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Let Γ n be the symmetric group on { 1 , 2 , … , n } and S be the generating set of Γ n . The corresponding Cayley graph is denoted by Γ n ( S ) . If all elements of S are transpositions, a simple way to depict S is to via a graph, called the transposition generating graph of S , denoted by A ( S ) (or say simply A ), where the vertex set of A is { 1 , 2 , … , n } , there is an edge in A between i and j if and only if the transposition ( i j ) ∈ S , and Γ n ( S ) is called a Cayley graph obtained from a transposition generating graph A . Conditional diagnosability, proposed by Lai et al, is a novel measure of diagnosability that adds the additional condition that any faulty set cannot contain all of the neighbors of any vertex in a system. There are two well-known diagnostic models, PMC model and MM* model. The conditional diagnosability under the PMC (resp. MM*) model of a graph G , denoted by t c P M C ( G ) (resp. t c M M ∗ ( G ) ), is the maximum value of t is the maximum value of t such that G is conditionally t -diagnosable under the PMC (resp. MM*) model. The conditional diagnosability of many well-known multiprocessor systems has been explored. In this paper, by exploring the structural properties of these Cayley graphs, suppose | E ( A ) | is the number of edges in the transposition generating graph A , we obtain the following results: (1) Under the MM* model, t c M M ∗ ( Γ n ( S ) ) = 3 | E ( A ) | − 6 , if A has a triangle; 3 | E ( A ) | − 5 , if A has no triangles and A is not a star. (2) Under the PMC model, t c P M C ( Γ n ( S ) ) = 4 | E ( A ) | − 9 , if A has a triangle; 4 | E ( A ) | − 7 , if A has no triangles and A is not a star. As corollaries, the conditional diagnosability of many kinds of graphs under the MM* model and the PMC model such as Cayley graphs generated by unicyclic graphs, wheel graphs, complete graphs, tree graphs etc. are obtained.

Let S be a semigroup and let G be a subset of S. A set G is a generating set G of S which is denoted by . The rank of S is the minimal size or the minimal cardinality of a generating set of S, i.e. rank . In last twenty years, the rank of semigroups is worldwide studied by many researchers. Then it lead to a new definition of rank that is called the relative rank of S modulo U is the minimal size of a subset such that generates S, i.e. rank . A set with is called generating set of S modulo U. The idea of the relative rank was generalized from the concept of the rank of a semigroup and it was firstly introduced by Howie, Ruskuc and Higgins in 1998. Let X be a finite chain and let Y be a subchain of X. We denote the semigroup of full transformations on X under the composition of functions. Let be the set of all transformations from X to Y which is so-called the transformation semigroup with restricted range Y. It was firstly introduced and studied by Symons in 1975. Many results in were extended to results in . In this paper, we focus on the relative rank of semigroup and the semigroup of all orientation-preserving transformations in . In Section 2.1, we determine the relative rank of modulo the semigroup of all order-preserving or order-reversing transformations. In Section 2.2, we describe the results of the relative rank of modulo the semigroup . In Section 2.3, we determine the relative rank of modulo the semigroup of all orientation-preserving or orientation-reversing transformations. Moreover, we obtain that the relative rank modulo and modulo are equal.

We study the parameterized complexity of computing nontrivial automorphisms of weight k for a given hypergraph $$X=(V,E)$$ , with k as fixed parameter, where the weight of a permutation $$\pi \in S_n$$ is the number of points moved by $$\pi$$ . Building on the earlier work of Schweitzer (in: Proceedings of 19th ESA, Springer, Berlin, 2011. https://doi.org/10.1007/978-3-642-23719-5_32 ), we show the following results: (1) Computing nontrivial automorphisms of weight at most k for d-hypergraphs (that is, with edge-size bounded by d) remains fixed parameter tractable, with d treated as a second fixed parameter. Likewise, finding isomorphisms of weight k between d-hypergraphs X and Y (both defined on vertex set [n]) remains fixed parameter tractable. (2) For dealing with the exact weight k version of the problem, we introduce a more general algorithmic problem PermCode: given a permutation group G by a generating set and a fixed parameter k, is there is a nontrivial element of G with support at most (or exactly) k? We give a method for shrinking large orbits of the given group G to obtain subgroups while maintaining existence of weight at most k elements in it. An application of this yields an FPT algorithm for finding exact weight k nontrivial automorphisms in d-hypergraphs, d as second fixed parameter. (3) For hypergraphs with edges of unbounded size, we show that the problem is in $$\textsf {FPT } ^{\textsc {GI}}$$ . (4) Computing d-hypergraph isomorphisms of weight exactly k is fixed parameter tractable. This requires a more complicated orbit shrinking technique.

For a given odd prime number p, in this paper we construct a minimal generating set for the mod-p cohomology of the Steinberg summand of a family of Thom spectra over the classifying space of an elementary p-abelian group. This resolves the remaining cases for odd prime numbers of a problem studied previously by M. Inoue (Contemporary Mathematics, vol. 293, pp. 125–139, 2002) and (J. Lond. Math. Soc. 75: 317–329, 2007) and by the author (J. Algebra 381: 164–175, 2013).

It is a classical result that the direct product $$A\times B$$ of two groups is finitely generated (finitely presented) if and only if A and B are both finitely generated (finitely presented). This is also true for direct products of monoids, but not for semigroups. The typical (counter) example is when A and B are both the additive semigroup $${\mathbb {P}}=\{1,2,3,\ldots \}$$ of positive integers. Here $${\mathbb {P}}$$ is freely generated by a single element, but $${\mathbb {P}}^2$$ is not finitely generated, and hence not finitely presented. In this note we give an explicit presentation for $${\mathbb {P}}^2$$ in terms of the unique minimal generating set; in fact, we do this more generally for $${\mathbb {P}}^K$$ , the direct product of arbitrarily many copies of $${\mathbb {P}}$$ .

We consider the natural $$\mathrm {SO}(3,\Bbbk )$$ linear representation, $$\Bbbk =\mathbb {C}$$ or $$\mathbb {R}$$ , on the $$\Bbbk$$ vector space $${\mathsf {V}}=n\mathbb {S}^{2}(\mathbb {R}^3)$$ of n second-order symmetric tensors, the associated invariant field $$\Bbbk ({\mathsf {V}})^{\mathrm {SO}(3,\Bbbk )}$$ being known to be a purely transcendental extension in the complex case. We give an explicit tensorial form of a minimal generating set of the field of invariants, in both the complex and the real cases, showing that the invariant field is also a purely transcendental extension in the real case. The present results rely on some octahedral polynomial invariants obtained from Clebsch–Gordan projectors defined by a fourth-order octahedral covariant. Thanks to Cartan’s map, we also obtain a minimal set of generators for the $$\mathrm {SL}(2,\mathbb {C})$$ -rational invariant field of n binary quartics.

In this paper we revisit the family of algebroid space curves defined by Moh and find an explicit minimal generating set for the defining ideal.

Sets of solutions to finite systems of equations in a free group, are equivalent to sets of homomorphisms from a fixed f.p. group into a free group. The latter can be encoded in a diagram, the construction of which is valid also for f.g. groups. The diagram is known to be canonical for a fixed f.g. group with a fixed generating set. In this paper we prove that the construction depends on the chosen generating set of the given f.g. group.

A well-known conjecture of Babai states that if G is a finite simple group and X is a generating set of G, then the diameter of the Cayley graph Cay ( G , X ) is bounded above by ( log ⁡ | G | ) c for some absolute constant c. The goal of this paper is to prove such a bound for the diameter of Cay ( G , X ) whenever G = S L ( n , p ) and X is a generating set of G which contains a transvection. A natural analogue of this result is also proved for G = S L ( n , K ) , where K can be any field.

We consider the symmetric group Sym Ω with Ω = { 1 , … , n } for any integer n ⩾ 2 and a set S = { ( 1 i ) , i ∈ { 2 , … , n } } . The Star graph S n = Cay ( Sym Ω , S ) is the Cayley graph over the symmetric group Sym Ω with the generating set S. For n ⩾ 3 , the spectrum of the Star S n is integral such that for each integer 1 ⩽ k ⩽ n − 1 , the values ± ( n − k ) are its eigenvalues; if n ⩾ 4 , then 0 is also an eigenvalue of S n . A family of PI-eigenfunctions of the Star graph S n , n ⩾ 3 , has been obtained recently for eigenvalues n − 2 , … , n − ⌊ n 2 ⌋ − 1 . We generalise the family of PI-eigenfunctions and present a family of eigenfunctions for all non-zero eigenvalues of this graph.

We study the differential polynomial identities of the algebra UTm(F) under the derivation action of the two dimensional metabelian Lie algebra, obtaining a generating set of the TL-ideal they constitute. Then we determine the Sn-structure of their proper multilinear spaces and, for the minimal cases m = 2, 3, their exact differential codimension sequence.

Let X be a projective K3 surface over C . We prove that its Cox ring has a generating set whose degrees are either classes of smooth rational curves, sums of at most three elements of the Hilbert basis of the nef cone, or of the form 2 ( f + f ′ ) , where f , f ′ are classes of smooth elliptic curves with f ⋅ f ′ = 2 . This result and techniques using Koszul's type exact sequences are then applied to determine a generating set for the Cox ring of all Mori dream K3 surfaces of Picard number three which is minimal in most cases. A presentation for the Cox ring is given in some special cases with few generators.

In this work, we find a system of generators for the Picard modular group $$SU(2,1,{\mathcal {O}}_{11})$$ . This system contains four transformations, two translations a rotation and an involution.

The rank of a point-line geometry G is usually defined as the generating rank of G, namely the minimal cardinality of a generating set. However, when the subspace lattice of G satisfies the Exchange Property we can also try a different definition: consider all chains of subspaces of G and take the least upper bound of their lengths as the rank of G. If G is finitely generated then these two definitions yield the same number. On the other hand, as we shall show in this paper, if infinitely many points are needed to generate G then the rank as defined in the latter way is often (perhaps always) larger than the generating rank. So, if we like to keep the first definition we should accordingly discard the second one or modify it. We can modify it as follows: consider only well ordered chains instead of arbitrary chains. As we shall prove, the least upper bound of the lengths of well ordered chains of subspaces is indeed equal to the generating rank. According to this result, the (possibly infinite) rank of a polar space can be characterized as the least upper bound of the lengths of well ordered chains of singular subspaces; referring to arbitrary chains would be an error., Comment: 18 pages

Given a finite group G, we denote by $$\Delta (G)$$ Δ ( G ) the graph whose vertices are the elements G and where two vertices x and y are adjacent if there exists a minimal generating set of G containing x and y. We prove that $$\Delta (G)$$ Δ ( G ) is connected and classify the groups G for which $$\Delta (G)$$ Δ ( G ) is a planar graph.

We prove that the word problem of the Brin-Thompson group nV over a finite generating set is coNP -complete for every n ≥ 2 . It is known that { n V : n ≥ 1 } is an infinite family of infinite, finitely presented, simple groups. We also prove that the word problem of the Thompson group V over a certain infinite set of generators, related to boolean circuits, is coNP -complete.

The aim of this paper is to investigate upper bounds for the maximum degree of the elements of any minimal Janet basis of an ideal generated by a set of homogeneous polynomials. The presented bounds depend on the number of variables and the maximum degree of the generating set of the ideal. For this purpose, by giving a deeper analysis of the method due to Dube (SIAM J Comput 19:750–773, 1990), we improve (and correct) his bound on the degrees of the elements of a reduced Grobner basis. By giving a simple proof, it is shown that this new bound is valid for Pommaret bases, as well. Furthermore, based on Dube’s method, and by introducing two new notions of genericity, so-called J-stable position and prime position, we show that Dube’s (new) bound holds also for the maximum degree of polynomials in any minimal Janet basis of a homogeneous ideal in any of these positions. Finally, we study the introduced generic positions by proposing deterministic algorithms to transform any given homogeneous ideal into these positions.

We find a constructive bound for the word length of a generating set for the centralizer of an element of the Mapping Class Group. As a consequence, we show that it is algorithmically decidable whether two postcritically finite branched coverings of the sphere are Thurston equivalent.

We assume to have information about the generating properties of the subsets of a finite group G. In particular, we consider the two following situations. We know, for every subset X of G, whether X is a generating set of G. We know the graph whose vertices are the subsets of G and in which there is an edge connecting X and Y if and only if $$X\cup Y$$ is a generating set of G. We discuss how this kind of information can be used to discover properties of the group G.

Computer minimization of generating subsets (subsets of generators) of cubic Fedorov groups listed in the International Tables for Crystallography has been performed. The structure and fuzziness of the obtained minimal subsets are discussed. It is established that the allowable number of elements for almost half the minimized subsets is not the least one, which suggests that the generating subsets of Fedorov groups in the International Tables for Crystallography should be improved.

The application of autonomous electrical complexes with parallel diesel generator sets based on synchronous generators is considered. The problem of the existence of exchange power fluctuations in such technical systems is described. A mathematical model has been developed that allows conducting research of an autonomous electrical complex. It is proposed to implement graphical visualization of the dependence of the amplitude of power exchange fluctuations on time constants in the speed control circuits of diesel generators in the form of configuration maps. An example of using the proposed method for a specific electrical complex is given. The possibility of reducing the amplitude of exchange power fluctuations by reducing the values of time constants of frequency control circuits is justified, which is the basis of a new method for eliminating exchange power fluctuations and makes it possible to improve the method, which is based on changing the transmission coefficients of frequency regulators.

Let $$G=\left\langle S|R_{A}\right\rangle $$ be a semigroup with generating set S and equivalences $$R_{A}$$ among S determined by a matrix A. This paper investigates the complexity of G-shift spaces by yielding the Petersen–Salama entropies [defined in Petersen and Salama (Theoret Comput Sci 743:64–71, 2018)]. After revealing the existence of Petersen–Salama entropy of G-shift of finite type (G-SFT), the calculation of Petersen–Salama entropy of G-SFT is equivalent to solving a system of nonlinear recurrence equations. The complete characterization of Petersen–Salama entropies of G-SFTs on two symbols is addressed, which extends (Ban and Chang in On the topological entropy of subshifts of finite type on free semigroups, 2018. arXiv:1702.04394 ) in which G is a free semigroup.

В данной работе мы строим систему образующих Гензеля - Шафаревича для формальных модулей Хонды над многомерным полем. В дальнейшем это даст возможность вычисления символа Гильберта в описанной ситуации.

Let $N$ balls of the same radius be given in a $d$-dimensional real normed vector space, i.e., in a Minkowski $d$-space. Then apply a uniform contraction to the centers of the $N$ balls without changing the common radius. Here a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. The main results of this paper state that a uniform contraction of the centers does not increase (resp., decrease) the volume of the union (resp., intersection) of $N$ balls in Minkowski $d$-space, provided that $N\geq 2^d$ (resp., $N\geq 3^d$ and the unit ball of the Minkowski $d$-space is a generating set). Some improvements are presented in Euclidean spaces.

Graphs considered in this paper are finite simple undirected graphs. Let G = (V(G), E(G)) be a graph with E(G) = {e1, e2,..., em}, for some positive integer m. The edge space of G, denoted by ℰ(G), is a vector space over the field ℤ2. The elements of ℰ(G) are all the subsets of E(G). Vector addition is defined as X+Y = X ∆ Y, the symmetric difference of sets X and Y, for X,Y ∈ ℰ(G). Scalar multiplication is defined as 1.X =X and 0.X = ∅ for X ∈ ℰ(G). Let H be a subgraph of G. The uniform set of H with respect to G, denoted by EH(G), is the set of all elements of ℰ(G) that induces a subgraph isomorphic to H. The subspace of ℰ(G) generated by ℰH(G) shall be denoted by ℰH(G). If EH(G) is a generating set, that is ℰH(G)= ℰ(G), then H is called a generator subgraph of G. This study determines the dimension of subspace generated by the set of all subsets of E(G) with even cardinality and the subspace generated by the set of all k-subsets of E(G), for some positive integer k, 1 ≤ k ≤ m. Moreover, this paper determines all the generator subgraphs of star graphs. Furthermore, it gives a characterization for a graph G so that star is a generator subgraph of G.

Given a finitely generated group with generating set $S$, we study the \emph{cogrowth} sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks in a Cayley graph with steps from $S$. We prove that the cogrowth sequence is not $P$-recursive when~$G$ is an amenable group of superpolynomial growth, answering a question of Garrabant and Pak.

For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. Since there exist efficient algorithms to construct this generating set in any copy of the group, our presentations can be used to verify claimed isomorphisms between representations of the classical group. The presentations are available in Magma .

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

Identifying finite non-group semigroups for every positive integer is significant because of many applications of such semigroups are functional in various branches of sciences such as computer science, mathematics and finite machines. The finite non-commutative monoids as a type of such semigroups were identified in 2014, for every positive integer. We here attempt to identify the finite commutative monoids and finite commutative non-monoids of a given integer $$n=p^\alpha q^\beta$$n=pαqβ, for every integers $$\alpha , \beta \ge 2$$α,β≥2 and different primes p and q. In order to recognize the commutative monoids, we present a class of 2-generated monoids of a given order, and for the commutative non-monoids of order $$n=p^\alpha q^\beta,$$n=pαqβ, we give the minimal generating set. Moreover, we prove that there are exactly $$(p^{\alpha }-2)(q^{\beta }-2)$$(pα-2)(qβ-2) non-isomorphic commutative non-monoids of order $$p^\alpha q^\beta$$pαqβ. The identification of non-group semigroups for the integers $$p^{2\alpha }$$p2α and $$2p^\alpha$$2pα is achieved. The automorphism groups of these groups are specified as well. As a result of this study, an interesting difference between the abelian groups and the commutative semigroups of order $$p^2$$p2 is presented.

A directed Cayley graph C a y ( Γ , X ) is specified by a group Γ and an identity-free generating set X for this group. Vertices of C a y ( Γ , X ) are elements of Γ and there is a directed edge from a vertex u to a vertex v in C a y ( Γ , X ) if and only if there is a generator x ∈ X such that u x = v . We study the metric dimension for the directed Cayley graphs C a y ( Γ s , { a , b } ) of general split metacyclic groups, and present the exact values of the metric dimension for the special split metacyclic groups Γ s = 〈 a , b | a n = b 2 s = 1 , b a = a − 1 b 〉 .

A subset $S$ of a group $G$ invariably generates $G$ if, when each element of $S$ is replaced by an arbitrary conjugate, the resulting set generates $G.$ An invariable generating set $X$ of $G$ is called minimal if no proper subset of $X$ invariably generates $G.$ We will address several questions related to the behaviour of minimal invariable generating sets of a finite group., This is a belated posting of a 2019 paper. 24 pp

The normaliser problem has as input two subgroups H and K of the symmetric group $$\mathrm {S}_n$$ S n , and asks for a generating set for $$N_K(H)$$ N K ( H ) : it is not known to have a subexponential time solution. It is proved in Roney-Dougal and Siccha (Bull Lond Math Soc 52(2):358–366, 2020) that if H is primitive, then the normaliser problem can be solved in quasipolynomial time. We show that for all subgroups H and K of $$\mathrm {S}_n$$ S n , in quasipolynomial time, we can decide whether $$N_{\mathrm {S}_n}(H)$$ N S n ( H ) is primitive, and if so, compute $$N_K(H)$$ N K ( H ) . Hence we reduce the question of whether one can solve the normaliser problem in quasipolynomial time to the case where the normaliser in $$\mathrm {S}_n$$ S n is known not to be primitive.

In this paper we work toward the HOMFLYPT skein module of $L(p, 1)$, $\mathcal{S}(L(p,1))$, via braids. Our starting point is the linear Turaev-basis, $\Lambda^{\prime}$, of the HOMFLYPT skein module of the solid torus ST, $\mathcal{S}({\rm ST})$, which can be decomposed as the tensor product of the "positive" ${\Lambda^{\prime}}^+$ and the "negative" ${\Lambda^{\prime}}^-$ sub-modules, and the Lambropoulou invariant, $X$, for knots and links in ST, that captures $S({\rm ST})$. It is a well-known result by now that $\mathcal{S}(L(p, 1))=\frac{\mathcal{S}(ST)}{}$, where bbm's (braid band moves) denotes the isotopy moves that correspond to the surgery description of $L(p, 1)$. Namely, a HOMFLYPT-type invariant for knots and links in ST can be extended to an invariant for knots and links in $L(p, 1)$ by imposing relations coming from the performance of bbm's and solving the infinite system of equations obtained that way. \smallbreak In this paper we work with a new basis of $\mathcal{S}({\rm ST})$, $\Lambda$, and we relate the infinite system of equations obtained by performing bbm's on elements in $\Lambda^+$ to the infinite system of equations obtained by performing bbm's on elements in $\Lambda^-$ via a map $I$. More precisely we prove that the solutions of one system can be derived from the solutions of the other. Our aim is to reduce the complexity of the infinite system one needs to solve in order to compute $\mathcal{S}(L(p,1))$ using the braid technique. Finally, we present a generating set and a potential basis for $\frac{\Lambda^+}{}$ and thus, we obtain a generating set and a potential basis for $\frac{\Lambda^-}{}$. We also discuss further steps needed in order to compute $\mathcal{S}(L(p,1))$ via braids., Comment: 26 pages, 6 figures. arXiv admin note: text overlap with arXiv:1802.09376, arXiv:1702.06290

We study the properties of groups that have presentations in which the generating set is a fixed set of involutions and all additional relations are of even length. We consider the parabolic subgroups of such a group and show that every element has a factorization with respect to a given parabolic subgroup. Furthermore, we give a counterexample, using a cluster group presentation, which demonstrates that this factorization is not necessarily unique. Publisher PDF

Yoshikawa moves were introduced at least quarter-century ago and are still actively used by researchers. For any marked graph diagram we will define its twisted diagram and its mirror cut surface. By using a surface-link group of a mirror cut surface of a twisted diagram we will prove the independence of Yoshikawa eighth move. As a consequence we will establish a minimal generating set of band moves for links with bands., 13 pages

Let $n>3$ be a natural number. By a 1975 result of H. Strietz, the lattice Part$(n)$ of all partitions of an $n$-element set has a four-element generating set. In 1983, L. Z\'adori gave a new proof of this fact with a particularly elegant construction. Based on his construction from 1983, the present paper gives a lower bound on the number $\nu(n)$ of four-element generating sets of Part$(n)$. We also present a computer assisted statistical approach to $\nu(n)$ for small values of $n$. In his 1983 paper, L. Z\'adori also proved that for $n\geq 7$, the lattice Part$(n)$ has a four element generating set that is not an antichain. He left the problem whether such a generating set for $n\in\{5,6\}$ exists open. Here we solve this problem in negative for $n=5$ and in affirmative for $n=6$. Finally, the main theorem asserts that the direct product of some powers of partition lattices is four-generated. In particular, by the first part of this theorem, Part$(n_1)\times$ Part$(n_2)$ is four-generated for any two distinct integers $n_1$ and $n_2$ that are at least 5. The second part of the theorem is technical but it has two corollaries that are easy to understand. Namely, the direct product Part$(n)$ $\times$ Part$(n+1)$ $\times\dots\times$ Part$(3n-14)$ is four-generated for each integer $n\geq 9$. Also, for every positive integer $u$, the $u$-th the direct power of the direct product Part$(n)$ $\times$ Part$(n+1)$ $\times\dots\times$ Part$(n+u-1)$ is four-generated for all but finitely many $n$. If we do not insist on too many direct factors, then the exponent can be quite large. For example, our theorem implies that the $10^{127}$-th direct power of Part$(1011)$ $\times$ Part$(1012)$ $\times \dots \times$ Part$(2020)$ is four-generated., Comment: 34 pages, 6 figures

In this work, we investigate DNA codes of odd length over the finite ring R = Z 4 +vZ 4 , v 2 = v. Firstly, we give a map Φ from R n to {A, T, G, C} 2n . Then cyclic codes of odd length satisfying the reverse and reverse-complement constraint are characterized over such ring. Moreover, when C is a reverse-complement cyclic code over R, we prove that Φ(C) is a cyclic DNA code by using the method different from all the previously known ones, which helps us finding a new DNA code with 256 codewords. Finally, we consider the GC content of code C over finite ring Z 4 + vZ 4 by means of the Gray images of the minimal generating set of C.

Let g be a simple, finite-dimensional complex Lie algebra, and let V k ( g ) denote the universal affine vertex algebra associated to g at level k . The Cartan involution on g lifts to an involution on V k ( g ) , and we denote by V k ( g ) Z 2 the orbifold, or fixed-point subalgebra, under this involution. Our main result is an explicit minimal strong finite generating set for V k ( g ) Z 2 for generic values of k .

We consider metric spaces over the ring of integers. For any generating set S of ℤ, it is shown that the metric dimension of the metric space X = X(ℤ, S) is not greater than 2 max S. The resolving ...

We describe an algorithm that, given a positive integer N, computes a Grobner basis of the ideal of polynomial relations among Dedekind η-functions of level N, i.e., among the elements of { η ( δ 1 τ ) , … , η ( δ n τ ) } where 1 = δ 1 δ 2 … δ n = N are the positive divisors of N. More precisely, we find a finite generating set (which is also a Grobner basis) of the ideal ker ⁡ ϕ where ϕ : Q [ E 1 , … , E n ] → Q [ η ( δ 1 τ ) , … , η ( δ n τ ) ] , E k ↦ η ( δ k τ ) , k = 1 , … , n .

In recent years many papers have derived polyhedral and non-polyhedral convex envelopes for different classes of functions. Except for the univariate cases, all these classes of functions share the property that the generating set of their convex envelope is a subset of the border of the region over which the envelope is computed. In this paper we derive the convex envelope over a rectangular region for a class of functions which does not have this property, namely the class of bivariate cubic functions without univariate third-degree terms.

According to [5] , the arithmetic mean is a function characterized by four features: it is non-decreasing, idempotent, symmetric and additive. The first three of them can be naturally converted to the theory of posets but the last one generally can not. Due to this problem, we will replace it with another suitable property, which is called transfer-stability. However, we do not get the exact arithmetic mean but some approximation. These functions will be called transfer-stable means. The first aim of the paper is to show that transfer-stable means on a finite chain form a lattice which is isomorphic to the direct power of a finite chain. The second goal is to create a generating set of transfer-stable means, i.e., means that can generate all other transfer-stable means of the same arity by classical composition of functions. The last goal deals with question of how to generate all transfer-stable means of any arity by binary transfer-stable means only. For this problem we define special transfer-stable means composition.

In this article, we show that, if $S\in \mathcal{B}(H)$ is irreducible and essential unitary, then $\{S,SS^*\}$ is a hyperrigid generator for the unital $C^*$-algebra $\mathcal{T}$ generated by $\{S,SS^*\}$. We prove that, if $T$ is an operator in $\mathcal{B}(H)$ that generates an unital $C^*$-algebra $\mathcal{A}$ then $\{T,T^*T,TT^*\}$ is a hyperrigid generator for $\mathcal{A}$. As a corollary it follows that, if $T\in \mathcal{B}(H)$ is normal then $\{T,TT^*\}$ is hyperrigid generator for the unital $C^*$-algebra generated by $T$ and if $T\in \mathcal{B}(H)$ is unitary then $\{T\}$ is hyperrigid generator for the $C^*$-algebra generated by $T$. We show that if $V\in \mathcal{B}(H)$ is an isometry (not unitary) that generates the $C^*$-algebra $\mathcal{A}$ then the minimal generating set $\{V\}$ is not hyperrigid for $\mathcal{A}$., Comment: 8 pages

The paper studies two numerical characteristics of matrix incidence algebras over finite fields associated with generating sets of such algebras: the minimal cardinality of a generating set and the length of an algebra. Generating sets are understood in the usual sense, the identity of the algebra being considered a word of length 0 in generators, and also in the strict sense, where this assumption is not used. A criterion for a subset to generate an incidence algebra in the strict sense is obtained. For all matrix incidence algebras, the minimum cardinality of a generating set and a generating set in the strict sense are determined as functions of the field cardinality and the order of the matrices. Some new results on the lengths of such algebras are obtained. In particular, the length of the algebra of “almost” diagonal matrices is determined, and a new upper bound for the length of an arbitrary matrix incidence algebra is obtained.

We confirm the Hanna Neumann conjecture for topologically finitely generated closed subgroups U and W of a nonsolvable Demushkin group G. Namely, we show that ∑ g ∈ U \ G / W d ¯ ( U ∩ g W g − 1 ) ≤ d ¯ ( U ) d ¯ ( W ) where d ¯ ( K ) = max ⁡ { d ( K ) − 1 , 0 } and d ( K ) is the least cardinality of a topological generating set for the group K.

We present a new algorithm that, given two matrices in $GL(n,Q)$, decides if they are conjugate in $GL(n,Z)$ and, if so, determines a conjugating matrix. We also give an algorithm to construct a generating set for the centraliser in $GL(n,Z)$ of a matrix in $GL(n,Q)$. We do this by reducing these problems respectively to the isomorphism and automorphism group problems for certain modules over rings of the form $\mathcal O_K[y]/(y^l)$, where $\mathcal O_K$ is the maximal order of an algebraic number field and $l \in N$, and then provide algorithms to solve the latter. The algorithms are practical and our implementations are publicly available in Magma.

A directed Cayley graph C(Γ, X) is specified by a group Γ and an identity-free generating set X for this group. Vertices of C(Γ, X) are elements of Γ and there is a directed edge from the vertex u to the vertex v in C(Γ, X) if and only if there is a generator x ∈ X such that ux = v. We study graphs C(Γ, X) for the direct product Zm × Zn of two cyclic groups Zm and Zn, and the generating set X = {(0, 1), (1, 0), (2, 0), …, (p, 0)}. We present resolving sets which yield upper bounds on the metric dimension of these graphs for p = 2 and 3.

In this note we compute the leading term with respect to the De Concini–Kac filtration of U q ( gl n ) of a generating set for the quantum Gelfand–Tsetlin subalgebra.

We establish an operator formula for the shift of a solution in the case where the summand in the Euler-Poisson-Darboux equation containing the first derivative of the solution is perturbed by a group generator.