Related papers: Gauss-Bonnet for multi-linear valuations

We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined graph and the spectrum of the connection Laplacian L of G determine each other: we prove that L-L^(-1) is similar to the Hodge Laplacian H of G…

Let $G(V,E)$ be a graph, and $\mathscr{H}:=\big\{H:H\subseteq G\big\}$ denote the collection of all possible subgraphs of $G$. Then for each non-negative function $w:\mathscr{H}\to\mathbb{R_+}$, the graph $G(V,E,w)$ is said to be a weighted…

Given a non-hyperelliptic curve $C\in\mathscr{M}_g$ and $2\leq n\leq g-2$, we prove that the generic fiber of the Gauss map on $W_n$ has one element and we characterize its multiple locus. Assuming that $C$ doesn't have a…

We give a generalization of the geometric estimate used by Hart and the second author in their 2008 work on sums and products in finite fields. Their result concerned level sets of non-degenerate bilinear forms over finite fields, while in…

The multiplicate form of Gould--Hsu's inverse series relations enables to investigate the dual relations of the Chu-Vandermonde-Gau{\ss}'s, the Pfaff-Saalsch\"utz's summation theorems and the binomial convolution formula due to Hagen and…

To a simple graph we associate a so-called graph series, which can be viewed as the Hilbert--Poincar\'e series of a certain infinite jet scheme. We study new $q$-representations and examine modular properties of several examples including…

Concerning the value distribution problem for generalized Gauss maps, we not only generalize Fujimoto's theorem to complete space-like stationary surfaces in Minkowski spacetime, but also estimate the upper bound of the number of…

We study a new set of duality relations between weighted, combinatoric invariants of a graph $G$. The dualities arise from a non-linear transform $\mathfrak{B}$, acting on the weight function $p$. We define $\mathfrak{B}$ on a space of…

Interior and exterior angle vectors of polytopes capture curvature information at faces of all dimensions and can be seen as metric variants of $f$-vectors. In this context, Gram's relation takes the place of the Euler--Poincar\'e relation…

Some eigenvalue matrix models possess an interesting property: one can manifestly define the basis where all averages can be explicitly calculated. For example, in the Gaussian Hermitian and rectangular complex models, averages of the Schur…

A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain…

The graph isomorphism problem looks deceptively simple, but although polynomial-time algorithms exist for certain types of graphs such as planar graphs and graphs with bounded degree or eigenvalue multiplicity, its complexity class is still…

In the present paper, the simplest scalar ODE case is studied for polynomials $$ \dot{w}=f(w)=(w-e_0)\cdot\ldots\cdot(w-e_{d-1}) $$ of degree $d$ with $d$ simple complex zeros. The explicit solution by separation of variables and explicit…

Let $V$ be a linear representation of a connected complex reductive group $G$. Given a choice of character $\theta$ of $G$, Geometric Invariant Theory defines a locus $V^{ss}_\theta(G) \subseteq V$ of semistable points. We give necessary,…

Let $G$ be a finite group acting on a small category $I$. We study functors $X \colon I \to \mathscr{C}$ equipped with families of compatible natural transformations that give a kind of generalized $G$-action on $X$. Such objects are called…

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix…

For any finite abelian group G, the equivariant Gromov-Witten invariants of C^r/G can be viewed as a certain kind of abelian Hurwitz-Hodge integrals. In this note, we use Tseng's orbifold quantum Riemann-Roch theorem to express this kind of…

Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the `periodic constant' of the topological multivariable Poincar\'e series (zeta function).…

Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we study the gauge structure and duality properties of theories for free gauge fields transforming in arbitrary…

The Hopf sign conjecture states that a compact Riemannian 2d-manifold M of positive curvature has Euler characteristic X(M)>0 and that in the case of negative curvature X(M) (-1)^d >0. The Hopf product conjecture asks whether a positive…