DEPTS learns to forecast periodic time series with improved accuracy.
problem Forecasting periodic time series is challenging due to complex dependencies and diverse periods.
method DEPTS uses a decoupled formulation with an expansion module and a periodicity module to handle these challenges.
result DEPTS significantly improves forecasting accuracy, reducing errors by up to 20%.
Paper extends Schur's theorem to spherical curves via monotonicity.
problem Comparing chord lengths of convex and spherical curves.
method Monotonicity and expansion module approach.
result Schur's Theorem extended to spherical curves.
We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a…
Researchers compute dimensions of GLN-skein modules for genus-one mapping tori.
problem Computing dimensions of GLN-skein modules for mapping tori.
method Explicit Euler product expansion of the skein partition function.
result Explicit computation of dimensions and generating function.
New Langlands duality conjectures for 3-manifold skein modules.
problem Understanding Langlands duality in 3-manifold skein modules.
method Combining representation theory of double affine Hecke algebras and 1-form symmetry structure.
result Recent special cases confirmed, with detailed proofs.
Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.
problem Understanding HOMFLYPT polynomials and their geometric origins.
method Defining worldsheet skein module and D-module, considering skein valued open curve counts.
result Worldsheet skein D-module for Hopf link conormal is generated by three operator polynomials.
Constructs a sheaf of Bargmann-Fock modules on Kähler manifolds.
problem Deformation quantization on Kähler manifolds.
method Extends Fedosov's method to Kähler manifolds with compatible Fedosov abelian connections.
result Explicit construction of sheaf of flat sections as a module over deformation quantization algebras.
TEAFormers preserve multi-dimensional time series structures for better forecasting.
problem Traditional Transformers flatten multi-dimensional time series data, losing critical multi-dimensional relationships.
method Tensor-Augmented Transformer (TEAFormer) with Tensor-Augmentation (TEA) module.
result Significant performance enhancements in time series forecasting across benchmarks.
Quantum trace map defines invariants for knots and links, confirming a length conjecture.
problem Defining invariants for knots and links in hyperbolic 3-manifolds.
method Introducing a quantum trace map for ideally triangulated knot complements, combining with state-integral models.
result Perturbative invariants determine an asymptotic expansion of the Jones polynomial, confirming the length conjecture.
Paper proposes new methods for improving interatomic potentials.
problem Limitations of conventional SO(2) Linear architectures in MLIPs.
method Direct Cartesian construction, recursive Clebsch-Gordan construction, Edge Complex Product Basis, Radial Rotary Complex Attention.
result TECE-OAM-RRA-1.0 achieves SOTA performance on Matbench Discovery.
Researchers find isospectral but non-diffeomorphic nilmanifolds.
problem Identifying isospectral but non-diffeomorphic nilmanifolds.
method Analysis of pseudo H-type nilmanifolds and construction of isospectral nilmanifolds.
result Construction of isospectral but non-diffeomorphic nilmanifolds.
Proposes ContSup to boost local learning by supplying context between isolated modules.
problem Local learning's performance degrades with more isolated modules.
method Theoretical analysis and ContSup scheme to supply context between modules.
result Significant performance improvement with minimal overhead.
We give new explicit formulas for the representations of the mapping class group of a genus one surface with one boundary component which arise from Integral TQFT. Our formulas allow one to compute the h-adic expansion of the TQFT-matrix associated to a mapping class in a straightforward way. Truncating the h-adic expa…
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…
Study of 3d-3d correspondence involving q-Weyl algebra and 3d-index.
problem Understanding the action of a q-Weyl algebra on the 3d-index of knots. method Investigation of the q-Weyl algebra's module action on the 3d-index, conjecturing structural properties. result Bilinear factorization, pair of linear q-difference equations, and rational function matrix for the 3d-index determination. Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for A1 and A2 clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formula…
Let M be a Riemannian manifold. For p∈M, the tensor algebra of the negative part of the (complex) affinization of the tangent space of M at p has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over M with a connection. We …
The asymptotic concentration of the Fr{é}chet mean of IID random variables on a Rieman-nian manifold was established with a central limit theorem by Bhattacharya \& Patrangenaru (BP-CLT) [6]. This asymptotic result shows that the Fr{é}chet mean behaves almost as the usual Euclidean case for sufficiently concentrated di…
We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is intimately related to …
For a Catalan state C of a lattice crossing L(m,n) with no returns on one side, we find its coefficient C(A) in the Relative Kauffman Bracket Skein Module expansion of L(m,n). We show, in particular, that C(A) can be found using the plucking polynomial of a …
Residual networks' depth is mathematically equivalent to expanding an implicit ensemble size.
problem Understanding why deep residual networks are effective.
method Formal analysis of residual networks as ensembles of shallow models.
result Increasing network depth is equivalent to expanding the size of an implicit ensemble, revealing a hierarchical structure.
The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by TTQ. Le and the author that such sequences are q-holonomic, that is, they satisfy linear q-difference equations with coefficients Laurent polynomials in q and qn. We show from first principles that q-holonomic sequence…
Rescaling expansiveness proven for k*-expansive vector fields.
problem Proving rescaling expansiveness for k*-expansive vector fields.
method Introducing and exploring singular-expansive flows.
result Rescaling expansiveness established for k*-expansive vector fields.
The Habiro ring of a number field uses power series to study algebraic K-theory.
problem Analyzing algebraic K-theory of number fields.
method Introduces Habiro ring and modules graded by K3(K). result Establishes connections between Habiro ring and Chern-Simons theory.
The paper derives expansions for Green's operators and resolvents using Hadamard methods.
problem Analyzing normally hyperbolic operators and their Green's functions.
method Hadamard expansions for powers of Green's operators and resolvents.
result Derives expansions involving Hadamard coefficients for advanced/retarded Green's operators.
Develops a martingale expansion for stochastic volatility models.
problem Approximating marginal distributions of stochastic volatility models.
method Martingale expansion framework for continuous stochastic volatility models.
result First-order perturbation expansions for small volatility-of-volatility and fast mean-reversion models.
Taylor expansions improve reinforcement learning policies.
problem Improving reinforcement learning policy optimization.
method Taylor expansion policy optimization.
result Taylor expansions enhance performance of distributed algorithms.
Analytic torsion expansions for symmetric and complex homogeneous spaces.
problem Calculating the full asymptotic expansion of analytic torsion for various spaces.
method Explicit calculation and comparison with existing results.
result Explicit full asymptotic expansions for symmetric and complex homogeneous spaces.
Proved cyclotomic expansion for double twist knots' HOMFLY-PT invariants.
problem Proving cyclotomic expansion for colored HOMFLY-PT invariants of double twist knots.
method Cyclotomic expansion formula for double twist knots.
result Confirmed cyclotomic expansion conjecture for SU(N)-invariants.
Study of hypersurfaces with specific expansion properties.
problem Existence and properties of hypersurfaces with prescribed null expansion.
method Adapted Eichmair's Perron approach for existence.
result Topology theorem for hypersurfaces with prescribed null expansion.
A new hypergraph expansion method treats vertices and hyperedges equally, improving node classification.
problem Information loss in hypergraph expansions on either vertex or hyperedge level.
method Proposes a new hypergraph formulation named line expansion (LE) that treats vertices and hyperedges symmetrically.
result The proposed line expansion method outperforms state-of-the-art baselines on five hypergraph datasets.
Classifies modules of surface-knots in terms of their properties.
problem Characterizing modules of surface-knots in terms of their properties.
method Using homology and covering spaces, the reduced first module is characterized.
result The reduced first module for every genus g is characterized in terms of properties of a finitely generated module.
Asymmetric expansion preserves convexity in hyperbolic geometry.
problem Maintaining convexity in hyperbolic geometry under asymmetric expansions.
method Generalizing earlier results on radial expansion to asymmetric expansion.
result Asymmetric expansion of hyperbolic convex sets remains convex.
Curvature defined for Hilbert modules and Kasparov modules.
problem Defining and studying curvature in Hilbert modules and Kasparov modules.
method Introduced curvature for densely defined universal connections on Hilbert C∗-modules relative to spectral triples. result Curvature only depends on the represented form of the universal connection modulo junk forms.
The paper calculates asymptotic expansions for specific types of oscillatory integrals.
problem Analyzing oscillatory integrals with complex phase functions.
method Using asymptotic expansions of simpler phase functions to derive results for more complex cases.
result Explicit computation of coefficients in asymptotic expansions for certain integrals.
Dropout increases the generalization of neural networks by expanding the weight space.
problem Understanding and improving the generalization of neural networks.
method Introducing weight expansion and showing that dropout leads to it.
result Dropout increases the generalization of neural networks by expanding the weight space.
Proves finiteness and holonomicity of skein modules for 3-manifolds.
problem Finiteness and holonomicity of skein modules for 3-manifolds.
method Defining skein transfer bimodules and using q-analogues of D-module theory.
result Internal skein modules are holonomic modules over the internal skein algebra of the boundary.
Defines super projective modules and explores their properties.
problem Exploring the geometric-algebraic link in super geometry.
method Defined and explored super projective modules over supersmooth functions.
result Module of vector fields over a supersphere is a super projective module.
New derivation of knot invariants from universal invariant.
problem Deriving knot invariants from universal invariant.
method Using Hopf algebra D and a Mathematica implementation to compute ZD(K). result Derivation of large-color expansion from universal invariant.
This work explores functional expansions to handle path dependence in various fields.
problem Path dependence and infinite-dimensional problems in non-Markovian systems.
method Generalizes Wiener series and functional Taylor expansion to handle static and dynamic functionals.
result Elegant separation of functionals from future trajectories in dynamic cases.
In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus correctio…
A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is …
We introduce higher skein modules of links generalizing the Conway skein module. We show that these modules are closely connected to the HOMFLY polynomial.
Neural Module Networks, originally proposed for the task of visual question answering, are a class of neural network architectures that involve human-specified neural modules, each designed for a specific form of reasoning. In current formulations of such networks only the parameters of the neural modules and/or the or…
Paper calculates third coefficient in Kaehler-Einstein metric expansion.
problem Understanding Kaehler-Einstein metrics and their epsilon functions.
method Computes the third coefficient in the TYCZ-expansion of the epsilon function.
result Discovers the vanishing of the third coefficient's significance.
Paper compares skein modules to Kauffman bracket modules.
problem Comparing skein modules to Kauffman bracket modules.
method Using skein relations and Reshetikhin-Turaev model.
result Resolved the problem of comparing skein modules to Kauffman bracket modules.
Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach "algebra situs." When looking at the panorama of skein modules we see, past the rolling hills of homologies and homotopies, distant mountains - the Kauffman bracket skein mo…
We describe the first known mean-field study of landing probabilities for random walks on hypergraphs. In particular, we examine clique-expansion and tensor methods and evaluate their mean-field characteristics over a class of random hypergraph models for the purpose of seed-set community expansion. We describe paramet…