Cauchy's Theorem Research Articles

THE PROPERTIES OF THE VOLUME POTENTIAL FOR ONE PARABOLIC EQUATION WITH GROWING LOWEST COEFFICIENTS

The class of equations considered in the paper is a combination of two classes of equations: a degenerate parabolic equation of the Kolmogorov type and a parabolic equation with increasing coefficients in the group of younger members. Such a combination occurs in the problems of the theory of stochastic processes where, in the case of a normal Markov process, the Kolmogorov-Fokker-Planck equation has a similar form. The coefficients of this equations are constant in the group of principal terms and ones are increasing functions in the group of lowest terms. The article is devoted to the study of the properties of the volume potential, the kernel of which is the fundamental solution of the Cauchy problem for such an equation. Estimates of the fundamental solution of the Cauchy problem have a more complex structure than in the case of the classical Kolmogorov equation. These properties concern the existence of the derivatives included in the equation. They are used to establish theorems on the integral representations of solutions of the Cauchy problem and theorems on the correct solvability of the Cauchy problem in the corresponding classes of functional spaces. Such studies are carried out in this work. The obtained results are new and published for the first time.

Особливості алгоритмізації процесів мінімізації похибок апроксимації при вирішенні прикладних задач

The paper defines the mathematical features arising during the algorithmization of the minimi-zation of errors that appear as a result of the approximation of functions in applied environmen-tal safety problems, the creation of automated personnel selection systems, and the improve-ment of systems with a recommendation mechanism. It has been proved that when performing the algorithmization of any process, it is necessary to take into account the ignoring of some part of the information in the process of formalization. When transforming information, any event in the process of functioning of a complex system over a certain period of time can be considered as the occurrence of a specific situation at some specified point, to which the re-quirements for ensuring the properties of information are put forward. The work is based on the basic postulates of the Cauchy theorem and Dr. Tesler's works on some assertions on the de-composition of functions by incoherence. Using the approach to solving problems with complex variants of approximation of functions and existing methods of minimizing errors in approxi-mations, there is proposed an algorithm that in the case of finding a solution with a large num-ber of iterations allows for carrying out a number of transformations that will let obtain the re-sult through expansion in the Taylor series by degrees. It has been found that in the absence of a clear solution to the given problem, it is best to use an approximation with some incoherence. That means choosing a parameter that acts as an element of adaptation to the specified condi-tions of the task. Of course, this method of problem solving requires functional transformations, but in the end, it allows you to use basic trigonometric functions that simplify algorithmization. An example of a practical implementation of the algorithm with the minimization of approxi-mation errors has been considered. Based on it, it has been proved that it is more rational to use tables of a number of widely used functions and decompose functions into series by errors.

A Determinantal Identity for the Permanent of a Rank 2 Matrix

We prove an identity relating the permanent of a rank 2 matrix and the determinants of its Hadamard powers. When viewed in the right way, the resulting formula looks strikingly similar to an identity of Carlitz and Levine, suggesting the possibility that these are actually special cases of some more general identity (or class of identities) connecting permanents and determinants. The proof combines some basic facts from the theory of symmetric functions with an application of a famous theorem of Binet and Cauchy in linear algebra.

Bergman theory for the inhomogeneous Cimmino system

We first prove a Cauchy's integral theorem and a Cauchy-type formula for certain inhomogeneous Cimmino system from quaternionic analysis perspective. The second part of the paper directs the attention towards some applications of the mentioned results, dealing in particular with four kinds of weighted Bergman spaces, reproducing kernels, projection and conformal invariant properties.

A Formalization of Finite Group Theory

Previous formulations of group theory in ACL2 and Nqthm, based on either "encapsulate" or "defn-sk", have been limited by their failure to provide a path to proof by induction on the order of a group, which is required for most interesting results in this domain beyond Lagrange's Theorem (asserting the divisibility of the order of a group by that of a subgroup). We describe an alternative approach to finite group theory that remedies this deficiency, based on an explicit representation of a group as an operation table. We define a "defgroup" macro for generating parametrized families of groups, which we apply to the additive and multiplicative groups of integers modulo n, the symmetric groups, arbitrary quotient groups, and cyclic subgroups. In addition to a proof of Lagrange's Theorem, we present an inductive proof of the abelian case of a theorem of Cauchy: If the order of a group G is divisible by a prime p, then G has an element of order p.

Smearing of causality by compositeness divides dispersive approaches into exact ones and precision-limited ones

Scattering off the edge of a composite particle or finite-range interaction can precede that off its center. An effective theory treatment with pointlike particles and contact interactions must find that the scattered experimental wave is slightly advanced, in violation of causality (the fundamental underlying theory being causal). In practice, partial-wave or other projections of multivariate amplitudes exponentially grow with Im(E), so that analyticity is not sufficient to obtain a dispersion relation for them, but only for a slightly modified function (the modified relations additionally connect different J). This can limit the precision of certain dispersive approaches to compositeness based on Cauchy's theorem. Awareness of this may be of interest to some dispersive tests of the Standard Model with hadrons, and to unitarization methods used to extend electroweak effective theories. Interestingly, the Inverse Amplitude Method is safe (as the inverse amplitude has the opposite, convergent behavior allowing contour closure). Generically, one-dimensional sum rules such as for the photon vacuum polarization, form factors or the Adler function are not affected by this uncertainty; nor are fixed-t dispersion relations, cleverly constructed to avoid it and whose consequences are solid.

Boundary behavior for the heat equation on the half‐line

The initial‐boundary value problem for the heat equation on with nonzero Dirichlet boundary data is studied rigorously, with emphasis on the behavior of the solution along the boundary of the spatiotemporal domain, that is, in the limits or , including the case . Our results specify how the regularity of the solution at the boundary is dictated by the level of smoothness of the relevant initial and boundary data. To the best of our knowledge, for unbounded domains like the half‐line , such results are not available in the literature. The corresponding analysis for the derivatives of the solution is also performed and yields additional compatibility conditions on the initial and boundary data which guarantee the extension of the solution to a function up to the boundary, that is, on . The analysis relies on the explicit solution formula provided by the unified transform, also known as the Fokas method, whose rigorous derivation is presented. The main advantage of this formula, namely, its uniform convergence at the boundary, largely reduces the proof of our results to fundamental tools from real and complex analysis such as Lebesgue's dominated convergence theorem and Cauchy's theorem. Moreover, thanks to uniform convergence, the unified transform solution formula can actually be directly evaluated at the boundary to yield the prescribed initial and boundary data.

Seismic resilient steel substation with BI-TMDI: A theoretical model for optimal design

The based-isolation with tuned mass damper inerter (BI-TMDI) is utilized to enhance the seismic response of steel High-voltage substation switches mounted on frame (SMF). An analytical model with three degree-of-freedom was explored for the optimization analysis and seismic design. Firstly, the displacement variance of this SMF-BI-TMDI system in the whole frequency spectrum was obtained, using statistical Cauchy's residue theorem, which provides an explicit expression of the optimization objective function. Further, a comprehensive parametric analysis was performed, covering a wide spectrum of influential factors. Subsequently, the effectiveness of the proposed analytical model was confirmed by time history response analysis of prototype structures subjected to an ensemble of earthquake motions. The model was also compared with conventional analytical models for structural seismic optimization, showing obviously improved accuracy. This innovative control method and optimization analysis in BI-TMDI enlighten its potential with enhanced vibration control effect for protection of steel substation under earthquake motions.

Micropolar mechanics of product fractal media

Motivated by the abundance of fractals in the natural world, this paper further develops continuum-type models for product-like fractals. The theory is based on a version of the non-integer dimensional space approach, in which global balance laws written for fractal media are expressed in terms of conventional (integer-order) integrals. Key relations of calculus for finite strain kinematics of fractal media are obtained, especially clarifying the fractal Jacobian and the fractal Reynolds transport theorem. The local forms are then written in terms of partial differential equations with derivatives of integer order. Hence, fractal versions of local continuity, linear and angular momenta and energy balance are derived. The angular momentum balance implies that the approximating continuum is micropolar rather than classical. Accordingly, the Cauchy postulate, lemma and theorem for Cauchy force-stress and couple-stress are re-formulated. The corresponding partial differential equations for finite as well as infinitesimal elasticity are given explicitly in both the displacement and stress formulations. The invariance of the stress field in planar fractal elastic media is shown to hold just like the one in planar micropolar elasticity.

Oscillatory Path Integrals for Radio Astronomy

We introduce a new method for evaluating the oscillatory integrals which describe natural interference patterns. As an illustrative example of contemporary interest, we consider astrophysical plasma lensing of coherent sources like pulsars and fast radio bursts in radioastronomy. Plasma lenses are known to occur near the source, in the interstellar medium, as well as in the solar wind and the earth's ionosphere. Such lensing is strongest at long wavelengths hence it is generally important to go beyond geometric optics and into the full wave optics regime. Our computational method is a spinoff of new techniques two of us, and our collaborators, have developed for defining and performing Lorentzian path integrals. Cauchy's theorem allows one to transform a computationally fragile and expensive, highly oscillatory integral into an exactly equivalent sum of absolutely and rapidly convergent integrals which can be evaluated in polynomial time. We require only that it is possible to analytically continue the lensing phase, expressed in the integrated coordinates, into the complex domain. We give a first-principles derivation of the Fresnel-Kirchhoff integral, starting from Feynman's path integral for a massless particle in a refractive medium. We then demonstrate the effectiveness of our method by computing the interference patterns of Thom's caustic catastrophes, both in their normal forms and within a variety of more realistic, local lens models, over all wavelengths. Our numerical method, implemented in a freely downloadable code, provides a fast, accurate tool for modeling interference patterns in radioastronomy and other fields of physics.

Qualitative analysis and dynamical behavior of a Lassa haemorrhagic fever model with exposed rodents and saturated incidence rate

We have proposed an unprecedented deterministic model of Lassa Hemorrhagic fever (LHF) model with nonlinear force of LHF infection to capture the transmission dynamics and long-term effects of the disease. The Qualitative analyses we have conveyed on this model using well-established methods viz: Cauchy's differential theorem, Birkhoff & Rota's theorems verify and reveal the well-posedness of the model respectively. We established that an LHF-free equilibrium termed the disease-free equilibrium (DFE) exists for this model and this equilibrium, however, from our stability analyses, tends to be stable when the basic reproduction number computed via the next generation matrix method is less than unity (one); and unstable otherwise. Furthermore, we have carried out a sensitivity analysis to check for the variation effects of the model parameters when increased or decreased using the normalized forward-sensitivity index; unraveling the most sensitive parameters which requires the attention of the healthcare workers as; the effective contact rates β1,β2andβ3, and the rodents’ recruitment rate ΩR. After which numerical simulations of the model were carried out to verify our qualitative analyses (Stability and sensitivity analysis) and to study the dynamical behavior of the model; showing that the presence of saturation instantaneously causes the system to approach a DFE/LHF-Free equilibrium. From these qualitative analyses and numerical simulation results, we recommend early intervention and early treatment of Lassa hemorrhagic virus infection (LAHV) with Ribavirin on the infected, maximum hygiene practices and periodic evacuation of rodents in households in order to curb the recruitment of wild/rodents.

Using Cauchy theorem to compute complicated real integrals

As the central object in the theory of complex analysis, Holomorphic functions have many elegant mathematical properties. Holomorphic functions are extremely valuable because, on the one hand, they are unexpectedly common, and on the other hand, they may be used to establish extremely powerful theorems. For example, To establish theorems like the prime number theorem, analytic number theorists commonly create holomorphic or meromorphic functions that hold number-theoretic information, such as the Riemann zeta function. Given knowledge about a holomorphic function in a relatively small part of its domain, one may extract information about the function’s behavior in other a priori unrelated sections of its domain, according to Cauchy’s integral formula and the identity theorem (and this is what allows things like contour integration to work). Due to difficulties in obtaining primitives of some real function, the fundamental theorem of Calculus does not work in most cases. In this article, we review the Cauchy theorem and use it as a tool to compute several real integrals.

No Cauchy horizon theorem for nonlinear electrodynamics black holes with charged scalar hairs

We prove a no Cauchy horizon theorem for general nonlinear electrodynamics black holes with charged scalar hairs. By constructing a radially conserved charged, we show that there is no inner Cauchy horizon for both spherical and planar symmetric cases, independent of the form of scalar potential and nonlinear electrodynamics. After imposing the null energy condition, we are also able to rule out the existence of the Cauchy horizon for the hyperbolic black holes. We take the Born-Infeld black hole as a concrete example to study the interior dynamics beyond the event horizon. When the contribution from the scalar potential can be neglected, the asymptotic near-singularity takes a universal Kasner form. We also confirm that the intricate interior dynamics is closely associated with the instability of the inner Cauchy horizon triggered by scalar hairs.

Theory of fluid saturated porous media with surface effects

Fluid saturated porous media are abundant in nature and engineering. Existing constitutive theories of fluid saturated porous media have largely ignored surface effects, whereas recent experimental results demonstrate the importance of accounting for surface effects in constitutive modeling at micro- and nano-scales (such as brain tissue, cytoplasm, soil, oil shale, etc.), thus raising two critical issues: (1) is classical poromechanics (e.g., the Biot theory) valid at sufficiently small scales, and (2) how do surface effects influence the mechanical behaviors of saturated porous media? To squarely address these issues, establishing the mechanics of solid-fluid interfaces within the porous media becomes a necessity. Built upon the homogenization assumption in the mixture theory and making use of Cauchy's theorem, we first prove mathematically that stresses on solid-fluid interfaces can be expressed by a second order tensor defined at each point of a saturated porous medium with surface effects, and then establish a rigorous theoretical framework to characterize its mechanical behaviors. Restrictions on various constitutive laws of saturated porous media developed in this framework are discussed. As an application of the developed theoretical framework, we prove that, at small deformation, the classical Biot theory still holds, at least in format, for porous media with surface effects; however, relevant parameters (e.g., effective moduli) appearing in Biot's constitutive laws are dependent upon surface effects. This proof not only addresses the first issue, but also lays a solid foundation for theoretical and experimental studies of porous media with surface effects. To address the second issue, we employ the microstructures of a macromolecular network and an elastic matrix embedded with liquid inclusions as prototypes to demonstrate how their mechanical behaviors are related to the properties of three constituents (i.e., solid, fluid and solid-fluid interface) and determine explicitly the parameters appearing in their constitutive relations.

Analytical field model of segmented Halbach array permanent magnet machines considering iron nonlinearity

The permeability of the iron is usually assumed to be infinite by the linear subdomain technique, leading to the neglect of magnet saturation. This study presents an analytical model of a segmented Halbach array permanent magnet machine by considering the iron B-H curve and saturation effects. Firstly, the entire field problem is divided into six regions in the radial direction according to the excitation source. Then, Poisson's and Laplace's equations are solved in each region to obtain the magnetic vector potential. Specifically, in our approach, the teeth and slots' permeability variation is solved by Cauchy's product theorem. An iterative algorithm incorporates the nonlinearity of the iron. Subsequently, the electromagnetic performance, such as air-gap magnetic flux density, electromagnetic torque, and unbalanced magnetic force, is predicted by the analytical method. The analytical results are shown in line with those obtained from finite element analysis, and a prototype was fabricated to further verify the accuracy of the analytical method. Besides, the proposed method can also be used to evaluate the performance of the Halbach array machines with any number of segmented magnets.

A series representation for Riemann's zeta function and some interesting identities that follow

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions $\zeta^{\pm}(s)$ and $\eta^{\pm}(s)$ are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints.

Fuzzy complex numbers: Representations, operations, and their analysis

As a key theoretical basis of fuzzy complex analysis, a new representation of fuzzy complex numbers is proposed in this paper which unifies the exponential form, trigonometric form and algebraic form of fuzzy complex numbers, and some arithmetic operations are investigated. The concepts of strong sum and strong difference are defined in order to simplify the addition and subtraction operations of complex fuzzy numbers and establish the calculus theory of fuzzy complex valued functions. The metric between two fuzzy complex numbers, the derivative, differentiability and analyticity of fuzzy complex-valued functions are defined and characterized. The necessary and sufficient conditions of the analyticity of fuzzy complex-valued functions are investigated. Furthermore, the integral of fuzzy complex-valued functions and Cauchy integral theorem are given and discussed.

General time-dependent Green's functions of line forces in a two-dimensional, anisotropic, elastic, and infinite solid

In this paper, we derive analytical time-dependent Green's functions in a two-dimensional, anisotropic elastic, and infinite solid. It is based on the Stroh formalism combined with application of the Cauchy's residue theorem. Final expressions of the Green's function are in terms of simple finite line integral from 0 to 2π. The time-dependence of the line forces can be impulsive, Heaviside, or within a given time duration. The space-dependence of the line force is very general, including concentrated or uniformly distributed sources within a given line interval. Green's functions of both displacements and stresses are derived in analytical forms, and are verified against existing results. Numerical examples are presented to demonstrate the effect of source types and material anisotropy on the Green's functions.

The Concept of Development and Implementation of High-Speed Transport

The article discusses the issues of implementation and organization of high-speed transport. The objective of the article is to consider possible options for implementing highspeed (HS) motion systems using the principle of magnetic levitation, which will ensure high speeds for delivery of goods and carrying people over long distances. To achieve this objective, it is necessary to develop an engine and technical solutions for design of HS rolling stock, make decisions on energy supply infrastructure and the HS track, address safety issues and new control systems considering the state of the infrastructure and its design elements. The article discusses several options for implementation of high-speed transport systems, differing in the power supply system, current collection and track based on the magnetic levitation approach. An original approach is proposed in implementation of magnetic levitation transport using the technology of electromagnetic guns designed to implement traction forces of a magnetic levitation vehicle. The advantage of this approach is that it opens the possibility of maneuvering for the vehicle while driving. This allows to abandon switch turnouts, now significantly limiting the use of magnetic levitation transport. A mathematical model describing interaction of an electromagnetic gun and supermagnets located on the track is considered. In constructing the model, methods of the theory of electromagnetic field and interaction of magnetic bodies were used, and when constructing a model of interaction of rolling stock with a magnetic track, methods of mathematical algebra and the Cauchy theorem were used. The article discusses various principles of organization of movement using the magnetic levitation for urban, suburban, and intercity transport.

Ergodicity and stability for (b, l)-regularized resolvent operator families

This paper is concerned with the ergodicity and strong stability for (b, l)-regularized resolvent operator families. First, we study Abel ergodicity and Cesaro ergodicity of (b, l)-regularized resolvent operator families by using the methods of operator theory and complex Tauberian theorem. And then, by constructing a new operator-valued function, we obtain some sufficient conditions on the strong stability of bounded (b, l)-regularized resolvent operator families by means of Cauchy theorem and Riemann–Lebesgue lemma. Our results generalize the related conclusions for operator semigroups and resolvent operator families.