Related papers: Functional Analysis
In this article, we impose a new class of fractional analytic functions in the open unit disk. By considering this class, we define a fractional operator, which is generalized Salagean and Ruscheweyh differential operators. Moreover, by…
The unprecedented pace of machine learning research has lead to incredible advances, but also poses hard challenges. At present, the field lacks strong theoretical underpinnings, and many important achievements stem from ad hoc design…
By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations -- the d'Alembert equation, the Wilson…
Interval approaches for the reachability analysis of initial value problems for sets of classical ordinary differential equations have been investigated and implemented by many researchers during the last decades. However, there exist…
Density functional theory (DFT) is shown to provide a novel conceptual and computational framework for entanglement in interacting many-body quantum systems. DFT can, in particular, shed light on the intriguing relationship between quantum…
Theorems and explicit examples are used to show how transformations between self-similar sets (general sense) may be continuous almost everywhere with respect to stationary measures on the sets and may be used to carry well known flows and…
This is the Habilitation Thesis manuscript presented at Besan\c{c}on on January 5, focusing on Matrix Analysis, Matrix Inequalities and Matrix Decompositions. There are also some topics in (Hilbert space) Operator Theory. The text should be…
This book provides an introduction to the mathematical analysis of deep learning. It covers fundamental results in approximation theory, optimization theory, and statistical learning theory, which are the three main pillars of deep neural…
There have been many proposed forms of fractional calculus, which can be grouped into a few broad classes of operators. By replacing the kernel of the power function with another kernel function, the traditional Riemann-Liouville formula…
Density functional theory (DFT) has transformed our ability to investigate and understand electronic ground states. In its original formulation, however, DFT is not suited to addressing (e.g.) degenerate ground states, mixed states with…
We survey a few classes of analytic functions on the disk that have real boundary values almost everywhere on the unit circle. We explore some of their properties, various decompositions, and some connections these functions make to…
This book provides a comprehensive introduction to the foundational concepts of machine learning (ML) and deep learning (DL). It bridges the gap between theoretical mathematics and practical application, focusing on Python as the primary…
Finite temperature density functional theory requires representations for the internal energy, entropy, and free energy as functionals of the local density field. A central formal difficulty for an orbital-free representation is…
We propose an in-silico experiment to introduce classical density functional theory (cDFT). Den- sity functional theories, whether quantum or classical, rely on abstract concepts that are non- intuitive. However, they are at the heart of…
A review of fundamentals and physical applications of fractional quantum mechanics has been presented. Fundamentals cover fractional Schr\"odinger equation, quantum Riesz fractional derivative, path integral approach to fractional quantum…
The results summarized here are intended as rigorous mathematical statements on various physical models coming from condensed matter physics, statistical mechanics (classical and quantum), quantum field theory and cold atoms physics. The…
The density-functional (DF) theory provides a simple method for calculating the properties of an interacting system under an external potential by associating it with a corresponding non-interacting system. Here, we find some relations in…
This dissertation focuses on developing a new construction of a functional calculus using Henstock-Kurzweil integration methods. The assignment of a functional calculus will be applied to self-adjoint operators. We will address both the…
A Density Matrix Functional theory is constructed semi-empirically for the two-level Lipkin model. This theory, based on natural orbitals and occupation numbers, is shown to provide a good description for the ground state energy of the…
Density Functional Theory (DFT) is widely used for atomistic simulations. However, its reach stays limited due to several limitations such as lack of accurate exchange-correlation functional, requirement of costly O(N 3) diagonalization…