相关论文: On Cannon cone types and vector-valued multiplicat…
This paper presents a new methodology to count the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius…
Let $g\geq3$ and $n\geq0$, and let ${\mathcal{M}}_{g,n}$ be the mapping class group of a surface of genus $g$ with $n$ boundary components. We prove that ${\mathcal{M}}_{g,n}$ contains a unique subgroup of index $2^{g-1}(2^{g}-1)$ up to…
This note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms…
In this paper, we present a new method for computing the f-vector of a marked order polytope. Namely, given an arbitrary (polyhedral) subdivision of an arbitrary convex polytope, we construct a cochain complex (over the two-element field…
We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard…
The equality between the number of odd spin structures on a Riemann surface of genus g, with $2^g - 1$ being a Mersenne prime, and the even perfect numbers, is an indication that the action of the modular group on the set of spin structures…
For a complex connected reductive group G, we classify the simple modules whose cone of primitive vectors admits a nontrivial G-invariant deformation. We relate this classification to that of simple Jordan algebras, and to that (due to…
Extending the method of the paper [FS3] we prove three structure theorems for vector valued modular forms, where two correspond to 4-dimensional cases (two hermitian modular groups, one belonging to the field of Eisenstein numbers, the…
Modulo a prime number, we define semi-primitive roots as the square of primitive roots. We present a method for calculating primitive roots from quadratic residues, including semi-primitive roots. We then present progressions that generate…
Given a graph $G$, its genus polynomial is $\Gamma_G(x) = \sum_{k\geq 0} g_k(G)x^k$, where $g_k(G)$ is the number of 2-cell embeddings of $G$ in an orientable surface of genus $k$. The Log-Concavity Genus Distribution (LCGD) Conjecture…
In a previous paper we studied ``weakly primitive axial algebras'' with respect to more general fusion rules, for which at least one axis satisfies the fusion rules. In this continuation, a concise description is provided of the…
We prove that the description of cubic functors is a wild problem in the sense of the representation theory. On the contrary, we describe several special classes of such functors (2-divisible, weakly alternative, vector spaces and torsion…
The factorial hull of the projective variety X (or its cone) is a graded algebra R(X) that can be used in some situations to consider simultaneously all divisor classes on X. Associated with X is a certain cone H in the divisor class group…
Orthogonal polynomials and the Fourier orthogonal series on a cone of revolution in $\mathbb{R}^{d+1}$ are studied. It is shown that orthogonal polynomials with respect to the weight function $(1-t)^\gamma (t^2-\|x\|^2)^{\mu-\frac12}$ on…
A method of constructing specific polynomial representations $f(x)$ over the finite field $\mathbb{F}_p$ of the square roots function modulo a prime $p = 2^kn + 1$, $n$ odd, is presented. The formulas for the cases $k = 2$, $3$ and $4$ are…
We discover a simple construction of a four-dimensional family of smooth surfaces of general type with $p_g(S)=q(S)=0$, $K^2_S=3$ with cyclic fundamental group $C_{14}$. We use a degeneration of the surfaces in this family to find…
We investigate the nef cone spanned by the diagonal and the fibre classes of second symmetric product of a curve of genus $g$. This 2-dimensional nef cone gives a characterization of double covers of curves of genus $\le \frac{g-1}{8}$.…
Affine structures on a Lie groupoid, including affine $k$-vector fields, $k$-forms and $(p,q)$-tensors are studied. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the…
Let $\Gamma_g$ be the fundamental group of a closed connected orientable surface of genus $g\geq2$. We introduce a combinatorial structure of "core surfaces", that represent subgroups of $\Gamma_g$. These structures are (usually)…
We study singularities obtained by the contraction of the maximal divisor in compact (non kaehlerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein or…