On the set of limit points of the partial sums of series rearranged by a given divergent permutation

We give a new characterization of divergent permutations. We prove that for any divergent permutation p, any closed interval I of (the 2-point compactification of ) and any real number sI, there exists a series ∑a_n of real terms convergent to s such that I=σa_p(n) (where σa_p(n) denotes the set of limit points of the partial sums of the series ∑a_p(n)). We determine permutations p of for which there exists a conditionally convergent series ∑a_n such that ∑a_p(n)=+∞. If the permutation p of possesses the last property then we prove that for any and there exists a series ∑a_n convergent to α and such that σa_p(n)=[β,+∞]. We show that for any countable family P of divergent permutations there exist conditionally convergent series ∑a_n and ∑b_n such that any series of the form ∑a_p(n) with pP is convergent to the sum of ∑a_n, while for every pP.     

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.     

Uniqueness of limit cycles for polynomial first-order differential equations

We study the uniqueness of limit cycles (periodic solutions that are isolated in the set of periodic solutions) in the scalar ODE in terms of {i_k}, {j_k}, {n_k}. Our main result characterizes, under some additional hypotheses, the exponents {i_k}, {j_k}, {n_k}, such that for any choice of the equation has at most one limit cycle. The obtained results have direct application to rigid planar vector fields, thus, planar systems of the form x^′=y+xR(x,y), y^′=−x+yR(x,y), where . Concretely, when the set has at least three elements (or exactly one) and another technical condition is satisfied, we characterize the exponents {i_k}, {j_k} such that the origin of the rigid system is a center for any choice of and also when there are no limit cycles surrounding the origin for any choice of .     

Positive periodic solutions of periodic neutral Lotka–Volterra system with state dependent delays

By using a fixed point theorem of strict-set-contraction, some new criteria are established for the existence of positive periodic solutions of the following periodic neutral Lotka–Volterra system with state dependent delays

where (i,j=1,2,…,n) are ω-periodic functions and (i=1,2,…,n) are ω-periodic functions with respect to their first arguments, respectively.     

Partial regularity for solutions of a nonlinear elliptic equation with singular nonlinearity

We consider the following nonlinear elliptic equation with singular nonlinearity:

where α>β>1, a>0, and Ω is an open subset of , n2. Let uH¹(Ω) with and be a nonnegative stationary solution. If we denote the zero set of u by

we shall prove that the Hausdorff dimension of Σ is less than or equal to .     

Analyticity of functions analytic on circles

Let Δ be the open unit disc in, let pbΔ, and let f be a continuous function on which extends holomorphically from each circle in centered at the origin and from each circle in which passes through p. Then f is holomorphic on Δ.     

A pseudo-Boolean consensus approach to nonlinear 0–1 optimization

It is proved that any pseudo-Boolean function f can be represented as , where z is the minimum of f and φ is a polynomial with positive coefficients in the original variables x_i and in their complements . A non-constructive proof and a constructive one are given. The latter, which is based on a generalization to pseudo-Boolean functions of the well-known Boolean-theoretical operation of consensus, provides a new algorithm for the minimization of pseudo-Boolean functions.     

Numerical boundaries for some classical Banach spaces

Globevnik gave the definition of boundary for a subspace . This is a subset of Ω that is a norming set for . We introduce the concept of numerical boundary. For a Banach space X, a subset BΠ(X) is a numerical boundary for a subspace if the numerical radius of f is the supremum of the modulus of all the evaluations of f at B, for every f in . We give examples of numerical boundaries for the complex spaces X=c₀, and d_*(w,1), the predual of the Lorentz sequence space d(w,1). In all these cases (if K is infinite) we show that there are closed and disjoint numerical boundaries for the space of the functions from B_X to X which are uniformly continuous and holomorphic on the open unit ball and there is no minimal closed numerical boundary. In the case of c₀, we characterize the numerical boundaries for that space of holomorphic functions.     

Intersective polynomials and the polynomial Szemerédi theorem

Let be a family of polynomials such that , i=1,…,r. We say that the family P has the PSZ property if for any set with there exist infinitely many such that E contains a polynomial progression of the form {a,a+p₁(n),…,a+p_r(n)}. We prove that a polynomial family P={p₁,…,p_r} has the PSZ property if and only if the polynomials p₁,…,p_r are jointly intersective, meaning that for any there exists such that the integers p₁(n),…,p_r(n) are all divisible by k. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If are jointly intersective integral polynomials, then for any finite partition of , there exist i{1,…,k} and a,nE_i such that {a,a+p₁(n),…,a+p_r(n)}E_i.     

More on the revised GCH and the black box

We strengthen the revised GCH theorem by showing, e.g., that for , for all but finitely many regular κ<_ω, it holds that “λ is accessible on cofinality κ” in some weak sense (see below).As a corollary, λ=2^μ=μ⁺>_ω implies that the diamond holds on λ when restricted to cofinality κ for all but finitely many .We strengthen previous results on the black box and the middle diamond: previously it was established that these principles hold on for sufficiently large n; here we succeed in replacing a sufficiently large _n with a sufficiently large _n.The main theorem, concerning the accessibility of λ on cofinality κ, Theorem 3.1, implies as a special case that for every regular λ>_ω, for some κ<_ω, we can find a sequence such that , , and we can fix a finite set of “exceptional” regular cardinals θ<_ω so that if Aλ satisfies |A|<_ω, there is a pair-coloring so that for every -monochromatic BA with no last element, letting δ:=supB it holds that —provided that is not one of the finitely many “exceptional” members of .     

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 .     

Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory

In this paper we study extensions of the arithmetic operators +, -, ·, ÷ to the lattice of closed subintervals of the unit interval. Starting from a minimal set of axioms that these operators must fulfill, we investigate which properties they satisfy. We also investigate some classes of t-norms on which can be generated using these operators; these classes provide natural extensions of the Łukasiewicz, product, Frank, Schweizer–Sklar and Yager t-norms to .     

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.     

Improved variance estimation under sub-space restriction

For the linear regression model , we assume that for a given positive definite scale matrix , the error vector has a multivariate normal distribution and has the inverted Wishart distribution. For under an orthogonal sub-space restriction , we propose restricted unbiased, preliminary test and Stein-type estimators of variance of the error term, for when the scale of the inverse Wishart distribution is assumed to be unknown. We compare the weighted quadratic risks of the underlying estimators and propose dominance pictures for them.     

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.     

Quadratic and cubic invariants of unipotent affine automorphisms

Let K be an arbitrary field of characteristic zero, P_n:=K[x₁,…,x_n] be a polynomial algebra, and , for n2. Let σ^′Aut_K(P_n) be given by

It is proved that the algebra of invariants, , is a polynomial algebra in n−1 variables which is generated by quadratic and cubic (free) generators that are given explicitly.Let σAut_K(P_n) be given by

It is well known that the algebra of invariants, , is finitely generated (theorem of Weitzenböck [R. Weitzenböck, Über die invarianten Gruppen, Acta Math. 58 (1932) 453–494]), has transcendence degree n−1, and that one can give an explicit transcendence basis in which the elements have degrees 1,2,3,…,n−1. However, it is an old open problem to find explicit generators for F_n. We find an explicit vector space basis for the quadratic invariants, and prove that the algebra of invariants is a polynomial algebra over in n−2 variables which is generated by quadratic and cubic (free) generators that are given explicitly.The coefficients of these quadratic and cubic invariants throw light on the ‘unpredictable combinatorics’ of invariants of affine automorphisms and of SL₂-invariants.     

On a class of degenerate and singular elliptic systems in bounded domains

This paper deals with the nonexistence and multiplicity of nonnegative, nontrivial solutions to a class of degenerate and singular elliptic systems of the form

where Ω is a bounded domain with smooth boundary ∂Ω in , N2, and , , h_i (i=1,2) are allowed to have “essential” zeroes at some points in Ω, (F_u,F_v)=F, and λ is a positive parameter. Our proofs rely essentially on the critical point theory tools combined with a variant of the Caffarelli–Kohn–Nirenberg inequality in [P. Caldiroli, R. Musina, On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Equations Appl. 7 (2000) 189–199].     

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.     

The Fréchet and limiting subdifferentials of integral functionals on the spaces

A new approach to computing the Fréchet subdifferential and the limiting subdifferential of integral functionals is proposed. Thanks to this way, we obtain formulae for computing the Fréchet and limiting subdifferentials of the integral functional , uL₁(Ω,E). Here is a measured space with an atomless σ-finite complete positive measure, E is a separable Banach space, and . Under some assumptions, it turns out that these subdifferentials coincide with the Fenchel subdifferential of F.     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,