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This article is concerned with the question of uniqueness of self-similar profiles for Smoluchowski’s coagulation equation which exhibit algebraic decay (fat tails) at infinity. More precisely, we consider a rate kernel Establishing uniqueness of self-similar profiles for Smoluchowski’s coagulation equation is generally considered to be a difficult problem which is still essentially open. Concerning fat-tailed self-similar profiles this article actually gives the first uniqueness statement for a non-solvable kernel.

The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth. In this present paper we consider the normal matrix model with cubic plus linear potential. In order to regularize the model, we follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain We also study in detail the mother body problem associated to To construct the mother body measure, we define a quadratic differential Following previous works of Bleher & Kuijlaars and Kuijlaars & López, we consider multiple orthogonal polynomials associated with the normal matrix model. Applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain strong asymptotic formulas for these polynomials. Due to the presence of the linear term in the potential, there are no rotational symmetries in the model. This makes the construction of the associated

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.

Laplace transform has many applications in applied mathematics, science, engineering, and technology. Its further study will play an important role in developing new theories with applications. This paper deals with new results on the Laplace-Complex EE (Emad-Elaf) integral transform (LCEE) of two variables. Starting with definitions and standard results, we obtained various properties, including the shifting property and change of scale, etc. We developed new theorems on the Laplace-Complex EE integral transform of partial derivatives. Further, we applied our results to solve non-homogeneous telegraph equations. Finally, we illustrate our results with examples.

In this paper, we introduce a new integral transform called the Formable integral transform, which is a new efficient technique for solving ordinary and partial differential equations. We introduce the definition of the new transform and give the sufficient conditions for its existence. Some essential properties and examples are introduced to show the efficiency and applicability of the new transform, and we prove the duality between the new transform and other transforms such as the Laplace transform, Sumudu transform, Elzaki transform, ARA transform, Natural transform and Shehu transform. Finally, we use the Formable transform to solve some ordinary and partial differential equations by presenting five applications, and we evaluate the Formable transform for some functions and present them in a table. A comparison between the new transform and some well-known transforms is made and illustrated in a table.

We present a novel proof, using group theory, for a Meijer transform formula. This proof reveals the formula as a specific case of a broader generalized result. The generalization is achieved through a linear operator that intertwines two representations of the connected component of the identity of the group SO(2,2). Using this same approach, we derive a formula for the sum of three double integral transforms, where the kernels are represented by Bessel functions. It is particularly noteworthy that the group SO(2,2) is connected to symmetry in several significant ways, especially in mathematical physics and geometry.

Accurate estimation techniques are crucial in statistical modeling and reliability analysis, which have significant applications across various industries. The three-parameter Weibull distribution is a widely used tool in this context, but traditional estimation methods often struggle with outliers, resulting in unreliable parameter estimates. To address this issue, our study introduces a robust estimation technique for the three-parameter Weibull distribution, leveraging the probability integral transform and specifically employing the Weibull survival function for the transformation, with a focus on complete data. This method is designed to enhance robustness while maintaining computational simplicity, making it easy to implement. Through extensive simulation studies, we demonstrate the effectiveness and resilience of our proposed estimator in the presence of outliers. The findings indicate that this new technique significantly improves the accuracy of Weibull parameter estimates, thereby expanding the toolkit available to researchers and practitioners in reliability data analysis. Furthermore, we apply the proposed method to real-world reliability datasets, confirming its practical utility and effectiveness in overcoming the limitations of existing estimation methodologies in the presence of outliers.

In this paper, we introduce a novel integral transform to address certain ordinary differential equations, and it has been demonstrated that this transform possesses unique and highly advantageous properties. We examine this fascinating integral transform and its effectiveness in solving linear ordinary and partial differential equations. Also, we provide a graphical depiction of the solution to the respective differential equations.

This work presents three new Fourier-type Banach algebras generated by the distinct convolution multiplications. In particular, the first algebra is based on the Hartley convolution. For the second and third algebras, a main ingredient is the appropriate use of a group of four previously constructed convolutions associated with the Fourier-cosine and Fourier-sine integral transforms defined on the whole space R d and half-axis R + , respectively. With the practical problems, Sect.  3 considers the applications for two large enough classes of the integral equations. Namely, the solvability of the equations is completely investigated, and the explicit solutions are obtained. With respect to approximate computations in the practical problems, every solution can be expressed in terms of a Neumann functional series, and of course, they belong to the constructed algebra.

This study presents a generalized integral residual power series method (GIRPSM) for finding semi-analytical solutions to the nonlinear fractional Zakharov–Kuznetsov equation (FZKE). This method combines the residual power series method with a new general integral transform to improve accuracy and convergence. The effectiveness of this method is demonstrated by the robustness of the numerical results. The results demonstrate that GIRPSM is highly accurate and reliable in solving nonlinear fractional partial differential equations, including those modeling plasma wave propagation.