We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties. For knots with point singularities, we obtain a classification of these polynomial…
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Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
problem Analyzing locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
method Examines Riemannian and Finslerian surfaces, providing necessary and sufficient conditions for locally symmetric fourth root metrics in 2D and more complex conditions for higher dimensions.
result Formulates conditions for positive definiteness of locally symmetric polynomial metrics in Finslerian surfaces and provides explicit examples.
By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the i…
Local polynomial regression (Fan and Gijbels 1996) is an important class of methods for nonparametric density estimation and regression problems. However, straightforward implementation of local polynomial regression has quadratic time complexity which hinders its applicability in large-scale data analysis. In this pap…
New knot models analyze local entanglement for robust curve analysis.
problem Lack of local structural information in classical knot theory.
method Proposed multiscale and persistent Jones polynomials.
result Models are stable to small perturbations, robust for real-world applications.
Study local moduli of Sasaki-Einstein metrics on specific polynomial links.
problem Understanding the local moduli of Sasaki-Einstein metrics on links of invertible polynomials.
method Analyzing Sasaki-Einstein metrics on links of invertible polynomials of cycle type and Thom-Sebastiani sums.
result For polynomials of cycle type, local moduli spaces are zero-dimensional. For Thom-Sebastiani sums, dimensions are positive.
In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These \emph{virtual graphs} are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, t…
Continuity of roots of hyperbolic polynomials with smooth coefficients.
problem Continuity of the solution map for hyperbolic polynomials.
method Proving continuity of the solution map from hyperbolic polynomials of degree d with C^d coefficients to their increasingly ordered roots.
result Continuity of the solution map for hyperbolic polynomials with C^d coefficients.
We prove that the set of non-properness of a polynomial mapping of the three dimensional space which is a local homeomorphism cannot be homeomorphic to the real line
Homological algebra used to study local equivalence of complex rings.
problem Local equivalence of bounded complexes over polynomial rings.
method Homological algebra approach
result Results have been proved in many places in the literature.
Proposes a new regression method using -norms for non-Gaussian noise.
problem Non-Gaussian noise in residuals affects the performance of local least squares regression.
method Introduces local polynomial -norm regression, replacing weighted least squares with weighted -norm estimation.
result Demonstrates superior performance over local least squares in one-dimensional data and higher dimensions.
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.
Local high-degree polynomial integrals of geodesic flows and the generalized hodograph methodmath.DS
Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
problem Integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
method Semi-Hamiltonian systems of PDEs and generalized hodograph method.
result Construction of many local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5.
As we have proved in [L], the geodesic flows associated with the flat metrics on T^2 minimize the polynomial entropy. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated to flat metrics are local strict minima for the polynomial entropy.…
Paper proposes a robust LPR method using similarity kernels.
problem Outliers and high-leverage points affect traditional LPR's accuracy.
method Integrates predictor and response variables in weighting mechanism using a conditional density kernel.
result Lower empirical bias compared to iterative robust LOWESS.
Shallow neural networks can represent polynomials efficiently.
problem Representing polynomials using shallow neural networks.
method Using shallow neural networks of width to represent -variate polynomials of degree .
result Derives minimax optimal convergence rate for shallow networks to unknown univariate regression functions.
We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We recover the classical torsionness and divisibility results, which say that, unde…
A well-known identity (Alex+) - (Alex-)=(t^{1/2}-t^{-1/2}) (Alex0) holds for three 1-links L+, L-, and L0 which satisfy a famous local-move-relation. We prove a new local-move-identity for the Z[t,t^{-1}]-Alexander polynomials of 2-links, which is a 2-dimensional analogue of the 1-dimensional one. In the 1-dimensional …
Study local perturbations of vector bundles with polynomial curvature solutions.
problem Existence and stability of solutions to geometric PDEs under deformations.
method Geometric invariant theory, moment map framework, polystability conditions.
result Existence and uniqueness of solutions under local polystability conditions.
Study groups with polynomial growth, finding structure and applications.
problem Understanding groups with polynomial growth structure.
method Structure theorem for locally compact groups of polynomial growth.
result Applications on various growth functions and relations to FC-G series.
The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momentamath.DG
Any smooth geodesic flow is locally integrable with smooth integrals. We show that generically this fails if we require, in addition, that the integrals are polynomial (or, more generally, analytic) in momenta. Consequently we obtain that a generic real-analytic metric does not admit, even locally, a real-analytic inte…
New proof of Alesker's Irreducibility Theorem using localization techniques.
problem Representing polynomial valuations on convex bodies.
method Introducing a localization technique for polynomial valuations and reducing to a representation problem for differential forms.
result Smooth and translation invariant valuations are representable by integration with the normal cycle.
Study shows polynomial-width neural networks can closely approximate infinite-width networks in polynomial time.
problem Approximating dynamics of polynomial-width neural networks with infinite-width networks.
method Bounding approximation gap through a differential equation governed by mean-field dynamics, considering local Hessian.
result Polynomially many neurons are sufficient to closely approximate mean-field dynamics.
This study shows the moment-SOS hierarchy converges in polynomial optimization over product of spheres.
problem Minimizing multihomogeneous polynomials over product of spheres.
method Moment-SOS hierarchy, local optimality conditions, differential geometry, Morse theory.
result The moment-SOS hierarchy has finite convergence for generic multihomogeneous objective functions.
New estimator adapts to various error distributions.
problem Adapting to different error distributions in nonparametric regression.
method Introduces outrigger local polynomial estimator with modified weighted least squares.
result Minimax optimal over Hölder classes with multiplicative factor.
New local-search methods close the gap in sparse tensor PCA.
problem Sparse tensor PCA underperforms compared to other methods.
method Proposes new local-search methods including greedy and random-threshold variants.
result Proves local-search methods close the gap to best known polynomial-time procedures.
In this paper we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the -character variety. We also discuss similar things for the higher dimensional twisted Alexander polynomial and the Reidemeister torsion.
Almost sharp local Bernstein estimates for Laplace eigenfunctions on compact Riemannian manifoldsmath.AP
The study sharpens local Bernstein estimates for Laplace eigenfunctions on compact manifolds.
problem Understanding local growth properties of Laplace eigenfunctions on compact Riemannian manifolds.
method Refined Donnelly-Fefferman method based on --Carleman estimates, combined with elliptic regularity and patching of local Carleman estimates.
result Almost sharp local --Bernstein inequalities for .
For any virtual link that may be decomposed into a pair of oriented -tangles and , an oriented local move of type is a replacement of with the -tangle in a way that preserves the orientation of . After developing a general decomposition for the Jones polynomial of …
We study a class of 2-variable polynomials called exact polynomials which contains -polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the …
New geometric object for polynomials simplifies complex data.
problem Understanding the combinatorial and geometric properties of polynomials.
method Introducing a compact planar 2-complex for polynomials with distinct roots.
result Extracts combinatorial data from a geometric structure of polynomials.
The paper explores algebraic properties of Alexander polynomials and Reidemeister torsions for torus knots.
problem Investigating algebraic properties of Alexander polynomials and Reidemeister torsions for torus knots.
method Analyzing twisted Alexander polynomials and Reidemeister torsions of torus knots associated with irreducible -representations.
result Proves that coefficients of twisted Alexander polynomials are locally constant functions on the -character variety.
Proves curvature of conference graphs and finds local matchings.
problem Proving precise values of curvature in conference graphs.
method Combining parameter relations and combinatorial approach.
result Existence of local perfect matchings in broader classes of graphs.
Given a homomorphism from a knot group to a fixed group, we introduce an element of a -group, which is a generalization of (twisted) Alexander polynomials. We compare this -class with other Alexander polynomials. In terms of semi-local rings, we compute the -classes of some knots and show their non-trivi…
New approach to adaptively select bandwidths in nonparametric regression.
problem Adaptive bandwidth selection in nonparametric regression.
method Inspired by -norms of interval projections, introduces a new bandwidth selection procedure.
result Obtains non-asymptotic risk bounds for local polynomial regression methods that adapt to local Hölder exponent.
New polynomial invariants for virtual links are stronger than F-polynomials.
problem Defining new invariants for virtual links.
method Introducing weight functions and a recurrent construction for new invariants.
result New polynomial invariants are stronger than F-polynomials.
The space C of conservative vertex colorings (over a field F) of a countable, locally finite graph G is introduced. The subspace of based colorings is shown to be isomorphic to the bicycle space of the graph. For graphs G with a free Z^d-action by automorphisms, C is a finitely generated module over the polynomial ring…
Exposes two methods for constructing flat surfaces in 4D spaces.
problem Building locally flat embedded surfaces in 4-manifolds.
method Direct methods and surgery theory.
result Every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat embedded torus.
Covariance pooling is a feature pooling method with good classification accuracy. Because covariance features consist of second-order statistics, the scale of the feature elements are varied. Therefore, normalizing covariance features using a matrix square root affects the performance improvement. When pooling methods …
A pseudo-Riemannian manifold is called CSI if all scalar polynomial invariants constructed from the curvature tensor and its covariant derivatives are constant. In the Lorentzian case, the CSI spacetimes have been studied extensively due to their application to gravity theories. It is conjectured that a CSI spacetime i…
On Acceleration of Gradient-Based Empirical Risk Minimization using Local Polynomial Regressionmath.OC
Accelerates ERM problems with LPI-GD and improved oracle complexity.
problem Empirical Risk Minimization (ERM) problems with strong convexity and smoothness.
method Local Polynomial Interpolation-based Gradient Descent (LPI-GD) and accelerated methods.
result Oracle complexity improved to $ ilde{O}\left(\sqrtσ m^d \log(1/\varepsilon)
ight)$.
Ancient caloric functions on manifolds with polynomial growth are studied under volume doubling barrier.
problem Analyzing ancient caloric functions on manifolds beyond volume doubling.
method Time polynomial structure result on ancient caloric functions with polynomial growth.
result Finiteness result for ancient caloric functions is essentially sharp, except for multi-end cases.
New methods test discrete distributions faster with local privacy constraints.
problem Testing discrete distributions under local differential privacy constraints.
method Efficient randomized algorithms and test procedures, both non-interactive and interactive.
result Faster separation rates in interactive privacy mechanisms.
We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples…
We give a new proof of Gromov's theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.
MixCIT tests conditional independence for mixed data types efficiently and reliably.
problem Testing conditional independence for mixed data types, especially when at least one is continuous.
method Graph-based test statistic comparing kernel similarities, debiased local-polynomial approach for continuous variables.
result Unified, efficient, and statistically guaranteed solution across heterogeneous data types.
Polynomial delay algorithm tests causal models with hidden variables.
problem Testing causal models with hidden variables in polynomial delay.
method c-component local Markov property (C-LMP) and polynomial delay algorithm.
result First algorithm for poly-delay testing of CIs in causal graphs with hidden variables.
Paper addresses ERM in LDP, reducing sample complexity for smooth and convex losses.
problem Achieving error α in ERM with non-interactive LDP, especially for high-dimensional data.
method Developed algorithms using Bernstein polynomial and polynomial approximation techniques.
result For smooth and convex losses, sample complexity is linear in dimensionality.