Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs | Paper Citation Record

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This paper cites Uniqueness and bifurcation branches for planar steady N avier- S tokes equations under N avier boundary conditions.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Uniqueness and bifurcation branches for planar steady N avier- S tokes equations under N avier boundary conditions

Reference 1

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This paper cites Integration of dissipative partial differential equations: a case study.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Integration of dissipative partial differential equations: a case study

Reference 2

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Non-radial solutions for some semilinear elliptic equations on the disk

Reference 3

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Periodic and quasiperiodic waves on the sphere

Reference 4

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This paper cites Rounding error bounds for the C lenshaw and F orsythe algorithms for the evaluation of orthogonal polynomial series.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Rounding error bounds for the C lenshaw and F orsythe algorithms for the evaluation of orthogonal polynomial series

Reference 5

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This paper cites Constructive proofs for some semilinear PDE s on H ^2(e^ |x|²/4 , R ^d).

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Constructive proofs for some semilinear PDE s on H ^2(e^ |x|²/4 , R ^d)

Reference 6

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This paper cites Smooth imploding solutions for 3D compressible fluids.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Smooth imploding solutions for 3D compressible fluids

Reference 7

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Julia: A fresh approach to numerical computing

Reference 8

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This paper cites Angular momentum in quantum physics: theory and application.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Angular momentum in quantum physics: theory and application

Reference 9

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This paper cites Iteration-free computation of gauss--legendre quadrature nodes and weights.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Iteration-free computation of gauss--legendre quadrature nodes and weights

Reference 10

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This paper cites C hebyshev and F ourier spectral methods.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs C hebyshev and F ourier spectral methods

Reference 11

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This paper cites Dedalus: A flexible framework for numerical simulations with spectral methods.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Dedalus: A flexible framework for numerical simulations with spectral methods

Reference 12

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This paper cites Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light

Reference 13

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 14

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This paper cites Zernile P olynomials.jl.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Zernile P olynomials.jl

Reference 15

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This paper cites Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis

Reference 16

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This paper cites Computer-assisted proof of heteroclinic connections in the one-dimensional O hta- K awasaki M odel.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Computer-assisted proof of heteroclinic connections in the one-dimensional O hta- K awasaki M odel

Reference 17

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This paper cites Determining the source of period-doubling instabilities in spiral waves.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Determining the source of period-doubling instabilities in spiral waves

Reference 18

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This paper cites A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations

Reference 19

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This paper cites Forsythe.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Forsythe

Reference 20

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This paper cites Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDE s.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDE s

Reference 21

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 22

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This paper cites Computer-assisted proofs in PDE : a survey.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Computer-assisted proofs in PDE : a survey

Reference 23

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Golub and John H

Reference 24

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This paper cites Potential singularity formation of incompressible axisymmetric euler equations with degenerate viscosity coefficients.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Potential singularity formation of incompressible axisymmetric euler equations with degenerate viscosity coefficients

Reference 25

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This paper cites Dynamic stability of the three-dimensional axisymmetric N avier- S tokes equations with swirl.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Dynamic stability of the three-dimensional axisymmetric N avier- S tokes equations with swirl

Reference 26

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 27

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This paper cites Fast and accurate computation of G auss-- L egendre and G auss-- J acobi quadrature nodes and weights.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Fast and accurate computation of G auss-- L egendre and G auss-- J acobi quadrature nodes and weights

Reference 28

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This paper cites Zernike expansion of derivatives and L aplacians of the Z ernike circle polynomials.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Zernike expansion of derivatives and L aplacians of the Z ernike circle polynomials

Reference 29

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This paper cites A proof of J ones' conjecture.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs A proof of J ones' conjecture

Reference 30

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This paper cites Banach algebra related to disk polynomials.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Banach algebra related to disk polynomials

Reference 31

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This paper cites Spiral breakup in model equations of action potential propagation in cardiac tissue.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Spiral breakup in model equations of action potential propagation in cardiac tissue

Reference 32

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This paper cites Electrical alternans and spiral wave breakup in cardiac tissue.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Electrical alternans and spiral wave breakup in cardiac tissue

Reference 33

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This paper cites Nakao, Yoshitaka Watanabe, and Takaaki Nishida.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Nakao, Yoshitaka Watanabe, and Takaaki Nishida

Reference 34

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This paper cites Positivity proofs for linearization and connection coefficients of orthogonal polynomials satisfying an addition formula.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Positivity proofs for linearization and connection coefficients of orthogonal polynomials satisfying an addition formula

Reference 35

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This paper cites Computer-assisted proofs in analysis and programming in logic: a case study.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Computer-assisted proofs in analysis and programming in logic: a case study

Reference 36

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This paper cites Lanford, III.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Lanford, III

Reference 37

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This paper cites Nakao, and Shin'ichi Oishi.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Nakao, and Shin'ichi Oishi

Reference 38

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This paper cites Verified eigenvalue evaluation for the L aplacian over polygonal domains of arbitrary shape.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Verified eigenvalue evaluation for the L aplacian over polygonal domains of arbitrary shape

Reference 39

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This paper cites On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe

Reference 40

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T11:24:46.652099Z

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Observation 6bae777d-2163-4f9c-9e2e-ddc55a19d92c · outbound

This paper cites An efficient spectral-projection method for the N avier-- S tokes equations in cylindrical geometries: I.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs An efficient spectral-projection method for the N avier-- S tokes equations in cylindrical geometries: I

Reference 41

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T11:24:46.633696Z

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This paper cites Vorticity and incompressible flow.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Vorticity and incompressible flow

Reference 42

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T11:24:46.614911Z

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This paper cites A spectral method for polar coordinates.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs A spectral method for polar coordinates

Reference 43

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Observation d1248a14-7797-4e53-8baa-ac506130b8f2 · outbound

This paper cites Chaos in the L orenz equations: a computer-assisted proof.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Chaos in the L orenz equations: a computer-assisted proof

Reference 44

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 45

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 46

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 47

Resolution: unresolved; raw_fallback, observed 2026-08-12T11:24:46.525835Z

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 48

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This paper cites Nakao, Michael Plum, and Yoshitaka Watanabe.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Nakao, Michael Plum, and Yoshitaka Watanabe

Reference 49

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This paper cites The NIST Handbook of Mathematical Functions.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs The NIST Handbook of Mathematical Functions

Reference 50

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 51

Resolution: unresolved; raw_fallback, observed 2026-08-12T11:24:46.442375Z

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Observation f6bdd94b-efb1-4875-9472-4065e4ef12a4 · outbound

This paper cites Sanders and Luis Benet.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Sanders and Luis Benet

Reference 52

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This paper cites A new fast C hebyshev-- F ourier algorithm for P oisson-type equations in polar geometries.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs A new fast C hebyshev-- F ourier algorithm for P oisson-type equations in polar geometries

Reference 53

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This paper cites Bifurcations and dynamics of spiral waves.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Bifurcations and dynamics of spiral waves

Reference 54

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This paper cites Verified calculation of the nodes and weights for gaussian quadrature formulas.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Verified calculation of the nodes and weights for gaussian quadrature formulas

Reference 55

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This paper cites Spectral methods: algorithms, analysis and applications , volume 41.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Spectral methods: algorithms, analysis and applications , volume 41

Reference 56

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This paper cites Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains

Reference 57

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This paper cites The L orenz attractor exists.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs The L orenz attractor exists

Reference 58

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T11:24:46.338986Z

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This paper cites A rigorous ODE S olver and S male's 14th P roblem.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs A rigorous ODE S olver and S male's 14th P roblem

Reference 59

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This paper cites Validated numerics.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Validated numerics

Reference 60

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This paper cites Tensor calculus in polar coordinates using jacobi polynomials.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Tensor calculus in polar coordinates using jacobi polynomials

Reference 61

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This paper cites Computer-assisted proofs for radially symmetric solutions of pdes.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Computer-assisted proofs for radially symmetric solutions of pdes

Reference 62

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Spontaneous periodic orbits in the N avier- S tokes flow

Reference 63

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This paper cites Rotation invariant patterns for a nonlinear L aplace- B eltrami equation: A T aylor- C hebyshev series approach.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Rotation invariant patterns for a nonlinear L aplace- B eltrami equation: A T aylor- C hebyshev series approach

Reference 64

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This paper cites A proof of W right's conjecture.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs A proof of W right's conjecture

Reference 65

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T11:24:46.220069Z

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This paper cites Rigorous numerics in dynamics.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Rigorous numerics in dynamics

Reference 66

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T11:24:46.206591Z

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 67

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 68

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 69

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This paper cites Computer-assisted Existence Proofs for N avier- S tokes Equations on an Unbounded Strip with Obstacle.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Computer-assisted Existence Proofs for N avier- S tokes Equations on an Unbounded Strip with Obstacle

Reference 70

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T11:24:46.131525Z

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This paper cites A geometric method for infinite-dimensional chaos: symbolic dynamics for the K uramoto- S ivashinsky PDE on the line.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs A geometric method for infinite-dimensional chaos: symbolic dynamics for the K uramoto- S ivashinsky PDE on the line

Reference 71

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T11:24:46.113373Z

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 72

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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Unresolved cited work

Reference 73

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This paper cites Rigorous numerics for partial differential equations: the K uramoto- S ivashinsky equation.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs Rigorous numerics for partial differential equations: the K uramoto- S ivashinsky equation

Reference 74

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