On Sharp Sobolev Embedding and The Logarithmic Sobolev Inequality

The purpose of this note is to give a short proof of the Grosslogarithmic Sobolev inequality using the asymptotics of thesharp L² Sobolev constant and the product structure of Euclideanspace. Let FL^r(Rⁿ) for some positive r with ||F||_r=1. 1991 MathematicsSubject Classification 58G35.     

Integer Points of Entire Functions

A result is proved which implies the following conjecture ofOsgood and Yang from 1976: if f and g are non-constant entirefunctions, such that T(r, f) = O(T(r,g)) as r and such thatg(z Z implies that f(z) Z, then there exists a polynomialG with coefficients in Q, such that G(Z) Z and f = G g. 2000Mathematics Subject Classification 30D20, 30D35.     

Peripheral Spectra of Order-Preserving Normal Operators

Let X be a real Banach space. A set K X is called a total coneif it is closed under addition and non-negative scalar multiplication,does not contain both x and –x for any non-zero xX, andis such that K–K:= {x–y:x, yK} is dense in X. Supposethat T is a bounded linear operator on X which leaves a closedtotal cone K invariant. We denote by (T) and r(T) the spectrumand spectral radius of T. Krein and Rutman [5] showed that if T is compact, r(T) >0 and K is normal (that is, inf{||x + y||: x, y K, ||x|| =||y|| = 1} > 0), then r(T) is an eigenvalue of T with aneigenvector in K. This result was later extended by Nussbaum[6] to any bounded operator T such that r_e(T)<r(T), wherer_e(T) denotes the essential spectral radius of T, without thehypothesis of normality. The more general question of whetherr(T) (T) for all bounded operators T was answered in the negativeby Bonsall [1], who as well as giving counterexamples describeda property of K called the bounded decomposition property, whichis sufficient to guarantee that r(T) (T). More recently, Toland [8] showed that if X is a separable Hilbertspace and T is self-adjoint, then r(T) (T), without any extrahypotheses on K. In this paper we extend Toland's results tonormal operators on Hilbert spaces, removing in passing theseparability hypothesis. 1991 Mathematics Subject Classification47B65.     

Well matched systems

The product (3.10) on page 33 is incorrectly called a cartesianproduct on pages 33 and 35. This misnomer in effect amountsto a wrong definition. The product (3.10) should be definedso that the right-hand member of (3.10) is the set of all sums_f=1 f_j (not the set of all ordered q-tuples) such that f₁ F(m₁,d₁), ..., f_q F(m_q, d_q).     

Littlewood's One Circle Problem

It is shown that Littlewood's one circle problem has a negativeanswer, that is, there exists a continuous bounded functionf on the unit disk U such that f is not harmonic, but neverthelessfor every x U the equality holds for some r(x) with 0 < r(x) < 1 – ||x||.     

Spectral Inclusion and Analytic Continuation

Let a be an element in a complex Banach algebra with unit, andlet r be a nonnegative number. The Gelfand spectral radius formulaimplies that the spectrum of a is included in the disk {z C:|z| r} if and only if 1991Mathematics Subject Classification 46H99, 47A10, 30B40.     

Reaping Numbers of Boolean Algebras

A subset A of a Boolean algebra B is said to be (n,m)-reapedif there is a partition of unity p B of size n such that |{b p:b a 0}| m for all a A. The reaping number r_n,m (B) ofa Boolean algebra B is the minimum cardinality of a set A B\{0}which cannot be (n,m)-reaped. It is shown that for each n, thereis a Boolean algebra B such that r_n+1,2(B) r_n,2(B). Also, {r_n,m(B):mn } consists of at most two consecutive cardinals. The existenceof a Boolean algebra B such that r_n,m (B) r_n',m' (B) is equivalentto a statement in finite combinatorics which is also discussed.     

Minimal vanishing sums of roots of unity with large coefficients

A vanishing sum , where_n is a primitive nth root of unity and the a_is are non-negativeintegers is called minimal if the coefficient vector (a₀, ..., a_n–1) does not properly dominate the coefficient vectorof any other such non-zero sum. We show that for every c thereis a minimal vanishing sum of nth roots of unity with its greatestcoefficient equal to c, where n is of the form 3pq for odd primesp, q. This solves an open problem posed by Lenstra Jr.     

Profinite Groups with Multiplicative Probabilistic Zeta Function

To a finitely generated profinite group G, a formal Dirichletseries P_G(s)=_na_n/n^s is associated, where a_n = _|G:H|=n µ_G(H).It is proved that G is prosoluble if and only if the sequence{a_n}_nN is multiplicative, that is, a_rs = a_ra_s for any pairof coprime positive integers r and s. This extends the analogousresult on the probabilistic zeta function of finite groups.     

Sharp Results Concerning the Expression of Functions as Sums of Finite Differences

Let * denote convolution and let _x denote the Dirac measureat a point x. A function in L²(R)) is called a difference oforder 1 if it is of the form g-_x * g for some x R and g L²(R)).Also, a difference of order 2 is a function of the form for some x R and g L²(R)). In fact,the concept of a ‘difference of order s’ may bedefined in a similar manner for each s 0. If f denotes the Fouriertransform of f, it is known that a function f in L²(R)) is afinite sum of differences of order s if and only if , and the vector space of all suchfunctions is denoted by D_s (L²(R)). Every function in D_s (L²(R))is a sum of int(2s) + 1 differences of order s, where int(t)denotes the integer part of t. Thus, every function in D₁ (L²(R))is a sum of three first order differences, but it was provedin 1994 that there is a function in D₁ (L(R)) which is neverthe sum of two first order differences. This complemented, forthe group R, the corresponding result for first order differencesobtained by Meisters and Schmidt in 1972 for the circle group.The results show that there is a function in L₂ R such that,for each s 1/2, this function is a sum of int (2s) + 1 differencesof order s but it is never the sum of int (2s) differences oforder s. The proof depends upon extending to higher dimensionsthe following result in two dimensions obtained by Schmidt in1972 in connection with Heilbronn's problem: if x₁, x__n arepoints in the unit square, Following on from the work of Meisters and Schmidt, this workfurther develops a connection between certain estimates in combinatorialgeometry and some questions of sharpness in harmonic analysis.2000 Mathematics Subject Classification 42A38 (primary), 52A40(secondary).     

Julia Sets of Polynomials are Uniformly Perfect

Let f be a polynomial of degree at least two. We shall showthat the Julia set J(f) of f is uniformly perfect. This meansthat there is a constant c(0,1) depending on f only such thatwhenever zJ(f) and 0 < r < diam J(f) then J(f) intersectsthe annulus {w:cr |w — z| r}.     

A reverse Denjoy theorem

Suppose that C₁ and C₂ are two simple curves joining 0 to ,non-intersecting in the finite plane except at 0 and enclosinga domain D which is such that, for all large r, has measure at most 2, where 0 < < .Suppose also that u is a non-constant subharmonic function inthe plane such that u(z) = B(|z|, u) for all large z C₁ C₂.Let A_D(r, u) = inf { u(z):z D and | z | = r }. It is shownthat if A_D(r, u) = O(1) (or A_D(r, u) = o(B(r, u))), then lim_r B(r, u)/r^/2 > 0 (or lim_r log B(r, u)/log r /2).     

Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows

Endre Süli ^{We develop a posteriori upper and lower error bounds for mixedfinite-element approximations of a general family of steady,viscous, incompressible quasi-Newtonian flows in a bounded Lipschitzdomain ; thefamily includes degenerate models such as the power law model,as well as non-degenerate ones such as the Carreau model. Theunified theoretical framework developed herein yields residual-baseda posteriori bounds which measure the error in the approximationof the velocity in the W1, r() norm and that of the pressurein the Lr'() norm, 1/r + 1/r' = 1, r (1, ).}     

The Second Dual Algebra of the Measure Algebra of a Compact Group

It is shown that for every compact group G, L¹(G)^ is uniqueand minimal among all the closed subsets I of M(G)** such thatI is a proper (0, M(G)**) algebraic ideal, and such that I issolid with respect to absolute continuity; that is, n L¹(G)^whenever n M(G)** and n << µ L¹(G)^. 1991 MathematicsSubject Classification 43A20, 43A22.     

Congruences De Sommes De Chiffres De Valeurs Polynomiales

Let m, g, q N with q 2 and (m, q – 1) = 1. For n N,denote by s_n(n) the sum of digits of n in the q-ary digitalexpansion. Given a polynomial f with integer coefficients, degreed 1, and such that f(N) N, it is shown that there exists C= C(f, m, q) > 0 such that for any g Z, and all large N, In the special case m = q = 2 and f(n)= n², the value C = 1/20 is admissible. 2000 Mathematics SubjectClassification 11B85 (primary), 11N37, 11N69 (secondary).     

Bifurcations of Periodic Points of Holomorphic Maps From C2 into C2

Let F:Cⁿ Cⁿ be a holomorphic map, F^k be the kth iterate ofF, and p Cⁿ be a periodic point of F of period k. That is,F^k(p) = p, but for any positive integer j with j < k, F^j(p) p. If p is hyperbolic, namely if DF^k(p) has no eigenvalue ofmodulus 1, then it is well known that the dynamical behaviourof F is stable near the periodic orbit = {p, F(p),..., F^k–1(p)}.But if is not hyperbolic, the dynamical behaviour of F near may be very complicated and unstable. In this case, a veryinteresting bifurcational phenomenon may occur even though may be the only periodic orbit in some neighbourhood of : forgiven M N\{1}, there may exist a C^r-arc {F_t: t [0,1]} (wherer N or r = ) in the space H(Cⁿ) of holomorphic maps from Cⁿinto Cⁿ, such that F₀ = F and, for t (0,1], F_t has an Mk-periodicorbit _t with as t 0. Theperiod thus increases by a factor M under a C^r-small perturbation!If such an F_t does exist, then , as well as p, is said to beM-tupling bifurcational. This definition is independent of r. For the above F, there may exist a C^r-arc in H(Cⁿ), with t [0,1], such that and, for t (0,1], has two distinct k-periodic orbits _t,1 and _t,2 with d(_t,i, ) 0 as t 0 for i = 1,2. If such an does exist, then , as well as p, is said to be 1-tupling bifurcational. In recent decades, there have been many papers and remarkableresults which deal with period doubling bifurcations of periodicorbits of parametrized maps. L. Block and D. Hart pointed outthat period M-tupling bifurcations cannot occur for M >2 in the 1-dimensional case. There are examples showing thatfor any M N, period M-tupling bifurcations can occur in higher-dimensionalcases. An M-tupling bifurcational periodic orbit as defined here actsas a critical orbit which leads to period M-tupling bifurcationsin some parametrized maps. The main result of this paper isthe following. Theorem. Let k N and M N, and let F: C² C² be a holomorphicmap with k-periodic point p. Then p is M-tupling bifurcationalif and only if DF^k(p) has a non-zero periodic point of periodM. 1991 Mathematics Subject Classification: 32H50, 58F14.     

Eigenvectors of Order-Preserving Linear Operators

Suppose that K is a closed, total cone in a real Banach spaceX, that A:XX is a bounded linear operator which maps K intoitself, and that A' denotes the Banach space adjoint of A. Assumethat r, the spectral radius of A, is positive, and that thereexist x₀0 and m1 with A^m(x₀)=r^mx₀ (or, more generally, thatthere exist x₀(–K) and m1 with A^m(x₀)r^mx₀). If, in addition,A satisfies some hypotheses of a type used in mean ergodic theorems,it is proved that there exist uK–{0} and K'–{0}with A(u)=ru, A'()=r and (u)>0. The support boundary of Kis used to discuss the algebraic simplicity of the eigenvaluer. The relation of the support boundary to H. Schaefer's ideasof quasi-interior elements of K and irreducible operators Ais treated, and it is noted that, if dim(X)>1, then thereexists an xK–{0} which is not a quasi-interior point.The motivation for the results is recent work of Toland, whoconsidered the case in which X is a Hilbert space and A is self-adjoint;the theorems in the paper generalize several of Toland's propositions.     

On the Fitting Height of a Soluble Group that is Generated by a Conjugacy Class

Let G be a finite group and suppose that P is a soluble {2,3}'-subgroup of G. The reader will lose only a little by assumingthat P is a subgroup of prime order p > 3. _G(P)={AG|A is soluble and A=P,P^a for some a A This set is partially ordered by inclusion and we let denote the set of maximal members of _G(P).     

Multiple and Polynomial Recurrence for Abelian Actions in Infinite Measure

The (C,F)-construction from a previous paper of the first authoris applied to produce a number of funny rank one infinite measurepreserving actions of discrete countable Abelian groups G with‘unusual’ multiple recurrence properties. In particular,the following are constructed for each p N{}:

1. a p-recurrent actionT=(T_g)_gG such that (if p) no one transformationT_g is (p+1)-recurrentfor every element g of infinite order;
2. an action T=(T_g)_gGsuch that for every finite sequence g1,...,g_rGwithout torsionthe transformation T_g1x...x T_gr is ergodic,p-recurrent but(if p) not (p+1)-recurrent;
3. a p-polynomially recurrent (C,F)-transformationwhich (if p)is not (p+1)-recurrent.

-recurrence here meansmultiple recurrence. Moreover, it is shown that there existsa (C,F)-transformation which is rigid (and hence multiply recurrent)but not polynomially recurrent. Nevertheless, the subset ofpolynomially recurrent transformations is generic in the groupof infinite measure preserving transformations endowed withthe weak topology.     

Heat Kernel Estimates and LP Spectral Independence of Elliptic Operators

Let be an open subset of R^d, and let T_p for p[1, ) be consistentC₀-semigroups given by kernels that satisfy an upper heat kernelestimate. Denoting their generators by A_p, we show that thespectrum (A_p) is independent of p[1, ). We also treat the caseof weighted L^p-spaces for weights that satisfy a subexponentialgrowth condition. An example shows that independence of thespectrum may fail for an exponential weight. 1991 MathematicsSubject Classification 47D06, 47A10, 35P05.