Left zero covering homomorphisms of laminated nearrings

Nearrings here are right nearrings. LetN be a nearring and fix an element α εN. Form another nearring N_α by taking addition to be the same as the addition inN but define the productx ⊗y of two elementsx, y ε N_α byx ⊗y =xay. The nearring N_α is referred to as a laminated nearring ofN andN is referred to as the base nearring. The element α is called the laminating element or the laminator. An elementx of a nearingN is a left zero ifxy =x for ally εN. A homomorphismϕ from a nearringN ₁ into a nearringN ₂ is a left zero covering homomorphism if for each left zeroy εN ₂,ϕ(x) =y for somex εN ₁. The left zero covering homomorphisms from one laminated nearring into another are investigated where the base nearring is the nearring of all continuous selfmaps of the Euclidean group ℝ² under pointwise addition and composition and the laminators are complex polynomials. Finally, it is shown that one can determine whether or not two such laminated nearrings are isomorphic simply by inspecting the coefficients of the two laminating polynomials.     

Parabolic analytic functions

Let bev=x+αy where α#0, α² = 0,x andy real are elements of a commutative ring. So inR(1, 3) ife ₀,e ₁,e ₂,e ₃ form a canonical framee ₀²=1,e ₁²=e ₂²=e ₃²=−1 the vector α=e ₀+e ₁ is different from zero while

Asymptotic behavior of solutions for a class of retarded difference equations

Consider the retarded difference equationx _n −x _n−1 =F(−f(x _n )+g(x _n−k )), wherek is a positive integer,F,f,g:R→R are continuous,F andf are increasing onR, anduF(u)>0 for allu≠0. We show that whenf(y)≥g(y) (resp. f(y)≤g(y)) fory∈R, every solution of (*) tends to either a constant or −∞ (resp. ∞) asn→∞. Furthermore, iff(y)≡g(y) fory∈R, then every solution of (*) tends to a constant asn→∞. Project supported by NNSF (19601016) of China and NSF (97-37-42) of Hunan     

The 2-circle and 2-disk problems on trees

Our purpose here is to consider on a homogeneous tree two Pompeiutype problems which classically have been studied on the plane and on other geometric manifolds. We obtain results which have remarkably the same flavor as classical theorems. Given a homogeneous tree, letd(x, y) be the distance between verticesx andy, and letf be a function on the vertices. For each vertexx and nonnegative integern let Σ _n f(x) be the sum Σ _{d(x, y)=n} f(y) and letB _n f(x)=Σ _{d(x, y)≦n} f(y). The purpose is to study to what extent Σ _n f andB _n f determinef. Since these operators are linear, this is really the study of their kernels. It is easy to find nonzero examples for which Σ _n f orB _n f vanish for one value ofn. What we do here is to study the problem for two values ofn, the 2-circle and the 2-disk problems (in the cases of Σ _n andB _n respectively). We show for which pairs of values there can exist non-zero examples and we classify these examples. We employ the combinatorial techniques useful for studying trees and free groups together with some number theory.     

A lower bound for linear forms in values of a continued fraction

Let y = y(x) be a function defined by a continued fraction. A lower bound for |Λ| = |β ₁ y ₁ + β ₂ y ₂ + α| is given, where y ₁ = y(x ₁), y ₂ = y(x ₂), x ₁ and x ₂ are positive integers, α, β ₁ and β ₂ are algebraic irrational numbers.     

Symmetric block bases of sequences with large average growth

We show that if 0<ε≦1, 1≦p<2 andx ₁, …,x _n is a sequence of unit vectors in a normed spaceX such thatE ‖∑ _l ⁿ ε_i x _l‖≧n ^1/p, then one can find a block basisy ₁, …,y _m ofx ₁, …,x _n which is (1+ε)-symmetric and has cardinality at leastγn ^2/p-1(logn)⁻¹, where γ depends on ε only. Two examples are given which show that this bound is close to being best possible. The first is a sequencex ₁, …,x _n satisfying the above conditions with no 2-symmetric block basis of cardinality exceeding 2n ^2/p-1. This sequence is not linearly independent. The second example is a sequence which satisfies a lowerp-estimate but which has no 2-symmetric block basis of cardinality exceedingCn ^2/p-1(logn)^4/3, whereC is an absolute constant. This applies when 1≦p≦3/2. Finally, we obtain improvements of the lower bound when the spaceX containing the sequence satisfies certain type-condition. These results extend results of Amir and Milman in [1] and [2]. We include an appendix giving a simple counterexample to a question about norm-attaining operators.     

Criteria for selection of response variables and the asymptotic properties in a multivariate calibration

Summary Let a set ofp responsesy=(y ₁,...y _p )′ has a multivariate linear regression on a set ofq explanatory variablesx=(x ₁,...x _q )′. Our aim is to select the most informative subset of responses for making inferences about an unknownx from an observedy. Under normality ony, two selection methods, based on the asymptotic mean squared error and on the Akaike's information criterion, are proposed by Fujikoshi and Nishii (1986,Hiroshima Math. J.,16, 269–277). In this paper, under a mild condition we will derive the cross-validation criterion and obtain the asymptotic properties of the three procedures.     

The complexity of computing the tutte polynomial on transversal matroids

The complexity of computing the Tutte polynomialT(M,x,y) is determined for transversal matroidM and algebraic numbersx andy. It is shown that for fixedx andy the problem of computingT(M,x,y) forM a transversal matroid is #P-complete unless the numbersx andy satisfy (x−1)(y−1)=1, in which case it is polynomial-time computable. In particular, the problem of counting bases in a transversal matroid, and of counting various types of “matchable” sets of nodes in a bipartite graph, is #P-complete.     

Limit T-spaces

Let F be a field of prime characteristic p and let V _p be the variety of associative algebras over F without unity defined by the identities [[x, y], z] = 0 and x ⁴ = 0 if p = 2 and by the identities [[x, y], z] = 0 and x ^p = 0 if p > 2 (here [x, y] = xy − yx). Let A/V _p be the free algebra of countable rank of the variety V _p and let S be the T-space in A/V _p generated by x ₁² x ₂² ⋯ x _k² + V ₂, where k ∈ ℕ if p = 2, and by {ie4170-01}, where k ∈ ℕ and α ₁, …, α _2k ∈ {0, p − 1} if p > 2. As is known, S is not finitely generated as a T-space. In the present paper, we prove that S is a limit T-space, i.e., a maximal nonfinitely generated T-space. As a corollary, we have constructed a limit T-space in the free associative F-algebra without unity of countable rank. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 135–159, 2007.     

On the average distance property and certain energy integrals

One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceR ⁿ andr(X, d ₂) the rendezvous number ofX, whered ₂ denotes the Euclidean distance inR ⁿ . (The rendezvous numberr(X, d ₂) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x ₁,x ₂,...,x _n inX, there exists somex inX such that .) Then there exists some regular Borel probability measure μ₀ onX such that the value of ∫ _X d ₂(x, y)dμ₀ (y) is independent of the choicex inX, if and only ifr(X, d ₂) = sup_μ ∫ _X ∫ _X d ₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX.     

On the number of y-smooth natural numbers ≤x representable as a sum of two integer squares

LetB(x,y) be the sum taken over alln, 1≤n≤x, such that n can be represented as a sum of two squares of integers andn has no prime factors exceedingy. It is shown foru smaller than about .5log logx/log log logx thatB(x,x ^1/u)≈cxlog^-1/2 xσ(u), where σ(u satisfies a delay differential equation similar to the one satisfied by the Dickman function andc is a positive constant.     

Weak-Geodetically Closed Subgraphs in Distance-Regular Graphs

 Let Γ=(X,R) denote a distance-regular graph with diameter D≥2 and distance function δ. A (vertex) subgraph Ω⊆X is said to be weak-geodetically closed whenever for all x,y∈Ω and all z∈X,

Excess of a Lattice

 A pair (x,y) of elements in a lattice is mismatching if x is join-irreducible, y is meet-irreducible, and xy. The excess of a lattice L is defined by ex(L):=|L|− min{|V _x |+|I _y |:(x,y) is mismatching}, where V _x (I _y ) is the principal filter (ideal) generated by x(y), respectively. In the first half of this paper, it is shown that a lattice has excess zero (one) if and only if it is isomorphic to a Boolean lattice (one of a chain of length two, a diamond, and a pentagon), respectively. In the second half, we show that for each relatively complemented lattice which is not Boolean, its size is at most five times its excess. Moreover we determine extremal configurations. Received: September 2, 1999 Final version received: July 25, 2000 Acknowledgments. The author thanks the referee for constructive comments. Present address: Bureau of Human Resources, National Personnel Authority, 1-2-3 Kasumigaseki, Chiyoda-ku, Tokyo 100-8913, Japan. e-mail: abetet@jinji.go.jp     

Difference Families in Z2d+1⊕Z2d+1 and Infinite Translation Designs in Z⊕Z

We analyse 3-subset difference families of Z_2d+1⊕Z_2d+1 arising as reductions (mod 2d+1) of particular families of 3-subsets of Z⊕Z. The latter structures, namely perfect d-families, can be viewed as 2-dimensional analogues of difference triangle sets having the least scope. Indeed, every perfect d-family is a set of base blocks which, under the natural action of the translation group Z⊕Z, cover all edges {(x,y),(x′,y′)} such that |x−x′|, |y−y′|≤d. In particular, such a family realises a translation invariant (G,K₃)-design, where V(G)=Z⊕Z and the edges satisfy the above constraint. For that reason, we regard perfect families as part of the hereby defined translation designs, which comprise and slightly generalise many structures already existing in the literature. The geometric context allows some suggestive additional definitions. The main result of the paper is the construction of two infinite classes of d-families. Furthermore, we provide two sporadic examples and show that a d-family may exist only if d≡0,3,8,11 (mod 12).     

On some characterization of probability distributions in Hilbert spaces

Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x₁, x₂, …, x_n be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals f_k(t) = E[e^i(t,x _k)], (k=1, 2, …, n): let y₁=x₁ + x_n, y₂=x₂ + x_n, …, y_n−1=x_n−1 + x_n. If the characteristic functional f(t₁, t₂, …, t_n−1) of the random variables (y₁, y₂, …, y_n−1) does not vanish, then the joint distribution of (y₁, y₂, …, y_n−1) determines all the distributions of x₁, x₂, …, x_n up to change of location.     

A Property About Minimum Edge- and Minimum Clique-Cover of a Graph

 Let G be a graph with n vertices, and denote as γ(G) (as θ(G)) the cardinality of a minimum edge cover (of a minimum clique cover) of G. Let E (let C) be the edge-vertex (the clique-vertex) incidence matrix of G; write then P(E)={x∈ℜ ⁿ :Ex≤1,x≥0}, P(C)={x∈ℜ ⁿ :Cx≤1,x≥0}, α _E (G)=max{1 ^T x subject to x∈P(E)}, and α _C (G)= max{1 ^T x subject to x∈P(C)}. In this paper we prove that if α _E (G)=α _C (G), then γ(G)=θ(G). Received: May 20, 1998?Final version received: April 12, 1999     

Positive linear operators generated by analytic functions

Let φ be a power series with positive Taylor coefficients {a _k } _k=0^∞ and non-zero radius of convergence r ≤ ∞. Let ξ _x , 0 ≤ x < r be a random variable whose values α _k , k = 0, 1, …, are independent of x and taken with probabilities a _k x ^k /φ(x), k = 0, 1, …. The positive linear operator (A _φ f)(x):= E[f(ξ _x )] is studied. It is proved that if E(ξ _x ) = x, E(ξ _x ²) = qx ² + bx + c, q, b, c ∈ R, q > 0, then A _φ reduces to the Szász-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupaş operator in the case q > 1.     

Multilinear polynomials and cocentralizing conditions in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, let Z(R) be its center, let U be the Utumi quotient ring of R, let C be the extended centroid of R, and let f(x ₁,..., x _n ) be a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all evaluations of f(x ₁, …, xn) on R. If F and G are generalized derivations of R such that [[F(x), x], [G(y), y]] ∈ Z(R) for any x, y ∈ f(R), then one of the following holds:

Left Annihilators of Commutators with Derivation on Right Ideals

Abstract

Let R be a prime ring of characteristic different from 2, d a non-zero derivation of R, I a non-zero right ideal of R, a ∈ R, S ₄(x ₁,…, x ₄) the standard polynomial in 4 variables. Suppose that, for any x, y ∈ I, a[d([x, y]), [x, y]] = 0. If S ₄(I, I, I, I)I ≠ 0, then aI = ad(I) = 0.     

Admissible functions with multiple discontinuities

LetT be the unit circle, α irrational andF: T → R a step function. A necessary and sufficient condition for the skew of the α-rotation byf (considered as taking values mod 1) to be minimal is given. Also, the boundedness of Σ _i=1 ⁿ f(x+iα asn → α is resolved.