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From resolvent bounds to semigroup bounds  [PDF]
Bernard Helffer,Johannes Sjoestrand
Mathematics , 2010,
Abstract: The purpose of this note is to revisit the proof of the Gearhardt-Pr\"uss-Hwang-Greiner theorem for a semigroup S(t), following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on the norm of S(t) in terms of bounds on the resolvent of the generator.
Lower Bounds for Representation Growth  [PDF]
David A Craven
Mathematics , 2009,
Abstract: This article examines lower bounds for the representation growth of finitely generated (particularly profinite and pro-p) groups. It also considers the related question of understanding the maximal multiplicities of character degrees in finite groups, and in particular simple groups.
Linear growth for semigroups which are disjoint unions of finitely many copies of the free monogenic semigroup  [PDF]
Nabilah Abughazalah,Pavel Etingof
Mathematics , 2015,
Abstract: We show that every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) has linear growth. This implies that the the corresponding semigroup algebra is a PI algebra.
Semigroup graded algebras and codimension growth of graded polynomial identities  [PDF]
Alexey Sergeevich Gordienko
Mathematics , 2014,
Abstract: We show that if $T$ is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative $T$-graded algebra over a field of characteristic $0$ such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic $0$ with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if $T$ is a left or right zero band and the $T$-graded algebra is unital, or $T$ is a cancellative semigroup, then the $T$-graded algebra satisfies the graded analog of Amitsur's conjecture, i.e. there exists an integer graded PI-exponent. Moreover, in the first case it turns out that the ordinary and the graded PI-exponents coincide. In addition, we consider related problems on the structure of semigroup graded algebras.
New lower bounds on subgroup growth and homology growth  [PDF]
Marc Lackenby
Mathematics , 2005, DOI: 10.1112/plms/pdn032
Abstract: We establish new strong lower bounds on the (subnormal) subgroup growth of a large class of groups. This includes the fundamental groups of all finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic groups. The lower bound is nearly exponential, which should be compared with the fastest possible subgroup growth of any finitely generated group. This is achieved by free non-abelian groups and is slightly faster than exponential. As a consequence, we obtain good estimates on the number of covering spaces of a hyperbolic 3-manifold with given covering degree. We also obtain slightly weaker information on the number of covering spaces of closed 4-manifolds with non-positive Euler characteristic. The results on subgroup growth follow from a new theorem which places lower bounds on the rank of the first homology (with mod p coefficients) of certain subgroups of a group. This is proved using a topological argument.
Lower bounds of Growth of Hopf algebras  [PDF]
D. -G. Wang,J. J. Zhang,G. Zhuang
Mathematics , 2011,
Abstract: Some lower bounds of GK-dimension of Hopf algebras are given.
The growth of a C_0-semigroup characterised by its cogenerator  [PDF]
Tanja Eisner,Hans Zwart
Mathematics , 2008,
Abstract: We characterise contractivity, boundedness and polynomial boundedness for a C_0-semigroup on a Banach space in terms of its cogenerator V (or the Cayley transform of the generator) or its resolvent. In particular, we extend results of Gomilko and Brenner, Thomee and show that polynomial boundedness of a semigroup implies polynomial boundedness of its cogenerator. As is shown by an example, the result is optimal. For analytic semigroups we show that the converse holds, i.e., polynomial boundedness of the cogenerators implies polynomial boundedness of the semigroup. In addition, we show by simple examples in (C^2,\|\cdot\|_p), p \neq 2, that our results on the characterisation of contractivity are sharp. These examples also show that the famous Foias-Sz.-Nagy theorem on cogenerators of contraction semigroups on Hilbert spaces fails in (C^2,\|\cdot\|_p) for p\neq 2.
Linear bounds on growth of associated primes  [PDF]
Karen E. Smith,Irena Swanson
Mathematics , 2002,
Abstract: We find explicit bounds on the primary components and on the Castelnuovo-Mumford regularity of powers of monomial ideals. We also analyze the primary decompositions of Katzman's example.
Volume growth and bounds for the essential spectrum for Dirichlet forms  [PDF]
Sebastian Haeseler,Matthias Keller,Rados?aw K. Wojciechowski
Mathematics , 2012, DOI: 10.1112/jlms/jdt029
Abstract: We consider operators arising from regular Dirichlet forms with vanishing killing term. We give bounds for the bottom of the (essential) spectrum in terms of exponential volume growth with respect to an intrinsic metric. As special cases we discuss operators on graphs. When the volume growth is measured in the natural graph distance (which is not an intrinsic metric) we discuss the threshold for positivity of the bottom of the spectrum and finiteness of the bottom of the essential spectrum of the (unbounded) graph Laplacian. This threshold is shown to lie at cubic polynomial growth.
Shape-dependent bounds on cell growth rates  [PDF]
Jonathan Landy
Quantitative Biology , 2013, DOI: 10.1209/0295-5075/105/68002
Abstract: I consider how cell shape and environmental geometry affect the rate of nutrient capture and the consequent maximum growth rate of a cell, focusing on rod-like species like \textit{E.\ coli}. Simple modeling immediately implies that it is the elongated profiles of such cells that allows for them to grow -- as observed -- at exponential rates in nutrient-rich media. Growth is strongly suppressed when nutrient capture is diffusion-limited: In three dimensions, the length is bounded by $\log L \lesssim t^{1/2}$, and in lower dimensions growth is algebraic. Similar bounds are easily obtained for other cell geometries, groups of cells, \textit{etc}. Fits of experimental growth curves to such bounds can be used to estimate various quantities of interest, including generalized metabolic rates.
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