Polynomial Dehn Functions and the Length of Asynchronously Automatic Structures

We extend the range of observed behaviour among length functionsof optimal asynchronously automatic structures. We do so bymeans of a construction that yields asynchronously automaticgroups with finite aspherical presentations where the Dehn functionof the group is polynomial of arbitrary degree. Many of thesegroups can be embedded in the automorphism group of a free group.Moreover, the fact that the groups have aspherical presentationsmakes them useful tools in the search to determine the spectrumof exponents for second order Dehn functions. We contributeto this search by giving the first exact calculations of groupswith quadratic and superquadratic exponents. 2000 Mathematical Subject Classification: 20F06, 20F65, 20F69.     

On the Isomorphism of Some Groups with One Defining Relation

Each of the groups in the class under consideration in the paper is presented by one defining relation depending on three integral parameters and is an HNN-extension of some Baumslag—Solitar group. Necessary and sufficient conditions for two groups of this class to be isomorphic and also for two nonisomorphic groups to be homomorphic images of each other are obtained. A consequence of this fact is the negative answer to Question 3.33 in “The Kourovka Notebook.”     

On the orders of finite semisimple groups

The aim of this paper is to investigate the order coincidences among the finite semisimple groups and to give a reasoning of such order coincidences through the transitive actions of compact Lie groups. It is a theorem of Artin and Tits that a finite simple group is determined by its order, with the exception of the groups (A₃(2), A₂(4)) and(B _n (q), C _n (q)) forn ≥ 3,q odd. We investigate the situation for finite semisimple groups of Lie type. It turns out that the order of the finite group H( ) for a split semisimple algebraic groupH defined over , does not determine the groupH up to isomorphism, but it determines the field under some mild conditions. We then put a group structure on the pairs(H ₁,H ₂) of split semisimple groups defined over a fixed field such that the orders of the finite groups H₁( ) and H₂( ) are the same and the groupsH _i have no common simple direct factors. We obtain an explicit set of generators for this abelian, torsion-free group. We finally show that the order coincidences for some of these generators can be understood by the inclusions of transitive actions of compact Lie groups.     

Minimal length elements of Thompson’s groups <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">p</Emphasis>)

We describe a method for determining the minimal length of elements in the generalized Thompson’s groups F(p). We compute the length of an element by constructing a tree pair diagram for the element, classifying the nodes of the tree and summing associated weights from the pairs of node classifications. We use this method to effectively find minimal length representatives of an element.      

Stochastic modeling of the growth process

In this paper we estimate the upper boundary of the number of trees in the selection stand determined for harvesting in a future. At the same time we showed that the present resource of the number of trees in selection stand is sustained. This is achieved by stochastic modeling of the number of trees and the number of felled trees and by solving the partial differential equation. The same problem is solved in the papers, Mitrović (Stochastic modeling of the number of trees and the number of felled trees in selection stands, YUJOR, vol 14(1), pp 57–64, 2004; Stochastic modeling of the number of felled trees in selection stands, Computational and Applied Mathematics, vol 24(2), pp 285–292, 2005). In this paper the modified mathematical model is represented.      

On the solvability of equations of a nonlinear viscous-ideal fluid

A model of the one-dimensional motion of a viscous compressible fluid is considered. A class of nonlinear stressed-state equations for which the initial boundary-value problem has global (in time) solutions in the class of generalized solutions satisfying the energy identity is described. In particular, media exhibiting viscous properties only for large strain rates are studied. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 13–23, July, 2000.     

On generalized measures of entropy and dependence

Various authors have studied extensions of Shannon’s entropy but their inferential properties and applications in applied sciences have not invited proper attention from researchers. In the present paper we explore the motivation and implication of using various classes of the generalized entropies and conditional entropies. We evaluate β-class and (α, β)-class entropies for multivariate normal density function. We also obtain the measures of dependence in terms of the classes of generalized entropies.      

On estimates for integral solutions of linear inequalities

Recently, Bombieri and Vaaler obtained an interesting adelic formulation of the first and the second theorems of Minkowski in the Geometry of Numbers and derived an effective formulation of the well-known “Siegel’s lemma” on the size of integral solutions of linear equations. In a similar context involving linearinequalities, this paper is concerned with an analogue of a theorem of Khintchine on integral solutions for inequalities arising from systems of linear forms and also with an analogue of a Kronecker-type theorem with regard to euclidean frames of integral vectors. The proof of the former theorem invokes Bombieri-Vaaler’s adelic formulation of Minkowski’s theorem.     

On a possibility of generalization of the Hopf lemma to the case of the Navier-Stokes system with nonzero flows

We examine the Hopf lemma (Leray inequality) which is used in proving the existence of a solution to a nonhomogeneous boundary value problem for the stationary Navier-Stokes equations of an incompressible fluid in a bounded domain. We study a possibility of generalization of a weakened variant of the lemma to the case of nonzero flows through the connected components of the boundary of the domain.     

On pseudorandom properties of some Dirichlet characters

In this paper, we try to generalize the Mauduit–Sárközy pseudorandom properties of the Legendre symbol to Dirichlet characters. We prove that in our construction the well-distribution and correlation measures are as “small” as in the case of the Legendre symbol. Furthermore we provide with sufficient conditions to ensure the coincidence of two binary sequences using the Legendre symbol.     

On the convergence of the secant method under the gamma condition

We provide sufficient convergence conditions for the Secant method of approximating a locally unique solution of an operator equation in a Banach space. The main hypothesis is the gamma condition first introduced in [10] for the study of Newton’s method. Our sufficient convergence condition reduces to the one obtained in [10] for Newton’s method. A numerical example is also provided.      

A homotopy operator for Spencer’s sequence in the C ∞-case

The main result of the formal theory of overdetermined systems of differential equations says that any regular system Au = f with smooth coefficients on an open set U ⊂ ℝ ⁿ admits a solution in smooth sections of the bundle of formal power series provided that f satisfies a compatibility condition in U. Our contribution consists in detailed study of the dependence of formal solutions on the point of the base U of the bundle. We also parameterize these solutions by their Cauchy data. In doing so, we prove that, under absence of topological obstructions, there is a formal solution which smoothly depends on the point of the base. This leads to a concept of a finitely generated system (do not mix up it with holonomic or finite -type systems) for which we then prove a C ^∞-Poincaré lemma. The text was submitted by the authors in English.     

On the distribution of the sequence of fractional parts of a slowly increasing exponential function

In this paper for a slowly increasing exponential function we study the degree of deviation of the distribution of the sequence of its fractional parts from the “ideal” uniform distribution. An estimate is given which can appropriately be called the measure of irregularity of the distribution of a given sequence. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 148–152, January, 1999.     

Asymptotics of derivatives of orthogonal polynomials on the real line

We show that uniform asymptotics of orthogonal polynomials on the real line imply uniform asymptotics for all their derivatives. This is more technically challenging than the corresponding problem on the unit circle. We also examine asymptotics in the L ₂ norm. Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353.     

On a theorem of Wigner on products of positive matrices

A simple algebraic proof of a theorem due to Wigner on the product of three positive matrices is given. It is shown that the theorem holds for four matrices under an additional condition. The proofs are valid in the more general case of operators in a Hilbert space.     

On the rate for uniform strong consistency of empirical distributions of independent nonidentically distributed multivariate random variables

Let X_j = (X_1j ,…, X_pj), j = 1,…, n be n independent random vectors. For x = (x₁ ,…, x_p) in R^p and for α in [0, 1], let F_j${}^1$(x) = αI(X_1j < x₁ ,…, X_pj < x_p) + (1 ? α) I(X_1j ≤ x₁ ,…, X_pj ≤ x_p), where I(A) is the indicator random variable of the event A. Let F_j(x) = E(F_j${}^1$(x)) and D_n = sup_{x, α} max_{1 ≤ N ≤ n} |Σ₀ⁿ(F_j${}^1$(x) ? F_j(x))|. It is shown that P[D_n ≥ L] < 4pL exp{?2(L²n^?1 ? 1)} for each positive integer n and for all L² ≥ n; and, as n → ∞, $\text{D}{}_{\mathrm{n}}\text{= 0((n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ with probability one.