A generalization of the Goldston–Pintz–Yildirim prime gaps result to number fields

D. A. Goldston, J. Pintz and C. Y. Y?ld?r?m [3] proved the existence of infinitely many prime pairs whose difference is arbitrarily small compared to the average gap, namely $$ \liminf_{n\to\infty} \frac{p_{n+1}-p_n}{\log p_n}=0. $$ In the present work we generalize their result to totally real number fields. We prove that if ω ₀ and ω ₁ run over distinct prime elements of the number field, then $$ \liminf_{\omega_0,\omega_1} \left|\frac{{\mathbf {N}}(\omega_1-\omega_0)}{{\mathbf {N}}\omega_0}\right|=0. $$      

A generalization of the Runge–Kutta iteration

Iterative solvers in combination with multi-grid have been used extensively to solve large algebraic systems. One of the best known is the Runge–Kutta iteration. We show that a generally used formulation [A. Jameson, Numerical solution of the Euler equations for compressible inviscid fluids, in: F. Angrand, A. Dervieux, J.A. Désidéri, R. Glowinski (Eds.), Numerical Methods for the Euler Equations of Fluid Dynamics, SIAM, Philadelphia, 1985, pp. 199–245] does not allow to form all possible polynomial transmittance functions and we propose a new formulation to remedy this, without using an excessive number of coefficients.     

A generalization of the Dedekind–MacNeille completion

In this paper we introduce the notion of \(Z_{\delta }\)-continuity as a generalization of precontinuity, complete continuity and \(s_{2}\)-continuity, where Z is a subset selection. And for each poset P, a closure space \(Z^{c}_{\delta }(P)\) arises naturally. For any subset system Z, we define a new type of completion, called \(Z_{\delta }\)-completion, extending each poset P to a Z-complete poset. The main results are: (1) if a subset system Z is subset-hereditary, then \(cl_{Z}(\Psi (P))\), the Z-closure of all principal ideals \(\Psi (P)\) of poset P in \(Z^{c}_{\delta }(P)\), is a \(Z_{\delta }\)-completion of P and \(Z^{c}_{\delta }(P) \cong Z^{c}_{\delta }(cl_{Z}(\Psi (P)))\); (2) let Z be an HUL-system and P a \(Z_{\delta }\)-continuous poset, then the \(Z_{\delta }\)-completion of P is also \(Z_{\delta }\)-continuous, and a Z-complete poset L is a \(Z_{\delta }\)-completion of P iff P is an embedded \(Z_{\delta }\)-basis of L; (3) the Dedekind–MacNeille completion is a special case of the \(Z_{\delta }\)-completion.     

A generalization of the Hall–Witt identity

Let Y → X be a finite normal cover of a wedge of n ≥ 3 circles. We prove that for any nonzero v ∈ H ₁(Y; Q) there exists a lift \(\widetilde F\) to Y of a basepoint-preserving homotopy equivalence F: X → X such that the set of iterates \(\left\{ {{{\widetilde F}^d}\left( v \right)} \right\}:d \in \mathbb{Z} \subseteq {H_1}\left( {Y,\mathbb{Q}} \right)\) is infinite. The main achievement of this paper is the use of representation theory to prove the existence of a purely topological object that seems to be inaccessible via topology.     

A generalization of the Burkholder–Davis–Gundy inequalities

In this paper, by applying some improved inequalities, we extend the Burkholder–Davis–Gundy inequalities for α ∈ (0,1) to more general functions and submartingales. Moreover, a series of inequalities for a logarithmic function are also obtained correspondingly. Finally, we give an application to a stopped Brownian motion.     

A generalization of the Jaffard–Ohm–Kaplansky Theorem

The well-known Jaffard–Ohm–Kaplansky Theorem states that every abelian ?-group can be realized as the group of divisibility of a commutative Bézout domain. To date there is no realization (except in certain circumstances) of an arbitrary, not necessarily abelian, ?-group as the group of divisibility of an integral domain. We show that using filters on lattices we can construct a nice quantal frame whose “group of divisibility” is the given ?-group. We then show that our construction when applied to an abelian ?-group gives rise to the lattice of ideals of any Prüfer domain assured by the Jaffard–Ohm–Kaplansky Theorem. Thus, we are assured of the appropriate generalization of the Jaffard–Ohm–Kaplansky Theorem.     

A generalization of the Sherman–Morrison–Woodbury formula

In this paper, we develop conditions under which the Sherman–Morrison–Woodbury formula can be represented in the Moore–Penrose inverse and the generalized Drazin inverse forms. These results generalize the original Sherman–Morrison–Woodbury formula.     

A generalization of the Bombieri–Pila determinant method

The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper, we give a generalization of that method to varieties of higher dimension, yielding a proof of Heath-Brown’s “Theorem 14” by real-analytic considerations alone. Bibliography: 11 titles.     

A new generalization of the Pareto–geometric distribution

In this paper we introduce a new distribution called the beta Pareto–geometric. We provide a comprehensive treatment of the mathematical properties of the proposed distribution and derive expressions for its moment generating function and the rth generalized moment. We discuss estimation of the parameters by maximum likelihood and obtain the information matrix that is easily numerically determined. We also demonstrate its usefulness on a real data set.     

A generalization of the category O of Bernstein–Gelfand–Gelfand

In the study of simple modules over a simple complex Lie algebra, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this note, we define a family of categories which generalizes the BGG category. We classify the simple modules for some of these categories. As a consequence we show that these categories are semisimple.     

A generalization of the Pólya–Vinogradov inequality

In this paper we consider an approach of Dobrowolski and Williams which leads to a generalization of the Pólya–Vinogradov inequality. We show how the Dobrowolski–Williams approach is related to the classical proof of Pólya–Vinogradov using Fourier analysis. Our results improve upon the earlier work of Bachman and Rachakonda (Ramanujan J. 5:65–71, 2001). In passing, we also obtain sharper explicit versions of the Pólya–Vinogradov inequality.     

A generalization of the original Jordan–von Neumann theorem

The problem of representability of quadratic functionals (acting on modules over unital complex ∗-algebras), by sesquilinear forms, is generalized by weakening the homogeneity equation. The corresponding representation theorem can be considered as a generalization of (the original form of) the classical Jordan–von Neumann characterization of complex inner product spaces.     

A generalization of the Egalitarian and the Kalai–Smorodinsky bargaining solutions

We characterize the class of weakly efficient n-person bargaining solutions that solely depend on the ratios of the players’ ideal payoffs. In the case of at least three players the ratio between the solution payoffs of any two players is a power of the ratio between their ideal payoffs. As special cases this class contains the Egalitarian and the Kalai–Smorodinsky bargaining solutions, which can be pinned down by imposing additional axioms.     

A generalization of k-Cohen–Macaulay simplicial complexes

For a positive integer k and a non-negative integer t, a class of simplicial complexes, to be denoted by k-CM _t , is introduced. This class generalizes two notions for simplicial complexes: being k-Cohen–Macaulay and k-Buchsbaum. In analogy with the Cohen–Macaulay and Buchsbaum complexes, we give some characterizations of CM _t (=1?CM _t ) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CM _t property under the operation of join of two simplicial complexes. We show that a complex is k-CM _t if and only if the links of its non-empty faces are k-CM _t?1. We prove that for an integer s≤d, the (d?s?1)-skeleton of a (d?1)-dimensional k-CM _t complex is (k+s)-CM _t . This result generalizes Hibi’s result for Cohen–Macaulay complexes and Miyazaki’s result for Buchsbaum complexes.     

A number of stable matchings in models of the Gale–Shapley type

We give some upper bounds on the maximum number of stable matchings in the Gale–Shapley marriage model with n$n$ men and n$n$ women. We also characterize, with the use of some graph-theoretical notions, the exact number of such matchings, assuming that the preferences of men and women are given.     

A new generalization of Hardy–Berndt sums

In this paper, we construct a new generalization of Hardy–Berndt sums which are explicit extensions of Hardy–Berndt sums. We express these sums in terms of Dedekind sums s _r (h, k : x, y|λ) with x?=?y?=?0 and obtain corresponding reciprocity formulas.