Finite 𝒟C-groups

2022 ◽  
Author(s):  
Dandan Zhang ◽  
Haipeng Qu ◽  
Yanfeng Luo
Keyword(s):  
Semidirect Product ◽  
Upper Bound ◽  
Set Inclusion ◽  
A Chain

Let [Formula: see text] be a group and [Formula: see text]. [Formula: see text] is said to be a [Formula: see text]-group if [Formula: see text] is a chain under set inclusion. In this paper, we prove that a finite [Formula: see text]-group is a semidirect product of a Sylow [Formula: see text]-subgroup and an abelian [Formula: see text]-subgroup. For the case of [Formula: see text] being a finite [Formula: see text]-group, we obtain an optimal upper bound of [Formula: see text] for a [Formula: see text] [Formula: see text]-group [Formula: see text]. We also prove that a [Formula: see text] [Formula: see text]-group is metabelian when [Formula: see text] and provide an example showing that a non-abelian [Formula: see text] [Formula: see text]-group is not necessarily metabelian when [Formula: see text]. In particular, [Formula: see text] [Formula: see text]-groups are characterized.

scholarly journals On the explosion of chain-thermal reactions

1982 ◽  
Vol 24 (2) ◽  
pp. 194-202 ◽  
Author(s):  
R. O. Ayeni
Keyword(s):  
Activation Energy ◽  
Asymptotic Analysis ◽  
Upper Bound ◽  
Homogeneous Model ◽  
Chain Reaction ◽  
Isothermal Reaction ◽  
Thermal Reactions ◽  
Spatially Homogeneous ◽  
Branched Chain ◽  
A Chain

AbstractA chain reaction of oxygen (reactant) and hydrogen (active intermediary) with mtrosyl chloride (sensitizer) as a catalyst may be modelled mathematically as a non-isothermal reaction. In this paper we present an asymptotic analysis of a spatially homogeneous model of a non-isothermal branched-chain reaction. Of particular interest is the so-called explosion time and we provide an upper bound for it as a function of the activation energy which can vary over all positive values. We also establish a bound on the temperature when the activation energy is finite.

The Family of Lodato Proximities Compatible With a Given Topological Space

1974 ◽  
Vol 26 (02) ◽  
pp. 388-404 ◽  
Author(s):  
W. J. Thron ◽  
R. H. Warren
Keyword(s):  
Lower Bound ◽  
Topological Space ◽  
Distributive Lattice ◽  
Structural Properties ◽  
Upper Bound ◽  
Set Inclusion ◽  
The Family

Let (X, ) be a topological space. By we denote the family of all Lodato proximities on X which induce . We show that is a complete distributive lattice under set inclusion as ordering. Greatest lower bound and least upper bound are characterized. A number of techniques for constructing elements of are developed. By means of one of these constructions, all covers of any member of can be obtained. Several examples are given which relate to the lattice of all compatible proximities of Čech and the family of all compatible proximities of Efremovič. The paper concludes with a chart which summarizes many of the structural properties of , and .

Some Properties of the Lattice of Path Sets of a Connected Graph

2018 ◽  
Vol 7 (4.10) ◽  
pp. 310
Author(s):  
Girishkumara R ◽  
Lavanya S
Keyword(s):  
Connected Graph ◽  
Inclusion Relation ◽  
Single Vertex ◽  
Partially Ordered ◽  
Cut Vertex ◽  
Set Inclusion ◽  
A Chain

It is known that the set of all path sets of a finite connected graph G together with empty set partially ordered by set inclusion relation forms a lattice denoted by PATH(G). In this paper we studied some properties of PATH(G). In fact, it has been shown that an element of PATH(G) is doubly irreducible if and only if it contains a single vertex which is not a cut vertex of G. Also it is proved that PATH(G) is planar if and only if G is a chain of three or more blocks.  

scholarly journals Brill-Noether theory for curves of a fixed gonality

2021 ◽  
Vol 9 ◽  
Author(s):  
David Jensen ◽  
Dhruv Ranganathan
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Closed Formula ◽  
Noether Theorem ◽  
Stable Maps ◽  
Technical Result ◽  
General Curve ◽  
Divisor Theory ◽  
Arbitrary Genus ◽  
A Chain

Abstract We prove a generalisation of the Brill-Noether theorem for the variety of special divisors $W^r_d(C)$ on a general curve C of prescribed gonality. Our main theorem gives a closed formula for the dimension of $W^r_d(C)$ . We build on previous work of Pflueger, who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne. A crucial technical result is a new realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of Speyer on genus $1$ curves to arbitrary genus.

scholarly journals The EA-Dimension of a Commutative Ring

ISRN Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Mosbah Eljeri
Keyword(s):  
Commutative Ring ◽  
Upper Bound ◽  
Zero Divisors ◽  
A Chain

An elementary annihilator of a ring A is an annihilator that has the form (0:a)A; a∈R∖(0). We define the elementary annihilator dimension of the ring A, denoted by EAdim(A), to be the upper bound of the set of all integers n such that there is a chain (0:a0)⊂⋯⊂(0:an) of annihilators of A. We use this dimension to characterize some zero-divisors graphs.

TYPE-A SEMIGROUPS WHOSE FULL SUBSEMIGROUPS FORM A CHAIN UNDER SET INCLUSION

2008 ◽  
Vol 01 (03) ◽  
pp. 359-367 ◽  
Author(s):  
Xiaojiang Guo ◽  
Fusheng Huang ◽  
K. P. Shum
Keyword(s):  
Type A ◽  
Inverse Semigroups ◽  
Set Inclusion ◽  
A Chain ◽  
Characterization Theorems ◽  
Inverse Subsemigroups

A semigroup S is called a ∇-semigroup if the set of all its full subsemigroups forms a chain under set inclusion. In this paper, we investigate some properties of such kind of semigroups and establish several characterization theorems of type-A ∇-semigroups. Our theorems generalize the known results of P. R. Jones obtained in 1981 on inverse semigroups whose full inverse subsemigroups form a chain.

scholarly journals On Saturated $k$-Sperner Systems

10.37236/4136 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Natasha Morrison ◽  
Jonathan A. Noel ◽  
Alex Scott
Keyword(s):  
Lower Bound ◽  
Minimum Size ◽  
Set Inclusion ◽  
Sperner System ◽  
Polynomial Factor ◽  
A Chain

Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\varepsilon>0$ such that for every $k$ and $|X| \geq n_0(k)$ there exists a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ with cardinality at most $2^{(1-\varepsilon)k}$.A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an oversaturated $k$-Sperner system if, for every $S\in\mathcal{P}(X)\setminus\mathcal{F}$, $\mathcal{F}\cup\{S\}$ contains more chains of length $k+1$ than $\mathcal{F}$. Gerbner et al. proved that, if $|X|\geq k$, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements. We show that if $|X|\geq k^2+k$, then the lower bound is best possible, up to a polynomial factor.

scholarly journals On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

2021 ◽  
Author(s):  
Angkana Rüland ◽  
Antonio Tribuzio
Keyword(s):  
Sobolev Space ◽  
Upper Bound ◽  
Scaling Law ◽  
Approximate Solutions ◽  
Bootstrap Argument ◽  
Perturbation Problem ◽  
Energy Scaling ◽  
A Chain ◽  
Negative Sobolev Space ◽  
Numer Math

AbstractIn this article we derive an (almost) optimal scaling law for a singular perturbation problem associated with the Tartar square. As in Winter (Eur J Appl Math 8(2):185–207, 1997), Chipot (Numer Math 83(3):325–352, 1999), our upper bound quantifies the well-known construction which is used in the literature to prove the flexibility of the Tartar square in the sense of the flexibility of approximate solutions to the differential inclusion. The main novelty of our article is the derivation of an (up to logarithmic powers matching) ansatz free lower bound which relies on a bootstrap argument in Fourier space and is related to a quantification of the interaction of a nonlinearity and a negative Sobolev space in the form of “a chain rule in a negative Sobolev space”. Both the lower and the upper bound arguments give evidence of the involved “infinite order of lamination”.

Observation of Atom Rows in Thorium Pyromellitate Using a Field Emission STEM

1977 ◽  
Vol 35 ◽  
pp. 80-81
Author(s):  
H. Todokoro ◽  
S. Nomura ◽  
T. Komoda
Keyword(s):  
Field Emission ◽  
Amorphous Carbon ◽  
Carbon Film ◽  
Dark Field Image ◽  
Dark Field ◽  
Amorphous Carbon Film ◽  
Atomic Size ◽  
Wide Range ◽  
A Chain

It is interesting to observe polymers at atomic size resolution. Some works have been reported for thorium pyromellitate by using a STEM (1), or a CTEM (2,3). The results showed that this polymer forms a chain in which thorium atoms are arranged. However, the distance between adjacent thorium atoms varies over a wide range (0.4-1.3nm) according to the different authors.The present authors have also observed thorium pyromellitate specimens by means of a field emission STEM, described in reference 4. The specimen was prepared by placing a drop of thorium pyromellitate in 10-3 CH3OH solution onto an amorphous carbon film about 2nm thick. The dark field image is shown in Fig. 1A. Thorium atoms are clearly observed as regular atom rows having a spacing of 0.85nm. This lattice gradually deteriorated by successive observations. The image changed to granular structures, as shown in Fig. 1B, which was taken after four scanning frames.

Time-Resolved Cryo-Electron Microscopy of Oscillating Microtubules

1990 ◽  
Vol 48 (1) ◽  
pp. 502-503
Author(s):  
Eva-Maria Mandelkow ◽  
Ron Milligan
Keyword(s):  
Electron Microscopy ◽  
Dynamic Instability ◽  
Entire Solution ◽  
Reaction Scheme ◽  
Time Resolved ◽  
X Ray ◽  
X Ray Scattering ◽  
A Chain ◽  
Ray Scattering

Microtubules form part of the cytoskeleton of eukaryotic cells. They are hollow libers of about 25 nm diameter made up of 13 protofilaments, each of which consists of a chain of heterodimers of α-and β-tubulin. Microtubules can be assembled in vitro at 37°C in the presence of GTP which is hydrolyzed during the reaction, and they are disassembled at 4°C. In contrast to most other polymers microtubules show the behavior of “dynamic instability”, i.e. they can switch between phases of growth and phases of shrinkage, even at an overall steady state [1]. In certain conditions an entire solution can be synchronized, leading to autonomous oscillations in the degree of assembly which can be observed by X-ray scattering (Fig. 1), light scattering, or electron microscopy [2-5]. In addition such solutions are capable of generating spontaneous spatial patterns [6].In an earlier study we have analyzed the structure of microtubules and their cold-induced disassembly by cryo-EM [7]. One result was that disassembly takes place by loss of protofilament fragments (tubulin oligomers) which fray apart at the microtubule ends. We also looked at microtubule oscillations by time-resolved X-ray scattering and proposed a reaction scheme [4] which involves a cyclic interconversion of tubulin, microtubules, and oligomers (Fig. 2). The present study was undertaken to answer two questions: (a) What is the nature of the oscillations as seen by time-resolved cryo-EM? (b) Do microtubules disassemble by fraying protofilament fragments during oscillations at 37°C?

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