scholarly journals One-relator groups and algebras related to polyhedral products

2021 ◽  
pp. 1-20
Author(s):  
Jelena Grbić ◽  
George Simmons ◽  
Marina Ilyasova ◽  
Taras Panov
Keyword(s):  
Homology Group ◽  
Homotopy Theory ◽  
Commutator Subgroup ◽  
Homology Groups ◽  
Homological Characterization ◽  
One Relator Groups ◽  
Combinatorial Condition ◽  
The Moment ◽  
Necessary And Sufficient

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

A Homological characterization of certain Abelian groups

1961 ◽  
Vol 57 (2) ◽  
pp. 256-264 ◽  
Author(s):  
A. J. Douglas
Keyword(s):  
Abelian Group ◽  
Homology Group ◽  
Abelian Groups ◽  
Unit Element ◽  
Homology Theory ◽  
Homology Groups ◽  
Complete Answer ◽  
Homological Characterization

Let G be a monoid; that is to say, G is a set such that with each pair σ, τ of elements of G there is associated a further element of G called the ‘product’ of σ and τ and written as στ. In addition it is required that multiplication be associative and that G shall have a unit element. The so-called ‘Homology Theory’† associates with each left G-module A and each integer n (n ≥ 0) an additive Abelian group Hn (G, A), called the nth homology group of G with coefficients in A. It is natural to ask what can be said about G if all the homology groups of G after the pth vanish identically in A. In this paper we give a complete answer to this question in the case when G is an Abelian group. Before describing the main result, however, it will be convenient to define what we shall call the homology type of G. We write Hn(G, A) ≡ 0 if Hn(G, A) = 0 for all left G-modules A.

On semi-projective modules and their endomorphism rings

2018 ◽  
Vol 11 (02) ◽  
pp. 1850029 ◽  
Author(s):  
Manoj Kumar Patel ◽  
Varun Kumar ◽  
A. J. Gupta
Keyword(s):  
Local Ring ◽  
Projective Module ◽  
Sufficient Conditions ◽  
Projective Modules ◽  
Endomorphism Rings ◽  
Large Classes ◽  
Counter Example ◽  
Homological Characterization ◽  
Necessary And Sufficient

This paper provides the several homological characterization of perfect rings and semi-simple rings in terms of semi-projective modules. We investigate whether Hopkins–Levitzki Theorem extend to semi-projective module i.e. whether there exists an artinian semi-projective module which are noetherian. Unfortunately, the answer we have is negative; counter example is given. However, it is shown that the answer is positive for certain large classes of semi-projective modules in Proposition 2.26. We have discussed the summand intersection property, summand sum property for semi-projective modules. Apart from this, we have introduced the idea of [Formula: see text]-hollow modules, also several necessary and sufficient conditions are established when the endomorphism rings of a semi-projective modules is a local ring.

scholarly journals Mating Type in Chlamydomonas Is Specified by mid, the Minus-Dominance Gene

Genetics ◽  
1997 ◽  
Vol 146 (3) ◽  
pp. 859-869 ◽  
Author(s):  
Patrick J Ferris ◽  
Ursula W Goodenough
Keyword(s):  
Transcription Factor ◽  
Mating Type ◽  
Codon Bias ◽  
Leucine Zipper ◽  
Laboratory Strain ◽  
Leucine Zipper Motif ◽  
Type Locus ◽  
Diploid Cells ◽  
Necessary And Sufficient

Diploid cells of Chlamydomonas reinhardtii that are heterozygous at the mating-type locus (mt  +/mt  –) differentiate as minus gametes, a phenomenon known as minus dominance. We report the cloning and characterization of a gene that is necessary and sufficient to exert this minus dominance over the plus differentiation program. The gene, called mid, is located in the rearranged (R) domain of the mt  – locus, and has duplicated and transposed to an autosome in a laboratory strain. The imp11 mt  – mutant, which differentiates as a fusion-incompetent plus gamete, carries a point mutation in mid. Like the fus1 gene in the mt  + locus, mid displays low codon bias compared with other nuclear genes. The mid sequence carries a putative leucine zipper motif, suggesting that it functions as a transcription factor to switch on the minus program and switch off the plus program of gametic differentiation. This is the first sex-determination gene to be characterized in a green organism.

scholarly journals On cryptographic properties of (n + 1)-bit S-boxes constructed by known n-bit S-boxes

2020 ◽  
Vol 15 (1) ◽  
pp. 258-265
Author(s):  
Yu Zhou ◽  
Daoguang Mu ◽  
Xinfeng Dong
Keyword(s):  
Sufficient Conditions ◽  
Sum Of Squares ◽  
Basic Component ◽  
Necessary And Sufficient Conditions ◽  
Autocorrelation Functions ◽  
Walsh Spectrum ◽  
Cryptographic Algorithms ◽  
Necessary And Sufficient ◽  
Relationship Of

AbstractS-box is the basic component of symmetric cryptographic algorithms, and its cryptographic properties play a key role in security of the algorithms. In this paper we give the distributions of Walsh spectrum and the distributions of autocorrelation functions for (n + 1)-bit S-boxes in [12]. We obtain the nonlinearity of (n + 1)-bit S-boxes, and one necessary and sufficient conditions of (n + 1)-bit S-boxes satisfying m-order resilient. Meanwhile, we also give one characterization of (n + 1)-bit S-boxes satisfying t-order propagation criterion. Finally, we give one relationship of the sum-of-squares indicators between an n-bit S-box S0 and the (n + 1)-bit S-box S (which is constructed by S0).

On representation of Poisson mixtures as Poisson sums and a characterization of the gamma distribution

1977 ◽  
Vol 82 (2) ◽  
pp. 297-300 ◽  
Author(s):  
A. V. Godambe
Keyword(s):  
Gamma Distribution ◽  
Exponential Type ◽  
Similar Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Poisson Mixture ◽  
Mixing Distribution ◽  
Poisson Mixtures ◽  
Necessary And Sufficient

AbstractA necessary and sufficient condition for a Poisson mixture with an exponential type mixing distribution to be equivalently represented as a Poisson sum is obtained. The problem of deriving a similar condition under any mixing distribution on (0, ∞) is discussed. Finally, a characterization of the gamma distribution is obtained.

La structure de classes québécoise au 19e siècle et le problème de l'articulation des modes de production

1981 ◽  
Vol 14 (3) ◽  
pp. 487-518
Author(s):  
Gérald Bernier
Keyword(s):  
Census Data ◽  
New France ◽  
Class Structure ◽  
Capitalist Development ◽  
Single Class ◽  
The Moment ◽  
Empirical Demonstration ◽  
Work Done ◽  
The Impact

The study of social classes in the nineteenth century requires the development of conceptual tools able to explain the impact of the Conquest on the pre-existant social structures in determining transformations of the class structure during the subsequent decades.This article examines the work done on this question by Marxist writers. The author criticizes certain conclusions which have been drawn and which suggest deficiencies at a theoretical level. The objections relate to the marked tendency of these conclusions to perceive the structural effects of the Conquest in terms of the formation of a double-class structure characterized by “ethnic origins.” Specifically, the author challenges the notion of the division itself, as well as the criterion on which the division is based.The author proposes that an analysis centred upon the concepts relating to a problem of the transition and linkage of different modes of production permits a more satisfying interpretation, if accompanied by a certain number of considerations of the “upside” and “downside” of the Conquest. To this end, the argument is based on a characterization of New France in terms of the domination of the relations of production of the feudal type and on an analysis of metropolitan centres with intent to evaluate their level of capitalist development at the moment of their respective colonial penetration in Canada. The results of this approach permit one to posit the existence of a single-class structure, characterized principally by the existence of elements connecting diverse modes and forms of production, whose origin reflects the unequal state of economic development in the two metropolitan centres.The empirical demonstration rests on the census data of 1851–1852 and on the complementary information drawn from the works of historians.

A characterization of the Poisson process

1974 ◽  
Vol 11 (1) ◽  
pp. 72-85 ◽  
Author(s):  
S. M. Samuels
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complete Proof ◽  
Necessary And Sufficient

Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.

Separability criteria based on the Bloch representation of density matrices

2007 ◽  
Vol 7 (7) ◽  
pp. 624-638
Author(s):  
J. de Vicente
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Pure Product ◽  
Separability Problem ◽  
Arbitrary Dimensions ◽  
Necessary And Sufficient ◽  
Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

scholarly journals An Application of Spherical Geometry to Hyperkähler Slices

2020 ◽  
pp. 1-30
Author(s):  
Peter Crooks ◽  
Maarten van Pruijssen
Keyword(s):  
Sufficient Conditions ◽  
Compact Subgroup ◽  
Spherical Geometry ◽  
Spherical Subgroup ◽  
Cotangent Bundles ◽  
Complex Semisimple ◽  
Reductive Subgroup ◽  
Necessary And Sufficient

Abstract This work is concerned with Bielawski’s hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group  $G$ , a reductive subgroup $H\subseteq G$ , and a Slodowy slice $S\subseteq \mathfrak{g}:=\text{Lie}(G)$ , defining it to be the hyperkähler quotient of $T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of  $G$ . This hyperkähler slice is empty in some of the most elementary cases (e.g., when $S$ is regular and $(G,H)=(\text{SL}_{n+1},\text{GL}_{n})$ , $n\geqslant 3$ ), prompting us to seek necessary and sufficient conditions for non-emptiness. We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$ -regularity of $(G,H)$ . This $\mathfrak{a}$ -regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$ . We also provide a classification of the $\mathfrak{a}$ -regular pairs $(G,H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces.

Characterization of Low-pass Filters on Local Fields of Positive Characteristic

2016 ◽  
Vol 59 (3) ◽  
pp. 528-541 ◽  
Author(s):  
Qaiser Jahan
Keyword(s):  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass ◽  
Necessary And Sufficient ◽  
Low Pass Filters ◽  
The Scaling Function

AbstractIn this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field K of positive characteristic associated with the scaling function for multiresolution analysis of L2(K). We use probability and martingale methods to provide such a characterization.

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