arXiv research

Formulates Hilbert reciprocity law on 3-manifolds.

We show some fundamental results concerning $3$-dimensional foliated dynamical systems (FDS$^3$ for short) introduced by Deninger. Firstly, we give a decomposition theorem for an FDS$^3$, which yields a classification of FDS$^3$'s. Secondly, for each type of the classification, we construct concrete examples of FDS$^3$…

Reciprocity laws for line bundles on circle fibrations over complex manifolds.

This paper gives a new definition of the Contou-Carrere symbol in terms of an exponential of a Chen iterated integral and proves the corresponding reciprocity law.

Reciprocal processes are acausal generalizations of Markov processes introduced by Bernstein in 1932. In the literature, a significant amount of attention has been focused on developing dynamical models for reciprocal processes. In this paper, we provide a probabilistic graphical model for reciprocal processes. This le…

We prove that the Hilbert geometry of a convex domain in ${\mathbb R}^n$ has bounded local geometry, i.e., for a given radius, all balls are bilipschitz to a euclidean domain of ${\mathbb R}^n$. As a consequence, if the Hilbert geometry is also Gromov hyperbolic, then the bottom of its spectrum is strictly positive. We…

We compute the average Tristram---Levine signature of any graph link with positive weights in a three sphere, generalizing the results of Kirby and Melvin. The main tools are the Neumann's algorithm for computing the equivariant signatures of graph links and the Reciprocity Law for Dedekind sums.

The role of kernels is central to machine learning. Motivated by the importance of power-law distributions in statistical modeling, in this paper, we propose the notion of power-law kernels to investigate power-laws in learning problem. We propose two power-law kernels by generalizing Gaussian and Laplacian kernels. Th…

The article consist of two main parts: an analog of the Leray Theory for Singular Varieties and its application to the Theory of Parshin's Residues. The first part is independent from the second. It uses the theory of Whitney stratifications. The second part is an application of the first. In particular, a geometric an…

Classifies reciprocal elements in Hecke groups, generalizing Sarnak's work.

The paper develops divergences for Gaussian processes and RKHS settings.

Reciprocal processes are acausal generalizations of Markov processes introduced by Bernstein in 1932. In the literature, a significant amount of attention has been focused on developing dynamical models for reciprocal processes. Recently, probabilistic graphical models for reciprocal processes have been provided. This …

Estimates growth of reciprocal classes in Hecke groups.

The paper models social networks with varying levels of reciprocity.

The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.

Model shows feature learning can improve neural scaling laws for hard tasks.

The paper defines a category of Lagrangian correspondences in super Hilbert spaces and constructs a functorial field theory.

Minimal spectral radii found for specific matrix types.

Study on geodesics and dihedral groups in lattices.

We study a topological analogue of idèlic class field theory for 3-manifolds, in the spirit of arithmetic topology. We firstly introduce the notion of a very admissible link $\mathcal{K}$ in a 3-manifold $M$, which plays a role analogous to the set of primes of a number field. For such a pair $(M,\mathcal{K})$, we intr…

Study uses Bayes Hilbert framework to recover probability measure flows from sensors.

The paper models reciprocity in interbank markets using a statistical null model.

Reciprocal learning unifies various machine learning algorithms.

Study geodesics entering a fixed cusp neighborhood multiple times.

New method reconstructs interbank networks enforcing reciprocity to improve stability and risk prediction.

A cone projection maps complex plane arcs to conic sections with fixed focus and directrix.

We interpret magnetic billiards as Finsler ones and describe an analog of the string construction for magnetic billiards. Finsler billiards for which the law "angle of incidence equals angle of reflection" are described. We characterize the Finsler metrics in the plane whose geodesics are circles of a fixed radius. Thi…

Growth rates of geodesics on modular orbifolds are studied.

The paper proves generalization bounds and stopping rules for self-selected data in reciprocal learning.

The abstract discusses conjectures about Chern-Simons invariants of 3-manifolds.

Given a knot and an SL(n,C) representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given of a knot group and SL(3,Z) representation that is not conjugate to its dual for …

We obtain an exact modularity relation for the $q$-Pochhammer symbol. Using this formula, we show that Zagier's modularity conjecture for a knot $K$ essentially reduces to the arithmeticity conjecture for $K$. In particular, we show that Zagier's conjecture holds for hyperbolic knots $K\neq 7_2$ with at most seven cros…

We reformulate Lehmer's question from 1933 and a question due to Schinzel and Zassenhaus from 1965 in terms of a comparison of the Mahler measures and the houses, respectively, of monic integer reciprocal and skew-reciprocal polynomials of the same degree. This entails that understanding the difference between orientat…

We show that the characteristic series for the greedy normal form of a Coxeter group is always a rational series, and prove a reciprocity formula for this series when the group is right-angled and the nerve is Eulerian. As corollaries we obtain many of the known rationality and reciprocity results for the growth series…

We prove a reciprocity formula between Gauss sums that is used in the computation of certain quantum invariants of 3-manifolds. Our proof uses the discriminant construction applied to the tensor product of lattices.

Paper proves existence of isometric immersions for negatively curved surfaces with unbounded second fundamental form.

We prove that under certain linear reciprocal transformation, an evolutionary PDE of hydrodynamic type that admits a bihamiltonian structure is transformed to a system of the same type which is still bihamiltonian.

We reveal connections between RBMs and Bosons, explaining symmetry breaking in their energy landscapes.

We link SVEs to SPDEs and derive Kolmogorov equations for singular kernels.

The study examines the growth of reciprocal classes in Hecke groups, proving an asymptotic formula.

This paper characterizes hierarchical clustering methods that abide by two previously introduced axioms -- thus, denominated admissible methods -- and proposes tractable algorithms for their implementation. We leverage the fact that, for asymmetric networks, every admissible method must be contained between reciprocal …

We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade network and show that a strong embedding in …

Paper tackles division difficulty, proposing new methods to improve accuracy.

Study on TQFT signatures converging to modular form.

New model accounts for scale variation and noise in pairwise comparisons.

Study how past eon's matter affects present eon in Penrose's cyclic cosmology.

Empirical study shows carriers ignore past shippers' behavior, focusing only on current actions.

This article investigates local properties of the further generalized Weierstrass relations for a spin manifold $S$ immersed in a higher dimensional spin manifold $M$ from viewpoint of study of submanifold quantum mechanics. We show that kernel of a certain Dirac operator defined over $S$, which we call submanifold Dir…