Gagliardo–Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions

We consider a triple of N-functions (M,H,J) that satisfy the Δ^′-condition, and suppose that an additive variant of interpolation inequality holds

where , is an arbitrary set invariant with respect to external and internal dilations. We show that the above inequality implies its certain nonlinear variant involving the expressions and . Various generalizations of this inequality to the more general class of N-functions, measures and to higher order derivatives are also discussed and the examples are presented.     

Strong uniform continuity

Let be an ideal of subsets of a metric space X,d. This paper considers a strengthening of the notion of uniform continuity of a function restricted to members of which reduces to ordinary continuity when consists of the finite subsets of X and agrees with uniform continuity on members of when is either the power set of X or the family of compact subsets of X. The paper also presents new function space topologies that are well suited to this strengthening. As a consequence of the general theory, we display necessary and sufficient conditions for continuity of the pointwise limit of a net of continuous functions.     

Pointwise universal trigonometric series

A series is called a pointwise universal trigonometric series if for any , there exists a strictly increasing sequence of positive integers such that converges to f(z) pointwise on . We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if as |n|→∞ for some ε>0, then the series S_a cannot be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series S_a with as |n|→∞.     

Conditional limiting distribution of beta-independent random vectors

The paper deals with random vectors in , possessing the stochastic representation , where R is a positive random radius independent of the random vector and is a non-singular matrix. If is uniformly distributed on the unit sphere of , then for any integer m<d we have the stochastic representations and , with W≥0, such that W² is a beta distributed random variable with parameters m/2,(d−m)/2 and (U₁,…,U_m),(U_m+1,…,U_d) are independent uniformly distributed on the unit spheres of and , respectively. Assuming a more general stochastic representation for in this paper we introduce the class of beta-independent random vectors. For this new class we derive several conditional limiting results assuming that R has a distribution function in the max-domain of attraction of a univariate extreme value distribution function. We provide two applications concerning the Kotz approximation of the conditional distributions and the tail asymptotic behaviour of beta-independent bivariate random vectors.     

Weak and point-wise convergence in for filter convergence

We study those filters on for which weak -convergence of bounded sequences in C(K) is equivalent to point-wise -convergence. We show that it is sufficient to require this property only for C[0,1] and that the filter-analogue of the Rainwater extremal test theorem arises from it. There are ultrafilters which do not have this property and under the continuum hypothesis there are ultrafilters which have it. This implies that the validity of the Lebesgue dominated convergence theorem for -convergence is more restrictive than the property which we study.     

All-derivable points in continuous nest algebras

Let be an operator algebra on a Hilbert space. We say that an element is an all-derivable point of for the strong operator topology if every strong operator topology continuous derivable linear mapping φ at G (i.e. φ(ST)=φ(S)T+Sφ(T) for any with ST=G) is a derivation. Let be a continuous nest on a complex and separable Hilbert space H. We show in this paper that every orthogonal projection operator P(M) () is an all-derivable point of for the strong operator topology.     

On fractional maximal function and fractional integral on the Laguerre hypergroup

Let be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and the fractional integral operator on the Laguerre hypergroup from the spaces to the spaces and from the spaces to the weak spaces .     

Spaces of holomorphic functions in regular domains

Let Ω be a regular domain in the complex plane , . Let be the linear space over of the holomorphic functions f in Ω such that f⁽ⁿ⁾ is bounded in Ω and is continuously extendible to the closure of Ω, n=0,1,2,… . We endow , in a natural manner, with a structure of Fréchet space and we obtain dense subspaces F of , with good topological linear properties, also satisfying that each function f of F, distinct from zero, does not extend holomorphically outside Ω.     

Sobolev–Hardy space with general weight

In this paper, it is defined the kth order Sobolev–Hardy space with norm

Then the corresponding Poincaré inequality in this space is obtained, and the results are given that this space is embedded in with weight and in with weight ^q/2 for 1q<2. Moreover, we prove that the constant of k-improved Hardy–Sobolev inequality with general weight is optimal. These inequalities turn to be some known versions of Hardy–Sobolev inequalities in the literature by some particular choice of weights.     

Plancherel–Polya-type inequalities for entire functions of exponential type in

The goal of the paper is to prove generalizations of the classical Plancherel–Polya inequalities in which point-wise sampling of functions (δ-distributions) is replaced by more general compactly supported distributions on . As an application it is shown that a function , 1p∞, which is an entire function of exponential type is uniquely determined by a set of numbers {Ψ_j(f)}, , where {Ψ_j}, , is a countable sequence of compactly supported distributions. In the case p=2 a reconstruction method of a Paley–Wiener function f from a sequence of samples {Ψ_j(f)}, , is given. This method is a generalization of the classical result of Duffin–Schaeffer about exponential frames on intervals.     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

Linearizability conditions of time-reversible cubic systems

In this paper we present the necessary and sufficient conditions for linearizability of the planar time-reversible cubic complex system , . From these conditions, the necessary and sufficient conditions for the origin to be an isochronous center of the time-reversible cubic real system , can be obtained. Thus, the isochronous center problem of time-reversible cubic systems is solved completely.     

There's something about the diameter

We study diameter preserving linear bijections from onto where X, Y are compact Hausdorff spaces and V, Z are Banach spaces. For instance, we obtain that if X has at least four points, Z is linearly isometric to V and either Z is a space or Z^* is strictly convex or smooth, then there is a diameter preserving linear bijection from onto if and only if X is homeomorphic to Y. We also consider the case when X and Y are not compact but locally compact spaces.     

Configurations in abelian categories. II. Ringel–Hall algebras

This is the second in a series on configurations in an abelian category . Given a finite poset (I,), an (I,)-configuration (σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in satisfying some axioms, where J,KI. Configurations describe how an object X in decomposes into subobjects.The first paper defined configurations and studied moduli spaces of (I,)-configurations in , using the theory of Artin stacks. It showed well-behaved moduli stacks of objects and configurations in exist when is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod- of representations of a quiver Q.Write for the vector space of -valued constructible functions on the stack . Motivated by the idea of Ringel–Hall algebras, we define an associative multiplication * on using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that is a -algebra. We also study representations of , the Lie subalgebra of functions supported on indecomposables, and other algebraic structures on .Then we generalize all these ideas to stack functions , a universal generalization of constructible functions, containing more information. When Extⁱ(X,Y)=0 for all and i>1, or when for P a Calabi–Yau 3-fold, we construct (Lie) algebra morphisms from stack algebras to explicit algebras, which will be important in the sequels on invariants counting τ-semistable objects in .     

Inheritance of hyper-duality in imprimitive Bose–Mesner algebras

We prove the following result concerning the inheritance of hyper-duality by block and quotient Bose–Mesner algebras associated with a hyper-dual pair of imprimitive Bose–Mesner algebras. Let and denote Bose–Mesner algebras. Suppose there is a hyper-duality ψ from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to . Also suppose that is imprimitive with respect to a subset of Hadamard idempotents, so is dual imprimitive with respect to the subset of primitive idempotents, where is the formal duality associated with ψ. Let denote the block Bose–Mesner algebra of on the block containing p, and let denote the quotient Bose–Mesner algebra of with respect to . Then there is a hyper-duality from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to .     

Two canonical passive state/signal shift realizations of passive discrete time behaviors

A discrete time invariant linear state/signal system Σ with a Hilbert state space and a Kren signal space has trajectories (x(),w()) that are solutions of the equation , where F is a bounded linear operator from into with a closed domain whose projection onto is all of . This system is passive if the graph of F is a maximal nonnegative subspace of the Kren space . The future behavior of a passive system Σ is the set of all signal components w() of trajectories (x(),w()) of Σ on with x(0)=0 and . This is always a maximal nonnegative shift-invariant subspace of the Kren space , i.e., the space endowed with the indefinite inner product inherited from . Subspaces of with this property are called passive future behaviors. In this work we study passive state/signal systems and passive behaviors (future, full, and past). In particular, we define and study the input and output maps of a passive state/signal system, and the past/future map of a passive behavior. We then turn to the inverse problem, and construct two passive state/signal realizations of a given passive future behavior , one of which is observable and backward conservative, and the other controllable and forward conservative. Both of these are canonical in the sense that they are uniquely determined by the given data , in contrast earlier realizations that depend not only on , but also on some arbitrarily chosen fundamental decomposition of the signal space . From our canonical realizations we are able to recover the two standard de Branges–Rovnyak input/state/output shift realizations of a given operator-valued Schur function in the unit disk.     

On ternary Kloosterman sums modulo 12

Let K(a) denote the Kloosterman sum on . It is easy to see that for all . We completely characterize those for which , and . The simplicity of the characterization allows us to count the number of the belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov.     

A characterization of the natural embedding of the split Cayley hexagon in by intersection numbers

In this paper, we prove that a set of q⁵+q⁴+q³+q²+q+1 lines of with the properties that (1) every point of is incident with either 0 or q+1 elements of , (2) every plane of is incident with either 0, 1 or q+1 elements of , (3) every solid of is incident with either 0, 1, q+1 or 2q+1 elements of , and (4) every hyperplane of is incident with at most q³+3q²+3q members of , is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in .     

Properties of multilevel block -circulants

In a previous paper we characterized unilevel block α-circulants , , 0mn-1, in terms of the discrete Fourier transform of , defined by . We showed that most theoretical and computational problems concerning A can be conveniently studied in terms of corresponding problems concerning the Fourier coefficients F₀,F₁,…,F_n-1 individually. In this paper we show that analogous results hold for (k+1)-level matrices, where the first k levels have block circulant structure and the entries at the (k+1)-st level are unstructured rectangular matrices.     

Oscillation of second-order damped dynamic equations on time scales

The study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which may be an arbitrary closed subset of the reals. This way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. In this paper, by employing the Riccati transformation technique we will establish some oscillation criteria for second-order linear and nonlinear dynamic equations with damping terms on a time scale . Our results in the special case when and extend and improve some well-known oscillation results for second-order linear and nonlinear differential and difference equations and are essentially new on the time scales , h>0, for q>1, , etc. Some examples are considered to illustrate our main results.