In this paper, we investigate uncertainty principles and stable signal reconstruction in the framework of the generalized Dunkl transform. Extending classical results of Donoho and Stark, we establish quantitative lower bounds linking the concentration of a function and its generalized Dunkl transform on measurable sets. We introduce the concept of strongly annihilating pairs adapted to this transform and prove stability results for the reconstruction of bandlimited functions from incomplete and noisy data. Furthermore, we develop Tikhonov regularization methods and iterative algorithms to address ill-posed recovery problems, providing explicit error bounds and convergence analysis. Our work broadens the scope of harmonic analysis uncertainty principles to singular differential-difference operators and contributes novel tools for signal processing applications within this setting.
Krylov methods have been successfully used to solve large matrix equations with sparse coefficients. In this paper, we are interested on solving the low-rank Lyapunov equation whether in the continuous or discrete case. We propose to modify the block Arnoldi process so that the projection Krylov subspace contains some additional blocks that involves the inverse of the square coefficient of this equation. Our goal with this approach is to reduce the number of iterations and convergence time required by the classical block Arnoldi method. To show the effectiveness of the proposed method, some numerical experiments are carried out.
In this paper, we enhance the oncolytic virotherapy model examined by [23] by incorporating multi-fractional derivatives of the Caputo type, assigning distinct derivative orders to each model variable. We establish the Hadamard well-posedness of the new model and confirm that the solutions remain within the positive half-plane, aligning with the criteria of the modeled problem. Furthermore, we conduct a stability analysis for the equilibrium solutions of these multi-fractional dynamics, demonstrating their dependence on the derivative orders specified in the dynamics. The theoretical insights are supported by numerical simulations.
We obtain new results on the relationship between uniqueness sets and stable sampling sets in function quasinormed spaces. The concept of perturbation of a uniformly discrete set plays an essential role. We also state and prove a result on stability for sampling sets and interpolation sets in a wide class of function quasinormed spaces.
In this study, we prove the existence of solutions to the nonlinear elliptic boundary value problem described by the equation −div a(x, u,∇u) + Ψ(x, u) = f where f, Ψ(x, u) are elements of L1(Ω), and where no monotonicity condition will be supposed on the function a(x, s, ξ).
Suppose κ is a regular cardinal. We prove that if the set of Hλ+-reflecting cardinals λ < κ is ineffable, then κ is an Hκ+-reflecting cardinal. Similarly, we also prove that if the set of Woodin cardinals/cardinals having the stationary reflection property below κ is ineffable, then κ is a Woodin cardinal/cardinal having the stationary reflection property.
In this paper, we introduce inertial Tseng’s method and Halperntype algorithm for solving monotone variational inequality and fixed point problems in 2-uniformly convex and 2-uniformly smooth real Banach spaces. We establish strong convergence of our proposed method under some assumptions on parameters without knowledge of the operator norm. Finally, we give numerical experiments to illustrate the efficiency of our main result.
Let A be a prime ring, Z(A) its center, Q its right Martindale quotient ring, C its extended centroid, ψ a non-zero b-generalized derivation of A with associated map ξ. In this article, we prove that: (i) If [ψ(x), ψ(y)] = 0 for all x, y ∈ A, then A is either commutative or there exists q ∈ Q such that ξ = ad(q), ψ(x) = -bxq, and qb = 0. (ii) If ψ(x) ◦ ψ(y) = 0 for all x, y ∈ A, then A is either commutative with char(A) = 2 or there exists q ∈ Q such that ψ(x) = -bxq and qb = 0. Additional results are established for cases involving [ξ(x), ψ(x)] = 0 or ξ(x)◦ψ(x) = 0, where char(A) = 2. Furthermore, we give some examples that show the importance of the hypotheses of our theorems.
In this article we study Kummer's $\mathcal D$-groupoid, which is the groupoid of symmetries of a meromorphic projective structure. We give necessary and sufficient conditions for its minimality, in the sense of not having infinite sub-$\mathcal D$-groupoids. The condition that we find turns out to be equivalent to the strong minimality of the non-linear Schwarzian equation and the non-integrability by means of Liouvillian functions of the linear Schwarzian equation.
For k ≥ 2, let (P(k)n)n≥2−k be the k-generalized Pell sequence which starts with 0, · · · , 0, 1 (k terms) and each term afterwards is given by the linear recurrence P(k)n = 2P(k)n-1 + P(k)n-2 + · · · + P(k)n-k, for all n ≥ 2. An integer n is said to be close to a positive integer m if n satisfies |n−m| < √m. In this paper, we solve the Diophantine inequality |P(k)n - 2m| < 2m/2, in positive unknowns k, n, and m.
La congruencia de fase es una técnica de procesamiento de imágenes relativamente desconocida y potente para la segmentación. No obstante, una limitación de este método es su alta sensibilidad al ruido; en ese sentido, para evitar que el ruido afecte los resultados de la segmentación, es necesaria una buena estimación de su nivel, teniendo en cuenta que en la congruencia de fase, esta estimación se realiza a partir de la imagen de la energía local. Por lo tanto, con el fin de mejorar los resultados de la técnica, es indispensable realizar una buena detección del umbral de ruido. Por esta razón, en este trabajo se introduce un método eficiente para la estimación de los parámetros de una distribución Weibull, empleada para modelar el ruido de la imagen de energía de la congruencia de fase.
In this paper, we prove existence results for entropy solutions of a nonlinear boundary value problems represented by a class of nonlinear elliptic anisotropic equations with variable exponents and natural growth terms. The functional setting involves variable exponents anisotropic Sobolev spaces.
Recently Hossein Jafari introduced a new general integral transform called Jafari transform to solve higher order initial value problems and integral equations. The main objective of this paper is to modify this integral transform that we call the ρ-Jafari transform and to study its properties. Then, we present interesting results and apply them to solve linear and nonlinear generalized fractional differential equations. The results obtained confirmed that the ρ-Jafari transform acts as a powerful tool for generalized fractional problems. As a result, we assert that in the future, the modified transform can be applied to many generalized fractional differential equations that arise in applied science and engineering.
This work introduces a geometric mean algorithm for positive definite matrices using generalized eigenvalue problems and Cholesky factorization. The geometric mean of a finite set of positive definite matrices minimizes the sum of square distances to all the matrices where the distance is an affine-invariant Riemannian metric in the manifold of the symmetric positive definite matrices SN++. In order to compute numerical approximations of the geometric mean several algorithms have been proposed. Some of these algorithms require the computation of several diagonalizations in each iteration. We show that by rewriting the iterations in terms of generalized eigenvalue problems, it is possible to omit some of the diagonalizations at the cost of introducing much less generalized eigenvalue problems that can be solved using Cholesky factorizations. We numerically compare the performance of classical methods and the modified algorithms that use generalized eigenvalue problems. The resulting method is applied to video analysis using the mean of covariance matrices as a compact descriptor for actions classification. The proposed mean descriptor with just 105 scalar values achieved an average accuracy of 75% over a publicaction video dataset.
La congruencia de fase es una técnica de detección de bordes de imágenes que, mediante el análisis de la fase de los componentes de frecuencia de una señal, permite encontrar la ubicación de los bordes. Según la función matemática utilizada para la cuantificación de la congruencia de fase, el resultado de la segmentación presenta variabilidad, lo cual puede resultar en una mejora potencial en la detección. Por ello, en este trabajo se realiza un estudio de la función exponencial, boxcar y cuártica utilizando dos métricas para la evaluación de técnicas de segmentación, el índice de Dice-Sorensen y la figura de mérito de Pratt. Para este estudio, se introduce una pequeña base de datos que incluye 30 imágenes originales y sus correspondientes imágenes de referencia. Además, para comparar la congruencia de fase con las técnicas basadas en el gradiente, se compararon los resultados con los obtenidos con el método de detección Canny, encontrando que la congruencia de fase permite una mejor detección de bordes en la mayoría de los casos.
We prove boundedness of a discrete version of Vainikko operator on discrete Morrey spaces. We also show that the commutator of this Vainikko operator with a multiplication operator by an element of a discrete version of BMO is bounded on these spaces.
The present paper focus on Lagrangian distributions with complex phase. We provide an alternative construction of their principal symbol map, which allows us to compute the principal symbol after clean composition of Fourier integral operators with complex phase.
In this paper, we study new results regarding the existence, uniqueness and convergence of the solution of nonlinear fractional Volterra integrodifferential equations via Caputo-Fabrizio operator. The main results of this paper are based on the Banach contraction principle. Furthermore, we investigate the approximate analytical solutions of the proposed problem using a new combination method called Khalouta decomposition method. Some illustrated examples of our results are provided with some numerical simulations of the solutions.
In his celebrated article of 1956, John Milnor established the existence of smooth structures on the 7-dimensional sphere that differs from the usual one. These so-called "exotic" structures have been of great interest ever since. The purpose of this article is to give a clear exposition of the different tools that Milnor used in order to provide an almost self-contained construction of exotic structures on the 7-dimensional sphere and then to show that they are not diffeomorphic to the standard sphere.
