Free products with amalgamation and HNN-extensions of uniformly exponential growth

We prove that, with rare exceptions, groups that are free products with amalgamation or HNN-extensions are of uniformly exponential growth. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 811–815, June, 2000.     

Uniform Growth of Polycyclic Groups

We prove that polycyclic groups have uniform exponential or polynomial growth.     

Large hyperbolic lattices

For a fundamental group of a compact orientable manifold, a condition is specified that is sufficient to guarantee the presence of a “virtual” epimorphism onto a free non-Abelian group. A consequence is deriving a strong Tits alternative. An arbitrary noncompact finitely generated discrete subgroup in PO(3, 1) either is large or is virtually Abelian. An application is provided to the problem of uniform exponential growth for lattices in a 3-dimensional hyperbolic space and of growth of Betti numbers for lattices in a hyperbolic n-dimensional space, where n is an odd number. Supported by RFBR (project No. 08-01-00067), by DFG grant Gr 627-11, and by Forschergruppe “Spektrale Analysis, asymptotical Verteilungen und stochastische Dynamiken,” Billfold University. (G. A. Noskov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 174–189, March–April, 2009.     

Attainability of the Exponent of Exponential Growth in Free Products of Cyclic Groups

In the paper, the set of exponents of exponential growth (growth exponents) for a finitely generated group with respect to all possible generators of this group is studied. It is proved that the greatest lower bound of this set is attained for the free products of a cyclic group of prime order and a free group of finite rank.     

Counting Growth Types of Automorphisms of Free Groups

Given an automorphism of a free group Fn, we consider the following invariants: e is the number of exponential strata (an upper bound for the number of different exponential growth rates of conjugacy classes); d is the maximal degree of polynomial growth of conjugacy classes; R is the rank of the fixed subgroup. We determine precisely which triples (e, d, R) may be realized by an automorphism of Fn. In particular, the inequality e ￡ \frac3n-24{{e \leq \frac{3n-2}{4}}} (due to Levitt–Lustig) always holds. In an appendix, we show that any conjugacy class grows like a polynomial times an exponential under iteration of the automorphism.     

Groups of Exponential Algebras

In this paper we study groups corresponding a free exponential algebra, a free product of exponential algebras and a HNN-extension of exponential algebras over a field F of characteristic 0.     

Growth tightness of free and amalgamated products

We show that every nontrivial free product, different from the infinite dihedral group, is growth tight with respect to any algebraic distance: that is, its exponential growth rate is strictly greater than the corresponding growth rate of any of its proper quotients. A similar property holds for the amalgamated product of residually finite groups over a finite subgroup. As a consequence, we provide examples of finitely generated groups of uniform exponential growth whose minimal growth is not realized by any generating set.     

Macpherson's chern classes of singular algebraic varieties

Sacerdote [Sa] has shown that the non-Abelian free groups satisfy precisely the same universal-existential sentences Th(F₂)??? in a first-order language Lo appropriate for group theory. It is shown that in every model of Th(F₂)??? the maximal Abelian subgroups are elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two classes of groups are interpolated between the non-Abelian locally free groups and Remeslennikov’s ?-free groups. These classes are the almost locally free groups and the quasi-locally free groups. In particular, the almost locally free groups are the models of Th(F₂)??? while the quasi-locally free groups are the ?-free groups with maximal Abelian subgroups elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two principal open questions at opposite ends of a spectrum are: (1.) Is every finitely generated almost locally free group free? (2.) Is every quasi-locally free group almost locally free? Examples abound of finitely generated quasi-locally free groups containing nontrivial torsion in their Abelianizations. The question of whether or not almost locally free groups have torsion free Abelianization is related to a bound in a free group on the number of factors needed to express certain elements of the derived group as a product of commutators     

Optimal Disturbance Growth in Watertable Flow

The linear stability of a liquid flow down an inclined plane is investigated. The equations governing the evolution of the disturbance are written in vector form where the dependent variables are the normal velocity and the normal vorticity. Similar to other shear flows, it is shown that there can be transient growth in the energy of a disturbance followed by an exponential decay although all eigenvalues predict decay only. Parameter studies reveal that the maximum amplification occur for waves with no streamwise dependence and with a spanwise wavenumber of (1). The mechanism involved in this growth is analyzed. A free surface parameter (S) can be identified that is related to the extent gravity and surface tension influence the free surface. A scaling of the equations is studied which revealed that the maximum transient growth scales with the Reynolds number as Re² if k2 S Re² is kept constant, where k is the absolute value of the wavenumber vector. For small values of S exponential growth of free-surface modes also exists. In general, however, we have found that for moderate times the transient growth dominates over the exponential growth and that its characteristics are similar to the transient growth found in other shear flows, e.g., plane Poiseuille flow.     

On Growth Types of Quotients of Coxeter Groups by Parabolic Subgroups

The principal objects studied in this note are infinite, non-affine Coxeter groups W. A well-known result of de la Harpe asserts that such groups have exponential growth. We study the growth type of quotients of W by parabolic subgroups and by a certain class of reflection subgroups. Our main result is that these quotients have exponential growth as well.     

Intermediate Growth of Solvable Lie Superalgebras

Finitely generated solvable Lie algebras have an intermediate growth between polynomial and exponential. Recently the second author suggested the scale to measure such an intermediate growth of Lie algebras. The growth was specified for solvable Lie algebras F(A ^q , k) with a finite number of generators k, and which are free with respect to a fixed solubility length q. Later, an application of generating functions allowed us to obtain more precise asymptotic. These results were obtained in the generality of polynilpotent Lie algebras. Now we consider the case of Lie superalgebras; we announce that main results and describe the methods. Our goal is to compute the growth for F(A ^q , m, k), the free solvable Lie superalgebra of length q with m even and k odd generators. The proof is based upon a precise formula of the generating function for this algebra obtained earlier. The result is obtained in the generality of free polynilpotent Lie superalgebras. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.     

Centers in domains with quadratic growth

Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.     

Growth of relatively hyperbolic groups

We show that a finitely generated group that is hyperbolic relative to a collection of proper subgroups either is virtually cyclic or has uniform exponential growth.

     

Multiple solutions for a class of fractional quasi-linear equations with critical exponential growth in ℝN

This paper deals with a class of non-local equations involving the fractional p-Laplacian, where the non-linear term is assumed to have critical exponential growth. More specifically, by applying variational methods together with a suitable Trudinger-Moser inequality for fractional Sobolev space, we obtain the existence of at least two positive weak solutions.     

Enumeration of algebras close to absolutely free algebras and binary trees

We study three classes of algebras: absolutely free algebras, free commutative non-associative, and free anti-commutative non-associative algebras. We study asymptotics of the growth for free algebras of these classes and for their subvarieties as well. Mainly, we study finitely generated algebras, also the codimension growth for varieties in theses classes is studied. For these purposes we use ordinary generating functions as well as exponential generating functions. The following subvarieties are studied in these classes: solvable, completely solvable, right-nilpotent, and completely right-nilpotent subvarieties. The obtained results are equivalent to an enumeration of binary labeled and unlabeled rooted trees that do not contain some forbidden subtrees. We enumerate these trees using generating functions. For solvable and right-nilpotent algebras the generating functions are algebraic. For completely solvable and completely right-nilpotent algebras the respective functions are rational. It is known that these three varieties of algebras satisfy Schreier's property, i.e., subalgebras of free algebras are free. For free groups, there is Schreier's formula for the rank of a subgroup of a free group. We find analogues of this formula for these varieties. They are written in terms of series. As an application, we study invariants of finite groups acting on absolutely free algebras.     

带临界指数增长的拟线性椭圆型方程的多解性

本文利用山路定理、Ekeland变分准则结合Trudinger-Moser不等式,证明了一类非线性项带临界指数增长的拟线性方程至少存在两个正的弱解.     

Solving four-dimensional surgery problems using controlled theory

In this paper, the controlled surgery sequence of Ranicki, Pedersen, and Quinn is applied to the solution of surgery problems in dimension four when the fundamental group is not known to be good. Our examples concern free non-Abelian fundamental groups, surface fundamental groups, and special knot groups. Using results from our earlier paper (joint with Spaggiari), we state a general result from which our examples follow. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 221–236, 2005.     

一种能够提高建模精度的含参变量函数变换及其应用

根据传统预测模型适宜低增长、近指数序列这一基本事实出发,提出用原始序列本身的级比偏差来比较低增长的程度,用原始序列所对应的级比序列的级比偏差比较原始序列接近指数序列的程度.并通过此种方法找到了一种含参变量函数变换缩小级比偏差的范围,从而提高了建模精度,又因为此种变换能够保证还原相对误差保持不变,以至于最终达到了提高建模精度的目的.并通过实例验证了方法的可行性及其结果的正确性、优越性.     

Numerical differentiation inspired by a formula of R.P. Boas

First, we briefly discuss three classes of numerical differentiation formulae, namely finite difference methods, the method of contour integration, and sampling methods. Then we turn to an interpolation formula of R.P. Boas for the first derivative of an entire function of exponential type bounded on the real line. This formula may be classified as a sampling method. We improve it in two ways by incorporating a Gaussian multiplier for speeding up convergence and by extending it to higher derivatives. For derivatives of order s, we arrive at a differentiation formula with N^′ nodes that applies to all entire functions of exponential type without any additional restriction on their growth on the real line. It has an error bound that converges to zero like e^-αN/N^m as N→∞, where α>0 and N^′=2N, m=3/2 for odd s while N^′=2N+1, m=5/2 for even s. Comparable known formulae have stronger hypotheses and, for the same α, they have m=1/2 only. We also deduce a direct (error-free) generalization of Boas’ formula (Corollary 5). Furthermore, we give a modification of the main result for functions analytic in a domain and consider an extension to non-analytic functions as well. Finally, we illustrate the power of the method by examples.     

Asymptotic behaviour of the loss probability of theM/G/1/K andG/M/1/K queues

This paper investigates the asymptotic behaviour of the loss probability of theM / G/1/K and G/M/1/K queues as the buffer size increases. It is shown that the loss probability approaches its limiting value, which depends on the offered load, with an exponential decay in essentially all cases. The value of the decay rate can be easily computed from the main queue parameters. Moreover, the close relation existing between the loss behaviour of the two examined queueing systems is highlighted and a duality concept is introduced. Finally some numerical examples are given to illustrate on the usefulness of the asymptotic approximation.