Multiplicative Cauchy Research Articles

The multiplicative Cauchy equation on [0,1] and its application to the quasi-homogeneity equation

The (extended) homogeneity plays important, and often crucial role in economics, decision making and image processing. When studying the quasi-homogeneity equation, the multiplicative Cauchy equation on [0,1] naturally appears. However, after a detailed analysis of its existing solutions it turned out that some solution of (extended) homogeneity equation was missing. The aim of the paper is to present a complete list of solutions of the multiplicative Cauchy equation and then to give the missing solution of quasi-homogeneity equation from the previous work. For a triangular norm T, we introduce a new concept of T-quasi-homogeneity and, for a strict t-norm T, we clarify the link between the quasi-homogeneity and the T-quasi-homogeneity.

Least-squares reverse time migration with a multiplicative Cauchy constraint

One of reverse time migration’s main limitations is that an unscaled adjoint operator is prone to produce images with low resolution, inaccurate amplitudes, and even artifacts. Least-squares reverse time migration (LSRTM) has been introduced to mitigate this inadequacy via the use of an approximation to an inverse operator. LSRTM suffers from its own limitations, most importantly from poor condition, which often manifests itself as image artifacts. One approach to ameliorate this issue is to constrain the optimization problem by introducing a penalty term to the cost function. Penalizing estimated parameters for sparsity is one such constraint that has been shown to be effective. A drawback of this technique is that it introduces a trade-off between data fitting and image sparsity. Furthermore, if using the Cauchy constraint, an additional trade-off is introduced due to the requirement to estimate a hyperparameter. We introduce an alternative approach that mitigates these trade-offs by combining a multiplicative cost function with an effective means for determining the Cauchy hyperparameter. We also introduce a new formulation of the multiplicative cost function that avoids over-penalization by the constraint via the introduction of a relaxation term. Finally, we seek to improve the computational efficiency by introducing a new approach for computing the step length. As such, our method introduces three novel aspects to constrained LSRTM: (1) a relaxed multiplicative cost function, (2) semiautomatic estimation of the Cauchy hyperparameter, and (3) efficient computation of the step length. We discuss the theory and implementation, followed by application to three synthetic data sets and a real ultrasonic data set. Given the presence of large salt bodies, elasticity, and noise, along with the directivity of piezoelectric ultrasonic transducers, these data sets provide a challenging test of the approach outlined. Results demonstrate that our method is robust in handling the challenges imposed by these scenarios.

Limit theorems for Bajraktarević and Cauchy quotient means of independent identically distributed random variables

We derive strong laws of large numbers and central limit theorems for Bajraktarević, Gini and exponential- (also called Beta-type) and logarithmic Cauchy quotient means of independent identically distributed (i.i.d.) random variables. The exponential- and logarithmic Cauchy quotient means of a sequence of i.i.d. random variables behave asymptotically normal with the usual square root scaling just like the geometric means of the given random variables. Somewhat surprisingly, the multiplicative Cauchy quotient means of i.i.d. random variables behave asymptotically in a rather different way: in order to get a non-trivial normal limit distribution a time dependent centering is needed.

Matrix homomorphism equations on a class of monoids and non Abelian groupoids

Let α,β∈R. The aim of this paper is to give an explicit description of the solutions f:R×R→M2(C) of the following parametric functional equationsf(x1x2+αy1y2,x1y2+βx2y1)=f(x1,y1)f(x2,y2),that arise from number theory. Depending on the value of β, we present different methods for solving these equations. The solutions of a matrix multiplicative Cauchy functional equation on abelian regular semigroups are given. These results apply to more general equations.

Fast and High-Order Accuracy Numerical Methods for Time-Dependent Nonlocal Problems in $${\pmb {\mathbb {R}}}^2$$

In this paper, we study the Crank-Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such discretization are that the proposed numerical method is unconditionally stable and its global truncation error is of $\mathcal{O}\left(\tau^2+h^{4-\gamma}\right)$ with $0<\gamma<1$, where $\tau$ and $h$ are the discretization sizes in the temporal and spatial dimensions respectively. Also we develop the conjugate gradient squared method to solving the resulting discretized nonsymmetric and indefinite systems arising from time-dependent nonlocal problems including two-dimensional cases. By using additive and multiplicative Cauchy kernels in non-local problems, structured coefficient matrix-vector multiplication can be performed efficiently in the conjugate gradient squared iteration. Numerical examples are given to illustrate our theoretical results and demonstrate that the computational cost of the proposed method is of $O(M \log M)$ operations where $M$ is the number of collocation points.

Multiplicative operator functions and abstract Cauchy problems

Multiplicative operator functions and abstract Cauchy problems

Approximate Solution of a Singular Integral Equation with a Cauchy Kernel on the Euclidean Plane

Abstract Multidimensional singular integral equations (SIEs) play a key role in many areas of applied science such as aerodynamics, fluid mechanics, etc. Solving an equation with a singular kernel can be a challenging problem. Therefore, a plethora of methods have been proposed in the theory so far. However, many of them are discussed in the simplest cases of one–dimensional equations defined on the finite intervals. In this study, a very efficient method based on trigonometric interpolating polynomials is proposed to derive an approximate solution of a SIE with a multiplicative Cauchy kernel defined on the Euclidean plane. Moreover, an estimation of the error of the approximated solution is presented and proved. This assessment and an illustrating example show the effectiveness of our proposal.

Remarks on the Cauchy functional equation and variations of it

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.

A generalization of the symmetrized multiplicative Cauchy equation

Let S be a semigroup, and let \({\sigma,\tau \in {\rm Hom}(S,S)}\) satisfy \({\tau\circ\tau = \sigma\circ\sigma = \rm{id}}\). We determine the solutions \({f : S \to \mathbb{C}}\) of the functional equation $$f(x\sigma(y)) + f(\tau(y)x) = 2f(x)f(y),\quad x,y \in S,$$ in terms of multiplicative functions on S.

Reconsidering a functional equation arising from number theory

In a recent paper Chavez and Sahoo considered the functional equation f(ux− vy, uy + v(x+ y)) = f(x, y)f(u, v), which arose in a number theoretical context. Unfortunately one of their results is incorrect. Here we reconsider the equation on various domains. We observe that it is in fact a multiplicative Cauchy equation in disguise. We also point out some remaining open problems.

Multiplicative Cauchy functional equation on symmetric cones

We solve the logarithmic Cauchy functional equation in the symmetric cone with respect to two different multiplication algorithms. We impose no regularity assumptions on respective functions.

A note on the definition of a generalized fuzzy normed space

We reconsider the definition of a generalized fuzzy normed space from [I. Goleţ, On generalized fuzzy normed spaces and coincidence point theorems, Fuzzy Sets and Systems 161 (2010) 1138–1144], showing that the mapping used in the homogeneity axiom must satisfy the multiplicative Cauchy functional equation. Therefore, generalized fuzzy normed spaces are, in fact, fuzzy p-normed spaces with p>0.

Solution of a class of the first kind singular integral equation with multiplicative Cauchy kernel

In the present paper, we give the exact solutions of a singular equation with logarithmic singularities in two classes of functions and construct formulae for the approximate solutions.

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

Multiplicative Cauchy functional equation and the equation of ratios on the Lorentz cone

It is proved that the solution of the multiplicative Cauchy functional equation on the Lorentz cone of dimension greater than two is a power function of the determinant. The equation is solved in full generality, i.e. no smoothness assumptions on the unkn

Multiplicative Cauchy functional equation in the cone of positive-definite symmetric matrices

We solve the multiplicative Cauchy equation for real functions of symmetric positive definite matrices under the differentiability restriction. The specialty of the problem lies in the symmetry of the multiplication.