Bezout Rings,Polynomials, and Distributivity

Let A be a ring, be an injective endomorphism of A, and let be the right skew polynomial ring. If all right annihilator ideals of A are ideals, then R is a right Bezout ring is a right Rickartian right Bezout ring, (e)=e for every central idempotent eA, and the element (a) is invertible in A for every regular aA. If A is strongly regular and n 2, then R/x ⁿ R is a right Bezout ring R/x ⁿ R is a right distributive ring R/x ⁿ R is a right invariant ring (e)=e for every central idempotent eA. The ring R/x ² R is right distributive R/x ⁿ R is right distributive for every positive integer n A is right or left Rickartian and right distributive, (e)=e for every central idempotent eA and the (a) is invertible in A for every regular aA. If A is a ring which is a finitely generated module over its center, then A[x] is a right Bezout ring A[x]/x ² A[x] is a right Bezout ring A is a regular ring.     

Mahler measure and entropy for commuting automorphisms of compact groups

We compute the joint entropy ofd commuting automorphisms of a compact metrizable group. LetR _d = [u ₁ ^±1 ,...,[d ₁ ^±1 ] be the ring of Laurent polynomials ind commuting variables, andM be anR _d -module. Then the dual groupX _M ofM is compact, and multiplication onM by each of thed variables corresponds to an action _M of ^d by automorphisms ofX _M . Every action of ^d by automorphisms of a compact abelian group arises this way. IffR _d , our main formula shows that the topological entropy of is given by

On Singular Functional Differential Inequalities

Classical theorems on differential inequalities [1, 2, 3] are generalized for initial value problems of the kind and where is a singular Volterra operator, is continuous and positive on ]a, b], is a norm in R ⁿ, and [u]₊ and [u]_– are respectively the positive and the negative part of the vector u R ⁿ.     

A characterization of Chebyshev curves inR n

Given a totally ordered setT containing at leastn+1 elements (say a subset ofR ¹), the graph of the functiona:TR ⁿ is called a Chebyshev curve (inR ⁿ) if the determinant of the matrix (a(t ₁),a(t ₂), ...,a(t _n)) is either positive whenevert ₁<t ₂<...<t _n or negative whenevert ₁<t ₂<...<t _n. For finiteT a characterization of these curves (sequences) has been given by the author.In this paper the result is extended to non-finiteT. The characterization proved here is an improved (reformulated) version of that given by the author for infiniteT.     

Q-Splines

The classical weighted spline introduced by Ph. Cinquin (1981), (see also K. Salkauskas (1984) and T.A. Foley (1986)) consists in minimizing _a ^b w(t)(x(t))² dt under the conditionsx(t _i )=y _i ,i=1,...,n, where the functionw is piecewise constant on the subdivisiona<t ₁<t ₂<...<t _n <b. The solution is a cubic spline, but it is notC ². We consider here the minimization of

L^2（R^n） Boundedness for a Class of Multilinear Singular Integral Operators

The L^2(R^n) boundedness for the multilinear singular integral operators defined by TAf(x)=∫R^nΩ（x-y）/|x-y|^n 1(A(x)-A(y)-△↓A(y)(x-y))f(y)dy is considered,where Ω is homogeneous of degree zero,integrable on the unit sphere and has vanishing moment of order one,A has derivatives of order one in BMO(R^n） boundedness for the multilinear operator TA is given.     

Numerical results on the Goldbach conjecture

The number of Goldbach partitions has been computed for all even numbers 350,000 and compared to well-known theoretical estimates. The random fluctuations are slowly decreasing and less than ± 5 per cent at the upper end of the interval. The number of partitions is given explicitly for 2 ⁿ ,n=3(1)22. Further, if for a givenN the smallest prime in all partitions of 2N isa=a(2N) we have also determineda ₁(2N ₁)<a ₂(2N ₂)<... withN ₁<N ₂<... such thatn<N _k impliesa(2n)<a _k (2N _k ) up to 2n=40,000,000.     

The law of the iterated logarithm for normalized empirical distribution function

This paper deals with the law of the iterated logarithm and its analogues for sup , where the sup is taken on an interval of the form (a _n ,b _n ),(0a _n <b _n 1). Under certain conditions on a _n and b _n the corresponding lim sup results will be proved.     

Asymptotics for Amitsur's Capelli-type polynomials and verbally prime PI-algebras

We consider associativePI-algebras over a field of characteristic zero. The main goal of the paper is to prove that the codimensions of a verbally prime algebra [11] are asymptotically equal to the codimensions of theT-ideal generated by some Amitsur's Capelli-type polynomialsE _M,L ^* [1]. We recall that two sequencesa _n,b _nare asymptotically equal, and we writea _n ≃b _n,if and only if lim _n→∞(a _n/b _n)=1.In this paper we prove that % MathType!End!2!1!, whereG is the Grassmann algebra. These results extend to all verbally primePI-algebras a theorem of A. Giambruno and M. Zaicev [9] giving the asymptotic equality % MathType!End!2!1! between the codimensions of the matrix algebraM _k(F) and the Capelli polynomials. The second author is partially supported by grants RFFI 04-01-00739a, E02-2.0-26.     

Maximal submodules and locally perfect rings

Rings over which every nonzero right module has a maximal submodule are calledright Bass rings. For a ringA module-finite over its centerC, the equivalence of the following conditions is proved:

Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II

In Part I of the paper, we have proved that, for every α > 0 and a continuous function f, which is either convex (s = 0) or changes convexity at a finite collection Y _s = {y _i } ^s _i=1 of points y _i ∈ (-1, 1),

Finite Element Matrix Sequences: the Case of Rectangular Domains

In the present paper, we consider a preconditioning strategy for Finite Element (FE) matrix sequences {A _n (a)} _n discretizing the elliptic problem $$\left\{ \begin{gathered} A_a u \equiv ( - )^k \nabla ^k [a(x,y)\nabla ^k u(x,y)] = f(x,y),{ }(x,y) \in \Omega = (0,1)^2 , \hfill \\ \left. {\left( {\frac{{\partial ^s }}{{\partial v^s }}u(x,y)} \right)} \right|_{\partial \Omega } \equiv 0,{ }s = 0,...,k - 1,{ }^{^{^{^{^{^{(1)} } } } } } \hfill \\ \end{gathered} \right.$$ with a(x,y) being a uniformly positive function and ν denoting the unit outward normal direction. More precisely, in connection with preconditioned conjugate gradient (PCG) like methods, we define the preconditioning sequence: {P _n (a)} _n , P _n (a):= $$\widetilde D$$ _n ^1/2(a)A _n (1) $$\widetilde D$$ _n ^1/2(a), where $$\widetilde D$$ _n (a) is the suitable scaled main diagonal of A _n (a). In fact, under the mild assumption of Lebesgue integrability of a(x), the weak clustering at the unity of the corresponding preconditioned sequence is proved. Moreover, if a(x,y) is regular enough and if a uniform triangulation is considered, then the preconditioned sequence shows a strong clustering at the unity so that the sequence {P _n (a)} _n turns out to be a superlinear preconditioning sequence for {A _n (a)} _n . The computational interest is due to the fact that the computation with A _n (a) is reduced to computations involving diagonals and two-level Toeplitz structures {A _n (1)} _n with banded pattern. Some numerical experimentations confirm the efficiency of the discussed proposal.     

The double sign of a real division algebra of finite dimension greater than one

For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a ∈ A?{0} are shown to form an invariant of A, called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category $\mathbb {D}_nFor any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a ∈ A?{0} are shown to form an invariant of A, called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category $\mathbb {D}_n$ of all n‐dimensional real division algebras to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of $\mathscr {D}_n$.     

Strong uniform consistency of nonparametric regression function estimates

Let (X, Y) be a ^dx-valued random vector and let r(t)=E(Y/X=t) be the regression function of Y on X that has to be estimated from a sample (X _i, Y_i), i=1,..., n. We establish conditions ensuring that an estimate of the form

On the hopficity of the polynomial rings

In this paper we prove that if a ringR satisfies the condition that for some integern > 1,a ⁿ =a for everya inR, thenR a hopfian ring implies that the ringR [T] of polynomials is also hopfian. This generalizes a recent result of Varadarajan which states that ifR is a Boolean hopfian ring then the ringR[T] is also hopfian. We show furthermore that there are numerous ringsR satisfying the hypothesis of our theorem which are neither Boolean nor Noetherian.     

Division Algebras with Left Algebraic Commutators

Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a ₀ + a ₁ x + ? + a _n x ⁿ (resp. right polynomial a ₀ + x a ₁ + ? + x ⁿ a _n ) over K such that a ₀ + a ₁ a + ? + a _n a ⁿ = 0 (resp. a ₀ + a a ₁ + ? + a ⁿ a _n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions.     

On a characterization of complete matrix rings

Peter R. Fuchs established in 1991 a new characterization of complete matrix rings by showing that a ringR with identity is isomorphic to a matrix ringM _n (S) for some ringS (and somen 2) if and only if there are elementsx andy inR such thatx ^n–1 0,x ⁿ=0=y ²,x+y is invertible, and Ann(x ^n–1)Ry={0} (theintersection condition), and he showed that the intersection condition is superfluous in casen=2. We show that the intersection condition cannot be omitted from Fuchs' characterization ifn3; in fact, we show that if the intersection condition is omitted, then not only may it happen that we do not obtain a completen ×n matrix ring for then under consideration, but it may even happen that we do not obtain a completem ×m matrix ring for anym2.     

Asymptotic properties for median cross-validated nearest neighbor median estimate in nonparametric regression

Consider the nonparametric regression model , whereg is an unknown function to be estimated on [0, 1], are the fixed design points in the interval [0, 1] and is a triangular array of row iid random variables having median zero. The nearest neighbor median estimator is taken as the estimator of the unknown functiong(x). Median cross validation (mev) criterion is employed to select the smoothing parameterh. Leth _π ^* be the smoothing parameter chosen by mev criterion. Under mild regularity conditions, the upper and lower bounds ofh _π ^* , the rate of convergence and the weak consistency of the median cross-validated estimate are obtained. Project supported by the National Natural Science Foundation of China and the Doctoral Foundation of Education of China.     

Erdos-Ko-Rado Theorems of Labeled Sets

For k = (k1, ··· , kn) ∈ Nn, 1 ≤ k1 ≤···≤ kn, let Lkr be the family of labeled r-sets on k given by Lkr := {{(a1, la1), ··· , (ar, lar)} : {a1, ··· , ar} ■[n],lai ∈ [kai],i = 1, ··· , r}. A family A of labeled r-sets is intersecting if any two sets in A intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.     

An Inertial Proximal Algorithm with Dry Friction: Finite Convergence Results

Let H be a Hilbert space and A, B: H ⇉ H two maximal monotone operators. In this paper, we investigate the properties of the following proximal type algorithm: