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Making new methods for quantum problems often relies on using basic operations in linear algebra. Often these routines are hidden behind well-known libraries that have been optimized over decades. Attempting to improve on those basic…
This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving…
We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it contains…
Quantum computing is a promising way to systematically solve the longstanding computational problem, the ground state of a many-body fermion system. Many efforts have been made to realise certain forms of quantum advantage in this problem,…
This paper introduces a formalism that aims to describe the intricacies of quantum computation by establishing a connection with the mathematical foundations of tensor theory and multilinear maps. The focus is on providing a comprehensive…
This paper focuses on the derivations and automorphism groups of certain finite-dimensional associative algebras over the field of complex numbers. Using classification results for algebras of dimensions two, three, and four, along with…
The primary goal of this paper is to present a unified way to transform the syntax of a logic system into certain initial algebraic structure so that it can be studied algebraically. The algebraic structures which one may choose for this…
Conformal field theory, describing systems with scaling symmetry, plays a crucial role throughout physics. We describe a quantum algorithm to simulate the dynamics of conformal field theories, including the action of local conformal…
Complexified spacetime algebra is defined as the geometric (Clifford) algebra of spacetime with complex coefficients, isomorphic $\mathcal{G}_{1,4}$. By resorting to matrix representation by means of Dirac-Pauli gamma matrices, the paper…
We introduce the Non-commutative Subset Convolution - a convolution of functions useful when working with determinant-based algorithms. In order to compute it efficiently, we take advantage of Clifford algebras, a generalization of…
I describe a method for computer algebra that helps with laborious calculations typically encountered in theoretical microhydrodynamics. The program mimics how humans calculate by matching patterns and making replacements according to the…
The main purpose of this book is to propose an introduction to the modern tools of algebraic complexity. To remain as simple as possible while providing meaningful examples, we chose to focus on effective linear algebra; this is certainly…
Some idea, which leads to a non-trivial solution of the quantum four-simplex equation, is exposed in this paper. We call this idea "pentagonal algebra". Few examples of the realisation of this idea are given here, and thus few examples of…
Multi-task indoor scene understanding is widely considered as an intriguing formulation, as the affinity of different tasks may lead to improved performance. In this paper, we tackle the new problem of joint semantic, affordance and…
Calcium is a C library for real and complex numbers in a form suitable for exact algebraic and symbolic computation. Numbers are represented as elements of fields $\mathbb{Q}(a_1,\ldots,a_n)$ where the extensions numbers $a_k$ may be…
Transformers are effective and efficient at modeling complex relationships and learning patterns from structured data in many applications. The main aim of this paper is to propose and design NLAFormer, which is a transformer-based…
We report on the computation of invariants, covariants, and contravariants of cubic surfaces. All algorithms are implemented in the computer algebra system magma.
We present here algorithms for efficient computation of linear algebra problems over finite fields.
In this work, we propose a novel framework for defining the dual structure of a spinor. This construction relies on the basis elements of the Clifford algebra, leading to a covariant structure that embeds the dual. The formulation includes…
We propose a quantum algorithm to solve systems of nonlinear algebraic equations. In the ideal case the complexity of the algorithm is linear in the number of variables $n$, which means our algorithm's complexity is less than $O(n^{3})$ of…