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Deep learning has generated diverse perspectives in astronomy, with ongoing discussions between proponents and skeptics motivating this review. We examine how neural networks complement classical statistics, extending our data analytical…
Neural networks functions are supposed to be able to encode the desired solution of an inverse problem very efficiently. In this paper, we consider the problem of solving linear inverse problems with neural network coders. First we…
In the field of mathematics, a purely combinatorial equivalent to a simplicial complex, or more generally, a down-set, is an abstract structure known as a family of sets. This family is closed under the operation of taking subsets, meaning…
A spread code is a set of vector spaces of a fixed dimension over a finite field Fq with certain properties used for random network coding. It can be constructed in different ways which lead to different decoding algorithms. In this work we…
It has been discovered that linear codes may be described by binomial ideals. This makes it possible to study linear codes by commutative algebra and algebraic geometry methods. In this paper, we give a decoding algorithm for binary linear…
Quantum error correction, which utilizes logical qubits that are encoded as redundant multiple physical qubits to find and correct errors in physical qubits, is indispensable for practical quantum computing. Surface code is considered to be…
Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all…
This paper studies the large-scale subspace clustering (LSSC) problem with million data points. Many popular subspace clustering methods cannot directly handle the LSSC problem although they have been considered as state-of-the-art methods…
Coded computation is an emerging research area that leverages concepts from erasure coding to mitigate the effect of stragglers (slow nodes) in distributed computation clusters, especially for matrix computation problems. In this work, we…
We discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schr\"{o}dinger equation in…
The Schubert vanishing problem is a central decision problem in algebraic combinatorics and Schubert calculus, with applications to representation theory and enumerative algebraic geometry. The problem has been studied for over 50 years in…
Quantum computing is emerging as a new computing resource that could be superior to conventional computing for certain classes of optimization problems. However, in principle, most existing approaches to quantum optimization are intended to…
Semigroup theory is a branch of abstract algebra, and it provides mathematical tools for the theory of computation. Finite semigroups can describe state transition systems and thus they model physically realizable computers. Engineering…
The multiplicative-additive finite-field matrix channel arises as an adequate model for linear network coding systems when links are subject to errors and erasures, and both the network topology and the network code are unknown. In a…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
Simulating quantum circuits using classical computers lets us analyse the inner workings of quantum algorithms. The most complete type of simulation, strong simulation, is believed to be generally inefficient. Nevertheless, several…
The cyclic code is a subclass of linear codes and has applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open…
Separable coordinate systems are introduced in the complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also…
This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main…
This is a comprehensive review on fault-tolerant topological quantum computation with the surface codes. The basic concepts and useful tools underlying fault-tolerant quantum computation, such as universal quantum computation, stabilizer…