A posteriori joint detection of a recurring tuple of reference fragments in a quasi-periodic sequence

The problem of joint detection of a recurring tuple of reference fragments in a noisy numerical quasi-periodic sequence is solved in the framework of the a posteriori (off-line) approach. It is assumed that (i) the total number of fragments in the sequence is known, (ii) the index of the sequence member corresponding to the beginning of a fragment is a deterministic (not random) value, and (iii) a sequence distorted by an additive uncorrelated Gaussian noise is available for observation. It is shown that the problem consists of testing a set of simple hypotheses about the mean of a random Gaussian vector. A specific feature of the problem is that the cardinality of the set grows exponentially as the vector dimension (i.e., the length of the observed sequence) and the number of fragments in the sequence increase. It is established that the search for a maximum-likelihood hypothesis is equivalent to the search for arguments that maximize a special auxiliary objective function with linear inequality constraints. It is shown that this function is maximized by solving the basic extremum problem. It is proved that this problem is solvable in polynomial time. An exact algorithm for its solution is substantiated that underlies an algorithm guaranteeing optimal (maximum-likelihood) detection of a recurring tuple of reference fragments. The results of numerical simulation demonstrate the noise stability of the detection algorithm.     

The behavior of a finite difference approximation to a singular initial boundary value problem

We develop difference approximations to a singular parabolic initial-boundary value problem and its corresponding steady-state problem. A critical value for the existence of nonnegative solutions to the discrete steady state system is established. Convergence of the computed critical values is obtained. The long time behavior for the approximated solution of the parabolic problem is investigated. It is shown that the behavior of the discrete system is consistent with that of the continuous one     

Third order duality in nonlinear programming problems

Third order dual of a primal nonlinear programming problem is established which involves the third order derivatives of the functions constituting the primal problem. Desired duality theorems are provided for the pair of primal and the corresponding third order dual problem. Numerical examples are illustrated to justify the efficiency of the proposed method. It is also observed that some of the existing results are obtained as special cases.     

Optimal detection of a recurring tuple of reference fragments in a quasiperiodic sequence

An a posteriori (off-line) approach to solving the problem of maximum-likelihood detection of a recurring tuple containing reference fragments in a numerical quasiperiodic sequence is studied. The case is analyzed where (1) the total number of fragments in a sequence is unknown; (2) the index of a sequence term corresponding to the beginning of a fragment is a deterministic (not random) value; (3) a sequence distorted by additive uncorrelated Gaussian noise is available for observation. It is shown that the problem under consideration is reduced to testing a set of simple hypotheses about the mean of a random Gaussian vector. The cardinality of this totality grows exponentially as the vector dimension (i.e., the length of the sequence under study) increases. It is established that searching for a maximum-likelihood hypothesis is equivalent to finding arguments that yield a maximum for an auxiliary objective function. It is shown that maximizing the objective function reduces to solving a special optimization problem, which is proved to be solvable in polynomial time. An exact algorithm for solving this problem, which underlies the optimal (maximum-likelihood) detection algorithm for a recurring tuple, is substantiated. The kernel of the exact algorithm is an algorithm for solving a special (basic) optimization problem. Results of numerical simulations are presented.     

Law of large numbers and central limit theorem for randomly forced PDE's

We consider a class of dissipative PDE's perturbed by an external random force. Under the condition that the distribution of perturbation is sufficiently non-degenerate, a strong law of large numbers (SLLN) and a central limit theorem (CLT) for solutions are established and the corresponding rates of convergence are estimated. It is also shown that the estimates obtained are close to being optimal. The proofs are based on the property of exponential mixing for the problem in question and some abstract SLLN and CLT for mixing-type Markov processes.     

A non-linear wear-contact problem for a Winkler foundation with an increasing contact area

A wear-contact problem for a Winkler foundation is considered in the case when the rate of wear depends nonlinearly on the contact pressure and the contact area increases. The corresponding integral and differential equations are obtained. A successive approximation procedure is proposed which enables an exact solution of the problem in the space of continuous functions to be found. The property of non-negativity of the contact pressure when it has non-negative initial values is established. It is shown, using a qualitative analysis and calculations, that the non-linearity of the wear law can have a considerable effect on the behaviour of the contact pressure during the wear process.     

On a Nonlinear Eigenvalue Problem Related to the Theory of Propagation of Electromagnetic Waves

The eigenvalue problem is studied for a quasilinear second-order ordinary differential equation on a closed interval with Dirichlet’s boundary conditions (the corresponding linear problem has an infinite number of negative and no positive eigenvalues). An additional (local) condition imposed at one of the endpoints of the closed interval is used to determine discrete eigenvalues. The existence of an infinite number of (isolated) positive and negative eigenvalues is proved; their asymptotics is specified; a condition for the eigenfunctions to be periodic is established; the period is calculated; and an explicit formula for eigenfunction zeroes is provided. Several comparison theorems are obtained. It is shown that the nonlinear problem cannot be studied comprehensively with perturbation theory methods.     

Initial boundary and inverse problems for the inhomogeneous equation of a mixed parabolic-hyperbolic equation

A problem with inhomogeneous boundary and initial conditions is studied for an inhomogeneous equation of mixed parabolic-hyperbolic type in a rectangular domain. The solution is constructed as the sum of an orthogonal series. A criterion for the uniqueness of the solution is established. It is shown that the uniqueness of the solution and the convergence of the series depend on the ratio of the sides of the rectangle from the hyperbolic part of the mixed domain. On the basis of this problem, inverse problems for finding the factors of the time-dependent right-hand sides of the original equation of mixed type are stated and studied for the first time. The corresponding uniqueness theorems and the existence of solutions are proved using the theory of integral equations for inverse problems.     

The trace identity,a powerful tool for constructing the hamiltonian structure of integrable systems (II)

An isospectral problem with four potentials is discussed. The corresponding hierarchy of nonlinear evolution equations is derived. It is shown that the AKNS, Levi,D-AKNS hierarchies and a new one are reductions of the above hierarchy. In each case the relevant Hamiltonian form is established by making use of the trace identity.The project supported by National Natural Science Foundation Committee through Nankai Institute of Mathematics.     

Two types of hierarchies of evolution equations associated with the extended Kaup–Newell spectral problem with an arbitrary smooth function

In this paper we consider an extended Kaup–Newell (EKN) isospectral problem with an arbitrary smooth function and the corresponding two kinds of Lax integrable hierarchies by introducing two types of auxiliary spectral problems. The Hamiltonian structure of the second hierarchy is established. It is shown that the Hamiltonian system are integrable in Liouville’s sense and the set of Hamiltonian functions is the conserved densities of the second hierarchy, as well as they are in involutive in pairs under the Poisson bracket.     

Remote monitoring of environmental particulate pollution: a problem in inversion of first-kind integral equations

The possibility of remotely sensing the optical properties of scattering particulates from the variations in either the angular or the spectral characteristics, or both, of the radiation they transmit or scatter, is a problem of fundamental importance in the monitoring of environmental particulate pollution. It is shown that the corresponding problem is (or can be brought to) one of inverting first kind Fredholm integral equations. The solution to this problem would also enable one to follow the dynamical evolution of the polluted environment if it can be obtained in a time scale that is comparable to, or shorter than, the time constant of the physical measurements.The direct problem of how given physical parameters of such particles affect the transmission and scattering of incident radiation is first analyzed on the basis of the corresponding radiative transfer problem, including single and multiple scattering, and polarization induced on scattering. The various available methods for reconstructing the size distribution from the observed directly transmitted or scattered light are reviewed, particularly with regard to their main advantages and shortcomings. For direct light transmission, these include: library, iterative, and least-squares methods; the trial-and-error method; the matrix inversion method with smoothing constraint (MIM); the resolution accuracy trade-off method; and the analytical method. A minimization search method with smoothing constraint, an essential modification to the MIM, is also proposed. The corresponding methods for singly and multiply scattered light are likewise reviewed. The proposed forward-scattering method is shown to provide excellent reconstructions from either angular or spectral light measurements under proper experimental conditions. It can also be coupled with a minimization search in order to provide simultaneously the complex refractive index of the particles. The potentialities of other methods for the complete multiple scattering problem—the minimization search method, quasilinearization method, and small-angle Gaussian approximation method—are also studied.     

Global smooth solutions for a one-dimensional nonisentropic hydrodynamic model with non-constant lattice temperature

In this paper, a one-dimensional nonisentropic hydrodynamic model for semiconductors with non-constant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. The existence and uniqueness of the corresponding stationary solutions are investigated carefully under proper conditions. Then, global existence of the smooth solutions for the Cauchy problem with initial data, which are perturbations of stationary solutions, is established. It is shown that these smooth solutions tend to the stationary solutions exponentially fast as t → ∞.      

A posteriori joint detection of reference fragments in a quasi-periodic sequence

The problem of joint detection of quasi-periodic reference fragments (of given size) in a numerical sequence and its partition into segments containing series of recurring reference fragments is solved in the framework of the a posteriori approach. It is assumed that (i) the number of desired fragments is not known, (ii) an ordered reference tuple of sequences to be detected is given, (iii) the index of the sequence member corresponding to the beginning of a fragment is a deterministic (not random) value, and (iv) a sequence distorted by an additive uncorrelated Gaussian noise is available for observation. It is established that the problem consists of testing a set of hypotheses about the mean of a random Gaussian vector. The cardinality of the set grows exponentially as the vector dimension (i.e., the sequence length) increases. It is shown that the search for a maximum-likelihood hypothesis is equivalent to the search for arguments that minimize an auxiliary objective function. It is proved that the minimization problem for this function can be solved in polynomial time. An exact algorithm for its solution is substantiated. Based on the solution to an auxiliary extremum problem, an efficient a posteriori algorithm producing an optimal (maximum-likelihood) solution to the partition and detection problem is proposed. The results of numerical simulation demonstrate the noise stability of the algorithm.     

On minimax cardinal spline interpolation

Statistical Inference for Stochastic Processes - It is shown that in the problem of cardinal interpolation, spline interpolants of various degrees are R-minimax, with respect to corresponding...     

On the index of the boundary value problem for a homogeneous Hamiltonian system of differential equations

The index of the homogeneous self-adjoint boundary value problem for the Hamiltonian systems of ordinary differential equations is introduced. It is assumed that the system has a nontrivial solution. The relationship between the index of an eigenvalue of the nonlinear eigenvalue problem and the index of the corresponding homogeneous problem is established. Properties of the index of the problem and those of the eigenvalue are examined.     

The essential norm of an operator is not self-dual

It is established by an example that the distance of a bounded linear operatorS from the class of compact operators on a Banach space is not always uniformly comparable with that of its adjointS′. This provides a negative solution to an old problem. It is also shown that the seminorms due to Schechter and Weis, that measure the deviation from strict singularity or strict cosingularity of an operator, are not uniformly comparable with the corresponding distance functions. Both results rely on a general construction related to certain approximation properties that are associated with closed ideals of operators.     

Solution of the unconditional extremum problem for a linear-fractional integral functional on a set of probability measures

A new, generalized and strengthened, form of an assertion about an extremum of a linear-fractional integral functional given on a set of probability measures is presented. It is shown that the solution of the extremal problem for such a functional is completely determined by the extremal properties of the so-called test function, which is the ratio of the integrands of the numerator and the denominator. On the basis of this assertion, a theorem on an optimal strategy for controlling a semi-Markov process with a finite set of states is proved. In particular, it is established that if the test function of the objective functional of a control problem attains a global extremum, then an optimal control strategy exists, is deterministic, and is determined by the point of global extremum. The corresponding assertions are also obtained for the case where the test function does not attain the global extremum.     

The Dirichlet Problem for a Hyperboli-Type Equation with Power Degeneracy in a Rectangular Domain

The first boundary value problem is studied for a linear hyperbolic equation in a rectangle with a power degeneracy on one of its sides for different degrees of degeneracy. A uniqueness criterion is established. The solution is constructed as the sum of a Fourier series. The problem of small denominators arises when justifying the convergence of this series. In this connection, some estimates for the separation of the small denominators from zero are established with the corresponding asymptotics specified. It is these asymptotics that have made it possible to substantiate the existence of a regular problem solution.     

Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells

The natural vibrations of orthotropic shells are considered in a three-dimensional formulation for different versions of the boundary conditions on the faces: rigid clamping rigid clamping, rigid clamping free surface, and mixed conditions. Asymptotic solutions of the corresponding dynamic equations of the three-dimensional problem of the theory of elasticity are obtained. The principal values of the frequencies of natural vibrations are determined. It is shown that three types of natural vibrations occur in the shell: two shear vibrations and a longitudinal vibration, which are due solely to the boundary conditions on the faces. It is proved that each boundary layer has its own natural frequency. The boundary-layer functions are determined and the rates at which they decrease with distance from the faces inside the shell are established.     

Transport equations with boundary conditions of reverse reflection type

A study is performed of transport equations on arbitrary three-dimensional domains with boundary conditions of reverse reflection type. The existence of the dominant eigenvalue of the criticality problem is proved and its independence of the functional setting and its continuous dependence on a variety of data are established. The corresponding time-dependent problem is shown to be well-posed, also for a conservative boundary. The relationship between the criticality and the time-dependent problem is given explicitly.