Related papers: Haruspicy 2: The anisotropic generating function o…
We construct families of rational functions $f \colon \bP^1_k \to \bP^1_k$ of degree $d \geq 2$ over a perfect field $k$ whose associated fixed-point processes fail to be martingales. Conversely, for any normal variety $X \subset…
We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ${\mathbb{Z}}^d$. The two-point functions of these models are respectively the generating function for self-avoiding walks from…
It was proved by Tien-Cuong Dinh and me that there is a smooth complex projective surface whose automorphism group is discrete and not finitely generated. In this paper, we will show that there is a smooth projective surface, birational to…
Based on an assumption on the Hessian of the Green function, we derive some monotonicity formulas on nonparabolic manifolds. This assumption is satisfied on manifolds that meet certain conditions including bounds on the sectional curvature…
We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z^2, and always stay in the quadrant x >= 0, y >= 0. We first give a criterion which guarantees that the length generating…
There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences,…
We present a self-contained analysis of theories of discrete 2D gravity coupled to matter, using geometric methods to derive equations for generating functions in terms of free (noncommuting) variables. For the class of discrete gravity…
Let (M,w,L) be a symplectic manifold endowed with a lagrangian foliation L. Liberman and Weinstein have shown that the leaves of L are endowed with an affine structure. In this paper we provide links between the theories of affine manifolds…
We show that for non-conjugate subgroups $G_1$ and $G_2$ of a finite group $G$ there exists an extension of $G$ (by a finite group) in which the pre-images of $G_1$ and $G_2$ are not isomorphic. This allows us to show that $\mathbb Z$-coset…
We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These statements…
This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy types of their classifying spaces. Double categories (Ehresmann, 1963) have well-understood geometric realizations, and…
We consider inhomogeneous lattice walk models in a half-space and in the quarter plane. For the models in a half-space, we show by a generalization of the kernel method to linear systems of functional equations that their generating…
In this paper we apply the anholonomic frames method developed in refs. [1-4] to construct and study anisotropic vacuum field configurations in 5D gravity. Starting with an off--diagonal 5D metric, parameterized in terms of several ansatz…
We construct a sequence of subset partition graphs satisfying the dimension reduction, adjacency, strong adjacency, and endpoint count properties whose diameter has a superlinear asymptotic lower bound. These abstractions of polytope graphs…
In a previous paper: A. Paszkiewicz, T. Sobieszek, Additive Entropies of Partitions, we have given a description of additive partition entropies that is real functions $I$ on the set of finite partitions that are additive on stochastically…
In a recent preprint, Lai showed that the quotient of generating functions of weighted lozenge tilings of two "half hexagons with lateral dents", which differ only in width, factors nicely, and the same is true for the quotient of…
We present an effective medium theory that can predict the effective permittivity and permeability of a geometrically anisotropic two-dimensional metamaterial composed with a rectangular array of elliptical cylinders. It is possible to…
The present paper contains new geometric theorems in mixed characteristic case. We derive a bunch of cohomological consequences using these geometric theorems. Among them an isotropy result for quadratic spaces, a purity result for…
In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric…
We prove a few results concerning the notions of finite dimensionality of mixed Tate motives in the sense of Kimura and O'Sullivan. It is shown that being oddly or evenly finite dimensional is equivalent to vanishing of certain cohomology…