Inverse Theorem Research Articles

The structure of arbitrary Conze–Lesigne systems

Let Γ \Gamma be a countable abelian group. An (abstract) Γ \Gamma -system X \mathrm {X} - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of Γ \Gamma - is said to be a Conze–Lesigne system if it is equal to its second Host–Kra–Ziegler factor Z 2 ( X ) \mathrm {Z}^2(\mathrm {X}) . The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian Γ \Gamma , namely that they are the inverse limit of translational systems G n / Λ n G_n/\Lambda _n arising from locally compact nilpotent groups G n G_n of nilpotency class 2 2 , quotiented by a lattice Λ n \Lambda _n . Results of this type were previously known when Γ \Gamma was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers U 3 ( G ) U^3(G) norm for arbitrary finite abelian groups G G .

Theorems on direct and inverse approximation by algebraic polynomials and piecewise polynomials in the spaces H<sup>m</sup>(a, b) and B<sup>s</sup><sub>2,q</sub>(a, b)

The best estimates for the approximation error of functions, defined on a finite interval, by algebraic polynomials and piecewise polynomial functions are obtained in the case when the errors are measured in the norms of Sobolev and Besov spaces. We indicate the weighted Besov spaces, whose functions satisfy Jackson-type and Bernstein-type inequalities and, as a consequence, direct and inverse approximation theorems. In a number of cases, exact constants are indicated in the estimates.

On some generalizations of Hadamard's inversion theorem beyond differentiability

A recognized trend of research investigates generalizations of the Hadamard's inversion theorem to functions that may fail to be differentiable. In this vein, the present paper explores some consequences of a recent result about the existence of global Lipschitz continuous inverse by translating its assumptions of metric nature in terms of nonsmooth analysis constructions. This exploration focuses on continuous, but possibly not locally Lipschitz mappings, acting in finite-dimensional Euclidean spaces. As a result, sufficient conditions for global invertibility are formulated by means of strict estimators, ∗ -difference of convex compacta and regular/basic coderivatives. These conditions qualify the global inverse as a Lipschitz continuous mapping and provide quantitative estimates of its Lipschitz constant in terms of the above constructions.

Lp$$ {L}^p $$‐theory of linear canonical transforms and related uncertainty principles

In this paper, we revisit some fundamental properties of linear canonical transform (abbreviated as LCT). In particular, we prove the additive property rigorously for LCT in the higher dimensional case (abbreviated as MLCT). We also consider the ‐theory of MLCT with . Specifically, the inversion theorem of MLCT by the related Gauss and Abel means is studied, and the pointwise convergence of approximate identities with respect to convolution for MLCT is also obtained. As applications, we study the ‐type Heisenberg‐Pauli‐Weyl uncertainty principles and the ‐type Donoho‐Stark uncertainty principles for MLCT.

Inverse approximation theorem using Bernstein’s Durrmeyer operator in terms of 2nd modulus of smoothness

Many inverse theorems have been proven using the ordinary modulus of smoothness but no one proved the invers theorems for approximation by Bernstein’s Durrmeyer operator in terms 2nd modulus of smoothness.

Insensitizing control problems for the stabilized Kuramoto–Sivashinsky system

In this work, we address the existence of insensitizing controls for a nonlinear coupled system of fourth- and second-order parabolic equations known as the stabilized Kuramoto–Sivashinsky model. The main idea is to look for controls such that some functional of the states (the so-called sentinel) is locally insensitive to the perturbations of the initial data. Since the underlying model is coupled, we shall consider a sentinel in which we may observe one or two components of the system in a localized observation set. By some classical arguments, the insensitizing problem can be reduced to a null-controllability one for a cascade system where the number of equations is doubled. Upon linearization, the null-controllability for this new system is studied by means of Carleman estimates but unlike other insensitizing problems for scalar models, the election of the Carleman tools and the overall control strategy depends on the initial choice of the sentinel due to the (lack of) couplings arising in the extended system. Finally, the local null-controllability of the extended (nonlinear) system (and thus the insensitizing property) is obtained by applying the inverse mapping theorem.

Approximation by double periodic functions in the class C^r_A

In this work we will consider approximation by polynomials in doubly periodic Weierstrass functions for functions that are analytic in a domain and continuous in its closure. This problem is closely related to the approximation by holomorphic polynomials in two variables of a function that is holomorphic in a domain on an elliptic curve. We assume that at the boundary of the region on the plane the length of the are is commensurable with length of the chord. This condition also applies to the region on the elliptic curve. The possibility of obtaining an approximation estimate that is consistent with the so-called the inverse theorem. i. e., with the restoration of the smoothness of the function by the speed of approximation, it was previously obtained for classed of function analytic in a domain for which in the closure of the domain the derivative of a given order has a modulus of continuity of the Holder type, with an order less than unity. The approximation method used earlier did not make it possible to study classes of analytic functions whose derivative of some order is limited. In this paper, we use a different approximation method to approximate by polynomials in doubly periodic Weierstrass functions the functions that are analytic in a domain and whose derivative of a given order is bounded in the domain.

Neuro-BERT: Rethinking Masked Autoencoding for Self-Supervised Neurological Pretraining.

Deep learning associated with neurological signals is poised to drive major advancements in diverse fields such as medical diagnostics, neurorehabilitation, and brain-computer interfaces. The challenge in harnessing the full potential of these signals lies in the dependency on extensive, high-quality annotated data, which is often scarce and expensive to acquire, requiring specialized infrastructure and domain expertise. To address the appetite for data in deep learning, we present Neuro-BERT, a self-supervised pre-training framework of neurological signals based on masked autoencoding in the Fourier domain. The intuition behind our approach is simple: frequency and phase distribution of neurological signals can reveal intricate neurological activities. We propose a novel pre-training task dubbed Fourier Inversion Prediction (FIP), which randomly masks out a portion of the input signal and then predicts the missing information using the Fourier inversion theorem. Pre-trained models can be potentially used for various downstream tasks such as sleep stage classification and gesture recognition. Unlike contrastive-based methods, which strongly rely on carefully hand-crafted augmentations and siamese structure, our approach works reasonably well with a simple transformer encoder with no augmentation requirements. By evaluating our method on several benchmark datasets, we show that Neuro-BERT improves downstream neurological-related tasks by a large margin.

Extended inverse theorems for $h$-fold sumsets in integers

Let $h \geq 2$, $k \geq 5$ be integers and $A$ be a nonempty finite set of $k$ integers. Very recently, Tang and Xing studied extended inverse theorems for $hk-h+1 < \left|hA\right| \leq hk+2h-3$. In this paper, we extend the work of Tang and Xing and study all possible inverse theorems for $hk-h+1<\left|hA\right| \leq hk+2h +1$. Furthermore, we give a range of $|hA|$ for which inverse problems are not possible.

Approximation Properties of Parametric Kantorovich-Type Operators on Half-Bounded Intervals

The main purpose of this paper is to introduce a new family of parametric Kantorovichtype operators on the half-bounded interval. The convergence properties of these new operators are investigated. The Voronovskaja-type weak inverse theorem and the rate of uniform convergence are obtained. Furthermore, we obtain some shape preserving properties of these operators, including monotonicity, convexity, starshapeness, and semi-additivity preserving properties. Finally, some numerical illustrative examples show that these new operators have a better approximation performance than the classical ones.

Programmable repulsive potential for tight-binding from Chen-Möbius inversion theorem

Programmable repulsive potential for tight-binding from Chen-Möbius inversion theorem

Обернена теорема для узагальненої похідної в банахових просторах

Establishing the properties of the approximation characteristics of the studied functions is one of the main tasks of the theory of approximations. If, based on information about the behavior of the generalized derivative of some function f, it is possible to predict the behavior of the sequence of the best approximations of this function by polynomials, then we are talking about stating and proving direct theorems of the theory of approximations. If the properties of the function f Î X itself and its generalized derivatives are studied, relying on the behavior of the sequence best approximations, i.e., the differential-difference characteristics of the function f are established based on the study of the behavior of the sequence of its best approximations, then we speak of the proof of inverse theorems of approximation theory. The study of direct and inverse theorems begins with the works of Bernstein, Valle Poussin, Jackson and others in 1910-1912. They were continued by many scientists (N. I. Ahiezer, M. G. Crane, J. Favar, B. V. Stechkin, S. M. Nikolskyi, A. F. Timan, A. Zygmund, V. K. Dzya­dyk, O. I. Stepanets). There are still many important and unsolved problems in the theory of approximations, in particular, such as extending direct and inverse theorems to new classes of functions and establishing the best values of constants in the corresponding inequalities. At the same time, it becomes possible to formulate new problems, in particular, problems of mathematical modeling already for whole classes of functions, that describe the studied processes. This article considers the inverse theorem – based on the properties of the sequence of best approximations, a conclusion is made about the properties of the element f of some Banach space X and its generalized derivatives. As well as the relations between Szego constants for different equivalent systems of elements of the Banach space are established

A Note on Clarke’s Generalized Jacobian for the Inverse of Bi-Lipschitz Maps

Clarke’s inverse function theorem for Lipschitz mappings states that a bi-Lipschitz mapping f is locally invertible about a point x0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0$$\\end{document} if the generalized Jacobian ∂f(x0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial f(x_0)$$\\end{document} does not contain singular matrices. It is shown that under these assumptions the generalized Jacobian of the inverse mapping at f(x0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x_0)$$\\end{document} is the convex hull of the set of matrices that can be obtained as limits of sequences Jf(xk)-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_f(x_k)^{-1}$$\\end{document} with f differentiable in xk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_k$$\\end{document} and xk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_k$$\\end{document} converging to x0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0$$\\end{document}. This identity holds as well if f is assumed to be locally bi-Lipschitz at x0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0$$\\end{document}.

Operators that are Weakly Coercive and a Compact Perturbation

ABSTRACT Let X, Y be two Banach spaces over K = ℝ \[\mathbb{K}=\mathbb{R}\] or ℂ \[\mathbb{C}\] , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C 1-compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C 1-diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f(x) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems.

Hadamard inverse function theorem proved by variational analysis

We present a proof of Hadamard Inverse Function Theorem by the methods of Variational Analysis, adapting an idea of I. Ekeland and E. Séré.

Insensitizing control problem for the Hirota–Satsuma system of KdV–KdV type

This paper is concerned with the existence of insensitizing controls for a nonlinear coupled system of two Korteweg–de Vries (KdV) equations, typically known as the Hirota–Satsuma system. The idea is to look for controls such that some functional of the states (the so-called sentinel) is insensitive to the small perturbations of initial data. Since the system is coupled, we consider a sentinel in which we observe both components of the system in a localized observation set. By some classical argument, the insensitizing problem is then reduced to a null-control problem for an extended system where the number of equations is doubled. We study the null-controllability for the linearized model associated to that extended system by means of a suitable Carleman estimate which is proved in this paper. Finally, the local null-controllability of the extended (nonlinear) system is obtained by applying the inverse mapping theorem, and this implies the required insensitizing property for the concerned model.

A wide-beam NCS algorithm for multi-receiver SAS based on azimuth spectrum superposition

The existing multi-receiver synthetic aperture sonar (SAS) imaging algorithms are suitable for narrow-beam width, which will lead to a decrease in imaging quality under wide-beam condition and are not in line with the development needs of SAS. We propose a non-linear chirp scaling algorithm (NCSA) for wide beam multi-receiver SAS. Firstly, the point target reference spectrum (PTRS) of each receiver is obtained by the Lagrange inversion theorem (LIT), and then the under-sampled signal in the azimuth frequency domain is obtained through azimuth spectrum extension; Then, considering the cubic term of range frequency in the PTRS and the linear variation of equivalent frequency modulation slope with range, each receiver is imaged using the NCSA, and coherent superposition is performed in the azimuth frequency domain to eliminate spectrum aliasing caused by azimuth spectrum extension; Finally, the azimuth inverse transform is performed on the superimposed signal to obtain the focusing imaging. Computer simulation experiments and field data verify that this method is superior to the existing SAS imaging algorithm, improving the quality of wide-beam imaging, avoiding the interpolation operation of the traditional range-Doppler algorithm, and saving computation cost.

Neutrosophic hyperbolic programming strategy for uncertain multi-objective transportation problem

This paper suggests a neutrosophic hyperbolic programming approach based on the hyperbolic membership function to get an efficient solution for an uncertain multi-objective transportation problem where the uncertain normal distribution presents uncertainty. This technique provides flexibility to the decision-maker to select various confidence levels according to his/her own choice and analyze the satisfaction percentage of the obtained optimal compromise solution along with its truth, indeterminacy, and falsity values. In this approach, the parameters are changed from uncertain to crisp by using the inverse measure theorem, and the neutrosophic non-linear hyperbolic membership functions are used to convert the whole model into a non-linear programming problem in a neutrosophic environment. The advantages of our recommended method in real-world applications are effectively illustrated by a numerical illustration. This illustration also makes it easier to compare the results attained using our suggested strategy to those obtained using the methodologies put forward by Uddin et al. (2021) and Miah et al. (2022). Also, MATLAB software (version R2019a,64bit(win64)) is used to develop the code for the solution of this non-linear uncertain multi-objective transportation problem.

Herglotz type conservation laws for nonconservative nonholonomic systems

The Herglotz variational principle offers an effective method for studying nonconservative system dynamics. The aim of this paper is to study the conservation laws of nonholonomic systems by using the Herglotz type generalized variational principle and establish Noether’s theorem and its inverse theorem for this system. In deriving the equations of motion, we use the Suslov definition of the reciprocity relation between differential and variational operations. First, the Herglotz type generalized variational principle is listed, and the Herglotz type Chaplygin equations for nonconservative nonholonomic systems are deduced. Second, Noether’s theorem and Noether’s inverse theorem are established, and the Herglotz type conservation laws are given. Finally, an example is provided to illustrate the practical implementation of the findings.

Approximation by multivariate Baskakov–Durrmeyer operators in Orlicz spaces

Employing some properties of multivariate Baskakov–Durrmeyer operators and utilizing modified K-functional and a decomposition technique, the authors obtain the direct theorem and weak type inverse theorem in the Orlicz spaces.