spaces of matrix constraints","p_539edcf190e39c01":"Leading error term of the midpoint rule","p_c57d7afd2859d06a":"Sylow action and solvability","p_67b8b4d007448397":"Intersection of splitting fields and the Galois group","p_f95de809beaff434":"Unique factorization lost under localization","p_d25dc52935bf46cd":"Embedding of a torus and rank","p_856488493b6df55e":"Pullback of a two-form and zero set","p_0e785034f3f51667":"Deformation retractions of two-frame spaces","p_06bb3eb7c7e103bd":"Frame spaces modulo finite groups","p_9d0b574f45f1046a":"Weighted Schur estimates","p_904aa5c06e9e0889":"Compactness of non-Hilbert–Schmidt kernels","p_db1b425a25fbb017":"Infinite first moment and Laplace transform","p_953c5fe3fa3517cd":"Uniqueness of zeros and infinite right derivatives","p_1fade46e9b7c3006":"Conservation of the mean in zero-flux diffusion","p_a3ae8a74934a899d":"Energy identity and spectral decay","p_d27b8601b2a99e16":"Transition from time average to uniform norm","p_55417bd53130254b":"Nonexistence of steady-state solutions for degenerate diffusion","p_388bd638415335ca":"Determination of the diffusion scale","p_5c47b97afefe1665":"Similarity equations and boundary conditions","p_000bf1308a43ee03":"Integral representation of a similarity solution and boundary flux","p_d766d26dc6e0ec42":"Comparison predicate by primitive recursion","p_7ac13b8356d02803":"Recursive construction of bounded minimization","p_c31e08f2acf3546d":"Lower bound on exchanges in a spanning tree containing a specified edge","p_fc4e9b66189269a6":"Variation of a single edge weight and the optimal-value function","p_94d2642fc9f78c0c":"Hamilton equations with external forces","p_fff6d6a09066e058":"Ermakov scaling and transformation to constant frequency","p_897466d22c0b2379":"Schrödinger equation under translation and scaling transformations","p_daf8be3514fa79ff":"Linear operators on quotient spaces","p_ae777366da1cd64c":"Minimal polynomial and Jordan canonical form","p_78fa648dc36ecb4b":"Branches of the logarithm and residues in the exterior region","p_d921e9f28d9a7423":"Presentation of finite abelian groups by integer matrices","p_7bea9526e8423d11":"Uniform limits of distance functions and connectedness","p_9e5f9ea68d0bafe3":"First-order correction to an endpoint-concentrated kernel","p_7ca4d6be7a3ae54f":"Anisotropic improper integrals and critical logarithmic factors","p_5316ca3496681785":"Parameter dependence of cyclic operators","p_9a73fd1d106ad281":"Rank and image space from singular values","p_46b6fab9462e993b":"Isotropic complements of finite abelian groups","p_8341e2d143e4abdc":"Minimization of the mean distance to a curve","p_9c0d4dbc37d4ba69":"Regularized oscillatory integrals","p_ad4478cca793fe23":"Level sets of a matrix and compact manifolds","p_40ff3a748ea1d045":"Number of semisimple matrices over a finite field","p_31585eb474f0c377":"Degree of a splitting field over a function field","p_ff3d0fc09e2837c2":"Generators and structure of the Galois group","p_224f1d325a372859":"Invariant ring of a cyclic group action","p_906b2d87550b56dd":"Distinguishing cotangent spaces from polynomial rings","p_849f7695f5fa33e7":"Inversion map torus","p_330a4bf41faceb26":"Immersion of the circle and fibers","p_84e854ab1b0e7742":"Identification of a non-orientable circle bundle","p_9081acc61102d880":"Gluing solid tori and homology with integer coefficients","p_fd5e23dea91753c2":"Transfer of twisted homology to product spaces","p_dc1a0cf37e9918d8":"Second-order error of the binomial approximation and dominated convergence","p_356bc0784c56c4f5":"Compact diagonal operator and weak convergence","p_e9ae3eff15b8b369":"Characterization of compactness by small-norm terms","p_9fe47e0c2e985af0":"Uniform energy estimates for reaction-diffusion equations","p_37f3ad911c3bfe50":"Strong convergence and error rates for singular perturbations","p_e8296043cc0c94d6":"Nonhomogeneous differential equations derived from variations","p_51b4c6d345dc8921":"Boundary value solutions by power series and minimality","p_736d758aff76167c":"Cyclic Gray order with fixed weight","p_6d0da6a8f1a88706":"Gas constant measurement eliminating pressure gauge zero offset","p_d60e73f4a147adee":"Miller indices, direction indices and interplanar spacing of a square lattice","p_35f4a63edd7bab83":"Structure factor and intensity ratio for a basis of different atoms","p_d3c4987b6673ef73":"Compensation doping and charge neutrality","p_318e63dfc664789f":"Ground term and Landé factor of a two-electron configuration","p_dd8ffb5bd7a3e062":"Nomenclature of complexes including coordination spheres and configuration","p_c00b077d0dd9c45e":"Reason for the weakness of spin-allowed transitions","p_5198e220cec927f7":"Vibrational spectra and ligand-field splitting","p_69f03059574b4d57":"Harmonic oscillator with a momentum shift","p_bf6148cea13bf4bb":"Displacement under a constant force and ground-state overlap","p_8368f572e37f8f7b":"Two-degree-of-freedom oscillator with different masses","p_1480b0b5ee156aaa":"Mass-weighted normal coordinates","p_bf3fdbc833bcbf46":"External force response and zero-point energy of a coupled system","p_f534255678fca989":"Weighted two-mode constrained statistical model","p_18bd7901ffd5b5dc":"Entropy at the critical temperature including degeneracy","p_ac697124763c056f":"Multiplicity of pairs of occupation numbers with equal energy","p_0f63a402fdaa1332":"Finite partition functions with even-odd structure","p_8d8c35bcb5ff55b4":"Finite-size average at the critical point","p_6328975516729022":"Cost fluctuation of a single mode","p_e9ddecf08a9dca1f":"Probability and conditional mean of a constrained subgroup","p_206d156d0e8f8b90":"Central forces and periapsis precession","p_dda89f75325ddebd":"Problem 1: weighted triple integrals","p_930de059379e2649":"Problem 2: interpolation conditions and subspaces","p_068535229d7576b3":"Problem 3: cyclic vectors and commuting maps","p_e05d7320345e1ff9":"Problem 4: first-order correction to a concentrating integral kernel","p_f41e79dcaeb4ab8b":"Problem 5: limits of differential equations and tails at infinity","p_8e268e850197b00d":"Question 6: regular surfaces and horizontal sections","p_618bdc994f0dbc20":"Problem 1: finite groups generated by inversion","p_8b90ba8b6144f3fd":"Problem 2: local ring of a plane curve","p_ab2609aac3bc5e83":"Problem 3: normal map and orientation","p_deab7d4fa2258e64":"Problem 4: disk attachment and integral homology","p_ef3637abf42d9dd3":"Problem 5: second-order correction to an infinite product","p_1d1635e5c871868f":"Problem 6: integral operator and boundary conditions","p_f0521bf4961ebed0":"Problem 7: periodic Poincaré inequality and advection-diffusion","p_79aedb11986e2d84":"Question 8: reversible systems and homoclinic orbits","p_72b534f7667c2a09":"Problem 9: invariant of synchronous label propagation","p_d6a548e2491935e5":"English exposition on mathematics","p_81157febf5e90157":"Incident ion in crossed electromagnetic fields","p_93dcd8fd60636937":"Thermodynamics/1A(a)","p_5f71ddb5e4b8580b":"Thermodynamics / 1A(b)","p_d087a4c26d1e1fe9":"Thermodynamics/1A(c)","p_5db875572b0dedb4":"Thermodynamics: 1A(d)","p_1b86b15676356980":"Thermodynamics/1A(e)","p_2f89ab32581a9bfd":"Physical chemistry/1B(a)","p_2ec33a0771c212dc":"Physical chemistry/1B(b)","p_5c22544e814fd743":"Physical chemistry / 1B(c)","p_9531966825417b28":"Physical chemistry/1B(d)","p_0349a4c05429eb8e":"Physical chemistry/1B(e)","p_1bdc739dc4c636bd":"Physical chemistry/2A","p_46b04099fd1c303c":"Physical chemistry/2B","p_f260211594458ccd":"Physical chemistry: 2C","p_e83a488bae41594f":"Physical chemistry/2D","p_0fefd8af7cbefb48":"Physical chemistry: 2E","p_eda6d1ad07879403":"Quantum theory/2F","p_fcf2b9b38d2e4e18":"Quantum theory / 2G","p_cdb65c843bb11b5d":"Quantum theory/2H","p_13fe4610e39c78f6":"Optics / 1A","p_2de1bb29408b3aaf":"Optics / 1B","p_3294364081279c35":"Optics / 1C","p_d9e442e4d18931c1":"Optics/1D","p_97d464b7622ef794":"Optics/1E","p_ab1794da2d006a68":"Optics: 1F","p_083c3b17a37366b4":"Statistical mechanics/2A","p_93fbd9cd21ca8051":"Statistical mechanics/2B","p_74703a3d7b5c06f4":"Statistical mechanics / 2C","p_eb92770bf4224775":"Statistical mechanics: 2D","p_a693b6d79c755f86":"Statistical mechanics - 2E","p_73bf11e90ba28963":"Statistical mechanics / 2F","p_7b120d0f21356758":"Statistical mechanics: 2G","p_445fe08444934240":"Question 1: calculus and real analysis","p_70fee7a53745f70b":"Question 2: linear algebra","p_fe6de702e8ea5b50":"Linear algebra / Question 3","p_733d31aa33fb95dc":"Question 4(1): calculus and real analysis","p_9d7e166a310c0311":"Calculus and real analysis/Question 4(2)","p_596e95518c8f8bda":"Calculus and real analysis / Question 4(3)","p_b87302484a4a9c5d":"Complex analysis/Question 5","p_1fdd4230c55b37a6":"Geometry and topology/Question 6","p_210aade4a40a0cfe":"Linear algebra / Question 7","p_b6b80bafca22109a":"Algebra/Question 1(1)","p_4ef7455b55a51eea":"Algebra/Question 1(2)","p_7e11a19904995024":"Question 1(3): algebra","p_48640f545cd590b7":"Algebra/Question 2","p_83a91eb6abce86d3":"Algebra/Question 3","p_5b634bfbefee7d7b":"Geometry and topology/Question 4(1)","p_dabe971ac8ce3ee2":"Geometry and topology/Question 4(2)","p_1469a7aa09899def":"Geometry and topology/Question 4(3)","p_231022b861c89935":"Geometry and topology/Question 5(1)","p_4f48d1e83872a0e8":"Geometry and topology/Question 5(2)","p_69bcc2f5522a02fd":"Functional analysis / Question 6(1)","p_80da708be43a337a":"Functional analysis/Question 6(2)","p_c2fca8759bc0e362":"Question 6(3): functional analysis","p_7d7d81d9f7bfb5a5":"Question 7(1): functional analysis","p_e12f88fe2cadf50a":"Functional analysis, Question 7(2)","p_0159c5d20e226971":"Functional analysis / Question 7(3)","p_91e28a6b331f282d":"Partial differential equations / Question 8","p_33f860034e532c24":"Ordinary differential equations/Question 9(1)","p_11f90c39e6977d8c":"Ordinary differential equations / Question 9(2)","p_277658f314187728":"Question 9(3): ordinary differential equations","p_007bc1896ec8f44b":"Partial differential equations / Question 10(1)","p_fe5890f530bcda25":"Partial differential equations / Question 10(2)","p_930814d6c4a0d99f":"Partial differential equations / Question 10(3)","p_a8d8acb51ac4f0d7":"Partial differential equations/Question 10(4)","p_ff6e3ec4bce75edf":"Algorithm correctness / Question 11","p_33d5e69fede92ccd":"Discrete mathematics/Question 12","p_e97ce5a9c6e00653":"Ordinary differential equations/Question 13(1)","p_256100ec17b17793":"Question 13(2): quantum theory","p_2f988e286b99cad3":"Quantum theory/Question 13(3)","p_89b784614b6810c3":"Electromagnetism: A","p_639f4bb037a97992":"Electromagnetism: B","p_9c462e513669a761":"Electromagnetism/C","p_326d0bc79956b888":"Ordinary differential equations / D","p_811f7b36cbb14528":"Mechanics/E","p_5d9f00d602f10536":"Statistical mechanics","p_e5917704b0211ee8":"Fluids","p_4cce170b34fa2064":"Optimization and mathematical programming","p_d1e65d96b4568e3c":"English (essay)","p_1c2724125661df64":"One-dimensional scattering and transfer matrix","p_b5068c19a1b3c8ae":"Canonical distribution and second virial coefficient","p_f9e623003013b010":"Biot–Savart law, mutual induction and vector potential","p_5afe35c228e87327":"Thermodynamic identities and Reversible adiabatic process","p_9fbc14a8ea38460f":"Molecular orbitals, point groups and selection rules","p_854b082f36f23b65":"Nondegenerate perturbation theory and symmetry","p_7b91ebd923838201":"Improper integrals and the logarithmic test at endpoints","p_b2257eaff8e320df":"Parameter matrices, kernels and nilpotency","p_cd41f8d04ad6e265":"Jordan chains and commutative operators","p_6c3dfe4733d8ef71":"Limsup, liminf and realizability","p_fc17850080c976cb":"Harmonic functions, simple poles and residues","p_cbd915163d29d0b3":"Regular values, critical points and Hessian","p_b45c08df073eabb2":"Intersecting families, complete graphs and automorphisms","p_1aa0d4e249236546":"Integral domains, nonprincipal ideals and singular curves","p_aafa2cfd0ab1b0a0":"Galois theory and normal closure","p_8bc70abbb3b82cb6":"Finite fields, conjugacy classes and normalizer groups","p_0b6fe305a8677feb":"Differential forms, orientation and Stokes theorem","p_f83cc8df79234487":"Mayer–Vietoris, product spaces and inclusion-induced maps","p_52478b9e6924477a":"Variation of complex measures and partition by phase","p_399440ee7740af86":"Finite-rank approximations, compact operators and eigenvalues","p_7fe13d0270a9b0b2":"Robin boundary, weighted heat flow and log-convexity","p_cfddbe31f8aa76fa":"Invariant regions, return maps and uniqueness of periodic solutions","p_1e53e0fa7228bfcc":"Optimization of coupled waves, viscous diffusion and damping","p_a416f3ab1399ba2a":"Operational semantics, call by value and determinism","p_ecf29bd198fd55c6":"Cycle averages, difference constraints and potentials","p_4ac432ec2f2f8846":"Commutators, spectral sums and gap estimates","p_f4b8e830cb7037c4":"Mathematical explanation in English","p_ff05dd5caba4b646":"Planar motion, moving basis and dissipation","p_1aa70a233f368b68":"One-dimensional quantum systems and atomic orbitals","p_b404cfa8b8f1a543":"Fermi statistics and constraints","p_6443e53e8b4d397f":"Phase equilibrium and thermochemistry","p_937ab5976466ec62":"Improper integrals and uniform convergence","p_fc6b2e5dffd6e128":"Real diagonalization and linear maps","p_6f9b82211836157e":"Residues and real integrals","p_0c565f5bfd7f1fe0":"Regular values of maps and relative topology","p_33b7f1032f8d015b":"I Splitting field and intermediate field","p_410b5a21bc9af411":"II Nodal curves and integral closure","p_2e1ca2452c424c6f":"III Differential forms on a torus","p_f95328000154ac9b":"IV map torus and fiber product","p_167f5d5a6a602220":"V Weak boundedness and strength of convergence","p_09c9657c4a002df3":"VI Compact operators and strong convergence","p_89672cf499ad790a":"VII Unstable and neutral modes of the diffusion equation","p_8d7acbca6ff8173f":"VIII Rotating system with a saturation term","p_b15a58f309ab221a":"IX Nonuniform compressible flow and density transport","p_7f7516201a7dc09b":"X Rank and termination of multisets","p_229527772cec7a0f":"XI Weighted terminal-separating cut","p_4877d1a6853f7679":"XII Covariance of quantum states and equality conditions","p_dfc42317ed749a70":"English composition: computational experiments and proofs","p_a1baeac0f51c95e4":"I Intersection of subspaces and nearest point","p_d7b0f4ebbbaec5a2":"II nilpotent matrices and rectangular matrix spaces","p_825d3f5a63709943":"III function series and boundary layers","p_4b06619f234a5aca":"IV singular double integrals at two endpoints","p_fd06242622477b41":"I Weighted orthogonal complement","p_ba1ea686c13bcef7":"II Similarity test requiring more than first-order invariants","p_255a8f8f6edf3b39":"III Maximization by separating direction and radius","p_c0e8176f713b6338":"IV Logarithmic factor and convergence boundary","p_65c66b6eacf17578":"I Singularities of rank-one perturbations","p_65cf0ad0b28e10ae":"II Similarity conditions with two parameters","p_c2ca081743326b4d":"III: singular weighted moments over an elliptical region","p_ada08484e12870df":"IV Uniform convergence and noncommutativity of differentiation","p_857288838029ef81":"Intersection of subspaces","p_ba33b6709451bc66":"Second differences and asymptotic estimates","p_a6c9e704ba70bc1b":"Minimal polynomial and decomposition projection","p_e79106d2ee1d5f2d":"Integrable derivatives and behavior at infinity","p_f29d45b45f9de6d6":"Uniform approximation of moving averages","p_64b4a34a14382aa9":"Congruence subgroups over a nilpotent ring","p_f62086d0d7556d69":"Compactness of projective matrices","p_fe196dbb23d307a0":"Principal part of a pole","p_6138d6428ffda8e2":"Sign-dependent case split of Fourier-type improper integrals","p_921ae3f9bdc3f70c":"Commutative ring of cyclic nilpotent matrices","p_5bf2463182e63916":"Integration of dilated differences","p_5547d4d30d4dd84e":"Logarithmic weights and double magnification differences","p_b898d9d87e3e5292":"Finite abelian groups and subgroup counting","p_2025d7814c5ca917":"Rank of the differential on a sphere","p_f0090c36c7fabf44":"Oscillatory integrals using poles","p_22393639e12a611d":"Primary decomposition and trace of a rational matrix","p_9052cd8f8da3d0f2":"Integral limits with vanishing moments","p_1ec490650ef586f2":"Relative homotopy of double loops","p_39da6eb2f94e2b2b":"Commutativity of two connected-sum operations","p_cde61793a60395fe":"Commutativity of the double loop group","p_0f5d0b751784a95a":"Oscillatory integrals that cancel singularities","p_4e7a75b0a30a4b87":"Jordan form of a finite-difference operator","p_2fdd927096e2e052":"Logarithmic scale of differences and iterated averages","p_adb7d61d389a5d58":"Uniqueness of the positive semidefinite square root","p_fc5c98dc9a5d5c08":"Uniform convergence and termwise differentiation","p_3b9935921683997d":"Finitely generated subrings of an algebraic number field","p_d4d72aa87f7dcb32":"Lattice periodicity and affine increments","p_e22f0a292cdc81a3":"Rectangular contour and overlapping singularities","p_f9ae5e946ff35707":"Similarity classification of parameter matrices","p_5d47f685f320c340":"Singular derivative at an endpoint and continuous extension","p_9e892a552d86bc0d":"Adjugate matrix and rank-one update","p_770bc61c01df2dc1":"Damped integral and undamped limit","p_2b9d0b31d56e5349":"Algebraic independence of the exponential function","p_ebb5e3ca56ee6b8e":"Deformation and contraction of an elliptic ring","p_a79580f3b85b87f9":"Fundamental group and winding number","p_84960fabd8e42482":"Boundary values on an annulus and obstruction to a harmonic conjugate","p_30adf77a6fe8083b":"Totally positive units of cyclotomic fields","p_e41432139dfdde5a":"Unions of curves and nonreduced structures","p_bdf7653b877aeb75":"Two sheets and integer twists","p_302de28f5685fdd9":"Projective flow and zeros at infinity","p_eb6a2424da68cdde":"Compactness via interval averages","p_4301de156bd92787":"Dissipative periodic equation","p_44665af390fb3179":"Phase averages and compact operators","p_52ef523695a8ae4b":"Root configurations and quadratic subfields","p_e0306459d92a02d3":"Weighted invariants and fibers","p_f93392ba0f45ec1c":"Homology of join operations and complement spaces","p_1703acd5475ace95":"Variable structure coefficients and the Stokes obstruction","p_e6cea35ac70f37b9":"Fast averaging with a moving phase","p_64b989b13c026516":"Regularity of kernel sections and finite-rank approximation","p_9096d8a0c82facdd":"Stable modes of a nonlocal heat equation","p_6938c1f02b57ba11":"Full correctness of the binary lifting method","p_80469f8e8b14b182":"General fiber and flat extension","p_88a215afe1caf2e1":"Compositum of splitting fields and real fixed fields","p_75e044328367f2de":"Tangent vector quotient and twisted homology","p_5fe26be04aecfb65":"Connection and curvature integral of a circle bundle","p_cba011fa22c55c19":"Convergence in measure and norm preservation","p_4491b3dc5c02fe7b":"Largest eigenvalue of a symmetric integral kernel","p_a84d6600116f8304":"Vanishing moments and Laplace limit","p_f63dcb96cca12a9f":"Capacity decision and optimal number search","p_eea541616c7bdf0f":"Laurent ring over the integers and maximal ideals","p_19057ca52295a5fa":"Permutations of roots and semidirect products","p_c464b18e10f95175":"Classification of cubic intermediate fields","p_165a7661a5617345":"Torus actions and integer normal forms","p_7777544ce022551c":"Twisting of a mapping torus","p_fc95100ce1271c2e":"Regular values of indefinite Hermite forms","p_8c5fbce5f9ebc6e2":"Global coordinates from polar decomposition","p_8b13cc984f200bd7":"Concentrating signed functionals","p_e00a4e85ca5271d5":"Cancellation and estimates for continuous functions","p_d6be82428ccecfe8":"Difference between weak convergence and norm convergence","p_f7d434db24e4902d":"Mean defect of the Poisson equation","p_ea8327b3dfebeaa5":"Laplacian reconstruction from local averages","p_9bf3174906993417":"Resolvent of the bilateral shift","p_6ce47b44af948167":"Approximate eigenvectors using finite windows","p_88d4e34e97fcca87":"Compact perturbations and additional eigenvalues","p_97e0c37e429cb554":"Galois group determined by the number of real roots","p_09dbc1f1e20ea890":"Commutativity of low-dimensional local algebras","p_396f807cabe4cc20":"Classification of local algebras and number of ideals","p_81de9f1507759d2b":"Complement of a sphere and Alexander duality","p_80ec9575f77ca53d":"Complement homology formed by intersection periods","p_2e81541611d8ebc6":"Reversal of the normal field and parity of dimension","p_7be9a661fb9d3f65":"Local convergence in measure and weighted strong convergence","p_0106cefc10241f01":"Gaussian convolution operator norm","p_c858139c6336902c":"Noncompactness and localization under translation","p_3592443d6c0b65c8":"Green kernel for a boundary value problem with advection","p_52e77ab5c3f4f5d2":"Contextual preorder of nondeterministic programs","p_02fbed1277b3cc81":"Quartic number fields and permutations of roots","p_31a0796ae007248f":"Integral-domain property of nodal surfaces","p_08a3981ca837782b":"Rational parameters of singular surfaces","p_7e928a793a178bee":"Normalization and conductor","p_b2852da98aeb5082":"Degree and surjectivity of multivariable power maps","p_2243084d5f5c92b7":"Torsion from identifying points and attaching cells","p_7ce749e5d341152a":"Sharp almost-sure convergence estimates via exponential moments","p_7c4263cc0d90b30a":"Whole-plane resolvent of a second-order Volterra operator","p_429610cd05b2a56b":"Approximate identity for the periodic Poisson kernel","p_8930c704a7921bb6":"Bounded harmonic extension on a semicylinder","p_bc7df2a81547a5f5":"Simultaneous computation of powers and derivatives by binary iteration","p_f58194725ee927e2":"Laurent principal ideal over a discrete valuation ring","p_1acb12b8f082d90c":"Ramification of an inversion-symmetric rational map","p_180d411e515e6ed8":"Glued spaces and the fundamental group","p_96f3529c963760ee":"Integral cohomology of circle bundles","p_c8c2203893756fbd":"Complex null cone and tangent bundle with the zero section removed","p_3c2c83f724e08de7":"Uniform decay of quadratic-phase oscillatory integrals","p_6d8f1a74d3840a34":"Weak operator convergence and determining compactness","p_5fd9bc61f2a12c42":"Heat equation with time-dependent forcing and resonance","p_87c4ec97153fae02":"Least fixed point of a recursive functional","p_953b5b50d0d03a96":"Fixed-point fusion by strictly continuous maps","p_3a5e3af1420d1406":"Recursive transformation by commuting matrices","p_dc6ac5adb1788ca5":"Algebraic number field generated by two square roots","p_60af8430b29ea536":"Reflection-invariant formal power series","p_682d59d2962c2775":"Singularities of a central-inversion-invariant ring","p_1e49b9235b18dd2c":"Double mapping cylinder with an attached cover","p_3a336bf2371a1d58":"Compatibility conditions for periods specified on multiple closed curves","p_a1c6cc57d3c5a97a":"Pointwise specification by correction of an exact form","p_6ba1a3ed5eaac69a":"Zero-exponent derivative of power means and logarithmic variance","p_e46c5df692b4ebe2":"Inverse problems for quasiperiodic differential operators","p_0b873d2cd8dd7de3":"Discrete spectrum and compact resolvent","p_7f512f5f247fedc9":"Sharp long-time asymptotics for time-dependent damped waves","p_bda9f3604704e615":"Correctness of deletion operations in interval search trees","p_c79a674b727d09cd":"Complex conjugation and quaternionic extensions","p_853efaea63c4505c":"Minimal faithful representation of a semidirect product","p_00f2f76336c8c91e":"Homology of twisted gluings","p_011224bee707fb8d":"Product with a circle and twisted homology","p_ac9b6e9c4e68f94e":"Critical set of a normalized spherical map","p_ad2014e09764e88b":"2nd-order differences and strong convergence","p_3791b012910f7754":"Characterization of second-order regularity using differences","p_1ac150ccd10fa11c":"Hilbert ellipsoid and best approximation width","p_cc9503df090870fc":"Zero-mean condition and periodic energy estimate","p_6d5aba84feb46920":"Leaf replacement in ternary trees and structural induction","p_bf9dd94cad7fd9d2":"Number of unprocessed slots in prefix coding","p_8585c82ed6cca093":"Counting flags and nilpotent matrices over finite fields","p_a551a29d3f6a19f3":"Nonsemisimple matrix with a single eigenvalue","p_bba44c0af28afb8e":"Splitting field generated by real and nonreal roots","p_69ccf83afbbba07d":"Dihedral Galois group and 2 intermediate fields","p_26257def662cb924":"Local index of a parameterized map","p_ed84ca3c73346c59":"Degree of a boundary map and crossing of zeros","p_4086158eaab8b102":"3-dimensional manifold with three holes","p_1f80115ce68b2f03":"Cell attachment in different dimensions and twisting","p_5b0de7a6c3ca2034":"Sharp exponent range for singular weighted convergence","p_a58dcd809fd0cd7b":"Boundedness of integral operators and threshold for compactness","p_cc3eb34dc9a97206":"Sharp growth conditions for global existence of ordinary differential equations","p_2ac0b338d85a7e32":"Amortized complexity of a queue using recursive reversal","p_a3edf16700ed3b91":"Integral closure and conductor of a cusp ring","p_0983c0a889afd318":"Numerical semigroups and a general formula for the conductor","p_a72f61c19275c372":"Norms and ideals of finite field extensions","p_334db1957b557146":"An example of contraction disappearing in a transcendental extension","p_e780223d03c20868":"Prime ideal chains and dimension in integral extensions","p_66122f147f66e13e":"Vector fields without zeros on the torus","p_290647713148e143":"Homology of four-dimensional manifolds with boundary spheres","p_1aab2ec020764ce0":"Inclusion map and intersection form","p_132e5d6570104450":"Multiple boundary components and a unique relation","p_614a73b286bb25c7":"Infinite product of an entire function with zeros on the imaginary axis","p_167a6933aa3ad13c":"Monotonicity and surjectivity via logarithmic differentiation","p_3e99f23d8e77a0d4":"Uniqueness of the minimum point determined by an infinite product","p_abf323417cc285db":"Fourier transform of the tent kernel","p_66630e3aa1e88e8a":"Difference between continuous and integer translations","p_904f915bbb599dc0":"First-order regularity of the heat kernel integral","p_a37c274c4fdf03b9":"From first-order data to second-order regularity","p_5932b4dfadd34797":"From the heat semigroup to elliptic equations and uniqueness","p_267be0c451d1fad4":"From observations on a dense set to uniform convergence","p_6d3c81c1afc9b60b":"Determining the connectedness of the configuration space of two points","p_f612d892811cfa11":"Normalizer and connected components of matrix groups","p_1bcdeee6ec87f0bd":"Left and right basis transformations and operator similarity","p_8fe0a0f8e289ff8b":"Sylvester operator in the non-diagonalizable case","p_7edca5bd3159071e":"Growth rigidity of quotients of entire functions allowing zeros","p_9934cddfeb2f7545":"Criterion for extending the quotient of a subgroup to the whole group","p_91f4566f951b30a2":"Obstruction to extensions in dihedral groups","p_eec14da6c94080ea":"Family of smooth functions closed under scaling","p_0935c1e2941f28d4":"Maximal subgroups of finite groups and generation","p_63a27ae1b883bf82":"Cover of a rational function field and fixed group","p_842a5695b84e9d5d":"Rational curve with four points removed and quotient group action","p_b9297fb69443de09":"Critical set of an interpolation map","p_1099f9dddba18915":"Low-dimensional homotopy groups of circle bundles","p_274d17bb5f974d7d":"Integer homology of a circle bundle","p_54568aedf1d83cf7":"Third homotopy group of a circle bundle","p_302e15d30c471371":"Distinguishing homotopy types using cup products","p_24c495af7a083247":"Two-sided finite truncations of compact operators","p_1e6cc9a9444fd73a":"Sharp estimates of Laurent coefficients in an annulus","p_a32c01678f33f7f7":"Uniform continuity of the cross-correlation and vanishing at infinity","p_8ead9c618af323b1":"Moving fixed point and linear stability","p_e1fe6ab91fe52ce5":"Asymmetric period-doubling bifurcation","p_0574df5b9cfa1dc5":"Weighted diffusion matrix and unique solvability","p_d20846e6d6d0f35e":"Positive definiteness of the inverse matrix and the discrete maximum principle","p_53e693a8e46fa606":"Difference between energy stability and monotonicity","p_40edd28c2ad8de08":"Common probability space and CHSH inequality","p_581afb215934ef6c":"Two-qubit correlations with noise","p_ca53ad54d65f8a50":"Interpretation of quantum correlations and joint distributions","p_87b4fdbfa65660bd":"Quantization of a free Dirac field with mass mixing","p_994d90d526856d80":"Linear-time merging of two monotone generated sequences","p_0ea0d2cbc040ef88":"Time discretization of variable-coefficient diffusion","p_35c91c7f6d5a59ba":"Simultaneous canonical form of anticommuting involutions","p_18f8af493f5452b8":"Fixing a sequence of functions by weighted averages","p_2d1f4b98cd6f0541":"From simultaneous complex similarity to real orthogonal similarity","p_4cc90a51dec3383e":"Polynomial subring identifying two points","p_948f8d722542aed7":"Weighted homogeneity and regular level sets","p_540db996f2fbc1b4":"Convergence at dense points and Fourier series","p_0328bb715b78eb8e":"Two-scale analysis of multiple-root splitting and roots at infinity","p_3b53f83652a36a4d":"Normalization and conductor of a double cover ring","p_e27940afa0e0fc29":"Action on the projective line and embedding of an intermediate field","p_3d6a5331b9b3f46d":"Power maps and mapping degree in complex coordinates","p_f6264e8eaf9868b1":"Bilinear integral of an exact form and mapping degree","p_d3f8c22ab4466278":"1 Invariants from differential forms","p_70f4d6037e6fe14b":"2 Optimal constant for continuous point observation","p_91388a9a3da92834":"3(1) Compactness of smoothing operators","p_30e73ff26c9e34d8":"3(2) Sign of eigenvalues and orthogonality","p_6b1cab98a31efc78":"3(3) Trace and eigenvalue counting","p_d8fe3d3de79166f7":"4 Approximate concentration on small sets","p_74476c1b32f8a932":"Required: concentration rate of an elliptic equation","p_49c799c9780287cf":"Selection A(1) nonlocal damping and uniform estimates","p_67f97c4f92c37867":"Selection A(2): case analysis of the decay rate","p_1c0ff9f9883021fb":"Optional B(1): periodic integral operator","p_567873664941a031":"Selection B(2): fixed points and parameter stability","p_86a5fbdc01624887":"Choice B(3): invariant graphs and attraction","p_3472cadee4249ef2":"1(1) strongly convex distance minimization","p_305da1115b171a5d":"1(2) Variational inequalities and stability","p_e76f690733b40298":"2 Weak convergence and convergence in measure","p_f402674211afe9d3":"3(1) Logarithmic coordinates and uniqueness","p_e12a08e9bf9dfd96":"3(2) response coefficients including zero modes","p_b5a4585fe593dce3":"3(3) Construction using Fourier series","p_7f4623bfcd08719b":"1 Localization and saturation of ideals","p_cdace8d0485a6296":"2 Commutative subextensions and composita","p_49c95325f96c70df":"3(1) trace constraint and connected components","p_764fd778bbd48723":"3(2) Submanifold structure of an orbit","p_4d65f6199b43b512":"4 Quotients with free actions and fixed points","p_637e30e850009eca":"1 Rank jump and projection","p_08e703427d2d37b8":"2 Endpoint effects of an oscillatory integral","p_e903fe31270eb588":"3 Counting intersections over a finite field","p_671a80f8b7be9464":"4 Euler-type operator asymptotic inversion","p_220d264a912c24cb":"5 Interpolation avoiding the excluded set","p_6abdd90881f6cf44":"6 Closed graph of a set-valued map","p_3bcf260a12733991":"7 From partial convergence to a holomorphic limit","p_020e9a092e335357":"First-order factorization and potential","p_285ffa47b8260473":"Transformed solutions and scattering coefficients","p_e5a5b0547b38ab6d":"Complete classification of square-integrable eigenstates","p_278760ebecab8319":"Localization and gluing of radicals","p_2dbb9e25e7b8a6e1":"Abelian subextensions and derived series","p_8aa14721332aab92":"Connected components of pairs of orthogonal projections","p_e6d540e1c2856bd9":"Embedding of a flag manifold and tangent space","p_646fa34883803de8":"Global and local invariants of orbit spaces","p_0c9929c73f8f2f74":"Localization, quotient rings and elimination","p_c4f07161e09d15ea":"Real maximal ideals, residue fields and nilpotent elements","p_58d5f029d02ac991":"Galois action and double cosets","p_2354768712ee04c9":"Mapping degree and Poincaré duality","p_ee2dce38f54b17f3":"Realization of map degree and product maps","p_73fa13f49859bb2c":"Degree of a map on the boundary and Stokes' theorem","p_1f558aac25018e52":"Compactness of weighted difference operators","p_70d9834cf5a30ba8":"Compactness of the image and nonclosedness of the range","p_70da07e6d79f37cf":"Even moments and analytic continuation of the Fourier transform","p_c4f989aa7435faaf":"Density recovery from moments","p_c28558a0d0ae625e":"Residue, Schwarz lemma and range","p_3de779144ddc8d29":"Perturbed integral iterations and uniform convergence","p_3dc7eaaae45ab63a":"Construction of finite-rank kernels and periodic orbits","p_3d64658e53d9902b":"Factorization and intertwining of differential operators","p_76aa475c624a8d94":"Wronskian and uniqueness of monic operators","p_e5e8a649d6c5c41d":"Two-step transformation and exponential-type eigenfunctions","p_b3ea1f33d77da24c":"Quotient space graph and fibers of continuous maps","p_2fc81563db590164":"Weighted integer lattices and stabilizer groups","p_e3a86051cf7acdb5":"Separating families of functions, metrization and convergence criteria","p_dd307e8030cccbdb":"Determining a holomorphic function from discrete observations","p_5acabbcbce86c653":"Conjugate action and multiplicative eigenvalue sequences","p_68962a0a43539a2e":"Singular kernel and choice of integration method","p_65dd564438ff7d7f":"Inverse limits of tori and dense subgroups","p_2ec1131f1f5d3f42":"Polynomial rings acted on by rotation and exchange","p_2bb65c478db69cbe":"Indefinite functions on bundles and critical points","p_6fa6b6847a28eab6":"Graph mapping torus and cellular chain complex","p_071fec18dd872375":"Time-frequency concentration of multiple signals","p_782069cbb63302f5":"Positive convolution and dense translations","p_013850ce3d6ecc50":"Rigidity of bounded holomorphic functions under higher-order differential conditions","p_3463cd2f2149b49f":"Eigenvalue problem with endpoint masses","p_fbb592ff0b307db0":"Conserved quantities and convergence of the two-step difference method","p_980d60cd8e545fbc":"Nonuniform string and dynamic boundary conditions","p_d0c1137dae1f9c91":"Statistical physics and thermal fluctuations","p_bd252778b46cfc18":"Ordinary differential equations and conservative systems","p_8c509f333314972c":"Correctness of algorithms and programs","p_fec78f0fe35ab648":"Computational models and lower bounds","p_d2e6c8eb94c830e0":"English composition for academic purposes","p_d6d531ef0e15b04d":"Real analysis and asymptotic analysis","p_420b4e8a4cedbc1b":"Geometry and topology","p_cfa12da6cc3f1eb7":"Matrix q-difference equation and meromorphic connection","p_d46c7dfe021fba40":"Jordan-type degeneracy and coalescence of poles","p_0693d496ab77eed8":"Robin boundary value problems and energy estimates","p_89cf58a076d494e7":"Nonlocal semilinear boundary value problems","p_3f456165fdbbc836":"Recurrence points, divergence and periodic orbits in the plane","p_2ecee892e069c7e6":"Quantitative counterexamples to transience and recurrence","p_61b73323ceee28ae":"English-to-Japanese translation related to mathematical research","p_40394b8dd1f1a1c2":"Second-order error of an approximate identity operator","p_a9443a1b5d6f2f01":"Countable image and incommensurable periods","p_0c149c0bca4f3068":"Affine similarity relations and Jordan structure","p_742fb51b73c384bb":"Common eigenvalues of adjacent operators and the greatest common divisor of polynomials","p_115b8ae1824ad274":"Real closed fields and normal closures","p_a31b68a56725e349":"Automorphisms of normal closures and dihedral groups","p_818cdde8f2ac5e39":"Complete classification of quartic intermediate fields","p_d2c92489eaf02355":"Integer coefficient test using finitely many roots of unity","p_3d9857366ed70c7e":"Triangular derivations in characteristic two","p_b5f010800dce8631":"Generators and relations of the kernel of a derivation","p_77c1e6300d29602b":"Homology rings and nilpotent elements","p_8763b2bf0eb00a4a":"Split exact sequence of the free loop space","p_3e4fd1e9c70e1a2c":"Actions on covering spaces and the infinite dihedral group","p_50b397521f6ec775":"Distance to an algebraic hyperplane","p_3628a5f8f8458521":"Quantitative criteria for closed and dense kernels","p_3827d55767c639d3":"Injective holomorphic maps on a punctured domain","p_eae4ac5eb19410e4":"Complete classification by normalization and omitted values","p_be0d7f1201d4d08c":"Boundedness and compactness of thermal multipliers","p_99e5661593cecb7e":"Convergence rate from strong convergence and regularity","p_3f8a8c49a7b522f1":"Operator norms and moving frequency windows","p_2777dba908db6444":"Discrete maximum principle on nonuniform grids","p_b9d8e5c9b3120aed":"Lattice asymmetry and an upper bound on global error","p_874d7d656c1eb523":"Modal decomposition and uniqueness for nonlocal beam equations","p_4ab25648471b556c":"Tunneling scattering in different asymptotic media","p_8cab0ef8d6d36ff9":"Probability current conservation with point interactions and uniqueness of scattering solutions","p_481cb18561ad57fb":"Reconstruction of a massive vector field from a reduced Hamiltonian","p_145e719dd9e345e1":"Reduced canonical quantization and equal-time commutation relations","p_61129f7d5f817dd5":"Initial-value construction of field commutators on periodic space","p_5777c28d248aa1b1":"Maximum subarray sum with length constraints and monotonic queue","p_aa2d562f9e7df820":"Identification of an integer program's output specification","p_04d3e6f8b26994c8":"Invariants and termination of two-phase loops","p_34c6b9ce54426f4f":"Semantics of a remainder-operation program","p_3121f1bec18771e6":"Loop invariants by synthesis and total correctness","p_ab41ce1d9682ddde":"Nonvanishing of multivariate polynomials and zero estimates on finite grids","p_c13ad4b475de978e":"Self-homeomorphisms of spaces with branch points","p_097e835f86fe69d4":"Homogeneity of topological groups and local branching structure","p_0c792c440c4ea080":"Uniform boundedness of a family of nonnegative series and block construction","p_624034a1b03e3875":"Analyticity of the weighted Laplace transform of antiperiodic functions","p_26714018f32f3896":"Meromorphic connection and principal part of a double pole","p_50dda043a1bc3598":"numerical analysis","p_bcce92569ad7a117":"Sharp examples for weighted greedy algorithms","p_63ccb8f2d95f750e":"Approximation guarantees by the allocation method","p_c0e02874415070d0":"Termination and growth of recursive evaluation","p_6022971d27303a8b":"Translation of academic English","p_abc44de3af75b775":"Orthogonal decomposition and operator diagonalization","p_2de3a42ff7a94ef3":"Heat equation with boundary conditions","p_3fee84c7f8125a17":"Intermediate fields of cyclotomic fields and real subfields","p_4c3166b55e2b1dba":"Ramified extensions and discrete valuations","p_c5623f5ae0475959":"Geometry of determinant constraints and symmetric matrices","p_d299bec2c1d48776":"Nonlinear oscillation and dissipated energy","p_100f0965e7448e83":"ω-limit sets and invariance","p_e2af06860b7feecc":"Convergence to an equilibrium point and energy barriers","p_7d4438d320a97753":"English-to-Japanese translation: dependence between assumptions and conclusions","p_5b1e2559efc7cdb8":"Boundary dissipation and energy estimates for the diffusion equation","p_49aef3a93378b182":"Contractivity of time evolution","p_574d35e03582a475":"Existence, uniqueness and uniform attraction of periodic solutions","p_9e599f59d4407687":"Convergence and regularity of regularized integrals","p_0877095b72b5af38":"Elimination of branch terms using an Euler-type operator","p_9c03d03133fc4b74":"Quadratic residues and Wilson congruence","p_7962c4c299e6f6b0":"Congruence invariants of the extended Euclidean algorithm","p_83cdc48632fb0133":"Construction of sums of prime squares by threshold stopping","p_ec1db6a50ec37c53":"Decision algorithm and correctness for integer predicates","p_2652423ceae8434d":"Partial correctness of enumeration programs and termination on all inputs","p_411fadf32defca08":"Logical structure of English and academic translation into Japanese","p_cc19d0a4e5e094f8":"German indirect speech, comparative constructions and Japanese translation","p_1ce206ae9c032282":"Branches of the logarithm and jumps in boundary values","p_f1e8edf39ae60564":"Linear space consisting of invertible matrices","p_c19b663ac378916e":"Construction of a matrix space and an upper bound on its dimension","p_1a209b1c8b0419b0":"Order of a finite affine group","p_de7ddfb56424fe53":"Sylow subgroup order","p_5a50ce48fbff2b23":"Construction and center of affine p-groups","p_53bf204882942b38":"Periods of closed forms and local antiderivatives","p_9310801a43432b59":"Lebesgue measure and separated compact subsets","p_09fa44b5622c996f":"Automorphisms of splitting fields and dihedral groups","p_467a3432df75ed45":"Changes in groups caused by intermediate fields and base field extensions","p_817244ff86028328":"Continuous frames and lifting to the special orthogonal group","p_c886f3d228a0afbb":"Relative Euler characteristic and coefficient field","p_7441fdcc507c0532":"Boundary and duality of odd-dimensional manifolds","p_758a218f6577796f":"Constraints on Betti numbers from boundary components","p_5c9e026b591d6b54":"Weighted averages and orthogonal projections","p_04637fbde11d8491":"Norm estimate for weighted averages and equality","p_51b0ad64a20b1566":"L2 approximation by nonnested partitions","p_bc468840d0b81ca8":"Strong convergence and endpoints of relaxation iterations","p_6188779b82fc8f8a":"Difference between strong convergence and operator norm convergence","p_0534b6a8119054f0":"Exterior Cauchy integral and polynomial part","p_64f58de9ddbb9c90":"Elimination of transcendental factors in differential determinants","p_b381b0d0493fff89":"Moment vanishing and determination of constants","p_6d5597b3cc7f027e":"Neutral shielded charge of finite radius","p_d5e60788071c5dcb":"Correlated quantum oscillator","p_f73820b994c05e0c":"Normal-mode oscillation and partition function of a heteronuclear diatomic molecule","p_20ba40d15f963122":"Photon interactions and inverse problems","p_7c77ebe36643773a":"Gravitational scattering of a two-body system with finite masses","p_9523482aaaf110dc":"Two-stage operation of a two-level system","p_501289aa40cf3c0c":"Constrained mechanical systems","p_bc765193ded8fe2b":"Conservation laws in time-varying media","p_d352fa1f4eff0191":"Periodic boundary conditions and electromagnetic potential","p_a535fd8d49f32b09":"Shielded electrostatic field and dispersive transverse wave","p_b20736a88e5adcbb":"Quantum measurement and state estimation","p_6145506dabfd920e":"Coupled oscillators and statistical mechanics","p_e2d1d445213678aa":"Boundary conditions for static magnetic fields and magnetization","p_17f222eb95d91133":"Analytical mechanics: compound pendulum","p_dd4ca6ad4df5babc":"Electromagnetism: vector potential and magnetization boundary","p_57f39419ea364f4c":"Statistical mechanics: two oscillation scales and heat capacity","p_20d037a93bccb649":"Quantum mechanics: complex measurement bases and conditional states","p_b5fc3aa3a2db1e97":"Mechanics: asymmetric coupled torsional oscillations","p_92cbaba874f11ed0":"Mechanics: critical damping and constrained motion on a conical surface","p_a01013cffcd0dea3":"Damped rotational motion and conserved angular momentum","p_9732bf4e623d58b7":"Fields and energy in polarized media","p_93bc0d303ff502a3":"Circular polarization dispersion in magnetized plasma","p_ab9aa65b41eff021":"Radial harmonic oscillator","p_8e4c786c25daf584":"Classical gas in a bounded harmonic trap","p_ae4ad7581dbbeb78":"Local pressure in a rotating annular vessel","p_f3a1e7030957cea1":"Mathematics I: conservation laws and singular matrices","p_c42b241721c650d8":"Mathematics II: order of products and normal forms of semidirect product type","p_a214309287f52e07":"Mathematics III: weighted approximation and regularization","p_d25fc1fce2e7621e":"Physics I: total derivative term and canonical momentum","p_1440e5f779a92fa4":"Physics II: coaxial system with a graded permittivity","p_6bdcc69eb6690793":"Physics III: thermal activation of excited magnetic centers","p_cd8b9631bd03674d":"Chemistry I: trial functions, term symbols and molecular orbitals","p_46461d290ab8cf95":"Chemistry II: multiple metal bonds, conditional formation constants and molecular geometry","p_36adf26b31e961d7":"Chemistry III: carbon skeleton construction, acidity and basicity, and determination of substitution positions","p_c91f0297ec259d18":"Mathematics I: affine consistency of height data","p_4e1e5aae8cc672fc":"Mathematics II: positive operator approximation by interval averages","p_8740830e460fdb95":"Mathematics III: unit group and conjugacy classes of dihedral groups","p_cda1fd89ac3ea5f3":"Physics I: electrostatic self-energy of a hollow distribution","p_e95c4871eee59258":"Physics II: energy damping and mass-weighted oscillations","p_6ef7f19f57c142ed":"Physics III: dispersion of magnetic-field levels due to confinement","p_255a1d968a50130f":"Chemistry I: reverse Carnot cycle, molecular orbitals and isotope effects","p_b643ae93285376fc":"Chemistry II: electron configuration, three-center four-electron bond and coordination structure","p_90f4e33a04f34b9c":"Chemistry III: Two-step acid dissociation, reaction branching and stereochemistry","p_d1c8dc3a3acffaa2":"Mathematics I: orthogonal polynomials and differential operators","p_618816258495d143":"Mathematics II: finite topology and invariant closed sets","p_ef720cf2500ad9d8":"Mathematics III: residue classes and the kernel of a group action","p_700ef490e3c975c1":"Physics I: charged annular disk and axial stability","p_676327e35846fabb":"Physics II: quantum box with anisotropic mass and momentum kick","p_101995f220e7cf61":"Physics III: two types of magnetic moments and an alternating coupling chain","p_748e59aba7953ce0":"Chemistry I: freezing point, chemical potential and electronic structure","p_78fc99d90e41129d":"Chemistry II: fractions, common ions and two-stage titration","p_b2c4e7621fee9ab4":"Chemistry III: aromaticity, cyclization mechanism, symmetry and reaction rates","p_f9ae936c8fcc8359":"Multicenter bonding, coordination structure and magnetism","p_9f34b562bb056aa1":"Electron distribution in multicenter bonding","p_5a8744415c7f9e8e":"Geometric isomerism of octahedral chelate complexes","p_2c4e3e2c179eceb2":"Optical isomerism in chelate complexes","p_5c8d0dbc9751f479":"Oxidation numbers and spin states","p_8c290e4383bc1fe0":"Ligand field energy and mixing ratio","p_cda54ea3dccfe289":"Bonding and vibrations in metal carbonyls","p_943bd67f90502340":"Stereoisomerism and labeled cyclization reaction","p_b9003ed1e46e65f5":"Conjugate addition and carbon tracing","p_77b74fdba6cc56c8":"Intramolecular aldol cyclization","p_252fa0aed7900116":"Conjugate enolate formation and material balance","p_1c9b19c272ce55ac":"Reaction pathways, stereochemistry and functional group selectivity","p_46d93da689221a9e":"Isotope tracing in hydroboration","p_5ad92cb4ac171f94":"Reaction sites in aromatic nucleophilic substitution","p_2c07317f8dc6c860":"Conjugate addition of organocopper reagents","p_1e3bd00ac7a46093":"Chair conformation and E2 reaction","p_ddab7c2746bbd9e6":"Ring opening and reclosure of cyclic anhydrides","p_05ee1bd7868b7d7c":"Selectivity of reductive amination","p_3fa35b8d5b437522":"Synthetic routes and spectral interpretation","p_6551067f195b8f39":"Selection of protecting groups and reduction conditions","p_b871ec0fc41b6591":"Order of hydrolysis and protection operations","p_a12c84e1602c7f91":"Equivalent protons and first-order NMR","p_4265e70ca6b67ee0":"Ratio-of-ratios series test and orthogonal projection","p_cdd1fff37a547ce1":"Divergence by the two-step ratio test and adjacent ratios","p_cc001f8835dc0580":"Logarithmic factor at the boundary","p_b3045eabf9360425":"Polynomial inner product with point evaluation","p_36e50b8a48cddcbb":"Orthogonal projection and best linear approximation","p_01d0f3cc38245d8b":"Bessel inequality and equality conditions","p_46c5edf2890fd5ce":"Positive-term series and approximation by inner products","p_c9a0bc9aefe2e780":"Squares and square-root counts of permutations","p_7d1ffff046ddefb1":"Equation of motion and conservation laws for a torsional oscillation system","p_150e0dfb468f2a64":"Dispersion relation of a chain with alternating couplings","p_cfbd7e4c5f1a3cce":"Electrostatic field of three point charges and far-field expansion","p_8b23bbf4cafa471f":"Dipoles and force gradients in an electrostatic field","p_1f653924aa2e4b44":"One-dimensional scattering by a step and a thin barrier","p_de383d137fc85969":"Free-particle wave packets with phase correlations","p_6653822e591319d1":"Exchange symmetry and optical excitation in the box model","p_44f9f35ba566c6bb":"Steady-state rate of chain reactions with scavengers","p_d915e10aad618275":"Scattering by a finite barrier and design of transmission resonances","p_a464a9e499631709":"Plant diagnostic records and pathogen classification","p_90a445db5b3e8e91":"Waves in conducting media and inverse estimation of electrical constants","p_83725cc9ba9a384f":"Quantum oscillator with phase correlations","p_bb873f70de95060f":"Mixed damping of symmetric and antisymmetric modes","p_dae7e76453d6fcce":"Elimination of residual oscillation using two angular impulses","p_0c81eecbc18ed947":"Waves and energy flow in a uniaxially anisotropic medium","p_755fe83a02c7046f":"Reflection, transmission and absorption at an interface with a resistive sheet","p_4d725e9e7f600fc1":"Quantum mechanics: oscillator with mixed terms","p_61b98773708966cb":"Mechanics: equation of motion for an elastically supported string","p_d91aad13fa0f6bfc":"Mechanics: torsional mode of a free-end rod and initial angular velocity","p_4ab300f340e25359":"Electromagnetism: polarization trajectories of obliquely propagating waves","p_1d0267ae3e2b6dd8":"Optics: phase estimation and transmission through a two-stage grating","p_da6afe154ee53ee4":"Quantum mechanics: canonical transformations and harmonic oscillator","p_968ac11d7a6ce2cb":"Quantum mechanics: symmetry and simultaneous eigenstates of a two-mode system","p_363dd4a830581ce0":"Statistical mechanics: occupation and density of states in a two-dimensional two-branch Fermi system","p_003f285b3e8b89e4":"Statistical mechanics: low-temperature heat capacity at fixed particle number and inverse inference of the dispersion relation","p_0263a061c6695592":"Mechanics: waves and boundary conditions of a string with continuous elastic support","p_c89712d912e8770a":"Sampling from a finite population with pairs","p_b478b3c1622aee60":"Common tangent lines and integral quantities of quartic functions","p_a4ecbd2dc94abdc6":"Quadratic function on an interval and uniform approximation","p_d33fed1b403055cd":"Overlap of arithmetic progression sets and generating function","p_dbd88d67864a420f":"Integral equation determined by finitely many integral quantities","p_4366a1a39c6c5c73":"Analytical mechanics: constrained systems","p_a68eb72ebc71f940":"Cylindrically symmetric electrostatic field","p_2a5eead9fe78ee56":"Electrostatic potential and interface conditions","p_35df9ff36a37dcc3":"Electrostatic energy and virtual displacement","p_9bd953cf613e2e56":"Interfacial electric pressure","p_992706dda101fa3c":"Steady radial conduction","p_249d28b48192d710":"Free surface charge and conducting interfaces","p_0d94e65ce4f96988":"Poynting theorem","p_55f11249645a5169":"Time average of electromagnetic energy","p_2d4fd28613ae0bca":"Wave equation in a conducting medium","p_e98fd49b3450ea13":"Phase velocity, refractive index and weak loss","p_b9acff96d07c291f":"I, Question 1: small motions of a two-point-supported ring","p_e968937f476c3f46":"I, Question 2: recoil measurement of a stationary scattering target","p_b962aa3a12c2bf9b":"II, Question 1: coaxial electrodes with tilted permittivity","p_31aecd3179dea996":"II, Question 2: wave speed and impedance","p_6ca81474f5b28010":"I Question 1: three-degree-of-freedom oscillation with fixed ends","p_b642693c42c16bd4":"I Question 2: steady precession of a conical top","p_927a3d972cbc8d44":"II Question 1: electromagnetic energy in coaxial lines","p_971aa9a42fae34ba":"II, Question 2: transient induced current and mean force","p_52da355ef226ce0e":"II, Question 3: electromagnetic waves with finite conductivity","p_6cc38fadf7b748d2":"I Question 1: inverse-square potential and capture boundary","p_69b5bc545c3ffd6e":"I, Question 2: coupling of translation and torsion","p_d0cfcbe75ff5759b":"II, Question 1: connection and regrounding of coaxial electrodes","p_4fbe9ffadd2fb6f0":"II Question 2: potential and gauge freedom","p_1b7dbd0471a8c16a":"I, Question 1: coordinate transformation and damping of a simple pendulum","p_1a16afadc2067b10":"I, Question 2: cart with increasing mass","p_13fb737bb44f1486":"II, Question 1: virtual displacement of a coaxial dielectric interface","p_2d4537b163672366":"II, Question 2: energy balance of a time-modulated medium","p_2de05c0f8821e4a7":"Point well with a step","p_3320657f9df7ba51":"Superposition of angular momentum and current","p_6015d3c9f1eb8b54":"Adiabatic throttling with changing flow velocity","p_215c87adddb18bf0":"Degenerate finite two-level system","p_25e813e5fc891371":"Information content and constrained maximum entropy","p_c4125dd13b3d63be":"Rotational flow and vector identities","p_5029e64a865e3ec6":"Complex integration along an eccentric circle","p_d57c6785125b1981":"Weighted self-adjoint operator","p_245bf1ef125b01cb":"Periodic response of damped diffusion","p_c0b300ea8e668055":"Radiation balance of an artificial planet","p_f2ce29ad382b9601":"Chain reaction with an added scavenger","p_41bfea253b3f2bee":"Two-dimensional nucleation at fixed thickness","p_27d5ec5eec898363":"Two suspended rods with added mass","p_2303304661f7e6d4":"Relativistic scattering of an obliquely incident photon","p_7d3d0173770bda8d":"Radially nonuniform coaxial dielectric","p_661520becf36e16f":"Electromagnetic waves with a conducting sheet between them","p_cf56eb0b1b2a9d4a":"Two oppositely directed impulses and the harmonic oscillator","p_caced994bf0aebe7":"Resonant transmission through two point barriers","p_180dcddc131d31d3":"Partition function of a classical gas under gravity","p_72702fd2d3339363":"Two-sublattice mean field of an antiferromagnet","p_da25067e7c9fe279":"Heat engine with thermal resistance on both sides","p_2327aa5a5f0bb7cb":"Low-rank inverse matrices, resonance and integration","p_bfe88a545a39adbf":"Singular radial vector field","p_f23e69d040b821c1":"Fourier expansion of a quadratic function","p_990dbdc50ac1ffff":"Planets with internal heat and nonuniform gravity","p_9cc6045461c81a1a":"Quadrupole magnetic fields and pressure balance","p_890e574c6b8cee33":"Enzyme reactions under competitive inhibition","p_3135b782a8ca048a":"Curvature and condensation of charged droplets","p_6958ad8df0bfc077":"Landau states in a crossed electric field","p_47889b36bd0a8818":"Noncommuting two-axis spin pulses","p_62876ada38eeaf35":"Sudden switching to a spring load","p_ac1d62888b24d667":"Thermal fluctuation piston under harmonic loading","p_c5f5e1666754b861":"Two-level matrix with complex coupling","p_5d6901cf53349c44":"Regularized rotational vector field","p_bb7fccd49c8891d7":"Harmonic conjugate on the slit plane","p_999982f66a557bfd":"Fourier integrals with a double pole","p_99b4c723fccec637":"Rare-event limits with missed detections","p_ede0380766d597e7":"Diagnosis of a moist air parcel undergoing mixing","p_1dd5b1881d852b74":"Three-mass system with a long-range spring","p_e8a917da2e41e027":"Steady precession of an annular rigid body","p_a3dc3cb4ea8b47be":"Electromagnetic energy in coaxial lines","p_72cfa09122e896b1":"Average electromagnetic force between resonant circuits","p_8e3d491b5435cc8c":"Waves in a conducting medium retaining displacement current","p_c857e660b6433b81":"Oscillator in an external field and a two-particle system","p_c5e33f2a6c005ac7":"Perturbation and anisotropy of degenerate spaces","p_af715dfef457f531":"Reversible engine with background energy","p_b707ebe435c46ab0":"Selective effusion of a gas mixture","p_959255f0183579d4":"Rank-one matrix update","p_77643e6ce873de40":"Regularized radial field","p_25e16f2f15f313e9":"Complex integrals and singularities","p_50f45daba887d313":"Regular singular points and polynomial solutions","p_19e0ce55ba4bda04":"Binding of a rotating dilute outer layer","p_1a1ac7d22dea7e59":"Gas column of finite thickness","p_b924833f18bce727":"Radiative equilibrium with internal heat generation","p_0ede1889164d5324":"Scattering and capture by an absorbing center","p_07d59f475c4f7c07":"Translational and rotational coupled oscillations","p_d51f92f14ebad611":"Coaxial capacitance and discharge of a nonuniform dielectric","p_3ca8891315ebdbbe":"Gauge and transverse waves in a medium","p_02562da4616cfa84":"Quantum transmission through three asymmetric regions","p_52bfd02e3a331ae1":"Angular momentum algebra and two spins","p_b16564c74af6a103":"Recovering state variables from thermodynamic differentials","p_d36ff7bd87a263d8":"Solidification with a finite heat bath","p_a00ecc2b3e9ad733":"Quantum solid model with two oscillation frequencies","p_0a3f1ae79d665737":"Resonantly forced ordinary differential equation","p_d82bb651239a0c00":"Jordan-type recurrence","p_e7dd211c102bdaf3":"Product rule, flux and topology of a region","p_e2b8c50781142716":"Hyperbolic functions and poles of order two","p_e9bff344ff72b3cc":"Distance estimation using parallax, mixed light and extinction","p_c52a570fc50e385e":"Drift distribution and warm plasma","p_42c45ff566a7c7d4":"Periodic chain of alternating springs","p_427ef12a72fadeed":"Rigid frame supported by a unilateral constraint","p_6cfdb13023d6120e":"Grounded wedge region and image of a neutral conducting sphere","p_7d46e7e9001e7222":"TE mode in a dielectric-filled waveguide","p_a069985cc853b5b5":"Harmonic oscillator with phase","p_74eb3be56c891bc5":"Vibration and rotation and the exchange symmetry of fermions","p_3f15d55456cd9cb3":"Microcanonical statistics of degenerate excited states","p_f46914c31271edfb":"Excluded volume and a gas with attraction","p_eb27af4b0862bcaf":"Rotation and scaling of real matrices","p_458ae84df148f879":"Anisotropic gradients and transport conservation laws","p_5b02beb7f4e779f1":"Multivalued powers and a displaced contour","p_2ead3b708a48eadf":"Compositional data and inference of internal structure","p_f65582e634540441":"Nonisothermal atmosphere with a finite upper bound","p_143adcba4b39359c":"Conductive surface layer with a volumetric heat source","p_4ff82f4ab7bb3f06":"Rotation of a rigid body and torsion spring","p_e3b140435ec31f76":"Constrained motion and stability","p_052abdd936412e79":"Dielectrics and the energy of electromagnetic fields","p_2f0f702ca996c008":"Magnetic circuits and field energy","p_f49a21627ff0ec53":"Displaced oscillator and quantization","p_6341e421c931dc7e":"Constrained particle in a magnetic field","p_e28edabf0d2da5da":"Exact differentials in thermodynamics and nonideal gases","p_862200b05c9d443a":"Transfer matrix of an antiferromagnetic chain","p_ed70f4518d5752d5":"Low-rank structure of symmetric matrices","p_b50f96b546a1bde0":"Coupled differential equations for exchange and loss","p_5c115816772acbb5":"Vector analysis and surface integrals","p_d3504fc1bdb9be83":"Periodic integral using a double pole","p_6e3edbfae4cd575b":"Planetary radiation budget and observations","p_090dacb4525d48da":"Horizontal motion with thermal wind and friction","p_0ee5dd3c90834524":"Motion and impact of composite rigid bodies","p_1ad7a6916b5edea8":"Initial acceleration due to a constraint","p_e14f49c38bb5a27c":"Coupled oscillations and mass normalization","p_375dc5aece4150ee":"Conservation laws in a time-modulated medium","p_cf3c1da815b8ae94":"Magnetic field gradient and mutual induction","p_84c565a3f2148153":"Degenerate perturbation and crossing field","p_4c2b6b87492a370e":"Spin measurements and time evolution","p_9adebe4be9262408":"Variable heat capacity and regenerative heat engines","p_8da65a338a7c95aa":"Three-level system and level-crossing cooling","p_c4a532cb9aa9c7e1":"Rank-one perturbation and exceptions to diagonalization","p_17b5b04f9a941adb":"Cross product identities and conservation laws","p_96d0e96759ea0eeb":"Logarithmic polar coordinates and residues","p_0cb927eeb85f61d2":"Internal structure and elastic waves","p_57af30c1dc308098":"Wind-driven circulation and boundary layer","p_8f2e58d9bc8fa6cf":"Stellar scales and evolution","p_e11f1db889129438":"Motion on a nonlinear constraint surface","p_995d2541ec8b2954":"Horizontal circular motion on a curved surface","p_b64f9c9c6d144c96":"Constrained motion on a sphere","p_f1e5cd080353415c":"Horizontal circular motion and contact conditions","p_51c8dde911fae74b":"Dimensions and unit design for electromagnetic quantities","p_42d7b5097de3bc38":"Inverse problem for a smooth charge distribution","p_635bfe99d09a2c09":"Coaxial capacitor with tilted permittivity","p_28de1aeaa5310990":"Nonuniform magnetic cores and air gaps","p_36cdb568ce30e4db":"Rotated quantization axis","p_2c9d18681d580d7b":"Quadrupole-type angular wave function","p_56887e74fe20a33b":"Composition of different angular momenta","p_d8dc9e093e43ec33":"Viscosity measurement using two springs","p_5e261cf45700092f":"Elastically supported string","p_9febbee77f64687e":"Corrected central force and orbital precession","p_334bdd216024f71e":"Charged shell and nonuniform spherical dielectric","p_3088649e50ea667a":"Nonuniform current and magnetic field gradient","p_ed9678ece0bd1620":"Two counterpropagating electromagnetic pulses","p_bc921a3900eaa18b":"Elliptical orbits and contact maintenance","p_ec0d7e6f8d895780":"Nonidentical transverse dipole oscillations","p_47ecbc4d08916bf8":"Image charges for grounded conductors at right angles","p_d385567ceb849a84":"Displacement current and waves in a conducting medium","p_546577b3ba8ea8f8":"Extensible rotor with an inertial disk","p_f2c506c83669bb98":"Dielectric coaxial system divided along the axial direction","p_c6d8617bb8656057":"Transformer with a nonuniform magnetic circuit","p_6a5600704fce6b2b":"Electromagnetic waves in a resistive thin film","p_1edb20a8fd8599e4":"I Question 1: torsional oscillator coupled to a triangle","p_cc748da16e8dc23e":"I, Question 2: repulsive scattering from a finite distance","p_93f52299522747f0":"II Question 1: charged ring and quadrupole field","p_d2597653894e584f":"II Question 2: current shell and oppositely oriented circular coil","p_ccabd39885b85bdf":"II, Question 3: pulse response of a gapped magnetic core","p_9344c0bd3e7ed863":"I Question 1: support-point excitation and dissipative resonance","p_6bee99da001fbf3c":"I Question 2: isotropic harmonic motion viewed in a rotating frame","p_63ce8ebfc7d8771f":"II Question 1: spherical capacitor with two dielectric layers","p_6cd2f6a485903c88":"II Question 2: two-air-gap magnetic actuator","p_6bb3e0f26f154183":"II Question 3: energy transport in standing electromagnetic waves","p_05d3dd6d932de0ae":"I Question 1: grooved disk and movable point mass","p_b480c220bfa606eb":"I Question 2: dimensions and magnitude of the central force correction","p_eb9a0730cd24f4b3":"II Question 1: charge and pressure of a conductor with a cavity","p_f6125e3c8ae14f9e":"II Question 2: DC magnetic field of a hollow coaxial conductor","p_354abe49c90d2d73":"II Question 3: traveling pulse in a parallel-plate transmission line","p_7232de95261cfc0b":"I Question 1: pendulum and collision of an eccentric disk","p_651ab23238bb4426":"I Question 2: torsional oscillation system with periodic boundary conditions","p_f91c2175f36c9f97":"II Question 1: fields of distributed charge and finite current","p_20eccf4180f1cbec":"II Question 2: TE mode between parallel conducting plates","p_a3d5057134a9967a":"I Question 1: two-component central potential and orbits","p_7d215f9388a153c2":"I Question 2: slow and instantaneous changes in central force","p_990e28a4d226bd72":"II Question 1: charging and discharging of an annular plate and displacement current","p_5beccfe76ce26994":"II Question 2: nonuniform conductivity and energy inflow","p_c5f8b191b97f6a98":"I-1(1)","p_647507675b31a716":"I-1(2)","p_fb6a5c43313dbc8a":"I-2(1)","p_3371b3d5bee71bf1":"I-2(2)","p_a0742476876e81c6":"I-2(3)","p_1e1d8e0b549ef3f4":"I-2(4)","p_5d5c302ba858b47a":"I-3(1)","p_67ded5aabd26c954":"II-1(1)","p_1a1456f27fd4892f":"II-1(2)","p_1de292afabdceb75":"II-1(3)","p_017cd3cb8a74ab15":"II-1(4)","p_ec2943d27b9eb21e":"II-1(5)","p_cef0d980ce6f7b03":"II-2(1)","p_893e877fe4e0c3bd":"II-2(2)","p_ef13242f32334972":"II-2(3)","p_166038ba3232bc98":"II-2(4)","p_dcda01565afa53b1":"II-2(5)","p_e24f9cf1c1fdb399":"I-1","p_1f3b8b9f31a5453f":"I-3","p_486e07dfefc83b8c":"II-3(1)","p_76bb7f3cabd3a21d":"II-3(2)","p_27aa1daf2d572b6d":"II-3(3)","p_55aafb0fef5d41c4":"II-3(4)","p_e1f284fff7ada5e4":"II-3(5)","p_7e87051a322f22d2":"I-1(3)","p_f0d148ec01b551de":"I-1(4)","p_9ffb4f9a098ce6f5":"I-2(5)","p_98e205635b3817db":"I-2(6)","p_3edb418577ce36ae":"I-2(7)","p_219661321732b8d9":"I-3(2)","p_85cff05bbe2b5e7f":"I-3(3)","p_d7f3e02ea8a0e0a7":"I-3(4)","p_0e7ae12adb93bd94":"II-1(6)","p_cbcc99b7a6b446ff":"III-1(1-A)","p_e8869cd121c7fd1b":"III-1(1-B)","p_3612f99b3ce77138":"III-1(1-C)","p_d1cca078ddc5bb46":"III-1(1-D)","p_20cf85199e119fa2":"III-1(1-E)","p_71458fba3916948c":"III-1(1-F)","p_dc1bca7032de4bf1":"III-1(2)","p_7a30b6d02df0685e":"III-1(3)","p_ab01793a4d281f61":"III-1(4)","p_75d4fa8a6020a7d8":"III-2(1)","p_060af1c9076c988a":"III-2(2)","p_19183d5ef96955c5":"III-2(3)","p_bc0e0a48ab9b0787":"III-2(4)","p_bf7fd5214ed189c8":"III-2(5)","p_0100816d4dba5778":"III-2(6)","p_84abd50f96b348ed":"IV-1(1)","p_6372e3165b9bc3ef":"IV-1(2)","p_f829fa0e9bf5c2fb":"IV-1(3)","p_2a582299b4d99d23":"IV-2(1)","p_0d2868649a1d4af6":"IV-2(2)","p_b5d7d0178be5e16a":"IV-2(3)","p_f56577e5aa1f16d6":"IV-2(4)","p_6e1c365b50357954":"IV-2(5)","p_6840cc90b5ad1f90":"IV-2(6)","p_14bd45113523f416":"IV-2(7)","p_c602540d8916ccc6":"V-1(1)","p_b0334b64b4a80c4a":"V-1(2)","p_3159750e4133c364":"V-1(3)","p_7d542abf799d708a":"V-2(1)","p_97d80406090210f9":"V-2(2)","p_72ee251940230476":"V-2(3)","p_acd8c1e715977a1f":"V-2(4)","p_39125c0aa68677f6":"VI-1(1)","p_fb59da242472afc1":"VI-1(2)","p_c7eebd2e21f82fa5":"VI-1(3)","p_7e42b1483c26745f":"VI-1(4)","p_3379558dedb70011":"VI-1(5)","p_214c9deec9f028a2":"VI-2(1)","p_6ff66fc24906b9bc":"VI-2(2)","p_d1e160e4edd2e2db":"VI-2(3)","p_8a37f5463bc9621a":"VI-2(4)","p_d675e0e254b7d6f2":"VI-3","p_472153a38fc6a125":"III-1(1)","p_ef05543956515409":"III-1(5)","p_911f20531ba71a7c":"III-1(6)","p_89f642d4fc14f7f6":"IV-1(4)","p_d2cbc670cbf55a6a":"IV-1(5)","p_74a91146dc41c7a7":"V-3","p_7a1fce1a7bc92ab1":"VI-1","p_ed971a7a5e293735":"VI-2","p_472bf5665c0cee92":"VI-4","p_53f0a4ffcc44c105":"VI-5","p_4d8df5a4f6688e69":"VI-6","p_4cdbcd649c1f1fc3":"VII-1(1)","p_da93da15aef3bce7":"VII-1(2)","p_b84fea919542ca1d":"VII-1(3)","p_23c73f3017234ff2":"VII-1(4)","p_1a30c43789dcc57a":"VII-1(5)","p_f391fd4513257c6e":"VII-1(6)","p_e6510b7b2d487b18":"VII-2(1)","p_8e2f0a1d7ba04b70":"VII-2(2)","p_32df589f0856de47":"VII-2(3)","p_35a9719915918d5f":"VII-2(4)","p_2b054fc4986b1f4c":"A1: one-dimensional scattering with absorption","p_25214073aa98e649":"A2 Central harmonic field and angular momentum","p_881f87ae256972b0":"B1 State functions and finite heat reservoirs","p_e525db513a10e858":"B2: classical gas in an external field","p_345be099d03de951":"B3 | Two-dimensional Fermi gas with linear dispersion","p_cf70754a053fe935":"C1: Gram matrix and reciprocal basis","p_2c053fc001c1ec65":"C2: nonhomogeneous equation with regularity at the center","p_5bf6c8a2964e8676":"C3 | Integral transforms and regularization","p_fba710aba0d90ea4":"C4 | screened Green function","p_f1651a015702785b":"D1: gray model with internal heat and atmospheric absorption","p_9701aad36c0a8b6f":"D2 Axial deformation of a rod and dispersive waves","p_4ec6225fba2838ae":"D3 Reasoning that connects observations to causal mechanisms","p_4f3169db23d567ec":"E1 Inverse problem of rotational spectroscopy","p_3e4f504d809fc89a":"E2 | Branching and daughter-species loss in a clock","p_1f005771edef79ed":"A1: twisted boundary conditions and weak angular potential","p_04a8ce1bf0e9ed57":"A2: tilted spin and detuned rotating field","p_196675c48c7a0ecf":"B1 | Excluded volume and speed of sound","p_ba94a938a844633f":"B2 Exchange statistics and fixed particle number","p_6fe078ef9b06ab9d":"C1 Euler equation with repeated roots","p_0ec98d646edf9569":"C2: spectrum of a stochastic matrix and mixing time","p_af0f825a1608c9b8":"C3: smooth approximation of singular fields","p_320f1695dd6ab512":"C4: contour integrals and double poles","p_18248fc94df9dab8":"D1 | Thin atmosphere with a latitudinal distribution","p_b54768d7d98dfc23":"D2: orbit-averaged irradiation and radial velocity","p_aaa9503a61a952b0":"D3: second-order thermal boundary layer and cooling clock","p_762f1b6d8f47ac9e":"E1 Accumulation in sequential first-order reactions","p_4a7fdb3404630fd7":"E2: collision activation and competitive deactivation","p_367132aae6a09089":"E3 | wettability and nucleation barrier","p_27fb281571fa2db9":"A1: three-point coupling system with nonuniform mass","p_cc1dc31766830ace":"A2: central-force capture with short-range repulsion","p_7d096fa14da90806":"B1 Spherically symmetric electric field and multipoles","p_99b576d87a52199e":"B2: nonuniform current and compensation loop","p_1764006ecf987c10":"B3 | Finite-width toroid and voltage reversal","p_c29b53a3ecb9770f":"A1: two-step angular momentum transition","p_06af0d693f3eaf49":"A2 | orbital states of symmetric traceless tensors","p_d311f5f1b0b93ab4":"A complete basis and interference of different spins","p_58263c81316383c3":"Polynomial spaces with integral constraints","p_d6c8719ac9cb0bd5":"Repeated root condition and intersection of subspaces","p_ccf494f5d58ca283":"Monotonicity and error of nonlinear iteration","p_77574b77f2acf18a":"Residues and parameter integrals","p_e61a1c6569a6ad2e":"Weighted inner product and best approximation","p_7aab9ae2934977bb":"The image and kernel of differential operators and commutativity conditions","p_761d79b09f81da53":"Threshold for differentiability of oscillatory functions","p_d0f4beda6370a1d6":"Logarithmic periodic integral","p_156c227bf81de7a8":"Two constraints and orthogonal decomposition","p_dc915c729b5f2dbe":"Recovering a projection from its image and kernel","p_367e251cc0a7c8bb":"Functions defined by oscillatory integrals","p_6594081c34c45650":"Quadrature error and asymptotic normalization of the factorial","p_2866b92934f795bb":"Kernels under parameter variation","p_857099475a1da22f":"Matrix equations and symmetry conditions","p_f4789abd317c1dd8":"Refinement of the mean value inequality using curvature","p_431eb89159c8ace3":"Weighted integrals involving inverse functions","p_6570a9059f7e2249":"Canonical forms and exponentials of real matrices","p_21531d44756e225a":"Units of a quotient ring and invertible matrices","p_e4dfdc632e6d372a":"Criteria for ideals and ring isomorphisms","p_478f71bdfce9b302":"Compactness and closed sets","p_3de7026230a08fe4":"Tangent planes and families of lines on surfaces","p_bb525bc7e1e21171":"Cancellation of zeros and oriented closed-curve integral","p_8d68ff993af99d61":"Systems of differential equations with repeated eigenvalues","p_d6e63415d73714b2":"Fourth moment of a weighted sum of independent variables","p_4bb6f117699f60e0":"Simultaneous equations and minimum norm","p_888e8c5760cd6f03":"Direct sum of polynomial spaces","p_7b84fc870db98519":"Convexity and quantitative inequalities","p_52943fcfed5fb2aa":"Weighted inverse function integral","p_28fc624a101240eb":"Maps constructed by integration and endpoint evaluation","p_ae1630890b6d0763":"Diagonalization and spectral projection","p_20467dd9205069d9":"Power series with periodic coefficients","p_1aa26397a5fae62d":"Improper integrals over anisotropic regions","p_f6d1ed95c1ecba54":"Nonlinear observations and linear equations","p_3e38a147ca719993":"Reflections in a weighted inner product","p_f23da8afc7c1cbca":"Arc length and chord length of a parabola","p_cdf02141c1a378e8":"Linear constraints and extrema of multivariable functions","p_84b4279001c9b2a3":"Boundary conditions and linear transformations","p_c2060446d4092c1f":"Intersection of parameterized subspaces","p_8495799c3ecf4e4b":"Iterative methods for finding square roots","p_24ecd9b674bd28f3":"Multiple poles and parameter-dependent integrals","p_1b7e82d16d703d7f":"Orthogonal decomposition of polynomials","p_39f05167dab18a11":"Differential operators and commutation conditions","p_a1d369fe366b1ed6":"Antiderivatives involving oscillations","p_1f6ddb294432bb43":"Phase-shifted contour integral","p_6b53eba56ba56bff":"Weighted orthogonal decomposition and best approximation","p_346134165838d34e":"Rank-one operator on polynomials","p_5e4bc0b211895adf":"Regularity of an oscillatory integral at the origin","p_7ad77470f2aa924d":"Trapezoidal sums and asymptotic constants for convex functions","p_957ba3c2811c530b":"Consistency conditions and minimum-norm solutions","p_7d48759caa5c99ab":"Subspace degenerating with a parameter","p_7cea70b8aa1c1452":"Strong convexity and quantitative inequalities for means","p_1047a7de53e7b41b":"Weighted integrals of inverse functions","p_fa29154efa6091d3":"Multiplication on quotient spaces and matrix exponentials","p_b13828cf1930672f":"Matrix lifting over the dual number ring","p_1320d7b8ce79183b":"Properties of a ring inferred from relations","p_cb88d7251e60d4af":"Splitting field and intermediate field of a fourth root","p_84a0f30585b2a515":"Determining the conditions of topological assertions","p_ae2270e5ce9730d2":"Closed forms and periods of multiply connected domains","p_a86c699a2618b494":"Geodesics and shortest paths on a cylinder","p_4971dfe25a0cc0f2":"Correspondence between spherical zones and spherical shells by projection","p_6686fc7fe6730d24":"Completeness of the weighted uniform norm","p_3a2aec29e29ce24b":"Four convergence concepts for oscillating sequences","p_11f6c58ac4f041cf":"Linear systems subjected to resonant forcing","p_bd105aa1a25b4b3d":"Second and fourth moments of overlapping increments","p_16b3e1f4d2f80dda":"Faithful group actions on the projective line","p_f0427113bb991409":"Isomorphism test for quotient algebras with repeated roots","p_352f7d18ab389d1f":"Cyclic group representation on a quadratic form space","p_9be3b657d5ad31ee":"Quadratic extension recovered from a primitive element","p_c26790ce2048fbb1":"Orientation-reversing orthogonal transformation in three dimensions","p_c816552a5ad0afc1":"Reconstruction of a plane curve from curvature","p_bacf5e63e63ae270":"Local potential of an annular domain","p_a506a53adf38608d":"Space obtained by collapsing the boundary of a cylinder","p_47f1742cd88e3f18":"Constraints on entire functions from circle mean squares","p_a0295ca789cf8f67":"Elliptical limit cycles of nonlinear systems","p_353f924f4e0e4dfd":"Pointwise vanishing and concentration of integrals","p_22a70ace45df33cb":"Completeness of a distance involving derivatives","p_7bab655abf7952d3":"Continuous and discrete comparison of repeated roots and resonance","p_ff5473e83f1b9b5c":"Continuous limit sets of anisotropic gradient flows","p_cc944f6337adcbde":"Degeneracy of the transfer matrix and bounded iteration","p_3d4f3ba777d788d9":"The nine-element field and restrictions on determinants","p_aa066787ac148440":"Quotients of finite fields and ideals of subrings","p_7d0adb8a797218b2":"Homeomorphism, completeness and the closed map criterion","p_82ef116fc5e81649":"Tangent planes and a family of double lines on saddle surfaces","p_98c85f980e4f4868":"Complex parameters inferred from residues","p_f6f46303f070bdcf":"Three-component relaxation system preserving the mean","p_a6c25fd671ea33ee":"Fourth moment of sampling without replacement","p_632ba7b56c5ad2e8":"Matrix groups over finite fields and centralizer groups","p_4fef4b4b2ee4d178":"Quotient algebra and invariant subspaces","p_edc15118bdc26c53":"Orientation-reversing orthogonal transformation","p_35400709af792b04":"Quotient spaces, manifolds and point removal","p_7921d27ea6690432":"Radius and phase trajectories of nonlinear planar systems","p_7e05793cb6297c0c":"Resonant external force and continuous and discrete variation of constants","p_10a6147f6f5c5194":"Fourier coefficients and sum of squares","p_e562013413b7d193":"Measurement map and normalization of polynomials","p_1dcfa2799f51acfb":"Eigenspaces of differential operators and collision","p_4cb04325e43076c7":"Taylor series with periodic coefficients","p_5ef44e45fd69e9e5":"Integration over a cusp domain and critical exponents","p_e8aa221a27a6e062":"Symmetric coordinate decomposition and parameterized equations","p_f6540307de672850":"Composition of two reflections","p_261a7f48fd4d493e":"Asymptotic comparison of a tangent polyline and a circular arc","p_d111cf3ff80f0bfd":"Minima on a line and saddle points of functions of two variables","p_f82e2d7a2ced92f8":"Difference measurement and minimum-energy solution","p_b9f4754f0a5d1aef":"Cyclic vector and minimal polynomial","p_cce2b94c3a890c40":"Squares of inverse trigonometric functions and coefficient recurrence","p_fd2481e27d7064a9":"Integration at infinity and symmetric truncation","p_d7637eee84f965f5":"Spectrum of a matrix commutator","p_34fba50bee67f837":"Lattice rotations, translations and quotient groups","p_716506cb90e056d5":"Splitting field and intermediate fields of a quartic polynomial","p_9ea09cc0f1189d68":"Connectivity and cut vertices of graphs","p_4674f0ea02e8d666":"Gauss curvature of a tilted graph","p_e118541a3c709e94":"Hamiltonian flow of a weighted area form","p_45050a449f498a31":"Logarithmic derivatives and zeros on the boundary","p_ecfb9254ec459b22":"Weak convergence and uniform tail control in coordinate directions","p_59c651419cf16131":"One-sided kernel smoothing and differential equation","p_84a2afa7e6784e0d":"Approach to an equilibrium point with a multiple root","p_b05dc802a6712def":"Measurable structure generated by a countable partition","p_1ada4b7623a1b900":"Rank-1 perturbation and discrete-time evolution","p_5403b7908c7f23e3":"Commutators and quotient groups of affine groups","p_fceff57f8a2b8f8b":"Quotient rings and scalar extensions","p_cb0ab7cd0f80253e":"Primitive element and intermediate field","p_b085f284fbdaecde":"Compactness and closed graph","p_9e052630502ff7e5":"Classification of quadric surfaces by eigenvalues","p_b9f548aa1735cd3a":"Curve reconstruction from curvature","p_f6f0c9de4cb87ce3":"Nonlinear shear and exact differential","p_f64259a1d96c8543":"Laurent coefficients and real integrals","p_0498ef6f089a0333":"Convergence in density and total variation","p_ce66833988bb3613":"Slope constraint and uniform norm","p_ada35fe7e07485e4":"Rotational equation with damping","p_a00f4a4cd3065cfa":"Double-well Hamiltonian system","p_8f3922b4e1142b45":"Binomial samples and nonlinear statistics","p_3a904ff783617c28":"Chains and rank of a nilpotent map","p_5fd8f5ed2ac0baff":"Center of the Heisenberg group","p_391ad77ab5f694a5":"Real algebras with nilpotent elements","p_982066a210d22252":"Fixed field of a cyclotomic extension","p_38df244d72491581":"Quotient space obtained by gluing endpoints","p_74676d7fec31b738":"Degeneracy of indefinite quadratic forms","p_c708f94a8fefbbc4":"Parallel curves and singularities of curvature","p_cdfa58150b7a1686":"Forms and twisting on a punctured plane","p_3358e08f9c2a6c44":"Multiple poles and oscillatory integrals","p_a2b017100702cd1d":"Convergence in measure and uniform integrability","p_dc5bdae8c5a72497":"Hölder constraint and sharp estimates","p_5b2690c2de3460c7":"Time-dependent rotation and axial translation","p_bb16c05e0175785f":"Real phase plane of a pendulum","p_08abc4354dd5fdb7":"Limiting distribution of a square statistic","p_909403799a6a9386":"Rank of simultaneous conditions and orthogonal projection","p_95595460fdb008bd":"Constrained Rayleigh quotient","p_6d2004b19214faf1":"Asymptotic estimates, differentiation and change of variable","p_48fe59abcf1f6c5e":"Coefficient recurrence derived from the inverse function","p_e81553f3b1fcb784":"Polynomial identities and invariant decompositions","p_2ec0dcdea8b1e2df":"Continuity test using a distance function","p_64386cddd793cf17":"Anisotropic improper integral and double pole","p_2a2951381f099a53":"Nodal rings and integral closure","p_e11ac7e435af2716":"Nondegenerate fixed points and tori","p_1a98ba2a9335a8bc":"Decay of weighted Sobolev type","p_6226ba3d5af975ae":"Standard matrix of a polynomial operator","p_e5fb65de74117b6d":"Extrema of coupled quartic functions","p_af0b6b1353fd83ce":"Interpolation decomposition and dual space","p_f5e6a4ad007c592d":"Uniform tail estimates and differentiation of integrals","p_4b65fa9b312584bc":"Invariant subspaces and polynomial operators","p_9dde5f6cbccd07ee":"Linearity of matrix spaces","p_9c8b2edeab3bbac3":"Line integrals and coordinate transformations","p_9ce0c62efb9ab26b":"Corrected exponential function and series","p_1c78ce77a151a5a4":"Nilpotent chain of length three","p_567f5e8ff39c4b85":"Implicit functions and optimal values","p_667bccac529826f4":"Double pole and semicircular contour","p_5590bd83ef73dcee":"Accumulation points and sequences of distinct points","p_4465d9a71670b713":"Subspace determined by interpolation conditions","p_94509d2e02c8b27f":"Elementary transformations, degeneracy and tangent vectors","p_64ec4719ec0057ed":"Nonlinear iteration with conserved quantities","p_36ae64b6289db11c":"Transport equation and logarithmically weighted integral","p_61054696f3c542d4":"Resonance of polynomial differential operators","p_24938506e8c29595":"Linearization of a recurrence relation with external forcing","p_053a17be468a9bc6":"Reproducing formula on the circle","p_bc9e8dbffe495bce":"Distance to a closed set and subsequences","p_7ac4bc8e35e9498c":"Eigenvalue coalescence and diagonalization","p_3bcbda28895740c4":"Transpose, trace and direct sum decomposition","p_e79b1f2c30ad5b51":"Multiplicative coordinates and partial differential equations","p_0dce6cb0545457f6":"Convergence tests at both endpoints using cumulative variables","p_06b065e2c76e5f2b":"Parabolic coordinates and harmonic functions","p_139190fecc66d5b8":"Singular integrals and nonlinear change of variables","p_c46eff0ead586238":"Dimension of the image of a plane","p_e577a8427c40c1b7":"Iteration of asymmetric averaging","p_c12a5010844b9966":"Degenerate critical points along a curve","p_ce2f8e27912626ab":"Gaussian integrals and second moments by geometric estimation","p_b1d6b489a6c974a0":"Interpolation using function values and derivative values","p_a1cd73727516f7d3":"Images of powers and cyclic vectors","p_5ab145d20e1bfd73":"Variable-coefficient equations and boundedness at both endpoints","p_95cadc4e8b39f144":"Space decomposition from polynomial conditions","p_cdc3a3eee3515edb":"Three singular points and a uniform estimate","p_93fb4b72ea7ef28c":"Saturation of sets and correspondence of images","p_4bf4bb3cd2bfa84a":"Insertion of a dielectric into an isolated coaxial capacitor","p_1221cf90b66a2da1":"Finite-capacity liquid tank and electrical boundary conditions","p_8991dcf74ca3d13b":"Quantum confinement with an internal barrier","p_1c8944ebda7f343a":"Fermion filling in a two-hole system","p_7e4bf459139b5a8b":"Junction quantum wire and open-shell ring","p_44438ddaca3da4d0":"Dielectric-driven system in a series circuit","p_23ed3152885d9df2":"Central-force orbits with short-range repulsion","p_70c9c6e1c7935c4f":"Rigid rods and momentum constraints in a finite ring system","p_5fd34212087a79b5":"Anisotropic two-dimensional electron system and two subbands","p_a22dab94132f5072":"Fermi distribution, density of states and particle number conservation","p_d20ae91f3027218e":"Generalized coordinates and planar motion","p_5e9f787909e91fff":"Lagrangian formulation and conjugate momentum","p_3045bfd17f2bd5d1":"Euler-Lagrange equation","p_d2341f5bfdc2f734":"Direct proof of conservation laws","p_8112f0c8a69e6856":"One-dimensional reduction of a central force","p_9f9ad23eeefa1529":"Bound motion and schematic of the effective potential","p_05e3f320e7d59b25":"Circular motion and parameter identification","p_399ea212fa3f6ab8":"Small oscillations and orbit closure conditions","p_eec4f110be4c9d40":"Free energy and internal degrees of freedom","p_9a2229bc209facad":"State equation and reversible work","p_308a9d7affe415f1":"Canonical partition function and phase space integration","p_bf4a64b31840f95b":"Rate equations for parallel reactions and material balance","p_0f2963e957534bea":"Pseudo-first-order approximation and competing yields","p_b55ac64567384f01":"Visualization of rate data and background correction","p_9eb5e6fa75f2cda0":"Concentration dependence and identification of parallel processes","p_2faeac62023766b5":"Determination of rate constants and unit conversion","p_78fac396832530e8":"Temperature dependence and apparent activation energy","p_4c37c64965daeee6":"E1 dehydration, competitive elimination and isotope tracing","p_5e6399f27c835e3c":"Solvent capture, haloetherification and stereochemistry","p_4894306f05744fbe":"Acid-base equilibrium and resonance stabilization","p_665c9d92bf632c01":"Substituent effect and degree of dissociation","p_311111053b7a3dde":"Electrophilic aromatic substitution, directing effects and symmetry","p_59a90d81f3c2657f":"Intramolecular aromatic acylation and ring size","p_6dda47b4d1fe5d2b":"Ionic addition, radical addition and rearrangement","p_6764d3c7a52e3b2a":"Cyclic acetalization, equilibrium and catalysis","p_4962623bb2c96047":"Reaction rate equation","p_0a938889ccf615b0":"Pseudo-first-order reaction and measurement background","p_446bc0190e0ebdae":"Decay curves and half-life","p_6848de846024b7fa":"Curve comparison as concentration changes","p_99340b8689df072b":"Separation of the reaction constant and wall losses","p_d0fed79c895d24ae":"Arrhenius analysis","p_e07439bebd4616b3":"Dissolution equilibrium and complexation","p_83c6116bd11f5014":"Total concentration and free concentration","p_56b57154d71b5a4f":"Determining a constant from solubility","p_1ad2d0d6fe640484":"Identification of free ions in a supersaturated solution","p_b57b65d2d70966d0":"Competition between the common-ion effect and complexation","p_bdeb766004b93a59":"Acid dissociation and the equilibrium of water","p_636a1b480a7e6dd0":"Fraction of the acid component","p_7e3b88c73e52b7f1":"Electrical neutrality and mixing ratio","p_2670b1a86ba88210":"Approximation of pH and self-consistency","p_444a5c0a75fba752":"Molecular structure and environmental reactivity","p_1a67025a20ef71df":"Applications within the apparatus and environmental assessment","p_65c553ca70d3f99b":"Competition between transport and first-order loss","p_1412abbc837f63d1":"Catalyst regeneration and chain length","p_f3b7bb3b74d331d5":"Radiative efficiency and evaluation period","p_44e48196c21a6eb2":"Real square roots and polynomial representations","p_de969fc844b63cca":"Uniqueness of the symmetric cube root","p_1606e28e13daa115":"Weighted orthogonal complement","p_bff5e944589dc938":"Linear map with a specified kernel","p_2f6cf1a96bbdb66c":"Right inverse map on a complementary space","p_436c25a2634f0f83":"Projection and self-adjointness","p_dd429de4e06e7c68":"Direction-dependent limits","p_c07f3b012dfeca5d":"Threshold for continuity","p_4ef97095add8695e":"Total differentials and directional derivatives","p_19e78754f9496ab5":"Absolute integrability and symmetric cancellation","p_f1eeafa5e631a983":"Convexity and asymptotes","p_17d202e87b497c86":"Convergence of nonlinear iteration","p_b1f39f8adeba7cf7":"Weighted Euler identity","p_e7b724304591de1c":"Tangent line to an implicit function and second derivative","p_572ef038c8903a21":"Kernel, image and affine solution sets","p_68254e778cf70f3e":"Orthogonal decomposition from nonorthogonal normals","p_4230a5086a50aff8":"Shortest distance to an affine plane","p_d61bcf89e69d758c":"Damped conserved quantity","p_b56b2867835ff536":"Kernel of an odd-order skew-symmetric matrix","p_c838cc560458d22b":"Conservation laws along the axis","p_a6e63c9a0d85af12":"Distinguishing circular orbits from degenerate cases","p_4819bee3f80803ee":"Uniform estimates for exponential inequalities","p_62f9b93d227db0f8":"Limits at two endpoints","p_56ccfa58187adcc1":"Rigidity under two-sided conditions","p_df9e884a3c77a088":"Second-order approximation and elimination of cross terms","p_75cd61ab1b644ac9":"Characteristic coordinates and amplitude factors","p_22ed53d52a7232fd":"Integral test at two endpoints","p_5f593ad14343fcce":"Anisotropic transformation and double improper integrals","p_b4a12d1ae1c34be1":"Comprehensive mathematics A","p_391734625bdf8287":"Mathematics comprehensive B","p_c1123afbc407cd27":"Statistical mechanics of an oscillator with internal states","p_31044d1a8b75f8c3":"Three-mass system with closed-loop coupling","p_c6824c02034807dc":"Quantum well with a pointlike attractive force at an asymmetric position","p_e2e5f79db1b456d5":"Coupled rotational oscillations","p_07db35bc28a3cec1":"Canonical statistics of finite-level systems","p_cd7bd4230d35f12b":"Degenerate orbitals and Fermi statistics","p_f1161620388ba785":"Skin effect in a finite conducting plate","p_65a7e2f8495d29fc":"Bound states in a double-delta well","p_756b17b9ff469c7b":"Statistical mechanics of finite-level systems","p_2ab11f36fc659ae7":"Rate analysis of parallel reactions","p_80b89ae796653c27":"Reaction mechanisms and synthetic routes","p_18cbceb3ab979832":"Selective precipitation and buffer equilibrium","p_1f3fd638ca4e2234":"Environmental chemistry: carbon cycle and process assessment","p_fde15c1ea1b0f0bd":"I Sum and intersection of polynomial subspaces","p_7a9dcbf2d64c2e42":"II Anticommuting operators on matrix spaces","p_5f1a07af678dbffa":"III Expansion, change of variable and integration","p_6cfb503f433366bc":"IV Precise comparison of iterated logarithms and series","p_d0ca7c219939ad4f":"I Difference operator for polynomials","p_24d3195f30269506":"II Two kernels varying with a parameter","p_d066697fe663e9ba":"III Logarithmic coordinates and improper integrals","p_84ea5631f67587bc":"IV Uniform error and minimality of degree","p_4de9f269626e675f":"I Stabilization of images and Jordan decomposition","p_3a5b19cfaab440ad":"II extrema involving non-isolated stationary points","p_8c1449df74c5fe5a":"III Jumps across branches and parameter integrals","p_ba8d40ba14fc77b0":"IV Infinitely many constraints and compactness","p_08703d3fbc6e724f":"I Two anticommuting real linear transformations","p_cbb904bc3086f472":"II Resonant forcing and damped solutions","p_0226474cf41676bc":"III Square of the logarithm and double pole","p_ed50c0b6ff6a7016":"IV sequential criterion for uniform continuity","p_d404b13c869adbaf":"Bose distribution and particle-number constraint for two branches","p_868276ad435e44f7":"Infrared convergence criterion for Bose condensation","p_75541d872fd2a0b9":"Lagrangian with surface constraints","p_48a4ce0bd2026481":"Angular momentum and conservation of energy","p_39d31e0a80961f9a":"Effective potential and stable circular motion","p_d6aba785f76cb063":"Small oscillations at fixed angular momentum","p_308965909cc8ce64":"Angular velocity modulation due to small oscillations","p_9f4ca63e80280c56":"Electrostatic field of layered dielectrics","p_9a2a343aba0425b4":"Line capacitance of series dielectrics","p_3203dffb36f179ae":"Magnetic field of parallel current-carrying plates","p_e7d61704b24e575d":"Magnetic flux, magnetic field energy and line inductance","p_a822cac440161f4b":"Wave equation of a lossless transmission line","p_23755eb64bbdfeee":"Propagation speed and geometric dependence in composite dielectrics","p_ad822cf7a41c3796":"Characteristic impedance and two-condition matching","p_d18c88b1bdf6dd13":"Single-particle partition function for anisotropic dispersion","p_aabe2bbd2410f33e":"Identical classical particles and dilute conditions","p_208be96fd43d8a34":"Helmholtz free energy and extensivity","p_78e6d3ba843ebaf0":"Thermodynamic quantities and anisotropic dispersion","p_d086898e358496e1":"Recalculation using the grand canonical distribution and energy in each direction","p_cef7077233e42b27":"Density of states from anisotropic energy surfaces","p_f944f09e239c2ee5":"Pressure and directional energy of a Bose gas","p_214fecbb27d6a27b":"Eigenstates of preparation pulses for a two-level system","p_aeb1613630fc7a37":"Initial state with a phase from a preparation pulse","p_2cc3656fb557fed5":"Two-level Hamiltonian with complex coupling","p_d3403e8d41d8189d":"Normalized eigenstate including phase","p_3b6408f5f60b9f24":"Time evolution of a two-stage pulse","p_db1f09c48c308de3":"Measurement probabilities and diagram construction for two-stage interference","p_7d4b907cd5f6b4f4":"Phase matching condition for complete transitions","p_ab9cc7573ec9e101":"Ideal gas with anisotropic dispersion and internal levels","p_345d1f4144936b31":"Two excited branches and the infrared condition for condensation","p_828d6c01513e3941":"Quantum interference by a two-step Hamiltonian","p_6385fb094a4ade60":"Two-intermediate catalytic mechanism and rate saturation","p_e8d6000faf033978":"Reaction mechanism, synthesis sequence and stereoisomerism","p_4e7fd8d6682c6366":"Elementary reactions and dimensions of rate constants","p_13d5bb929e3a32fb":"Material balance of a consecutive catalytic mechanism","p_c78c9ebc9312c825":"Balance of free substrate and products","p_9502414582c9ab32":"Steady-state approximation and saturation velocity","p_15de91a415366491":"Mechanism identification using rate equations","p_7a0aa7a7d5b541ae":"Dissociation equilibrium with a common ion","p_057a30001e80ccfe":"Dissociation constant and buffer approximation","p_3ff3aae40eed8f24":"Recalculation of equilibrium after dilution","p_034bdc5a49263a59":"Real gases and low-density expansion","p_e31afaf17a885c4a":"Competitive adsorption equilibrium","p_b8eb44ac5326c12d":"Photochemical production and third bodies","p_d93ee55ebe8c4947":"Vertical distribution and source-sink budget","p_9ac9d309b9f83ff3":"Catalytic cycles and identification by observation","p_79fbbe3930cb8741":"Ultraviolet shielding and nonlinear response","p_8773af974d31d65e":"Assessment of residual sources and recovery","p_5b268420b75354b1":"Photochemical cycles and net production","p_cf59d731c9c8771c":"Linear algebra: orthogonal decomposition and generalized inverse map","p_db2276780f8b00e3":"Mathematical analysis: harmonic functions on an annulus","p_f4bc2f313b993e8d":"Complex analysis: logarithms and sector-shaped contours","p_9bd722e9ca6515d1":"Topological spaces: irreducibility and continuous images","p_2b4db15be1faea93":"Statistical mechanics: finite spin systems with couplings of opposite sign","p_f9ce34d476aa9ec6":"Analytical mechanics: elastically supported strings and the variational principle","p_739b5ee47b440bc9":"Electromagnetism: charged particles in a pulsed field and a rotating electric field","p_00034298642b1b44":"Variational principles of continua and dynamic boundary conditions","p_6e595b3b0adf2b47":"Separation and static response of a two-body system with unequal masses","p_5b529335a47acfc6":"Leakage processes, Exchange symmetry and conditional spin measurements","p_1bc53be0bcf1b8ee":"Finite spin ring with competing couplings","p_ab52ea2736787f00":"Collision activation and competitive formation pathways","p_127c366b25e75417":"Collision activation, branching reactions and pressure control","p_495114bcc0aa8e65":"Nomenclature of organic structures, cyclization and regioselectivity","p_9ea634d9ff360d48":"Thermal and magnetic responses of a three-state spin dimer","p_44f94bb780bde1ce":"Coordination equilibria and dual-wavelength spectrophotometry","p_41d9c68b25fd1060":"Radiation balance, time scale and carbon balance of synthetic fuels","p_fe348680695caec2":"Polynomial subspaces and evaluation maps","p_1748dfd6b41bf203":"Linear operators on the space of symmetric matrices","p_6d9adde42808d83d":"Change of variable, implicit function and series","p_f21329e73ced8a93":"Multivariable extrema with parameters","p_ebda2f0cff98e7c5":"Sampling map and orthogonal complement","p_e35263005e434bcc":"Inverse iteration and spectrum","p_20c77001e0bb005c":"Composition of functions and multiple integrals","p_fcd558d411beb398":"Global extrema of a quadratic polynomial and a Gauss factor","p_a7994494eebece7d":"Image of the commutator and orthogonal complement","p_3bc4d04412d10635":"Dominant eigenvalues of opposite signs and iteration","p_b10883504415fcb4":"Expansion, differentiation on curves and integral transforms","p_129df0eef1602dfe":"Global extrema of rational functions","p_d6466c441d9b782c":"Subspace determined by simultaneous constraints","p_21614f9e8cd92be9":"Differential operator on polynomials","p_cde67709b91c462f":"Polar coordinates, implicit functions and quotient expansion","p_3a2d1dcaa3121e04":"Maximization of a polynomial with parameters","p_8b139058f68845bc":"Commuting involutions and common eigenspaces","p_54f289a6eb2c6406":"Anisotropic singular integrals and logarithmic boundaries","p_331f170f28f2cf85":"Holomorphic infinite products with correction factors","p_ee86c73a59ba1d7e":"Sequential limits and criterion for continuity points","p_b560423708eb41e4":"A nilpotent operator and the inverse of a polynomial","p_24c9828c74e35b79":"B Anisotropic harmonic equation and quasiperiodic conditions","p_053369cc30b54393":"C: oscillatory integrals and comparison of frequencies","p_9a5d06bef74842d4":"D: saturation by images and left inverse maps","p_7f78e20f39412175":"A Three spectral components","p_ab2d7ca34f767b03":"B Angular average of a quadratic form and critical integral","p_62a3181a8d10ecc6":"C Infinite product with symmetric zeros","p_540f3559a2ce2df5":"D, double limit and set of points of continuity","p_21282baf6e243454":"A nilpotent operator on two connected spaces","p_03380f0d312eacbb":"B Harmonic functions with unidirectional periodicity and growth conditions","p_a63a1608b6991279":"C double pole and parameter differentiation","p_da74f40b6481d94e":"D inverse image operation, injective and surjective","p_0cf665673da5f07b":"Two-dimensional ideal Fermi gas with two internal levels","p_7275cb9ecbace1b8":"1 Free charge and polarization charge","p_dd50f206c41c8a9d":"2 Radially nonuniform coaxial capacitor","p_4738b90d44882c08":"Two-dimensional Fermi gas with two branches","p_7d7db50868486bda":"Radially designed dielectric and coaxial electrodes","p_ace296f82a65d2d6":"Motion on a paraboloid and small oscillations","p_f8beb4edd5fa04c6":"Scattering at an interface with contact interactions","p_1292d5de9c439ece":"Linear confinement with different slopes on the left and right","p_c6ef85a5b90081b3":"Quantum mechanics: spin scattering and conservation laws","p_5313a310d22f3abc":"Quantum mechanics / two-channel boundary conditions","p_edfea9d7cb07176f":"Linear algebra/kernel, image and rank","p_b04e6aee95dc6f3b":"Linear algebra: direct sum and oblique projection","p_f52c529c4e65b8e7":"Linear algebra: normals and inner products","p_4cca7b2d32b9d15d":"Linear algebra/matrix products and stability","p_31023fff9a027cf0":"Linear algebra: evaluation maps and counterexamples","p_7076a2d29e8228ac":"Analysis: parameter integrals and convergence","p_bcd3ebf4564249fa":"Analysis, integral estimates and best constants","p_0cb6a05025c157a8":"Analysis: angular transport and homogeneity","p_0c7da49c5aed6bac":"Analysis: region bounded by a hyperbola","p_a133c07384bc1e06":"Analysis: nonlinear change of variables","p_60ebc0712547c305":"Analysis: changes of variables in double integrals","p_054725d3d0aa3a21":"Linear algebra / null space","p_eb6e851e2b69a5b9":"Linear algebra: equations of subspaces","p_41fcfdae3d70ed5f":"Linear algebra: parameters and rank","p_f437f70f646c5709":"Linear algebra: intersection spaces and exceptional values","p_1d0485041ec74c97":"Linear algebra: complex spectrum of a cyclic map","p_54e2807c31e27f63":"Linear algebra: orthogonal spectral decomposition","p_c15de431acb3c82a":"Linear algebra: polynomial spectral projection","p_c5b95b1c59ebe224":"Analysis: full-order expansion in several variables","p_3d1e7329044e80df":"Analysis / degenerate stationary points and extrema","p_9ea009cdae5d8f99":"Analysis/elliptic scaling and polar coordinates","p_23c3b33f3a55b4ff":"Analysis: midpoint sums and error estimates","p_915da7dc64c7108f":"Analysis: limit of Riemann sums and leading error","p_fbeeff2956c16d07":"Analysis / convergence boundary of positive-term series","p_14ec61057ae804e7":"Linear algebra: kernel and image of cyclic differences","p_cd461fc6210ddf67":"Linear algebra: cyclic symmetry and projections","p_1bba24d4a72d9d47":"Linear algebra: outer product and inverse transpose","p_c5ac34ab31b4430c":"Linear algebra: matrix mean and Jordan components","p_d760ad3cb6f72d21":"Linear algebra: finitely many linear functionals and linear dependence","p_1cf657ab5c0be658":"Analysis: inversion transform and improper integrals","p_345c4e33ca024ab1":"Analysis/exponential function best inequality","p_9538479255eeb881":"Partial differential equations: regularity at the origin","p_408bf55799f1de68":"Analysis: sketching two connected components","p_2ab6746149c370e3":"Analysis/two-to-one transformation and Jacobian","p_12e3c615554d9c43":"Analysis / double integrals including multiplicity","p_9e8cffc2cb09cfb2":"Linear algebra/annihilator space","p_a6d4e20d682cb49c":"Linear algebra: graph of a linear map","p_54137353910a3f50":"Linear algebra/Case split of the dimension of intersections","p_ce86b79dbf3d403f":"Linear algebra: repeated roots and generalized eigenvectors","p_7e01cecda5d0516b":"Linear algebra / direct sum of generalized eigenspaces","p_0a6e0107a06e5599":"Linear algebra: spectral projections and iterated actions","p_6c361b172b39c6ed":"Analysis and Taylor expansion to all orders","p_c3841105f3b75f81":"Analysis: quartic forms and case analysis of extrema","p_391810e12452578b":"Analysis / polar coordinate integration over noncircular regions","p_0c3f7384e6181927":"Analysis: monotone Riemann sums and logarithmic mean","p_13bd9c8b4bef0cf9":"Analysis / limit and concavity of the geometric mean","p_322f92cd6698e2c8":"Analysis / difference of means and series convergence","p_aa1fe94fecd9a631":"Invariant bilinear forms and inverse spectrum","p_d2c739a911b62a65":"Area and centroid of a curve with nodes","p_65dab2b0c9bd0f5f":"Integral kernel with two poles","p_a9538413774f6bae":"Openness and closedness of distance maps","p_9cace594ade49acd":"Polynomial equations of a nonsemisimple matrix","p_8963894a91fa5390":"Regularity of oscillatory functions with a directional factor","p_d16561bc0306c3fc":"Continuous boundary values and obstructions to analytic continuation","p_fd9a8de1d53c12ba":"Set maps satisfying a cubic relation","p_741a45ed274c1adf":"Spectral interpolation and constrained square roots","p_903db1dbee40dfbe":"Anisotropic singularity and differentiability","p_1187770e58068e24":"Natural boundaries viewed through functional equations","p_c5bc21f2309db519":"Two idempotent maps and information recovery","p_8bb64ddc2e0c15cb":"Product of involutions and Jordan structure","p_98089b56e7c2b40c":"Algebraic curve with two leaves","p_7be22ad3a495e4cb":"Openness and closedness of product maps","p_211c178ee3906c39":"Two-body problem with central forces, and orbital precession and scattering","p_dd83be9d3a04f50a":"Magnetic field gradients and vector analysis","p_151b3805f1c3ff6d":"Magnetic field lines of a quadrupole field","p_b2922a93ddf634a4":"Interaction energy of magnetic dipoles","p_68b796353fde802f":"Magnetic force with fixed and free orientations","p_30c9e4ce80aae95d":"Electromagnetic induction in an oscillating pickup coil","p_a3a3e26bd22374ec":"Self-inductance and approximation error","p_ad4b34556b085cf0":"Canonical transformation of a quadratic Hamiltonian","p_a4a4b6202850e140":"Annihilation operator and energy lower bound","p_7001f01f1e324fa2":"Gaussian ground state with phase","p_f431aa7442253e84":"Normalization and position-momentum covariance","p_085e203eb40ad09c":"Inverse problems of the ground state","p_8dcf133a1dadbe72":"Binding conditions for a correlated two-particle system","p_28c1f7427e126960":"Hole density and Fermi distribution","p_afdcde89f5d05384":"Uniqueness of the charge neutrality point","p_86312d5bb848ed40":"Density of states for anisotropic dispersion","p_2755cd1ae1a49523":"Low- and high-temperature asymptotics of thermal excitation density","p_8f8c9dfde2032952":"Zeeman-split step-like density of states","p_c24a81b44d6e117a":"Fermi integral at fixed particle number","p_fc8c9adaf9d0046d":"Valley depopulation and chemical potential","p_1d189b8c6843a0ef":"Magnetization curve and stepwise saturation","p_dcde3cdafb87fd79":"Center-of-mass separation in a common harmonic field","p_60777e7efd3d794f":"Relative angular momentum and plane of motion","p_1bcc663f59ba1273":"Polar-coordinate Lagrangian in the relative plane","p_27560ad071af10e2":"Hamiltonian and canonical equations","p_21cfd5b7e09a9638":"Inverse-square interaction and stable circular orbits","p_2b72f47a31460c63":"Isochronous oscillation of the squared radial coordinate","p_96b4182f347e3dfd":"Periapsis shift and closed-orbit condition","p_7a6b949c04ca0321":"The two-body problem and center-of-mass separation","p_1cafa9d3b301efd7":"Central force and conserved angular momentum","p_bcd04a8226845afb":"Canonical variables and effective potential","p_f3b023461265cf61":"Orbit equation and precession","p_3f24784d92165bd0":"Bounded orbits and turning points","p_ed134afa145acc91":"Stability and oscillations of circular orbits","p_eb541805223bfdd5":"Central-force scattering and branch selection","p_b868f03f0e07a6aa":"Harmonic oscillator under external force","p_58ae1aaae62d5d4e":"Operator factorization and lower bound","p_34a1dd2428e0c60c":"Gaussian ground state","p_a64f086be01feefd":"Normalization and quantum variance","p_55e8aa0966db3fde":"Inverse estimation from quantum states and abrupt changes","p_07650fa3133f40c2":"Stability conditions for coupled oscillators of different masses","p_72fc8bc353355b6f":"Exchange antisymmetry and ground-state energy","p_3bdeddb47131b0b8":"First excited state and level crossing","p_bcddcd1a6fd16f2e":"Kernel and rank of polynomial spaces","p_f12abaa4bbc6ab65":"Intersection and sum of subspaces","p_f4d59e846f563c45":"Right inverse maps and singular limits","p_bb1f0d0777c52e04":"Stationary conditions for constrained quadratic forms","p_9103c7a232a92721":"Unique minimization of a positive semidefinite problem","p_459ed8d14c021cbc":"Nonlinear extrema under an elliptic constraint","p_abf4ac530410b5f4":"Removable singularities and Taylor expansion","p_4319005e564ae018":"Uniform tail estimates for oscillatory integrals","p_5e14754fa9401d4f":"Improper integrals with parameters","p_d8b73f5a8c5d1099":"Absolute convergence and conditional convergence","p_6f7f3ebfca6eb92b":"Affine plane and linear span","p_2fd0dd58220f792a":"Positive half-line and minimum norm","p_086f96ff2f9e1b50":"Representation matrices in nonorthogonal bases","p_f95cfb40325c2c48":"Eigenspaces and exceptions to diagonalization","p_522f0737e24a6109":"Changes of inner product and self-adjointness","p_02dcccf9a26ca52b":"Transformation to standard coordinates and verification","p_f17c5c450afa5989":"Higher-order limits using logarithmic transformations","p_e2604ab5f0ae2289":"Shear and nonlinear change of variables","p_9fea42827bab228c":"Multivariable Taylor expansion and remainder","p_cbc268b8001be627":"Cross-sections and sketches of surfaces","p_e98c761af3d7292b":"Numerical changes and guaranteed rounding","p_44a3c72fc3910a69":"Linear measurement and direct sum decomposition","p_c5695ed0b33042f4":"Regularized least squares and kernel","p_06991b3ff7d11450":"Constrained extrema and inverse function expansions","p_2a29fd1cd6d17669":"Improper integrals with nonlinear phase","p_e023fd5a7710fdb4":"Affine sets and directed half-lines","p_3caef19e2ad26792":"Weighted inner product and spectrum","p_f0d78d402481249c":"Limits and domain transformations","p_684873e8550500aa":"Surfaces, Taylor approximation and cancellation","p_6fccf7d097490665":"Jordan structure on polynomial spaces","p_fe10a3b404eb75fa":"Discrete averages and anisotropic differential conditions","p_d10e7643ccebdbb8":"Sector contour and double pole","p_c0926780ff5e8101":"Quartic level sets and parameter stability","p_f27675e7b297e26b":"Gauge transformations and third-order differential equations","p_2d3a1ca7e3a8f606":"Heavy-tailed concentration kernel","p_9c5b76100db11467":"Cauchy estimates and range constraints","p_2e86218a0f142cda":"Weighted distance and nearest point","p_267c4682f1a3da5d":"Reciprocal eigenvalues and powers of a matrix","p_71b3c2cc3e6ef982":"Circular average with a moving center","p_d3243d6d5e223932":"Semicircular contour and smoothness at the boundary","p_7de6a7a7ef5e7040":"Nonhomogeneous level sets and the boundary of boundedness","p_6a7fd0c7c212e971":"Euler operator and solution space","p_2a6a6faa75225ab9":"Concentration kernel on a finite interval","p_74aa5c0ef9505050":"Cauchy estimates and half-plane maps","p_8d2cf8a4c7c8645e":"Weighted distance and convex projection","p_04123b1b994d032f":"Nonuniform-inertia torsional oscillations and selective damping","p_b644b36a53893017":"Coaxial capacitance, finite rise time and moving charge","p_04798fa2607bba3b":"Spectrum of a ring waveguide and magnetic flux","p_fe8aaff9ec12b12e":"Binary dilute gas and linear external field","p_5e36b8776b7adaaa":"Remainder map of a polynomial and subspaces","p_1747ee10ce31a429":"Weighted inner products and simultaneous congruence diagonalization","p_50531a103d6b08ab":"Constrained extrema, change of variables and surface area","p_7ad1384418e00c1e":"Symmetric logarithmic expansion and convergence acceleration","p_02c6a4e1b38c9032":"Image and annihilator structure of a commutator operator","p_3edfd8a524eb31c8":"Noncommutative block elimination and output independence","p_05939c50e69328c5":"Parabolic coordinates and area of the image","p_3341e4460a82ff15":"Variable exponent and logarithmic boundary test","p_6454d29cda55cf0c":"Symmetrization operator and parameter singularity","p_846efe65bc255e82":"Realification of complex matrices and matrix fractional transformation","p_281f045b68eef145":"Logarithmic spiral coordinates and annular regions","p_8e7f48ea70e7bdd1":"Condensation criterion and critical exponent of the iterated logarithm","p_99a725dafc02faa4":"Statistical mechanics: normalization of nonequilibrium velocity distributions","p_597b796d2f9ac22d":"Statistical mechanics: mean flow velocity and velocity variance","p_d9f974e9c335b982":"Statistical mechanics and decomposition of kinetic energy","p_08bc0c47dc963ce3":"Statistical mechanics: adiabatic relaxation with recoil","p_66e378583a5d21f0":"Mechanics: logarithmic radial coordinates","p_df34dc3651946b70":"Mechanics: Lagrangian and conserved quantities","p_2c4a52a7be71c400":"Mechanics: radial equation and effective potential","p_12b34ca231071fb3":"Mechanics: gravitational potential of a hollow spherical shell","p_02b8e809ec8ccd6e":"Mechanics, circular orbits and radial stability","p_03fa39aabdde85a3":"Mechanics, rotating coordinate systems and tidal forces","p_6b1a083bd064d257":"Mechanics: relative equilibrium in a rotating system","p_cc0a20ef66b1cfd6":"Mechanics: natural frequencies of coupled oscillations","p_c5726b501ade517e":"Mechanics: instability of inner equilibrium","p_e78ad74b8b547521":"Electromagnetism: Stokes' theorem and circulation","p_75d7eb0a2728fc59":"Electromagnetism: magnetic field of a nonuniform coaxial return path","p_6b57157ca2d89955":"Electromagnetism: magnetic field of an asymmetric finite straight segment","p_45fbbe930e24714a":"Electromagnetism: taking the infinite-length limit","p_c719fda11e7308d5":"Electromagnetism: vector potential and coaxial fields","p_33fc6bfdc276c724":"Electromagnetism: conserved magnetic flux and magnetic torque","p_7f949a255481b50f":"Electromagnetism: self-magnetic field at the center of a coil","p_7c52086d08aeafc2":"Mechanics: moment of inertia from linear density","p_fba41d3118c6cecb":"Electromagnetism: oscillations and critical points under flux constraints","p_a5f9b306dd9130d4":"Steady magnetic field and boundary conditions","p_6b0df3b2c638ce68":"Steady magnetic field and distributed current","p_268aeb47e9aef43d":"Steady magnetic field and finite current distribution","p_0d9ecac4ee91f5a7":"Steady magnetic field, limits and error estimates","p_d1437024e29eefe0":"Vector potential and gauge","p_2eca04aae0b45a35":"One-dimensional quantum systems and connection conditions","p_83934a80ccda6414":"One-dimensional quantum systems and symmetry","p_2bfc30ea9683890c":"One-dimensional quantum systems, eigenvalues and waveforms","p_958a6344662937a1":"Identical particles and exchange symmetry","p_0d8e169fa0fec7f3":"Two-particle quantum systems and level mixing","p_86c33ebe184a79ee":"Identical fermions and the ground state","p_48e88d37c3994b19":"Identical fermions and excitation spectrum","p_b19af36a98663a8c":"Classical gas and mixed velocity distribution","p_edcf60840ffef960":"Classical gas and velocity moments","p_4090026f5e8666ee":"Classical gas and energy decomposition","p_3c64b12d96208121":"Classical gas, adiabatic relaxation and recoil","p_6001b2ab5876eba2":"Two-dimensional Fermi gas and ground-state energy","p_0d52c99b83f83da7":"Fermi gas and mechanism of low-temperature specific heat","p_22a147d78a4410c9":"Two-dimensional Fermi gas and density of states","p_a074125169295ee3":"Fermi gas and thermal energy","p_8bce1b405330fc7c":"Fermi gas and occupation numbers in a moving frame","p_514887e344a16e1e":"Fermi gas and flow energy","p_30a6503159058c4e":"Fermi gas, conservation laws and low-temperature relaxation","p_41bcf23747382313":"Differential operators on polynomials and solvability conditions","p_694988061f13b072":"Composition of two plane reflections","p_3e1de9eb21ed7bc0":"Integration by change of variable and quadratic approximation","p_ed950e3283c45d90":"Differentiation along a curve and nonlinear extrema","p_7307eb0b6d981a18":"Averaging and periodicity of star graphs","p_625a8dc926c332ee":"Rank of a parameterized vector family","p_9b79f6c4559145d8":"Moving anisotropic Gaussian kernels","p_b9247cba41d4a66a":"Global estimates for distance on a hyperbola","p_cf28fc1943cc66d0":"Kernel of a polynomial transformation and generalized eigenspaces","p_f8ff1e78f73bdd1d":"Reflection along an oblique direction","p_fec1175c0dd15b88":"Integration in hyperbolic coordinates and logarithmic approximation","p_912c4f2731531e97":"Implicit function paths and parameter extrema","p_a06064b7eb57c894":"Directed cyclic graph and mean limit","p_08031d610fba0009":"Linear independence of polynomial families and exceptional sets","p_5230638cc0497a07":"Diffusion kernel with mixed terms","p_06688d32de9ff9df":"Homogeneous algebraic curves and distance inequalities","p_96447a8906581777":"Nested quotient spaces and bases","p_c2ae014cd0b4c1ed":"Normalization to an ellipsoid and gradient","p_9cd88cb8354e41a4":"Zeros of exponential polynomials and coalescing poles","p_6366d38f988f8f27":"Gluing over a locally finite closed cover","p_8c1e8ac4734a4e11":"Inner product constraints and minimum norm","p_2b155348836a727f":"Anisotropic approximate identity operator","p_79983dab3cee5c81":"Poles and conditions for the vanishing of integrals","p_839b32bf8ae57d62":"Interior of a subspace and infinite intersection","p_ddf159d40e2e9f8b":"Interpolation and quotient vector spaces","p_5620c1a237e626f4":"Differentiation and metrics on surfaces","p_e9fe8a8d352cfc3c":"Coalescence of periodic zeros and residues","p_2fd73cb23d93009b":"Gluing using locally finite closed sets","p_ebe5a7dc33d8384c":"First-order perturbation of the inner product and dual basis","p_63772af1bf9042d4":"Signed smoothing kernel and error","p_1762db6dc7aad909":"Interference and coalescence of two poles","p_0d8e7760279783d3":"Closures and open dense sets","p_b82fa462ec8292e6":"Fields and precession of a two-layer charged shell","p_cccd8e579e46125a":"Generalized natural vibrations of a three-rotor system","p_d38726b76479681e":"Spherically symmetric electric field and boundary conditions","p_ce294a89081949ac":"Electrostatic energy of charges with opposite signs","p_de47cee9d9c1c8b9":"Central magnetic field of counter-rotating spherical shells","p_36ead8a733019a6d":"Magnetic dipole moment due to surface current","p_7287b0ca27621e67":"Distinction between static Poynting flow and radiation","p_2abed7a843c71e44":"Magnetogyric ratio of a system in which charge and mass distributions are not proportional","p_7ac9c76f3b4bcd62":"Torque in an external magnetic field and magnetic orientation energy","p_8e9c303a0a11a8d0":"Precession equation for a system with a negative magnetogyric ratio","p_a27c7ef1de476e29":"Initial conditions for precession solutions and self-consistency of the approximation","p_605b3c31ce7a562f":"Statistical mechanics: independent rotators","p_50ad9771413528fd":"I Polynomial evaluation conditions and subspaces","p_266d59bc748e6531":"II Inner products, positive operators and adjoints","p_259a83df2ece1844":"III Generating functions with poles and periodic components","p_5413419779997908":"IV, two equality constraints and extrema","p_37fa9cc423c3f4c7":"I matrix equations: resonance and intersection","p_66deb91677833135":"II Jordan structure of a difference operator","p_2355a7537996914c":"III Harmonicity, differentiation of integrals and growth of products","p_fc6934445e45d440":"IV Solids with cusps and improper integrals","p_0f5d0c44a489e49e":"I Polynomial interpolation with contact conditions","p_6fb53c4cb5238352":"II Reflection and second-derivative operator","p_4ad2d3fa6dc8ac44":"III nonlinear transformations, differential equations and boundary growth","p_12793568a43ea644":"IV Unbounded solid and logarithmic boundary","p_c4f7c4bd5d2de00c":"I Commutative matrices and subspaces of cancellation conditions","p_a18c95380f889e9b":"II Cayley transform of a skew-adjoint operator","p_a38737fe788e1560":"III Generating functions with logarithmic singularities","p_8fd6c48b8f1cac6a":"IV Quartic on the sphere and classification of stationary points","p_e62ec826e7baad36":"Electrostatics","p_27aca503a0500634":"Rigid body mechanics","p_b9a18c4c7c1817b4":"Nonlinear elastic energy","p_7a0ebdfdbbb5e54a":"Energy conservation of a rigid body","p_bb7413135e6a62a4":"Angular momentum and conserved torque","p_1660184f6c5aed06":"Rapid rotation and nonlinear precession","p_6b35673147739ede":"Gauss's law and nonuniform charge","p_43d9be543a5a67ac":"Integrals of the potential and boundary conditions","p_39edeaa368e5618b":"Image charge method and energy of a grounded system","p_4909f8ca0755289d":"Ampere's law and return current","p_81e5ce7bceb206f0":"Steady current and Joule heating","p_95532a021b3ed478":"Magnetic flux and external inductance","p_f374c69e579e33db":"Coaxial capacitance","p_af90fc99926f4e4b":"Stored electromagnetic energy in a constant-voltage circuit","p_e5e619c780285c45":"Maxwell stress and generalized forces","p_a7a49b6bac8fa427":"Classical partition function and internal degrees of freedom","p_20e9c5ea26dc95b2":"Internal energy and the heat capacity of a two-level system","p_039365613b96993e":"Equation of state and internal degrees of freedom","p_50cb6bbd4f857eca":"Classical momentum distribution and independence","p_d26b9e472f9c828d":"Two-branch Fermi occupation distribution","p_f52a4f6e814ced4f":"Anisotropic density of states and Fermi level","p_551030149b86e389":"Low-temperature chemical potential at fixed particle number","p_1e9778ae19ede501":"Potential profile and stability points","p_62d066e1d2860e55":"Local expansion and harmonic approximation","p_ff4c7872d3382abe":"Raising and lowering operators and zero-point energy","p_5e44fae7433b5766":"Harmonic eigenstate and range of applicability of the approximation","p_e178ac9e1761d207":"Ladder operators and compensation conditions","p_7d53cdf464ca2eed":"Parity and first-order perturbation","p_c90d579389b9fdb6":"Perturbation by a quartic anharmonic term","p_b173471650612999":"Second-order perturbation and interference terms","p_90931425cd1e6c7c":"Static force and mean position","p_b83837fb5efd229f":"Internal energy and heat capacity","p_602a47311eb3ffa6":"Equation of state and energy decomposition","p_f55bc55c5fa766e9":"Probability density of kinetic energy","p_9d940c1b094d130a":"Shape of the Fermi distribution","p_228e0719e7d87e54":"Density of states in a two-band system","p_21c226e2a82ebeb6":"Fixed particle number and low-temperature expansion at the band edge","p_ecd87cb5d2fbc51b":"Low-temperature thermodynamics of linear dispersion","p_f628fb242af1edda":"Grand partition function","p_9f2e8d0e83e4bd71":"Homogeneity of pressure and dispersion","p_f4d13c70a0afe3fa":"Angular selection and particle flux","p_edfdf9d5ef0f53c8":"Asymptotic expansion of thermally activated flux","p_0f1b48b3c6613509":"Intersection of affine graph sets","p_4e549ad72169e028":"Intersection and basis of graph subspaces","p_d6b0df72c856b5e4":"Direct sum decomposition and oblique projection","p_976eb70274fc7be9":"Linearity of conditional matrix polynomials","p_35c71403b1f71c1f":"Nilpotency index and dimension of the polynomial image","p_6e0fe00892cd745b":"Minimum dimension and Jordan canonical form","p_4abe7f9ca503b2ad":"Change of variables and mixed partial derivatives","p_e71324b7e5c48d3d":"Improper integrals and logarithmic critical exponents","p_fa352fd0ea95a116":"Multiple integrals over symmetric regions with boundary values","p_d38bc7a48969587e":"Concavity with error and corrected difference","p_9c834bbd825ec26c":"Nonnegative sequences and a lower bound for differences","p_020301603558324f":"Mean difference and asymptotic slope","p_97758fbbc3a84cd8":"Interpolation differences and linear maps","p_a3df4b1a5bf42347":"Representation matrix of a differential operator","p_13ba827d7beb4585":"Kernel of a parameterized operator","p_6e79a1484a0d5b0a":"Eigenvalue collisions and eigenspaces","p_96abbd1cf43eb0bc":"Jordan chains and minimal polynomial","p_e17eb318ee1b38bb":"Image space and inhomogeneous equations","p_9bb6b08a55964e0d":"Differential inequalities on curves","p_48c05f2ca05149f9":"Ordinary convergence and absolute convergence of oscillatory integrals","p_d58dbde211661ddd":"Kernel of repeated integration","p_6c67fe331c3702da":"Iterated kernel and uniform convergence","p_6d3f2f4bf5841e4e":"Scaling and conservation-form equation","p_30564f50ee3819c2":"Convergence of concentrating sequences of functions","p_658e61cee4308462":"Uniform estimates for nearby rescalings","p_4a883d3afcbe6b92":"Linear constraints and bases","p_274f93e2b39bb447":"Quotient spaces and dimension","p_b1113ccb8402dd41":"Constraint identification and minimum number of observations","p_a8efe3ebe595c88b":"Quasiperiodic functions and compactness","p_257c877892be7667":"Coercivity and minimization","p_38914d7f4da52552":"Estimation of Laurent coefficients","p_fdc0b907987b5e75":"Growth and determination of holomorphic functions","p_31cacde30f152cac":"Continuous images and distance minimization","p_26ca9741c59aa1d6":"Stability of the inverse map","p_c78b617561807d78":"Decreasing sequences of sets and limits of minima","p_6189bb3b82b3c9d4":"Polarization and boundary value problems","p_684f6ae6503662b3":"Anisotropic dielectric response","p_deb62473dc9b0146":"Constraint reaction forces and rotational symmetry","p_438af5470841576e":"Velocity of constrained coordinates","p_d2707061ee022dba":"Magnitude and components of angular momentum","p_d89355dd205e44fb":"Rotating coordinates and Hamiltonian","p_d773564916836afa":"Effective potential and stability","p_e019c4d7b685598e":"Competing central forces and barriers","p_c1ea58194758c747":"Energy threshold for capture","p_7a06dae4d53f5325":"Orbit integral of the incoming branch","p_4de3309ae0d071c8":"Instability near the critical circular orbit","p_c87eff0a0f45f574":"Exact integration of critical orbits","p_c379b4510e087a68":"Ground state of a classical system","p_c3d38d5a211541db":"Partition function and state probabilities","p_0612853ef4c19882":"Derivation of internal energy","p_9689793e4fcaaf2e":"Low- and high-temperature asymptotic expansions","p_e9efb68353342395":"Energy curves and heat capacity","p_29a3c1d0f7174421":"Position average and variance of a mixed state","p_49055d4768d40dab":"Transition width of the response curve","p_9c6d1064f9913e63":"Degenerate entropy and smooth crossing","p_6fdc7aa8fdfd9628":"Constrained motion on a conical surface","p_a7397c00f49191c7":"Electrostatic field of an eccentric cavity and a line charge","p_e1bd172ba8a3ad47":"Dipole-free condition for a coated dielectric sphere","p_20ae0cea7b8b19f1":"Classical harmonic system with internal states","p_3c525fdd0a175c5e":"Finite well with a delta barrier","p_ac78bb258181e13f":"Quantum mechanics: weak coupling and boundary conditions","p_6d2a5a396517e921":"Quantum mechanics / spherically symmetric delta shell","p_9b9ed2810373cbe1":"Quantum mechanics / critical coupling","p_20ff5c34ad1d52af":"Quantum mechanics / Fourier transforms and separable interactions","p_db0f2df4bae434b5":"Quantum mechanics/dimension and coupling threshold","p_1c5309e4861d542e":"Quantum mechanics: pairing near the Fermi surface","p_70e1ee18a11b380a":"Linear algebra: low-rank linear maps","p_922d3d1f96f44159":"Linear algebra / intersection of kernel and image","p_4cfa63a5e5857b87":"Linear algebra: nonorthogonal bases and dual systems","p_4335d10a9e578e79":"Linear algebra: oblique projection","p_9c6eac48a402239d":"Linear algebra: nondiagonalizability and matrix powers","p_13f44e5c275a0bdc":"Mathematical analysis / convergence classification of oscillatory integrals","p_1acaf62bdc837526":"Mathematical analysis: multiple integrals and change of variables","p_323c066adca20f55":"Mathematical analysis / function series and integral comparison","p_41f4782092c3a052":"Mathematical analysis / uniform convergence and boundary scale","p_731dea640c0fc430":"Linear algebra: operators on polynomial spaces","p_2bda61fc41e182c7":"Linear algebra/operator invertibility","p_1a7d83fe24e6bd01":"Linear algebra: case split by rank and kernel","p_467a51a3ed614788":"Linear algebra/cyclic vectors and commutative rings","p_bced91ae9da49f91":"Linear algebra: commuting matrices of companion form","p_f39d0175d4d01fd6":"Linear algebra: eigenvalue coalescence and dimension of a commutative space","p_f5893b4b8574d437":"Ordinary differential equations: change of variables and degenerate parameters","p_440f5def099fd9a1":"Mathematical analysis: total differentiability at anisotropic singularities","p_14db58a7a5138b0e":"Mathematical analysis: improper integrals over curved regions","p_042f9e0be8088231":"Mathematical analysis/monotone functions and dyadic intervals","p_fc204391bf44f08e":"Mathematical analysis / observations and monotonicity on subdomains","p_cd0c79e56951b74f":"Complex integration, residues and inverse conditions","p_3fcbab7cae4b20b6":"Laplace transforms and integral equations with exponential kernels","p_9c1f42bafc57cc52":"Improper integrals, differentiation with respect to a parameter and change of variables","p_5769336168059097":"Linear algebra: eigenspaces and convergence of iterates","p_98749a90ac7b0deb":"Vector calculus, differential operations and path integrals","p_8051682238ce9266":"Change of variable and improper integrals","p_21b980137638c456":"Tail integral estimates","p_6573d503b3f86a3c":"Scale transformation and lower endpoint parameter","p_edc34ae0dad13367":"Iterated integrals with cross terms","p_bb042033775401c5":"Conserved quantities of matrix iterations","p_864029482218b6f8":"Eigenvalues and eigenspaces","p_48224dd7a50359a9":"Eigenvector decomposition and general term","p_232fb76d7fe2e16c":"Convergence and oscillating errors","p_608b1b5e2008f27b":"Convolution and differential operator","p_e7c5b019dcd68afb":"Laplace transform and interchange of the order of integration","p_1dda8bf421d2b682":"Integral equation with a damped kernel","p_d9837238524aee53":"Gradient and directional derivative","p_4743f8a7833e9f02":"Divergence, curl and conservative fields","p_9dfecab17ec7897c":"Line integrals along curves and polygonal lines","p_11058bfb14dd1d11":"Winding number and Laurent coefficients","p_19f228ee5e1acef3":"Multiple poles and vanishing contour integrals","p_f9166929e2438f40":"Integral kernel of the Laplace transform","p_971949f6609be80a":"Transformation and convergence of damped oscillations","p_732136a3e2ee279b":"Transform and inverse transform of the memory equation","p_4f67efefd7310cb3":"Eigenfunctions of a differential operator","p_aef89499246656a3":"Distinct eigenvalues and infinite dimensionality","p_4bf2afbb8e812d44":"Triangular decomposition under a positive definite inner product","p_a21f8c4de6c332ee":"Mean curvature of a helicoidal surface","p_51339902ba6488f4":"Ampere's law","p_54b0a7fc7b08ed66":"Magnetic field of a nonuniform ring current","p_2ab139d84e97f5df":"Piecewise function and schematic of the magnetic field","p_92dcc29e5cb5b455":"Force balance in helical motion","p_087248a17cd9a534":"Numerical evaluation of current and units","p_796745fdc58fa37b":"Electric field wave equation including charge gradients","p_51f08708f2d58502":"Cylindrical waves of an azimuthal electric field","p_0332c0a897840eeb":"Nondimensionalization and 1st-order Bessel equation","p_fec4d574ef5717cb":"Magnetic field of cylindrical waves and asymptotic corrections","p_40235678522a511b":"Poynting flow and cylindrical surface area","p_e2dfbf0f5abc7caf":"Far-field approximation and energy conservation","p_33690ed076e58916":"Simple pendulum in an accelerating frame","p_332e1fb6b5bbea8a":"Initial value problem for a simple pendulum","p_40367c912a38399c":"Moment of inertia of a disk about its pivot","p_9c826cbdc9e5d02b":"Quadratic Lagrangian of a rigid pendulum","p_97914e7bacc24394":"Reduced length of a disk pendulum","p_64fa06e00855a8fc":"Coordinates and velocity of a constrained particle","p_c2d05c7cfa471e05":"Quadratic potential at an equilibrium point","p_807cb396fb7274ae":"Quadratic kinetic energy of constrained motion","p_9399f3e56698643e":"Independent natural vibrations of vertical extension","p_951f63e5ee8a9240":"Reference coordinates of a non-inertial frame","p_53f9eda4fb1790c6":"Stationary equation for an infinite well","p_438350c3b3797ef3":"Boundary conditions for an infinite potential wall","p_8e4be362aae5abc3":"Normalized energy eigenstate system","p_7c6aab2a06c68d35":"Expansion coefficients and measurement probabilities of a broken-line state","p_3ff6db4e88027777":"Energy expectation and existence of variance","p_d72b7be0003974f8":"Turning point and domain of an operator","p_ad82e271839d6754":"Time evolution with phase and recurrence","p_09d5fc42755bd874":"Stationary waves inside a point barrier","p_bfd7a06c85060cfc":"Reflection amplitude and phase shift","p_f7a0678ce57e1913":"Reflection probability and the barrier limit","p_c64c5a8f2a14b85a":"Cylindrically symmetric electromagnetic fields and waves","p_2c61482b85d01cf6":"Translation and rotation of a composite rigid body","p_8ea2167f85ff3244":"Fourier analysis and linear response","p_5e5f06a405eab3cf":"Three-state spin and mean-field transition","p_186cc13acc73175b":"Initial motion of a bent slender rod","p_fbbefb12680a2f54":"Coaxial lines and Maxwell equations","p_b7970649124c0739":"Quantum eigenstates in a triangular prism domain","p_bc1f88e98afdcc75":"Fourier series, transforms and boundary terms","p_2d82a737b71688d8":"Complex analysis: large semicircle estimates with rational factors","p_6d81fc766b85fefd":"Complex analysis, small arcs and uniform convergence","p_708dc92925ecc9a0":"Complex analysis, residues and weighted oscillatory integrals","p_b98e1047e9883095":"Probabilistic numerical computation and random number generation without modulo bias","p_972887f1ef302004":"Numerical integration and implementation of the antithetic variates method","p_6b7312f8411cdc9e":"Probability and statistics: variance comparison at equal computational complexity","p_88b84d2ec77bb8cd":"Numerical experiments, seed management and reproducibility","p_6602655699429801":"Effective stress and sliding along pre-existing surfaces in layered rock","p_529dd7ad3592e092":"Weathering, grain-size composition and sediment transport","p_a8b3ccd7d16ccc1c":"Cosmochemistry, nucleosynthesis and comparison of organic matter","p_c920c75d044d8ebf":"Isotope geochemistry: Sm–Nd geochronology","p_f89f22a10ec5ad1b":"Carbonate equilibrium, alkalinity and reaction pathway identification","p_15460f74db972c3b":"Meteoritics, early solar system chronology and parent body evolution","p_ddf1a8a7b97ce8c4":"Earth materials and internal structure","p_3d24e9927a620ea5":"Gravity, elasticity and dissipation","p_151c58ecfe629591":"Coupled oscillation","p_9414581899ca3f56":"Coaxial current sheet and rotating conductor","p_f89edb65f44419fc":"Narration and editing in Chinese vernacular novels","p_cb5f76838b37555f":"Two-center contact interaction and exchange symmetry","p_a46076e0f1db9644":"Partition function of coupled oscillators and two-step excitation","p_0dd07a6aa7909a52":"Classification, design and interpretation of psychological tests","p_0682da2dcee2d989":"Philosophy: concepts, ethics, aesthetics and subjectivity","p_3a4863ad784c2219":"Chinese literature: term explanations and comparative essays","p_63874651122c9ff5":"Japanese linguistics: analysis of phenomena using terminology","p_ea14031e3f9ae86e":"Classical literature: expression, tradition and interpretation","p_57a2e974f7a14e2d":"Japanese language education: theory, lesson design and evaluation","p_d542a7bea9a02284":"Analytical concepts in Chinese-language literature","p_864ee1c4a5342db1":"Comparative interpretation of Chinese-language literature","p_49728927eeb5af53":"Literature, language and cultural concepts in the German-speaking world","p_8e364718823e8642":"Application of sociological concepts to cases","p_fd71534857b9a770":"Selection and regression to the mean","p_d3848faa6f769252":"Interview design: role experience","p_de8ee3ef99efd801":"Individual interviews with minors and data management","p_29ae6536bba786a6":"Distinctions among measurement, sampling and nonresponse","p_d02adefbd4b6d36b":"Distinguishing nonresponse from the target of estimation","p_98f660d76e6de95c":"Interview design: process of determining task allocation","p_f7b19b40a01dd5d7":"Experimental design and mutual influence","p_fa2a64b88011556f":"Random assignment and spillover effects","p_7d3d70568612e799":"Interview design: experience of knowledge sharing","p_1d8d7e428d1de2ae":"School management and professional collaboration","p_3b4234950163029b":"Evaluation research on school improvement","p_b9d515d0a6aa3b90":"Definitions of educational concepts and applicable conditions","p_401038e1324c8332":"Social work and service user participation","p_71be77e118caa361":"Mechanical evaluation of sports movements","p_8a9770193e594d9c":"Design and evaluation of sports nutrition education","p_596aea220ee01ef9":"Adjustment of training load and interpretation of performance changes","p_d3dc629120582132":"1-factor analysis of variance and the distinction from repeated measures","p_7f9b2f8e6d629514":"Recovery methods and intervention evaluation","p_3796d3cdfb428eec":"Prioritization of life-support needs and interagency collaboration","p_a8728b71bbc6c685":"Social welfare: decision-making support","p_6e2d2a397a6578ce":"Modern and contemporary literature, narration and the visibility of labor","p_123f86dde4753807":"Modern and contemporary literature and analytical terminology","p_00857ad9414ffde1":"Japanese linguistics: analysis of linguistic examples using concepts","p_0222ca2aebbdcc8e":"Japanese language education: learning processes and interaction","p_7508c2c9d41cd057":"Modern and contemporary literature, the uncertainty of memory and invisible judgment","p_c3130e9d3c6f0c49":"Modern and contemporary literature: critical concepts of time and narration","p_fe7eba407864a90e":"Japanese language education, assessment and lesson design","p_86a32945b89b9c64":"Concepts in Japanese language education and instructional judgment","p_04934740e83001ff":"Basic concepts of social welfare and their application to support settings","p_0b7d9454443bba7d":"Basic concepts of linguistics and analysis of examples","p_84d326691fd1ada7":"Conditions for using welfare systems and gender","p_321812b21c8a1f7f":"Training welfare professionals and regional comparison","p_083359b0b2f5f9e4":"Social welfare policy, gender and access opportunities","p_748fd4ff9984950f":"Orienting response, habituation and physiological measures","p_231d9cf7cec57843":"Bystander effect and diffusion of responsibility","p_695db10688b307b9":"Working memory and multicomponent model","p_78d8dbf1985b4b85":"Speech acts and functional classification","p_410381dcf447894a":"Orthogonal diagonalization of symmetric matrices and singular equation","p_f5c3ad56bd319fed":"Integral Gram matrix and positive definiteness","p_65d96850c91d6304":"Unique reconstruction of a polynomial from integral moments","p_8496ccfe8029bd97":"Comparison of logarithmic functions, existence and uniqueness of zeros","p_55c0a5e37df57b41":"Functional equations of composite functions and number of real solutions","p_d4d0dfd33245e8fb":"Double integrals and Volume of an elliptic region","p_0d86501001973315":"Affine groups, conjugates and centralizers","p_683de5a9499a471f":"Conjugate homomorphism and the First isomorphism theorem","p_7b2cf1e5bb779630":"Automorphisms of finite additive groups and general linear groups","p_814ae6072859477a":"Subgroups of finite groups and set stabilizers","p_68ffec338fa0fa94":"Ring theory: quotient rings and ideals","p_1d106f5633b6324a":"Complex analysis: removable singularities and residues","p_133c5411dbe3fc0a":"Differential geometry: shape operator and classification of surfaces","p_a203446a7a296a5b":"Ordinary differential equations: nonlinear damping and stability sets","p_bc915e551799dec8":"Algebraic topology: cellular homology and induced homomorphisms","p_27e873351c3633cf":"Topology: homology of finite graphs","p_ba465eeb0405481b":"Topology: gluing along a circle and induced map","p_ff8168163ef46cba":"Complex analysis: multiple poles and vanishing residues","p_d50aff3873cb43a0":"Complex analysis: choice of contour and parameter dependence","p_78128b364814c297":"Ordinary differential equations and equilibrium points","p_284b7c1eaabee350":"Ordinary differential equations: separation of variables","p_d39633f057796af4":"Ordinary differential equations and first integrals","p_495f63fbbd9c572f":"Algebra: polynomial quotient rings","p_6bc797753511ba8b":"Algebra: partial fractions and symmetry","p_49ceb95c7d65846e":"Differential geometry: curve curvature","p_d1d832858f1c4a5a":"Differential geometry, mean curvature and Gauss curvature","p_75a52971e388a470":"Complex analysis: residues and removable singularities","p_f45d4f958e65f51f":"Complex analysis and improper integrals","p_1ec2374d76252f7b":"Mechanics: constrained motion","p_4865d9ad42b472b3":"Image method for a conducting sphere and sign-selection fill-in-the-blank","p_9aa70907ab7a357b":"Energy of electrostatic induction","p_147f727a65f17c2d":"Electric field at a spherical boundary","p_33931048c7cc681a":"Gaussian cylinder and written fill-in-the-blank","p_99bd326f539285e3":"Surface integrals and constraints on an isolated conductor","p_36b3a7d20cb54459":"Opposing coil pair and magnetic field gradient","p_cef2fd5e84f81af8":"Opposing-face currents and Ampere's law","p_10a4937fb2bffbb4":"Phase boundary conditions and eigenvalues","p_3266d2ba0f26bbb7":"Orthogonal complete systems and probability current","p_2c27661cd8174491":"Nondegenerate perturbation, selection rules and second-order correction","p_a9a5fc22eac76bca":"Canonical partition function and high- and low-temperature limits","p_8882b08fdb20c9b9":"Energy moments and the two-level approximation","p_9c15f38c74eb97d1":"Relation between heat capacity and fluctuations","p_5988a27b1ad3e56d":"Kinematics of a constrained compound pendulum","p_8b9bbbddab8a68a6":"Moment of inertia and kinetic energy","p_50c0fd7e71cff0b6":"Equilibrium point shifts and the Lagrangian","p_13fd76c2a9ce17cf":"Nonlinear Lagrange equation","p_8fdcf62e41b78109":"Second variation and coupled matrix","p_ba8801de420ef2bd":"Natural vibrations, positive definiteness and separation limit","p_af9d9865f5784fc7":"Exchange interaction and conserved quantities","p_0b9138f9a9dccb6b":"Two-level mixing and diagonalization","p_f5f98a93b6a6da12":"Total spin and quantum mixture","p_90145f2776673836":"Uniform field and ground-state level crossing","p_cad87ba417ab0e2c":"Partition function of a finite set of levels","p_890f79d0873f2c03":"Magnetization, the low-temperature limit and degeneracy","p_686fb576c2b2d131":"Paleoenvironment and relative sea level","p_093b1fc7d186a055":"Large fossils and the validity of correlation","p_46a0ecf0ec4ae26c":"Description and origin of melange","p_9efa83edb51db581":"Origin and formation age of rock masses","p_c16fbea1be9adbee":"Three criteria for stratigraphic subdivision","p_2737956152e63e12":"Integration of microfossil stratigraphy and age intervals","p_7d06607dcb94cab2":"Repetition of accretionary stratigraphic sequences and structural restoration","p_24a6d5c1e4d1f7df":"Crystal geometry, diffraction and vacancy occupancy","p_1e6c54fae890d603":"Changes in paleoenvironmental indicators and developmental timing","p_51e2f7a1e01fed63":"Geological inference integrating lithology, age and structure","p_533296e02be2a6b5":"Zoning in solid solutions and temperature history","p_9f64196ce01dcac1":"Energy of electromagnetic fields and nonuniform media","p_1fc16144cf4ad43e":"Microcanonical distribution with a particle number constraint","p_5efbf8615738a082":"Sudden removal of a partition and quantum interference","p_215cad1dd2e5f897":"Rolling of a spool and the direction of friction","p_d7e0e9de323a8bd7":"Essay on turning research disagreements into a validation plan","p_ca67587ab5b5d252":"Central forces and the inverse problem of orbits","p_7d9064ca004d13ec":"Electrostatics: spherical boundary value problems","p_0bff2381b101830b":"Quantum operators, the virial relation and energy lower bound","p_6de307c8c61022c7":"Quantum wells, sudden approximation and phase interference","p_0d7fe327b1c420e7":"Rank-one perturbation and eigenvalues","p_c79b23ffe1019bbf":"Normalization of eigenvectors","p_88b26bd7aed5d6ef":"Orthogonal diagonalization","p_aa240333e1a43e26":"Block matrices and positive definiteness","p_1203bccce9bca765":"Moment of inertia of a composite rigid body","p_375b2e7ae1212707":"Nonlinear equation of motion for a rigid-body pendulum","p_c16e25ed8061ee79":"Small oscillations and identification of the torsion constant","p_ebdc799ffa374b98":"Period correction due to finite amplitude","p_9c2b4bac8f89ad93":"Rotational viscous resistance","p_aab2237c02560f53":"Damped period and logarithmic decrement","p_a24ed193f1bcf9bd":"The stationary Schrodinger equation in a central force field","p_93851c137fd83bd3":"Verification and normalization of Gaussian eigenstates","p_de7aa79281143e1a":"Probability with simultaneous radial and angular constraints","p_ee28bfda07dfcd01":"Gauss's law for a spherically symmetric volume charge","p_2467a643827a3e10":"Electric fields and induced charges in conductors","p_7dc194414aaab53a":"External electric field and total charge","p_32fd9f1832506155":"External potential and reference","p_e4c05c850a7a0191":"Inferring conductor charge from potential conditions","p_fe22ec6214b3f55e":"Matching of the interior potential","p_596d818de9fa7251":"Potential graph over the entire domain","p_459b254d946f8e28":"Electric flux density and polarization in a nonuniform dielectric","p_caef6d263da54490":"Bound surface charge and bound volume charge","p_c8a8df758687f338":"Inverse problem for a central force field and spherically symmetric wave function","p_7034645004411fa7":"Angular momentum, real orbits and complex superposition","p_1b08a68d86d5128e":"Electrostatic field and band bending in a double-layer insulating-film MOS structure","p_e45b26826c07e59c":"Semiconductors: transient response of MOS capacitance","p_cce4797d04152b82":"Semiconductors: relaxation of the deep-depletion state","p_b64263884a7ec30d":"Wave optics: phase difference and spatial frequency","p_a78a819a843a9807":"Wave optics: visibility and amplitude estimation","p_b4924e114171b345":"Wave optics: fringe shifts and cancellation of phase gradients","p_1c6f1d2cf6135c64":"Analysis: principal values of inverse trigonometric functions","p_b9c86e64e7da416c":"Complex analysis: exponential form and domain constraints","p_25b6b797b7a5bd06":"Multivariable calculus: affine transformations and area","p_e075ab16f1b7e9e9":"Analysis: recurrence relations for improper integrals","p_30a6b63263172b14":"Multivariable calculus and integrals with boundary singularities","p_b91baa0448653a76":"Linear algebra: repeated eigenvalues and positive definiteness","p_f2c2643db2f52b44":"Linear algebra: orthonormal basis of a degenerate eigenspace","p_c31efb3935d0cb56":"Linear algebra: orthogonal diagonalization and projection of external forces","p_08a14e9e7c45773c":"Ordinary differential equations, mode separation and resonance","p_76cd989b48146e99":"Rigid-body dynamics: mass elements with nonuniform density","p_0cca5378f824b807":"Rigid-body dynamics: density normalization and moment of inertia","p_4791c7ab19287e03":"Rigid-body dynamics: moment of inertia of composite bodies","p_1e40fb88413931e6":"Rigid-body dynamics and comparative experiments on a frictionless inclined plane","p_f16669aef82e69b7":"Rigid-body dynamics: pure rolling and the limit of static friction","p_b64d2b213ea834a0":"Rigid-body dynamics: identifying mass distribution from arrival times","p_8da4149d013fca36":"Electromagnetism: layered dielectrics and oppositely directed circular currents","p_a2bd6d1f49c5143d":"Quantum mechanics: periodic boundaries and probability current","p_57884a4ae6d94ed9":"Optics: optical path and phase in a parallel plate","p_f35ca0f6a8516c53":"Semiconductor physics: junction with an intrinsic layer","p_0aca6b8ae055d8e8":"Artificial intelligence: engineering validation and research plan","p_8c4d0308b48abeaf":"Essay on materials design and reusability","p_e0f1ee86937be452":"Short essay verifying claims in materials science","p_7a2e9fcabbad5e1b":"Quantum states on a ring and probability conservation","p_a71a8e90601a2105":"Falling motion of a drum with nonuniform density","p_a350e865492159d6":"Static field of layered dielectrics and a pair of coils","p_495a61a2e3d51353":"Quantum well with a point barrier and probability transport","p_3534d2e554109a9b":"Depletion layer and capacitance measurement in a linear graded junction","p_f0237ac38bbe7515":"Uniqueness of the light path through a parallel plate and thickness sensitivity","p_5432ba87895df823":"Layered spherical capacitor and oppositely oriented coil pair","p_15784b622f144dd5":"Validation design for predictors and generators in thin-film measurement","p_1d2f4a10852bd266":"Mathematical analysis","p_1024dd382fdf575b":"Discrete mathematics","p_b5de519e211abae6":"Earth science: the solid Earth","p_1195b149ef8bb0c1":"Statistics: frequency models","p_e16472f3e30b98fc":"Crystals and solid-state physics","p_9d5fd8db48c46f03":"Crystal symmetry and ordering","p_5e9befcf8d4e5c77":"Paleoenvironmental inference from geological records","p_a093a2042a432a55":"Correction, uncertainty and distribution in chemical analysis","p_824d056b1cf0dbe5":"Functional equations and singular integrals","p_cec57d7715c8aca6":"Operators on polynomials and commutators","p_639c89581c1b6620":"Composite rigid-body system constrained by orthogonal guides","p_f9b3a3caf5ff7724":"Bifurcations and small oscillations of an inverted pendulum with a torsion spring","p_ea79e7575251596f":"AC response of two fluids and penetration of the electromagnetic field","p_16dc9cf42a87e9ed":"Shielding currents in a superconducting slab of finite thickness","p_4b85488d19304a46":"Calorimetry and chemical potential of two crystal phases","p_e2889d8ab9500503":"Phase equilibrium and chemical potential","p_c21211e81df3dd56":"Differential relation for the coexistence curve","p_fb73a72daca93612":"Integration of the coexistence curve including the heat capacity difference","p_15c3bde071f9a970":"Comparison of finite temperature changes and tangent approximations","p_2f340abd699f0571":"Forced quantum harmonic oscillator","p_c161913ffdad26be":"Commutator and driving term","p_b278513924714920":"Heisenberg equation of motion","p_e77b211db402ab74":"Translation of the annihilation operator","p_d350c71b3d3750c3":"Inhomogeneous time evolution of an operator","p_7d4f4d9da217b3ff":"Interference of even excitations and position variance","p_e1987826798caffa":"Variance minimization under normalization constraints","p_8be0d58f36334201":"Mean position and mean momentum under driving","p_50ccaa560a8c9d8c":"Separation of displacement and quantum fluctuations","p_e1a2f69006d76e48":"Mineral composition and modal classification of plutonic rocks","p_86bf8bebd63b4ebb":"Mass balance of reaction rims and decarbonation","p_f73b086d8764a814":"Variation in metamorphic reaction temperature with fluid composition","p_86fe99edd058b461":"Mass and density of spherical celestial bodies","p_73083d68d193ae78":"Internal gravity and the location of the maximum","p_cf5b936ad698ed70":"Hydrostatic pressure in a nonuniform celestial body","p_a028e038fd72a9ee":"Comparative scales of pressure and intensity","p_eb0eae7df2fd577b":"Self-gravity and nonspherical retention","p_e58b080a781eb682":"Brittle and ductile strength and geothermal gradient","p_fe8a4a0275ab0203":"Unit cell occupancy and mineral composition","p_548715b87f431fe5":"Mineral classification based on chemical composition","p_beb30262cade4a12":"Crystal basis and coordination geometry","p_26e10c1282b94733":"Point groups and polarity constraints","p_c8ddc5c733ba571a":"Quantitative construction using stereographic projection","p_3124a81b97c5222c":"Point group action and equivalent poles","p_120887446dfc9724":"Energy balance and temperature in Raman scattering","p_05d86f474347a4d0":"Phase equilibrium and thermodynamics","p_633c6a094ae75571":"Quantum harmonic oscillator and time evolution","p_7c0454816080f407":"Integral form of Gauss's law","p_4919dfcfca782437":"Electric field inside a uniformly charged sphere","p_584b7fccd610ebb2":"Representing a cavity as a negative virtual charge","p_2923e293728aedfc":"Reconstructing a cavity from the internal electric field and total charge","p_163db5b938e9fe0d":"External potential of a hollow sphere","p_7e26c9d6a3956b44":"Far-field expansion of a nonneutral charge distribution","p_601749ad152867c3":"Pure rolling constraint on an inclined plane","p_ee7d4adefc6708f6":"Translational kinetic energy of a disk","p_141dcd9a5932efef":"Derivation of the moment of inertia of a uniform disk","p_af5d44f70f34b098":"Rotational kinetic energy of a disk","p_4f18d8b196c082e0":"Height of the center of mass of an eccentric rod","p_9d4308bea28a1337":"Gravitational potential energy of a disk and a rod","p_61f9d5fbe10d0e1f":"Velocity and rotational inertia of an eccentric rod","p_7de0cc86bb9107cf":"Energy conservation equation for a composite rigid body","p_66dc3917548760af":"Determination of angular velocity and degenerate limit","p_57ee47111149f327":"Gauss's law and boundary conditions","p_4f7961bf96762739":"Spherically symmetric charge distribution","p_b9412ffb358c354d":"Superposition and domain determination","p_1addd4c5037b6228":"Spatial variation of the electric field inside a cavity","p_24ee4f47dad1a1aa":"Neutralization and external potential","p_8b05a6f539d0bc87":"Multipole expansion and vanishing of the leading term","p_a20e3cdf3a88e1ec":"Additivity of the number of states and entropy","p_b039fd195f7a1d80":"Derivation of the canonical distribution from a heat bath","p_34ff674ae4861cb3":"Degenerate partition function and inverse estimation","p_2082393f2390b40c":"Differentiation of the partition function and expectation values","p_497473027776f972":"State counting of independent localized units","p_0222953708c068b5":"Internal energy and degeneracy","p_9f1e875451c9e805":"Energy fluctuations and response","p_5b338c288fd514ed":"Constant-volume heat capacity","p_4623170bd90dd1ea":"Asymptotic form of heat capacity and calibration values","p_520a100e3069a43f":"Complex matrices and spin eigenstates","p_228987c020c2577d":"Spectral projection of spin","p_4134cfc57564057b":"Normalization and global phase","p_35cf4ff45cdf38d2":"Sequential projective measurements","p_6d25eb33e9e88014":"Normalization and orthogonality of eigenstates","p_263baf0984fb689f":"Relative phase and spin expectation value","p_e752f496b862e65b":"Spin operator in an arbitrary direction","p_bbe718d27aadde0c":"Direct proof of spin eigenstates","p_038f7e9fccf71821":"Nonselective measurement and disappearance of interference","p_6cd5244839250036":"Diatomic basis and diffraction selection rules","p_0c4ebe6a959bf93b":"Diffraction indexing and zone axis","p_10060eeb6f80c022":"Electron microscopy, selected-area diffraction and diffraction contrast","p_c74203f222d7c700":"Regular structures, structure factor, partial order and diffraction patterns","p_9c214ab5c74f4f1d":"Inorganic chemistry","p_7e16f36a1c359519":"Physical chemistry","p_60259b6edb028237":"Analytical chemistry","p_09603c15a4f80d87":"Geophysics","p_4fa8826fbe72e6f9":"Earth science","p_ec142bbc4f520ce2":"Mineralogy and materials chemistry","p_32f8e4a2b0693861":"Crystal structure","p_899e7b6c8a7c3b8f":"Fluids and sedimentation","p_6a4759445eaefd62":"Order of rotations and low-rank updates","p_eabc4c195a960fbf":"Geometry and topology","p_375800c68798618c":"Local computation and global conditions","p_bc3dfef4e8d2ccc1":"Short scientific writing based on observations"}}}}}