General Characteristic Function Research Articles

A generalization of a result of Báez-Duarte, Balazard, Landreau, and Saias

It is known that the Riemann hypothesis is true if and only if ▪ belongs to the Nyman-Beurling space, where ▪ denotes the characteristic function on the unit interval. Moreover, Báez-Duarte, Balazard, Landreau, and Saias provided a lower bound for the distance between ▪ and the subspaces of the Nyman-Beurling space. In this paper, we extend the result of Báez-Duarte et al. by considering more general characteristic functions. In addition, we give some applications and raise several questions related to the Riemann hypothesis.

Well-posedness problem of an anisotropic parabolic equation

In this paper, we are concerned with well-posedness of an anisotropic parabolic equation with the convection term. When some diffusion coefficients are degenerate on the boundary ∂Ω and the others are positive on Ω‾, we propose a novel partial boundary value condition to study the stability of the solutions for the anisotropic parabolic equation. A new concept, the general characteristic function of the domain Ω, is introduced and applied. The existence and stability of the solutions is established under the given partial boundary value conditions.

The entropy solution of a reaction\u2013diffusion equation on an unbounded domain

The degenerate parabolic equations from the reaction–diffusion problems are considered on an unbounded domain varOmegasubsetmathbb {R}^{N}. It is expected that only a partial boundary should be imposed the homogeneous boundary value, but how to give the analytic expression of this partial boundary seems very difficult. A new method, which is called the general characteristic function method, is introduced in this paper. By this new method, a reasonable analytic expression of the partial boundary value condition is found. Moreover, the stability of the entropy solutions is established based on this partial boundary value condition.

A Note on Strategic Stability of Cooperative Solutions for Multistage Games

The problem of strategic stability of cooperative solutions for multistage games is studied. The sufficient conditions related to discount factors are presented, which guarantee the existence of Nash or strong Nash equilibria in such games and therefore guarantee the strategic stability of cooperative solutions. The deviating payoffs of players are given directly, which are related closely to these conditions and avoid the loss of super-additivity of a class of general characteristic functions. As an illustration, Nash and strong Nash equilibria are found for the repeated infinite stage Prisoner’s dilemma game.

Extended smoothlets: An efficient multi-resolution adaptive transform

As a family of multiresolution adaptive transforms and a generalization of wedgelets, smoothlets are more efficient in representing images with various sharp edges than other “X-lets”. Smoothlets use a horizon function to model an edge and define transition in a fixed axis direction. Furthermore, the conventional elliptical smoothlets (ES) use half of an ellipse to model edges. Various sharp edges of images may not be expressed well by half of an ellipse with transition only in a fixed axis direction. In this paper, we propose extended smoothlets (ExSmoothlets) framework by using more general characteristic functions with adaptive transition directions. Specifically, two methods of ExSmoothlets, the elliptical ExSmoothlets (EES) and homocentric elliptical ExSmoothlets (HEES) are developed and evaluated in the experiments against the ES. The results show that the proposed EES and HEES can effectively improve the image quality in image approximation and denoising in terms of PSNR.

Monge extensions of cooperation and communication structures

Cooperation structures without any a priori assumptions on the combinatorial structure of feasible coalitions are studied and a general theory for marginal values, cores and convexity is established. The theory is based on the notion of a Monge extension of a general characteristic function, which is equivalent to the Lovász extension in the special situation of a classical cooperative game. It is shown that convexity of a cooperation structure is tantamount to the equality of the associated core and Weber set. Extending Myerson’s graph model for game theoretic communication, general communication structures are introduced and it is shown that a notion of supermodularity exists for this class that characterizes convexity and properly extends Shapley’s convexity model for classical cooperative games.

Some inequalities for characteristic functions. II

Recently we established Matysiak and Szablowski's conjecture [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711–713] about a lower bound of real-valued characteristic functions. In this paper, applying an alternative approach we are able to give explicitly the ranges of argument for which the obtained inequalities hold true for general characteristic functions.