Bigravity in the Kuchař Hamiltonian formalism: The general case

We develop the Hamiltonian formalism of bigravity and bimetric theories for the general form of the interaction potential of two metrics. When studying the role of lapse and shift functions in theories with two metrics, we naturally use the Kucha? formalism in which these functions are independent of the choice of the space-time coordinate system. We find conditions on the potential necessary and sufficient for the existence of four first-class constraints. These constraints realize a well-known hypersurface deformation algebra in the framework of the formalism of Dirac brackets constructed on the base of all second-class constraints. Fixing one of the metrics, we obtain a bimetric theory not containing first-class constraints. Conserved quantities corresponding to symmetries of the background metric can then be expressed ultralocally in terms of the metric interaction potential.     

On the extended Hardy-Hilbert's inequality

In this paper, by introducing three parameters A, B, and λ, and estimating the weight coefficient, we give a generalization of the extended Hardy-Hilbert's inequality with a best possible constant factor, involving the β function. We also consider its equivalent inequality and the associated double series form.     

Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws

We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave-long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α=o(ε1/2). We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled.     

Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.     

Further results on the reverse order law for the Moore-Penrose inverse in rings with involution

We present some equivalent conditions of the reverse order law for the Moore-Penrose inverse in rings with involution, extending some well-known results to more general settings. Then we apply this result to obtain a set of equivalent conditions to the reverse order rule for the weighted Moore-Penrose inverse in C^∗-algebras.     

Algebraic logoi

We introduce normal cores, as well as the more general action cores, in the context of a semi-abelian category, and further generalise those to split extension cores in the context of a homological category. We prove that, if the category is moreover well-powered with (small) joins, then the existence of split extension cores is equivalent to the condition that the change-of-base functors in the fibration of points are geometric. We call a finitely complete category that satisfies this condition an algebraic logos. We give examples of such categories, compare them with algebraically coherent ones, and study equivalent conditions as well as stability under common categorical operations.     

Projection iterative methods for extended general variational inequalities

In this paper, we introduce and study a new class of variational inequalities involving three operators, which is called the extended general variational inequality. Using the projection technique, we show that the extended general variational inequalities are equivalent to the fixed point and the extended general Wiener-Hopf equations. This equivalent formulation is used to suggest and analyze a number of projection iterative methods for solving the extended general variational inequalities. We also consider the convergence of these new methods under some suitable conditions. Since the extended general variational inequalities include general variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems.     

Algorithms for solutions of extended general mixed variational inequalities and fixed points

In this paper, we introduce and study a new class of extended general nonlinear mixed variational inequalities and a new class of extended general resolvent equations and establish the equivalence between the extended general nonlinear mixed variational inequalities and implicit fixed point problems as well as the extended general resolvent equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear mixed variational inequalities. Applying the aforesaid equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S, we construct some new resolvent iterative algorithms for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is the unique solution of the problem of extended general nonlinear mixed variational inequalities. We study convergence analysis of the suggested iterative schemes under some suitable conditions. We also suggest and analyze a class of extended general resolvent dynamical systems associated with the extended general nonlinear mixed variational inequalities and show that the trajectory of the solution of the extended general resolvent dynamical system converges globally exponentially to the unique solution of the extended general nonlinear mixed variational inequalities. The results presented in this paper extend and improve some known results in the literature.     

Three-step iterative methods for general variational inclusions in L P spaces

In this paper, we show that the general variational inclusions are equivalent to the fixed point problem. We use this equivalence to discuss the existence of the variational inclusions in L ^p spaces. Using the technique of the updating solution, we suggest some three-step iterative methods for solving the general variational inclusion. We also consider the convergence analysis of the proposed iterative methods under some mild conditions. Since the general variational inclusions include several classes of variational inequalities and optimization problems as special cases, results proved in this paper continue to hold for these problems.     

Un modèle du calcul de Dirac dans la théorie relative des ensembles

The aim of this Note is a new justification of the pertinence of the Dirac calculus (see [1]). We give a model of it in a relatively consistent set theory,the Relative Set Theory (see [2]). We have obtained this result by introducing a derivation of step functions.     

Iterative behaviour of generalized majority functions

Let us consider a society of n persons {P₁,…,P_n} with interaction coefficient a_ij between P_i and P_j. Let {θ₁,…,θ_p} be the set of opinions which may be assumed by any person. We study the dynamical behaviour of such a society under local majority rules.The results presented in this paper generalize many partial results previously obtained by several authors (Tchuente, 1977; Goles and Olivos, 1980; Poljak and Sura, 1982). Furthermore, by introducing algebraic invariants, analogous to those exhibited in Goles (1980) for the study of Boolean threshold functions, we provide a tool for a general approach of such dynamical models.     

Pattern avoidance in binary trees

This paper considers the enumeration of trees avoiding a contiguous pattern. We provide an algorithm for computing the generating function that counts n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this counting mechanism, we study the analogue of Wilf equivalence in which two tree patterns are equivalent if the respective n-leaf trees that avoid them are equinumerous. We investigate the equivalence classes combinatorially. Toward establishing bijective proofs of tree pattern equivalence, we develop a general method of restructuring trees that conjecturally succeeds to produce an explicit bijection for each pair of equivalent tree patterns.     

The Carathéodory topology for multiply connected domains II

We continue our exposition concerning the Carathéodory topology for multiply connected domains which we began in [Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340] by introducing the notion of boundedness for a family of pointed domains of the same connectivity. The limit of a convergent sequence of n-connected domains which is bounded in this sense is again n-connected and will satisfy the same bounds. We prove a result which establishes several equivalent conditions for boundedness. This allows us to extend the notions of convergence and equicontinuity to families of functions defined on varying domains.     

Estimates of the dilatation function of Beurling-Ahlfors extension

In this paper we estimate the dilatation function of the Beurling-Ahlfors extension in the most general case. By introducing ?h,m-function, we obtain an inequality which is sharp up to a constant.     

Exactness of direct limits for abelian categories with an injective cogenerator

We prove that the exactness of direct limits in an abelian category with products and an injective cogenerator J is equivalent to a condition on J which is well-known to characterize pure-injectivity in module categories, and we describe an application of this result to the tilting theory. We derive our result as a consequence of a more general characterization of when inverse limits in the Eilenberg–Moore category of a monad on the category of sets preserve regular epimorphisms.     

Matrices permutation equivalent to primitive matrices

We give a necessary and sufficient condition for an n×n (0,1) matrix (or more generally, an n×n nonnegative matrix) to be permutation equivalent to a primitive matrix. More precisely, except for two simple permutation equivalent classes of n×n (0,1) matrices, each n×n (0,1) matrix having no zero row or zero column is permutation equivalent to some primitive matrix. As an application, we use this result to characterize the subsemigroup of B_n (B_n is the multiplicative semigroup of n×n Boolean matrices) generated by all the primitive matrices and permutation matrices. We also consider a more general problem and give a necessary and sufficient condition for an n×n nonnegative matrix to be permutation equivalent to an irreducible matrix with given imprimitive index.     

The general facility location problem with connectivity on trees

In this note we study the general facility location problem with connectivity. We present an O(np ²)-time algorithm for the general facility location problem with connectivity on trees. Furthermore, we present an O(np)-time algorithm for the general facility location problem with connectivity on equivalent binary trees.     

A Hilbert-type integral inequality with the mixed kernel of multi-parameters

In this paper, by means of the weight function and the technique of real analysis, and introducing the Γ-function and the Riemann ζ-function to jointly characterize the constant factor, a Hilbert-type integral inequality with the mixed kernel of multi-parameters and its equivalent form are given; their constant factors are proved to be the best possible. By selecting special parameter values, some meaningful results are obtained.     

Characteristic properties of equivalent structures in compositional models

Compositional model theory serves as an alternative approach to multidimensional probability distribution representation and processing. Every compositional model over a finite non-empty set of variables N is uniquely defined by its generating sequence - an ordered set of low-dimensional probability distributions. A generating sequence structure induces a system of conditional independence statements over N valid for every multidimensional distribution represented by a compositional model with this structure.The equivalence problem is how to characterise whether all independence statements induced by structure P are induced by a second structure P^′ and vice versa. This problem can be solved in several ways. A partial solution of the so-called direct characterisation of an equivalence problem is represented here. We deduce and describe three properties of equivalent structures necessary for equivalence of the respective structures. We call them characteristic properties of classes of equivalent structures.     

Closure of dilates of shift-invariant subspaces

Let V be any shift-invariant subspace of square summable functions. We prove that if for some A expansive dilation V is A-refinable, then the completeness property is equivalent to several conditions on the local behaviour at the origin of the spectral function of V, among them the origin is a point of A*-approximate continuity of the spectral function if we assume this value to be one. We present our results also in a more general setting of A-reducing spaces. We also prove that the origin is a point of A*-approximate continuity of the Fourier transform of any semiorthogonal tight frame wavelet if we assume this value to be zero.