Imbedding Technique Research Articles

Highly accurate compact difference scheme for fourth order parabolic equation with Dirichlet and Neumann boundary conditions: Application to good Boussinesq equation

In this work, a three-level implicit compact difference scheme for the generalised form of fourth order parabolic partial differential equation is developed. The discretization is derived by approximating the lower order derivative terms using the governing differential equation with the imbedding technique and is fourth order accurate in space and second order accurate in time. The current approach is advantageous since the boundary conditions are completely satisfied and no further approximations are required to be carried out at the boundaries. The ability of the proposed scheme in handling linear singular problems is examined. The value of first order space derivative is computed alongwith the solution so it does not have to be estimated using the calculated value of the solution. The method successfully works for the highly nonlinear good Boussinesq equation for which more accurate solutions are obtained for the single and the double-soliton solutions in comparison with the existing numerical methods.

Boundary Crossing Probabilities of Jump Diffusion Processes to Time-Dependent Boundaries

The finite Markov chain imbedding technique has been used to compute the boundary crossing probabilities of one and higher-dimensional Brownian motion. The idea is to cast the boundary crossing probabilities as the limiting probabilities of a finite Markov chain entering a set of absorbing states induced by the boundaries. In this manuscript, we extend the technique to compute the boundary crossing probabilities of a class of jump diffusion processes to time-dependent boundaries. We allow the jump sizes to have general distributions and the boundaries to be non-linear. Numerical examples are given to illustrate our theoretical results.

Some theoretical comments regarding the run-length properties of the synthetic and runs-rules monitoring schemes – part 2: steady-state

In this paper, the long term (also known as the steady-state mode) run-length theoretical properties of the four different types of synthetic and runs-rules monitoring schemes, that were empirically analysed in part 1 of this research work. That is, using the Markov chain imbedding technique (thoroughly discussed in Part 1 of this work), the closed-form expressions of the steady-state initial probabilities and average run-length (ARL) vectors are derived; so that the corresponding steady-state ARL and overall performance expressions, of each of the four different types of the synthetic and runs-rules monitoring schemes, may be formulated. Since there is very little literature on steady-state analysis of the synthetic and runs-rules charts, the closed-form expressions derived here will ease the understanding and implementation of the different types synthetic and runs-rules schemes in practice and in further academic research.

Invariant Imbedding and the Radiation Transfer in a Plane-Parallel Inhomogeneous Atmosphere

The invariant imbedding technique is applied to the problems of radiation transfer in a plane-parallel inhomogeneous atmosphere. All the parameters which describe the elementary event of scattering and the distribution of the energy sources are allowed to vary with depth. Mathematically, the considered standard problems of the theory are reduced to initial-value problems which are better adapted to capabilities of the modern high speed computers. The reflectance of an atmosphere is shown to play a prominent role in describing the diffusion process since all the other characteristics of the radiation field are expressed through it. Three transfer problems frequently encountered in astrophysical applications are discussed: the radiation diffusion in the source-free medium, in a medium with arbitrarily distributed energy sources, as well as the problem of finding the statistical mean quantities, characteristics of the multiple scattering in the atmosphere.

Joint distribution of rank statistics considering the location and scale parameters and its power study

The ranking method used for testing the equivalence of two distributions has been studied for decades and is widely adopted for its simplicity. However, due to the complexity of calculations, the power of the test is either estimated by a normal approximation or found when an appropriate alternative is given. Here, via the Finite Markov chain imbedding technique, we are able to establish the marginal and joint distributions of the rank statistics considering the shift and scale parameters, respectively and simultaneously, under two different continuous distribution functions. Furthermore, the procedures of distribution equivalence tests and their power functions are discussed. Numerical results of a joint distribution of rank statistics under the standard normal distribution and the powers for a sequence of alternative normal distributions with means from −20 to 20 and standard deviations from 1 to 9 and their reciprocal are presented. In addition, we discuss the powers of the rank statistics under the Lehmann alternatives.

First exit times and diffusion approximations for storage models with Poisson inputs, Poisson outputs and deterministic release rule

In this paper we discuss a number of finite storage problems with random inputs and random outputs together with linear release policy. This class of problems forms one dimensional master equations with separable kernel. For this class of problems the first passage time for overflow or emptiness can be transformed into ordinary differential equations. Closed form analytical solutions are obtained for first passage times for emptiness and overflow, treating the barriers as absorbing or reflecting. The imbedding technique is used to obtain the closed form solution for all the models. The results are derived in the form of third order differential equations for Laplace Transform functions for the first passage times. Diffusion approximation for these models is also obtained using suitable statistical conditions.

Recursive equations in finite Markov chain imbedding

In this paper, recursive equations for waiting time distributions of r-th occurrence of a compound pattern are studied via the finite Markov chain imbedding technique under overlapping and non-overlapping counting schemes in sequences of independent and identically distributed (i.i.d.) or Markov dependent multi-state trials. Using the relationship between number of patterns and r-th waiting time, distributions of number of patterns can also be obtained. The probability generating functions are also obtained. Examples and numerical results are given to illustrate our theoretical results.

On the Distribution of the Number of Occurrences of an Order-Preserving Pattern of Length Three in a Random Permutation

Recently, a considerable number of papers in computer science and mathematics examined the number of permutations containing exactly s occurrences of a prescribed order-preserving pattern (or forbidden pattern). It is well known that, mathematically, this is an NP-hard problem. Even in the simple case where the length of an order-preserving pattern is three, the number of permutations of size n containing s (s ≥ 3) order-preserving patterns remains unknown (see Fulmek Adv Appl Math 30(4):607–632, 2003). This manuscript provides a probabilistic approach to enumerate the number of permutations that contain exactly s occurrences of an order-preserving pattern of length three. The method is based on the insertion procedure of the finite Markov chain imbedding technique. Numerical results are provided to illustrate the theoretical results.

Distribution of the length of the longest common subsequence of two multi-state biological sequences

The length of the longest common subsequence (LCS) among two biological sequences has been used as a measure of similarity, and the application of this statistic is of importance in genomic studies. Even for the simple case of two sequences of equal length and composed of binary elements with equal state probabilities, the exact distribution of the length of the LCS remains an open question. This problem is also known as an NP-hard problem in computer science. Apart from combinatorial analysis, using the finite Markov chain imbedding technique, we derive the exact distribution for the length of the LCS between two multi-state sequences of different lengths. Numerical results are provided to illustrate the theoretical results.

On the Normal Approximation for the Distribution of the Number of Simple or Compound Patterns in a Random Sequence of Multi-state Trials

Distributions of numbers of runs and patterns in a sequence of multi-state trials have been successfully used in various areas of statistics and applied probability. For such distributions, there are many results on Poisson approximations, some results on large deviation approximations, but no general results on normal approximations. In this manuscript, using the finite Markov chain imbedding technique and renewal theory, we show that the number of simple or compound patterns, under overlap or non-overlap counting, in a sequence of multi-state trials follows a normal distribution. Poisson and large deviation approximations are briefly reviewed.

On exact Type I and Type II errors of Cochran's test

In this article, we obtain exact Type I and Type II error probabilities for Cochran's test comparing K different treatments using independent blocks with dichotomous responses. A finite Markov chain imbedding technique is used to construct a non-homogeneous Markov chain, and then the exact Type I and Type II error probabilities are cast in terms of the transition probabilities of the imbedded Markov chain. An example from Lehmann [1974. Nonparametrics: Statistical Methods Based on Ranks. Holden-Day Inc., San Francisco, p. 268] is given to illustrate our theorem.

The Exact and Limiting Distributions for the Number of Successes in Success Runs Within a Sequence of Markov-Dependent Two-State Trials

The total number of successes in success runs of length greater than or equal to k in a sequence of n two-state trials is a statistic that has been broadly used in statistics and probability. For Bernoulli trials with k equal to one, this statistic has been shown to have binomial and normal distributions as exact and limiting distributions, respectively. For the case of Markov-dependent two-state trials with k greater than one, its exact and limiting distributions have never been considered in the literature. In this article, the finite Markov chain imbedding technique and the invariance principle are used to obtain, in general, the exact and limiting distributions of this statistic under Markov dependence, respectively. Numerical examples are given to illustrate the theoretical results.

COMPARISONS BETWEEN CERTAIN QUEUING SYSTEMS

We use a Poisson imbedding technique to investigate the possibilities of generalizing some pathwise and stochastic monotonicity results from the M/M/k queues to systems with monotone failure rate service time distribution. A dichotomy between the decreasing failure rate (DFR) and the increasing failure rate (IFR) cases is revealed: In the DFR case, we achieve that the number of customers in the systems is stochastically increasing if it is idle at time 0, whereas in the IFR case, there is an alternating character that renders a bound between two identical systems that have different initial conditions. We also explore how our methods work in a comparison between systems with different numbers of stations but the same maximal capacity.

A hierarchical optimization neural network for large-scale dynamic systems

A recurrent neural network for dynamical hierarchical optimization of nonlinear discrete large-scale systems is presented. The proposed neural network consists of hierarchically structured sub-networks: one coordination sub-network at the upper level and several local optimization sub-networks at the lower level. In particular, the coordination sub-network and the local optimization sub-networks work simultaneously. This feature makes the proposed method outperform in computational efficiency the conventional iterative algorithms where there usually exists an alternately waiting time during the coordination and local optimization processes. Moreover, the state equations of the subsystems of the large-scale system are imbedded into their corresponding local optimization sub-networks. This imbedding technique not only overcomes the difficulty in treating the constraints imposed by the state equations, but also leads to significant reduction in the network size. We present stability analysis to prove that the neural network is asymptotically stable and this stable state corresponds to the optimal solution to the original optimal control problem. Finally, we illustrate the performance of the proposed method by an example.

Reconstruction of relaxation modulus for viscoelastic medium with complete data

The inverse scattering problem for viscoelastic medium with wave impedance mismatch at the rear boundary is studied in this paper. The propagation operators of the viscoelastic medium are defined first and the integro-differential equations of these operators are derived via the invariant imbedding technique. For the inverse scattering problem, a new inversion procedure to reconstruct the relaxation modulus of the viscoelastic medium is developed, which utilizes a complete set of data, namely the one-side reflection data for one round trip through the viscoelastic slab. These data are complete in the sense that they can be extended to arbitrary time t in the inversion procedure. This inverse method is implemented numerically on several problems at the end of the paper. The more general case of a medium consisting of a stack of homogeneous viscoelastic medium layers is also considered.

The exact distribution of the continuity of care measure NOP

The number of different providers (NOP) visited by one individual during a follow-up period can serve as a measure of concentration, and has formed the basis for a number of indices of continuity of care in the health care sector. In this article, the exact distribution of NOP, under the assumption of i.i.d. as well as under Markov dependence, is derived using the finite Markov chain imbedding technique. Numerical examples are given to illustrate the main results.

Time domain inversion of a viscoelastically restrained Timoshenko beam

This paper deals with the inverse scattering problem for a homogeneous Timoshenko beam suspended on a semi-infinite viscoelastic layer. The purpose is to derive the properties of the suspension from knowledge of the reflection data. The method used is the imbedding technique, which render solutions that are independent of the excitation. From a numerical point of view, the reflection equation in its usual form is inappropriate when solving the inverse problem. In order to bypass such complications, the reflection equation is modified and solved. In the numerical results, the properties of the layer are calculated using noisy reflection data.

Imbeddings of the tensor product of graphs where the second factor is a complete graph

In this paper we present genus result for the tensor product of graphs where the second factor is the complete graph K m and m is a power of 2. We are going to view the second factor as a Cayley graph and we will use an imbedding technique that combines surgery and voltage graph theory. In this imbedding technique we start with a suitable imbedding of H on some surface and then modify the edges of H according to the nature of K m to get a voltage graph H ∗ . The covering graph of this voltage graph H ∗ will be the desired tensor product of H with K m . The genus of H ⊗ K m , where H is a graph with a quadrilateral imbedding and also H is either, bipartite, with bichromatic dual, or with straight-ahead imbedding is given

THE IMBEDDING EQUATIONS FOR THE TIMOSHENKO BEAM

Wave reflection in a Timoshenko beam is treated, using wave splitting and the imbedding technique. The beam is assumed to be inhomogeneous and restrained by a viscoelastic suspension. The viscoelasticity is characterized by constitutive relations that involve the past history of deflection and rotation of the beam through memory functions of the suspension. By applying wave splitting, the propagating fields are decomposed into left-and right-moving parts. An integral representation of the split fields in impulse responses is presented. This representation gives the reflected and transmitted fields as convolutions of the incident field with the reflection and transmission kernels, respectively. The kernels are independent of the incident field and depend only on the material properties. Invariant imbedding is used to obtain equations for these kernels. In general, the kernels contain discontinuities for which transport equations are derived and solved. Some numerical solutions are presented for the reflection by a homogeneous beam suspended on two separated, semi-infinite layers of continuously distributed, viscoelastically damped, local acting springs.

Surface-wave mode coupling for efficient forward modelling and inversion of body-wave phases

SUMMARY The importance of surface-wave mode coupling in the modelling of body-wave phases by surface-wave mode summation is studied by means of sensitivity kernels obtained with the Born approximation and exact solutions of the Invariant Imbedding Technique. It is shown that, independent of the character of the lateral heterogeneity, surface-wave mode coupling is required to model body-wave phase perturbations and that neglecting intermode coupling, as in the WKBJ method for surface waves, can lead to large biases. Because methods which describe surfacewave mode coupling in an exact fashion are computationally too expensive to use in inversion schemes, the Scalar Exponent Approximation (SEA) is presented, which is a computationally efficient method and takes mode coupling into account. Since, instead of Earth normal modes, surface-wave modes are used, the summation over the angular order l is carried out analytically. This means that the number of modes and mode interactions needed is significantly reduced which assures an efficient manner of modelling. It is shown that the SEA is accurate in modelling body-wave phase perturbations for geophysically realistic configurations. Because, in contrast to the WKBJ sensitivity kernels, mode coupling introduces sensitivity kernels which also depend on the position along the source-receiver path, the SEA requires a larger model parameter set in inversions. A procedure is presented which reorganizes the model parameter set and leads to a reduced set of physically relevant model parameters. The combination of the SEA and the reorganization of the model parameters can be used efficiently in large-scale 3-D inversions which incorporate the important effects of surface-wave mode coupling.