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Enumerative geometry, the art and science of counting geometric objects satisfying geometric conditions, has seen a resurgence of activity in recent years due to an influx of new techniques that allow for enriched computations. This paper…
While an explicit basis is common in the study of Euclidean spaces, it is usually implied in the study of inertial relativistic systems. There are some conceptual advantages to including the basis in the study of special relativistic…
Transcript of G.J. Chaitin's 2 March 2000 Carnegie Mellon University School of Computer Science Distinguished Lecture. The notion of randomness is taken from physics and applied to pure mathematics in order to shed light on the…
The article treats the geometrical theory of partial differential equations in the absolute sense, i.e., without any additional structures and especially without any preferred choice of independent and dependent variables. The equations are…
It is a consequence of the Jacobi Inversion Theorem that a line bundle over a Riemann surface M of genus g has a meromorphic section having at most g poles, or equivalently, the divisor class of a divisor D over M contains a divisor having…
We give a selective survey of topics in algebraic deformation theory ranging from its inception to current times. Throughout, the numerous contributions of Murray Gerstenhaber are emphasized, especially the common themes of cohomology,…
We consider three modifications of our involutive algorithm for computing Janet bases. These modifications are related to degree compatible monomial orders and specify selection strategies for non-multiplicative prolongations. By using the…
We investigate using Clifford algebra methods the theory of algebraic dotted and undotted spinor fields over a Lorentzian spacetime and their realizations as matrix spinor fields, which are the usual dotted and undotted two component spinor…
We develop a $D-$module approach to various kinds of solutions to several classes of important differential equations by long divisions of different differential operators. The zeros of remainder maps of such long divisions are handled by…
This paper is a survey on the study of the behaviour of the composition of polynomials on the computation of Gr\"obner bases. This survey brings together some works published between 1995 and 2007. The authors of these papers gave answers…
We introduce a new framework called linear algebraic number theory (LANT) that reformulates the number-theoretic problem as a regression model and solves it using matrix algebra. This framework restricts all computations to log space,…
We present a formalization of Gr\"obner basis theory in Lean 4, built on top of Mathlib's infrastructure for multivariate polynomials and monomial orders. Our development covers the core foundations of Gr\"obner basis theory, including…
This paper is devoted to a fundamental system of equations in Linear Elasticity Theory: the famous Lam\'e-Navier system. The Clifford algebra language allows us to rewrite this system in terms of the euclidean Dirac operator, which at the…
The article associates two fundamental lattice constructions with each regular unital real ordered Banach space (function system). These are used to establish certain results in the theory of operator algebras, specifically relating the…
Modern neural network architectures still struggle to learn algorithmic procedures that require to systematically apply compositional rules to solve out-of-distribution problem instances. In this work, we focus on formula simplification…
Decomposing an algebraic variety into irreducible or equidimensional components is a fundamental task in classical algebraic geometry and has various applications in modern geometry engineering. Several researchers studied the problem and…
During the whole of 1874, Camille Jordan and Leopold Kronecker quar- relled vigorously over the organisation of the theory of bilinear forms. That theory promised a "general" and "homogeneous" treatment of numerous questions arising in…
Much recent work has addressed the solution of a family of partial differential equations by computing the inverse operator map between the input and solution space. Toward this end, we incorporate function-valued reproducing kernel Hilbert…
We investigate a class of algebras on $\mathbb{R}^3$ arising and generalized from the algebraic structure of magnetic gradient fields induced by systems of synchronous magnets with identical dipole moments (i.e.,…
We present a geometric approach to defining an algebra $\hat{\mathcal G}(M)$ (the Colombeau algebra) of generalized functions on a smooth manifold $M$ containing the space ${\mathcal D}'(M)$ of distributions on $M$. Based on differential…