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Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
The emergence of Transformer-based Large Language Models (LLMs) has substantially augmented the capabilities of Natural Language Processing (NLP), thereby intensifying the demand for computational resources. Therefore, enhancing efficiency…
In this article we focus our attention on the relation between a singular linear discrete time system and a singular linear system of fractional nabla difference equations whose coefficients are square constant matrices. By using matrix…
Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…
For many-particle systems defined on lattices we investigate the global structure of effective Hamiltonians and observables obtained by means of a suitable basis transformation. We study transformations which lead to effective Hamiltonians…
A popular and efficient discretization of evolutions involving the singular $p$-Laplace operator is based on a factorization of the differential operator into a linear part which is treated implicitly and a regularized singular factor which…
In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the…
With rapid progress in deep learning, neural networks have been widely used in scientific research and engineering applications as surrogate models. Despite the great success of neural networks in fitting complex systems, two major…
Large language models (LLMs) have revolutionized natural language processing (NLP) by excelling at understanding and generating human-like text. However, their widespread deployment can be prohibitively expensive. SortedNet is a recent…
A new model for lamellar surfaces formed by anisotropic molecules is proposed. The molecules have internal degrees of freedom, associated with their flexible section of length $N$ at zero temperature. We obtain a 2D non-standard six vertex…
The formal system lambda-delta is a typed lambda calculus that pursues the unification of terms, types, environments and contexts as the main goal. lambda-delta takes some features from the Automath-related lambda calculi and some from the…
In this article, we study the following fractional-Laplacian system with singular nonlinearity \begin{equation*} (P_{\lambda,\mu}) \left\{ \begin{array}{lr} (-\Delta)^s u = \lambda f(x) u^{-q}+ \frac{\alpha}{\alpha+\beta}b(x) u^{\alpha-1}…
We report on three improvements in the context of Feynman integral reduction and $\varepsilon$-factorised differential equations: Firstly, we show that with a specific choice of prefactors, we trivialise the $\varepsilon$-dependence of the…
We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When…
This work extends the applications of Anderson-type Hamiltonians to include transport characterized by anomalous diffusion. Herein, we investigate the transport properties of a one-dimensional disordered system that employs the discrete…
The main objective of this article is to present $\nu$-fractional derivative $\mu$-differentiable functions by considering 4-parameters extended Mittag-Leffler function (MLF). We investigate that the new $\nu$-fractional derivative…
A boundary element method (BEM) simulation is used to compare the efficiency of numerical inverse Laplace transform strategies, considering general requirements of Laplace-space numerical approaches. The two-dimensional BEM solution is used…
A typical system of k difference (or differential) equations can be compressed, or folded into a difference (or ordinary differential) equation of order k. Such foldings appear in control theory as the canonical forms of the controllability…
We study the scaling properties of latent diffusion models (LDMs) with an emphasis on their sampling efficiency. While improved network architecture and inference algorithms have shown to effectively boost sampling efficiency of diffusion…
Here we define a Caputo like discrete nabla fractional difference and we produce discrete nabla fractional Taylor formulae for the first time. We estimate their remaiders. Then we derive related discrete nabla fractional Opial, Ostrowski,…