Intervals of 1-Lipschitz aggregation operators, quasi-copulas, and copulas with given affine section

Best lower and upper bounds for 1-Lipschitz aggregation operators with a given affine section are given. These are used to determine best bounds for quasi-copulas and copulas with a given affine section. However, in general there is no greatest copula with a given non-decreasing affine section. These results are used to study (quasi-)copulas with arbitrary affine sections.     

The central limit theorem for Markov chains with normal transition operators, started at a point

The central limit theorem and the invariance principle, proved by Kipnis and Varadhan for reversible stationary ergodic Markov chains with respect to the stationary law, are established with respect to the law of the chain started at a fixed point, almost surely, under a slight reinforcing of their spectral assumption. The result is valid also for stationary ergodic chains whose transition operator is normal. Received: 28 March 2000 / Revised version: 25 July 2000 /?Published online: 15 February 2001     

Symmetric Graphs and Flag Graphs

 With any G-symmetric graph Γ admitting a nontrivial G-invariant partition , we may associate a natural “cross-sectional” geometry, namely the 1-design in which for and if and only if α is adjacent to at least one vertex in C, where and is the neighbourhood of B in the quotient graph of Γ with respect to . In a vast number of cases, the dual 1-design of contains no repeated blocks, that is, distinct vertices of B are incident in with distinct subsets of blocks of . The purpose of this paper is to give a general construction of such graphs, and then prove that it produces all of them. In particular, we show that such graphs can be reconstructed from and the induced action of G on . The construction reveals a close connection between such graphs and certain G-point-transitive and G-block-transitive 1-designs. By using this construction we give a characterization of G-symmetric graphs such that there is at most one edge between any two blocks of . This leads to, in a subsequent paper, a construction of G-symmetric graphs such that and each is incident in with vertices of B. The work was supported by a discovery-project grant from the Australian Research Council. Received April 24, 2001; in revised form October 9, 2002 Published online May 9, 2003     

Operator-semistable,operator semi-selfdecomposable probability measures and related nested classes on <Emphasis Type="Italic">p</Emphasis>-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

Classification of locally homogeneous affine connections with arbitrary torsion on 2-dimensional manifolds

The aim of this paper is to classify (locally) all locally homogeneous affine connections with arbitrary torsion on two-dimensional manifolds. Herewith, we generalize the result given by B. Opozda for torsion-less case in [(2004) Classification of locally homogeneous connections on 2-dimensional manifolds. Diff Geom Appl 21: 173–198]. Authors’ addresses: Teresa Arias-Marco, Department of Geometry and Topology, University of Valencia, Vicente Andrés Estellés 1, 46100 Burjassot, Valencia, Spain; Oldřich Kowalski, Faculty of Mathematics and Physics of the Charles University, Sokolovská 83, 18600 Praha 8, Czech Republic     

Multivariate copulas, quasi-copulas and lattices

We investigate some properties of the partially ordered sets of multivariate copulas and quasi-copulas. Whereas the set of bivariate quasi-copulas is a complete lattice, which is order-isomorphic to the Dedekind-MacNeille completion of the set of bivariate copulas, we show that this is not the case in higher dimensions.     

Parabolic Type Decay Rates for 1-D-Thermoelastic Systems with Time-Dependent Coefficients

 The purpose of this paper is to derive L ^p − L ^q decay estimates for linear thermoelastic systems with time-dependent coefficient in one space variable. When all coefficients in the system have the same growth speed with small oscillations, we obtain a parabolic type decay estimate. For the system with time-dependent coefficients, we need to investigate the delicate asymptotic behaviour of characteristic roots and the remainder of diagonalization, which will be treated by dividing the phase space into three regions. Received September 15, 2001; in revised form April 20, 2002     

The Expected Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Cube

We calculate E[V ₄(C)], the expected volume of a tetrahedron whose vertices are chosen randomly (i.e. independently and uniformly) in the interior of C, a cube of unit volume. We find

The Typical Growth of the kth Excess in a Random Integer Partition

 For a partition , of a positive integer n chosen uniformly at random from the set of all such partitions, the kth excess is defined by if . We prove a bivariate local limit theorem for as . The whole range of possible values of k is studied. It turns out that ρ and η _k are asymptotically independent and both follow the doubly exponential (extreme value) probability law in a suitable neighbourhood of . Received February 6, 2001; in revised form February 25, 2002 Published online August 5, 2002     

Cycles for Rational Maps of Good Reduction Outside a Prescribed Set

Let K be a number field and S a fixed finite set of places of K containing all the archimedean ones. Let R _S be the ring of S-integers of K. In the present paper we study the cycles in for rational maps of degree ≥2 with good reduction outside S. We say that two ordered n-tuples (P ₀, P ₁,… ,P _n−1) and (Q ₀, Q ₁,… ,Q _n−1) of points of are equivalent if there exists an automorphism A ∈ PGL₂(R _S ) such that P _i = A(Q _i ) for every index i∈{0,1,… ,n−1}. We prove that if we fix two points , then the number of inequivalent cycles for rational maps of degree ≥2 with good reduction outside S which admit P ₀, P ₁ as consecutive points is finite and depends only on S and K. We also prove that this result is in a sense best possible.     

On the construction of copulas and quasi-copulas with given diagonal sections

We study a method, which we call a copula (or quasi-copula) diagonal splice, for creating new functions by joining portions of two copulas (or quasi-copulas) with a common diagonal section. The diagonal splice of two quasi-copulas is always a quasi-copula, and we find a necessary and sufficient condition for the diagonal splice of two copulas to be a copula. Applications of this method include the construction of absolutely continuous asymmetric copulas with a prescribed diagonal section, and determining the best-possible upper bound on the set of copulas with a particular type of diagonal section. Several examples illustrate our results.     

A Local Limit Theorem on Certain p-AdicGroups and Buildings

 We present a local limit theorem for a measure on the special linear group with entries in a local field. (Received 20 March 2000; in revised form 8 September 2000)     

An Edge Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Let f be a function periodic in its two variables x, y with period 2 and having at the origin an edge point jump discontinuity (its graph may be roughly described by a cliff with a smooth but possibly inclined rim). The Fourier series of f exhibits a Gibbs phenomenon as in the one-dimensional case but with an affine horizontal deformation depending on the compass direction of the rim at the origin.Received February 7, 2002; in revised form May 24, 2002 Published online April 4, 2003     

When Does a Submodule of (ℝ[x 1 ⋯, x k]) n Contain a Positive Element?

We characterize for modules consisting of tuples of Laurent polynomials with real coefficients whether such a module contains a positive element. The two conditions needed are numerical and directional positivity. The proof applies universal Gröbner bases.     

On Fractional Part Sums: A Mean-Square Asymptotics over Short Intervals

 This article is concerned with sums 𝒮(t) = ∑ _n  ψ(tf(n/t)) where ψ denotes, essentially, the fractional part minus ?, f is a C ⁴-function with f″ ≠ 0 throughout, summation being extended over an interval of order t. We establish an asymptotic formula for ∫ _T−Λ ^T+Λ (𝒮(t))²dt for any Λ = Λ(T) growing faster than log T. Received April 30, 2001; in revised form February 15, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Operator Semi-Selfdecomposable Measures and Related Nested Subclasses of Measures on Vector Spaces

T-semi-selfdecomposability, operator selfdecomposability, and subclasses L_m(b, (T_t)) and of measures on complete separable metric vector spaces are introduced and basic properties are proved. In particular, we show that is T-semi-selfdecomposable if and only if =T() where is infinitely divisible and is operator selfdecomposable if and only if L₀(b, (T_t)) for all 0<b<.     

Inequalities of Fejér-Jackson Type

 We complement, extend, and sharpen some known inequalities for sine sums. Our main result is the following refinement of the classical Fejér-Jackson inequality: For all integers n ⩾ 2 and real numbers x ∈ (0,π) we have

Plurisubharmonic Defining Functions,Good Vector Fields,and Exactness of a Certain One Form

 We show that the approaches to global regularity of the -Neumann problem via the methods listed in the title are equivalent when the conditions involved are suitably modified. These modified conditions are also equivalent to one that is relevant in the context of Stein neighborhood bases and Mergelyan type approximation. Received 14 May 2001; in revised form 8 November 2001     

On the regularity of the distribution of the maximum of one-parameter Gaussian processes

The main result in this paper states that if a one-parameter Gaussian process has C ^2k paths and satisfies a non-degeneracy condition, then the distribution of its maximum on a compact interval is of class C ^k . The methods leading to this theorem permit also to give bounds on the successive derivatives of the distribution of the maximum and to study their asymptotic behaviour as the level tends to infinity. Received: 14 May 1999 / Revised version: 18 October 1999 / Published online: 14 December 2000     

A superatomic Boolean algebra with few automorphisms

Assuming GCH, we prove that for every successor cardinal μ > ω₁, there is a superatomic Boolean algebra B such that |B| = 2^μ and |Aut B| = μ. Under ◊_ω1, the same holds for μ = ω₁. This answers Monk's Question 80 in [Mo]. Received: 1 January 1998 / Revised version: 18 May 1999 / Published online: 21 December 2000