Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).     

On Finite and InfiniteCk-A-Determinacy

Let f: (Rⁿ,0) (R^p,0) be a C map-germ. We define f to be finitely,or -, A-determined, if there exists an integer m such that allgerms g with j^mg(0) = j^mf(0), or if all germs g with the sameinfinite Taylor series as f, respectively, are A-equivalentto f. For any integer k, 0 k < , we can consider A' sC^kcounterpart (consisting of C^k diffeomorphisms) A^(k), and wecan define the notion of finite, or -,A^(k)-determinacy in asimilar manner. Consider the following conditions for a C germf: (a_k) f is -A^(k)-determined, (b_k) f is finitely A^(k)-determined,(t) , (g) there exists a representative f : U R^p defined on some neighbourhood U of 0 in Rⁿ such thatthe multigerm of f is stable at every finite set , and (g') every f' with j f'(0)=j f(0) satisfiescondition (g). We also define a technical condition which willimply condition (g) above. This condition is a collection ofp+1 Lojasiewicz inequalities which express that the multigermof f is stable at any finite set of points outside 0 and onlybecomes unstable at a finite rate when we approach 0. We willdenote this condition by (e). With this notation we prove thefollowing. For any C map germ f:(Rⁿ,0) (R^p,0) the conditions(e), (t), (g') and (a) are equivalent conditions. Moreover,each of these conditions is equivalent to any of (a_k) (p+1 k < , (b_k) (p+1 k < ). 1991 Mathematics Subject Classification:58C27.     

On the Pair Correlation of Zeros of the Riemann Zeta-Function

To study the distribution of pairs of zeros of the Riemann zeta-function,Montgomery introduced the function where is real and T 2, and ' denote the imaginary parts ofzeros of the Riemann zeta-function, and w(u) = 4/(4 + u²). Assumingthe Riemann Hypothesis, Montgomery proved an asymptotic formulafor F() when || 1, and made the conjecture that F() = 1 + o(1)as T for any bounded with || 1. In this paper we use anapproximation for the prime indicator function together witha new mean value theorem for long Dirichlet polynomials andtails of Dirichlet series to prove that, assuming the GeneralizedRiemann Hypothesis for all Dirichlet L-functions, then for any > 0 we have uniformlyfor and all T T₀(). 1991Mathematics Subject Classification: primary 11M26; secondary11P32.     

Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball

Weighted Triebel–Lizorkin and Besov spaces on the unitball B^d in ^d with weights w_µ(x)=(1–|x|²)^µ–1/2,µ0, are introduced and explored. A decomposition schemeis developed in terms of almost exponentially localized polynomialelements (needlets) {}, {} and it is shown that the membershipof a distribution to the weighted Triebel–Lizorkin orBesov spaces can be determined by the size of the needlet coefficients{f, } in appropriate sequence spaces.     

On Lp boundedness of wave operators for 4-dimensional Schrodinger operators with threshold singularities

Let H=–+V(x) be a Schrödinger operator on L²(R⁴),H₀=–. Assume that |V(x)|+| V(x)|C x ^– for some>8. Let be the wave operators. It is known that W_± extend to bounded operators in L^p(R⁴)for all 1p, if 0 is neither an eigenvalue nor a resonance ofH. We show that if 0 is an eigenvalue, but not a resonance ofH, then the W_± are still bounded in L^p(R⁴) for all psuch that 4/3<p<4.     

Sums of Connectivity Functions on Rn

A function f: Rⁿ R is a connectivity function if the graphof its restriction f|C to any connected C Rⁿ is connected inRⁿ x R. The main goal of this paper is to prove that every functionf: Rⁿ R is a sum of n + 1 connectivity functions (Corollary2.2). We will also show that if n > 1, then every functiong: Rⁿ R which is a sum of n connectivity functions is continuouson some perfect set (see Theorem 2.5) which implies that thenumber n + 1 in our theorem is best possible (Corollary 2.6). Toprove the above results, we establish and then apply the followingtheorems which are of interest on their own. For every dense G-subset G of Rⁿ there are homeomorphisms h₁,..., h_n of Rⁿ such that Rⁿ = G h₁(G) ... h_n(G) (Proposition2.4). For every n > 1 and any connectivity function f: Rⁿ R, ifx Rⁿ and > 0 then there exists an open set U Rⁿ such thatx U Bⁿ(x, ), f|bd(U) is continuous, and |(x) – f(y)|< for every y bd(U) (Proposition 2.7). 1991 MathematicsSubject Classification: 26B40, 54C30, 54F45.     

Vinogradov's Integral and Bounds for the Riemann Zeta Function

The main result is an upper bound for the Riemann zeta functionin the critical strip: with A = 76.2 and B = 4.45, valid for 1 and |t| 3. The previousbest constant B was 18.5. Tools include a variant of the Korobov–Vinogradovmethod of bounding exponential sums, an explicit version ofT. D. Wooley's bounds for Vinogradov's integral, and explicitbounds for mean values of exponential sums over numbers withoutsmall prime factors, also using methods of Wooley. An auxiliaryresult is the exponential sum bound , where N is a positive integer, t is a real number, = log (t)/(logN) and 2000 Mathematical Subject Classification: primary 11M06, 11N05,11L15; secondary 11D72, 11M35.     

On the laplace equation with non-linear dynamical boundary conditions

The main part of the paper deals with local existence and globalexistence versus blow-up for solutions of the Laplace equationin bounded domains with a non-linear dynamical boundary condition.More precisely, we study the problem consisting in: (1) theLaplace equation in (0, ) x ; (2) a homogeneous Dirichlet condition(0, ) x ₀; (3) the dynamical boundary condition ; (4) the initial condition u(0, x) = u₀ (x) on . Here is a regular and bounded domain in Rⁿ, with n 1, and₀ and ₁ endow a measurable partition of . Moreover, m>1,2 p < r, where r = 2 (n – 1) / (n – 2) whenn 3, r = when n = 1,2, and u₀ H^1/2 , u₀ = 0 on ₀. The final part of the paper deals with a refinement of a globalnon-existence result by Levine, Park and Serrin, which is appliedto the previous problem. 2000 Mathematics Subject Classification35K55 (primary), 35K90, 35K77 (secondary).     

Fock Space and the Poisson Process

Using the Wiener–Poisson isomorphism, we show that if(F_t)_{0 t 1} is a real, bounded, predictable process adaptedto the filtration of a compensated Poisson process (X_t)_{0 t 1}, and if is the operator corresponding to multiplication by , then for any regular self-adjoint quantum semimartingale , the essentially self-adjoint quantumsemimartingale satisfies the quantum Ito formula. We also introduce a generalisation of the Poisson process toa measure space (M, M, µ) as an isometry I: L² (M, M,µ) L²(, F, P) and give a new construction of the generalisedWiener–Poisson isomorphism W_I: F₊ (L²(M)) L² (, F, P)using exponential vectors. Using C^*-algebra theory, given anymeasure space we construct a canonical generalised Poisson process.Unlike other constructions, we make no a priori use of Poissonmeasures. 2000 Mathematics Subject Classification 60G20, 60G35,46L53, 81S25.     

Spectra of graph neighborhoods and scattering

Let (G)_>0 be a family of ‘-thin’ Riemannian manifoldsmodeled on a finite metric graph G, for example, the -neighborhoodof an embedding of G in some Euclidean space with straight edges.We study the asymptotic behavior of the spectrum of the Laplace–Beltramioperator on G, as 0, for various boundary conditions. We obtaincomplete asymptotic expansions for the kth eigenvalue and theeigenfunctions, uniformly for kC^–1, in terms of scatteringdata on a non-compact limit space. We then use this to determinethe quantum graph which is to be regarded as the limit object,in a spectral sense, of the family (G). Our method is a directconstruction of approximate eigenfunctions from the scatteringand graph data, and the use of a priori estimates to show thatall eigenfunctions are obtained in this way.     

Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus

The functional Ito formula, in the form df() = f( + d ) –f(),is formulated and proved in the context of a Lie algebra L associatedwith a quantum (non-commutative) stochastic calculus. Here fis an element of the universal enveloping algebra U of L, andf() + d() – f() is given a meaning using the coproductstructure of U even though the individual terms of this expressionhave no meaning. The Ito formula is equivalent to a chaoticexpansion formula for f() which is found explicitly. 1991 MathematicsSubject Classification: primary 81S25; secondary 60H05; tertiary18B25.     

On the Asphericity of Length Five Relative Group Presentations

We investigate asphericity of the relative group presentation G,t |atbtctdtet=1 and prove it aspherical provided thesubgroupof G generated by ab^–1, bc^–1, cd^–1, de^–1is neither finite cyclic nor a finite triangle group. We alsoprove a similar result for the closely related relative grouppresentation G,s,t | sßst=1=tts^–1. 2000 MathematicsSubject Classification: 20F05, 57M05.     

Strongly minimal expansions of (C, +) definable in o-minimal fields

We characterize those functions f: definable in o-minimalexpansions of the reals for which the structure (,+, f) is stronglyminimal: such functions must be complex constructible, possiblyafter conjugating by a real matrix. In particular we prove aspecial case of the Zilber Dichotomy: an algebraically closedfield is definable in certain strongly minimal structures whichare definable in an o-minimal field.     

The Approximation Numbers of Hardy-Type Operators on Trees

The Hardy operator T_a on a tree is defined by Properties of T_a as a map from L^p() into itselfare established for 1 p . The main result is that, with appropriateassumptions on u and v, the approximation numbers a_n(T_a) ofT_a satisfy for a specified constant _p and 1 p < . This extends results of Naimark, Newmanand Solomyak for p = 2. Hitherto, for p 2, (*) was unknowneven when is an interval. Also, upper and lower estimates forthe l^q and weak-l^q norms of a_n(T_a) are determined. 2000 MathematicalSubject Classification: 47G10, 47B10.     

On the Fredholm Alternative for the p-Laplacian in One Dimension

We investigate the existence of a weak solution u to the quasilineartwo-point boundary value problem We assume that 1 < p < p ¬ = 2, 0 < a < , andthat f L¹(0,a) is a given function. The number _k stands forthe k-th eigenvalue of the one-dimensional p-Laplacian. Let_{p p} x/a) denote the eigenfunction associated with ₁; then _p(k_p x/a) is the eigenfunction associated with _k. We show the existenceof solutions to (P) in the following cases. (i) When k=1 and f satisfies the orthogonality condition the set of solutions is bounded. (ii) If k=1 and f_t L¹(0,a) is a continuous family parametrizedby t [0,1], with then there exists some t^* [0,1] such that (P) has a solutionfor f = f_t^*. Moreover, an appropriate choice of t_* yields asolution u with an arbitrarily large L¹(0,a)-norm which meansthat such f cannot be orthogonal to _pp x/a. (iii) When k 2 and f satisfies a set of orthogonality conditionsto _p(k p x/a) on the subintervals , again, the set of solutions is bounded. is a continuous family satisfying either or another related condition, then there exists some t^* [0,1]such that (P) has a solution for f = f_t*. Prüfer's transformation plays the key role in our proofs.2000 Mathematical Subject Classification: primary 34B16, 47J10;secondary 34L40, 47H30.     

Dynamics of projective morphisms having identical canonical heights

Let , :^{N N} be morphisms of degree at least 2 whose canonicalheights and are identical. We draw various conclusions aboutthe Green functions, Julia sets, and canonical local heightsof and . We use this information to completely characterize and in the following cases: (i) and are polynomial mapsin one variable; (ii) is the dth-power map; (iii) is a Lattèsmap.     

Convex Regions in the Plane and their Domes

We make a detailed study of the relation of a euclidean convexregion C Dome (). The dome is the relative boundary, in theupper halfspace model of hyperbolic space, of the hyperbolicconvex hull of the complement of . The first result is to provethat the nearest point retraction r: Dome () is 2-quasiconformal.The second is to establish precise estimates of the distortionof r near . 2000 Mathematics Subject Classification 30C75,30F40, 30F45, 30F60.     

Base sizes for simple groups and a conjecture of Cameron

Let G be a permutation group on a finite set . A base for Gis a subset B with pointwise stabilizer in G that is trivial;we write b(G) for the smallest size of a base for G. In thispaper we prove that b(G) 6 if G is an almost simple group ofexceptional Lie type and is a primitive faithful G-set. Animportant consequence of this result, when combined with otherrecent work, is that b(G) 7 for any almost simple group G ina non-standard action, proving a conjecture of Cameron. Theproof is probabilistic and uses bounds on fixed point ratios.     

Sub- and Superadditive Properties of Fejer's sine Polynomial

Let be Fejér's sine polynomial. We prove the following statements.

1. The inequality holds for all x, y (0, ) with x + y < if and only if 0 and + rß 1.
2. The converse of the above inequality is valid for allx, y (0, ) with x + y < if and only if 0 and + rß 1.
3. For all n N and x, y [0, ] we have . Both bounds are best possible.

2000 Mathematics Subject Classification 42A05, 26D05 (primary),39B62 (secondary).