scholarly journals Mil nor's problem on the growth of groups and its consequences

2014 ◽  
Author(s):  
Rostislav Grigorchuk
Keyword(s):  
Asymptotic Behavior ◽  
Random Walks ◽  
Exponential Growth ◽  
Main Part ◽  
Polynomial Growth ◽  
Open Problems ◽  
Growth Of Groups ◽  
Significant Interest ◽  
Groups Of Polynomial Growth

This chapter presents a survey of results related to John Milnor's problem on group growth. The notion of group growth first appeared in 1955 in a paper of A. S. Schwarz, but it remained virtually unnoticed for over a decade. The situation changed after Milnor's papers from 1968, which sparked significant interest in this area. Particularly influential were two problems raised in these papers: the characterization of groups of polynomial growth and the question of the existence of groups of intermediate growth. The chapter discusses the cases of polynomial growth and exponential but not uniformly exponential growth; the main part of this chapter is devoted to the intermediate (between polynomial and exponential) growth case. A number of related topics (growth of manifolds, amenability, asymptotic behavior of random walks) are considered, and a number of open problems are suggested.

scholarly journals Combining Losing Games into a Winning Game

2019 ◽  
Vol 18 (01) ◽  
pp. 1950003 ◽  
Author(s):  
Bruno Rémillard ◽  
Jean Vaillancourt
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Random Walks ◽  
Random Environment ◽  
Regime Switching ◽  
Random Environments ◽  
Full Characterization ◽  
Parrondo’S Paradox ◽  
Random Subspaces

Parrondo’s paradox is extended to regime switching random walks in random environments. The paradoxical behavior of the resulting random walk is explained by the effect of the random environment. Full characterization of the asymptotic behavior is achieved in terms of the dimensions of some random subspaces occurring in Oseledec’s theorem. The regime switching mechanism gives our models a richer and more complex asymptotic behavior than the simple random walks in random environments appearing in the literature, in terms of transience and recurrence.

Countable Amenable Identity Excluding Groups

2004 ◽  
Vol 47 (2) ◽  
pp. 215-228 ◽  
Author(s):  
Wojciech Jaworski
Keyword(s):  
Probability Measure ◽  
Unitary Representation ◽  
Discrete Group ◽  
Polynomial Growth ◽  
Irreducible Unitary Representation ◽  
Compact Groups ◽  
Groups Of Polynomial Growth ◽  
Identity Representation ◽  
Dimensional Identity

AbstractA discrete group G is called identity excluding if the only irreducible unitary representation of G which weakly contains the 1-dimensional identity representation is the 1-dimensional identity representation itself. Given a unitary representation π of G and a probability measure μ on G, let Pμ denote the μ-average ∫π(g)μ(dg). The goal of this article is twofold: (1) to study the asymptotic behaviour of the powers , and (2) to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure μ on an identity excluding group and every unitary representation π there exists and orthogonal projection Eμ onto a π-invariant subspace such that for every a ∈ supp μ. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of FC-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic.

scholarly journals Electromagnetic Gauge Choice for Scattering of Schrödinger Particle

2019 ◽  
Vol 20 (11) ◽  
pp. 3633-3650
Author(s):  
Andrzej Herdegen
Keyword(s):  
Asymptotic Behavior ◽  
External Electromagnetic Field ◽  
Asymptotic Completeness ◽  
Free Evolution ◽  
Wave Operators ◽  
Radiation Fields ◽  
Mechanical Evolution ◽  
Scattering Wave ◽  
Free Fields

Abstract We consider a Schrödinger particle placed in an external electromagnetic field of the form typical for scattering settings in the field theory: $$F=F^\mathrm {ret}+F^\mathrm {in}=F^\mathrm {adv}+F^\mathrm {out}$$ F = F ret + F in = F adv + F out , where the current producing $$F^{\mathrm {ret}/\mathrm {adv}}$$ F ret / adv has the past and future asymptotes homogeneous of degree $$-3$$ - 3 , and the free fields $$F^{\mathrm {in}/\mathrm {out}}$$ F in / out are radiation fields produced by currents with similar asymptotic behavior. We show that with appropriate choice of electromagnetic gauge the particle has ‘in’ and ‘out’ states reached with no further modification of the asymptotic dynamics. We use a special quantum mechanical evolution ‘picture’ in which the free evolution operator has well-defined limits for $$t\rightarrow \pm \infty $$ t → ± ∞ , and thus the scattering wave operators do not need the free evolution counteraction. The existence of wave operators in this setting is established, but the proof of asymptotic completeness is not complete: more precise characterization of the asymptotic behavior of the particle for $$|\mathbf {x}|=|t|$$ | x | = | t | would be needed.

On the number of times where a simple random walk reaches its maximum

1992 ◽  
Vol 29 (02) ◽  
pp. 305-312 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Probability And Statistics

Let Qn denote the number of times where a simple random walk reaches its maximum, where the random walk starts at the origin and returns to the origin after 2n steps. Such random walks play an important role in probability and statistics. In this paper the distribution and the moments of Qn , are considered and their asymptotic behavior is studied.

scholarly journals Secret Sharing Schemes on Sparse Homogeneous Access Structures with Rank Three

10.37236/1825 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Jaume Martí-Farré ◽  
Carles Padró
Keyword(s):  
Secret Sharing ◽  
Access Structure ◽  
Open Problems ◽  
Access Structures ◽  
The Family ◽  
Ideal Access Structures ◽  
Optimal Information ◽  
Made In ◽  
The Ideal

One of the main open problems in secret sharing is the characterization of the ideal access structures. This problem has been studied for several families of access structures with similar results. Namely, in all these families, the ideal access structures coincide with the vector space ones and, besides, the optimal information rate of a non-ideal access structure is at most $2/3$. An access structure is said to be $r$-homogeneous if there are exactly $r$ participants in every minimal qualified subset. A first approach to the characterization of the ideal $3$-homogeneous access structures is made in this paper. We show that the results in the previously studied families can not be directly generalized to this one. Nevertheless, we prove that the equivalences above apply to the family of the sparse $3$-homogeneous access structures, that is, those in which any subset of four participants contains at most two minimal qualified subsets. Besides, we give a complete description of the ideal sparse $3$-homogeneous access structures.

scholarly journals Evidence that the TRPV1 S1-S4 Membrane Domain Contributes to Thermosensing

2019 ◽  
Author(s):  
Minjoo Kim ◽  
Nicholas J. Sisco ◽  
Jacob K. Hilton ◽  
Camila M. Montano ◽  
Manuel A. Castro ◽  
...  
Keyword(s):  
Conformational Changes ◽  
Solution Nmr ◽  
Membrane Domain ◽  
Temperature Dependent ◽  
Nmr Characterization ◽  
Significant Interest ◽  
Pore Domain ◽  
Coupling Mechanisms

AbstractSensing and responding to temperature is crucial in biology. The TRPV1 ion channel is a well-studied heat-sensing receptor that is also activated by vanilloid compounds including capsaicin. Despite significant interest, the molecular underpinnings of thermosensing have remained elusive. The TRPV1 S1-S4 membrane domain couples chemical ligand binding to the pore domain during channel gating. However, the role of the S1-S4 domain in thermosensing is unclear. Evaluation of the isolated human TRPV1 S1-S4 domain by solution NMR, Far-UV CD, and intrinsic fluorescence shows that this domain undergoes a non-denaturing temperature-dependent transition with a high thermosensitivity. Further NMR characterization of the temperature-dependent conformational changes suggests the contribution of the S1-S4 domain to thermosensing shares features with known coupling mechanisms between this domain with ligand and pH activation. Taken together, this study shows that the TRPV1 S1-S4 domain contributes to TRPV1 temperature-dependent activation.

scholarly journals Besov algebras on Lie groups of polynomial growth

2012 ◽  
Vol 212 (2) ◽  
pp. 119-139 ◽  
Author(s):  
Isabelle Gallagher ◽  
Yannick Sire
Keyword(s):  
Lie Groups ◽  
Polynomial Growth ◽  
Groups Of Polynomial Growth

On Global Inverse Theorems of Szász and Baskakov Operators

1979 ◽  
Vol 31 (2) ◽  
pp. 255-263 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Baskakov Operators ◽  
Inverse Theorems ◽  
Image Position ◽  
Necessary And Sufficient

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

scholarly journals Local and Nonlocal Singular Liouville Equations in Euclidean Spaces

2019 ◽  
Author(s):  
Ali Hyder ◽  
Gabriele Mancini ◽  
Luca Martinazzi
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Finite Volume ◽  
Existence Of Solutions ◽  
Euclidean Spaces ◽  
Open Problems ◽  
Supercritical Regime

AbstractWe study the metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation \begin{equation*} (-\Delta)^{\frac{n}{2}}w=e^{nw}-c\delta_{0} \ \textrm{on}\ {\mathbb{R}}^n, \end{equation*}under a finite volume condition. We analyze the asymptotic behavior at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.

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