Limit laws in vector lattices

Abstract

We prove a law of large numbers and a central limit theorem with respect to the order convergence topology in vector lattices.     

Exact solutions and conservation laws in dissipative fluid dynamics

In this paper the 1D Euler system with a source term is considered in the case of isentropic flows. Classes of exact solutions of the equations under interest have been determined within the framework of the differential constraint method and a Riemann problem was solved. Finally Conservation Laws of the system here considered have been determined following the Direct Method.

     

Conservation laws for the relativistic P-system

Abstract

This note is devoted to the existence of rigorous asymptotic expansions for some boundary layer problems. We follow ideas of geometric optics and show that, generically, the study of such expansions is linked to the kernel and range of suitable projectors. We apply this remark to some classical geophysical systems, and recover in particular the results of (Grenier, E., Masmoudi, N. (1997). Ekman layers of rotating fluids, the case of well prepared initial data. Comm. Partial Differential Equations 22(5–6):953–975) with some improvements.     

Conservation laws and growth of discontinuities in a reactive polytropic gas

Summary We shall consider a conservation law associated with a non-equilibrium gas flow with chemical reaction. Some results on the growth of discontinuities and the occurrence of breakdown of the solution are pointed out.

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Sommario Si prende in esame una legge di conservazione che viene associata alle equazioni che governano un gas in non equilibrio in cui avviene una reazione chimica. Si mettono in evidenza alcuni risultati sull'evoluzione delle discontinuita' e sulla eventuale perdita di regolarita' della soluzione.

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This work was partially supported by the G.N.F.M. (Gruppo Nazionale Fisica Matematica) of C.N.R.     

Operator-stable laws

Sharpe investigated the structure of full operator-stable measures μ on a vector group V and obtained decompositions, $\text{μ = μ}{}_1\text{1 μ}{}_2$ and V = V₁ ⊕ V₂, in terms of the Gaussian component μ₁ and the Poisson component μ₂. The subspaces V₁ and V₂ are here identified in terms of an exponent B for μ. Sharpe also pointed out that the Lévy measure M of μ is a mixture of Lévy measures concentrated on single orbits of t^B. Here, an explicit representation is obtained for M as such a mixture by constructing a measure on the unit sphere. Also, necessary and sufficient conditions are given that a Lévy measure be the Lévy measure of a full operator-stable measure. The final result deals with full Gaussian measures μ and establishes the connection between its covariance operator and the class of all exponents of μ.     

Combined sinh-cosh-Gordon equation: Symmetry reductions,exact solutions and conservation laws

Abstract

In this paper we study the combined sinh-cosh-Gordon equation, which arises in mathematical physics and has a wide range of scientific applications that range from chemical reactions to water surface gravity waves. We employ Lie symmetry analysis along with the simplest equation method to obtain exact solutions based on the optimal systems of one-dimensional subalgebras for the combined sinh-cosh-Gordon equation. Furthermore, conservation laws for the combined sinh-cosh-Gordon equation are derived by employing two different methods; the direct method and new conservation theorem.     

Max-semistable laws

In this paper we give a characterization of the classMSS of all one-dimensional d.f.'s G satisfying the functional equation G^α(x)=G(L(x)), where α∈(0, 1] andL(x) is strictly monotone. Supported by the Bulgarian Ministry of Education and Science (grant MM-17/1993). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

General max-stable laws

We review the work on max-stable laws and their max domains of attraction introduced by Pancheva (Lect Notes Math 1155:284–309, 1984). We introduce the concept of general max domain of strict attraction of the general max-stable laws, a subclass of the general max domain of attraction and prove new results. Some interesting examples also are discussed.      

Degenerate multidimensional dispersion laws

A study is made of the degeneracy of multidimensional dispersion laws (k) that increase unboundedly as k and satisfy some additional conditions. Under the assumption that the corresponding degeneracy functionf(k) satisfies a certain condition [Eq. (4)], it is shown that only two-dimensional dispersion laws of the form (p,q)=p ³(q/p)+cp(q/p)(|p|,|q|1), wherep(q/p)=f(p, q) is the corresponding unique degeneracy function, can be degenerate with respect to a 12 process. Some conditions that the function () must satisfy are obtained. The explicit form of a degenerate dispersion law with functionp ³(q/p) of polynomial form is found.Institute of Physics, Georgian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 20–33, April, 1993.     

Stochastic scalar conservation laws

We introduce a notion of stochastic entropic solution à la Kruzkov, but with Ito's calculus replacing deterministic calculus. This results in a rich family of stochastic inequalities defining what we mean by a solution. A uniqueness theory is then developed following a stochastic generalization of L¹ contraction estimate. An existence theory is also developed by adapting compensated compactness arguments to stochastic setting. We use approximating models of vanishing viscosity solution type for the construction. While the uniqueness result applies to any spatial dimensions, the existence result, in the absence of special structural assumptions, is restricted to one spatial dimension only.     

Limit laws for record values

{X_n,n?1} are i.i.d. random variables with continuous d.f. F(x). X_j is a record value of this sequence if X_j>max{X₁,…,X_j?1}. Consider the sequence of such record values {X_Ln,n?1}. Set R(x)=-log(1?F(x)). There exist B_n > 0 such that . in probability (i.p.) iff i.p. iff $\text{{R(kx)?R(x)}}\text{R}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(kx)}$ → ∞ as x→∞ for all k>1. Similar criteria hold for the existence of constants A_n such that X_Ln?A_n → 0 i.p. Limiting record value distributions are of the form N(-log(-logG(x))) where G(·) is an extreme value distribution and N(·) is the standard normal distribution. Domain of attraction criteria for each of the three types of limit laws can be derived by appealing to a duality theorem relating the limiting record value distributions to the extreme value distributions. Repeated use is made of the following lemma: If $\text{P}\text{{X}{}_{\mathrm{n}}\text{?x}=1?e}{}^{\mathrm{-x}}\text{,x?0}$, then X_Ln=Y₀+…+Y_n where the Y_j's are i.i.d. and $\text{P}\text{{Y}{}_{\mathrm{j}}\text{?x}=1?e}{}^{\mathrm{-x}}$.     

Free moment measures and laws

In [3], it was shown that convex, almost everywhere continuous functions coordinatize a broad class of probability measures on ${\mathbb{R}}^n$ by the map $U?{\left(\mathrm{?}U\right)}_{\mathrm{\#}}{e}^{?U}dx$. We consider whether there is a similar coordinatization of non-commutative probability spaces, with the Gibbs measure $e^{?U}dx$ replaced by the corresponding free Gibbs law. We call laws parameterized in this way free moment laws. We first consider the case of a single (and thus commutative) random variable and then the regime of n non-commutative random variables which are perturbations of freely independent semi-circular variables. We prove that free moment laws exist with little restriction for the one dimensional case, and for small even perturbations of free semi-circle laws in the general case.     

Distributive laws for concept lattices

We study several kinds of distributivity for concept lattices of contexts. In particular, we find necessary and sufficient conditions for a concept lattice to be

Generalized reciprocity laws

The aim of this paper is to give an abstract formulation of the classical reciprocity laws for function fields that could be generalized to the case of arbitrary (non-commutative) reductive groups as a first step to finding explicit non-commutative reciprocity laws. The main tool in this paper is the theory of determinant bundles over adelic Sato Grassmannians and the existence of a Krichever map for rank vector bundles.

     

On formal quantum group laws

There exist natural generalizations of the concept of formal groups laws for noncommutative power series. This is a note on formal quantum group laws and quantum group law chunks. Formal quantum group laws correspond to noncommutative (topological) Hopf algebra structures on free associative power series algebras ká áx₁,...,x_m ? ?k\langle\! \langle x_1,\dots,x_m \rangle\! \rangle , k a field. Some formal quantum group laws occur as completions of noncommutative Hopf algebras (quantum groups). By truncating formal power series, one gets quantum group law chunks. ¶If the characteristic of k is 0, the category of (classical) formal group laws of given dimension m is equivalent to the category of m-dimensional Lie algebras. Given a formal group law or quantum group law (chunk), the corresponding Lie structure constants are determined by the coefficients of its chunk of degree 2. Among other results, a classification of all quantum group law chunks of degree 3 is given. There are many more classes of strictly isomorphic chunks of degree 3 than in the classical case.