Hafedh Rguigui

Umm Al-Qura University · Departement of Mathematics

It have been proved in Accardi and Dhahri (J Math Phys 51:2, 2010) that the set of the exponential vectors Φ(g),g∈K:=L2(Rd)∩L∞(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d...

Using a biorthogonal technique (Appell system), the foremost aim of this study is to develop and highlight specific aspects of a new polynomial sequence known as Fractional Krawtchouk Appell polynomials associated with the fractional Pascal probability distribution. Additionally, the connection that exists between brand-new polynomials and the orth...

Using an infinite-dimensional nuclear space, we introduce the quantum fractional number operator (QFNO) and the associated quantum fractional Ornstein–Uhlenbeck (O–U) semigroups. Then, we solve the Cauchy problems associated with the QFNO and show that its solutions can be expressed in terms of the aforementioned semigroups. Besides, we prove that...

This paper is a fundamental exploration of quantum theory within the quadratic Fock space in consistency with the quadratic quantization pro�gram, with a particular focus on two sets of operators that hold immense significance: the quadratic creation and preservation operators. In this paper, we highlight a critical contribution to the quadratic qu...

Based on an infinite dimensional distributions space, we study the solution of the generalized stochastic Clairaut equation using a suitable convolution calculus. The solution of such equation is shown to be positive and its integral representation with respect to the Radon measure is given. Moreover, the contractivity property is studied. Finally,...

UDC 519.21 Based on the topological dual space ℱ θ * ( S ' ℂ ) of the space of entire functions with θ -exponential growth of finite type, we introduce the generalized stochastic Bernoulli–Wick differential equation (or the stochastic Bernoulli equation on the algebra of generalized functions) by using the Wick product of elements in ℱ θ * ( S ' ℂ...

In this article, we discuss the existence and uniqueness of proportional Itô-Doob stochastic fractional order systems (PIDSFOS) by using the Picard iteration method. The paper provides new results using the proportional fractional integral and stochastic calculus techniques. We have shown the convergence of the solution of the averaged PIDSFOS to t...

This article is devoted to showing the existence and uniqueness (EU) of a solution with non-Lipschitz coefficients (NLC) of fractional Itô-Doob stochastic differential equations driven by countably many Brownian motions (FIDSDECBMs) of order ϰ∈(0,1) by using the Picard iteration technique (PIT) and the semimartingale local time (SLT).

The Liouville–Caputo and Riemann–Liouville fractional derivatives have been two of the most useful operators for modeling nonlocal behaviors by fractional differential equations. In terms of Mittag–Leffler function and convolution product, using the Laplace transform, we give the exact values of the solutions of the Liouville–Caputo and Riemann–Lio...

Our goal in this work is to demonstrate the existence and uniqueness of the solution to a class of Hadamard Fractional Itô–Doob Stochastic integral equations (HFIDSIE) of order φ∈(0,1) via the fixed point technique (FPT). Hyers–Ulam stability (HUS) is investigated for HFIDSIE according to the Gronwall inequality. Two theoretical examples are provid...

In this paper, we investigate the existence and uniqueness theorem (EUT) of Pantograph fractional stochastic differential equations (PFSDE) using the Banach fixed point theorem (BFPT). We show the Ulam–Hyers stability (UHS) of PFSDE by the generalized Gronwall inequalities (GGI). We illustrate our results by two examples.

In order to introduce the Riemann–Liouville (RL) and Caputo (C) fractional potentials, we use the Laplace transform and the Mittag–Leffler function to solve, in infinite dimension, the C time fractional diffusion equation and RL time fractional diffusion equation with respect to the number operator acting on a distribution space. Then, we show that...

To reduce the freezing time for a unit with curved walls which filled with water, nanoparticles has been dispersed. The shapes of CuO nano-powder can affect the conductivity of mixture and this factor efficacy has been scrutinized in current article. Freezing process is mostly affected by condition and due to negligible magnitude of velocity, only...

Based on the distributions space on [Formula: see text] (denoted by [Formula: see text]) which is the topological dual space of the space of entire functions with exponential growth of order [Formula: see text] and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements...

By means of the Laplace transform, we give the solution of the generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations in infinite dimensions associated to the number operator. These solutions are given in terms of the Mittag-Leffler function and the convolution product.

In this paper, we provide a new reformulation of the quadratic analogue of the Weyl relations. Especially, we offer some adjustments [10] on these relations and the corresponding group law, i.e., the quadratic Heisenberg group law. We provide a much more transparent description of the underlying manifold and we give a connection with the projective...

Using the Wick derivation operator and the Wick product of elements in a distribution space Fθ∗(SC′), we introduce the generalized Riccati Wick differential equation as a distribution analogue of the classical Riccati differential equation. The solution of this new equation is given. Finally, we finish this paper by building some applications.

In this paper, we introduce a space of \(\theta \)-admissible distributions denoted by \({\mathcal {A}}_\theta ^*\) as well as the notion of \(\theta \)-admissible operators. We study the regularity properties of the classical conditional expectation acting on \({\mathcal {A}}_\theta ^*\) and acting on \({\mathcal {L}}({\mathcal {A}}_\theta ,{\math...

This paper reports on the characterization of the quantum white noise (QWN) Gross Laplacian based on nuclear algebra of white noise operators acting on spaces of entire functions with \(\theta \)-exponential growth of minimal type. First, we use extended techniques of rotation invariance operators, the commutation relations with respect to the QWN-...

In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and...

The quantum white noise (QWN) Gaussian kernel operators (with operators parameters) acting on nuclear algebra of white noise operators is introduced by means of QWN- symbol calculus. The quantum-classical correspondence is studied. An integral representation in terms of the QWN- derivatives and their adjoints is obtained. Under some conditions on t...

Based on nuclear algebra of operators acting on spaces of entire functions with θ-exponential growth of minimal type, we introduce the quantum generalized Fourier–Gauss transform, the quantum second quantization as well as the quantum generalized Euler operator of which the quantum differential second quantization and the quantum generalized Gross...

Corresponding author at: High School of Sciences and Technology of Hammam Sousse, University of Sousse, Tunisia.

In this paper we introduce a new notion of λ −order homogeneous operators on the nuclear algebra of white noise operators. Then, we give their Fock expansion in terms of quantum white noise (QWN) fields {at,at∗;t∈ℝ}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}...

In this paper we study the homogeneous Wick differential equation associated to the quantum white noise (𝚀𝚆𝙽) Euler operator Δ E g , Q $\Delta _E^{g, Q}$ acting on generalized operators. Δ E g , Q $\Delta _E^{g, Q}$ is defined as sum of the extension of the 𝚀𝚆𝙽-Gross Laplacian and the 𝚀𝚆𝙽-conservation operator. It is shown that the operator Δ E g ,...

We introduce a new product of two test functions denoted by f□g (where f and g in the Schwartz space 𝒮(ℝ)). Based on the space of entire functions with θ-exponential growth of minimal type, we define a new family of infinite dimensional analytical operators using the holomorphic derivative and its adjoint. Using this new product f□g, such operators...

Based on the finding that the quantum white noise (QWN) conservation operator is a Wick derivation operator acting on white noise operators, we characterize the aforementioned operator by using an extended techniques of rotation invariance operators in a first place. In a second place, we use a new idea of commutation relations with respect to the...

We introduce a new notion of quantum white noise (QWN) first-order Wick differential operators with variable coefficients on the topological nuclear algebra ℒ(ℱθ(N'), ℱθ*(N')) endowed with the Wick product and we give their chaos expansions in terms of the QWN-derivatives {Dz-, Dz+ : z ∈ N}. We also study a generalized transport equation associated...

Using the quantum Ornstein–Uhlenbeck (O–U) semigroups (introduced in Rguigui [21]) and based on nuclear infinite dimensional algebra of entire functions with a certain exponential growth condition with two variables, the quantum -potential and the generalised quantum -potential appear naturally for . We prove that the solution of Poisson equations...

we develop an operator theory on a nuclear algebra of white noise operators in terms of the quantum white noise (QWN) derivatives and their dual adjoints. Using an adequate definition of a QWN-symbol transformation, we discuss QWN-integral-sum kernel operators which give the Fock expansion of the QWN-operators (i.e. the linear operators acting on n...

In this paper we introduce a quantum white noise (QWN) convolution calculus over a nuclear algebra of operators. We use this calculus to discuss new solutions of some linear and non-linear differential equations.

We introduce a new operator obtained from the quantum white noise (QWN) derivatives which satisfies new important commutation relations generalizing those of the renormalized power white noise Lie algebra introduced by Accardi, Boukas and Franz (see Ref. 5) without using renormalization conditions.

Based on nuclear infinite-dimensional algebra of entire functions with a certain exponential growth condition with two variables, we define a class of operators which gives in particular three semigroups acting on continuous linear operators, called the quantum Ornstein-Uhlenbeck (O-U) semigroup, the left quantum O-U semigroup and the right quantum...

The main objective of this paper is to investigate an extension of the "Volterra-Gross" Laplacian on nuclear algebra of generalized functions. In so doing, without using the renormalization procedure, this extension provides a continuous nuclear realization of the square white noise Lie algebra obtained by Accardi–Franz–Skeide in Ref. 2. An extende...

The quantum white noise (QWN)-Euler operator Δ E Q is defined as the sum Δ G Q +N Q , where Δ G Q and N Q stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that Δ E Q has an integral representation in terms of the QWN-derivatives {D t - ,D t + ;t∈ℝ} as a kind of functional integra...

Based on nuclear algebra of entire functions, we extend some results about operator-parameter transforms involving the Fourier-Gauss and Fourier-Mehler transforms. We investigate the solution of a initial-value problem associated to infinitesimal generators of these transformations. In particular, by using convolution product, we show to what exten...