On families of distributions characterized by certain properties of ordered random variables

In this work, we obtain new characterizations of certain probability distributions by relations with different ordered random variables. Such variables include order statistics, sequential maxima, and records. We consider relations that include not only upper, but also lower record values. The presented ordered objects are based on sequences of independent random variables with a common continuous distribution function. We also investigate equalities in the distribution of sequential maxima exposed by various random shifts. These shifts (one-sided or two-sided) have exponential distributions. Certain theorems and their corollaries present corresponding characterizations of distributions by relations of such a type. In addition, we consider exponentially shifted order statistics such that simple relations among them also characterize certain probability distributions. All of the presented results yield a set of characterizations of various distributions. For particular cases, we present the relations that characterize families of classical exponential and logistic distributions.     

Reference classes and relational learning

This paper studies the connections between relational probabilistic models and reference classes, with specific focus on the ability of these models to generate the correct answers to probabilistic queries. We distinguish between relational models that represent only observed relations and those which additionally represent latent properties of individuals. We show how both types of relational models can be understood in terms of reference classes, and that learning such models correspond to different ways of identifying reference classes. Rather than examining the impact of philosophical issues associated with reference classes on relational learning, we directly assess whether relational models can represent the correct probabilities of a simple generative process for relational data. We show that models with only observed properties and relations can only represent the correct probabilities under restrictive conditions, whilst models that also represent latent properties avoids such restrictions. As such, methods for acquiring latent-property models are an attractive alternatives to traditional ways of identifying reference classes. Our experiments on synthetic as well as real-world domains support the analysis, demonstrating that models with latent relations are significantly more accurate than those without latent relations.     

Dirichlet-Neumann conditions and the orthogonality of three-dimensional guided waves in layered solids

Orthogonality relations for homogeneous waves in layered plates are obtained, and they are generalized to the case of a contact with fluid layers. For layers of infinite thickness, it is shown that the homogeneous waves of the discrete spectrum are orthogonal to each other and to the waves of the continuous spectrum. For finite-size sources, exact formulas are derived for the coefficients multiplying the modes. Based on the orthogonality relations, a nonlocal radiation principle is proposed such that the infinite domain in the numerical solution of diffraction problems for layered plates can be replaced by a virtual cylinder.     

Neural fuzzy point processes

Synaptic events in neural systems were described as generated by an apparatus @ possessing memory and encoding a fuzzy point process (the presynaptic discharge) into another N (the postsynaptic discharge). @ was considered to be a fuzzy automata, for which state membership is dependent on input membership and distribution as well as on a control exercised by other neural structures. In such a device, irregular input distributions favour a direct monotonic codification, whereas regular ones induce discontinuous and inverse relations between both fuzzy point processes. Both behaviors favour analogic and membership relations between the fuzzy input and output. However, there exist intermediate grades of irregularities which result in a context-free encoding, where similitude and equivalence relations predominate. The importance of such findings to neurophysiology is discussed.     

Quasi-Fredholm and Semi-B-Fredholm Linear Relations

Quasi-Fredholm and semi-B-Fredholm linear relations in Banach spaces are defined in terms of conditions on their ranges and null spaces. We analyze the behavior of the powers of such linear relations and prove that these classes of linear relations define a spectrum that satisfies the polynomial spectral mapping property. Our results are used to calculate the quasi-Fredholm and the semi-B-Fredholm spectra of some linear relations.     

Comparison of Existence Results for Efficient Points

Existence results of maximal points with respect to general binary relations were stated by Hazen and Morin (Ref. 1) and by Gajek and Zagrodny (Ref. 2). In this paper, we point out that the natural framework for this problem is that of transitive and reflexive relations (preorders). The aim of this paper is to discuss existence results for maximal points with respect to general transitive relations in such a way that, when considering them for preorders defined by convex cones, we are able to recover most known existence results for efficient points; the quasi-totality of them, with their (short) proofs, is presented, too.     

Functional relations in lattice statistical mechanics, enumerative combinatorics, and discrete dynamical systems

We recall some non-trivial, non-linear functional relations appearing in various domains of mathematics and physics, such as lattice statistical mechanics, quantum mechanics, or enumerative combinatorics. We focus, more particularly, on the analyticity properties of the solutions of these functional relations. We then consider discrete dynamical systems corresponding to birational transformations. The rational expressions for dynamical zeta functions obtained for a particular two-dimensional birational mapping, depending on two parameters, are recalled, as well as some non-trivial functional relations satisfied by these dynamical zeta functions. We finally give some functional equations corresponding to some singled out orbits of this two-dimensional birational mapping for particular values of the two parameters. This example shows that functional equations associated with curves, for real values of the variables, are actually compatible with a chaotic dynamical system.     

A formula on the weight distribution of linear codes with applications to AMDS codes

The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singleton's defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations.     

Foliations of polynomial growth are hyperfinite

We define a class of equivalence relations with polynomial growth and show that such relations always support finite invariant measures and are hyperfinite. In particular, foliations of polynomial growth define hyperfinite equivalence relations with respect to any family of finite invariant measures on transversals. We also extend a result of Dye for countable groups to show that if a locally compact second countable groupG acts freely on a Lebesgue spaceX with finite invariant measure, so that the orbit relation onX is hyperfinite, thenG is amenable.     

Normal extensions of subnormal linear relations via quaternionic Cayley transforms

A quaternionic Cayley transform for linear relations is introduced and some of its properties are exhibited. We emphasize the role played by the linear relations whose quaternionic Cayley transforms are unitary operators, which happen to be normal relations, and investigate the class of those linear relations which extend to such normal relations.     

Learning intransitive reciprocal relations with kernel methods

In different fields like decision making, psychology, game theory and biology, it has been observed that paired-comparison data like preference relations defined by humans and animals can be intransitive. Intransitive relations cannot be modeled with existing machine learning methods like ranking models, because these models exhibit strong transitivity properties. More specifically, in a stochastic context, where often the reciprocity property characterizes probabilistic relations such as choice probabilities, it has been formally shown that ranking models always satisfy the well-known strong stochastic transitivity property. Given this limitation of ranking models, we present a new kernel function that together with the regularized least-squares algorithm is capable of inferring intransitive reciprocal relations in problems where transitivity violations cannot be considered as noise. In this approach it is the kernel function that defines the transition from learning transitive to learning intransitive relations, and the Kronecker-product is introduced for representing the latter type of relations. In addition, we empirically demonstrate on two benchmark problems, one in game theory and one in theoretical biology, that our algorithm outperforms methods not capable of learning intransitive reciprocal relations.     

An approach to group decision making based on interval multiplicative and fuzzy preference relations by using projection

This paper presents a consensus model for group decision making with interval multiplicative and fuzzy preference relations based on two consensus criteria: (1) a consensus measure which indicates the agreement between experts’ preference relations and (2) a measure of proximity to find out how far the individual opinions are from the group opinion. These measures are calculated by using the relative projections of individual preference relations on the collective one, which are obtained by extending the relative projection of vectors. First, the weights of experts are determined by the relative projections of individual preference relations on the initial collective one. Then using the weights of experts, all individual preference relations are aggregated into a collective one. The consensus and proximity measures are calculated by using the relative projections of experts’ preference relations respectively. The consensus measure is used to guide the consensus process until the collective solution is achieved. The proximity measure is used to guide the discussion phase of consensus reaching process. In such a way, an iterative algorithm is designed to guide the experts in the consensus reaching process. Finally the expected value preference relations are defined to transform the interval collective preference relation to a crisp one and the weights of alternatives are obtained from the expected value preference relations. Two numerical examples are given to illustrate the models and approaches.     

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (non-self-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C^?-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the wot-closures of these algebras as well.     

Integral relations for heun-class special functions

New integral relations are obtained for eigenfunctions produced by Heun-class equations. These relations demonstrate the duality property of the eigenfunctions with different behaviors at singularities, the eigenfunctions being defined at different intervals. The obtained relations form two hierarchies, such that in each of them, the equations are produced one from another by a confluence of singular points.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 388–396, June, 1996.     

Symmetries,coverings, and decomposition of systems and trajectory generation

We derive relations between the notions of symmetry, covering, and decomposition of systems and trajectory generation. We show that any decomposition of a system determines a finite-dimensional covering of that system and is determined by it. We present conditions on vector fields under which any covering is obtained by factorization along the Lie algebra of such fields. On the basis of these relations, we study whether a point-to-point steering problem can be transformed into a set of boundary value problems of lower dimension.     

Almost, Asymptotic and Fredholm Representations of Discrete Groups

An almost representation of a group is a map from this group into the unitary group of a Hilbert space, such that the group relations hold only approximately. We give a survey of the recent results on almost representations and on their relations to asymptotic representations. Applications to K-theory of classifying spaces are also discussed.