Class Open Image Research Articles

On the random max-closure for heavy-tailed random variables

In this paper, we study the random max-closure property for not necessarily identically distributed real-valued random variables X1,X2, . . . , which states that, given distributions \( {F}_{X_1} \),\( {F}_{X_2} \), . . . from some class of heavy-tailed distributions, the distribution of the random maximum X(η) := max{0,X1, . . . , Xη} or random maximum S(η) := max{0, S1, . . . , Sη} belongs to the same class of heavy-tailed distributions. Here, Sn = X1 + · · · + Xn, n ≥ 1, and η is a counting random variable, independent of {X1,X2, . . . }. We provide the conditions for the random max-closure property in the case of classes Open image in new window and Open image in new window.

The class of minimax groups is countably recognizable

A class Open image in new window of groups is said to be countably recognizable if a group belongs to Open image in new window whenever all its countable subgroups lie in Open image in new window. It is proved here that the class of minimax groups is countably recognizable.

Countable recognizability and nilpotency properties of groups

A class Open image in new window of groups is said to be countably recognizable if a group belongs to Open image in new window whenever all its countable subgroups lie in Open image in new window. It is proved here that most of the relevant group classes of generalized nilpotent groups are countably recognizable. Moreover, some further countably recognizable group classes are exhibited.

An analog of Rudin’s theorem for continuous radial positive definite functions of several variables

Let Open image in new window be the class of radial real-valued functions of m variables with support in the unit ball \(\mathbb{B}\) of the space ℝ m that are continuous on the whole space ℝ m and have a nonnegative Fourier transform. For m ≥ 3, it is proved that a function f from the class Open image in new window can be presented as the sum \(\sum {f_k \tilde *f_k } \) of at most countably many self-convolutions of real-valued functions f k with support in the ball of radius 1/2. This result generalizes the theorem proved by Rudin under the assumptions that the function f is infinitely differentiable and the functions f k are complex-valued.

On the relation between the darlington realizations of matrix functions from the Carathéodory class and their J p,r -inner SI-dilations

The paper is devoted to establishing a two-sided relation between the Darlington realizations of matrix functions from the Caratheodory class Open image in new window and their J p,r -inner SI-dilations.

Quasi-equational bases for graphs of semigroups, monoids and groups

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. Let denote by Open image in new window the class of all graphs of algebras from a class Open image in new window. We prove that if Open image in new window is a class of semigroups possessing a nontrivial member with a neutral element, then Open image in new window does not have finite quasi-equational basis. We deduce that, for a class Open image in new window of monoids or groups with a nontrivial member, Open image in new window also does not have finite quasi-equational basis.

On domination problem of non-negative distributions

The domination relationship between non-negative distributions is an important question in applied probability. It has important applications in the fields of finance, insurance and risk theory. In this paper, based on class ℳ, we find the sufficient condition of dominating all light-tailed distributions and also discuss its necessity. Almost all heavy-tailed distributions often used in risk theory satisfy this condition. We also consider the domination problem between heavy-tailed distributions, and show that classes Open image in new window and ℳ* have many good properties on domination problems.

C-spaces and simplicial complexes

Given a class Open image in new window of simplicial complexes G, we introduce the notion of a Open image in new window -C-space. In the definition of a C-space, open disjoint families vi refine coverings ui. The nerves of these families are zero-dimensional complexes. In our definition, the nerve of a family vi must embed in the complex Gi of the class Open image in new window . We give a complete characterization of bicompact Open image in new window -C-spaces.

Weakly infinite-dimensional spaces modulo simplicial complexes

The classes of spaces Open image in new window -wid and ℒ-wid are introduced for the class Open image in new window of finite simplicial complexes and the class ℒ of compact polyhedra. If Open image in new window , then Open image in new window -wid = wid, ℒ-wid = S-wid. It is proved that S-wid ⊂ ℒ-wid and ℒ-wid = S-ℒτ-wid for any triangulation τ of the class ℒ.

A note on the degree conjecture for the Selberg class

The degree conjecture for the Selberg class Open image in new window of L-functions states that the degree dF of every F ∈ Open image in new window is an integer. Moreover, it is expected that every F ∈ Open image in new window has polynomial Euler product, and that the degree ∂F of such an Euler product coincides with dF. In this note we prove that a suitable continuity assumption on the degree dF implies that ∂F = dF for all F ∈ Open image in new window with polynomial Euler product.

Connecting Solutions of the Lorentz Force Equation do Exist

Recent results on the maximization of the charged-particle action Open image in new window in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of Open image in new window over a given causal homotopy class Open image in new window of curves connecting two causally related events x0≤x1. Action Open image in new window is proved to admit a maximum on Open image in new window, and also one in the adherence of each timelike homotopy class C. Moreover, the maximum σ0 on Open image in new window is timelike if Open image in new window contains a timelike curve (and the degree of differentiability of all the elements is at least C2).

Selector processes on classes of sets

Consider the random subset X of ℕ obtained by selecting independently each integer with a probability δ. Consider a finite class Open image in new window of finite sets. We describe a combinatorial quantity that is of the same order as Open image in new window We then give a related result allowing to compute the supremum of the empirical process on a class of sets.

Fonctions plurisousharmoniques et courants positifs de type (1,1) sur une variété presque complexe

If (X, J) is an almost complex manifold, then a function u is said to be plurisubharmonic on X if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is conjectured that plurisubharmonicity is equivalent to the positivity of the (1,1)-current Open image in new window, (the (1,1)-current Open image in new window need not be closed here!). The conjecture is trivial if u is of class Open image in new window The result is elementary in the complex integrable case because the operator Open image in new window can be written as an operator with constant coefficients in complex coordinates. Hence the positivity of the current is preserved by regularising with usual convolution kernels. This is not possible in the almost complex non integrable case and the proof of the result requires a much more intrinsic study. In this chapter we prove the necessity of the positivity of the (1,1)-current Open image in new window. We prove also the sufficiency of the positivity in the particular case of an upper semi-continuous function f which is continuous in the complement of the singular locus f−1(−∞).

Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases.

In [HT2] Haagerup and Thorbjo rnsen prove the following extension of Voiculescu’s random matrix model (cf. [V2, Theorem 2.2]): For each n ∈ ℕ, let X1(n),..., Xr(n) be r independent complex self-adjoint random matrices from the class Open image in new window and let x1,...,xr be a semicircular system in a C*-probability space. Then for any polynomial p in r non-commuting variables the convergence

On the numerical inversion of the Laplace transform by the use of an optimized Legendre polynomials

A method for inverting the Laplace transform is presented, using a finite series of the classical Legendre polynomials. The method recovers a real valued function f(t) in a finite interval of the positive real axis when f(t) belongs to a certain class Open image in new window and requires the knowledge of its Laplace transform F(s) only at a finite number of discrete points on the real axis s>0. The choice of these points will be carefully considered so as to improve the approximation error as well as to minimize the number of steps needed in the evaluations. The method is tested on few examples, with particular emphasis on the estimation of the error bounds involved.

On some classes of contact metric manifolds

We define four new classes Open image in new window of contact metric manifoulds using Tanaka connection Open image in new window and Jacobi operators. We prove that a contact metric manifold with the structure vector field ξ belonging to thek-nullity distribution is contact metric locally ϕ-symmetric (in the sense of D. B. Blair) if and only if the manifold is a Open image in new window and Open image in new window space. Also, we prove that a 3-dimensional contact metric Open image in new window and Open image in new window is locally ϕ-symmetric (in the sense of D. E. Blair) and give counter-examples of the converse.

Motions of vortex patches

An evolution equation describing the motion of vortrex patches is established. The existence of steady solutions of this equation is proved. These solutions arem-fold symmetric regions of constant vorticity ω0 and are uniformly rotating with angular velocity Ω in the range $$\tilde \Omega _{m - 1}< \tilde \Omega \leqslant \tilde \Omega _m (\tilde \Omega = \Omega /\omega _0 ,m \geqslant 2)$$ where\(\tilde \Omega _m = (m - 1)/2m\). We call this class, ofm-fold symmetric rotating regionsD, the class Open image in new window of them-waves of Kelvin. Any Open image in new window may be regarded as a simply connected region which is a stationary configuration of the Euler equations in two dimensions. If Open image in new window then any magnification, rotation or reflection is also in Open image in new window with the same angular velocity Ω ofD. The angular velocity\(\Omega _m = \tilde \Omega _m \omega _0 \) corresponds only to the circle solution, which is a trivial member of every class Open image in new window ,m⩾2. The class Open image in new window corresponds to the rotating ellipses of Kirchoff. Other properties of the class Open image in new window are established.

Movability relative to various classes of spaces

This article is in answer to a question posed by K. Borsuk [1]. There exists a locally connected continuum X which is movable relative to the class of all spheres, but which is not 2-movable. We shall prove that the classes Open image in new window of movable compacta coincide for the following Open image in new window : 1) all polyhedra of dimension ⩽n, 2) all compacta of dimension ⩽n, and 3) all compacta of fundamental dimension ⩽n. We shall also prove that the movability of a compactum X is equivalent to its movability relative to the class of all polyhedra.