<Emphasis Type="Italic">p</Emphasis>-Geodesic transformations and their groups in second-order tangent bundles induced by concircular transformations of bases

We investigate the flattening properties of the Lie group G _r ^II of transformations of a second-order tangent bundle T ²(M) equipped with the lift ∇ ^II of an affine connection ∇ and the lift g ^II of a metric g on the base of M induced by the Lie group G _r of concircular transformations of the base of M. The obtained results reveal certain geometric features of the induced group G _r ^II within the framework of the theory of p-geodesic mappings.     

p-Adic dynamical systems of Chebyshev polynomials

We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ? _p , enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of T _p : they are simple and the nonzero ones lie in the unit circle of the unramified extension of ? _p , (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of T _p in the field ? _p , (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of T _n is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.     

Spectral stability of the Robin Laplacian

We consider the Robin Laplacian in two bounded regions Ω₁ and Ω₂ of ℝ ^N with Lipschitz boundaries and such that Ω₂ ⊂ Ω₁, and we obtain two-sided estimates for the eigenvalues λ _n,2 of the Robin Laplacian in Ω₂ via the eigenvalues λ _{n, 1} of the Robin Laplacian in Ω₁. Our estimates depend on the measure of the set difference Ω\Ω₂ and on suitably defined characteristics of vicinity of the boundaries ∂Ω₁ and ∂Ω₂, and of the functions defined on ∂Ω₁ and on ∂Ω₂ that enter the Robin boundary conditions.     

The moduli space of curves of genus two covering elliptic curves

We consider the moduli spaceS _n of curvesC of genus 2 with the property:C has a “maximal” mapf of degreen to an elliptic curveE. Here, the term “maximal” means that the mapf∶C→E doesn't factor over an unramified cover ofE. By Torelli mapS _n is viewed as a subset of the moduli spaceA ₂ of principally polarized abelian surfaces. On the other hand the Humbert surfaceH _Δ of invariant Δ is defined as a subvariety ofA ₂(C), the set of C-valued points ofA ₂. The purpose of this paper is to releaseS _n withH _Δ.     

Maximizing Voronoi Regions of a Set of Points Enclosed in a Circle with Applications to Facility Location

In this paper we introduce an optimization problem which involves maximization of the area of Voronoi regions of a set of points placed inside a circle. Such optimization goals arise in facility location problems consisting of both mobile and stationary facilities. Let ψ be a circular path through which mobile service stations are plying, and S be a set of n stationary facilities (points) inside ψ. A demand point p is served from a mobile facility plying along ψ if the distance of p from the boundary of ψ is less than that from any member in S. On the other hand, the demand point p is served from a stationary facility p _i  ∈ S if the distance of p from p _i is less than or equal to the distance of p from all other members in S and also from the boundary of ψ. The objective is to place the stationary facilities in S, inside ψ, such that the total area served by them is maximized. We consider a restricted version of this problem where the members in S are placed equidistantly from the center o of ψ. It is shown that the maximum area is obtained when the members in S lie on the vertices of a regular n-gon, with its circumcenter at o. The distance of the members in S from o and the optimum area increases with n, and at the limit approaches the radius and the area of the circle ψ, respectively. We also consider another variation of this problem where a set of n points is placed inside ψ, and the task is to locate a new point q inside ψ such that the area of the Voronoi region of q is maximized. We give an exact solution of this problem when n = 1 and a (1 − ε)-approximation algorithm for the general case.     

Trace cocharacters and the Kronecker products of Schur functions

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.     

Multiphasic Individual Growth Models in Random Environments

The evolution of the growth of an individual in a random environment can be described through stochastic differential equations of the form dY _t  = β(α − Y _t )dt + σdW _t , where Y _t  = h(X _t ), X _t is the size of the individual at age t, h is a strictly increasing continuously differentiable function, α = h(A), where A is the average asymptotic size, and β represents the rate of approach to maturity. The parameter σ measures the intensity of the effect of random fluctuations on growth and W _t is the standard Wiener process. We have previously applied this monophasic model, in which there is only one functional form describing the average dynamics of the complete growth curve, and studied the estimation issues. Here, we present the generalization of the above stochastic model to the multiphasic case, in which we consider that the growth coefficient β assumes different values for different phases of the animal’s life. For simplicity, we consider two phases with growth coefficients β ₁ and β ₂. Results and methods are illustrated using bovine growth data.     

Inner Radius of Univalence for a Strongly Starlike Domain

 The inner radius of univalence of a domain D with Poincaré density ρ _D is the possible largest number σ such that the condition ∥ S _f ∥ _D  = sup _{w∈ D} ρ _D (w) ⁻²∥ S _f (z) ∥ ≤ σ implies univalence of f for a nonconstant meromorphic function f on D, where S _f is the Schwarzian derivative of f. In this note, we give a lower bound of the inner radius of univalence for strongly starlike domains of order α in terms of the order α. The author was partially supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 11740088. A part of this work was carried out during his visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS. Received November 26, 2001; in revised form September 24, 2002 Published online May 9, 2003     

Unit groups of group algebras of some small groups

Let FG be a group algebra of a group G over a field F and U (FG) the unit group of FG. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group G with order 21 over any finite field of characteristic 3 is established. We also characterize the structure of the unit group of FA ₄ over any finite field of characteristic 3 and the structure of the unit group of FQ ₁₂ over any finite field of characteristic 2, where Q ₁₂ = 〈x, y; x ⁶ = 1, y ² = x ³, x ^y = x ^?1〉.     

Arithmetical properties of the number of t-core partitions

Let Λ={λ ₁≥⋅⋅⋅≥λ _s ≥1} be a partition of an integer n. Then the Ferrers-Young diagram of Λ is an array of nodes with λ _i nodes in the ith row. Let λ _j ′ denote the number of nodes in column j in the Ferrers-Young diagram of Λ. The hook number of the (i,j) node in the Ferrers-Young diagram of Λ is denoted by H(i,j):=λ _i +λ _j ′−i−j+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2 ^t -core partitions of n are obtained. A simple convolution identity for t-cores is also given.      

Certain Homology Cycles of the Independence Complex of Grids

Let G be an infinite graph such that the automorphism group of G contains a subgroup K ?? ^d with the property that G/K is finite. We examine the homology of the independence complex Σ(G/I) of G/I for subgroups I of K of full rank, focusing on the case that G is the square, triangular, or hexagonal grid. Specifically, we look for a certain kind of homology cycles that we refer to as “cross-cycles,” the rationale for the terminology being that they are fundamental cycles of the boundary complex of some cross-polytope. For the special cases just mentioned, we determine the set Q(G,K) of rational numbers r such that there is a group I with the property that Σ(G/I) contains cross-cycles of degree exactly r?|G/I|?1; |G/I| denotes the size of the vertex set of G/I. In each of the three cases, Q(G,K) turns out to be an interval of the form [a,b]∩?={r∈?:a≤r≤b}. For example, for the square grid, we obtain the interval $[\frac{1}{5},\frac{1}{4}]\cap \mathbb{Q}Let G be an infinite graph such that the automorphism group of G contains a subgroup K ≅ℤ ^d with the property that G/K is finite. We examine the homology of the independence complex Σ(G/I) of G/I for subgroups I of K of full rank, focusing on the case that G is the square, triangular, or hexagonal grid. Specifically, we look for a certain kind of homology cycles that we refer to as “cross-cycles,” the rationale for the terminology being that they are fundamental cycles of the boundary complex of some cross-polytope. For the special cases just mentioned, we determine the set Q(G,K) of rational numbers r such that there is a group I with the property that Σ(G/I) contains cross-cycles of degree exactly r⋅|G/I|−1; |G/I| denotes the size of the vertex set of G/I. In each of the three cases, Q(G,K) turns out to be an interval of the form [a,b]∩ℚ={r∈ℚ:a≤r≤b}. For example, for the square grid, we obtain the interval [\frac15,\frac14]?\mathbbQ[\frac{1}{5},\frac{1}{4}]\cap \mathbb{Q}.     

Uniqueness of analytic continuation: Necessary and sufficient conditions

Necessary and sufficient conditions for uniqueness of analytic continuation are investigated for a system of m ? 1 first-order linear homogeneous partial differential equations in one unknown, with complex-valued $\text{b}$^∞ coefficients, in some connected open subset of $\text{R}$^k, k ? 2. The type of system considered is one for which there exists a real k-dimensional, $\text{b}$^∞, connected C-R submanifold M^k of $\text{C}$ⁿ, for k, n ? 2, such that the system may be identified with the induced Cauchy-Riemann operators on M^k. The question of uniqueness of analytic continuation for a system of partial differential equations is thus transformed to the question of uniqueness of analytic continuation for C-R functions on the manifold M^k ? Cⁿ. Under the assumption that the Levi algebra of M^k has constant dimension, it is shown that if the excess dimension of this algebra is maximal at every point, then M^k has the property of uniqueness of analytic continuation for its C-R functions. Conversely, under certain mild conditions, it is shown that if M^k has the property of uniqueness of analytic continuation for all $\text{b}$^∞ C-R functions, and if the Levi algebra has constant dimension on all of M^k, then the excess dimension must be maximal at every point of M^k.     

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class of all rational functions of order (n,m) are considered. Let ρ _n,m = ρ _n,m (f;E) be the distance of f in the uniform metric on E from the class . We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation { ρ _n,m(n) } _n=0 ^∞ , m(n)/n → θ ∈ (0,1] as n → ∞ . In particular, we give an upper estimate for the liminf _{n →∞} ρ _n,m(n) ^1/(n+m(n)) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators. June 16, 1997. Date revised: December 1, 1997. Date accepted: December 1, 1997. Communicated by Ronald A. DeVore.     

Structure of H-classes in the semigroup of nonnegative matrices

This paper investigates the structure of the $\text{H}$-classes in the semigroup N_n of nonnegative matrices. We obtain two sets of equivalent conditions for any two matrices A,B to satisfy $\text{A}\text{H}\text{B in N}{}_{\mathrm{n}}$. We establish a one-to-one and onto correspondence between the $\text{H}$-class $\text{H}$_A and the group W_A0 of the greatest cone independent submatrix A₀ of A. We find W_A0 can be made up from the groups of the connective submatrices of A₀.     

An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree

Let ? be the genealogical tree of a supercritical multitype Galton–Watson process, and let Λ be the limit set of ?, i.e., the set of all infinite self-avoiding paths (called ends) through ? that begin at a vertex of the first generation. The limit set Λ is endowed with the metric d(ζ, ξ) = 2 ⁻ⁿ where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence Φ(ζ) of types of the vertices of ζ. Let Ω be the space of all such sequences. For any ergodic, shift-invariant probability measure μ on Ω, define Ω_μ to be the set of all μ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency μ(Ω(v)), where Ω(v) is the set of all ω′?Ω that begin with the word v. Then the Hausdorff dimension of Λ∩Φ⁻¹ (Ω_μ) in the metric d is

Idempotent lattices, Renner monoids and cross section lattices of the special orthogonal algebraic monoids

Let MSO_n (n is even) be the special orthogonal algebraic monoid, T a maximal torus of the unit group, and the Zariski closure of T in the whole matrix monoid M_n. In this paper we explicitly determine the idempotent lattice , the Renner monoid , and the cross section lattice Λ of MSO_n in terms of the Weyl group and the concept of admissible sets (see Definition 3.1). It turns out that there is a one-to-one relationship between and the admissible subsets, and that is a submonoid of  , the Renner monoid M_n. Also Λ is a sublattice of Λ_n, the cross section lattice of M_n.     

Stability conditions for cyclic quivers

Let Δ_n be a cyclic quiver of n vertices and let 𝒯_n denote the category of nilpotent representations of Δ_n over a field. In the present paper, we study the stability conditions on 𝒯_n and describe the semistable subcategories of 𝒯_n with a fixed slope. This provides an alternative characterization of the thick subcategories of 𝒯_n. Further, we obtain a relationship between the Harder–Narasimhan system for Δ_n and the Ringel–Hall algebra of Δ_n.     

2-Normalization of lattices

Let τ be a type of algebras. A valuation of terms of type τ is a function v assigning to each term t of type τ a value v(t) ⩾ 0. For k ⩾ 1, an identity s ≈ t of type τ is said to be k-normal (with respect to valuation v) if either s = t or both s and t have value ⩾ k. Taking k = 1 with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called k-normal (with respect to the valuation v) if all its identities are k-normal. For any variety V, there is a least k-normal variety N _k (V) containing V, namely the variety determined by the set of all k-normal identities of V. The concept of k-normalization was introduced by K. Denecke and S. L. Wismath in their paper (Algebra Univers., 50, 2003, pp.107–128) and an algebraic characterization of the elements of N _k (V) in terms of the algebras in V was given in (Algebra Univers., 51, 2004, pp. 395–409). In this paper we study the algebras of the variety N ₂(V) where V is the type (2, 2) variety L of lattices and our valuation is the usual depth valuation of terms. We introduce a construction called the 3-level inflation of a lattice, and use the order-theoretic properties of lattices to show that the variety N ₂(L) is precisely the class of all 3-level inflations of lattices. We also produce a finite equational basis for the variety N ₂(L). This research was supported by Research Project MSM6198959214 of the Czech Government and by NSERC of Canada.     

Continuity of iteration and approximation of iterative roots

Motivated by computing iterative roots for general continuous functions, in this paper we prove the continuity of the iteration operators T_n, defined by T_nf=fⁿ. We apply the continuity and introduce the concept of continuity degree to answer positively the approximation question: If lim_m→∞F_m=F, can we find an iterative root f_m of F_m of order n for each m∈N such that the sequence (f_m) tends to the iterative root of F of order n associated with a given initial function? We not only give the construction of such an approximating sequence (f_m) but also illustrate the approximation of iterative roots with an example. Some remarks are presented in order to compare our approximation with the Hyers-Ulam stability.     

On the Hardness of Computing Intersection,Union and Minkowski Sum of Polytopes

For polytopes P ₁,P ₂⊂ℝ ^d , we consider the intersection P ₁∩P ₂, the convex hull of the union CH(P ₁∪P ₂), and the Minkowski sum P ₁+P ₂. For the Minkowski sum, we prove that enumerating the facets of P ₁+P ₂ is NP-hard if P ₁ and P ₂ are specified by facets, or if P ₁ is specified by vertices and P ₂ is a polyhedral cone specified by facets. For the intersection, we prove that computing the facets or the vertices of the intersection of two polytopes is NP-hard if one of them is given by vertices and the other by facets. Also, computing the vertices of the intersection of two polytopes given by vertices is shown to be NP-hard. Analogous results for computing the convex hull of the union of two polytopes follow from polar duality. All of the hardness results are established by showing that the appropriate decision version, for each of these problems, is NP-complete.