scholarly journals On the Exponent of the NK0-Groups of Virtually Infinite Cyclic Groups

2002 ◽  
Vol 45 (2) ◽  
pp. 180-195 ◽  
Author(s):  
Francis X. Connolly ◽  
Stratos Prassidis
Keyword(s):  
Large Class ◽  
The Other ◽  
Main Difficulty ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Other Hand

AbstractIt is known that the K-theory of a large class of groups can be computed from the K-theory of their virtually infinite cyclic subgroups. On the other hand, Nil-groups appear to be the obstacle in calculations involving the K-theory of the latter. The main difficulty in the calculation of Nil-groups is that they are infinitely generated when they do not vanish. We develop methods for computing the exponent of NK0-groups that appear in the calculation of the K0-groups of virtually infinite cyclic groups.

Arithmetic functions monotonic at consecutive arguments

2014 ◽  
Vol 51 (2) ◽  
pp. 155-164
Author(s):  
Jean-Marie Koninck ◽  
Florian Luca
Keyword(s):  
Large Class ◽  
The Other ◽  
Arithmetic Functions ◽  
Arbitrary Integer ◽  
Other Hand ◽  
Positive Integers

For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities f(n + 1) < f(n + 2) < … < f(n + κ) holds for in-finitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying σ2(n − 1) < σ2 < σ2(n + 1), where σ2(n) = ∑d|nd2. On the other hand, we prove that for the function f(n) := ∑p|np2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.

scholarly journals Explicit isogenies in quadratic time in any characteristic

2016 ◽  
Vol 19 (A) ◽  
pp. 267-282 ◽  
Author(s):  
Luca De Feo ◽  
Cyril Hugounenq ◽  
Jérôme Plût ◽  
Éric Schost
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Large Class ◽  
The Other ◽  
Base Field ◽  
Other Hand ◽  
Previous Algorithm ◽  
The Cost ◽  
Interesting Alternative ◽  
Refined Version

Consider two ordinary elliptic curves$E,E^{\prime }$defined over a finite field$\mathbb{F}_{q}$, and suppose that there exists an isogeny$\unicode[STIX]{x1D713}$between$E$and$E^{\prime }$. We propose an algorithm that determines$\unicode[STIX]{x1D713}$from the knowledge of$E$,$E^{\prime }$and of its degree$r$, by using the structure of the$\ell$-torsion of the curves (where $\ell$ is a prime different from the characteristic $p$of the base field). Our approach is inspired by a previous algorithm due to Couveignes, which involved computations using the$p$-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of $\tilde{O} (r^{2})p^{O(1)}$base field operations. On the other hand, the cost of our algorithm is$\tilde{O} (r^{2})\log (q)^{O(1)}$, for a large class of inputs; this makes it an interesting alternative for the medium- and large-characteristic cases.

Situational Irony

2017 ◽  
Author(s):  
G. R. F. Ferrari
Keyword(s):  
Large Class ◽  
The Other ◽  
Other Hand

When a situation feels like a story, we call it ironic. It seems audience-directed, though we know it is not. On the other hand, a large class of situations that strike people as ironic are simple incongruities, with no apparent connection to how stories work. This chapter therefore proposes a model that applies equally to ironies of simple incongruity and story-like ironies of peripety. It is the play of the mundane against the surprising (and not surprise alone) that makes a situation ironic. This characteristic of situational irony bears comparison with how, in stories, surprises that a plot holds in store must emerge from a plausible nexus of events, rather than springing from nowhere. It turns out, however, that ironies of simple incongruity manifest the same pattern: what makes their incongruity exquisite is that it shows up against a mundane background. Situational ironies ‘speak’ to their audiences, then, in the manner of a three-quarters-on intimation.

LOWER TYPE DIMENSIONS OF SOME MORAN SETS

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950082
Author(s):  
JIAOJIAO YANG
Keyword(s):  
Large Class ◽  
The Other ◽  
Moran Set ◽  
Lower Type ◽  
Fractal Sets ◽  
Lipschitz Mappings ◽  
Other Hand ◽  
Lower Spectrum

In this paper, we discuss the lower type dimensions for some Moran sets. On one hand, for Moran set [Formula: see text] with [Formula: see text], we prove that [Formula: see text], where the supremum is taken over all quasi-Lipschitz mappings [Formula: see text]. On the other hand, we obtain the lower spectrum formula for homogeneous Moran sets. In the proof a lower spectrum formula for a large class of fractal sets is established.

scholarly journals On the minimal ramification problem for ℓ-groups

2010 ◽  
Vol 146 (3) ◽  
pp. 599-606 ◽  
Author(s):  
Hershy Kisilevsky ◽  
Jack Sonn
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Fields ◽  
Symmetric Groups ◽  
Wreath Products ◽  
The Other ◽  
Classical Groups ◽  
Direct Products ◽  
Cyclic Groups ◽  
Other Hand

AbstractLet ℓ be a prime number. It is not known whether every finite ℓ-group of rank n≥1 can be realized as a Galois group over ${\Bbb Q}$ with no more than n ramified primes. We prove that this can be done for the (minimal) family of finite ℓ-groups which contains all the cyclic groups of ℓ-power order and is closed under direct products, (regular) wreath products and rank-preserving homomorphic images. This family contains the Sylow ℓ-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not ℓ. On the other hand, it does not contain all finite ℓ-groups.

scholarly journals BOUNDED GENERATION AND LINEAR GROUPS

2003 ◽  
Vol 13 (04) ◽  
pp. 401-413 ◽  
Author(s):  
MIKLÓS ABÉRT ◽  
ALEXANDER LUBOTZKY ◽  
LÁSZLÓ PYBER
Keyword(s):  
Positive Characteristic ◽  
Structure Theorem ◽  
The Other ◽  
Irreducible Representations ◽  
Linear Groups ◽  
Profinite Completion ◽  
Cyclic Subgroups ◽  
Other Hand ◽  
Bounded Generation

A group Γ is called boundedly generated (BG) if it is the set-theoretic product of finitely many cyclic subgroups. We show that a BG group has only abelian by finite images in positive characteristic representations.We use this to reprove and generalize Rapinchuk's theorem by showing that a BG group with the FAb property has only finitely many irreducible representations in any given dimension over any field. We also give a structure theorem for the profinite completion G of such a group Γ.On the other hand, we exhibit boundedly generated profinite FAb groups which do not satisfy this structure theorem.

scholarly journals Branes, quivers and wave-functions

2021 ◽  
Vol 10 (2) ◽  
Author(s):  
Taro Kimura ◽  
Milosz Panfil ◽  
Yuji Sugimoto ◽  
Piotr Sułkowski
Keyword(s):  
Large Class ◽  
Wave Functions ◽  
The Other ◽  
Partition Functions ◽  
Other Hand ◽  
Generating Series

We consider a large class of branes in toric strip geometries, both non-periodic and periodic ones. For a fixed background geometry we show that partition functions for such branes can be reinterpreted, on one hand, as quiver generating series, and on the other hand as wave-functions in various polarizations. We determine operations on quivers, as well as SL(2,\mathbb{Z})SL(2,ℤ) transformations, which correspond to changing positions of these branes. Our results prove integrality of BPS multiplicities associated to this class of branes, reveal how they transform under changes of polarization, and imply all other properties of brane amplitudes that follow from the relation to quivers.

scholarly journals A Positive Answer for 3IM+1CM Problem with a General Difference Polynomial

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Huicai Xu ◽  
Shugui Kang ◽  
Qingcai Zhang
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Analytic Functions ◽  
Positive Answer ◽  
The Other ◽  
Main Difficulty ◽  
New Methods ◽  
Other Hand ◽  
Difference Polynomial ◽  
General Difference

In this paper, the 3IM+1CM theorem with a general difference polynomial L z , f will be established by using new methods and technologies. Note that the obtained result is valid when the sum of the coefficient of L z , f is equal to zero or not. Thus, the theorem with the condition that the sum of the coefficient of L z , f is equal to zero is also a good extension for recent results. However, it is new for the case that the sum of the coefficient of L z , f is not equal to zero. In fact, the main difficulty of proof is also from this case, which causes the traditional theorem invalid. On the other hand, it is more interesting that the nonconstant finite-order meromorphic function f can be exactly expressed for the case f ≡ − L z , f . Furthermore, the sharpness of our conditions and the existence of the main result are illustrated by examples. In particular, the main result is also valid for the discrete analytic functions.

scholarly journals Products of locally cyclic groups

2021 ◽  
Author(s):  
Bernhard Amberg ◽  
Yaroslav Sysak
Keyword(s):  
Finite Number ◽  
Finite Groups ◽  
Complete Classification ◽  
The Other ◽  
Supersoluble Group ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Prüfer Rank ◽  
Torsion Free

AbstractWe consider groups of the form $${G} = {AB}$$ G = AB with two locally cyclic subgroups A and B. The structure of these groups is determined in the cases when A and B are both periodic or when one of them is periodic and the other is not. Together with a previous study of the case where A and B are torsion-free, this gives a complete classification of all groups that are the product of two locally cyclic subgroups. As an application, it is shown that the Prüfer rank of a periodic product of two locally cyclic subgroups does not exceed 3, and this bound is sharp. It is also proved that a product of a finite number of pairwise permutable periodic locally cyclic subgroups is a locally supersoluble group. This generalizes a well-known theorem of B. Huppert for finite groups.

Sprung Rhythm

PMLA ◽  
1959 ◽  
Vol 74 (4-Part1) ◽  
pp. 418-425
Author(s):  
Paull F. Baum
Keyword(s):  
Large Class ◽  
The Other ◽  
English Poetry ◽  
Strong Case ◽  
Other Hand ◽  
The Right

In a long essay apropos of Yeats's Oxford Book of Modern Verse, G. M. Young works himself around to modern versification and to G. M. Hopkins. What should have been a development, he says, has turned out to be a catastrophe. “It is common, too, I find to look on Hopkins as the chief legislator of the new mode. For Hopkins as a poet I have the greatest admiration, but his theories on metre seem to me to be as demonstrably wrong as those of any speculator who has ever led a multitude into the wilderness to perish. Unfortunately they have been used as a justification for the cacophonies which naturally result when the metrically deaf write verse, and the metrically deaf are a very large class.” Young is not sure of the right meaning of counterpoint, but “using it as Hopkins did,” he says: “You must counterpoint to avoid monotony, but you must not silence the pattern. You can only work within limits, and if you go beyond them the result is prose. It is no use saying like the Pharisees: ‘It is Corban, a sprung rhythm’: it will not be verse.” That Young uses Hopkins as a stick to beat the moderns is merely amusing; they had their own bit of chaos and needed a form in which to express it. But when he casts a doubt on Hopkins' theories on meter and forth-rightly accuses Hopkins of “an ignorance of his subject so profound that he was not aware there was anything to know”—that is, as he would say, certainly a bone for the dog. And when, on the other hand, Sir Herbert Read, lending his authority to the defense, says that “Hopkins shows that he understood the technique of English poetry as no poet since Dryden had understood it,” it is time to apply a few tests. A beginning was made ten years ago by Yvor Winters, rather more extreme than what follows here. But granted that both Young and Winters have a strong case, it still seems best to deal patiently with Hopkins, if only because he was an impetuous novice in prosody, too impatient to think his theories through before he began to explain them.

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