Mappings of acyclic and parking functions

The two functions in question are mappings: [n]→[n], with [n] = {1, 2,?,n}. The acyclic function may be represented by forests of labeled rooted trees, or by free trees withn + 1 points; the parking functions are associated with the simplest ballot problem. The total number of each is (n + 1) ^n-1. The first of two mappings given is based on a simple mapping, due to H. O. Pollak, of parking functions on tree codes. In the second, each kind of function is mapped on permutations, arising naturally from characterizations of the functions. Several enumerations are given to indicate uses of the mappings.     

Turbulences and strict return trajectory types of interval mappings

In this note, we investigate the number of the strict return trajectory types with order n which are turbulent of interval mappings and show that the probability that a strict return trajectory type with order n is turbulent converges to 1 as n goes to infinity.     

On some classes of bounded univalent mappings in several complex variables

Let B ⁿ be the Euclidean unit ball in C ⁿ . In this paper, we study several properties of strongly starlike mappings of order α (0 < α < 1) and bounded convex mappings on B ⁿ . We prove that K-quasiregular strongly starlike mappings of order α on B ⁿ have continuous and univalent extensions to ${\overline{B}^n}$ . We show that bounded convex mappings on B ⁿ are strongly starlike of some order α. We give a coefficient estimate for K-quasiregular strongly starlike mappings of order α on B ⁿ . Finally, we give examples of strongly starlike mappings of order α and bounded convex mappings on B ⁿ .     

Solution of a Loewner chain equation in several complex variables

We find a solution to the Loewner chain equation in the case when the infinitesimal generator satisfies h(0,t)=0, Dh(0,t)=A for any A∈L(Cn,Cn) with m(A)>0. We also study the related classes of spirallike mappings, mappings with parametric representation and asymptotically spirallike mappings.     

On the number of A-mappings

Suppose that $ \mathfrak{S} $ _n is the semigroup of mappings of the set of n elements into itself, A is a fixed subset of the set of natural numbers ?, and V _n (A) is the set of mappings from $ \mathfrak{S} $ _n whose contours are of sizes belonging to A. Mappings from V _n (A) are usually called A-mappings. Consider a random mapping σ _n , uniformly distributed on V n(A). Suppose that ν _n is the number of components and λ _n is the number of cyclic points of the random mapping σ _n . In this paper, for a particular class of sets A, we obtain the asymptotics of the number of elements of the set V _n (A) and prove limit theorems for the random variables ν _n and λ _n as n → ∞.     

On the number of transversals in Cayley tables of cyclic groups

It is well known that if n is even, the addition table for the integers modulo n (which we denote by B_n) possesses no transversals. We show that if n is odd, then the number of transversals in B_n is at least exponential in n. Equivalently, for odd n, the number of diagonally cyclic latin squares of order n, the number of complete mappings or orthomorphisms of the cyclic group of order n, the number of magic juggling sequences of period n and the number of placements of n non-attacking semi-queens on an n×n toroidal chessboard are at least exponential in n. For all large n we show that there is a latin square of order n with at least (3.246)ⁿ transversals.We diagnose all possible sizes for the intersection of two transversals in B_n and use this result to complete the spectrum of possible sizes of homogeneous latin bitrades.We also briefly explore potential applications of our results in constructing random mutually orthogonal latin squares.     

Cascade search principle and its applications to the coincidence problems of n one-valued or multi-valued mappings

The problem of the construction of a multi-cascade with a given limit subset A is considered in a metric space X. A multi-cascade is a discrete multi-valued dynamic system with the translation semigroup (Z?0,+). The cascade search principle using so-called search functionals is suggested. It gives a solution of the problem. Also, an estimation is obtained for the distance between any initial point x and every correspondent limit point. Several applications of one-valued and multi-valued versions of the mentioned cascade search principle are given for the cases when the limit subset A is (1) the full (or expanded) preimage of a closed subspace under a mapping from X to another metric space; (2) the coincidence set (or expanded coincidence set) of n mappings from X to another metric space (n>1); (3) the common preimage (or the expanded one) of a closed subspace under n mappings; and (4) the common fixed point set of n mappings of the space X into itself (n?1). Generalizations of the previous authors results are obtained. And, in particular cases, generalizations of some recent results by A.V. Arutyunov on coincidences of two mappings and a generalization of Banach fixed point principle are obtained.     

Reconstructing edge-disjoint paths

For an undirected graph G=(V,E), the edge connectivity values between every pair of nodes of G can be succinctly recorded in a flow-equivalent tree that contains the edge connectivity value for a linear number of pairs of nodes. We generalize this result to show how we can efficiently recover a maximum set of disjoint paths between any pair of nodes of G by storing such sets for a linear number of pairs of nodes. At the heart of our result is an observation that combining two flow solutions of the same value, one between nodes s and r and the second between nodes r and t, into a feasible flow solution of value f between nodes s and t, is equivalent to solving a stable matching problem on a bipartite multigraph.Our observation, combined with an observation of Chazelle, leads to a data structure, which takes O(n^3.5) time to generate, that can construct the maximum number λ(u,v) of edge-disjoint paths between any pair (u,v) of nodes in time O(α(n,n)λ(u,v)n) time.     

Generalizations and applications of the nowhere zero linear mappings in network coding

In [2], Alon and Tarsi proposed a conjecture about the nowhere-zero point in linear mappings. In this paper, we first study some generalizations of this problem, and obtain necessary and sufficient conditions for the existence of nowhere point in these generalized problems under the assumption |F|?n+2, where n is the number of rows of the matrix A. Then we apply the results in these generalizations to give a polynomial time algebraic construction of the acyclic network codings.     

Approximation of coincidence points and common fixed points of a collection of mappings of metric spaces

On a complete metric space X, we solve the problem of constructing an algorithm (in general, nonunique) of successive approximations from any point in space to a given closed subsetA. We give an estimate of the distance from an arbitrary initial point to the corresponding limit points. We consider three versions of the subset A: (1) A is the complete preimage of a closed subspace H under a mapping from X into the metric space Y; (2) A is the set of coincidence points of n (n > 1) mappings from X into Y; (3) A is the set of common fixed points of n mappings of X into itself (n = 1, 2, …). The problems under consideration are stated conveniently in terms of a multicascade, i.e., of a generalized discrete dynamical system with phase space X, translation semigroup equal to the additive semigroup of nonnegative integers, and the limit set A. In particular, in case (2) for n = 2, we obtain a generalization of Arutyunov’s theorem on the coincidences of two mappings. In case (3) for n = 1, we obtain a generalization of the contraction mapping principle.     

Independent permutations,as related to a problem of Moser and a theorem of Pólya

We introduce the notion of a set of independent permutations on the domain {0, 1,… n ? 1}, and obtain bounds on the size of the largest such set. The results are applied to a problem proposed by Moser in which he asked for the largest number of nodes in a d-cube of side n such that no n of these nodes are collinear. Independent permutations are also related to the problem of placing n non-capturing superqueens (chess queens with wrap-around capability) on an n × n board. As a special case of this treatment we obtain Pólya's theorem that this problem can be solved if and only if n is not a multiple of 2 or 3.     

On locally biholomorphic finitely valent mappings from multi-connected domains onto the open disc

In this paper we offer new results of research presented in the referred papers of J.E. Fornaess and E.L. Stout, E. Ligocka and the authors, and concerning the existence of m-valent locally biholomorphic mappings from product domains of Cn onto n-dimensional complex manifolds. In particular, we confirm an own conjecture about the estimation of the valentness m of locally biholomorphic mappings from multi-connected domains onto the open unit disc.     

Differentiability of Mappings of the Sobolev Space <Emphasis Type="Italic">W</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>−1</Subscript><Superscript>1</Superscript> with Conditions on the Distortion Function

We define two scales of the mappings that depend on two real parameters p and q, with n?1 ≤ q ≤ p < ∞, as well as a weight function θ. The case q = p = n and θ ≡ 1 yields the well-known mappings with bounded distortion. The mappings of a two-index scale are applied to solve a series of problems of global analysis and applications. The main result of the article is the a.e. differentiability of mappings of two-index scales.     

A single-copy minimal-time simulation of a torus of automata by a ring of automata

We consider cellular automata on Cayley graphs and we simulate the behavior of a torus of n×m automata (nodes) by a ring of n·m automata (cells). Our simulation technique requires the neighborhood of the nodes to be preserved. We achieve this constraint by copying the contents of nodes on the cells. We consider the problem of minimizing the number of the copies. We prove that it is possible to simulate the behavior of a torus on a ring with a single copy on each cell if and only if n and m satisfy a given condition. In that case we propose a time-optimal algorithm. We thus improve a previous work done by Martin where two copies were requested. When the condition on n and m is not fulfilled one can use the previous algorithm.     

On the lower order of mappings with finite length distortion

We study the problem of the so-called lower order for one kind of mappings with finite distortion, actively investigated in the recent 15–20 years.We prove that mappings with finite length distortion f: D → ? ⁿ , n ≥ 2, whose outer dilatation is integrable to the power α > n ? 1 with finite asymptotic limit have lower order bounded from below.     

Basics of the Quasiconformal Analysis of a Two-Index Scale of Spatial Mappings

We define a scale of mappings that depends on two real parameters p and q, n?1 ≤ q ≤ p < ∞ and a weight function θ. In the case of q = p = n, θ ≡ 1, we obtain the well-known mappings with bounded distortion. Mappings of a two-index scale inherit many properties of mappings with bounded distortion. They are used for solving a few problems of global analysis and applied problems.     

Rectifiable sets and coarea formula for metric-valued mappings

We study Lipschitz mappings defined on an Hn-rectifiable metric space with values in an arbitrary metric space. We find necessary and sufficient conditions on the image and the preimage of a mapping for the validity of the coarea formula. As a consequence, we prove the coarea formula for some classes of mappings with Hk-σ-finite image. We also obtain a metric analog of the Implicit Function Theorem. All these results are extended to large classes of mappings with values in a metric space, including Sobolev mappings and BV-mappings.     

Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics

We consider holomorphic mappings sending a given Levi-nondegenerate pseudoconcave hypersurface M in Cn+1 into a nondegenerate hyperquadric of the same signature in PCN+1 and show that if M is sufficiently close to a hyperquadric in a certain sense, then any two such mappings differ only by an automorphism of the hyperquadric.     

Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k−1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface.As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.     

Pluriharmonic mappings and linearly connected domains in ℂ n

In this paper we obtain certain sufficient conditions for the univalence of pluriharmonic mappings defined in the unit ball \(\mathbb{B}^n \) of ? ⁿ . The results are generalizations of conditions of Chuaqui and Hernández that relate the univalence of planar harmonic mappings with linearly connected domains, and show how such domains can play a role in questions regarding injectivity in higher dimensions. In addition, we extend recent work of Hernández and Martín on a shear type construction for planar harmonic mappings, by adapting the concept of stable univalence to pluriharmonic mappings of the unit ball \(\mathbb{B}^n \) into ? ⁿ .