Discrete Algebra Research Articles

A Class of Binary Codes Using a Specific Automorphism Group

In this article, we showcase PSL(3,4) as the automorphism group for a specific class of three linear binary codes, C1, C2 and C3, with dimension 9. The demonstration involves leveraging the action of the group PSL(3,4), represented by invertible matrices of size 9 by 9 up to isomorphism, on the vector space F29. Additionally, we establish that these codes exhibit a three-weight self-orthogonal property. All computations presented in this paper were performed using the guava package of GAP (Groups, Algorithms, Programming) a system designed for computational discrete algebra.

ПАКЕТ GAP ДЛЯ РОЗЩЕПЛЕННЯ ЛІНІЙНИХ СИСТЕМ

There are a number of problems, the solution of which requires the breakdown of the initial system of equations using algebraic decoupling methods. This means reducing the matrix of coefficients to block-diagonal (or block-triangular) form by means of substitution of variables. The main computational tasks when using such methods are finding the centralizer of several matrices or compiling the algebra generated by these matrices. For calculations, it is convenient to use the GAP computer algebra system because the system itself is designed for discrete algebra calculations. The problem is that the GAP program does not support calculations with real numbers. For practical problems you can try to replace them (with some accuracy) by rational numbers. At the same time, the decision may turn out to be excessively cumbersome. On the other hand, the advantage of GAP is the complete absence of rounding errors.

Classifying Recollements of Derived Module Categories for Derived Discrete Algebras

Classifying Recollements of Derived Module Categories for Derived Discrete Algebras

Partitions by subgeometries of projective plane over a galois field of order 3n, n = 1, 2, 3

The purpose of this thesis is to split the projective plane of PG (2,9) into seven disjoint projective subplanes PG (2,3) and thirteen disjoint complete arcs of degree two and size seven. The projective line of order twenty-seven PG (1,27) has been partitioned into seven disjoint projective sublines PG (1,3) and the number of inequivalent -sets which are unordered sets of four points is classified. The group action on projective lines PG (1,3n) and projective PG (1,3n), n = 1, 2, 3 planes is explained and we have introduced theorems and examples and the subspaces of PG (1,3n) and PG (2,3n), n = 1, 2, 3 are shown. Each of these partitions gives rise to an error-correcting code that corrects the maximum possible number of errors for its length. GAP-Groups–Algorithms, Programming a system for computational discrete algebra has been applied.

Supercharacters of discrete algebra groups

Abstract The supercharacter theory of a finite group was introduced by Diaconis and Isaacs as an alternative to the usual irreducible character theory, and illustrated with a construction in the case of finite algebra groups. We extend this construction to arbitrary countable discrete algebra groups, where superclasses and indecomposable supercharacters play the role of conjugacy classes and indecomposable characters, respectively. However, we adopt an ergodic theoretical point of view. The theory is then illustrated with a characterization of standard supercharacters of the group of upper unitriangular matrices over an algebraically closed field of prime characteristic.

SU(2) Lie-Poisson algebra and its descendants

In this paper, a novel discrete algebra is presented which follows by combining the SU(2) Lie-Poisson bracket with the discrete Frenet equation. Physically, the construction describes a discrete piecewise linear string in R3. The starting point of our derivation is the discrete Frenet frame assigned at each vertix of the string. Then the link vector that connect the neighbouring vertices assigns the SU(2) Lie-Poisson bracket. Moreover, the same bracket defines the transfer matrices of the discrete Frenet equation which relates two neighbouring frames along the string. The procedure extends in a self-similar manner to an infinite hierarchy of Poisson structures. As an example, the first descendant of the SU(2) Lie-Poisson structure is presented in detail. For this, the spinor representation of the discrete Frenet equation is employed, as it converts the brackets into a computationally more manageable form. The final result is a nonlinear, nontrivial and novel Poisson structure that engages four neighbouring vertices.

Computing 3-dimensional groups: Crossed squares and cat2-groups

The category XSq of crossed squares is equivalent to the category Cat2 of cat2-groups. Functions for computing with these structures have been developed in the package XMod written using the GAP computational discrete algebra programming language. This paper contains a table listing the numbers ιG2 of isomorphism classes of cat2-groups on groups G of order at most 30 – a total of 1007 cat2-groups. Secondly, it contains general formulae for ιG2 in a number of special cases.

Seeking SUSY fixed points in the 4 \u2212 \u03f5 expansion

We use the 4 − ϵ expansion to search for fixed points corresponding to 2 + 1 dimensional mathcal{N} =1 Wess-Zumino models of NΦ scalar superfields interacting through a cubic superpotential. In the NΦ = 3 case we classify all SUSY fixed points that are perturbatively unitary. In the NΦ = 4 and NΦ = 5 cases, we focus on fixed points where the scalar superfields form a single irreducible representation of the symmetry group (irreducible fixed points). For NΦ = 4 we show that the S5 invariant super Potts model is the only irreducible fixed point where the four scalar superfields are fully interacting. For NΦ = 5, we go through all Lie subgroups of O(5) and use the GAP system for computational discrete algebra to study finite subgroups of O(5) up to order 800. This analysis gives us three fully interacting irreducible fixed points. Of particular interest is a subgroup of O(5) that exhibits O(3)/Z2 symmetry. It turns out this fixed point can be generalized to a new family of models, with NΦ = frac{mathrm{N}left(mathrm{N}-1right)}{2} − 1 and O(N)/Z2 symmetry, that exists for arbitrary integer N≥3.

Quasi-symmetries in complex networks: a dynamical model approach

Abstract The existence of symmetries in complex networks has a significant effect on network dynamic behaviour. Nevertheless, beyond topological symmetry, one should consider the fact that real-world networks are exposed to fluctuations or errors, as well as mistaken insertions or removals. Therefore, the resulting approximate symmetries remain hidden to standard symmetry analysis—fully accomplished by discrete algebra software. There have been a number of attempts to deal with approximate symmetries. In the present work we provide an alternative notion of these weaker symmetries, which we call ‘quasi-symmetries’. Differently from other definitions, quasi-symmetries remain free to impose any invariance of a particular network property and they are obtained from the phase differences at the steady-state configuration of an oscillatory dynamical model: the Kuramoto–Sakaguchi model. The analysis of quasi-symmetries unveils otherwise hidden real-world networks attributes. On the one hand, we provide a benchmark to determine whether a network has a more complex pattern than that of a random network with regard to quasi-symmetries, namely, if it is structured into separate quasi-symmetric groups of nodes. On the other hand, we define the ‘dual-network’, a weighted network (and its corresponding binnarized counterpart) that effectively encodes all the information of quasi-symmetries in the original network. The latter is a powerful instrument for obtaining worthwhile insights about node centrality (obtaining the nodes that are unique from that act as imitators with respect to the others) and community detection (quasi-symmetric groups of nodes).

Quantum hypothesis testing with group structure

The problem of discriminating between many quantum channels with certainty is analyzed under the assumption of prior knowledge of algebraic relations among possible channels. It is shown, by explicit construction of a novel family of quantum algorithms, that when the set of possible channels faithfully represents a finite subgroup of SU(2) (e.g., $C_n, D_{2n}, A_4, S_4, A_5$) the recently-developed techniques of quantum signal processing can be modified to constitute subroutines for quantum hypothesis testing. These algorithms, for group quantum hypothesis testing (G-QHT), intuitively encode discrete properties of the channel set in SU(2) and improve query complexity at least quadratically in $n$, the size of the channel set and group, compared to na\"ive repetition of binary hypothesis testing. Intriguingly, performance is completely defined by explicit group homomorphisms; these in turn inform simple constraints on polynomials embedded in unitary matrices. These constructions demonstrate a flexible technique for mapping questions in quantum inference to the well-understood subfields of functional approximation and discrete algebra. Extensions to larger groups and noisy settings are discussed, as well as paths by which improved protocols for quantum hypothesis testing against structured channel sets have application in the transmission of reference frames, proofs of security in quantum cryptography, and algorithms for property testing.

Using Jupyter for Reproducible Scientific Workflows

Literate computing has emerged as an important tool for computational studies and open science, with growing folklore of best practices. In this work, we report two case studies-one in computational magnetism and another in computational mathematics-where domain-specific software was exposed to the Jupyter environment. This enables high level control of simulations and computation, interactive exploration of computational results, batch processing on HPC resources, and reproducible workflow documentation in Jupyter notebooks. In the first study, Ubermag drives existing computational micromagnetics software through a domain-specific language embedded in Python. In the second study, a dedicated Jupyter kernel interfaces with the GAP system for computational discrete algebra and its dedicated programming language. In light of these case studies, we discuss the benefits of this approach, including progress toward more reproducible and reusable research results and outputs, notably through the use of infrastructure such as JupyterHub and Binder.

Exact and flexible solution approach to a critical chain project management problem

An important component in critical chain project management is a scheduling methodology to support resource constraints. Given task duration estimates and precedence relations along with resource designation for each task, one seeks to adhere to a target due date as well as to shorten the estimated makespan. Key are two types of time buffers to absorb potential delays accumulated from preceding tasks, and this makes the problem complex and hard to solve. The “good enough” strategy has been prevalent in research, focusing on the development of heuristic solution methods. There has been only one attempt to minimize estimated makespan, wherein discrete algebra is utilized. The method is, however, inflexible as you must compose a search space. Even the slightest extension requires a major recast of the approach. Hence, we develop a new approach to find an exact optimal solution in an efficient manner. Given a project instance, an optimization problem is constructed as a mixed-integer linear-program, for which an exact solution can be found using an off-the-shelf solver. While the problem is NP hard, larger-scale project instances, including up to 30 tasks, can be solved within a practical time standard. Extension with respect to resource assignment is easily accomplished. Experimental results highlight the significance of the developed framework.

Eventually homological isomorphisms in recollements of derived categories

Eventually homological isomorphisms are introduced between derived categories of algebras, and these lead to a special recollement situation, that is, (DB,DA,DC) with the functor j⁎:DA→DC being an eventually homological isomorphism. This recollement turns out to be a very good context to compare the algebras A and C with respect to Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology. The results are applied to stratifying ideals, triangular matrix algebras and derived discrete algebras.

Homology of Free Nilpotent Lie Rings

Some results obtained by the calculations of the integral homology of free nilpotent Lie algebras Hi (L(x1, x2,. . . , xr)/γN+1) in the system for computational discrete algebra GAP are presented. The attention is focused on the occurrence of unexpected torsion in this homology, similar to one that arises for a 4-generated free nilpotent group of class 2. The main result is that even for the case of two generators, the torsion occurs in the fourth integral homology when the nilpotency class is 5. Moreover, only 7-torsion occurs, and no others. Namely, there is an isomorphism H4(L(x1, x2)/γ6) ≅ ℤ85 ⨁ ℤ /7ℤ. We also give an explicit mathematical proof of the fact that H4(L(x1, x2, x3, x4)/γ3) has nontrivial 3-torsion. Bibliography: 6 titles.

Finding Cliques in Projective Space: A Method for Construction of Cyclic Grassmannian Codes

In general, the construction of subspace codes or, in particular, cyclic Grassmannian codes in some projective space P q (n) is highly mathematical and requires substantial computational power for the resulting searches. In this paper, we present a new method for the construction of cyclic Grassmannian codes. To do that was designed and implemented a series of algorithms using the GAP System for Computational Discrete Algebra and Wolfram Mathematica software. We also present a classification of such codes in the space P q (n), with n at most 9. The fundamental idea to construct and classify the cyclic Grassmannian codes is to endow the projective space P q (n) with a graph structure and then find cliques.

Constructions of cyclic and quasi-cyclic Grassmannian codes

In this paper we construct, using GAP System for Computational Discrete Algebra and Wolfram Mathematica, some cyclic subspace codes, specially an optimal code over the finite field F210 . Further we introduce the q-analogous of theclassic quasi cyclic block codes over finite fields.

On the representation type of subcategories of derived categories

Let [Formula: see text] be a finite-dimensional [Formula: see text]-algebra. In this paper, we mainly study the representation type of subcategories of the bounded derived category [Formula: see text]. First, we define the representation type and some homological invariants including cohomological length, width, range for subcategories. In this framework, we provide a characterization for derived discrete algebras. Moreover, for a finite-dimensional algebra [Formula: see text], we establish the first Brauer–Thrall type theorem of certain contravariantly finite subcategories [Formula: see text] of [Formula: see text], that is, [Formula: see text] is of finite type if and only if its cohomological range is finite.

Visible and Dark Groups of Spacetime

A remarkable correspondence exists between lattices generated by discrete Jordan algebras and symmetries of superstrings, strongly suggesting that all known superstring theories are related and descend from a more general theory related to the Conway–Sloane transhyperbolic group. Cartan tori has visible spaces and $${G \mathord{\left/ {\vphantom {G H}} \right. \kern-0em} H}$$ Cartan generators as dark builders of $$G$$ and determined by root lattices. $${{E}_{{10}}}$$ is shown as the dark group of the visible (9 + 1) space time with the Lorentz group $$O(9 + 1).$$ Embedding of the higher exceptional groups will also be presented.

Projective modules and orbit space of unimodular rows over Discrete Hodge algebras over a non-Noetherian ring

For any commutative ring $R$ of Krull dimension zero and for any discrete Hodge algebra $D$ over $R$, it is proven that, if $n\geq 3$, the group $E_n(D)$ of $n\times n$ elementary matrices acts transitively on $Um_n(D)$, the set of unimodular rows of length $n$ over $D$.

Discrete triangulated categories

We introduce and study several homological notions which generalise the discrete derived categories of D. Vossieck. As an application, we show that Vossieck discrete algebras have this property with respect to all bounded t-structures. We give many examples of triangulated categories regarding these notions.