Absorption spectrum of a one-dimensional chain with frenkel’s excition under diagonal disorder represented by hyperbolic defects

A method is proposed for calculating the absorption spectrum of a long one-dimensional closed-into-a-ring chain with Frenkel’s exciton under diagonal disorder as a function of atomic absorption. This disorder is represented by the hyperbolic singularities of atomic fission. These defects are shown to lead to a wing in the exciton zone of a chain without defects. The form of the wing does not depend on the relative positions or number of the defects and its value is proportional to the sum of the amplitudes of the defects. The proposed method uses only the continual approximation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 331–337, November, 1997.     

Specific quantum mechanical solution of difference equation of hyperbolic type

Difficulties connected to solving difference equations of hyperbolic type were analyzed in this work and discussed in detail. The results are compared to those of the standard wave equation and certain similarities were established. The method of solving the equation is generalized by means of kernel expanded into separable polynomials. The analysis was inspired by some new ideas concerning quantization of time. Two examples are given: excitons and phonons in thin crystalline films. The advanced methodology of Green’s function method and the application of this new methodology resulted in a set of interesting conclusions concerning thin film properties. The significance of the obtained spatial dependence of exciton concentration was discussed and it was concluded, on the basis of the found spatial dependence of exciton concentration, that such boundary conditions of a thin molecular film which lead to high exciton concentrations can be determined. It was also concluded that thin films possess high superconductive properties, that physical characteristics of thin films are spatially dependent and that the spatial dependence can be the basis for widening the field of application of nanostructures.     

Collective vibrations in a three-band superconductor

Collective vibrations in a superconductor with three overlapping energy bands on the Fermi surface are investigated. It is shown that for certain relationships between the parameters of the theory there can, in addition to the acoustic branch, be two different branches of exciton type, which arise because of fluctuations of the phases of the order parameters.Institute of Applied Physics, Moldova Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 101–110, April, 1993.     

Stochastic algorithm for solving transient diffusion equations with a precise accounting of reflection boundary conditions on a substrate surface

A new random walk based stochastic algorithm for solving transient diffusion equations in domains where a reflection boundary condition is imposed on a plane part of the boundary is suggested. The motivation comes from the field of exciton transport and recombination in semiconductors where the reflecting boundary is the substrate plane surface while on the defects and dislocations an absorption boundary condition is prescribed. The idea of the method is based on the exact representations of the first passage time and position distributions on a parallelepiped (or a cube) with a reflection condition on its bed face lying on the substrate. The algorithm is meshfree both in space and time, the particle trajectories are moving inside the domain in accordance with the Random Walk on Spheres (RWS) process but when approaching the reflecting surface they switch to move on parallelepipeds (or cubes). The efficiency of the method is drastically increased compared with the standard RWS method. For illustration, we present an example of exciton flux calculations in the cathodoluminescence imaging method in semiconductors with a set of threading dislocations.     

On the asymptotic solutions of the coupled quasiparticle-oscillator system

A coupled quasiparticle-oscillator system is considered for an arbitrary number of excitons. The exciton dynamics is described in terms of the second quantization (i.e. by means the bosonic operators). As a consequence a radius of a Bloch sphere is obtained different to the previous results. Some integrals of motion are obtained that allowed to reduce the system of equations of motion to a single nonlinear ordinary differential equation of the fourth order. This equation contains the energy of the system as a parameter. The fixed points are found as a functions of the energy of the system, and its stability properties are investigated. It is demonstrated that a bifurcation is presented for the energies H<−1/2p. An asymptotic quasiclassical solution around fixed point for the case H>−1/2p is obtained. The solutions around other stable fixed points can be obtained analogously. The expression for the evolution operator of the quasiparticle-oscillator system is obtained as a functional on the classical solutions.     

Collective oscillations in doped two-band superconductors

A study is made of the effect of a nonmagnetic impurity on the collective oscillations in two-band superconductors due to fluctuations in the phases of the order parameters of the different bands. A system of equations for the vertex functions that is valid for any carrier concentration is obtained. On the basis of this system, an equation is obtained for the frequency of collective excotations of a two-band supercodductor for phonon and nonphonon mechanisms of superconductivity. It is shown that at a low impurity concentration in two-band system two frequencies of exciton type determined by the presence of interband interaction and the impurity arise.Institute of Applied Physics, Moldovan Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 2, pp. 312–327, May, 1995.     

First passage Monte Carlo algorithms for solving coupled systems of diffusion–reaction equations

We suggest in this letter a new Random Walk on Spheres (RWS) stochastic algorithm for solving systems of coupled diffusion–reaction equations where the random walk is living both on the randomly walking spheres and inside the relevant balls. The method is mesh free both in space and time, and is well applied to solve high-dimensional problems with complicated domains. The algorithms are based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for balls and spheres. They can be conveniently used not only for the solutions, but also for a direct calculation of fluxes to any part of the boundary without calculating the whole solution in the domain. Some applications to exciton flux calculations in the diffusion imaging method in semiconductors are discussed.     

Center-of-mass tomography and probability representation of quantum states for tunneling

We develop a representation of quantum states in which the states are described by fair probability distribution functions instead of wave functions and density operators. We present a one-random-variable tomography map of density operators onto the probability distributions, the random variable being analogous to the center-of-mass coordinate considered in reference frames rotated and scaled in the phase space. We derive the evolution equation for the quantum state probability distribution and analyze the properties of the map. To illustrate the advantages of the new tomography representations, we describe a new method for simulating nonstationary quantum processes based on the tomography representation. The problem of the nonstationary tunneling of a wave packet of a composite particle, an exciton, is considered in detail.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 371–387, February, 2005.     

Short-Time Nonlinear Effects in the Exciton-Polariton System

In the exciton-polariton system, a linear dispersive photon field is coupled to a nonlinear exciton field. Short-time analysis of the lossless system shows that, when the photon field is excited, the time required for that field to exhibit nonlinear effects is longer than the time required for the nonlinear Schrödinger equation, in which the photon field itself is nonlinear. When the initial condition is scaled by \(\epsilon ^\alpha \), it is found that the relative error committed by omitting the nonlinear term in the exciton-polariton system remains within \(\epsilon \) for all times up to \(t=C\epsilon ^\beta \), where \(\beta =(1-\alpha (p-1))/(p+2)\). This is in contrast to \(\beta =1-\alpha (p-1)\) for the nonlinear Schrödinger equation. The result is proved for solutions in \(H^s(\mathbb {R}^n)\) for \(s>n/2\). Numerical computations indicate that the results are sharp and also hold in \(L^2(\mathbb {R}^n)\).     

Using implicitly filtered RKS for generalised eigenvalue problems

The rational Krylov sequence (RKS) method can be seen as a generalisation of Arnoldi's method. It projects a matrix pencil onto a smaller subspace; this projection results in a small upper Hessenberg pencil. As for the Arnoldi method, RKS can be restarted implicitly, using the QR decomposition of a Hessenberg matrix. This restart comes with a projection of the subspace using a rational function. In this paper, it is shown how the restart can be worked out in practice. In a second part, it is shown when the filtering of the subspace basis can fail and how this failure can be handled by deflating a converged eigenvector from the subspace, using a Schur-decomposition.     

Causal compositional models in valuation-based systems with examples in specific theories

We show that Pearl's causal networks can be described using causal compositional models (CCMs) in the valuation-based systems (VBS) framework. One major advantage of using the VBS framework is that as VBS is a generalization of several uncertainty theories (e.g., probability theory, a version of possibility theory where combination is the product t-norm, Spohn's epistemic belief theory, and Dempster–Shafer belief function theory), CCMs, initially described in probability theory, are now described in all uncertainty calculi that fit in the VBS framework. We describe conditioning and interventions in CCMs. Another advantage of using CCMs in the VBS framework is that both conditioning and intervention can be easily described in an elegant and unifying algebraic way for the same CCM without having to do any graphical manipulations of the causal network. We describe how conditioning and intervention can be computed for a simple example with a hidden (unobservable) variable. Also, we illustrate the algebraic results using numerical examples in some of the specific uncertainty calculi mentioned above.     

Multivariate process capability via Löwner ordering

Probability bounds can be derived for distributions whose covariance matrices are ordered with respect to Löwner partial ordering, a relation that is based on whether the difference between two matrices is positive definite. One example is Anderson’s Theorem. This paper develops a probability bound that follows from Anderson’s Theorem that is useful in the assessment of multivariate process capability. A statistical hypothesis test is also derived that allows one to test the null hypothesis that a given process is capable versus the alternative hypothesis that it is not capable on the basis of a sample of observed quality characteristic vectors from the process. It is argued that the proposed methodology is viable outside the multivariate normal model, where the p-value for the test can be computed using the bootstrap. The methods are demonstrated using example data, and the performance of the bootstrap approach is studied empirically using computer simulations.     

Self-Replicating Tiles and Their Boundary

All self-replicating lattice tilings of the plane can be constructed using special iterated function systems (IFS). Certain self-replicating curves can be constructed using the recurrent set method (RS). A bijection between the IFS parameters and the RS parameters is such that the curve K produced by the RS parameters is the boundary of the tile T produced by the IFS parameters. The correspondence is algorithmic in that K can be drawn from the IFS data using turtle graphics and T can be drawn from the RS data using an IFS iteration. Received April 15, 1997, and in revised form November 13, 1997, and April 6, 1998.     

Deterministic optimal control,given only a partial observation of the initial state

The deterministic linear-system, quadratic-cost optimal control problem is considered when the only state information available is a partial linear observation of the initial statex ₀. Thus, it is only known that the initial condition belongs to a particular linear variety. A control function is found which is optimal, in the sense (roughly) that (i) it can be computed using available information aboutx ₀ and (ii) no other control function which can be found using that information gives lower cost than it does for every initial condition that could have given rise to the information. The optimal control can be found easily from the conventional Riccati equation of optimal control. Applications are considered in the presence of unknown exponential disturbances and to the case with a sequence of partial state observations.     

Indecomposable totally positive numbers in real quadratic orders

In a totally real number field, every totally positive integral number is a finite sum of (additively) indecomposable totally positive integral numbers, and up to multiplication by totally positive units, there exist only finitely many indecomposables. In the paper it is shown that in quadratic fields all these numbers can be listed in a very efficient way by using the so-called intermediate convergents of a certain quadratic irrationality. The method can be viewed as a simple extension of the standard method of calculating the fundamental unit by using continued fractions. As an application it is shown that for instance in Z|√d| a number is decomposable if its norm is >d. It is remarkable that this bound does not depend on the size of the fundamental unit.     

“I can name that Bayesian network in two matrixes!”

For a number of situations, a Bayesian network can be split into a core network consisting of a set of latent variables describing the status of a system, and a set of fragments relating the status variables to observable evidence that could be collected about the system state. This situation arises frequently in educational testing, where the status variables represent the student proficiency and the evidence models (graph fragments linking competency variables to observable outcomes) relate to assessment tasks that can be used to assess that proficiency. The traditional approach to knowledge engineering in this situation would be to maintain a library of fragments, where the graphical structure is specified using a graphical editor and then the probabilities are entered using a separate spreadsheet for each node. If many evidence model fragments employ the same design pattern, a lot of repetitive data entry is required. As the parameter values that determine the strength of the evidence can be buried on interior screens of an interface, it can be difficult for a design team to get an impression of the total evidence provided by a collection of evidence models for the system variables, and to identify holes in the data collection scheme. A Q-matrix - an incidence matrix whose rows represent observable outcomes from assessment tasks and whose columns represent competency variables - provides the graphical structure of the evidence models. The Q-matrix can be augmented to provide details of relationship strengths and provide a high level overview of the kind of evidence available. The relationships among the status variables can be represented with an inverse covariance matrix; this is particularly useful in models from the social sciences as often the domain experts’ knowledge about the system states comes from factor analyses and similar procedures that naturally produce covariance matrixes. The representation of the model using matrixes means that the bulk of the specification work can be done using a desktop spreadsheet program and does not require specialized software, facilitating collaboration with external experts. The design idea is illustrated with some examples from prior assessment design projects.     

Pairings for cryptographers

Many research papers in pairing-based cryptography treat pairings as a “black box”. These papers build cryptographic schemes making use of various properties of pairings. If this approach is taken, then it is easy for authors to make invalid assumptions concerning the properties of pairings. The cryptographic schemes developed may not be realizable in practice, or may not be as efficient as the authors assume.The aim of this paper is to outline, in as simple a fashion as possible, the basic choices that are available when using pairings in cryptography. For each choice, the main properties and efficiency issues are summarized. The paper is intended to be of use to non-specialists who are interested in using pairings to design cryptographic schemes.     

Multiple scattering and modified Green's functions

Exterior boundary-value problems for the Helmholtz equation can be reduced to boundary integral equations. It is known that the simplest of these fail to be uniquely solvable at certain ‘irregular frequencies.’ For a single smooth scatterer, it is also known that irregular frequencies can be eliminated by using a modified fundamental solution, one that has additional singularities inside the scatterer. This approach is extended to treat the three-dimensional exterior Neumann problem for any finite number of disjoint smooth scatterers, using a fundamental solution that has additional singularities inside every scatterer.     

Finding the K best policies in a finite-horizon Markov decision process

Directed hypergraphs represent a general modelling and algorithmic tool, which have been successfully used in many different research areas such as artificial intelligence, database systems, fuzzy systems, propositional logic and transportation networks. However, modelling Markov decision processes using directed hypergraphs has not yet been considered.In this paper we consider finite-horizon Markov decision processes (MDPs) with finite state and action space and present an algorithm for finding the K best deterministic Markov policies. That is, we are interested in ranking the first K deterministic Markov policies in non-decreasing order using an additive criterion of optimality. The algorithm uses a directed hypergraph to model the finite-horizon MDP. It is shown that the problem of finding the optimal policy can be formulated as a minimum weight hyperpath problem and be solved in linear time, with respect to the input data representing the MDP, using different additive optimality criteria.