Search

The inverse scattering problem has numerous significant applications, including in geophysical explorations, medical imaging, and radar imaging. To achieve better performance of the imaging system, theoretical knowledge of the resolution of the algorithm is required for most of these applications. However, analytical investigations about the resolution presently feel inadequate. In order to estimate the achievable resolution, we address the point spread function (PSF) evaluation of the scattered field for a single frequency and the multi-view case both for the near and the far fields and the scalar case when the angular domain of the incident field and observation ranges is a round angle. Instead of the common free space condition, an inhomogeneous background medium, consisting of a homogeneous dielectric cylinder with a circular cross-section in free space, is assumed. In addition, since the exact evaluation of the PSF can only be accomplished numerically, an analytical approximation of the resolution is also considered. For the sake of its comparison, the truncated singular value decomposition (TSVD) algorithm can be used to implement the exact PSF. We show how the behavior of the singular values and the resolution change by varying the permittivity of the background medium. The usefulness of the theoretical discussion is demonstrated in localizing point-like scatterers within a dielectric cylinder, so mimicking a scenario that may occur in breast cancer imaging. Numerical results are provided to validate the analytical investigations. [ABSTRACT FROM AUTHOR]

In linear inverse scattering, the performance of the imaging system is sometimes evaluated in terms of its resolution, i.e., its capability to reconstruct a point-like scatterer. However, there is still a lack of analytical studies on the achievable resolution. To address this, we consider the point spread function (PSF) evaluation of the scattered near field for the single frequency and multi-view/multi-static case in homogeneous medium. Instead of numerically computing the PSF, we propose and discuss an approximate closed form under series expansions according to the angular ranges of both source and receiver location. In order to assess the effectiveness of the proposed approximation, we consider two cases including both full and limited view angles for the incident field and observation ranges. In addition, we provide a localization application to show the usefulness of the theoretical discussion. Numerical results confirmed the analytical investigations. [ABSTRACT FROM AUTHOR]

The solution to an ill-posed linear inverse problem requires the use of regularization methods to achieve a stable approximation solution. One powerful approach is the truncated singular value decomposition (TSVD), but it requires an appropriate choice of the truncation level. One suitable option is to take into account the number of degrees of freedom (NDF) of the scattered field, which is defined by the step-like behavior of the singular values of the relevant operator. Then, the NDF can be estimated as the number of singular values preceding the knee or the exponential decay. Therefore, an analytical estimation of the NDF is significant for obtaining a stable, regularized solution. This paper addresses the analytical estimation of the NDF of the field scattered by the surface of a cube geometry for a single frequency and the multi-view case in the far-zone. In addition, a method is proposed to find the minimum numbers of plane waves and their directions to achieve the total estimated NDF. The main results are that the NDF is related to the measure of the surface of the cube and can be achieved by only considering a limited number of impinging plane waves. The efficiency of the theoretical discussion is demonstrated through a reconstruction application for microwave tomography of a dielectric object. Numerical examples are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Field sampling should be devised to preserve the information required for the knowledge of the radiation of an antenna. In this paper, we introduce a sampling scheme based on an inverse source problem approach to the far field radiated by a conformal current source. The regularized solution of the problem requires the computation of the Singular Value Decomposition (SVD) of the relevant linear operator, leading to introduce the Point Spread Function in the observation domain, which can be related to the capability of the source to radiate a focusing beam. Then, the application of the Kramer generalized sampling theorem allows introducing a non-uniform discretization of the angular observation domain, tailored to each source geometry. The nearly optimal property of the scheme is compared with the best approximation achievable under a regularized inversion of the pertinent SVD. Numerical results for different two-dimensional curve sources show the effectiveness of the approach with respect to standard sampling approaches with uniform spacing, since it allows to reduce the number of sampling points of the far field.

Inverse scattering problems stand at the center of many important imaging applications, such as geophysical explorations, radar imaging, and synthetic-aperture radar (SAR). Several methods have been proposed to solve them when the full data are available, usually providing satisfactory reconstructions. However, it is impossible to acquire the full data in many practical circumstances, such as target detection and ground penetrating radar (GPR); consequently, only limited data are available. Thus, this paper focuses on the mathematical analysis and some numerical simulations to estimate the achievable resolution in reconstructing an object from the knowledge of the scattered far-field when only limited data are available, with multi-view excitations at a single frequency. We focus on 2D full rectangular geometry as the investigation domain (ID). We also examine the number of degrees of freedom (NDF) and evaluate the point spread function (PSF). In particular, the NDF of the considered geometry can be estimated analytically. An approximated closed-form evaluation of the PSF is recalled, discussed, and compared with the exact one. Moreover, receiving, transmission, and angle sensing modes are considered to apply the analysis to more realistic scenarios to highlight the difference between the corresponding NDF and the resulting resolution performances. Finally, interesting numerical applications of the resolution analysis for the localization of a collection of point-like scatterers are presented to illustrate how it matches the expectations. [ABSTRACT FROM AUTHOR]

The role of the source geometry in the radiation of focusing beams by conformal antennas is examined by the comparison of their directivity functions at different maximum directions. An inverse source problem approach is adopted, where solutions stable with respect to data uncertainties are to be found by relying on the analysis of the pertinent operator by the Singular Values Decomposition. This general framework allows to connect the mean value of the maximum directivity function to the Number of Degrees of Freedom of the conformal source, which depends only on its electrical length. For each source geometry the focusing far field ensuring the maximum directivity for every pointing direction and the corresponding source current are obtained. The comparison with the ones achieved by the usual phase compensation technique of the source current reveals their optimal behavior. The usefulness of the approach as a tool in antenna synthesis is shown by comparing different geometries for those applications where identical beams are required to be radiated for the coverage of a large angular domain.

This paper is centred on the spectral study of a random Fourier matrix, that is an n × n matrix A with (j, k) entries given by exp (2 i π m Y j Z k) , where Y j and Z k are two i.i.d sequences of random variables and 1 ≤ m ≤ n is a real number. When the Y j and Z k are uniformly distributed on a bounded symmetric interval, this may be seen as a random discretization of the finite Fourier transform, whose spectrum has been extensively studied in relation with band-limited functions. Moreover, this particular case of random finite Fourier matrix has been proposed in wireless telecommunication in order to approach the singular values of some channel fading matrices. We first compare in the ℓ 2 -norm, the spectrum of the matrix A ∗ A with the spectrum of its associated integral operator. We show that the classical methods of concentration inequalities for kernel random matrices are well adapted for the spectral analysis of random Fourier matrices. We then concentrate on uniform distributions for the laws of Y j ’s and Z k ’s, for which the integral operator is the well-known Sinc-kernel operator with parameter m. We translate to random Fourier matrices the knowledge that we have on the spectrum of this operator. In particular, we study some asymptotic and non-asymptotic behaviours of the set of the eigenvalues of A ∗ A. This study is done in the spirit of recent work on the Sinc-kernel integral operator. As applications, we give fairly good approximations of the number of degrees of freedom, as well as an estimate of the capacity of a MIMO wireless communication network approximation model. Finally, we provide the reader with some numerical examples that illustrate the theoretical results of this paper. [ABSTRACT FROM AUTHOR]

Regularization of the Bose-Einstein distribution using a parastatistical correction, i.e., by means of the Gentile statistics, is carried out. It is shown that the regularization result asymptotically coincides with the Erdős formula obtained by using Ramanujan's formula for the number of variants of the partition of an integer into summands. TheHartley entropy regarded as the logarithm of the number of variants defined by Ramanujan's exact formula asymptotically coincides with the polylogarithm associated with the entropy of the Bose-Einstein distribution. The fact that these formulas coincide makes it possible to extend the entropy to the domain of the Fermi-Dirac distribution with minus sign. Further, the formulas for the distribution are extended to fractional dimension and also to dimension 1, which corresponds to the Waring problem. The relationship between the resulting formulas and the liquid corresponding to the case of nonpolar molecules is described and the law of phase transition of liquid to an amorphous solid under negative pressure is discussed. Also the connection of the resulting formulas with the gold reserve in economics is considered. [ABSTRACT FROM AUTHOR]

In the paper 'The relationship between the Fermi-Dirac distribution and statistical distributions in languages' (This issue of 'Math. Notes'), the Bose-Einstein and Fermi-Dirac distributions are considered in connectionwith frequency distributions in languages in the context of the author's approach to thermodynamics and number theory problems. The present article presents certain clarifications of certain notions used in that approach, in particular, the identity of particles, the Poisson adiabat, thematching of number theory formulas with those given by the Bose-Einstein distribution, nonstandard analysis, and others. [ABSTRACT FROM AUTHOR]

In this article, we study, from the mathematical point of view, the analogies between language and multi-particle systems in thermodynamics. We attempt to introduce an appropriate mathematical apparatus and the technical tools of statistical physics to descriptions of language. In particular, we apply the notions of number of degrees of freedom, Bose condensate, phase transition and others to linguistics objects. On the basis of a statistical analysis of dictionaries and statistical distributions in languages, we conjecture that the transition from the semiotic communication system of the higher primates to human language can be described as a phase transition of the first kind. We show that the number of words appearing with frequency 1 in a corpus of texts is equal to the number of ones in the corresponding Fermi-Dirac distribution, while the high frequency of stop-words corresponds to the large number of particles in the Bose condensate, when the number of degrees of freedom is less than two, provided there is a gap in the spectrum. The presented considerations are illustrated by examples from the Russian language. Some of the illustrative examples are untranslatable into English, and so they were replaced in translation by similar examples from the English language. [ABSTRACT FROM AUTHOR]

This article aims at revisiting, with the aid of simple and neat numerical examples, some of the basic features of macroscopic irreversibility, and, thus, of the mechanical foundation of the second principle of thermodynamics as drawn by Boltzmann. Emphasis will be put on the fact that, in systems characterized by a very large number of degrees of freedom, irreversibility is already manifested at a single-trajectory level for the vast majority of the far-from-equilibrium initial conditions—a property often referred to as typicality . We also discuss the importance of the interaction among the microscopic constituents of the system and the irrelevance of chaos to irreversibility, showing that the same irreversible behaviors can be observed both in chaotic and non-chaotic systems. [ABSTRACT FROM AUTHOR]

A generalization of tropical geometry to the case of mixtures of ideal liquids and an increase in the number of prohibitions is presented. In addition to the well-known notions of skeleton and ameba, new notions of circulatory system and joints (angles in the skeleton) are introduced. [ABSTRACT FROM AUTHOR]

In the region of negative pressures P, the addend γ corresponding to the number of degrees of freedom D = 2 γ + 2 is less than zero. The energy equal to E = V P is also less than zero. Therefore, it is natural to introduce a negative relative temperature as the mean energy. The negative pressure on the negative spinodal tends to minus infinity as T → 0. Accordingly, the negative relative temperature will also approach minus infinity. As the chemical potential μ→−∞, μ/ T → 0, the asymptotic behavior of the thermodynamic quantities gives rise to amebas of special form in tropical (idempotent) analysis. It is shown that the theoretical isotherms obtained in this paper coincide with the isotherms of van derWaals gas up to the third decimal place. For theoretical distributions, transitions to amorphic solids (glass) occurs. [ABSTRACT FROM AUTHOR]

The notion of ideal liquid for the number of degrees of freedom less than 2, i.e., γ < 0, is introduced. The values of the pressure P and of the compressibility factor Z on the spinodal in the negative pressure region for the van der Waals equation determine the value of γ, γ( T) < 0, for μ = 0. For $$T \leqslant \frac{{{3^3}}}{{{2^5}}}{T_c}$$, a relationship with the van der Waals equation is established. [ABSTRACT FROM AUTHOR]

The article offers information on the concept of old thermodynamics and new thermodynamics and mentions the differences between them. Topics discussed include concerns of old thermodynamics with the gas-liquid phase transition, concerns of new thermodynamics on metastable states and explanation given by physicists on metastability of live human beings.

We construct gas and liquid binodals assuming that both the gas and the liquid branch of the spinodal are given. To determine how the number of degrees of freedom depends on temperature, we use the main formula expressing the compressibility factor via the ratio of two Riemann functions. The behavior of the second virial coefficient is studied for a given spinodal, which serves as an analog of caustics. The gas- and liquid-amorphous solid transitions are considered on the second sheet. An argument in favor of the mapping of the second sheet onto the negative quadrant {- P,- Z} is presented. [ABSTRACT FROM AUTHOR]

An example of a Gibbs ensemble for a system of N Brownian particles is given, and the relation with Kolmogorov complexity is considered. The irreversibility of the process is analyzed and shown to be a consequence of the loss of numbering of the particles. The dependence of the measure in the semiclassicalGibbs distribution on temperature is studied in examples. [ABSTRACT FROM AUTHOR]

The article discusses the possibility of the application of the UD-statistics to the revolutionary processes in society. It relates the teaching of mathematician Gottfried Wilhelm Leibniz to regarding the application of mathematics to law, politics, and other branches of humanities. It adds that the critical point is one of the important points associated to UD-statistics.

In the present paper, we describe an approach to thermodynamics that does not involve Bogolyubov chains or Gibbs ensembles. We present isotherms, isochores, and isobars of various pure gases, as well as binodals, i.e., lines along which gas becomes liquid, and spinodals (endpoints of isotherms). We study supercritical phenomena for values of temperature and pressure above the critical ones. A lot of attention is paid to the region of negative pressures. The superfluid component for supercritical phenomena is described, as well as the thermodynamics of nanostructures and superfluidity in nanotubes. [ABSTRACT FROM AUTHOR]

It is well known that the supercritical state of a gas has great dissolving capacity. In this paper, the mathematical reason for this phenomenon is studied in great detail. [ABSTRACT FROM AUTHOR]

The evaluation of measurement uncertainty for quantities expressed in decibels is discussed. The coverage factor for a fractional number of degrees of freedom is calculated and discreteness is taken into account for estimates of measurement uncertainty in multiple observations. [ABSTRACT FROM AUTHOR]

In the paper by E. M. Apfelbaum and V. S. Vorobiev [1], it was shown that the critical isotherms of real pure gases coincide with the corresponding isotherms of the classical distribution of Bose-Einstein type for fractional dimension (fractional number of degrees of freedom). For isotherms with T < T, this coincidence no longer occurs, and for some gases the discrepancy is considerable. In the paper, the law of preference of cluster formation over passage to liquid state is used to deduce equations that give a good coincidence for these isotherms. [ABSTRACT FROM AUTHOR]

We develop the ideology of unbounded probability theory in which the notion of elementary event is not used and present a new (more general) definition of independent events. The proposed construction is based on natural sequences 1, 2, ..., N and distributions of the type 'partitio numerorum.' Essential use is made of the mathematical phenomenon of 'Bose-Einstein condensate' type, a familiar notion in quantum statistical physics. The derived construction can be applied to thermodynamics and multistep relaxation processes. [ABSTRACT FROM AUTHOR]

Consideration of an example corresponding to the partition of integers results in revision of certain physical concepts. A substantial term is added to the Bose-Einstein distribution. The notion of 'fractional dimension' is introduced. Some effects considered earlier as pure quantum effects are explained from the classical standpoint. A distribution depending on three fixed points is given. [ABSTRACT FROM AUTHOR]

The well-known empiric Van der Waals equation contains two constants characterizing a given gas, so that, in general, it depends on the characteristic interaction of particles of the given gas. In this paper, equations depending on three defining constants. are constructed on the basis of the mathematically rigorous results obtained by the author in recent papers. [ABSTRACT FROM AUTHOR]

Abstract: With the aid of numerical simulations of the Fermi-Pasta-Ulam (FPU) system, we compare the different definitions of dynamical temperature for Hamiltonian systems. We have shown that each definition gives different values of temperature for a system with a small number of degrees of freedom (DOF). Only for systems with a sufficiently large number of DOF, do all the definitions of dynamical temperature approach the same value. [Copyright &y& Elsevier]

This paper aims at discussing the resolution achievable in the reconstruction of both circumference sources from their radiated far-field and circumference scatterers from their scattered far-field observed for the 2D scalar case. The investigation is based on an inverse problem approach, requiring the analysis of the spectral decomposition of the pertinent linear operator by the Singular Value Decomposition (SVD). The attention is focused upon the evaluation of the Number of Degrees of Freedom (NDF), connected to singular values behavior, and of the Point Spread Function (PSF), which accounts for the reconstruction of a point-like unknown and depends on both the NDF and on the singular functions. A closed-form evaluation of the PSF relevant to the inverse source problem is first provided. In addition, an approximated closed-form evaluation is introduced and compared with the exact one. This is important for the subsequent evaluation of the PSF relevant to the inverse scattering problem, which is based on a similar approximation. In this case, the approximation accuracy of the PSF is verified at least in its main lobe region by numerical simulation since it is the most critical one as far as the resolution discussion is concerned. The main result of the analysis is the space invariance of the PSF when the observation is the full angle in the far-zone region, showing that resolution remains unchanged over the entire source/investigation domain in the considered geometries. The paper also poses the problem of identifying the minimum number and the optimal directions of the impinging plane waves in the inverse scattering problem to achieve the full NDF; some numerical results about it are presented. Finally, a numerical application of the PSF concept is performed in inverse scattering, and its relevance in the presence of noisy data is outlined. [ABSTRACT FROM AUTHOR]

The article focuses on some of the reasons for the supercomputer's fall-out from the World Wide Web. It states that the polynomial growth of a random variable affects the transition from an open system. It says that Web developers no longer use Shannon entropy in passing to the communication packages through the Web. It adds that was a lesser growth on the values of the random variable in Bose liquid.

Solving inverse scattering problems by numerical methods requires investigating the number of independent pieces of information that can be reconstructed stably. To this end, we address the evaluation of the Number of Degrees of Freedom (NDF) of far-zone scattered fields for some strip geometries under the first-order Born approximation. The analysis is performed by employing the Singular Value Decomposition (SVD) of the scattering operator in the two-dimensional scalar geometry of one or more strips illuminated by a TM polarized plane wave. It is known that investigating the scattering scene at different incident plane waves (multi-view configuration) enhances the NDF. Therefore we mean to examine the minimum number of incident plane waves providing the NDF of the scattered fields both by theoretical estimations and numerical verifications. [ABSTRACT FROM AUTHOR]