Filtration of components of processes of random evolution

The problem of estimation of a nonobservable component θ_t for a two-dimensional process (θ_t, ξ_t) of random evolution (θ _t,ξ_t);x_t, 0≤t≤T, is investigated on the basis of observations of ξ_s. s≤t, where x _t is a homogeneous Markov process with infinitesimal operator Q. Applications to stochastic models of a (B,S)-market of securities is described under conditions of incomplete market. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1701–1705, December, 1998.     

Rates of Convergence of Means of Euclidean Functionals

Let L be the Euclidean functional with p-th power-weighted edges. Examples include the sum of the p-th power-weighted lengths of the edges in minimal spanning trees, traveling salesman tours, and minimal matchings. Motivated by the works of Steele, Redmond and Yukich (Ann. Appl. Probab. 4, 1057–1073, 1994, Stoch. Process. Appl. 61, 289–304, 1996) have shown that for n i.i.d. sample points {X ₁,…,X _n } from [0,1] ^d , L({X ₁,…,X _n })/n ^(d−p)/d converges a.s. to a finite constant. Here we bound the rate of convergence of EL({X ₁,…,X _n })/n ^(d−p)/d . Y. Koo supported by the BK21 project of the Department of Mathematics, Sungkyunkwan University. S. Lee supported by the BK21 project of the Department of Mathematics, Yonsei University.     

Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm(G) denote the number of perfect matchings of G.We denote the path with m vertices by P_m and the Cartesian product of graphs G and H by G×H. In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175-188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results:1. Let T be a tree and let C_n denote the cycle with n vertices. Then Pm(C₄×T)=∏(2+α²), where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm(C₄×T) is always a square or double a square.2. Let T be a tree. Then Pm(P₄×T)=∏(1+3α²+α⁴), where the product ranges over all non-negative eigenvalues α of T.3. Let T be a tree with a perfect matching. Then Pm(P₃×T)=∏(2+α²), where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm(C₄×T)=[Pm(P₃×T)]².     

Continuum cardinality of the set of solutions of one class of equations that contain the function of frequency of ternary digits of a number

We study the equation ν ₁(x) = x, where ν ₁(x) is the function of frequency of the digit 1 in the ternary expansion of x. We prove that this equation has a unique rational root and a continuum set of irrational solutions. An algorithm for the construction of solutions is proposed. We also describe the topological and metric properties of the set of all solutions. Some additional facts about the equations ν _i (x) = x, i = 0, 2, are given. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1414–1421, October, 2008.     

Minima of initial segments of infinite sequences of reals

Suppose that 〈x_k〉_k∈? is a countable sequence of real numbers. Working in the usual subsystems for reverse mathematics, RCA₀ suffices to prove the existence of a sequence of reals 〈u_k〉_k∈? such that for each k, u_k is the minimum of {x₀, x₁, …, x_k}. However, if we wish to prove the existence of a sequence of integer indices of minima of initial segments of 〈x_k〉_k∈?, the stronger subsystem WKL₀ is required. Following the presentation of these reverse mathematics results, we will derive computability theoretic corollaries and use them to illustrate a distinction between computable analysis and constructive analysis. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Limits of chromatic zeros of some families of maps

Two families of maps are considered, one consisting of maps with two pentagons separated by n 5-rings, the other of maps with two n-gons separated by two n-rings. For each family, a homogeneous linear recursion is derived for the corresponding family of chromatic polynomials. It is shown that B₅, B₇, and B₁₀ are limits of sequences of zeros from one or another of the families, where $\text{B}{}_{\mathrm{n}}\text{= 2(1+}\text{cos}\text{2π}\text{n}\text{)}$.     

Completeness of hyperspaces of compact subsets of quasi-metric spaces

We study the hyperspace K ₀(X) of non-empty compact subsets of a Smyth-complete quasi-metric space (X, d). We show that K ₀(X), equipped with the Hausdorff quasi-pseudometric H _d forms a (sequentially) Yoneda-complete space. Moreover, if d is a T ₁ quasi-metric, then the hyperspace is algebraic, and the set of all finite subsets forms a base for it. Finally, we prove that K ₀(X), H _d ) is Smyth-complete if (X, d) is Smyth-complete and all compact subsets of X are d ⁻¹-precompact.     

Stability of Solution of a Class of Quasilinear Elliptic Equations

In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.     

The geometry of products of minors

We consider the Newton polytope Σ(m,n) of the product of all minors of an m× n matrix of indeterminates. Using the fact that this polytope is the secondary polytope of the product Δ _m-1 ×Δ _n-1 of simplices, and thus has faces corresponding to coherent polyhedral subdivisions of Δ _m-1 ×Δ _n-1 , we study facets of Σ(m,n) , which correspond to the coarsest, nontrivial such subdivisions. We make use of the relation between secondary and fiber polytopes, which in this case gives a representation of Σ(m,n) as the Minkowski average of all m × n transportation polytopes. <lsiheader> <onlinepub>7 August, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>20n2p231.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>no <sectionname> </lsiheader> Received August 7, 1996, and in revised form April 4, 1997.     

Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

We study mathematical models of the structure of nilpotent subsemigroups of the semigroup PTD(B _n ) of partial contracting transformations of a Boolean, the semigroup TD(B _n ) of full contracting transformations of a Boolean, and the inverse semigroup ISD(B _n ) of contracting transformations of a Boolean. We propose a convenient graphical representation of the semigroups considered. For each of these semigroups, the uniqueness of its maximal nilpotent subsemigroup is proved. For PTD(B _n ) and TD(B _n ) , the capacity of a maximal nilpotent subsemigroup is calculated. For ISD(B _n ), we construct estimates for the capacity of a maximal nilpotent subsemigroup and calculate this capacity for small n. For all indicated semigroups, we describe the structure of nilelements and maximal nilpotent subsemigroups of nilpotency degree k and determine the number of elements and subsemigroups for some special cases.     

Classification of regular embeddings of hypercubes of odd dimension

By a regular embedding of a graph into a closed surface we mean a 2-cell embedding with the automorphism group acting regularly on flags. Recently, Kwon and Nedela [Non-existence of nonorientable regular embeddings of n-dimensional cubes, Discrete Math., to appear] showed that no regular embeddings of the n-dimensional cubes Q_n into nonorientable surfaces exist for any positive integer n>2. In 1997, Nedela and Škoviera [Regular maps from voltage assignments and exponent groups, European J. Combin. 18 (1997) 807-823] presented a construction giving for each solution of the congruence a regular embedding M_e of the hypercube Q_n into an orientable surface. It was conjectured that all regular embeddings of Q_n into orientable surfaces can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes Q_n into orientable surfaces for n odd, proving affirmatively the conjecture of Nedela and Škoviera for every odd n.     

Rings of quotients of f-rings by gabriel filters of ideals

The article examines the role of Gabriel filters of ideals in the ontext of semiprime f-rings. It is shown that for every 2-convex semiprime f-ring Aand every multiplicative filter B of dense ideals the ring of quotients of A by B, namely the direct limit of the Hom _A (I, A) over all I∈ B, is an l-subring of QA, the maximum ring of quotients. Relative to the category of all commutative rings with identity, it is shown that for every 2-convex semiprime f-ring A qA, the classical ring of quotients, is the largest flat epimorphic extension of A. If Ais also a Prüfer ring then it follows that every extension of Ain qA is of the form S ^-1A for a suitable multiplicative subset S. The paper also examines when a Utumi ring of quotients of a semiprime f-ring is obtained from a Gabriel filter. For a ring of continuous functions C(X), with Xcompact, this is so for each C(U) and C ^*(U), when Uis dense open, but not for an arbitrary direct limit of C(U),taken over a filter base of dense open sets. In conclusion, it is shown that, for a complemented semiprime f-ring A, the ideals of Awhich are torsion radicals with respect to some hereditary torsion theory are precisely the intersections of minimal prime ideals of A.     

Boundedness of solutions of a system of integro-differential equations

Let b: [?1, 0] →$\text{R}$ be a nondecreasing, strictly convex C²-function with b(? 1) = 0, and let g: $\text{R}$ⁿ → $\text{R}$ⁿ be a locally Lipschitzian mapping, which is the gradient of a function G: $\text{R}$ⁿ → $\text{R}$. Consider the following vector-valued integro-differential equation of the Levin-Nohel type

$\text{x}\text{?}\text{(t)=?∝}{}_{\mathrm{?1}}{}^0\text{b(θ)g(x(t + θ))dθ}$

. (E) This equation is used in applications to model various viscoelastic phenomena. By LaSalle's invariance principle, every bounded solution x(t) goes to a connected set of zeros of g, as time t goes to infinity. It is the purpose of this paper to give several geometric criteria assuring the boundedness of solutions of (E) or some of its components.     

Codimensions of products and of intersections of verbally primeT-ideals

By Kemer’s theory [9],T idealsJ ₁ ∪…∪J _r andJ ₁ …J _r, where eachJ _i is verbally prime, are of fundamental importance in the theory of P.I. algebras. We calculate, approximately and asymptotically, the codimensions of suchT-ideals, thereby extending the corresponding results about matrix algebras. In all such cases, the exponential growth of the codimensions is calculated; in particular, it is always an integer. Partially supported by NSF grant DMS 9303230. Partially supported by NSF grant DMS 9101488.     

A characterization of p-bases of rings of constants

We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.     

Structure of subsemigroups of factor-powers of finite symmetric groups

For the factor-powerFP(S _n ) of the symmetric groupS _n , we describe regular elements, maximal subgroups, isolated and fully isolated subsemigroups, and also maximal nilpotent subsemigroups whose zero elements coincide with the zero element of the semigroupFP(S _n ). Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 341–354, September, 1995. This research was partially supported by the Foundation for Fundamental Research of the State Committee for Science and Engineering of the Ukraine.     

On estimates of approximation characteristics of the Besov classes of periodic functions of many variables

We obtain order estimates for some approximate characteristics of the Besov classes B _p,ϑ ^r of periodic functions of many variables. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 9, pp. 1250–1261, September, 1997.     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.     

Systems of sets of lengths of Puiseux monoids

In this paper we study the system of sets of lengths of non-finitely generated atomic Puiseux monoids (a Puiseux monoid is an additive submonoid of ${\mathbb{Q}}_{\ge 0}$). We begin by presenting a BF-monoid M with full system of sets of lengths, which means that for each subset S of ${\mathbb{Z}}_{\ge 2}$ there exists an element $x\in M$ whose set of lengths $\mathsf{L}(x)$ is S. It is well known that systems of sets of lengths do not characterize numerical monoids. Here, we prove that systems of sets of lengths do not characterize non-finitely generated atomic Puiseux monoids. In a recent paper, Geroldinger and Schmid found the intersection of systems of sets of lengths of numerical monoids. Motivated by this, we extend their result to the setting of atomic Puiseux monoids. Finally, we relate the sets of lengths of the Puiseux monoid $P=〈1/p|p\text{is prime}〉$ with the Goldbach's conjecture; in particular, we show that $\mathsf{L}(2)$ is precisely the set of Goldbach's numbers.     

Convergence of distributions of integral functionals of extremal random functions

We study the convergence of distributions of integral functionals of random processes of the formU _n (t)=b _n (Z _n (t)-a _n G(t)),t⃛T, where {X=X(t), t⃛T} is a random process,X _n ,n≥1, are independent copies ofX, andZ _n (t)=max_1≤k≤n X _k (t). Ukrainian State Academy of Light Industry, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1201–1209, September, 1999.