Narrow Results

We have designed and built a special purpose processor with a very good performance to price ratio, which permits to propose a new way for parallel computing. A simple one spin flip Monte Carlo algorithm is realized in hardware, so the processor is suitable for studies of dynamic as well as thermodynamic properties of the two-dimensional Ising model with different types of inhomogeneities. The speed of the processor is defined completely by the speed of memories used in it: to perform an elementary Monte Carlo step the processor needs a time only several percent larger than one memory cycle time. So it realizes the fastest possible one spin flip Monte Carlo processor architecture.

We describe the architecture of the special purpose processor built to realize in hardware cluster Wolff algorithm, which is not hampered by a critical slowing down. The processor simulates two-dimensional Ising-like spin systems. With minor changes the same very effective architecture, which can be defined as a Memory Machine, can be used to study phase transitions in a wide range of models in two or three dimensions.

We describe a Special Purpose Processor, realizing the Wolff algorithm in hardware, which is fast enough to study the critical behaviour of 2D Ising-like systems containing more than one million spins. The processor has been checked to produce correct results for a pure Ising model and for Ising model with random bonds. Its data also agree with the Nishimori exact results for spin glass. Only minor changes of the SPP design are necessary to increase the dimensionality and to take into account more complex systems such as Potts models.

This paper presents the basic scheme and the log-normal, log-gamma and log-negative-inverted-gamma scenarios to establish the Rényi function for infinite products of homogeneous isotropic random fields on Rⁿ; in particular for random fields on the sphere in R³. The motivation of this paper is the test of (non-)Gaussianity on the Cosmic Microwave Background Radiation (CMBR) in Cosmology. In the presentation, we need to employ spherical harmonics for some concrete computations.

The defaultable term structure is modeled using stochastic differential equations in Hilbert spaces. This leads to an infinite dimensional model, which is free of arbitrage under a certain drift condition. Furthermore, the model is extended to incorporate ratings based on a Markov chain.

Gaussian random field on general ultrametric space is introduced as a solution of pseudodifferential stochastic equation. Covariation of the introduced random field is computed with the help of wavelet analysis on ultrametric spaces.

Notion of ultrametric Markovianity, which describes independence of contributions to random field from different ultrametric balls is introduced. We show that the random field under investigation satisfies this property.

Generation of random fields (GRF) for image segmentation represents partitioning an image into different regions that are homogeneous or have similar facets of the image. It is one of the most challenging tasks in image processing and a very important pre-processing step in the fields of computer vision, image analysis, medical image processing, pattern recognition, remote sensing, and geographical information system. Many researchers have presented numerous image segmentation approaches, but still, there are challenges like segmentation of low contrast images, removal of shadow in the images, reduction of high dimensional images, and computational complexity of segmentation techniques. In this review paper, the authors address these issues. The experiments are conducted and tested on the Berkely dataset (BSD500), Semantic dataset, and our own dataset, and the results are shown in the form of tables and graphs.

Solutions of quasi-linear stochastic Fokker–Planck equations for the number density of a system of solute particles in suspension are derived. The initial values and the solutions take values in a class of σ-finite Borel measures over R^d where d ≥ 1. The stochastic driving noise is defined by Itô differentials. For the special case of semi-linear stochastic Fokker–Planck equations, the solutions can be represented as solutions of first-order stochastic transport equations driven by Stratonovich differentials.

Lately, many phenomena in both applied and abstract mathematics and related disciplines have been expressed in terms of high order and fractional PDEs. Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) and connected it to a new system of fourth order interacting PDEs. The interaction in this multiparameter BTBS-PDEs connection is novel, leads to an intimately-connected linear system variant of the celebrated Kuramoto–Sivashinsky PDE, and is not shared with its one-time-parameter counterpart. It also means that these PDEs systems are to be solved for a family of functions, a feature exhibited in well known fluids dynamics models. On the other hand, the memory-preserving interaction between the PDE solution and the initial data is common to both the single- and the multi-parameter Brownian-time PDEs. Here, we introduce a new — even in the one-parameter case — proof that combines stochastic analysis with analysis and fractional calculus to simultaneously link BTBS to a new system of temporally half-derivative interacting PDEs as well as to the fourth-order system proved earlier and differently by Allouba. We then introduce a general class of random fields we call inverse-stable-Lévy-time Brownian sheets (ISLTBSs), and we link them to β-fractional-time-derivative systems of interacting PDEs for 0 < β < 1. When β = 1/ν, ν ∈ {2, 3, …}, our proof also connects an ISLTBS to a system of memory-preserving ν-Laplacian interacting PDEs. Memory is expressed via a sum of temporally-scaled k-Laplacians of the initial data, k = 1, …, ν - 1. Using a Fourier–Laplace-transform-fractional-calculus approach, we give a conditional equivalence result that gives a necessary and sufficient condition for the equivalence between the fractional and the high order systems. In the one-parameter case this condition automatically holds.

We obtain a necessary and sufficient condition for the orthomartingale-coboundary decomposition. We establish a sufficient condition for the approximation of the partial sums of a strictly stationary random fields by those of stationary orthomartingale differences. This condition can be checked under multidimensional analogues of the Hannan condition and the Maxwell–Woodroofe condition.

In this paper, we study the continuity in law of the solutions of two linear multiplicative SPDEs (the parabolic Anderson model and the hyperbolic Anderson model) with respect to the spatial parameter of the noise. The solution is interpreted in the Skorohod sense, using Malliavin calculus. We consider two cases: (i) the regular noise, whose spatial covariance is given by the Riesz kernel of order $\alpha \in (0,d)$, in spatial dimension $d\ge 1$; (ii) the rough noise, which is fractional in space with Hurst index $H<1/2$, in spatial dimension $d=1$. We assume that the noise is colored in time.

The effects of spatial variability of soil properties on the behaviour of a cohesive and a cohesionless soil profile subjected to seismic excitation are analysed. The input data for the random fields are derived from existing, extensive soil investigations, and Monte Carlo simulation technique, combining digital generation of non-Gaussian stochastic vector fields with dynamic, equivalent linear and nonlinear finite element analyses, is used for this purpose. It is found that the variability of shear wave velocity has small effect on the ground surface response spectrum. The motion intensity and the variation of soil properties in the deterministic description of the profile considerably affect the standard deviation of the surface spectral acceleration, which is successively compared to the uncertainty introduced in attenuation relationships, used for the construction of hazard-consistent design spectra.

In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to compute the expected Euler integral of a Gaussian random field using the Gaussian kinematic formula and obtain a simple closed form expression. This results in the first explicitly computable mean of a quantitative descriptor for the persistent homology of a Gaussian random field.

For a class of symmetric random matrices whose entries are martingale differences adapted to an increasing filtration, we prove that under a Lindeberg-like condition, the empirical spectral distribution behaves asymptotically similarly to a corresponding matrix with independent centered Gaussian entries having the same variances. Under a slightly reinforced condition, the approximation holds in the almost sure sense. We also point out several sufficient regularity conditions imposed to the variance structure for convergence to the semicircle law or the Marchenko–Pastur law and other convergence results. In the stationary case, we obtain a full extension from the i.i.d. case to the martingale case of the convergence to the semicircle law as well as to the Marchenko–Pastur one. Our results are well adapted to study several examples including nonlinear autoregressive conditional heteroscedastic random fields of infinite order.

Quenched random fields (RFs) are well-known to be a basic driving force of the relaxor behavior in disordered ferroelectrics containing either random charges as in PbMg_1/3Nb_2/3O₃ or random cation vacancies as in Sr_xBa_1−xNb₂O₆. They give rise to the formation of polar nanoregions in the paraelectric regime, which freeze into a nanodomain state at low temperatures thus precluding the ferroelectric phase transition. On the other hand, isovalent relaxors like bond-disordered BaTi_1−xZr_xO₃ prevalently experience random bonds (RBs), which are responsible for glassy features such as polydispersivity and non-ergodic aging and rejuvenation processes. Inspection shows, however, that all of the above characteristics apply generally to relaxor systems albeit with different weight. Subsequent formation of ferroic nanoregions via RFs followed by mesoscopic cluster glassy freezing via RBs upon cooling are universal signatures.

This chapter reviews and discusses various aspects of texture analysis. The concentration is on the various methods of extracting textural features from images. The geometric, random field, fractal, and signal processing models of texture are presented. The major classes of texture processing problems such as segmentation, classification, and shape from texture are discussed. The possible application areas of texture such as automated inspection, document processing, and remote sensing are summarized. A bibliography is provided at the end for further reading.

We describe a Special Purpose Processor, realizing the Wolff algorithm in hardware, which is fast enough to study the critical behaviour of 2D Ising-like systems containing more than one million spins. The processor has been checked to produce correct results for a pure Ising model and for Ising model with random bonds. Its data also agree with the Nishimori exact results for spin glass. Only minor changes of the SPP design are necessary to increase the dimensionality and to take into account more complex systems such as Potts models.

This chapter reviews and discusses various aspects of texture analysis. The concentration is on the various methods of extracting textural features from images. The geometric, random field, fractal, and signal processing models of texture are presented. The major classes of texture processing problems such as segmentation, classification, and shape from texture are discussed. The possible application areas of texture such as automated inspection, document processing, and remote sensing are summarized. A bibliography is provided at the end for further reading.