Related papers: The Unum Number Format: Mathematical Foundations, …

Turing machines and spin models share a notion of universality according to which some simulate all others. Is there a theory of universality that captures this notion? We set up a categorical framework for universality which includes as…

Many modern search domains comprise high-dimensional vectors of floating point numbers derived from neural networks, in the form of embeddings. Typical embeddings range in size from hundreds to thousands of dimensions, making the size of…

Selecting optimal intervals of checkpointing an application is important for minimizing the run time of the application in the presence of system failures. Most of the existing efforts on checkpointing interval selection were developed for…

We introduce and study non-uniform expansions of real numbers, given by two non-integer bases.

This paper presents a novel set-based computing method, called interval superposition arithmetic, for enclosing the image set of multivariate factorable functions on a given domain. In order to construct such enclosures, the proposed…

We explore the behavior and establish new properties of the cumulative-sum process (CUSUM) and its running maximum. The study includes precise expressions for CUSUM's moment generating function and moments, fast recursive computing…

Large foundation models have become central to modern machine learning, with performance scaling predictably with model size and data. However, training and deploying such models incur substantial computational and memory costs, motivating…

With the inflation of the data, clustering analysis, as a branch of unsupervised learning, lacks unified understanding and application of its mathematical law. Based on the view of fixed point, this paper restates the model-based clustering…

The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recently in [B. Tenner, arXiv:2007.06142]. We study this poset from the perspective of the…

Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex…

All but a few digital computers used for scientific computations have supported floating-point and digital arithmetic of rather limited numerical precision. The underlying assumptions were that the systems being studied were basically…

We discuss a formal system of mathematics. We use it to construct the natural numbers.

The numerical unitarity approach has been important for obtaining reliable QCD predictions for the LHC. Here I discuss the extension of the approach beyond the leading quantum corrections for computing multi-loop amplitudes. The numerical…

In this note, we consider a framework for the analysis of iterative algorithms which can described in terms of a structured set-valued operator. More precisely, at each point in the ambient space, we assume that the value of operator can be…

A brief introduction into Idempotent Mathematics and an idempotent version of Interval Analysis are presented. Some applications are discussed.

This paper presents a systematic study of the calculus of interval-valued functions and its application to interval differential equations. To this end, first, we introduce new interval arithmetic operations. Under new operations, the space…

Large scale atomistic simulations with suitable interatomic potentials are widely employed by scientists or engineers of different areas. Quick generation of high-quality interatomic potentials is of urgent need under present circumstances,…

Some concepts, such as non-compactness measure and condensing operators, defined on metric spaces are extended to uniform spaces. Such extensions allow us to locate, in the context of uniform spaces, some classical results existing in…

In this paper, a unified framework for representing uncertain information based on the notion of an interval structure is proposed. It is shown that the lower and upper approximations of the rough-set model, the lower and upper bounds of…

In this paper, we propose an architecture/methodology for making FPGAs suitable for integer as well as variable precision floating point multiplication. The proposed work will of great importance in applications which requires variable…