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Causal parametric representations of neutron-star equations of state are constructed here using Chebyshev polynomial based spectral expansions. The accuracies of these representations are evaluated for a collection of model equations of…
In this treatise I present the solutions of the third Clay Millennium problem in the computational complexity and the fourth Clay Millennium problem in classical fluid dynamics.
As in many other fields, the rapid rise of generative artificial intelligence is reshaping materials discovery by offering new ways to propose crystal structures and, in some cases, even predict desired properties. This review provides a…
We survey recent developments in the theory and applications of the broken ray transforms. Furthermore, we discuss some open problems.
Random matrices now play a role in many parts of computational mathematics. To advance these applications, it is desirable to have tools that are flexible, easy to use, and powerful. Over the last 25 years, researchers have developed a…
In this article, we introduce the notion of representations of polyadic groups and we investigate the connection between these representations and those of retract groups and covering groups.
This essay considers ways that recent uses of computers in mathematics challenge contemporary views on the nature of mathematical understanding. It also puts these challenges in a historical perspective and offers speculation as to a…
Crystal property prediction, governed by quantum mechanical principles, is computationally prohibitive to solve exactly for large many-body systems using traditional density functional theory. While machine learning models have emerged as…
A review is given of some mathematical contributions, ideas and questions concerning liquid crystals.
Representable implication algebras are known to be axiomatised by a finite number of equations (making the representation and finite representation problems decidable here). We show that this also holds in the context of unary (and binary)…
Graph Representation Learning (GRL) has experienced significant progress as a means to extract structural information in a meaningful way for subsequent learning tasks. Current approaches including shallow embeddings and Graph Neural…
We present a uniform construction of level 1 perfect crystals $\mathcal B$ for all affine Lie algebras. We also introduce the notion of a crystal algebra and give an explicit description of its multiplication. This allows us to determine…
The paper contains five examples of using cyclic bases of zero-curvature representations in studies of weak and strong Lax pairs, hierarchies of evolution systems, and recursion operators.
The unfolding problem formulation for correcting experimental data distortions due to finite resolution and limited detector acceptance is discussed. A novel validation of the problem solution is proposed. Attention is drawn to fact that…
We give a brief introduction to (upper) cluster algebras and their quantization using examples. Then we present several important families of bases for these algebras using topological models. We also discuss tropical properties of these…
Detection of crystal structures from particle positions of crystalline assemblies formed in computer simulations is an unsolved problem. The standard protocol, formulated in the reciprocal space, for structure determination from…
The history of the canonical basis and crystal basis of a quantized enveloping algebra and its representations is presented
The aim of this note is to give a gentle introduction to algebras of partial triangulations of marked surfaces, following the structure of a talk given during the 49th symposium on ring theory and representation theory, held in Osaka. This…
We use evaluation representations to give a complete classification of the finite-dimensional simple modules of twisted current algebras. This generalizes and unifies recent work on multiloop algebras, current algebras, equivariant map…
Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.