Generalized Functions Research Articles

Green’s Tensor of BIOT’s Equations by Stationary Oscillations

Various mathematical models of deformable solids mechanics are used in the study of seismic processes in the earth's crust. The processes of waves propagation are most studied in elastic media. But these models do not take into account many real properties of the ambient array. These are, for example, the presence of groundwater, which complicates the construction and operation of surface and underground structures, affect the magnitude and distribution of stresses. Models, which take into account the water saturation form the earth's crust structures, the presence of gas bubbles, etc., are multi-component medium. A variety of multicomponent media, the complexity of the processes associated with their deformation, lead to a large difference in the methods of analysis and modelling used in the solution of wave problems. In this paper the processes of wave propagation in a two-component Biot medium under the action of periodic forces of various forms are considered. Using the Fourier transform of generalized functions, fundamental solutions are constructed - the Green's tensor of the Biot equations and its properties are studied. This tensor describes the process of propagation of harmonic waves of a fixed frequency in spaces of dimension N = 1,2,3 under the action of power sources concentrated at the origin of coordinates, described by a singular delta function. On its basis, generalized solutions of these equations are constructed under the action of various sources of periodic disturbances, which are described by both regular and singular generalized functions. For regularly acting forces, integral representations of solutions are given, which can be used to calculate the stress-strain state of a porous water-saturated medium

Paley–Wiener theorems for the Bessel wavelet transformation

ABSTRACT In this paper, we defined a class of Colombeau-type generalized functions and established the Paley–Wiener theorem for the Bessel wavelet transformation on E μ , σ 2 and Colombeau-type generalized functions.

On H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${H}_\\infty $$\\end{document} Finite-Time Boundedness and Finite-Time Stability for Discrete-Time Neural Networks with Leakage Time-Varying Delay

This paper investigates the problems of H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_\\infty $$\\end{document} finite-time boundedness (FTB) and finite-time stability (FTS) for discrete-time neural networks (NNs) with both leakage delay and discrete delay, as well as different generalized activation functions. To this end, we construct suitable Lyapunov–Krasovskii (L–K) functionals and apply the extended reciprocally convex approach to derive delay-dependent criteria for the addressed problems. The obtained conditions are expressed as linear matrix inequalities (LMIs), which can be efficiently solved by the LMI Toolbox in Matlab. The paper also admits of two numerical examples to point up the advantages and reliability of the proposed method.

Thermodynamics of quark matter with multiquark clusters in an effective Beth-Uhlenbeck type approach

We describe multiquark clusters in quark matter within a Beth-Uhlenbeck approach in a background gluon field coupled to the underlying chiral quark dynamics using the Polyakov gauge. An effective potential for the traced Polyakov loop is used to establish the center symmetry of the SU(3) color which suppresses colored states and its dynamical breaking as an aspect of the confinement/deconfinement transition. Quark confinement is modeled by a large quark mass in vacuum which is motivated by a confining density functional approach. A multiquark cluster containing n quarks and antiquarks is described as a binary composite of smaller subclusters n1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_1$$\\end{document} and n2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_2$$\\end{document} (n1+n2=n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_1+n_2=n$$\\end{document}). It has a spectrum consisting of a bound state and a scattering state continuum. For the corresponding cluster-cluster phase shifts we use simple ansätze that capture the Mott dissociation of clusters as a function of temperature and chemical potential. We go beyond the simple “step-up-step-down” model that ignores continuum correlations and introduce an improved model that includes them in a generic form. In order to explain the model, we restrict ourselves here to the cases where the cluster size is 1≤n≤6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le n \\le 6$$\\end{document}. A striking result is the suppression of the abundance of colored multiquark clusters at low temperatures by the coupling to the Polyakov loop and their importance for a quantitative description of lattice QCD thermodynamics at non-vanishing baryochemical potentials. An important ingredient are Polyakov-loop generalized distribution functions of n-quark clusters which are derived here for the first time. Within our approach we calculate thermodynamic properties such as baryon density and entropy. We demonstrate that the limits of a hadron resonance gas at low temperatures and O(g2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(g^2)$$\\end{document} perturbative QCD at high temperatures are correctly reproduced. A comparison with lattice calculations shows that our model is able to give a unified, systematic approach to describe properties of the quark-gluon-hadron system.

Stochastic Clairaut Equation on Algebra of Generalized Functions

Stochastic Clairaut Equation on Algebra of Generalized Functions

Stochastic Bernoulli Equation on the Algebra of Generalized Functions

Stochastic Bernoulli Equation on the Algebra of Generalized Functions

Drought risk probabilistic models based on extreme value theory

The article presents original, probabilistic models for the quantification of drought risk, based on generalized distribution functions for the distribution of maximum values as well as the standardized precipitation index. Using the models, a probabilistic measurement of drought risk has been made for three regions over four 5-year periods for each region. Three measurement points located in Poland were selected as a case study. The study provides an innovative approach in the field of probabilistic models of quantification; moreover, drought risk levels have been calculated for the selected locations. Furthermore, the method may be used for creating indexes for any climatic region subject to study.

Asymptotic Almost Automorphy for Algebras of Generalized Functions

The paper aims to to study the concept of asymptotic almost automorphy in the context of generalized functions. We introduce an algebra of asymptotically almost automorphic generalized functions which contains the space of smooth asymptotically almost automorphic functions as a subalgebra. The fundamental importance of this algebra, is related to the impossibility of multiplication of distributions; it also contains the asymptotically almost automorphic Sobolev--Schwartz distributions as a subspace. Moreover, it is shown that the introduced algebra is stable under some nonlinear operations. As a by pass result, the paper gives a Seeley type result on extension of functions in the context of the algebra of bounded generalized functions and the algebra of bounded generalized functions vanishing at infinity, these results are used to prove the fundamental result on the uniqueness of decomposition of an asymptotically almost automorphic generalized function. As applications, neutral difference-differential systems are considered in the framework of the algebra of generalized functions.

Solving the Fractional Schrödinger Equation with Singular Initial Data in the Extended Colombeau Algebra of Generalized Functions

This manuscript aims to highlight the existence and uniqueness results for the following Schrödinger problem in the extended Colombeau algebra of generalized functions. 1 / ı ∂ / ∂ t u t , x − △ u t , x + v x u t , x = 0 , t ∈ R + , x ∈ R n , v x = δ x , u 0 , x = δ x , where δ is the Dirac distribution. The proofs of our main results are based on the Gronwall inequality and regularization method. We conclude our article by establishing the association concept of solutions.

One-Dimensional Spatial Boundary Value Problems of the Connected Thermoelasticity. Generalized Functions Method

Problems of definition of thermotension of a thermoelastic core with use of model of the connected thermoelasticity are considered. In this case in the equation of heat conductivity there is the divergence of speed of the movement of material points of the rod, and the elasticity equations contains the temperature gradient. On the basis of generalized functions method the generalized solutions of non-stationary and stationary boundary value problems have been solved at action of the power and thermal sources of various type including ones described singular generalized functions, under various boundary conditions on the ends of a core. Thermoshock waves which arise in such designs at action of impact loads and heat fluxes are considered, conditions on their fronts are received. The uniqueness of the set boundary tasks, including taking into account shock waves has been proved. Regular integral representation of the generalized solutions are given, which give the analytical solution of the tasks. Numerical implementation of solutions of a number of direct, return and semi-return boundary value problems of stationary fluctuations is carried out and results of computer experiments are presented

Infinite-Order Differential Operators in the Module of Formal Generalized Functions and in a Ring of Formal Power Series

Infinite-Order Differential Operators in the Module of Formal Generalized Functions and in a Ring of Formal Power Series

Report of the International Conference on Generalized Functions (Debrecen, Hungary, 1984)

Report of the International Conference on Generalized Functions (Debrecen, Hungary, 1984)

On the Leonardo's rule for the assessment of root profile

Leonardo's rule (Lrule) applied to below-ground systems defines a simple topological scheme that describes how the branches of root architectures develop within the soil. The approach does not consider the soil-climate-root interactions. From another hand, eco-hydrological approaches exploit physically-based formulations to derive the dynamic evolution of root profile based on soil and climate characteristics. In homogenous soil and simplified hydrological conditions, analytical solutions can be derived, as demonstrated by Laio's model, who proposed a simple exponential formulation to derive the Root Area (AR) profile. Apart from Laio's model, more generalized functions, i.e. derived by two and three parameters gamma distribution or others, can be efficiently used to derive the AR profile.This communication proposes a combination of the Lrule and eco-hydrological approaches to derive the AR profile, at given soil and climate conditions, allowing to identify a physical and theoretical meaning of the Lrule's parameters. A comprehensive root dataset from field measurements carried out in the region of Tuscany (Italy) is used. Results demonstrate that values of Lrule's parameters derived throughout the proposed mathematical relationships tend to constant values in case of exponential function, which is valid for homogenous soils. Moreover, in a realistic vegetated soil, where top-soil is different than deep-soil, functions derived from a two and three parameters gamma distribution may reproduce better root data observations.

Regularity of nonlinear generalized functions: A counterexample in the nonstandard setting

Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of G∞-regularity, which reduces to C∞-regularity when restricted to Schwartz distributions. Surprisingly, in the nonstandard version of the Colombeau algebras, this basic property of G∞-regularity does not hold.

Image Texture Analysis and Edge Detection Algorithm Based on Anisotropic Diffusion Equation

This paper uses partial differential equation image processing techniques to establish image texture analysis models based on nonlinear anisotropic diffusion equations for image denoising, image segmentation, and image decomposition. This paper proposes a class of denoising models based on the hybrid anisotropic diffusion equation from the characteristics of different noise types. The model exhibits anisotropic diffusion near the image boundary, which can protect the boundary well, and isotropic diffusion inside the image; so, it can remove the noise effectively. We use the immovable point theory to prove the uniqueness of the model solution and further discuss other properties such as asymptotics of the solution. We propose a class of image texture analysis algorithms based on anisotropic diffusion equations and discrete gray level sets. First, a class of nonconvex generalized functions is proposed to remove the noise from the original image to obtain a smooth image while sharpening the edges. Then, an energy generalization function based on the gray level set is proposed, and the existence of the global minimum of this energy generalization function is discussed. Finally, an equivalent form of this energy generalization is given in the discrete case, and an image texture analysis algorithm is designed based on the equivalent form. The algorithm is improved by initial position optimization, dynamic adjustment of parameters, and adaptive selection of thresholds so that the ants can search along the real edges. Experiments show that the improved algorithm for image edge detection can obtain more complete edges and better detection results. The energy generalization function is calculated directly on the discrete gray level set instead of solving the corresponding partial differential equation, which can avoid the selection of the initial level set and the reinitialization of the level set, thus greatly improving the segmentation efficiency. The new algorithm has a high improvement in segmentation efficiency and can efficiently handle large size complex images.

Linear Partial Differential Equations in Module of Formal Generalized Functions over Commutative Ring

Linear Partial Differential Equations in Module of Formal Generalized Functions over Commutative Ring

Implicit Linear Differential-Difference Equations in the Module of Formal Generalized Functions over a Commutative Ring

We study an implicit inhomogeneous linear differential-difference equation in the module of formal generalized functions over a commutative ring. We prove the well-posedness of the equation and find the fundamental solution. We obtain a representation of a unique solution to the equation in the form of the convolution of the fundamental solution and a given formal generalized function. We also consider the inhomogeneous second order differential equation over an arbitrary commutative ring.

Differential chain of algebras of generalized functions

Differential chain of algebras of generalized functions

A nonlinear theory of distributional geometry.

This paper builds on the theory of nonlinear generalized functions begun in Nigsch & Vickers (Nigsch, Vickers 2021 Proc. R. Soc. A 20200640 (doi:10.1098/rspa.2020.0640)) and extends this to a diffeomorphism-invariant nonlinear theory of generalized tensor fields with the sheaf property. The generalized Lie derivative is introduced and shown to commute with the embedding of distributional tensor fields and the generalized covariant derivative commutes with the embedding at the level of association. The concept of a generalized metric is introduced and used to develop a non-smooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalized metric with well-defined connection and curvature and that for C 2 metrics the embedding preserves the curvature at the level of association. Finally, we consider an example of a conical metric outside the Geroch-Traschen class and show that the curvature is associated to a delta function.

Bayesian cross-product quality control via transfer learning

ABSTRACT Quality control is essential for modern business success. The traditional statistical process control (SPC), however, lacks efficacy in current high-variety low-volume industrial practices since the historical reference data in Phase I are usually too scarce to infer the in-control process parameters accurately. To solve this ‘small data’ challenge, a novel Bayesian process monitoring scheme via transfer learning is proposed to facilitate a cross-product data sharing. In particular, a joint prior distribution is taken to explicitly capture the relatedness between the process data of two similar products, through which the process information can be transferred from one product (source domain) to improve the Bayesian inference for the other product (target domain). The posteriors can be derived analytically in closed forms by using generalised hypergeometric functions, thereby leading to a computationally efficient control chart for the online real-time monitoring in Phase II. A user-specified parameter is also provided to enable a better theoretical understanding of the transferability matter and a free practical control of the transferred information across domains. Extensive numerical simulations and real example studies of an assembly process validate the superiority of our proposed scheme in terms of both the false alarm rate and detection capability.