Inverse Theorem Research Articles

Triple Laplace Transform in Bicomplex Space with Application

In this article, we investigate bicomplex triple Laplace transform in the framework of bicomplexified frequency domain with Region of Convergence (ROC), which is generalization of complex triple Laplace transform. Bicomplex numbers are pairs of complex numbers with commutative ring with unity and zero-divisors, which describe physical interpretation in four dimensional spaces and provide large class of frequency domain. Also, we derive some basic properties and inversion theorem of triple Laplace transform in bicomplex space. In this technique, we use idempotent representation methodology of bicomplex numbers, which play vital role in proving our results. Consequently, the obtained results can be highly applicable in the fields of Quantum Mechanics, Signal Processing, Electric Circuit Theory, Control Engineering, and solving differential equations. Application of bicomplex triple Laplace transform has been discussed in finding the solution of third-order partial differential equation of bicomplex-valued function.

Global apparent resistivity definition for a high-performance TEM excitation source

AbstractThe high-performance transient electromagnetic method (TEM) excitation source is a new type of source that has been proposed for urban underground space exploration. This source is composed of two trapezoid plates. To ensure that the radiation field was focused in a certain direction, the two trapezoid plate-shaped antennas were arranged into a horn shape. This new source is characterised by high power, directional excitation and high resolution. The corresponding multi-component global apparent resistivity definition method is established for a high-performance transient electromagnetic excitation source. This method is studied using the inverse function theorem. Then, the monotonic relationship between components of the electromagnetic field and resistivity is analysed. For the fields that satisfy the monotonic relationship with half-space resistivity, the apparent resistivity can be calculated correctly to ignore the time period and location in the space. This means that this definition method can eliminate the limitation of early and late times, and the near and far zones. The apparent resistivity calculation results of the theoretical layered model reveal that global apparent resistivity curves show a regular change, which smoothly and comprehensively reflects the change of electrical information in the model. The experimental results of the 3D model show that the five-layer low-resistivity anomaly contained in the urban underground space designed in this paper exhibits an obvious response in the global apparent resistivity profile. It is concluded that a high-performance TEM excitation source possesses a high resolution, clearly reflecting all of the anomalies of a complex urban underground space model.

Sobolev mappings and moduli inequalities on Carnot groups

We study the mappings that satisfy moduli inequalities on Carnot groups. We prove that the homeomorphisms satisfying the moduli inequalities ($Q$-homeomor\-phisms) with a locally integrable function $Q$ are Sobolev mappings. On this base in the frameworks of the weak inverse mapping theorem, we prove that, on the Carnot groups $\mathbb G,$ the mappings inverse to Sobolev homeomorphisms of finite distortion of the class $W^1_{\nu,\loc}(\Omega;\Omega')$ belong to the Sobolev class $W^1_{1,\loc}(\Omega';\Omega)$.

Uniqueness of solutions to Lp-Christoffel-Minkowski problem for p < 1

Lp-Christoffel-Minkowski problem arises naturally in the Lp-Brunn-Minkowski theory. It connects both curvature measures and area measures of convex bodies and is a fundamental problem in convex geometric analysis. Since the lack of Firey's extension of Brunn-Minkowski inequality and constant rank theorem for p<1, the existence and uniqueness of Lp-Brunn-Minkowski problem are difficult problems. In this paper, we prove a uniqueness theorem for solutions to Lp-Christoffel-Minkowski problem with p<1 and constant prescribed data. Our proof is motivated by the idea of Brendle-Choi-Daskaspoulos's work on asymptotic behavior of flows by powers of the Gaussian curvature. One of the highlights of our arguments is that we introduce a new auxiliary function Z which is the key to our proof.

Continuous Selections of Solutions for Locally Lipschitzian Equations

This paper answers in the affirmative the long-standing question of nonlinear analysis, concerning the existence of a continuous single-valued local selection of the right inverse to a locally Lipschitzian mapping. Moreover, we develop a much more general result, providing conditions for the existence of a continuous single-valued selection not only locally, but rather on any given ball centered at the point in question. Finally, by driving the radius of this ball to infinity, we obtain the global inverse function theorem, essentially implying the well-known Hadamard’s theorem on a global homeomorphism for smooth mappings and the more general Pourciau’s theorem for locally Lipschitzian mappings.

Quaternionic inverse Fourier transforms on locally compact abelian groups

ABSTRACT In this paper, after introducing the concepts of quaternionic dual group and the quaternionic valued character on locally compact abelian group , the inverse of the quaternionic Fourier transform (QFT) on locally compact abelian groups is investigated. Due to the non-commutativity of multiplication of quaternions, there are different types of QFTs right, left and two-sided quaternionic Fourier transform. We focus on the right-sided quaternionic Fourier transform (RQFT) and two-sided quaternionic Fourier transform (SQFT). We establish the quaternionic Plancherel and inversion theorems for the square integrable quaternionic-valued signals on , the space , where G is a locally compact abelian group. Also RQFT on the space is studied. Furthermore relations between RQFT and SQFT are discussed. These results provide new proofs for the classical inverse Fourier transform, Plancherel theorem, etc. in .

Optimal dose calibration in radiotherapy

In this paper, the tools provided by the theory of Optimal Experimental Design are applied to a nonlinear calibration model. This is motivated by the need of estimating radiation doses using radiochromic films for radiotherapy purposes. The calibration model is in this case nonlinear and the explanatory variable cannot be worked out explicitly from the model. In this case an experimental design has to be found on the dependent variable. For that, the inverse function theorem will be used to obtain an information matrix to be optimized. Optimal designs on the response variable are computed from two different perspectives, first for fitting the model and estimating each of the parameters and then for predicting the proper dose. While the first is a common point of view in a general context of the Optimal Experimental Design, the latter is actually the main objective of the calibration problem for the practitioners and algorithms for computing these optimal designs are also provided.The optimal designs obtained have just three different points in their support, but practitioners usually demand for more support points. Thus, a methodology for computing space-filling designs is also provided when the support points are forced to follow some mathematical rule, such as arithmetic or geometric sequences. Cross efficiencies of all these designs are computed in order to show their ability for different goals.

On the Ramsey number of the Brauer configuration

We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions with the same colour as their common difference. Such a result has been obtained independently and in much greater generality by Sanders. Using Gowers' local inverse theorem, our bound is quintuple exponential in the length of the progression. We refine this bound in the colour aspect for three-term progressions, and combine our arguments with an insight of Lefmann to obtain analogous bounds for the Ramsey numbers of certain non-linear quadratic equations.

Alternative Solutions for Consolidation Problems in Isotropic Clay Stratum Using Laplace Heaviside’s Theorem

This study offers an alternative analytical solution, for one and two-dimensional consolidation problem, for functionally-homogenous isotropic clay layer. The fundamental analytical solution presented in this study is derived based on the Laplace transform which eliminates the time-dependent and Heaviside's inversion theorem to extract the inverse. The alternative solutions have been derived based on the governing available fundamental consolidation equations solution. The derived solutions have shown good estimation with the exact solution. The two dimensions of isotopic permeability expressed significant influence on the accumulated excess pore water pressure and approximate time for full consolidation. The analytical solution shows identical results with the numerical one.

On measures that improve $L^q$ dimension under convolution

The L^q dimensions, for 1 &lt; q &lt; \infty , quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the L^q dimension improve under convolution? This can be seen as a variant of the well-known L^p -improving property. Our main result asserts that uniformly perfect measures (which include Ahlfors-regular measures as a proper subset) have the property that convolving with them results in a strict increase of the L^q dimension. We also study the case q = \infty , which corresponds to the supremum of the Frostman exponents of the measure. We obtain consequences for repeated convolutions and for the box dimension of sumsets. Our results are derived from an inverse theorem for the L^q norms of convolutions due to the second author.

Weighted Simultaneous Approximation of the Linear Combinations of Baskakov Operators

In this paper, the approximation property of the linear combinations of Baskakov operators is investigated by using a Jacobi weight function. Moreover, both the positive and inverse theorems for the weighted simultaneous approximation of the linear combinations of Baskakov operators are proved.

ANALYTICALLY CLOSED-FORM SOLUTIONS FOR THE DISTRIBUTION OF A NUMBER OF CUSTOMERS SERVED DURING A BUSY PERIOD FOR SPECIAL CASES OF THE GEO/G/1 QUEUE

AbstractThis paper presents the distribution of the number of customers served during a busy period for special cases of the Geo/G/1 queue when initiated with m customers. We analyze the system under the assumptions of a late arrival system with delayed access and early arrival system policies. It is not easy to invert the functional equation for the number of customers served during a busy period except for the simple case Geo/Geo/1 queue, as stated by several researchers. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We find the distribution of the number of customers served during a busy period for various service-time distributions such as geometric, deterministic, binomial, negative binomial, uniform, Delaporte, discrete phase-type and interrupted Bernoulli process. We compute the mean and variance of these distributions and also give numerical results. Due to the clarity of the expressions, the computations are very fast and robust. We also show that in the limiting case, the results tend to the analogous continuous-time counterparts.

Inverse and implicit function theorems for noncommutative functions on operator domains

Classically, a noncommutative function is defined on a graded domain of tuples of square matrices. In this note, we introduce a notion of a noncommutative function defined on a domain Ω⊂B(H)d, where H is an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to injectivity.

Об одном доказательстве теоремы синусов

The author of the article outlines, that in school geometry course, the sine theorem and the cosine theorem are well known. In this course, they are proved by the authors of the textbook in a way different from the one that is presented in the article. The article considers the author's method of proving the sine theorem unknown in the literature sources and based on the vector-coordinate method; two more theorems are also proved, one of which concerns the calculation of the inscribed angle in the circle, and the other concerns the calculation of viewing angles of the chord of the circle; one task is proposed by the author for independent work, in which the object of research is the chord of a circle. Research results. A proof of the sine theorem, presented by the author on the basis of his technique, undoubtedly quicken public interest. The material can serve as a basis for organizing educational and research activities of students in mathematics. It is concluded that educational and research activities of students in mathematics can be effectively organized when: establishing the essential properties of concepts; identifying the relationship of this concept with other concepts; searching for other methods of proving theorems; formulating the inverse theorem and establishing its truth; classification of mathematical objects and relations between them; solving mathematical problems in various ways and methods; drawing up new tasks that result from already solved ones; providing examples and counterexamples that illustrate a particular fact, etc.

Social semiotics analysis of Palestinian mathematics textbooks for eighth grade

Mathematics textbook analysis can serve to understand the teaching and learning processes in the mathematics classroom. The present study utilizes a social semiotics framework to analyze the triangle unit of the Palestinian mathematics book for grade 8. The results of the study indicate that the authors utilized the representational aspect of the mathematical object to introduce those objects to the reader. Moreover, the nature of mathematics resulting from this unit is that of a subject that learners do not need material processes to discover, so it is enough to reason about it mentally to arrive at the mathematical objects and relations. The authors used a plural first-person pronoun to describe the need to engage with theorems and inverse theorems. They used the singular second-person pronoun to attract the attention of the reader to specific features of the mathematical objects. The authors did not use any pronoun when stating the theorem. Some of the connectors were verbs, nouns, and sentences, where the most used connector was the sentence, especially in reasoning. This use of the sentence in mathematical reasoning indicates that the authors wanted to advance the mathematical reasoning as a narrative to facilitate it for the reader.

A Sequential Approach to Mild Distributions

The Banach Gelfand Triple ( S 0 , L 2 , S 0 ′ ) ( R d ) consists of S 0 ( R d ) , ∥ · ∥ S 0 , a very specific Segal algebra as algebra of test functions, the Hilbert space L 2 ( R d ) , ∥ · ∥ 2 and the dual space S 0 ′ ( R d ) , whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce S 0 ( R d ) , ∥ · ∥ S 0 and hence ( S 0 ′ ( R d ) , ∥ · ∥ S 0 ′ ) , the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short ECmiCS). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra S 0 ( R d ) , ∥ · ∥ S 0 can be used to establish this natural identification.

An Inverse Mapping Theorem in Fréchet-Montel Spaces

Influenced by a recent note by M. Ivanov and N. Zlateva, we prove a statement in the style of Nash-Moser-Ekeland theorem for mappings from a Frechet-Montel space with values in any Frechet space (not necessarily standard). The mapping under consideration is supposed to be continuous and directionally differentiable (in particular Gateaux differentiable) with the derivative having a right inverse. We also consider an approximation by a graphical derivative and by a linear operator in the spirit of Graves’ theorem. Finally, we derive corollaries of the abstract results in finite dimensions. We obtain, in particular, sufficient conditions for the directional semiregularity of a mapping defined on a (locally) convex compact set in directions from a locally conic set; and also conditions guaranteeing that the nonlinear image of a convex set contains a prescribed ordered interval.

An inverse function theorem converse

We establish the following converse of the well-known inverse function theorem. Let g:U→V and f:V→U be inverse homeomorphisms between open subsets of Banach spaces. If g is differentiable of class Cp and f is locally Lipschitz, then the Fréchet derivative of g at each point of U is invertible and f must be differentiable of class Cp.

Direct and inverse approximation theorems in the weighted Orlicz-type spaceswith a variable exponent

In weighted Orlicz-type spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,\mu}}$ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is the best in a certain sense. Some applications of the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduli of smoothness are given. Equivalence between moduli of smoothness and certain Peetre $K$-functionals is shown in the spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,\mu}}$.

A sharpened form of the inverse function theorem

In this note we establish an advanced version of the inverse function theorem and study some local geometrical properties like starlikeness and hyperbolic convexity of the inverse function under natural restrictions on the numerical range of the underlying mapping.