Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non-locality (dispersion) associated with an internal length scale and localized dissipation due

In this paper computational aspects of the mathematical modelling of dynamic system evolution have been considered as a problem in information theory. The construction of such models is treated as a decision making process with limited available information. The solution of the problem is associated with a computational model based on heuristics of a Markov Chain in a discrete space-time of events. A stable approximation of the chain has been derived and the limiting cases are discussed. An intrinsic interconnection of constructive, sequential, and evolutionary approaches in related optimization problems provides new challenges for future work. Key words: decision making with limited information, optimal control theory, hyperbolicity of dynamic rules, generalized dynamic systems, Markov Chain approximation. 1 Introduction Many mathematical problems in information theory and optimal control related to dynamic system studies can be formulated in the following generic form. A decision...

This work contains review of original quantum Hierarchic theory of condensed matter, general for liquids and solids and its numerous branches. Computer program (copyright, 1997, Kaivarainen), based on new theory, was used for comprehensive simulations of water and ice physical properties. Condensed matter is considered as gas of 3D standing waves (collective excitations) of different nature: thermal de Broglie waves (waves B), IR photons and thermal phonons. Quantitative interrelation between microscopic, mesoscopic (as intermediate) and macroscopic properties of condensed matter are demonstrated. New theories of total internal energy, including contributions of kinetic and potential energy, heat capacity, surface tension, vapor pressure, thermal conductivity, viscosity and self-diffusion are described. Hierarchic theory of osmotic pressure, based on new state equation, new theories of light refraction, Brillouin light scattering and Mössbauer effect are presented also in article and compared with available experimental data for water and ice. Lot of hidden parameters, inaccessible for experiment, describing the dynamic and spatial properties of 24 quantum collective excitations of matter, can be calculated also, as demonstrated on examples of water and ice. Total number of physical parameters of liquids and solids in wide T-interval, including that of phase transitions, to be possible to evaluate using CAMP computer program, is about 300. The agreement between theoretical and available experimental results is very good. The evidence of high-T mesoscopic molecular Bose condensation (BC) in water and ice in form of coherent clusters is obtained. The new mechanisms of the 1st and 2nd order phase transitions, related to such clusters formation/melting, their assembly/disassembly and symmetry change is proposed. Theory unifies dynamics and thermodynamics on microscopic, mesoscopic and macroscopic scales in terms of quantum physics. The idea of new optoacoustic device: Comprehensive Analyzer of Matter Properties (CAMP) with huge informational possibilities, based on computer program, elaborated and its multisided applications are described. This work may be considered as a

In this thesis we consider two classical combinatorial problems arising in statistical mechanics: counting matchings and self-avoiding walks in lattice graphs. The first problem arises in the study of the thermodynamical properties of monomers and dimers (diatomic molecules) in crystals. Fisher, Kasteleyn and Temperley discovered an elegant technique to exactly count the number of perfect matchings in two dimensional lattices, but it is not applicable for matchings of arbitrary size, or in higher dimensional lattices. We present the first efficient approximation algorithm for computing the number of matchings of any size in any periodic lattice in arbitrary dimension. The algorithm is based on Monte Carlo simulation of a suitable Markov chain and has rigorously derived performance guarantees that do not rely on any assumptions. In addition, we show that these results generalize to counting matchings in any graph which is the Cayley graph of a finite group. The second problem is countin...

This paper studies the ensuing arithmetic classification in the simplest three-dimensional case, that is, monoatomic 2-lattices (triply periodic structures with two indistinguishable points in their unit cell). We show that there exist 29 distinct arithmetic types of monoatomic 2-lattices. As all monoatomic 2lattices are constituted by a single crystallographic orbit, these structures are also classified by the established criterion of Fischer and Koch (1975, 1996); the two classifications coincide. Among the 29 types we retrieve the 5 Strukturberichte that are themselves monoatomic 2-lattices, which indicates how many more such structures there exist in theory (to compare, recall that the arithmetic types of 1-lattices are the classical 14 Bravais types, among which 6 are listed as Strukturberichte) . 1. Introduction 1.1. Background

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In this paper we describe all the possibilities for symmetry breaking transformations in monoatomic 2-lattices (crystal structures with two identical points in their unit translational cell). This is done by establishing the symmetry hierarchies (partial ordering) for the arithmetic classes of symmetry groups of these crystals (Fig. 1). We also study the ‘Ericksen-Pitteri neighbourhoods ’ for them, thus making a local analysis of their configuration space. We give details about two physically relevant cases, analysing the neighbourhoods and the possible symmetry-breaking mechanisms for the diamond and the hexagonal close-packed structures (Figs. 2 and 3). 1

This paper deals with the theory of critical exponents and whether and how these exponents can be influenced if one includes disorder in a given pure system. Moreover, the renormalization group flow is examined in a more general framework. The paper itself consists of three parts. In the first part the well-known results for the Ising system, given by renormalization, are reviewed. After the introduction of the basic concepts and definitions of phase transitions and phenomenological Landau theory, an expression for the Landau free energy of the Ising system is derived. Once this result is obtained, the correspondence to the O(N) symmetric OE

This paper is divided in three parts. In the first part, we consider the functional a i,j (x) # i u # j u + Q(x) # (-1,1) (u) dx , for a i,j and Q periodic under integer translations. We assume that Q is bounded and bounded away from zero, and a i,j is a bounded elliptic matrix. We prove that there exists a universal constant M 0 , depending only on n, the bounds on a i,j and Q stated above, such that: given any # , there exists a class A minimizer u for the functional |#|]}.