Order Lipschitz Research Articles

Fractional order Lipschitz recurrent neural network with attention for long time series prediction

Fractional order Lipschitz recurrent neural network with attention for long time series prediction

A higher dimensional Marcinkiewicz exponent and the Riemann boundary value problems for polymonogenic functions on fractals domains

We use a high-dimensional version of the Marcinkiewicz exponent, a metric characteristic for non-rectifiable plane curves, to present a direct application to the solution of some kind of Riemann boundary value problems on fractal domains of Euclidean space Rn+1,n≥2 for Clifford algebra-valued polymonogenic functions with boundary data in classes of higher order Lipschitz functions. Sufficient conditions to guarantee the existence and uniqueness of solution to the problems are proved. To illustrate the delicate nature of this theory we described a class of hypersurfaces where the results are more refined than those that exist in literature.

Hardy decomposition of first order Lipschitz functions by Clifford algebra-valued harmonic functions

In this paper we solve the problem on finding a sectionally Clifford algebra-valued harmonic function, zero at infinity and satisfying certain boundary value condition related to higher order Lipschitz functions. Our main tool are the Hardy projections related to a singular integral operator arising in bimonogenic function theory, which turns out to be an involution operator on the first order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Hölder continuous functions on a simple closed curve in the complex plane.

Local Approximation Properties for the New Type q-Stancu-Durrmeyer Operators

In this article, we propose a new kind of q-Stancu-Durrmeyer operators (or shortly q-SD operators). We calculate first and second order moments, for and we give an estimation for the second order central moment, . Furthermore, through the use of the second modulus of continuity, we invent approximation properties of the q-SD operators in the local sense and by using order Lipschitz kind maximal function, we give a direct estimation for the q-SD operators again in the local sense.

On the Dirichlet problem for second order elliptic systems in the ball

In this paper we study the Dirichlet problem in the ball for the so-called inframonogenic functions, i.e. the solutions of the sandwich equation ∂x_f∂x_=0, where ∂x_ stands for the Dirac operator in Rm. The main steps in deriving our results are the establishment of some interior estimates for the first order derivatives of harmonic Hölder continuous functions and the proof of certain invariance property of the higher order Lipschitz class under the action of the Poisson integral. Using Mathematica we also implement an algorithm to find explicitly the solution of such a Dirichlet problem for a much wider class of partial differential equations in the ball of R3 with polynomial boundary data.

An arbitrary-order exact differentiator with predefined convergence time bound for signals with exponential growth bound

There is a growing interest in differentiation algorithms that converge in fixed time with a predefined Upper Bound on the Settling Time (UBST). However, existing differentiation algorithms are limited to signals having an nth order Lipschitz derivative. Here, we introduce a general methodology based on time-varying gains to circumvent this limitation, allowing us to design nth order differentiators with a predefined UBST for the broader class of signals whose (n+1)th derivative is bounded by a function with bounded logarithmic derivative. Unlike existing methods whose time-varying gain tends to infinity, our approach yields a time-varying gain that remains bounded at the convergence time. We show how this last property maintains exact convergence using bounded gains when considering a compact set of initial conditions and improves the algorithm’s performance under measurement noise.

On a Riemann--Hilbert boundary value problem for (ϕ,ψ)-harmonic functions in ℝ m

Abstract The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for ( φ , ψ ) {(\varphi,\psi)} -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m {\mathbb{R}^{m}} . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω ⊂ ℝ m {\Omega\subset\mathbb{R}^{m}} with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ⁡ ( 1 + α , Γ ) {\operatorname{Lip}(1+\alpha,\Gamma)} .

Hilbert and Poincaré-Bertrand Formulas in Polyanalytic Function Theory Involving Higher Order Lipschitz Classes

Hilbert and Poincaré-Bertrand Formulas in Polyanalytic Function Theory Involving Higher Order Lipschitz Classes

A Riemann jump problem for biharmonic functions in fractal domains

Biharmonic functions are the solutions of the fourth order partial differential equation $$\Delta \Delta \omega =0$$ . The purpose of this paper is to solve a kind of Riemann boundary value problem for biharmonic functions assuming higher order Lipschitz boundary data. We approach this problem making use of generalized Teodorescu transforms for obtaining the explicit expression of its solution in a Jordan domain $$\Omega \subset \mathbb {R}^2$$ with fractal boundary.

Polyanalytic Hardy decomposition of higher order Lipschitz functions

This paper is concerned with the problem of decomposing a higher order Lipschitz function on a closed Jordan curve Γ into a sum of two polyanalytic functions in each open domain defined by Γ. Our basic tools are the Hardy projections related to a singular integral operator arising in polyanalytic function theory, which, as it is proved here, represents an involution operator on the higher order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Hölder continuous functions on the boundary of a domain.

Semi-Local Analysis and Real Life Applications of Higher-Order Iterative Schemes for Nonlinear Systems

Our aim is to improve the applicability of the family suggested by Bhalla et al. (Computational and Applied Mathematics, 2018) for the approximation of solutions of nonlinear systems. Semi-local convergence relies on conditions with first order derivatives and Lipschitz constants in contrast to other works requiring higher order derivatives not appearing in these schemes. Hence, the usage of these schemes is improved. Moreover, a variety of real world problems, namely, Bratu’s 1D, Bratu’s 2D and Fisher’s problems, are applied in order to inspect the utilization of the family and to test the theoretical results by adopting variable precision arithmetics in Mathematica 10. On account of these examples, it is concluded that the family is more efficient and shows better performance as compared to the existing one.

Stabilization of Arbitrary Switched Nonlinear Fractional Order Dynamical Systems: Application to Francis Hydro-Turbine Governing System

This paper is a theoretical and practical study on the stabilization of fractional order Lipschitz nonlinear systems under arbitrary switching. The investigated system is a generalization of both switched and fractional order dynamical systems. Firstly, a switched frequency distributed model is introduced as an equivalent for the system. Subsequently, a sufficient condition is obtained for the stabilizability of the system based on the Lyapunov approach. Finally, the results are extended to synthesis mode-dependent state feedback controller for the system. All the results are expressed in terms of coupled linear matrix inequalities, which are solvable by optimization tools and directly reducible to the conditions of the integer order nonlinear switching systems as well as the conventional non-switched nonlinear fractional order systems. The proposed method has various practical implications. As an example, it is utilized to control Francis hydro-turbine governing system. This system is represented as a switching structure and supposed to supply a load suffering abrupt changes driven by an arbitrary switching mechanism. The simulation results support the usefulness of the method.

On the Plemelj–Privalov theorem in Clifford analysis involving higher order Lipschitz classes

The main purpose of this work is to prove that the higher order Lipschitz classes behave invariant under the action of a singular integral operator which arise naturally in polymonogenic Clifford algebra valued function theory.

Characterizations of Bloch-Type Spaces of Harmonic Mappings

We study the Banach space BHα (α>0) of the harmonic mappings h on the open unit disk D satisfying the condition supz∈D⁡(1-z2)α(hzz+hz¯z)<∞, where hz and hz¯ denote the first complex partial derivatives of h. We show that several properties that are valid for the space of analytic functions known as the α-Bloch space extend to BHα. In particular, we prove that for α>0 the mappings in BHα can be characterized in terms of a Lipschitz condition relative to the metric defined by dH,α(z,w)=sup⁡{hz-hw:h∈BHα,hBHα≤1}. When α>1, the harmonic α-Bloch space can be viewed as the harmonic growth space of order α-1, while for 0<α<1, BHα is the space of harmonic mappings that are Lipschitz of order 1-α.

A Bimonogenic Cauchy Transform on Higher Order Lipschitz Classes

We prove that a Cauchy transform naturally arising for bimonogenic Clifford algebra valued functions theory behaves invariant on the higher order Lipschitz classes and obtain a Plemelj–Privalov theorem for the related singular integral transform. Moreover, we obtain an upper bound for the norm of such an operator in the pathological case of fractal boundaries.

Compactness of embedding of generalized higher order Lipschitz classes

In this paper sufficient conditions for continuous and compact embeddings of generalized higher order Lipschitz classes on a compact subset of n-dimensional real Euclidean spaces are obtained.

Q-Szász–Durrmeyer Type Operators Based on Dunkl Analogue

The aim of present article is to introduce the q-Szasz–Durrmeyer operators based on Dunkl analogue. We gave basic estimates with the help of q-calculus and then discussed basic convergence theorems. Next, we studied pointwise approximation results in terms of Peetre’s K-functional, second order modulus of continuity, Lipschitz type space and s th order Lipschitz type maximal function. Lastly, weighted approximation results and statistical approximation theorems are proved.

The Plemelj–Privalov theorem in polyanalytic function theory

In this paper we prove that the higher order Lipschitz classes behave invariant under the action of a singular integral operator naturally arising in polyanalytic function theory. This result provides a generalization of the well-known theorem by Joseph Plemelj [16] and Ivan Privalov [17].

Second order smoothness of weak solutions of degenerate linear equations

Let m and n be positive integers and {Yk(x)}k=1m be a collection of m first order Lipschitz vector field derivatives in a bounded open set Ω⊂Rn. Given a real-valued function b in Ω, we find conditions guaranteeing that a weak solution u of ∑k=1mYk′Yku=b in Ω has some second order smoothness. More precisely, if Z is a first order vector field derivative, we study when the iterated weak derivatives ZYku exist in Ω and belong to Lloc2(Ω). The main theorem extends a result of the author for the special case when m=n and Z=Yi for some i. Corollaries include the existence of ZXju, j=1,…,n, for weak solutions u of divergence form equations div(Q∇u)=b if Q(x) is a nonnegative definite, symmetric n×n matrix and {Xj}j=1n are the derivatives corresponding to the rows of Q. The reverse orders YkZu and XjZu are also considered.

Singular Integral Operator Involving Higher Order Lipschitz Classes

In this paper, we investigate a singular integral operator with polyanalytic Cauchy kernel. In particular, we will prove that the higher order Lipschitz classes (of order \(1+\alpha \)) behave invariant under the action of that operator.