The energy method for constructing time-harmonic solutions to the maxwell equations

We propose some minimum principle for an energy functional in an elliptic boundary value problem that arises in constructing time-harmonic solutions to the Maxwell equations. We suggest the potentials other than the vector and scalar potentials, used in the mathematical modeling of electromagnetic fields since the operators of traditional problems are not sign definite, which complicates constructions of iterative solution methods. We consider the problem in a parallelepiped whose boundary is ideally conducting. For nonresonant frequencies we prove that the operator of the boundary value problem is positive definite, propose a minimum principle for a quadratic energy functional, and prove the existence and uniqueness of generalized solutions.     

A polynomial time upper bound for the number of contacts in the HP-model on the face-centered-cubic lattice (FCC)

Lattice protein models are a major tool for investigating principles of protein folding. For this purpose, one needs an algorithm that is guaranteed to find the minimal energy conformation in some lattice model (at least for some sequences). So far, there are only algorithm that can find optimal conformations in the cubic lattice. In the more interesting case of the face-centered-cubic lattice (FCC), which is more protein-like, there are no results. One of the reasons is that for finding optimal conformations, one usually applies a branch-and-bound technique, and there are no reasonable bounds known for the FCC. We will give such a bound for Dill's HP-model on the FCC, which can be calculated by a dynamic programming approach.     

An improved hybrid global optimization method for protein tertiary structure prediction

First principles approaches to the protein structure prediction problem must search through an enormous conformational space to identify low-energy, near-native structures. In this paper, we describe the formulation of the tertiary structure prediction problem as a nonlinear constrained minimization problem, where the goal is to minimize the energy of a protein conformation subject to constraints on torsion angles and interatomic distances. The core of the proposed algorithm is a hybrid global optimization method that combines the benefits of the αBB deterministic global optimization approach with conformational space annealing. These global optimization techniques employ a local minimization strategy that combines torsion angle dynamics and rotamer optimization to identify and improve the selection of initial conformations and then applies a sequential quadratic programming approach to further minimize the energy of the protein conformations subject to constraints. The proposed algorithm demonstrates the ability to identify both lower energy protein structures, as well as larger ensembles of low-energy conformations.     

On the wellposedness of constitutive laws involving dissipation potentials

We consider a material with memory whose constitutive law is formulated in terms of internal state variables using convex potentials for the free energy and the dissipation. Given the stress at a material point depending on time, existence of a strain and a set of inner variables satisfying the constitutive law is proved. We require strong coercivity assumptions on the potentials, but none of the potentials need be quadratic.

As a technical tool we generalize the notion of an Orlicz space to a cone ``normed' by a convex functional which is not necessarily balanced. Duality and reflexivity in such cones are investigated.

     

Multiexponential models of (1+1)-dimensional dilaton gravity and Toda-Liouville integrable models

We study general properties of a class of two-dimensional dilaton gravity (DG) theories with potentials containing several exponential terms. We isolate and thoroughly study a subclass of such theories in which the equations of motion reduce to Toda and Liouville equations. We show that the equation parameters must satisfy a certain constraint, which we find and solve for the most general multiexponential model. It follows from the constraint that integrable Toda equations in DG theories generally cannot appear without accompanying Liouville equations. The most difficult problem in the two-dimensional Toda-Liouville (TL) DG is to solve the energy and momentum constraints. We discuss this problem using the simplest examples and identify the main obstacles to solving it analytically. We then consider a subclass of integrable two-dimensional theories where scalar matter fields satisfy the Toda equations and the two-dimensional metric is trivial. We consider the simplest case in some detail. In this example, we show how to obtain the general solution. We also show how to simply derive wavelike solutions of general TL systems. In the DG theory, these solutions describe nonlinear waves coupled to gravity and also static states and cosmologies. For static states and cosmologies, we propose and study a more general one-dimensional TL model typically emerging in one-dimensional reductions of higher-dimensional gravity and supergravity theories. We especially attend to making the analytic structure of the solutions of the Toda equations as simple and transparent as possible.     

Zero-measure Cantor spectrum for Schrödinger operators with low-complexity potentials

We consider discrete one-dimensional Schrödinger operators whose potentials belong to minimal subshifts of low combinatorial complexity and prove for a large class of such operators that the spectrum is a Cantor set of zero Lebesgue measure. This is obtained through an analysis of the frequencies of the subwords occurring in the potential. Our results cover most circle map and Arnoux–Rauzy potentials.     

Statistical aspects of random fragmentations

Three random fragmentation of an interval processes are investigated. For each of them, there is a splitting probability and a probability not to split at each step of the fragmentation process whose overall effect is to stabilize the global number of splitting events. Some of their statistical features are studied in each case among which fragments’ size distribution, partition function, structure of the underlying random fragmentation tree, occurrence of a phase transition. In the first homogeneous model, splitting probability does not depend on fragments’ size at each step. In the next two fragmentation models, splitting probability is fragments’ length dependent. In the first such models, fragments further split with probability one if their sizes exceed some cutoff value only; in a second model considered, splitting probability of finite-size objects is assumed to increase algebraically with fragments’ size at each step. The impact of these dependencies on statistical properties of the resulting random partitions are studied. Several examples are supplied.     

Vacuum and Thermal Energies for Two Oscillators Interacting Through A Field

We consider a simple (1+1)-dimensional model for the Casimir–Polder interaction consisting of two oscillators coupled to a scalar field. We include dissipation in a first-principles approach by allowing the oscillators to interact with heat baths. For this system, we derive an expression for the free energy in terms of real frequencies. From this representation, we derive the Matsubara representation for the case with dissipation. We consider the case of vanishing intrinsic frequencies of the oscillators and show that the contribution from the zeroth Matsubara frequency is modified in this case and no problem with the laws of thermodynamics appears.     

Likelihood robust optimization for data-driven problems

We consider optimal decision-making problems in an uncertain environment. In particular, we consider the case in which the distribution of the input is unknown, yet there is some historical data drawn from the distribution. In this paper, we propose a new type of distributionally robust optimization model called the likelihood robust optimization (LRO) model for this class of problems. In contrast to previous work on distributionally robust optimization that focuses on certain parameters (e.g., mean, variance, etc.) of the input distribution, we exploit the historical data and define the accessible distribution set to contain only those distributions that make the observed data achieve a certain level of likelihood. Then we formulate the targeting problem as one of optimizing the expected value of the objective function under the worst-case distribution in that set. Our model avoids the over-conservativeness of some prior robust approaches by ruling out unrealistic distributions while maintaining robustness of the solution for any statistically likely outcomes. We present statistical analyses of our model using Bayesian statistics and empirical likelihood theory. Specifically, we prove the asymptotic behavior of our distribution set and establish the relationship between our model and other distributionally robust models. To test the performance of our model, we apply it to the newsvendor problem and the portfolio selection problem. The test results show that the solutions of our model indeed have desirable performance.     

Electrical Conductivity of Charged Particle Systems and Zubarev’s Nonequilibrium Statistical Operator Method

One of the fundamental problems in physics that are not yet rigorously solved is the statistical mechanics of nonequilibrium processes. An important contribution to describing irreversible behavior starting from reversible Hamiltonian dynamics was given by D. N. Zubarev, who invented the method of the nonequilibrium statistical operator. We discuss this approach, in particular, the extended von Neumann equation, and as an example consider the electrical conductivity of a system of charged particles. We consider the selection of the set of relevant observables. We show the relation between kinetic theory and linear response theory. Using thermodynamic Green’s functions, we present a systematic treatment of correlation functions, but the convergence needs investigation. We compare different expressions for the conductivity and list open questions.     

Exploring the dynamics of financial markets: from stock prices to strategy returns

Exploring the dynamics of financial time-series is an exciting and interesting challenge because of the many truly complex interactions that underly the price formation process. In this contribution we describe some of the anomalous statistical features of such time-series and review models of the price dynamics both across time and across the universe of stocks. In particular we discuss a non-Gaussian statistical feedback process of stock returns which we have developed over the past years with the particular application of option pricing. We then discuss a cooperative model for the correlations of stock dynamics which has its roots in the field of synergetics, where numerical simulations and comparisons with real data are presented. Finally we present summarized results of an empirical analysis probing the dynamics of actual trading strategy return streams.     

The application of the genetic algorithm to the minimization of potential energy functions

We adapted the genetic algorithm to minimize the AMBER potential energy function. We describe specific recombination and mutation operators for this task. Next we use our algorithm to locate low energy conformation of three polypeptides (AGAGAGAGA, A9, and [Met]-enkephalin) which are probably the global minimum conformations. Our potential energy minima are –94.71, –98.50, and –48.94 kcal/mol respectively. Next, we applied our algorithm to the 46 amino acid protein crambin and located a non-native conformation which had an AMBER potential energy 150 kcal/mol lower than the native conformation. This is not necessarily the global minimum conformation, but it does illustrate problems with the AMBER potential energy function. We believe this occurred because the AMBER potential energy function does not account for hydration.     

Higher Dimensional Aztec Diamonds and a (2d + 2)-Vertex Model

Motivated by the close relationship between the number of perfect matchings of the Aztec diamond graph introduced in [5] and the free energy of the square-ice model, we consider a higher dimensional analog of this phenomenon. For d 1, we construct d-uniform hypergraphs which generalize the Aztec diamonds and we consider a companion d-dimensional statistical model (called the 2d + 2-vertex model) whose free energy is given by the logarithm of the number of perfect matchings of our hypergraphs. We prove that the limit defining the free energy per site of the 2d + 2-vertex model exists and we obtain bounds for it. As a consequence, we obtain an especially good asymptotical approximation for the number of matchings of our hypergraphs.     

Recent advances on the interval distance geometry problem

We discuss a discretization-based solution approach for a classic problem in global optimization, namely the distance geometry problem (DGP). We focus our attention on a particular class of the DGP which is concerned with the identification of the conformation of biological molecules. Among the many relevant ideas for the discretization of the DGP in the literature, we identify the most promising ones and address their inherent limitations to application to this class of problems. The result is an improved method for estimating 3D structures of small proteins based only on the knowledge of some distance restraints between pairs of atoms. We present computational results showcasing the usefulness of the new proposed approach. Proteins act on living cells according to their geometric and chemical properties: finding protein conformations can be very useful within the pharmaceutical industry in order to synthesize new drugs.     

Statistical characteristics of continuous functions and statistically weakly invariant sets of controllable system

We continue the investigation of expansion of a concept of invariance for sets which consists in studying statistically invariant sets with respect to control systems and differential inclusions. We consider the statistical characteristics of continuous functions: Upper and lower relative frequency of containing for graph of a function in a given set. We obtain conditions under which statistical characteristics of two various asymptotical equivalent functions coincide; then by the value of one of them it is possible to calculate the value of another one. We adduce the equality for finding relative frequencies of hitting functions the given set in the case when the distance from the graph of one of functions to the given set is a periodic function. A consequence of these statements are conditions of statistically weak invariance of a set with respect to controlled system. For some almost periodic functions we obtain the formulas by which we can calculate the mean values and the statistical characteristics. We also consider the following problem. Let the number λ₀ ∈ [0, 1] be given. It is necessary to find the value c(λ₀) such that the upper solution z(t) of the Cauchy problem does not exceed c(λ₀) with the relative frequency being equal λ₀. Depending on statement of the problem, a value z(t) can be interpreted as the size of population, energy of a particle, concentration of substance, size of manufacture or the price of production.     

Convergence of analytic gradient-type systems with periodicity and its applications in Kuramoto models

We consider the convergence of gradient-type systems with periodic and analytic potentials. The main tool is the celebrated Łojasiewicz inequality which is valid for any analytic function. Our results show that the convergence of such systems with periodic and analytic potentials is unconditional to the initial data; in other words, any trajectory converges to some equilibrium. As direct applications, we can show that any trajectory converges to phase-locked state for the first- and second-order Kuramoto models on a symmetric network with attractive–repulsive forces and identical natural frequencies. In particular, the inertial Kuramoto model with identical oscillators converges to phase-locked state for any initial configuration.     

Capillary oscillations at a circular orifice

We compute the normal frequencies and normal modes for the oscillation of the free surface of a perfect incompressible fluid inside a semi-infinite container with a circular orifice. In doing that, a dual integral equation system involving the Bessel functions must be solved. We discuss the cases where the contact line between the free surface and the container is pinned as well as the case where it moves with a constant contact angle.     

Weakly Gibbsian representations for joint measures of quenched lattice spin models

Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spin-space and disorder-space be represented as (suitably generalized) Gibbs measures of an “annealed system”? - We prove that there is always a potential (depending on both spin and disorder variables) that converges absolutely on a set of full measure w.r.t. the joint measure (“weak Gibbsianness”). This “positive” result is surprising when contrasted with the results of a previous paper [K6], where we investigated the measure of the set of discontinuity points of the conditional expectations (investigation of “a.s. Gibbsianness”). In particular we gave natural “negative” examples where this set is even of measure one (including the random field Ising model). Further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint potential in terms of the decay of the disorder average over certain quenched correlations. We apply them to various examples. From this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero. Our proof uses a martingale argument that allows to cut (an infinite-volume analogue of) the quenched free energy into local pieces, along with generalizations of Kozlov's constructions. Received: 11 November 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000 RID="*" ID="*" Work supported by the DFG Schwerpunkt `Wechselwirkende stochastische Systeme hoher Komplexit?t'     

威布尔分布组与删失数据下最大似然估计的存在性

本文研究寿命服从威布尔分布,观测数据分组与可能删失的情况下,最大似然估计的存在性,针对所有数据类型,我们给出了最大似然估计存在性的一个充分必要条件,文章结尾讨论了仅一个失效数据时最大似然估计的计算。     

Distributions for the risk process with a stochastic return on investments

In this paper, we consider a risk model with stochastic return on investments. We mainly discuss the ruin probability, the surplus distribution at the time of ruin and the supremum distribution of the surplus before ruin. We prove some properties for these distributions and derive the integro-differential equations satisfied by them. We present the relation between the ruin probability and the supremum distribution before ruin.