相关论文: A package on formal power series

The growing capabilities of transformer models pave the way for solving increasingly complex NLP tasks. A key to supporting application-specific requirements is the ability to fine-tune. However, compiling a fine-tuning dataset tailored to…

We create a sequence version of calculus. First, we define equivalence, some fundamental operations, differential, and integral for sequences. Then, we propose sequence versions of identity function, power function, exponential function,…

We derive the general structure of the space of formal recursion operators of nonevolutionary equations~$q_{tt}=f(q,q_{x},q_t,q_{xx},q_{xt},q_{xxx},q_{xxxx})$. This allows us to classify integrable Lagrangian systems with a higher order…

We propose a two-step procedure to model and predict high-dimensional functional time series, where the number of function-valued time series $p$ is large in relation to the length of time series $n$. Our first step performs an…

The paper gives a detailed presentation of a framework, embedded into the simply typed higher-order logic and aimed at the support of sound and structured reasoning about various properties of models of imperative programs with interleaved…

The Residual Power Series Method (RPSM) provides a powerful framework for solving fractional differential equations. However, a significant computational bottleneck arises from the necessity of calculating the fractional derivatives of the…

This article discusses how the automation of tensor algorithms, based on A Mathematics of Arrays and Psi Calculus, and a new way to represent numbers, Unum Arithmetic, enables mechanically provable, scalable, portable, and more numerically…

We introduce the regular product for Cullen-regular quaternionic functions in a manner that does not depend upon a representation in power series but upon another, weaker kind of representation. The special case when the functions are…

We establish effective mean-value estimates for a wide class of multiplicative arithmetic functions, thereby providing (essentially optimal) quantitative versions of Wirsing's classical estimates and extending those of Hal\'asz. Several…

Linear superiorization (abbreviated: LinSup) considers linear programming (LP) problems wherein the constraints as well as the objective function are linear. It allows to steer the iterates of a feasibility-seeking iterative process toward…

This work presents a new approach for simulating the HHL linear systems of equations solver algorithm with tensor networks. First, a novel HHL in the qudits formalism, the generalization of qubits, is developed, and then its operations are…

Symbolic Mathematical tasks such as integration often require multiple well-defined steps and understanding of sub-tasks to reach a solution. To understand Transformers' abilities in such tasks in a fine-grained manner, we deviate from…

We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate…

The tensor t-function, a formalism that generalizes the well-known concept of matrix functions to third-order tensors, is introduced in [K. Lund, The tensor t-function: a definition for functions of third-order tensors, Numer. Linear…

Using integration by parts relations, Feynman integrals can be represented in terms of coupled systems of differential equations. In the following we suppose that the unknown Feynman integrals can be given in power series representations,…

In our recent publication we obtained a series expansion of the arctangent function involving complex numbers. In this work we show that this formula can also be expressed as a real rational function.

In this paper we introduce a finite field analogue of a Lauricella hypergeometric series. An integral formula for the Lauricella hypergeometric series and its finite field analogue are deduced. Transformation and reduction formulae and…

For a Riemann integrable function on an interval and for a point therein,we define 'Fourier series at the point on the interval' and bring out how and when the function element becomes expressible as Fourier series.In this process,we also…

Physically relevant field-theoretic quantities are usually derived from perturbation techniques. These quantities are solved in the form of an asymptotic series in powers of small perturbation parameters related to the physical system, and…

General theory of elliptic hypergeometric series and integrals is outlined. Main attention is paid to the examples obeying properties of the "classical" special functions. In particular, an elliptic analogue of the Gauss hypergeometric…