The Bergman projection as a singular integral operator

We show that the Bergman projection operator, associated to one of three classes of domains (all smoothly bounded)-a finite type domain ?²; a decoupled, finite type domain in ?ⁿ; or a convex, finite type domain in wfⁿ-may be viewed as a generalized Calderón-Zygmund operator. As an application of this observation, we show that the Bergman projector on any of these domains preserves the Lebesgue classesL ^p , 1 <p < ∞.     

Relationship between the posterior distributions of two regular Bayesian experiments related to a fixed family of sampling distributions

Let us consider twoRegular Bayesian Experiments (see [1]) related to a fixed family of sampling distributions in which the parameter space and the sample space are assumed to be Polish Spaces. In this paper we shall study the relationship between the posterior distributions of these two Bayesian Experiments considering all the different cases concerning the Lebesgue decomposition of the second prior distribution w.r.t. the first one.     

KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

We prove the existence of small amplitude quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of the linear Airy equation. For Hamiltonian or reversible nonlinearities we also prove their linear stability. The key analysis concerns the reducibility of the linearized operator at an approximate solution, which provides a sharp asymptotic expansion of its eigenvalues. For quasi-linear perturbations this cannot be directly obtained by a KAM iteration. Hence we first perform a regularization procedure, which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. At this point we implement a Nash–Moser iteration (with second order Melnikov non-resonance conditions) which completes the reduction to constant coefficients.     

Anti-mitre steiner triple systems

A mitre in a Steiner triple system is a set of five triples on seven points, in which two are disjoint. Recursive constructions for Steiner triple systems containing no mitre are developed, leading to such anti-mitre systems for at least 9/16 of the admissible orders. Computational results for small cyclic Steiner triple systems are also included.     

Implicative twist-structures

The twist-structure construction is used to represent algebras related to non-classical logics (e.g., Nelson algebras, bilattices) as a special kind of power of better-known algebraic structures (distributive lattices, Heyting algebras). We study a specific type of twist-structure (called implicative twist-structure) obtained as a power of a generalized Boolean algebra, focusing on the implication-negation fragment of the usual algebraic language of twist-structures. We prove that implicative twist-structures form a variety which is semisimple, congruence-distributive, finitely generated, and has equationally definable principal congruences. We characterize the congruences of each algebra in the variety in terms of the congruences of the associated generalized Boolean algebra. We classify and axiomatize the subvarieties of implicative twist-structures. We define a corresponding logic and prove that it is algebraizable with respect to our variety.     

Different capacities of a digraph

Sperner product is the natural generalization of co-normal product to digraphs. For every class of digraphs closed under Sperner product, the cardinality of the largest subgraph from the given class, contained as an induced subgraph in the co-normal powers of a graphG, has an exponential growth. The corresponding asymptotic exponent is the capacity ofG with respect to said class of digraphs. We derive upper and lower bounds for these capacities for various classes of digraphs, and analyze the conditions under which they are tight.     

Unslotted CSMA/CD protocol with the threshold control policy

We consider a single channelCSM A/CD system withD homogeneous stations and impeded buffer of infinite size. We find a sufficient condition for the model to be stable under the threshold control policy and derive the limiting distribution of the number of messages in the system at the moment of service completion. We also derive the limiting distribution of the number of messages in the system size at arbitrary time by using Markov regenerative processes. Some numerical examples and special cases are also treated.     

On geometric representations of modular ortholattices

A pre-orthogonality on a projective geometry is a symmetric binary relation, ⊥, such that for each point ${p, p^{\perp} = \{q | p \perp q \}}$ is a subspace. An orthogonality is a pre-orthogonality such that each p ^⊥ is a hyperplane. Such ⊥ is called anisotropic iff it is irreflexive. For projective geometries with an anisotropic pre-orthogonality, we show how to find a (large) projective subgeometry with a natural embedding for the lattices of subspaces and with an orthogonality induced by the given pre-orthogonality. We also discuss (faithful) representations of modular ortholattices within this context and derive a condition which allows us to transform a representation by means of an anisotropic pre-orthogonality into an anisotropic orthogeometry by means of an anisotropic orthogonality.     

On multiplicities for SL(n)

We use the strong Artin conjecture for Galois extensions of Heisenberg type to show that a cuspidal automorphic representation of SL(N)/F, forF a number field andN>2, can occur with multiplicity greater than one. We also exhibit two cuspidalL-packets (forF=Q andN prime) which are locally isomorphic for primesp different fromN, but which are disjoint atN, i.e. thatL-packets are not rigid.     

Binomial convolutions and hypergeometric identities

By means of formal power series calculus, some new recurrences on the generating functions for the generalized Abel and Gould coefficients are derived from the Gould's work (1956–1961), which yield equivalently several convolution formulas of binomial coefficients. Alternatively, some of these can be verified either through a pair of relations due to Gould and Hsu (1973), from which some strange hypergeometric evaluations including one of Gessel and Stanton (1982) may be produced mechanically. By associating the binomial convolutions investigated in this paper with Whipple's transform (1926) on very well-poised series, a new family of the₇ F ₆-hypergeometric identities are established in a unified way.     

Nonexistence and symmetry theorems for elliptic systems in R N

In this paper we study Liuoville type results for the system of elliptic equations $$ - \Delta u = F(u) u \geqslant 0, u \ne 0 in R^N ,$$ when the function $F:\bar R_ + ^m \to \bar R_ + ^m $ is regular, quasi-monotone, fully coupled and has suberitical growth. WhenF has a radial dependence on the space variable we also prove symmetry results. The nonexistence theorems of the nature studied here have consequences on existence results on bounded domains, through the blow-up technique and degree theory. Our results are proved using the moving planes method.     

Linear connections for reproducing kernels on vector bundles

We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism of the input kernel. The aforementioned correspondence turns out to be a canonical functor between categories of kernels and linear connections. A number of examples of linear connections including the ones associated to classical kernels, homogeneous reproducing kernels and kernels occurring in the dilation theory for completely positive maps are given, together with their covariant derivatives.     

On the growth of torsion in the cohomology of arithmetic groups

In this paper we consider certain families of arithmetic subgroups of $\mathrm{SO }^0(p,q)$ and $\mathrm{SL }_3(\mathbb {R})$ , respectively. We study the cohomology of such arithmetic groups with coefficients in arithmetically defined modules. We show that for natural sequences of such modules the torsion in the cohomology grows exponentially.     

On cohomology of finitely generated function groups

Let Γ be a non-elementary finitely generated Kleinian group with region of discontinuity Ω. Letq be an integer,q ≥ 2. The group Λ acts on the right on the vector space Π_2q?2 of polynomials of degree less than or equal to 2q ? 2 via Eichler action. We note by A_qq(Ω, Λ) the space of cusp forms for Λ of weight (?2q) and the space of parabolic cohomology classes by PH¹ (Λ, Π2q?2). Bers introduced an anti-linear map $$\beta _q^* :A^q \left( {\Omega ,\Gamma } \right) - - - \to PH^1 \left( {\Gamma ,\Omega _{2q - 2} } \right)$$ . We try to determine the class of Kleinian groups for which the Bers map is surjective. We show that the Bers map is surjective for geometrically finite function groups. We also obtain a characterization of geometrically finite function groups. As an application, we reprove theorems of Maskit on inequalities involving the dimension of the space of cusp forms supported on an invariant component and the dimension of the space of cusp forms supported on the other components for finitely generated function groups. We also show all these inequalities are equalities for geometrically finiteB-groups.     

Generic non-selfadjoint Zakharov–Shabat operators

In this paper we develop tools to study within a family of non-selfadjoint operators $L(\varphi )$ depending on a parameter $\varphi $ in a real Hilbert space, those with (partially) simple spectrum. As a case study we consider the Zakharov–Shabat operators $L(\varphi )$ appearing in the Lax pair of the focusing NLS on the circle. In particular, the main result implies that the set of potentials $\varphi $ of Sobolev class $H^N$ , $N\ge 0$ , so that all non real eigenvalues of $L(\varphi )$ are simple, is path connected and dense.     

Arithmetical progressions formed byk different Lehmer pseudoprimes

LetU _n=(αⁿ-β²)/(α-β) forn odd andU _n=(αⁿ-βⁿ)/(α²-β²) for evenn, where α and β are distinct roots of the trinomialf(z)=z ²-√Lz+Q andL>0 andQ are rational integers.U _n is then-th Lehmer number connected withf(z). A compositen is a Lehmer pseudoprime for the bases α and β ifU _n??(n)≡0 (modn), where?(n)=(LD/n) is the Jacobi symbol. IfD=L?4Q>0, U _n denotesn-th Lehmer number,p>3 and 2p?1 are primes,p(2p-1)+(α²-β²)², (α^2p-1±β^2p-1)/(α±β) are composite then the numbers (α^2p-1+β^2p-1)/(α+β), (α^2p-β^2p)/(α²-β²), (α^2p-1-β^2p-1)/(α-β) are lehmer pseudoprimes for the bases α and β and form an arithmetical progression. IfD>0 then from hypothesisH of A. Schinzel on polynomials it follows that for every positive integerk there exists infinitely many arithmetic progressions formed fromk different Lehmer pseudoprimes for the bases α and β.     

Domination problem for AM-compact abstract Uryson operators

The Boolean algebra of fragments of a positive abstract Uryson operator recently was described in M. Pliev (Positivity, doi:10.1007/s11117-016-0401-9, 2016). Using this result, we prove a theorem of domination for AM-compact positive abstract Uryson operators from a Dedekind complete vector lattice E to a Banach lattice F with an order continuous norm.     

Asymptotic normal distribution of multidimensional statistics of dependent random variables

A central limit theorem for multidimensional processes in the sense of [9], [10] is proved. In particular the asymptotic normal distribution of a sum of dependent random functions of m variables defined on the positive part of the integral lattice is established by the method of moments. The results obtained can be used, for example, in proving the asymptotic normality of different statistics of n₀-dependent random variables as well as to determine the asymptotic behaviour of the resultant of reflected waves of telluric type.     

Conditioned rates of convergence in the CLT for sums and maximum sums

An interesting recent result of Landers and Roggé (1977, Ann. Probability5, 595–600) is investigated further. Rates of convergence in the conditioned central limit theorem are developed for partial sums and maximum partial sums, with positive mean and zero mean separately, of sequences of independent identically distributed random variables. As corollaries we obtain a conditioned central limit theorem for maximum partial sums both for positive and zero mean cases.