Right ideals generated by an idempotent of finite rank

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let f(X₁,…,X_t) be an arbitrary and fixed polynomial over K in noncommuting indeterminates X₁,…,X_t with constant term 0 such that for some μK occurring in the coefficients of f(X₁,…,X_t). It is proved that a right ideal ρ of R is generated by an idempotent of finite rank if and only if the rank of f(x₁,…,x_t) is bounded above by a same natural number for all x₁,…,x_tρ. In this case, the rank of the idempotent that generates ρ is also explicitly given. The results are then applied to considering the triangularization of ρ and the irreducibility of f(ρ), where f(ρ) denotes the additive subgroup of R generated by the elements f(x₁,…,x_t) for x₁,…,x_tρ.     

Optimal Recovery of the Derivative of Periodic Analytic Functions from Hardy Classes

LetS_β{z : |Im z|<β}. For 2π-periodic functions which are analytic inS_βwithp-integrable boundary values, we construct an optimal method of recovery off′(ξ), ξS_β, using information about the valuesf(x₁), mldr;, f(x_n), x_j[0, 2π).     

Self-improving behaviour of inner functions as multipliers

Let X and Y be two spaces of analytic functions in the disk, with XY. For an inner function θ, it is sometimes true that whenever fX and fθY, the latter product must actually be in X. We discuss this phenomenon for various pairs of (analytic) smoothness classes X and Y.     

Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings

Let E be a real reflexive Banach space with uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed and convex subset of E. Let T:K→K be a strictly pseudo-contractive map and let L>0 denote its Lipschitz constant. Assume F(T){xK:Tx=x}≠0/ and let zF(T). Fix δ(0,1) and let δ^* be such that δ^*δL(0,1). Define , where δ_n(0,1) and limδ_n=0. Let {α_n} be a real sequence in (0,1) which satisfies the following conditions: . For arbitrary x₀,uK, define a sequence {x_n}K by x_n+1=α_nu+(1−α_n)S_nx_n. Then, {x_n} converges strongly to a fixed point of T.     

LOCAL TIME ANALYSIS OF ADDITIVE LEVY PROCESSES WITH DIFFERENT L(E)VY EXPONENTS

Let X1 XN be independent, classical Levy processes on R^d with Levy exponents ψ1,…, ψN, respectively. The corresponding additive Levy process is defined as the following N-parameter random field on R^d, X（t） △= X1（t1） ＋ ... ＋ XN（tN）, At∈N. Under mild regularity conditions on the ψi＇s, we derive estimate for the local and uniform moduli of continuity of local times of X = {X（t）; t ∈R^N}.     

Second-order boundary value problems with nonhomogeneous boundary conditions (II)

Sufficient conditions are given for the existence of solutions of the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition

We prove that the whole plane is divided by a “continuous decreasing curve” Γ into two disjoint connected regions Λ^E and Λ^N such that the above problem has at least one solution for (λ₁,λ₂)Γ, has at least two solutions for (λ₁,λ₂)Λ^EΓ, and has no solution for (λ₁,λ₂)Λ^N. We also find explicit subregions of Λ^E where the above problem has at least two solutions and two positive solutions, respectively.     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.     

Constant-sign solutions for systems of singular integral equations of Hammerstein type

We consider the system of Hammerstein integral equations

where T>0 is fixed, ρ_i’s are given functions and the nonlinearities f_i(t,x₁,x₂,…,x_n) can be singular at t=0 and x_j=0 where j{1,2,,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ_iu_i(t)≥0 for t[0,T] and 1≤i≤n, where θ_i{1,−1} is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.     

Density of Gabor Frames

A Gabor system is a set of time-frequency shifts S(g, Λ) ={e^{2 π ibx}g(x − a)}_{(a, b) Λ} of a function g L²(R^d). We prove that if a finite union of Gabor systems _{k = 1}^rS(g_k, Λ_k) forms a frame for L²(R^d) then the lower and upper Beurling densities of Λ = _{k = 1}^r Λ_k satisfy D⁻(Λ) ≥ 1 and D ⁺ (Λ) < ∞. This extends recent work of Ramanathan and Steger. Additionally, we prove the conjecture that no collection _{k = 1}^r{g_k(x − a)}_{a Γk} of pure translates can form a frame for L²(R^d).     

Operators Which Do Not Have the Single Valued Extension Property

In this paper we shall consider the relationships between a local version of the single valued extension property of a bounded operator T  L(X) on a Banach space X and some quantities associated with T which play an important role in Fredholm theory. In particular, we shall consider some conditions for which T does not have the single valued extension property at a point λ_o  C.     

On constants in some inequalities for intermediate derivatives on a finite interval

Let L_q (1q<∞) be the space of functions f measurable on I=[−1,1] and integrable to the power q, with normL_∞ is the space of functions measurable on I with normWe denote by AC the set of all functions absolutely continuous on I. For nN, q[1,∞] we setW_n,q={f:f⁽ⁿ⁻¹⁾AC, f⁽ⁿ⁾L_q}.In this paper, we consider the problem of accuracy of constants A, B in the inequalities (1)|| f^(m)||_qA|| f||_p+B|| f^(m+k+1)||_r, mN, kW; p,q,r[1,∞], fW_m+k+1,r.     

Local Hardy spaces and summability of Fourier transforms

New Wiener amalgam spaces are introduced for local Hardy spaces. A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from the amalgam space to W(L_p,ℓ_∞). This implies the almost everywhere convergence of the θ-means for all fW(L₁,ℓ_∞)L₁.     

About equivalent interval colorings of weighted graphs

Given a graph G=(V,E) with strictly positive integer weights ω_i on the vertices iV, a k-interval coloring of G is a function I that assigns an interval I(i){1,…,k} of ω_i consecutive integers (called colors) to each vertex iV. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0/ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χ_int(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω_i=1 for all vertices iV).Two k-interval colorings I₁ and I₂ are said equivalent if there is a permutation π of the integers 1,…,k such that ℓI₁(i) if and only if π(ℓ)I₂(i) for all vertices iV. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions.     

On some d-dimensional dual hyperovals in

Let d≥3. Let H be a d+1-dimensional vector space over GF(2) and {e₀,…,e_d} be a specified basis of H. We define Supp(t){e_t1,…,e_tl}, a subset of a specified base for a non-zero vector t=e_t1++e_tl of H, and Supp(0)0/. We also define J(t)Supp(t) if |Supp(t)| is odd, and J(t)Supp(t){0} if |Supp(t)| is even.For s,tH, let {a(s,t)} be elements of H(HH) which satisfy the following conditions: (1) a(s,s)=(0,0), (2) a(s,t)=a(t,s), (3) a(s,t)≠(0,0) if s≠t, (4) a(s,t)=a(s^′,t^′) if and only if {s,t}={s^′,t^′}, (5) {a(s,t)|tH} is a vector space over GF(2), (6) {a(s,t)|s,tH} generate H(HH). Then, it is known that S{X(s)|sH}, where X(s){a(s,t)|tH{s}}, is a dual hyperoval in PG(d(d+3)/2,2)=(H(HH)){(0,0)}.In this note, we assume that, for s,tH, there exists some x_s,t in GF(2) such that a(s,t) satisfies the following equation: Then, we prove that the dual hyperoval constructed by {a(s,t)} is isomorphic to either the Huybrechts’ dual hyperoval, or the Buratti and Del Fra’s dual hyperoval.     

Uniform packing dimension results for multiparameter stable processes

In this article, authors discuss the problem of uniform packing dimension of the image set of multiparameter stochastic processes without random uniform H(o)lder condition, and obtain the uniform packing dimension of multiparameter stable processes.If Z is a stable (N, d, α)-process and αN ≤ d, then the following holds with probability 1 Dim Z(E) = α DimE for any Borel setE ∈ B(R N),where Z(E) = {x: (E) t ∈ E, Z(t) = x}. Dim(E) denotes the packing dimension of E.     

On multipartite posets

A poset P=(X,) is m-partite if X has a partition X=X₁X_m such that (1) each X_i forms an antichain in P, and (2) xy implies xX_i and yX_j where i<j. In this article we derive a tight asymptotic upper bound on the order dimension of m-partite posets in terms of m and their bipartite sub-posets in a constructive and elementary way.     

Some exact rates in the functional law of the iterated logarithm

We find exact convergence rate in the Strassen's functional law of the iterated logarithm for a class of elements on the boundary of the limit set. Our result applies, in particular, to the power functions c_αx^α with α ]1/2,1[, thus solving a small ball estimate problem which was open for ten years.     

Fusion frames and distributed processing

Let {W_i}_iI be a (redundant) sequence of subspaces of a Hilbert space each being endowed with a weight v_i, and let be the closed linear span of the W_is, a composite Hilbert space. {(W_i,v_i)}_iI is called a fusion frame provided it satisfies a certain property which controls the weighted overlaps of the subspaces. These systems contain conventional frames as a special case, however they reach far “beyond frame theory.” In case each subspace W_i is equipped with a spanning frame system {f_ij}_jJi, we refer to {(W_i,v_i,{f_ij}_jJi)}_iI as a fusion frame system. The focus of this article is on computational issues of fusion frame reconstructions, unique properties of fusion frames important for applications with particular focus on those superior to conventional frames, and on centralized reconstruction versus distributed reconstructions and their numerical differences. The weighted and distributed processing technique described in this article is not only a natural fit to distributed processing systems such as sensor networks, but also an efficient scheme for parallel processing of very large frame systems. Another important component of this article is an extensive study of the robustness of fusion frame systems.     

Maps on spaces of symmetric matrices preserving idempotence

Suppose F is an arbitrary field. Let |F| be the number of the elements of F. Let M_n(F) be the space of all n×n matrices over F, and let S_n(F) be the subset of M_n(F) consisting of all symmetric matrices. Let V{S_n(F),M_n(F)}, a map Φ:V→V is said to preserve idempotence if A-λB is idempotent if and only if Φ(A)-λΦ(B) is idempotent for any A,BV and λF. It is shown that: when the characteristic of F is not 2, |F|>3 and n4, Φ:S_n(F)→S_n(F) is a map preserving idempotence if and only if there exists an invertible matrix PM_n(F) such that Φ(A)=PAP^-1 for every AS_n(F) and P^tP=aI_n for some nonzero scalar a in F.     

Best Approximation by Normal and Conormal Sets

The aim of the present paper is to develop a theory of best approximation by elements of so-called normal sets and their complements—conormal sets—in the non-negative orthant ^I₊ of a finite-dimensional coordinate space ^I endowed with the max-norm. A normal (respectively, conormal) set arises as the set of all solutions of a system of inequalities f_α(x)0 (αA), x ^I₊ (respectively, f_α(x)0 (αA), x ^I₊), where f_α is an increasing function and A is an arbitrary set of indices. We consider these sets as analogues (in a certain sense) of convex sets, and we use the so-called min-type functions as analogues of linear functions. We show that many results on best approximation by convex and reverse convex sets and corresponding separation theory (but not all of them) have analogues in the case under consideration. At the same time there are no convex analogues for many results related to best approximation by normal sets.