On a New Generalization of Hardy–Hilbert's Inequality and Its Applications

AbstractThis paper deals with a generalization of the Hardy–Hilbert inequality with best constant factor which involves the β function. As an application, we obtain a new equivalent form of the Hardy–Hilbert inequality

ESTIMATION OF THE CENTRAL MOMENTS OF A RANDOM VECTOR BASED ON THE DEFINITION OF THE POWER OF A VECTOR

The moments of a random vector based on the definition of the power of a vector, proposed by J. Tatar, are scalar and vector characteristics of a multivariate distribution. Analogously to the univariate case, we distinguish the uncorrected and the central moments of a random vector. Other characteristics of a multivariate distribution, i.e. an index of skewness and kurtosis, have been introduced by using the central moments of a random vector. For the application of the mentioned quantities for the analysis of multivariate empirical data, it appears desirable to construct their respective estimators. This paper presents the consistent estimators of the central moments of a random vector, for which essential characteristics have been found, such as a mean vector and a mean squared error. In these formulas, the relevant orders of approximation have been taken into account

A New Hilbert-type Integral Inequality with the Homogeneous Kernel of Real Degree Form and the Integral in Whole Plane

Abstract: In this paper,we build a new Hilbert's inequality with the homogeneous kernel of real order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using f unction

Integral expressions for Hilbert-type infinite multilinear form and related multiple Hurwitz-Lerch Zeta functions

The article deals with different kinds integral expressions concerning multiple Hurwitz-Lerch Zeta function (introduced originally by Barnes ), Hilbert-type infinite multilinear form and its power series extension. Here Laplace integral forms and multiple Mellin-Barnes type integral representation are derived for these special functions. As a special cases of our investigations we deduce the integral expressions for the Matsumoto's multiple Mordell-Tornheim Zeta function, that is, for Tornheim's double sum i.e. Mordell-Witten Zeta, for the multiple Hurwitz Zeta and for the multiple Hurwitz-Euler Eta function, recently studied by Choi and Srivastava

Ordered Incidence geometry and the geometric foundations of convexity theory

An Ordered Incidence Geometry, that is a geometry with certain axioms of incidence and order, is proposed as a minimal setting for the fundamental convexity theorems, which usually appear in the context of a linear vector space, but require only incidence, order (and for separation, completeness), and none of the linear structure of a vector space.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42995/1/22_2005_Article_BF01227810.pd

On the pigeonhole and the modular counting principles over the bounded arithmetic V0V^{0} (Theory and Applications of Proof and Computation)

The theorem of Ajtai ([1], improved by [11] and [12]), which shows a superpolynomial lower bound for AC⁰-Frege proofs of the pigeonhole principle, was a significant breakthrough of proof complexity and has been inspiring many other important works considering the strengths of modular counting principles and the pigeonhole principle. In terms of bounded arithmetics, the theorem implies that the pigeonhole principle is independent from the bounded arithmetic V⁰. Along the stream of researches, [7] gave the following conjectures and showed some sufficient conditions to prove them: ・V⁰ ＋ UCP[l, d k] ∀ PHP[n+1 n].・For any prime number p other than 2, V⁰ ＋ oddtownk ∀ Count[p n].・For any integer p ≥ 2, V⁰ ＋ FIEk ∀ Count[p n]. Here, injPHP[n+1 n] is a formalization of the pigeonhole principle for injections, UCP[l, d k] is the uniform counting principle defined in [7], Count[p n] is the modular counting principle mod p, oddtownk is a formalization of odd town theorem, and FIEk is a formalization of Fisher's inequality. In this article, we give a summary of the work of [7], supplement both technical parts and motivations of it, and propose the future perspective

Stochastic control liaisons: Richard Sinkhorn meets gaspard monge on a schr\uf6dinger bridge

In 1931-1932, Erwin Schr\uf6dinger studied a hot gas Gedankenexperiment (an instance of large deviations of the empirical distribution). Schr\uf6dinger's problem represents an early example of a fundamental inference method, the so-called maximum entropy method, having roots in Boltzmann's work and being developed in subsequent years by Jaynes, Burg, Dempster, and Csisz\ue1r. The problem, known as the Schr\uf6dinger bridge problem (SBP) with "uniform"prior, was more recently recognized as a regularization of the Monge-Kantorovich optimal mass transport (OMT) problem, leading to effective computational schemes for the latter. Specifically, OMT with quadratic cost may be viewed as a zerotemperature limit of the problem posed by Schr\uf6dinger in the early 1930s. The latter amounts to minimization of Helmholtz's free energy over probability distributions that are constrained to possess two given marginals. The problem features a delicate compromise, mediated by a temperature parameter, between minimizing the internal energy and maximizing the entropy. These concepts are central to a rapidly expanding area of modern science dealing with the so-called Sinkhorn algorithm, which appears as a special case of an algorithm first studied in the more challenging continuous space setting by the French analyst Robert Fortet in 1938-1940 specifically for Schr\uf6dinger bridges. Due to the constraint on end-point distributions, dynamic programming is not a suitable tool to attack these problems. Instead, Fortet's iterative algorithm and its discrete counterpart, the Sinkhorn iteration, permit computation of the optimal solution by iteratively solving the so-called Schr\uf6dinger system. Convergence of the iteration is guaranteed by contraction along the steps in suitable metrics, such as Hilbert's projective metric. In both the continuous as well as the discrete time and space settings, stochastic control provides a reformulation of and a context for the dynamic versions of general Schr\uf6dinger bridge problems and of their zero-temperature limit, the OMT problem. These problems, in turn, naturally lead to steering problems for flows of one-time marginals which represent a new paradigm for controlling uncertainty. The zero-temperature problem in the continuous-time and space setting turns out to be the celebrated Benamou-Brenier characterization of the McCann displacement interpolation flow in OMT. The formalism and techniques behind these control problems on flows of probability distributions have attracted significant attention in recent years as they lead to a variety of new applications in spacecraft guidance, control of robot or biological swarms, sensing, active cooling, and network routing as well as in computer and data science. This multifaceted and versatile framework, intertwining SBP and OMT, provides the substrate for the historical and technical overview of the field given in this paper. A key motivation has been to highlight links between the classical early work in both topics and the more recent stochastic control viewpoint, which naturally lends itself to efficient computational schemes and interesting generalizations

Unification in Commutative Theories, Hilbert's Basis Theorem, and Gröbner Bases

Unification in a commutative theory E may be reduced to solving linear equations in the corresponding semiring S(E) (Nutt (1988)). The unification type of E can thus be characterized by algebraic properties of S(E). The theory of abelian groups with n commuting homomorphisms corresponds to the semiring Z[X1,...,Xn]. Thus Hilbert’s Basis Theorem can be used to show that this theory is unitary. But this argument does not yield a unification algorithm. Linear equations in Z[X1,..,Xn] can be solved with the help of Gröbner Base methods, which thus provide the desired algorithm. The theory of abelian monoids with a homomorphism is of type zero (Baader (1988)). This can also be proved by using the fact that the corresponding semiring, namely N[X], is not noetherian. An other example of a semiring (even ring), which is not noetherian, is the ring Z, where X1, ..., Xn ( n > 1 ) are non-commuting indeterminates. This semiring corresponds to the theory of abelian groups with n non-commuting homomorphisms. Surprisingly, by construction of a Gröbner Base algorithm for right ideals in Z, it can be shown that this theory is unitary unifying