Narrow Results

Let $R$ be a ring with identity. The unit (respectively, unitary Cayley) graph of $R$ is a simple graph $G(R)$ (respectively, $\mathrm{\Gamma }(R)$) with vertex set $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $x+y$ (respectively, $x-y$) is a unit of $R$. In this paper, we explore when $G(R)$ and $\mathrm{\Gamma }(R)$ are isomorphic for a finite ring $R$. Among other results, we prove that $G(R)$ is isomorphic to $\mathrm{\Gamma }(R)$ for a finite ring $R$ if and only if char($R/J(R)$)=2 or $R/J(R)\cong {\mathbb{Z}}_2\times S$, where $J(R)$ is the Jacobson radical of $R$ and $S$ is a finite ring.

Let ${𝒮}$ and $𝔅$ be unital ∗-algebras such that ${𝒮}$ has a nontrivial projection. In this paper, we demonstrate under certain restrictions that if a bijective map $\mathrm{\Delta }:{𝒮}\to 𝔅$ satisfies $\mathrm{\Delta }({[M◇N,W]}_{\ast })={[\mathrm{\Delta }(M)◇\mathrm{\Delta }(N),\mathrm{\Delta }(W)]}_{\ast }$ for all $M,N,W\in {𝒮}$, then $\mathrm{\Delta }$ is a ∗-preserving ring isomorphism. Also, this result is applied to factor von Neumann algebras.

The degree structure of complete sets under 2DFA reductions is investigated. It is shown that, for any class that is closed under log-lin reductions:

• All complete sets for the class under 2DFA reductions are also complete under one-one, length-increasing 2DFA reductions and are first-order isomorphic.

• The 2DFA-isomorphism conjecture is false, i.e., the complete sets under 2DFA reductions are not isomorphic to each other via 2DFA reductions.

An adinkra is a graph-theoretic representation of space–time supersymmetry. Minimal four-color valise adinkras have been extensively studied due to their relations to minimal 4D, ${𝒩}=1$ supermultiplets. Valise adinkras, although an important subclass, do not encode all the information present when a 4D supermultiplet is reduced to 1D. Eigenvalue equivalence classes for valise adinkra matrices exist, known as ${\chi }_{\mathrm{\text{o}}}$ equivalence classes, where valise adinkras within the same ${\chi }_{\mathrm{\text{o}}}$ equivalence class are isomorphic in the sense that adinkras within a ${\chi }_{\mathrm{\text{o}}}$-equivalence class can be transformed into each other via field redefinitions of the nodes. We extend this to non-valise adinkras, via Python code, providing a complete eigenvalue classification of “node-lifting” for all 36,864 valise adinkras associated with the Coxeter group $B{C}_4$. We term the eigenvalues associated with these node-lifted adinkras Height Yielding Matrix Numbers (HYMNs) and introduce HYMN equivalence classes. These findings have been summarized in a Mathematica notebook that can be found at the HEPTHools Data Repository on GitHub.

Cu₂Te single crystal was grown by the Bridgman method. X-ray diffraction (XRD) study of Cu₂Te single crystals in the temperature range of 293–893 K was performed and possible phase transitions in the mentioned range of temperature have been investigated. (Cu₂Te)${}_{2-x}$(ZnTe)${}_x$ single crystals also were grown with $x=0.00$, 0.05, 0.10 concentrations and structural properties of the obtained single crystals were investigated by the XRD method in the temperature range 293–893 K. Lattice parameters and possible phase transitions in the mention temperature range were determined for (Cu₂Te)${}_{2-x}$(ZnTe)${}_x$ single crystals for $x=0.00$, 0.05, 0.10 concentrations.

We have recently shown that one can obtain the numbers and sizes of modules of a software system from the eigenvectors of its modularity matrix symmetrized and weighted by an affinity matrix. However such a weighting still demands a suitable definition of an affinity. This paper offers an alternative way to obtain the same results by means of the eigenvectors of a Laplacian matrix, directly obtained from the modularity matrix without the need of weighting. These two formalizations stand in a mutual isomorphism. We call it bipartite isomorphism since it is most straightforwardly shown by deriving the Laplacian from the modularity matrix and vice versa through the intermediate bipartite graph between two separate sets: the structors’ and the functionals’ sets. This isomorphism is also demonstrated through the equation defining the Laplacian in terms of the modularity matrix, or by the direct mapping of the respective matrices’ eigenvectors. Both matrices and the bipartite graph reflect one central idea: modules are connected components with high cohesion. The Laplacian matrix technique, of which the Fiedler vector is of central importance, is illustrated by case studies. An important claim of this paper is that, independently of the modularity matrix- and Laplacian matrix-specific properties, behind these two alternative matrices there is just one unified algebraic theory of software composition — the Linear Software Models — here concerning the application of the matrices’ eigenvectors to software modularity.

In this paper, we consider two distinct problems related to complexity aspects of the visibility graphs of simple polygons. Recognizing visibility graphs is a long-standing open problem. It is not even known whether visibility graph recognition is in NP. That visibility graph recognition is in NP would be established if we could demonstrate that any n vertex visibility graph is realized by a polygon which can be drawn on an exponentially-sized grid. This motivates a study of the area requirements for realizing visibility graphs. In this paper, we prove:

• Θ(n³) area is necessary and sufficient to realize the complete visibility graph K_n.

• There exist visibility graphs which require exponential area to realize.

• Any maximal outerplanar graph of diameter d can be realized in O(d² · 2^d) area, which can be as small as O(n log² n) for a balanced mop. Linear maximal outer-planar graphs can be realized in O(n⁸) area.

The second part of this paper considers the complexity of specific optimization problems on visibility graphs. Given a polygon P, we show that finding a maximum independent set, minimum vertex cover, or maximum dominating set in the visibility graph of P are all NP-complete. Further we show that for polygons P₁ and P₂, the problem of testing if they have isomorphic visibility graphs is isomorphism-complete. These problems remain hard when given the visibility graphs as input.

Let L be a finitely generated free Lie algebra. We construct an example of two elements u and v of L such that the factor algebras L/(u) and L/(v) are isomorphic, where (u) and (v) are ideals of L generated by u and v, respectively, but there is no automorphism φ of L such that φ(u)=v.

We study the way in which the abstract structure of a small overlap monoid is reflected in, and may be algorithmically deduced from, a small overlap presentation. We show that every C(2) monoid admits an essentially canonical C(2) presentation; by counting canonical presentations we obtain asymptotic estimates for the number of non-isomorphic monoids admitting a-generator, k-relation presentations of a given length. We demonstrate an algorithm to transform an arbitrary presentation for a C(m) monoid (m at least 2) into a canonical C(m) presentation, and a solution to the isomorphism problem for C(2) presentations. We also find a simple combinatorial condition on a C(4) presentation which is necessary and sufficient for the monoid presented to be left cancellative. We apply this to obtain algorithms to decide if a given C(4) monoid is left cancellative, right cancellative or cancellative, and to show that cancellativity properties are asymptotically visible in the sense of generic-case complexity.

We consider the problem of computing $R(c,a)$, the number of unlabeled graded lattices of rank $3$ that contain $c$ coatoms and $a$ atoms. More specifically, we do this when $c$ is fairly small, but $a$ may be large. For this task, we describe a computational method that combines constructive listing of basic cases and tools from enumerative combinatorics. With this method, we compute the exact values of $R(c,a)$ for $c\le 9$ and $a\le 1000$.

We also show that, for any fixed $c$, there exists a quasipolynomial in $a$ that matches with $R(c,a)$ for all $a$ above a small value. We explicitly determine these quasipolynomials for $c\le 7$, thus finding closed form expressions of $R(c,a)$ for $c\le 7$.

We consider two multi-parameter classes of cyclically presented groups, introduced by Cavicchioli, Repovš and Spaggiari, that contain many previously considered families of cyclically presented groups of interest both for their algebraic and for their topological properties. Building on results of Bardakov and Vesnin, O’Brien and the previously named authors, we prove theorems that establish isomorphisms of groups within these families.

In this work, we extend an established isomorphism between group rings and a subring of the $n\times n$ matrices. This extension allows us to construct more complex matrices over the ring $R.$ We present many interesting examples of complex matrices constructed directly from our extension. We also show that some of the matrices used in the literature before can be obtained by a direct application of our extended isomorphism.

Let $P$ and $Q$ be finite posets and $R$ a commutative unital ring. In the case where $R$ is indecomposable, we prove that the $R$-linear isomorphisms between partial flag incidence algebras $I^3(P,R)$ and $I^3(Q,R)$ are exactly those induced by poset isomorphisms between $P$ and $Q$. We also show that the $R$-linear derivations of $I^3(P,R)$ are trivial.

We solve a problem posed by Goncharov and Knight (Problem 4 in [S. Goncharov and J. Knight, Computable structure and antistructure theorems, Algebra Logika41(6) (2002) 639–681, 757]). More specifically, we produce an effective Friedberg (i.e. injective) enumeration of computable equivalence structures, up to isomorphism. We also prove that there exists an effective Friedberg enumeration of all isomorphism types of infinite computable equivalence structures.

We introduce and study the quantum version of the differential operator algebra on Laurent polynomials and its associated Lie algebra over a field F of characteristic 0. The q-quantum torus F_q is the unital associative algebra over F generated by subject to the defining relations t_it_j = q_i,jt_jt_i, where q_i,i = 1, . Let D be a subspace of where ∂_i is the derivation on F_q sending to . Then, the quantum differential operator algebra is the associative algebra F_q[D].

Assume that F_q[D] is simple as an associative algebra. We compute explicitly all 2-cocycles of F_q[D], viewed as a Lie algebra. More precisely, we show that the second cohomology group of F_q[D] has dimension n if D = 0, dimension 1 if dim D = 1, and dimension 0 if dim D > 1.

We also determine all isomorphisms and anti-isomorphisms F_q[D] → F_q′[D′] of simple associative algebras, and all isomorphisms F_q[D]/F → F_q′[D′]/F of simple Lie algebras.

In this paper, we give a quantitative description of the Fucík spectrum of -Δ with several technical conditions and get some results which improve considerably [5, Theorems 2.6 and 3.3]. The isomorphism between C_*(J_{(a, b, f)}, ∞) and C_*(J_{(a, b)}, 0) are obtained provided (a,b) ∉ ∑, which is exploited as a tool to generalize a result given by [14] without assuming λ_l being simple. Two applications are discussed.

The isomorphism problem for modular abelian group algebras is solved for p-groups of countable p-length for p odd. For the prime p = 2, it is solved for all 2-groups.

The enhanced power graph ${{𝒢}}_e(G)$ of a group $G$ is the graph with vertex set $G$ such that two vertices $x$ and $y$ are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups $G$ for which $\mathrm{\text{Aut}}({{𝒢}}_e(G)$ is abelian, all finite groups $G$ with $|\mathrm{\text{Aut}}({{𝒢}}_e(G)|$ being prime power, and all finite groups $G$ with $|\mathrm{\text{Aut}}({{𝒢}}_e(G)|$ being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

In [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301] we introduced a new class of algebras, which we named quantum generalized Heisenberg algebras and which depend on a parameter $q$ and two polynomials $f,g$. We have shown that this class includes all generalized Heisenberg algebras (as defined in [E. M. F. Curado and M. A. Rego-Monteiro, Multi-parametric deformed Heisenberg algebras: A route to complexity, J. Phys. A: Math. Gen. 34(15) (2001) 3253; R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291, MR 3325233]) as well as generalized down-up algebras (as defined in [G. Benkart and T. Roby, Down-up algebras, J. Algebra 209(1) (1998) 305–344; T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279(1) (2004) 402–421, MR 2078408 (2005f:16051)]), but the parameters of freedom we allow for give rise to many algebras which are in neither one of these two classes. Having classified their finite-dimensional irreducible representations in [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301], in this paper, we turn to their classification by isomorphism, the description of their automorphism groups and the study of their ring-theoretical properties.

This paper investigates the tensor product of a finite dimensional nilpotent evolution algebra. Some properties that translate from tensor products to factors and vice versa have been investigated, including the index of nilpotency and annihilator. The index of nilpotency of the tensor product of two nilpotent evolution algebras with different indexes of nilpotency is determined. Moreover, we investigate the tensorially decomposable of the 4-dimensional nilpotent evolution algebra. In addition, the decomposable nilpotent evolution algebra with the maximal index of nilpotency has been carried out.