The existence of eigenvalues for operators acting in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We present conditions that allow us to prove the existence of eigenvalues and characteristic values for operator F(D) ? C(λ): L ²(R ^m ) → L ²(R ^m ), where F(D) is a pseudo-differential operator with a symbol F(iξ) and C(λ): L ²(R ^m ) → L ²(R ^m ) is a linear continuous operator.     

<Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">c</Emphasis> Queue with catastrophes and state-dependent control at idle time

We consider an M ^X /M/c queue with catastrophes and state-dependent control at idle time. Properties of the queues which terminate when the servers become idle are first studied. Recurrence, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no catastrophes. All of these properties and the first effective catastrophe occurrence time are then investigated for the case of resurrection and catastrophes. In particular, we obtain the Laplace transform of the transition probability for the absorbing M ^X /M/c queue.     

Processes of <Emphasis Type="Italic">r</Emphasis><Superscript><Emphasis Type="Italic">t</Emphasis><Emphasis Type="Italic">h</Emphasis></Superscript> largest

For integers n ≥ r, we treat the rth largest of a sample of size n as an \(\mathbb {R}^{\infty }\)-valued stochastic process in r which we denote as M^(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M^(r) as r → ∞, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of M^(r) and M^(r) itself, after norming and centering. In continuous time, an analogous process Y^(r) based on a two-dimensional Poisson process on \(\mathbb {R}_{+}\times \mathbb {R}\) is treated similarly, but we note that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the rth highest point up to time t for any t >?0. This necessitates a different approach to the asymptotics in this case.     

The Center of <Emphasis Type="Italic">Dist</Emphasis> (<Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">m</Emphasis>|<Emphasis Type="Italic">n</Emphasis>)) in Positive Characteristic

The purpose of this paper is to investigate central elements in distribution algebras D i s t(G) of general linear supergroups G = G L(m|n). As an application, we compute explicitly the center of D i s t(G L(1|1)) and its image under Harish-Chandra homomorphism.     

Periods and nonvanishing of central <Emphasis Type="Italic">L</Emphasis>-values for GL(2<Emphasis Type="Italic">n</Emphasis>)

Let π be a cuspidal automorphic representation of PGL(2n) over a number field F, and η the quadratic idèle class character attached to a quadratic extension E/F. Guo and Jacquet conjectured a relation between the nonvanishing of L(1/2, π)L(1/2, π ? η) for π of symplectic type and the nonvanishing of certain GL(n,E) periods. When n = 1, this specializes to a well-known result of Waldspurger. We prove this conjecture, and related global results, under some local hypotheses using a simple relative trace formula.We then apply these global results to obtain local results on distinguished supercuspidal representations, which partially establish a conjecture of Prasad and Takloo-Bighash.     

Local <Emphasis Type="Italic">Tb</Emphasis> theorem with <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> testing conditions and general measures: square functions

Local Tb theorems with L ^p type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. In the non-homogeneous world local Tb theorems have only been proved assuming scale invariant (L ^∞ or BMO) testing conditions. In this paper, for the first time, we overcome these obstacles in the non-homogeneous world, and prove a nonhomogeneous local Tb theorem with L ² type testing conditions. This paper is in the setting of the vertical and conical square functions defined using general measures and kernels. On the technique side, we demonstrate a trick of inserting Calderón–Zygmund stopping data of a fixed function into the construction of the twisted martingale difference operators. This built-in control of averages is an alternative to Carleson embedding.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-norm of partial sums of Fourier-Legendre series

Asymptotic estimates of L ^q -norms of partial sums of Fourier-Legendre series of functions from L ^p [?1; 1] are obtained.     

The strong <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-<Emphasis Type="Italic">greedy</Emphasis> property of the Walsh system

For any 0 < ? < 1 one can find a measurable set E ? [0, 1] with the measure |E| > 1 ? ? such that for each function f(x) ε L ¹ (0, 1) a function g(x) ε L ¹ (0, 1) exists such that it coincides with f (x) on E, its Fourier—Walsh series converges to it in the metric of L ¹ (0, 1), and all nonzero terms of the sequence of Fourier coefficients of the new function obtained by the Walsh system have the modulo decreasing order; consequently, the greedy algorithm for this function converges to it in the L ¹ (0, 1)-norm.     

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">P</Emphasis></Superscript>

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L² spaces and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of L ^p spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator to prove that they have a bounded holomorphic functional calculus in those L ^p spaces. We also obtain functional calculus results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining L ^p results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and L ^p bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L² extends to L ^p for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L² proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces and about the relationship between conical and vertical square functions.     

The <Emphasis Type="Italic">p</Emphasis>-Adic order of the <Emphasis Type="Italic">k</Emphasis>-Fibonacci and <Emphasis Type="Italic">k</Emphasis>-Lucas numbers

Let (F _k,n ) _n and (L _k,n )n be the k-Fibonacci and k-Lucas sequence, respectively, which satisfies the same recursive relation a _n+1 = ka _n + a _n?1 with initial values F _k,0 = 0, F _k,1 = 1, L _k,0 = 2 and L _k,1 = k. In this paper, we characterize the p-adic orders ν _p (F _k,n ) and ν _p (L _k,n ) for all primes p and all positive integers k.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-consistency of wavelet estimators

This paper deals with the L ^p -consistency of wavelet estimators for a density function based on size-biased random samples. More precisely, we firstly show the L ^p -consistency of wavelet estimators for independent and identically distributed random vectors in R ^d . Then a similar result is obtained for negatively associated samples under the additional assumptions d = 1 and the monotonicity of the weight function.     

The dual space of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of a vector measure

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L ^p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E _q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L ^p (ν) ^×  = E _q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L ¹ (ν) ^×  may be equal or not to E _∞ (μ).     

Largest <Emphasis Type="Italic">H</Emphasis>-eigenvalue of uniform <Emphasis Type="Italic">s</Emphasis>-hypertrees

The k-uniform s-hypertree G = (V,E) is an s-hypergraph, where 1 ≤ s ≤ k - 1; and there exists a host tree T with vertex set V such that each edge of G induces a connected subtree of T. In this paper, some properties of uniform s-hypertrees are establised, as well as the upper and lower bounds on the largest H-eigenvalue of the adjacency tensor of k-uniform s-hypertrees in terms of the maximal degree Δ. Moreover, we also show that the gap between the maximum and the minimum values of the largest H-eigenvalue of k-uniform s-hypertrees is just Θ(Δ ^s/k ).     

On the Dunford Property (<Emphasis Type="Italic">C</Emphasis>) for Bounded Linear Operators <Emphasis Type="Italic">RS</Emphasis> and <Emphasis Type="Italic">SR</Emphasis>

In this paper we show that if \({S\in L(X,Y)}\) and \({R\in L(Y,X),}\) X and Y complex Banach spaces, then the products RS and SR share the Dunford property (C). We also study property (C) for R, S, RS and \({SR \in L(X)}\) in the case that R and S satisfies the operator equations RSR = R ² and SRS = S ².     

On Shalika models and <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions

We use modular symbols to construct p-adic L-functions for cohomological cuspidal automorphic representations on GL(2n), which admit a Shalika model. Our construction differs from former ones in that it systematically makes use of the representation theory of p-adic groups.     

Intersections of Primary Subgroups in Nonsoluble Finite Groups Isomorphic to <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>)

Given a finite group G with socle isomorphic to L _n (2 ^m ), we describe (up to conjugacy) all ordered pairs of primary subgroups A and B in G such that A ∩ B ^g ≠ 1 for all g ∈ g.     

Note edge-colourings of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n,n</Emphasis></Subscript> with no long two-coloured cycles

Consider the set of all proper edge-colourings of a graph G with n colours. Among all such colourings, the minimum length of a longest two-coloured cycle is denoted L(n, G). The problem of understanding L(n, G) was posed by Häggkvist in 1978 and, specifically, L(n, K _n,n ) has received recent attention. Here we construct, for each prime power q ≥ 8, an edge-colouring of K _n,n with n colours having all two-coloured cycles of length ≤ 2q ², for integers n in a set of density 1 ? 3/(q ? 1). One consequence is that L(n, K _n,n ) is bounded above by a polylogarithmic function of n, whereas the best known general upper bound was previously 2n ? 4.     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.     

<Emphasis Type="Italic">C</Emphasis>*-simplicity of <Emphasis Type="Italic">n</Emphasis>-periodic products

The C*-simplicity of n-periodic products is proved for a large class of groups. In particular, the n-periodic products of any finite or cyclic groups (including the free Burnside groups) are C*-simple. Continuum-many nonisomorphic 3-generated nonsimple C*-simple groups are constructed in each of which the identity xⁿ = 1 holds, where n ≥ 1003 is any odd number. The problem of the existence of C*-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.     

<Emphasis Type="Italic">M</Emphasis>-slenderness

Analogues of Nunke’s theorem are proved which characterize variants of slenderness. For a bounded monotone subgroup M of ? ^ω , a torsion-free reduced abelian group G is M-slender if, and only if, there is no monomorphism from M into G. It is consistent relative to ordinary set theory (ZFC) that if M ≠ ? ^ω is an unbounded monotone subgroup of ? ^ω , then a torsion-free reduced abelian group G is M-slender if, and only if, there is no monomorphism from M into G.