On strongly asymptotic lp spaces and minimality

Let 1 p and let X be a Banach space with a semi-normalizedstrongly asymptotic _p basis (e_i). If X is minimal and 1 p <2, then X is isomorphic to a subspace of _p. If X is minimaland 2 p < , or if X is complementably minimal and 1 p , then (e_i) is equivalent to the unit vector basis of _p (orc₀ if p = ).     

Stability and Asymptotic Stability of Functional-Differential Equations

We investigate asymptotic behaviour of solutions of the functional-differentialequation where f and g arelocally Lipschitz functions, C is a continuous matrix and thesmooth lag function obeys 0 (t) t for t 0. We transformthe equation into a delay equation with an infinity of delaysand use a theorem of Söderlind to derive sufficient conditionsfor stability and for asymptotic stability in the case lim_t (t) = . The situation is qualitatively different when lim_t (t) = * < and we outline stability conditions for thatcase by employing direct techniques.     

Construction of p-1 Irreducible Modules with Fundamental Highest Weight for the Symplectic Group in Characteristic p

Let K be a field and let V be a vector space of dimension 2mover K. Let V denote the exterior algebra of V and ^kV its kthexterior power for 0k2m. Let f be a non-degenerate alternatingbilinear form defined on VxV. The symplectic group Sp_2m(K) isthe group of all isometries of f and it acts as a group of vectorspace automorphisms on ^kV. In the case that K is algebraicallyclosed and 1km, it is known that ^kV contains a composition factorcorresponding to the fundamental weight _k of a root system oftype C_m. We shall refer to the irreducible module for Sp_2m(K)given by this composition factor as a fundamental module.     

Some Remarks on the Cone of Completely Positive Maps between Von Neumann Algebras

The close relationship between the notions of positive formsand representations for a C*-algebra A is one of the most basicfacts in the subject. In particular the weak containment ofrepresentations is well understood in terms of positive forms:given a representation of A in a Hilbert space H and a positiveform on A, its associated representation is weakly containedin (that is, ker ker ) if and only if belongs to the weak*closure of the cone of all finite sums of coefficients of .Among the results on the subject, let us recall the followingones. Suppose that A is concretely represented in H. Then everypositive form on A is the weak* limit of forms of the typex ^k_{i=1 i}, x_i with the _i in H; moreover if A is a von Neumannsubalgebra of (H) and is normal, there exists a sequence (_i)_{i 1} in H such that (x) = _{i 1 i}, x_i for all x.     

Optimal Cardinals for Metrizable Barrelled Spaces

We seek the smallest or largest cardinals for which certainbasic results hold, as did Mazur when he proved that c is thesmallest infinite-dimensionality for a Fréchet space.As with Mazur, we make no axiomatic assumptions outside theusual ZFC model. We discover three instances in which the optimalcardinal is the dominating number and three in which it isthe bounding number b, apparently giving the first locally convexspace characterizations of these venerable and easily describedcardinals. Here are two samples: it is known that for any non-normablemetrizable locally convex space E, the minimal size _b(E) fora fundamental system of bounded sets must satisfy _{1 b}(E) c;we prove that _b(E) = . Again, it is known that if E is a non-normablemetrizable barrelled space of minimal dimension, then ₁ dim(E) c; we prove that dim(E) = b. The most important individualresult is the reconstruction of Tweddle's space without useof the Continuum Hypothesis (₁ = c). The reconstruction is vitalin the characterizations of b and in subsequent papers answeringopen questions about countable enlargements.     

Free Groups of Outer Commutator Varieties of Groups

If F is a free group, 1 < i j 2i and i k i + j + 1 thenF/[_j(F), _i(F), _k(F)] is residually nilpotent and torsion-free.This result is extended to 1 < i j 2i and i k 2i + 2j.It is proved that the analogous Lie rings, L/[L^j, Lⁱ, L^k] whereL is a free Lie ring, are torsion-free. Candidates are foundfor torsion in L/[L^j, Lⁱ, L^k] whenever k is the least of {i,j, k}, and the existence of torsion in L/[L^j, Lⁱ, L^k] is provedwhen i, j, k 5 and k is the least of {i, j, k}.     

Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations

Consider the following infinite dimensional stochastic evolutionequation over some Hilbert space H with norm |·|: It is proved that under certain mild assumptions, the strongsolution X_t(x₀)VHV*, t 0, is mean square exponentially stableif and only if there exists a Lyapunov functional (·,·):HxR₊R¹ which satisfies the following conditions: (i)c₁|x|²–k₁e^–µ₁t(x,t)c₂|x|²+k₂+k₂e^–µ₂t; (ii) L(x,t)–c₃(x,t)+k₃e^–µ₃t, xV, t0; where L is the infinitesimal generator of the Markov processX_t and c_i, k_i, µ_i, i = 1, 2, 3, are positive constants.As a by-product, the characterization of exponential ultimateboundedness of the strong solution is established as the nulldecay rates (that is, µ_i = 0) are considered.     

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={₁, ₂, ..., _n} be a set of fundamentalroots of , with ⁺ being the set of positive roots of with respectto . For and ⁺, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin ⁺ will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected.     

Merit Factors of Character Polynomials

Let q be a prime and be a non-principal character modulo q.Let where 1 t q is the character polynomial associated to (cyclicallypermuted t places). The principal result is that for any non-principaland non-real character modulo q and 1 t q, where the implicit constant is independent of t and q. Here||·||₄ denotes the L₄ norm on the unit circle. It follows from this that all cyclically permuted characterpolynomials associated with non-principal and non-real charactershave merit factors that approach 3. This complements and completesresults of Golay, Høholdt and Jensen, and Turyn (andothers). These results show that the merit factors of cyclicallypermuted character polynomials associated with non-principalreal characters vary asymptotically between 3/2 and 6. The averages of the L₄ norms are also computed. Let q be a primenumber. Then where the summation is over all characters modulo q.     

Homology of GLn over algebraically closed fields

In this paper we define higher pre-Bloch groups _n(F) of a fieldF. When the base field is algebraically closed, we study itsconnection to the homology of the general linear groups withcoefficients in /l , where l is a positive integer. As a resultof our investigation we give a necessary and sufficient conditionfor the natural map H_n(GL_n–1(F), /l ) H_n(GL_n(F), /l )to be bijective. We prove that this map is bijective for n4.We also demonstrate that a certain property of _n() is equivalentto the validity of the Friedlander–Milnor isomorphismconjecture for (n+1)th homology of GL_n().     

The Isolated Points of G{rho} and the w*-Strongly Exposed Points of P{rho}(G)0

Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C^*-algebraC^*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w^*-strongly exposed point of P(G)₀, where P(G)₀={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w^*-strongly exposed point of P(G)₀, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w^*-strongly exposed point of P(G)₀. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w^*-stronglyexposed point of P(G)₀, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F₂ on two generators), there isno closed point in its reduced dual space .     

Turan's Inequalities for Trigonometric Polynomials

We present here a technique for establishing inequalities ofthe form in the set of alltrigonometric polynomials of order n which have only real zeros.The function is assumed to be convex and increasing on [0,). As a corollary of the main result we get Turan's inequalities with the exact constantc(n, k, q) for each 1 q , n and k.     

Identity Theorems for Functions of Bounded Characteristic

Suppose that f(z) is a meromorphic function of bounded characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from [3, Lemma 6.7, p. 174 and the following])that where h₁(z), h₂(z) are holomorphic in and have positive realpart there, while ₁(z), ₂(z) are Blaschke products, that is, where p is a positive integer or zero, 0<|a_j|<1, c isa constant and (1–|a_j|)<. We note in particular that, if c0, so that f(z)0, (1.1) so that f(z)=0 only at the points a_j. Suppose now that z_j isa sequence of distinct points in such that |z_j|1 as j and (1–|z_j|)=. (1.2) If f(z_j)=0 for each j and fN, then f(z)0. N. Danikas [1] has shown that the same conclusion obtains iff(z_j)0 sufficiently rapidly as j. Let _j, _j be sequences of positivenumbers such that _j< and _j as j. Danikas then defines and proves Theorem A.     

Immanant Inequalities, Induced Characters, and Rank Two Partitions

If = {₁, ₂, ..., _s}, where _{1 2} ... _s > 0, is a partitionof n then denotes the associated irreducible character of S_n,the symmetric group on {1, 2, ..., n}, and, if cCS_n, the groupalgebra generated by C and S_n, then d_c(·) denotes thegeneralized matrix function associated with c. If c₁, c₂ CS_nthen we write c₁ c₂ in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCS_n and c(e) 0, wheree denotes the identity in S_n, then or denotes (c(e))^–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {₁, ₂, ..., _s, 1^t} isa partition of n such that s > 1 and _s = 2, and ' denotes{₁, ₂, ..., _s-1, 1^t} then (, _{2}) where denotes characterinduction from S_n–2 x S₂ to S_n. This in turn implies thatif = {₁, ₂, ..., _s, 1^t} with s > 1, _s = 2, and ßdenotes {₁ + 2, ₂, ..., _s-1, 1^t} then _ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that _ß and _ß are both false.     

Minimal Geodesics on Manifolds with Discontinuous Metrics

The paper describes some qualitative properties of minimizerson a manifold M endowed with a discontinuous metric. The discontinuityoccurs on a hypersurface disconnecting M. Denote by ₁ and₂ the open subsets of M such that M\ =₁₂. Assume that and are endowed with metrics ·, · ₍₁₎ and ·,·₍₂₎, respectively, such that (i=1, 2) is convex or concave. The existence of a minimizerof the length functional on curves joining two given pointsof M is proved. The qualitative properties obtained allows therefraction law in a very general situation to be described.     

The Consistency of Holt's Conjectures on Cohomological Dimension of Locally Finite Groups

Let G be a locally finite group of cardinality _n where n isa natural number. Let (G) be the set of primes p for which Ghas an element of order p. In [5], Holt conjectures that ifk is a finite field with char k (G) then (1) G has cohomological dimension n+1 over k; (2) Hⁿ⁺¹(G, kG) has cardinality 2^_n; (3) Hⁱ(G, kG) = 0 for 0 i n.     

Faithful Representations of Free Products

In 1940 Nisnevi published the following theorem [3]. Let (G) be a family of groups indexed by some set and (F) a family of fields of the same characteristic p0. Iffor each the group G has a faithful representation of degreen over F then the free product* G has a faithful representationof degree n+1 over some field of characteristic p. In [6] Wehrfritzextended this idea. If (G) GL(n, F) is a family of subgroupsfor which there exists ZGL(n, F) such that for all the intersectionGF.1_n=Z, then the free product of the groups *_ZG with Z amalgamatedvia the identity map is isomorphic to a linear group of degreen over some purely transcendental extension of F. Initially, the purpose of this paper was to generalize theseresults from the linear to the skew-linear case, that is, togroups isomorphic to subgroups of GL(n, D) where the D are divisionrings. In fact, many of the results can be generalized to ringswhich, although not necessarily commutative, contain no zero-divisors.We have the following.     

BENFORD'S LAW FOR THE 3x + 1 FUNCTION

Benford's law (to base B) for an infinite sequence {x_k : k 1} of positive quantities x_k is the assertion that {log_B x_k: k 1} is uniformly distributed (mod 1). The 3x + 1 functionT(n) is given by T(n) = (3n + 1)/2 if n is odd, and T(n) = n/2if n is even. This paper studies the initial iterates x_k = T^(k)(x₀)for 1 k N of the 3x + 1 function, where N is fixed. It showsthat for most initial values x₀, such sequences approximatelysatisfy Benford's law, in the sense that the discrepancy ofthe finite sequence {log_B x_k : 1 k N} is small.     

Movement and Separation of Subsets of Points Under Group Actions

Let G be a permutation group on a set , and let m and k be integerswhere 0<m<k. For a subset of , if the cardinalities ofthe sets ^g\, for gG, are finite and bounded, then is said tohave bounded movement, and the movement of is defined as move()=max_gG|^g\|. If there is a k-element subset such that move()m, it is shown that some G-orbit has length at most (k²–m)/(k–m).When combined with a result of P. M. Neumann, this result hasthe following consequence: if some infinite subset has boundedmovement at most m, then either is a G-invariant subset withat most m points added or removed, or nontrivially meets aG-orbit of length at most m²+m+1. Also, if move ()m for allk-element subsets and if G has no fixed points in , then either||k+m (and in this case all permutation groups on have thisproperty), or ||5m–2. These results generalise earlierresults about the separation of finite sets under group actionsby B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann,and groups in which all subsets have bounded movement (by theauthor).     

Herman Rings and Arnold Disks

For (,a) C* x C, let f_,a be the rational map defined by f_,a(z)= z² (az+1)/(z+a). If R/Z is a Brjuno number, we let D bethe set of parameters (,a) such that f_,a has a fixed Hermanring with rotation number (we consider that (e²ⁱ,0) D). Resultsobtained by McMullen and Sullivan imply that, for any g D, theconnected component of D(C* x (C/{0,1})) that contains g isisomorphic to a punctured disk. We show that there is a holomorphic injection F:DD such thatF(0) = (e²ⁱ ,0) and , where r is the conformal radius at 0 of the Siegel disk of the quadraticpolynomial z e²ⁱ z(1+z). As a consequence, we show that for a (0,1/3), if f_l,a has afixed Herman ring with rotation number and if m_a is the modulusof the Herman ring, then, as a0, we have e m_a=(r/a) + O(a). We finally explain how to adapt the results to the complex standardfamily z e^(a/2)(z-1/z).