Counting lattice chains and Delannoy paths in higher dimensions

Lattice chains and Delannoy paths represent two different ways to progress through a lattice. We use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in a lattice of arbitrary finite dimension. Specifically, fix nonnegative integers n₁,…,n_d, and let L denote the lattice of points (a₁,…,a_d)∈Z^d that satisfy 0≤a_i≤n_i for 1≤i≤d. We prove that the number of chains in L is given by where . We also show that the number of Delannoy paths in L equals Setting n_i=n (for all i) in these expressions yields a new proof of a recent result of Duchi and Sulanke [9] relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension.     

Extension of one-dimensional proximity regions to higher dimensions

Proximity regions (and maps) are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which gave rise to class cover catch digraph (CCCD) and was applied to pattern classification. In this article, we note some appealing properties of the spherical proximity map in compact intervals on the real line, thereby introduce the mechanism and guidelines for defining new proximity maps in higher dimensions. For non-spherical PCDs, Delaunay tessellation (triangulation in the real plane) is used to partition the region of interest in higher dimensions. We also introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of these maps and the resulting PCDs. We provide the distribution of graph invariants, namely, domination number and relative density, of the PCDs and characterize the geometry invariance of the distribution of these graph invariants for uniform data and provide some newly defined proximity maps in higher dimensions as illustrative examples.     

Paintings: A planar approach to higher dimensions

A painting in dimension n is an object induced by certain regular (n+1)-colored finite graphs. Some classes of paintings are shown to be plane universal models for closed n-manifolds. Ferri and Gagliardi's equivalence theorem (graph-theoretical counterpart for homeomorphisms) [5], and Ferri's strengthening of their result [3] are used to provide a surprisingly simple way to state the equivalence theorem: the restricted crystallization moves [3] become deletion and insertion of one edge in certain plane graphs. Various new properties of minimum 3-crystallizations are obtained in the framework of paintings. Two conjectures related to the recognition of the 3-sphere are included.This work was performed under the support of UFPE, FINEP and CNPq (contract no. 30.1103/80).     

Strange attractors in higher dimensions

We consider generic one-parameter families of diffeomorphisms on a manifold of arbitrary dimension, unfolding a homoclinic tangency associated to a sectionally dissipative saddle point (the product of any pair of eigenvalues has norm less than 1). We prove that such families exhibit strange attractors in a persistent way: for a positive Lebesgue measure set of parameter values. In the two-dimensional case this had been obtained in a joint work with L. Mora, based on and extending the results of Benedicks-Carleson on the quadratic family in the plane.     

Some Bonnesen-style inequalities for higher dimensions

We discuss the higher dimensional Bonnesen-style inequalities.Though there are many Bonnesen-style inequalities for domains in the Euclidean plane R2 few results for general domain in R n(n ≥ 3) are known.The results obtained in this paper are for general domains,convex or non-convex,in Rn.     

Matrix semigroup homomorphisms into higher dimensions

We study non-degenerate irreducible homomorphisms from the multiplicative semigroup of all n-by-n matrices over an algebraically closed field of characteristic zero to the semigroup of m-by-m matrices over the same field. We prove that every non-degenerate homomorphism from the multiplicative semigroup of all n-by-n matrices to the semigroup of (n + 1)-by-(n + 1) matrices when n ? 3 is reducible and that every non-degenerate homomorphism from the multiplicative semigroup of all 3-by-3 matrices to the semigroup of 5-by-5 matrices is reducible.     

Simplicial finite elements in higher dimensions

Over the past fifty years, finite element methods for the approximation of solutions of partial differential equations (PDEs) have become a powerful and reliable tool. Theoretically, these methods are not restricted to PDEs formulated on physical domains up to dimension three. Although at present there does not seem to be a very high practical demand for finite element methods that use higher dimensional simplicial partitions, there are some advantages in studying the methods independent of the dimension. For instance, it provides additional insights into the structure and essence of proofs of results in one, two and three dimensions. In this survey paper we review some recent progress in this direction. The second author was supported by Grant No. 112444 of the Academy of Finland. The third author was supported by grant No. 201/04/1503 of the Grant Agency of the Czech Republic.     

A monotonicity lemma in higher dimensions

Let K be a body of constant width in a Minkowski space (i.e., in a real finite dimensional Banach space). Then any hyperplane section S of K bounds two parts of K one of which has the same diameter as S. Furthermore, if we represent K as the union of hyperplane sections S(t), t ∈[0, 1], continuously depending on t, then the Minkowskian diameter of S(t) is a unimodal function of the variable t. These two statements (being the core of this note) can be considered as higher-dimensional extensions of the well-known monotonicity lemma from Minkowski geometry.     

The bisection method in higher dimensions

Is the familiar bisection method part of some larger scheme? The aim of this paper is to present a natural and useful generalisation of the bisection method to higher dimensions. The algorithm preserves the salient features of the bisection method: it is simple, convergence is assured and linear, and it proceeds via a sequence of brackets whose infinite intersection is the set of points desired. Brackets are unions of similar simplexes. An immediate application is to the global minimisation of a Lipschitz continuous function defined on a compact subset of Euclidean space.     

Extreme VaR scenarios in higher dimensions

For a sequence of random variables X ₁, ..., X _n , the dependence scenario yielding the worst possible Value-at-Risk at a given level α for X ₁+...+X _n is known for n=2. In this paper we investigate this problem for higher dimensions. We provide a geometric interpretation highlighting the dependence structures which imply the worst possible scenario. For a portfolio (X ₁,..., X _n ) with given uniform marginals, we give an analytical solution sustaining the main result of Rüschendorf (Adv. Appl. Probab. 14(3):623–632, 1982). In general, our approach allows for numerical computations.      

On formal variational calculus of higher dimensions

Based upon the definition of variational derivatives presented by Galindo and Martinez Alonso, a series of formulae on formal variational calculus of higher dimensions is developed.     

Existence of nonstationary bubbles in higher dimensions

We are interested in the existence of travelling-waves for the nonlinear Schrödinger equation in R^N with “ψ³−ψ⁵”-type nonlinearity. First, we prove an abstract result in critical point theory (a local variant of the classical saddle-point theorem). Using this result, we get the existence of travelling-waves moving with sufficiently small velocity in space dimension N4.     

Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions

In 2011, Nathanson proposed several questions on minimal complements in a group or a semigroup. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a part of a co-minimal pair. Further, the uncountable collection of sets can be chosen so that they satisfy certain algebraic properties.

     

A natural extension of the Pythagorean equation to higher dimensions

The Pythagorean equation is extended to higher dimensions via circulant matrices. This form allows for the set of solutions to be expressed in a clean yet non-trivial way. The cubic case, namely the equation x ³+y ³+z ³−3xyz=1, was studied by Ramanujan and displays many interesting properties. The general case highlights the use of circulant matrices and systems of differential equations. The structure of the solutions also allows parametrized solutions of the Fermat equation in degrees 3 and 4 to be given in terms of theta functions.