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Pull-tabbing is an evaluation technique for functional logic programs which computes all non-deterministic results in a single graph structure. Pull-tab steps are local graph transformations to move non-deterministic choices towards the…
Humans tame the complexity of mathematical reasoning by developing hierarchies of abstractions. With proper abstractions, solutions to hard problems can be expressed concisely, thus making them more likely to be found. In this paper, we…
Large language models (LLMs) demonstrate considerable potential in various natural language tasks but face significant challenges in mathematical reasoning, particularly in executing precise, multi-step logic. However, current evaluation…
Looping is one of the fundamental logical instructions used for repeating a block of code. It is used in programs across all programming languages. Traditionally, in languages like C, the for loop is used extensively for repeated execution…
Higher order coefficients of the inverse mass expansion of one-loop effective actions are obtained from a one-dimensional path integral representation. For the case of a massive scalar loop in the background of both a scalar potential and a…
We discuss the extension of the maximal-unitarity method to two loops, focusing on the example of the planar double box. Maximal cuts are reinterpreted as contour integrals, with the choice of contour fixed by the requirement that integrals…
The paper discusses several extensions of the recursive representation of the flow shop scheduling problem. It is shown that recursive functions make it possible to describe multiple extensions in a single problem. The paper considers…
We study the applicability of the Z-Sum approach to multi-loop calculations with massive particles in perturbative quantum field theory. We systematically analyze the case of one-loop scalar integrals, which represent the building blocks of…
A leading twist expansion in terms of bi-local operators is proposed for the structure functions of deeply inelastic scattering near the elastic limit $x \to 1$, which is also applicable to a range of other processes. Operators of…
While Large Language Models (LLMs) are fundamentally next-token prediction systems, their practical applications extend far beyond this basic function. From natural language processing and text generation to conversational assistants and…
Mathematics has many useful properties for developing of complex software systems. One is that it can exactly describe a physical situation of the object or outcome of an action. Mathematics support abstraction and this is an excellent…
An algorithm for the evaluation of the complex exponential function is proposed which is quasi-linear in time and linear in space. This algorithm is based on a modified binary splitting method for the hypergeometric series and a modified…
The ability to precisely derive mathematical objects is a core requirement for downstream STEM applications, including mathematics, physics, and chemistry, where reasoning must culminate in formally structured expressions. Yet, current LM…
Taylor expansions of analytic functions are considered with respect to two points. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these…
In program semantics and verification, reasoning about loops is complicated by the need to produce two separate mathematical arguments: an invariant, for functional properties (ignoring termination); and a variant, for termination (ignoring…
For two-loop two-point diagrams with arbitrary masses, an algorithm to derive the asymptotic expansion at large external momentum squared is constructed. By using a general theorem on asymptotic expansions of Feynman diagrams, the…
For a large class of two-loop selfenergy- and vertex-type diagrams with only one non-zero mass ($M$) and the vertices also with only one non-zero external momentum squared ($q^2$) the first few expansion coefficients are calculated by the…
A standard informal method for analyzing the asymptotic complexity of a program is to extract a recurrence that describes its cost in terms of the size of its input, and then to compute a closed-form upper bound on that recurrence. We give…
In this paper we present an abstraction-refinement approach to Satisfiability Modulo the theory of transcendental functions, such as exponentiation and trigonometric functions. The transcendental functions are represented as uninterpreted…
In this paper we study the complexity of the problems: given a loop, described by linear constraints over a finite set of variables, is there a linear or lexicographical-linear ranking function for this loop? While existence of such…