Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.
problem Calculating curvature operators and Poincaré polynomials for symmetric spaces.
method Explicit formulas derived using quantum numbers and eigenvalue analysis.
result Maximum eigenvalue of curvature operators bounded by Einstein constant, with equality for Hermitian spaces.
The paper simplifies the computation of a complex polynomial using Yang-Baxter operators.
problem Computing the homology of Yang-Baxter operators for arbitrary m.
method Reduced the computation to initial conditions and produced explicit formulas.
result Explicit formulas for the third and fourth homology.
Adjusts Yang-Baxter operators for HOMFLYPT polynomials.
problem Computing 2-cocycle invariants for links.
method Adjusts Yang-Baxter operators and computes second homology.
result Potential for 2-cocycle invariant for links.
The paper explores how polynomial roots and operator eigenvalues change with parameters.
problem How do roots of polynomials and eigenvalues of operators vary with parameter changes?
method Analyzes parameter dependence of polynomials and linear operators, covering real analytic to differentiable of finite order.
result Definitive optimal results for perturbation theory of polynomials and linear operators, including hyperbolic polynomials.
4-move kills Alexander polynomial
problem Whether the 4-move is an unknotting operation
method showing every knot can be reduced via 4-moves and isotopies
result every knot can be reduced to one with a trivial Alexander polynomial
Correspondence found between Askey-Wilson polynomials and genus-two handlebody skein module.
problem Understanding the genus-two handlebody skein module.
method Using q-difference operators for the genus-two skein algebra.
result Correspondence between reduced Askey-Wilson polynomials and genus-two handlebody skein module.
Study polynomial structures on generalized tangent bundles and their compatibility with operators.
problem Understanding polynomial structures on generalized tangent bundles.
method Analyzing skew-symmetric endomorphisms satisfying polynomial equations and their compatibility with de Rham and Courant-Dorfman operators.
result Conditions equivalent to integrability of generalized almost complex structures.
Study the spectrum of Poincaré operator in triaxial ellipsoids.
problem Spectrum of the Poincaré operator in triaxial ellipsoids.
method Microlocal analysis of partial differential equations and polynomial vector fields.
result Polynomial eigenvectors and large-degree asymptotics of the operator.
Paper revisits graph-CNNs using Laplace-Beltrami spectral filters and polynomials.
problem Improving spectral graph convolutional neural networks (graph-CNNs).
method Developed Laplace-Beltrami CNN (LB-CNN) by replacing graph Laplacian with LB operator and approximating spectral filters using Chebyshev, Laguerre, and Hermite polynomials.
result Classification accuracy of LB-CNN is not dependent on the type of polynomials or operators.
In this note, we consider the problem of constructing knot invariants from Yang-Baxter operators associated to (unitary associative) algebra structures. We first compute the enhancements of these operators. Then, we conclude that Turaev's procedure to construct knot invariants, as modified by Murakami, invariably produ…
FNOs learn solution operators of dissipative equations efficiently via spectral methods.
problem Learning and approximation of solution operators for dissipative equations.
method Introducing spectral methods and deriving FNO approximation bounds and sample complexity guarantees.
result Polynomial sample complexity guarantees for FNOs learning solution operators of dissipative equations.
Quillen proved that, if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventu…
NO approximates non-Markovian BSDEs with polynomial scaling in 1/ε.
problem Complexity of NO approximations for structured families of BSDEs.
method Identifying structured families of non-Markovian BSDEs, informing NO's inductive bias.
result Polynomial scaling in 1/ε for NO approximations of BSDE solution operators.
We give a criterion to detect whether the derivatives of the HOMFLY polynomial at a point is a Vassiliev invariant or not. In particular, for a complex number b we show that the derivative P_K^{(m,n)}(b,0)=d^m/da^m d^n/dx^n P_K(a,x)|(a, x) = (b, 0) of the HOMFLY polynomial of a knot K at (b,0) is a Vassiliev invariant …
Polynomial Chaos Expansion improves operator learning for PDEs.
problem Approximating mappings between infinite-dimensional functional spaces.
method Polynomial Chaos Expansion (PCE) for operator learning.
result PCE achieves strong performance in operator learning and uncertainty quantification.
The study describes Nijenhuis operators with specific properties.
problem Characterizing Nijenhuis operators with functional independence and determinant constraints.
method Proving the general form and describing the specific case of Nijenhuis operators.
result Complete description of Nijenhuis operators with nondegenerate determinant.
We give a method of decomposing bundle-valued polynomials compatible with the action of the Lie group Spin(n), where important tools are Spin(n)-equivariant operators and their spectral decompositions. In particular, the top irreducible component is realized as an intersection of kernels of these operators.
Survey on strong convergence in random matrices and its applications.
problem Understanding convergence of random matrices to operators.
method Analysis of operator norms of noncommutative polynomials.
result New insights and applications in random graphs, geometry, and operator algebras.
We provide new examples of diffusion operators in dimension 2 and 3 which have orthogonal polynomials as eigenvectors. Their construction rely on the finite subgroups of O(3) and their invariant polynomials.
In this paper, we investigate the Dirchlet eigenvalue problems of poly-Laplacian with any order and quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first k eigenvalues which improve the previous results.
Classifies 3D non-degenerate left-symmetric algebras.
problem Classifying left-symmetric algebras in 3D.
method Using Nijenhuis geometry and algebraic independence of coefficients in characteristic polynomial.
result Classification of differentially non-degenerate LSA in dimension 3.
A representation of the Jacobi algebra h1⋊su(1,1) by first order differential operators with polynomial coefficients on the manifold C×D1 is presented. The Hilbert space of holomorphic functions on which the holomorphic first order differential operators with …
Neural operators solve families of 2BSDEs efficiently.
problem Solving infinite families of 2BSDEs on bounded domains.
method Introduces a mild generative neural operator model to approximate solutions.
result Solution operators can be approximated by neural operators with polynomial parameters.
PolyNSD improves Neural Sheaf Diffusion with polynomial operators and spectral rescaling.
problem Limitations of common Neural Sheaf Diffusion implementations, including scalability and stability issues.
method Introduces Polynomial Neural Sheaf Diffusion (PolyNSD) with a degree-K polynomial propagation operator and spectral rescaling.
result PolyNSD achieves state-of-the-art results on both homophilic and heterophilic benchmarks with reduced runtime and memory requirements.
We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie-Yamaguti algebra. This identity is a consequence of the defining identities for Lie-Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit…
Proves a Gel'fand-Kolmogoroff type result for vector bundles and polynomial functions.
problem Characterizing vector bundles and polynomial functions using differential operators.
method Analyzes the associative structure of symbols of differential operators and their eigenvectors.
result Derives a Gel'fand-Kolmogoroff type result for the algebra of symbols of differential operators.
Numerical experiments support conjecture about opers and nonabelian Hodge.
problem Testing predictions of Gaiotto-Moore-Neitzke and Gaiotto conjectures.
method Numerical experiments on polynomial holomorphic differentials.
result Supports conjectural formulas for Stokes data and Hitchin metric tensor.
We illustrate from the viewpoint of braiding operations on WZNW conformal blocks how colored HOMFLY polynomials with multiplicity structure can detect mutations. As an example, we explicitly evaluate the (2,1)-colored HOMFLY polynomials that distinguish a famous mutant pair, Kinoshita-Terasaka and Conway knot.
This paper continues the work of our previous paper [8], where we generalize kth-powers of the Euclidean Dirac operator D_x to higher spin spaces in the case the target space is a degree one homogeneous polynomial space. In this paper, we reconsider the generalizations of D_x^3 and D_x^4 to higher spin spaces in the ca…
Automatic continuity of polynomial maps and cocycles proved.
problem Proving continuity of polynomial maps and cocycles.
method The approach involves proving continuity of polynomial maps and cocycles.
result Automatic continuity of polynomial maps and cocycles.
A complete description of Osserman four-manifolds whose Jacobi operators have a nonzero double root of the minimal polynomial is given.
This paper studies a particular class of higher order conformally invariant dif- ferential operators and related integral operators acting on functions taking values in particular finite dimensional irreducible representations of the Spin group. The differential operators can be seen as a generalization to higher spin …
A new polynomial invariant for links in a thickened torus exhibits volume conjecture behavior.
problem Defining and characterizing a new polynomial invariant for links in a thickened torus.
method Defining a new invariant JnT, proving properties, and providing constructions. result The invariant JnT exhibits volume conjecture behavior, providing the first example of this in a virtual link. Study cohomology spaces of sl(2) acting on n-ary differential operators.
problem Computing cohomology spaces for sl(2) action on n-ary differential operators.
method Analyzes polynomial μ-densities as sl(2) modules and computes cohomological spaces H^2.
result Computed cohomological spaces H^2 of sl(2) on n-ary differential operators.
New method learns low-dimensional models for systems with non-polynomial terms.
problem Modeling systems with non-polynomial nonlinear terms that are spatially local and given in analytic form.
method Non-intrusive model reduction method that learns operators for linear and polynomially nonlinear dynamics via a least-squares problem incorporating given non-polynomial terms.
result Comparable accuracy to intrusive methods that require full knowledge of governing equations.
We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras cgaℓ(d,C) with d=1 for any integer value ℓ∈N. The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type, and id…
The paper defines higher invariants for groups of polynomial growth and proves their convergence.
problem Defining and proving convergence of higher invariants for groups of polynomial growth.
method Using delocalized cyclic cocycles and a determinant map construction.
result A well-defined pairing between delocalized cyclic cocyles and K-theory classes of C*-algebraic secondary higher invariants.
Study efficient neural operator learning using variation spaces.
problem Operator learning using encoder-decoder neural networks.
method Introduce variation space for nonlinear operators, establish approximation bounds.
result Algebraic approximation and learning rates for polynomially decaying input and output encoding errors.
The paper connects curvature data to polynomial coefficients in gluing formulas.
problem Understanding polynomial coefficients in gluing formulas for zeta-determinants.
method Expressing coefficients of a polynomial in terms of scalar and principal curvatures of a 2D hypersurface.
result Coefficients of the polynomial are expressed in terms of curvature data.
Study satellite operations on knot invariant θ, proving additivity and distinguishing knots.
problem Behavior of knot invariant θ under satellite operations.
method Proved additivity, introduced computational tool, verified conjecture for 2977 knots.
result Distinguished knots using invariant θ and verified conjecture for 2977 primes.
Estimates for polynomial operators using determinant majorization and subharmonics.
problem Bounding solutions of polynomial operators on Euclidean domains.
method Combines Alexandrov estimate and determinant majorization, using subharmonics and semiconvex approximation.
result Includes classical Alexandrov-Bakelman-Pucci estimate for linear operators.
Study G2 skein algebra elements using Kuperberg webs and threading operations.
problem Identify and analyze central elements in G2 skein algebra. method Use Kuperberg webs, threading operations, and skein-theoretic arguments.
result Verify central elements and obtain uniqueness of transparent polynomials.
Study examines boundedness of oscillating singular integrals on specific Lie groups.
problem Investigating boundedness of oscillating singular integrals on Lie groups of polynomial growth.
method Presented kernel criteria in terms of sub-Riemannian structure and Fourier analysis.
result Extended classical oscillating conditions for boundedness of oscillating convolution operators.
Our goal is to compute the minimal-order recurrence of the colored Jones polynomial of the 7_4 knot, as well as for the first four double twist knots. As a corollary, we verify the AJ Conjecture for the simplest knot 7_4 with reducible non-abelian SL(2,C) character variety. To achieve our goal, we use symbolic summatio…
The paper proves an inequality for symmetric polynomials under a fixed point measure.
problem An inequality for elementary symmetric polynomials under a fixed point measure of permutations.
method Constructing differential operators to set up a monotone flow.
result The inequality is proven and is sharp.
Study disproves conjecture about low-degree polynomials in hypothesis testing.
problem Conjecture about limitations of polynomial-time algorithms in hypothesis testing.
method Used counterexamples to refute the conjecture and modified the conjecture to rule out the counterexample.
result Disproved conjecture about limitations of low-degree polynomials in hypothesis testing.
In this paper, we enlarge the space of uniformly supported pseudo-differential operators on some groupoids by considering kernels satisfying certain asymptotic estimates. We show that such enlarged space contains the compact parametrix, and the generalized inverse of uniformly supported operators with Fredholm vector r…
We determine the structure of conformal powers of the Dirac operator on Einstein {\it Spin}-manifolds in terms of the product formula for shifted Dirac operators. The result is based on the techniques of higher variations for the Dirac operator on Einstein manifolds and spectral analysis of the Dirac operator on the as…