Related papers: The case for algebraic biology: from research to e…
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different…
We study structured optimization problems with polynomial objective function and polynomial equality constraints. The structure comes from a multi-grading on the polynomial ring in several variables. For fixed multi-degrees we determine the…
Despite widespread calls for the incorporation of mathematical modeling into the undergraduate biology curriculum, there is lack of a common understanding around the definition of modeling, which inhibits progress. In this paper, we extend…
Born from years of teaching undergraduate and graduate algebra courses at Chongqing University, this text is designed to introduce Galois theory while minimizing prerequisites. It seeks to reconnect the abstract machinery of modern algeba:…
We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of…
Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but…
Elementary Algebraic Geometry can be described as study of zeros of polynomials with integer degrees, this idea can be naturally carried over to `polynomials' with rational degree. This paper explores affine varieties, tangent space and…
Mathematical models are increasingly being used to understand complex biochemical systems, to analyze experimental data and make predictions about unobserved quantities. However, we rarely know how robust our conclusions are with respect to…
I believe an atomic biology is needed to supplement present day molecular biology, if we are to design and understand proteins, as well as define, make, and use them. Topics in the paper are molecular biology and atomic biology.…
Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear…
This preprint is the introduction of my habilitation thesis for Paris7 university. It is a sumary of a collection of works on the 2 matrix model. In an introduction, 3 different and unequivalent definitions of matrix models are given…
Modeling and simulation are recognized as important aspects of the scientific method for more than 70 years but its adoption in biology has been slow. Debates on its representativeness, usefulness, and whether the effort spent on such…
Machine learning has a long collaborative tradition with several fields of mathematics, such as statistics, probability and linear algebra. We propose a new direction for machine learning research: $C^*$-algebraic ML $-$ a…
Given any square matrix or a bounded operator $A$ in a Hilbert space such that $p(A)$ is normal (or similar to normal), we construct a Banach algebra, depending on the polynomial $p$, for which a simple functional calculus holds. When the…
This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed…
The conception of multi-alphabetical genetics is represented. Matrix forms of the representation of the multi-level system of molecular-genetic alphabets have revealed algebraic properties of this system. These properties are connected with…
Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among…
Theoretical and computational frameworks of modern science are dominated by binary structures. This binary bias, seen in the ubiquity of pair-wise networks and formal operations of two arguments in mathematical models, limits our capacity…
Many combinatorial optimisation problems hide algebraic structures that, once exposed, shrink the search space and improve the chance of finding the global optimal solution. We present a general framework that (i) identifies algebraic…
Design of experiments and model selection, though essential steps in data science, are usually viewed as unrelated processes in the study and analysis of biological networks. Not accounting for their inter-relatedness has the potential to…