arXiv research

POUnets combine partitions of unity and monomials for efficient deep learning.

We construct new multivariate copulas on the basis of a generalized infinite partition-of-unity approach. This approach allows - in contrast to finite partition-of-unity copulas - for tail-dependence as well as for asymmetry. A possibility of fitting such copulas to real data from quantitative risk management is also p…

Optimizes Lipschitz estimates for partitions of unity and characterizes spaces with Assouad-Nagata dimension.

Enhances POU-Nets with probabilistic noise model for efficient spatial data clustering.

In the paper, the author studies properties of three functions relating to the exponential function and the existence of partitions of unity, including accurate and explicit computation of their derivatives, analyticity, complete monotonicity, logarithmically complete monotonicity, absolute monotonicity, and the like.

We present a constructive and self-contained approach to data driven general partition-of-unity copulas that were recently introduced in the literature. In particular, we consider Bernstein-, negative binomial and Poisson copulas and present a solution to the problem of fitting such copulas to highly asymmetric data.

We present a constructive and self-contained approach to data driven infinite partition-of-unity copulas that were recently introduced in the literature. In particular, we consider negative binomial and Poisson copulas and present a solution to the problem of fitting such copulas to highly asymmetric data in arbitrary …

Paper studies Transformer learning theory for Euclidean and Riemannian domains.

In this paper we discuss a natural extension of infinite discrete partition-of-unity copulas which were recently introduced in the literature to continuous partition of copulas with possible applications in risk management and other fields. We present a general simple algorithm to generate such copulas on the basis of …

We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition function at or around roots of unity $q=e^{2πi \frac{1}{K}}$ with rational level $K=\frac{r}{s}$ where $r$ and $s$ are coprime integers. From the exact expression for the $G=SU(2)$ Witten-Reshetikhin-Turaev invariants of Seifert manifolds at…

This paper is devoted to dualization of paracompactness to the coarse category via the concept of $R$-disjointness. Property A of G.Yu can be seen as a coarse variant of amenability via partitions of unity and leads to a dualization of paracompactness via partitions of unity. On the other hand, finite decomposition com…

A-manifolds and A-bundles are manifolds and vector bundles modelled on a projective finitely generated module over a topological algebra A. In this paper we investigate the conditions under which an A-bundle is provided with an A-valued hermitian structure and a compatible connection, in case A is a commutative complet…

Gromov \cite{Gr$_1$} and Dranishnikov \cite{Dr$_1$} introduced asymptotic and coarse dimensions of proper metric spaces via quite different ways. We define coarse and asymptotic dimension of all metric spaces in a unified manner and we investigate relationships between them generalizing results of Dranishnikov \cite{Dr…

For non-compact manifolds with boundary we prove that bounded geometry defined by coordinate-free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geomet…

We study the Chern-Simons partition function of orthogonal quantum group invariants, and propose a new orthogonal Labastida-Mariño-Ooguri-Vafa conjecture as well as degree conjecture for free energy associated to the orthogonal Chern-Simons partition function. We prove the degree conjecture and some interesting cases o…

Resurgent analysis reveals full partition function for 3-manifold invariants.

We show a Whitney Approximation Theorem for a continuous map from a manifold to a smooth CW complex. This enables us to show that a topological CW complex is homotopy equivalent to a smooth CW complex in a category of topological spaces. It is also shown that, for any open covering of a smooth CW complex, there exists …

Continuing the study of bounded geometry for Riemannian foliations, begun by Sanguiao, we introduce a chart-free definition of this concept. Our main theorem states that it is equivalent to a condition involving certain normal foliation charts. For this type of charts, it is also shown that the derivatives of the chang…

AMORE uses neural operators to efficiently predict multiple thermochemical states in stiff chemical kinetics.

Given an open cover of a paracompact topological space X, there are two natural ways to construct a map from the cohomology of the nerve of the cover to the cohomology of X. One of them is based on a partition of unity, and is more topological in nature, while the other one relies on the Mayer-Vietoris double complex, …

Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star, Lie derivative, pullback and interior product. Partitions of unity exist in thi…

We propose an extension of the recently-proposed volume conjecture for closed hyperbolic 3-manifolds, to all orders in perturbative expansion. We first derive formulas for the perturbative expansion of the partition function of complex Chern-Simons theory around a hyperbolic flat connection, which produces infinitely-m…

Generalized Stacey-Roberts lemma for Banach manifolds.

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using parti…

We propose a new model for pricing Quanto CDS and risky bonds. The model operates with four stochastic factors, namely: hazard rate, foreign exchange rate, domestic interest rate, and foreign interest rate, and also allows for jumps-at-default in the FX and foreign interest rates. Corresponding systems of PDEs are deri…

The study explores knot invariants using roots of unity.

A classical spin network consists of a ribbon graph (i.e., an abstract graph with a cyclic ordering of the vertices around each edge) and an admissible coloring of its edges by natural numbers. The standard evaluation of a spin network is an integer number. In a previous paper, we proved an existence theorem for the as…

Jones Polynomial shows unity in math.

The paper studies Nijenhuis operators with a unity and their connection to F-manifolds.

The paper extends ternary algebra concepts using cube roots of unity.

For an arbitrary positive integer n, we construct infinitely many one-cusped hyperbolic 3-manifolds where each manifold's A-polynomial detects every n-th root of unity. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to which roots of unity arise in this manner.

We construct an invariant J_M of integral homology spheres M with values in a completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at each root of unity ζgives the the SU(2) Witten-Reshetikhin-Turaev invariant τ_ζ(M) of M at ζ. Thus J_M unifies all the SU(2) Witten-Reshetikhin-Turaev invariants of…

Study centers of quantum tori and skein algebras for even roots of unity.

Jones polynomials have infinitely many roots of unity as zeros.

Generative models learn manifold structure; new approach uses atlas and geodesic interpolation.

Categorifies colored Jones polynomial at roots of unity.

This paper gives examples of hyperbolic 3-manifolds whose SL(2,C) character varieties have ideal points whose associated roots of unity are not 1 or -1. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to whether roots of unity other than 1 and -1 occur.

The paper develops mixed-integer formulations for neural networks using partitioning.

New identities link Frobenius elements to Jones-Wenzl projectors at roots of unity.

The paper proves a relation between four types of invariants.

Constructs maps on skein modules using non-semisimple quantum invariants.

For each finite dimensional, simple, complex Lie algebra $\mathfrak g$ and each root of unity $ξ$ (with some mild restriction on the order) one can define the Witten-Reshetikhin-Turaev (WRT) quantum invariant $τ_M^{\mathfrak g}(ξ)\in \mathbb C$ of oriented 3-manifolds $M$. In the present paper we construct an invariant…

Recent advances in artificial intelligence have been driven by the presence of increasingly realistic and complex simulated environments. However, many of the existing environments provide either unrealistic visuals, inaccurate physics, low task complexity, restricted agent perspective, or a limited capacity for intera…

Study on quantum invariants of twist knots at specific roots of unity.

Study on detecting hierarchical community structures in networks.

Study on quantum invariants of twist knots at specific roots of unity.

Constructs integrable hierarchies for generalized Frobenius manifolds with non-flat unity.

Improved neural network robustness certification through tighter convex relaxations.