The Duality Property of Positive R-Compact Operators

We study the duality of r-compact operator. We establish if an operator ${T:E\rightarrow F}$ is r-compact, then its adjoint ${T^{\prime}: F^{\prime }\rightarrow E^{\prime }}$ is also r-compact. We also provide some sufficient condition on the pair of Banach lattices E and F which guarantees that a regular operator ${T:E\rightarrow F}$ such that ${T^{\prime }:F^{\prime }\rightarrow E^{\prime }}$ is r-compact, must itself be r-compact.     

Continuity of solutions of a problem in the calculus of variations

We study the problem of minimizing ${\int_{\Omega} L(x,u(x),Du(x))\,{\rm d}x}$ over the functions ${u\in W^{1,p}(\Omega)}$ that assume given boundary values ${\phi}$ on ???. We assume that L(x, u, Du)?=?F(Du)?+?G(x, u) and that F is convex. We prove that if ${\phi}$ is continuous and ?? is convex, then any minimum u is continuous on the closure of ??. When ?? is not convex, the result holds true if F(Du)?=?f(|Du|). Moreover, if ${\phi}$ is Lipschitz continuous, then u is H?lder continuous.     

The Lower Weyl Spectrum of a Positive Operator

For the lower Weyl spectrum $$\sigma_{\rm w}^-(T) = \bigcap_{0 \le K \in \mathcal{K}(E) \le T} \sigma(T - K),$$ where T is a positive operator on a Banach lattice E, the conditions for which the equality ${\sigma_{\rm w}^-(T) = \sigma_{\rm w}^-(T^*)}$ holds, are established. In particular, it is true if E has order continuous norm. An example of a weakly compact positive operator T on ? _∞ such that the spectral radius ${r(T) \in \sigma_{\rm w}^-(T) {\setminus} (\sigma_{\rm f}(T) \cup \sigma_{\rm w}^-(T^*))}$ , where σ _f(T) is the Fredholm spectrum, is given. The conditions which guarantee the order continuity of the residue T _?1 of the resolvent R(., T) of an order continuous operator T ≥ 0 at ${r(T) \notin \sigma_{\rm f}(T)}$ , are discussed. For example, it is true if T is o-weakly compact. It follows from the proven results that a Banach lattice E admitting an order continuous operator T ≥ 0, ${r(T) \notin \sigma_{\rm f}(T)}$ , can not have the trivial band ${E_n^\sim}$ of order continuous functionals in general. It is obtained that a non-zero order continuous operator T : E → F can not be approximated in the r-norm by the operators from ${E_\sigma^\sim \otimes F}$ , where F is a Banach lattice, ${E_\sigma^\sim}$ is a disjoint complement of the band ${E_n^\sim}$ of E*.     

E-local pseudovarieties

Generalizing a property of the pseudovariety of all aperiodic semigroups observed by Tilson, we call E -local a pseudovariety V which satisfies the following property: for a finite semigroup, the subsemigroup generated by its idempotents belongs to V if and only if so do the subsemigroups generated by the idempotents in each of its regular $\mathcal{D}$ -classes. In this paper, we present several sufficient or necessary conditions for a pseudovariety to be E-local or for a pseudoidentity to define an E-local pseudovariety. We also determine several examples of the smallest E-local pseudovariety containing a given pseudovariety.     

Spaces of compact operators and their dual spaces

LetE andF be reflexive Banach spaces andC the space of all compact linear operators fromE toF. A representation of the dual space ofC is given and it is proved thatC is either reflexive or nonconjugate. Applications of these results are also given.     

Estimates of the Number of Eigenvalues of Self-Adjoint Operator Functions

We consider an operator function F defined on the interval $\user2{[}\sigma \user2{,}\tau \user2{]} \subset \mathbb{R}$ whose values are semibounded self-adjoint operators in the Hilbert space $\mathfrak{H}$ . To the operator function F we assign quantities $\mathcal{N}_\user1{F}$ and ν _F (λ) that are, respectively, the number of eigenvalues of the operator function F on the half-interval [σ,τ) and the number of negative eigenvalues of the operator F(λ) for an arbitrary λ ∈ [σ,τ]. We present conditions under which the estimate $\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}$ holds. We also establish conditions for the relation $\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}$ to hold. The results obtained are applied to ordinary differential operator functions on a finite interval.     

Sine, cosine transforms and classical function classes

We study the continuity and smoothness properties of functions f ∈ L ¹([0, ∞)) whose sine transforms $ \hat f_s $ and cosine tranforms $ \hat f_c $ belong to L ¹([0,∞)). We give best possible sufficient conditions in terms of $ \hat f_s $ and $ \hat f_c $ to ensure that f belongs to one of the Lipschitz classes Lip α and lip α for some 0 < α ≤ 1, or to one of the Zygmund classes Zyg α and zyg α for some 0 < α ≤ 2. The conditions given by us are not only sufficient, but also necessary in the case when the sine and cosine transforms are nonnegative. Our theorems are extensions of the corresponding theorems by Boas from sine and cosine series to sine and cosine transforms.     

On restricted representations of the extended special type Lie superalgebra {\bar{S}(m, n, 1)}

Let F be an algebraically closed field of prime characteristic p > 2, and let ${\mathfrak{g}=\bar{S}(m, n, {\bf 1})}$ be the extended special type Lie superalgebra over F. Simple restricted ${\mathfrak{g}}$ -modules are classified. Moreover, a sufficient and necessary condition is provided for restricted baby Kac modules to be simple.     

A characterization of the Wishart exponential families by an invariance property

E is the space of real symmetric (d, d) matrices, andS and \(\bar S\) are the subsets ofE of positive definite and semipositive-definite matrices. Let there be ap in $$\Lambda = \left\{ {\frac{1}{2},1,\frac{3}{2}, \ldots \frac{{d - 1}}{2}} \right\} \cup \left] {\frac{{d - 1}}{2}, + \infty } \right[$$ The Wishart natural exponential family with parameterp is a set of probability distributions on \(\bar S\) defined by $$F_p = \{ \exp [ - \tfrac{1}{2}Tr(\Gamma x)](det\Gamma )^p \mu _p (dx);\Gamma \in S\} $$ where μ_p is a suitable measure on \(\bar S\) . LetGL(?^d) be the subset ofE of invertible matrices. Fora inGL(?^d), define the automorphismg _a ofE byg _a(x)=^t axa, where ^t a is the transpose ofa. The aim of this paper is to show that a natural exponential familyF onE is invariant byg _a for alla inGL(?^d) if and only if there existsp in Λ such that eitherF=F _p, orF is the image ofF _p byx??x. (Theorem).     

Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

We prove that if ${U\subset \mathbb {R}^n}$ is an open domain whose closure ${\overline U}$ is compact in the path metric, and F is a Lipschitz function on ?U, then for each ${\beta \in \mathbb {R}}$ there exists a unique viscosity solution to the β-biased infinity Laplacian equation $$\beta |\nabla u| + \Delta_\infty u=0$$ on U that extends F, where ${\Delta_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}}$ . In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased ${\epsilon}$ -game as follows. The starting position is ${x_0 \in U}$ . At the kth step the two players toss a suitably biased coin (in our key example, player I wins with odds of ${\exp(\beta\epsilon)}$ to 1), and the winner chooses x _k with ${d(x_k,x_{k-1}) < \epsilon}$ . The game ends when ${x_k \in \partial U}$ , and player II pays the amount F(x _k ) to player I. We prove that the value ${u^{\epsilon}(x_0)}$ of this game exists, and that ${\|u^\epsilon - u\|_\infty \to 0}$ as ${\epsilon \to 0}$ , where u is the unique extension of F to ${\overline{U}}$ that satisfies comparison with β-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation.     

Porous surfaces

In fractal modeling, porous surfaces in the plane are usually described as the residual setE of a packing by connected open domains \(C_n\) . In the case whereE is nonempty, we investigate the relationships between the dimensionality ofE and the geometry of the complementary sets \(C_n\) . If they satisfy suitable regularity conditions, then the Bouligand dimension ofE is equal to the exponent of convergence of the series ∑(diam \(C_n\) ) ^α . We give here general conditions to obtain this equality, together with numerous examples and possible ways of developing this theory.     

The tail behaviour of a random sum of subexponential random variables and vectors

Let $\left\{ X,X_{i},i=1,2,...\right\} $ denote independent positive random variables having common distribution function (d.f.) F(x) and, independent of X, let ν denote an integer valued random variable. Using X ₀=0, the random sum Z=∑ _i=0 ^ν X _i has d.f. $G(x)=\sum_{n=0}^{\infty }\Pr\{\nu =n\}F^{n\ast }(x)$ where F ^n?(x) denotes the n-fold convolution of F with itself. If F is subexponential, Kesten’s bound states that for each ε>0 we can find a constant K such that the inequality $$ 1-F^{n\ast }(x)\leq K(1+\varepsilon )^{n}(1-F(x))\, , \qquad n\geq 1,x\geq 0 \, , $$ holds. When F is subexponential and E(1 +ε) ^ν <∞, it is a standard result in risk theory that G(x) satisfies $$ 1 - G{\left( x \right)} \sim E{\left( \nu \right)}{\left( {1 - F{\left( x \right)}} \right)},\,\,x \to \infty \,\,{\left( * \right)} $$ In this paper, we show that (*) holds under weaker assumptions on ν and under stronger conditions on F. Stam (Adv. Appl. Prob. 5:308–327, 1973) considered the case where $ \overline{F}(x)=1-F(x)$ is regularly varying with index –α. He proved that if α>1 and $E{\left( {\nu ^{{\alpha + \varepsilon }} } \right)} < \infty $ , then relation (*) holds. For 0<α<1, it is sufficient that Eν<∞. In this paper we consider the case where $\overline{F}(x)$ is an O-regularly varying subexponential function. If the lower Matuszewska index $\beta (\overline{F})<-1$ , then the condition ${\text{E}}{\left( {\nu ^{{{\left| {\beta {\left( {\overline{F} } \right)}} \right|} + 1 + \varepsilon }} } \right)} < \infty$ is sufficient for (*). If $\beta (\overline{F} )>-1$ , then again Eν<∞ is sufficient. The proofs of the results rely on deriving bounds for the ratio $\overline{F^{n\ast }}(x)/\overline{F} (x)$ . In the paper, we also consider (*) in the special case where X is a positive stable random variable or has a compound Poisson distribution derived from such a random variable and, in this case, we show that for n≥2, the ratio $\overline{F^{n\ast }}(x)/\overline{F}(x)\uparrow n$ as x↑∞. In Section 3 of the paper, we briefly discuss an extension of Kesten’s inequality. In the final section of the paper, we discuss a multivariate analogue of (*).     

Remarks on the descriptive metric characterization of singular sets of analytic functions

This work presents two remarks on the structure of singular boundary sets of functions analytic in the unit diskD: ¦z¦<1. The first remark concerns the conversion of the Plessner theorem. We prove that three pairwise disjoint subsetsE ₁,E ₂, andE ₃ of the unit circle Γ: ¦z¦=1, $ \cup _{i = 1}^3 E_i$ = Γ, are the setsI(?) of all Plessner points,F(?) of all Fatou points, andE(?) of all exceptional boundary points, respectively, for a function ? holomorphic inD if and only ifE ₁ is aG _δ-set andE ₃ is a $G_{\delta \sigma }$ -set of linear measure zero. In the second part of the paper it is shown that for any $G_{\delta \sigma }$ -subsetE of the unit circle Γ with a zero logarithmic capacity there exists a one-sheeted function onD whose angular limits do not exist at the points ofE and do exist at all the other points of Γ.     

Closure of some heavy-tailed distribution classes under random convolution*

In this paper, we consider the closure property of a random convolution $ \sum\nolimits_{n = 0}^\infty {{p_n}{F^*}^n} $ , where F is a heavy-tailed distribution on [0, ??), and p _n (n?=?0, 1, . . . ) are the local probabilities of a nonnegative integer-valued random variable. We obtain conditions under which the fact that distribution F belongs to the dominatedly varying-tailed class, long-tailed class, or to the intersection of these classes implies that $ \sum\nolimits_{n = 0}^\infty {{p_n}{F^*}^n} $ is in the same class.     

Faserungen,die aus Reguli mit gemeinsamer Berührprojektivität längs einer gemeinsamen Erzeugenden zusammengesetzt sind

Let Π_PP be a pappian projective 3-space and ? be a set of lines of Π_PP; we define:

u

a line g of Π_PP has the property R with respect to ?, if all lines of ? meeting g form a regulus

? has the property E ₃, if there exists a pencil \(\mathfrak{L}_0 \) of lines such that one line z of \(\mathfrak{L}_0 \) belongs to ? and all lines of { \(\mathfrak{L}_0 \backslash \left\{ z \right\}\) have the property R with respect to ?.

A spread with the property E ₃ (abbreviated E ₃-spread) is built up of reguli which have one line in common and the same tangent projectivity along their common line. We point out a method of constructing an E ₃-spread of Π_PP. This construction is applied to the real 3-space ?³ to generalize a result of D. Betten [2, S.327] and to prove that another result of D. Betten [3, S.140, Bsp. 2] yields E ₃-spreads. For each natural number n (∈?) we specify two E ₃-spreads \(\mathfrak{F}_n \) and \(\mathfrak{S}_n \) of ?³ such that two different elements of \(F: = \left\{ {\mathfrak{F}_n |n \in \mathbb{N}} \right\} \cup \left\{ {\mathfrak{S}_n |n \in \mathbb{N}} \right\}\) are not equivalent with respect to the collineation group of ?³ apart from \(\mathfrak{F}_1 \) each spread of F represents a 4-dimensional translation plane with a 6-dimensional collineation group. Finally, the properties R and E ₃ are used to characterize the elliptic linear line congruences of a pappian 3-space.     

On σ(·, W)-Holomorphic Functions and Theorems of Vitali-Type

The aim of this paper is to give some criterions for holomorphy of F-valued σ(F, W)-holomorphic functions which are bounded on bounded sets in a domain D of Fréchet spaces E (resp. ${\mathbb{C}^n}$ ) where ${W \subset F'}$ defines the topology of Fréchet space F. Base on these results we consider the problem on holomorphic extension of F-valued σ(F, W)-holomorphic functions from non-rare subsets of D and from subsets of D which determines uniform convergence in H(D). As an application of the above, some theorems of Vitali-type for a locally bounded sequence ${\{f_i\}_{i \in \mathbb{N}}}$ of Fréchet-valued holomorphic functions are also proved.     

On the nullity of conformal Killing graphs in foliated Riemannian spaces

We deal with entire conformal Killing graphs, that is, graphs constructed through the flow generated by a complete conformal Killing vector field V on a Riemannian space \({\overline{M}}\) , and which are defined over an integral leaf of the foliation \({V^\bot {\rm of} \overline{M}}\) orthogonal to V. Under a suitable restriction on the norm of the gradient of the function z which determines such a graph Σ(z), we establish sufficient conditions to ensure that Σ(z) is totally geodesic. Afterwards, when the ambient space \({\overline{M}}\) has constant sectional curvature, we obtain lower estimates for the index of minimum relative nullity of Σ(z).     

Asymptotic behavior of product of two heavy-tailed dependent random variables

Let X and Y be positive weakly negatively dependent(WND)random variables with finite expectations and continuous distribution functions F and G with heavy tails, respectively. The asymptotic behavior of the tail of distribution of XY is studied and some closure properties under some suitable conditions on ˉ F(x)= 1 F(x)and ˉ G(x)= 1 G(x)are provided. Moreover, subexponentiality of XY when X and Y are WND random variables is derived.