Mathematical Framework for Lattice Problems
نویسنده
چکیده
Abstract. We propose a mathematical framework to effectively study lattice materials with periodic and non-periodic structures over entire spaces in one, two, and three dimensions. The existence and uniqueness of solutions for periodic lattice problems with absolute terms are proved in discrete Sobolev spaces. By Fourier transform discrete lattice problems are converted to semi-discrete problems for which similar results are establish in semi-discrete Sobolev spaces. For lattice problems without absolute terms, additional conditions are imposed on data for the existence and uniqueness of solutions in discrete energy spaces in one,two and three dimensions. Two concrete examples are analyzed in the proposed mathematical framework. The mathematical framework, methodology and techniques in this paper can be utilized or generalized to non-periodic lattices on entire spaces and boundary value problems on lattices.
منابع مشابه
Alternating Regular Tree Grammars in the Framework of Lattice-Valued Logic
In this paper, two different ways of introducing alternation for lattice-valued (referred to as {L}valued) regular tree grammars and {L}valued top-down tree automata are compared. One is the way which defines the alternating regular tree grammar, i.e., alternation is governed by the non-terminals of the grammar and the other is the way which combines state with alternation. The first way is ta...
متن کاملTREE AUTOMATA BASED ON COMPLETE RESIDUATED LATTICE-VALUED LOGIC: REDUCTION ALGORITHM AND DECISION PROBLEMS
In this paper, at first we define the concepts of response function and accessible states of a complete residuated lattice-valued (for simplicity we write $mathcal{L}$-valued) tree automaton with a threshold $c.$ Then, related to these concepts, we prove some lemmas and theorems that are applied in considering some decision problems such as finiteness-value and emptiness-value of recognizable t...
متن کاملA COMMON FRAMEWORK FOR LATTICE-VALUED, PROBABILISTIC AND APPROACH UNIFORM (CONVERGENCE) SPACES
We develop a general framework for various lattice-valued, probabilistic and approach uniform convergence spaces. To this end, we use the concept of $s$-stratified $LM$-filter, where $L$ and $M$ are suitable frames. A stratified $LMN$-uniform convergence tower is then a family of structures indexed by a quantale $N$. For different choices of $L,M$ and $N$ we obtain the lattice-valued, probabili...
متن کاملKato's chaos and P-chaos of a coupled lattice system given by Garcia Guirao and Lampart which is related with Belusov-Zhabotinskii reaction
In this article, we further consider the above system. In particular, we give a sufficient condition under which the above system is Kato chaotic for $eta=0$ and a necessary condition for the above system to be Kato chaotic for $eta=0$. Moreover, it is deduced that for $eta=0$, if $Theta$ is P-chaotic then so is this system, where a continuous map $Theta$ from a compact metric space $Z$ to itse...
متن کاملINTRODUCTION AND DEVELOPMENT OF SURROGATE MANAGEMENT FRAMEWORK FOR SOLVING OPTIMIZATION PROBLEMS
In this paper, we have outlined the surrogate management framework for optimization of expensive functions. An initial simple iterative method which we call the “Strawman” method illustrates how surrogates can be incorporated into optimization to stand in for the most expensive function. These ideas are made rigorous by incorporating them into the framework of pattern search methods. The SMF al...
متن کامل