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.     

On a variance associated with the distribution of primes in arithmetic progressions

As is usual in prime number theory, write It is well known that when q is close to x the averagevalue of is about xlog q,and recently Friedlander and Goldston have shown that if then the first moment of V(x,q)-U(x,q)is small. In this memoir it is shown that the same is true forall moments. 2000 Mathematics Subject Classification: 11N13.     

De Rham Cohomology and Hodge Decomposition For Quantum Groups

Let be one of the N²-dimensionalbicovariant first order differential calculi for the quantumgroups GL_q(N), SL_q(N), SO_q(N), or Sp_q(N), where q is a transcendentalcomplex number and z is a regular parameter. It is shown thatthe de Rham cohomology of Woronowicz' external algebra coincides with the de Rham cohomologiesof its left-coinvariant, its right-coinvariant and its (two-sided)coinvariant subcomplexes. In the cases GL_q(N) and SL_q(N) thecohomology ring is isomorphic to the coinvariant external algebra and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in thesecases. The main technical tool is the spectral decompositionof the quantum Laplace-Beltrami operator. 2000 MathematicalSubject Classification: 46L87, 58A12, 81R50.     

Differential Operators and the Steenrod Algebra

The Novikov-Landweber algebra and the Steenrod algebra are setup in terms of the primitive differential operators acting in the usual way on the integralpolynomial ring Z[x₁,... ,x_n,...]. A commutative wedge productV for differential operators is introduced and it is shown thatthe iterated wedge product is divisible by r! as an integral operator. The divided differentialoperator algebra D is generated over the integers by thedividedoperators under the wedge product. D is additively isomorphic to the abelian group ofsymmetric functions in the variables x_i. Furthermore D is closedunder composition of operators and admits a natural coproductwhich makes it a Hopf algebra in two ways, with respect to thecomposition and wedge products. Under composition D is isomorphicto the Landweber-Novikov algebra. A Hopf sub-algebra is generatedunder composition by the integral Steenrod squares and reduces mod 2 to the Steenrod algebra. An explicitproduct formula for two wedge expressions is developed and usedto derive Milnor's product formula for his basis elements inthe Steenrod algebra. The hit problem in the Steenrod algebrais reformulated in terms of partial differential operators.1991 Mathematics Subject Classification: 55S10.     

Almost Everywhere Convergence of Bochner-Riesz Means On The Heisenberg Group and Fractional Integration On The Dual

Let L denote the sub-Laplacian on the Heisenberg group H_n and the corresponding Bochner-Riesz operator. Let Q denote the homogeneous dimension and D the Euclideandimension of H_n. We prove convergence a.e. of the Bochner-Rieszmeans as r 0 for > 0and for all f L^p(H_n), provided that . Our proof is based on explicit formulas for the operators with a C, defined on the dual ofH_n by , which may be of independent interest. Here is given by for all (z,u) H_n. 2000 Mathematical Subject Classification: 22E30, 43A80.     

Asymptotic Behavior of Solutions of Differential Equations with Variable Delays

The authors consider the system of forced differential equationswith variable delays whereB_j(t) is a continuous n x n matrix on R⁺, F C(R⁺, Rⁿ) and C(R⁺, R⁺). Using Razumikhin-type techniques and Liapunov'sdirect method, they establish conditions to ensure the ultimateboundedness and the global attractivity of solutions of (*),and when F(t) = 0, the asymptotic stability of the zero solution.Under those same conditions, they also show that is a necessary and sufficient condition for allof the above properties to hold. 1991 Mathematics Subject Classification:34K15, 34C10.     

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.     

The Second Bounded Cohomology of a Group Acting on a Gromov-Hyperbolic Space

Suppose a group G acts on a Gromov-hyperbolic space X properlydiscontinuously. If the limit set L(G) of the action has atleast three points, then the second bounded cohomology groupof is infinite dimensional. For example, if M is a complete, pinched negatively curved Riemannianmanifold with finite volume, then is infinite dimensional. As an application, we show that ifG is a knot group with GZ, then is infinite dimensional. 1991 Mathematics Subject Classification:primary 20F32; secondary 53C20, 57M25.     

Finiteness Properties of Certain Metabelian Arithmetic Groups in the Function Field Case

Consider the group scheme where R is an arbitrary commutative ring with 1 0 and a unitx R* acts on R by multiplication. We will study the finiteness properties of subgroups of G(O_S)where O_S is an S-arithmetic subring of a global function field.The subgroups we are interested in are of the form where Q is a subgroup of O_S*. The finiteness propertiesof these metabelian groups can be expressed in terms of the-invariant due to R. Bieri and R. Strebel. Theorem A. Let S be a finite set of places of a global functionfield (regarded as normalized discrete valuations) and O_S thecorresponding S-arithmetic ring. Let Q be a subgroup of O_S*.Then Q is finitely generated and for all integers n 1 the followingare equivalent:

(1) O_S Q is of type FP_n;

(2) O_S is n-tameas a ZQ-module;

(3) each p S restricts to a non-trivial homomorphism and the set is n-tame.

If these conditions hold for at least one n 1 then the identity holds.} Theorem B. Let r denote the rank of Q. Then the followinghold:

(1) the group O_S Q is not of type FP_r+1};

(2) if Qhas maximum rank r = |S| –1 then the group O_S Q is oftype FP_r.

In particular, is of type FP_{|S| –1} but not of type FP_|S|. 1991 Mathematics SubjectClassification: 20E08, 20F16, 20G30, 52A20.     

Algebraic K-Theory In Low Degree and The Novikov Assembly Map

We prove that the Novikov assembly map for a group factorizes,in ‘low homological degree’, through the algebraicK-theory of its integral group ring. In homological degree 2,this answers a question posed by N. Higson and P. Julg. As adirect application, we prove that if is torsion-free and satisfiesthe Baum-Connes conjecture, then the homology group H₁(; Z)injects in and in , for any ring A such that . If moreover B is of dimension lessthan or equal to 4, then we show that H₂(; Z) injects in and in , where A is as before, and ₂ is generated by the Steinberg symbols{,}, for . 2000 Mathematical Subject Classification: primary 19D55, 19Kxx,58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.     

Weyl-Titchmarsh M-Function Asymptotics for Matrix-valued Schrodinger Operators

We explicitly determine the high-energy asymptotics for Weyl–Titchmarshmatrices corresponding to matrix-valued Schrödinger operatorsassociated with general self-adjoint m x m matrix potentials, where m N. More precisely,assume that for some N N and x₀R, for all c>x₀, and that x x₀ is a right Lebesgue point ofQ^(N–1). In addition, denote by I_m the mxm identity matrixand by C the open sector in thecomplex plane with vertex atzero, symmetry axis along the positive imaginary axis, and openingangle , with 0 < < . Then we prove the following asymptoticexpansion for any point M_+(z,x) of the unique limit point ora point of the limit disk associated with the differential expression in and a Dirichlet boundary condition at x=x₀: The expansion is uniform with respect to arg(z)for |z| in C and uniform in x as long as x varies in compactsubsets of R intersected with the right Lebesgue set of Q^(N–1).Moreover, the m x m expansion coefficients m_+,k(x) can be computedrecursively. Analogous results hold for matrix-valued Schrödinger operatorson the real line. 2000 Mathematics Subject Classification: 34E05,34B20, 34L40, 34A55.     

Bernstein-Type Inequalities for Linear Combinations of Shifted Gaussians

Let P_n be the collection of all polynomials of degree at mostn with real coefficients. A subtle Bernstein-type extremal problemis solved by establishing the inequality for all , and m= 1, 2, ..., where c is an absolute constant and Some related inequalities and direct and inversetheorems about the approximation by elements of in L_q (R) are also discussed. 2000 Mathematics SubjectClassification 41A17 (primary).     

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.     

Weak Covering Properties of Weak Topologies

We consider covering properties of weak topologies of Banachspaces, especially of weak or point-wise topologies of functionspaces C(K), for compact spaces K. We answer questions posedby A. V. Arkhangel'skii, S. P. Gul'ko, and R. W. Hansell. Ourmain results are the following. A Banach space of density atmost ₁ is hereditarily metaLindel of in its weak topology. Ifthe weight of a compact spaceK is at most ₁, then the spacesC_w(K) and C_p(K) are hereditarily metaLindel of. Let be the one-point compactificationof a treeT. Then the space is hereditarily -metacompact. If T is an infinitely branchingfull tree of uncountable height and of cardinality bigger thanc, then the weak topology of the unit sphere of is not -fragmented by any metric. The space C_p(rß₁)is neither metaLindel of nor -relatively metacompact. The spaceC_p(rß₂) is not -relatively metaLindel of. Under theset-theoretic axiom , there exists a scattered compact spaceK₁ such that the space C_p(K₁) is not -relatively metacompact,and under a related axiom , there exists a scattere compactspace K₂ such that the space C_p(K₂) is not -relatively metaLindelof. 1991 Mathematics Subject Classification: 54C35, 46B20, 54E20,54D30.     

Cohomology of Quantized Function Algebras at Roots of Unity

Let G be a simply-connected, semisimple algebraic group overk, an algebraically closed field of characteristic zero. Let O[G] be the quantized function algebra of G at a primitivelth root of unity , and let be the ‘restricted’ quantized function algebra at, a finite-dimensional k-algebra obtained from O[G] by factoringout a centrally generated ideal. It is known that is a Hopf algebra. We study the cohomology ring, a graded commutative algebra, and, for any finite-dimensional -module M, the -module . We prove that for sufficiently large l there isan isomorphism of graded algebras where each X_i is homogeneous of degree $2$, and $2N$ equalsthe number of roots associated to G. Moreover we show that inthis case is a finitely generated -module. We also show under much less restrictive conditions on l that continues to be a finitely generated graded commutativealgebra over which is a finitely generated module. 1991 Mathematics Subject Classification: 16W30,17B37, 17B56.     

Absolute, Relative, and Tate Cohomology of Modules of Finite Gorenstein Dimension

We study finitely generated modules M over a ring R, noetherianon both sides. If M has finite Gorenstein dimension G-dim_RMin the sense of Auslander and Bridger, then it determines twoother cohomology theories besides the one given by the absolutecohomology functors . Relative cohomology functors are defined for all non-negative integers n; they treat the modules of Gorensteindimension 0 as projectives and vanish for n > G-dim_RM. Tatecohomology functors are defined for all integers n; all groups vanish if M or N has finite projective dimension. Comparisonmorphisms and link these functors. We give a self-contained treatmentof modules of finite G-dimension, establish basic propertiesof relative and Tate cohomology, and embed the comparison morphismsinto a canonical long exact sequence . We show that these results provide efficient tools for computingold and new numerical invariants of modules over commutativelocal rings. 2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.     

Filtration-Closed Auslander-Reiten Components for Wild Hereditary Algebras

Let H=kQ be a finite-dimensional connected wild hereditary pathalgebra, over some field k. Denote by H-reg the category offinite-dimensional regular H-modules, that is, the categoryof modules M with for all integers m, where _H denotes the Auslander–Reiten translation.Call a filtration of a regular H-module M a regular filtration if all subquotients M_i/M_i+1are regular. Call a regular filtration (*) a regular compositionseries if it is strictly decreasing and has no proper refinement.A regular component C in the Auslander–Reiten quiver (H) of H-mod is called filtration closed if, for each M addC, the additive closure of C, and each regular filtration (*)of M, all the subquotients M_i/M_i+1 are also in add C. We showthat most wild hereditary algebras have filtration-closed Auslander–Reitencomponents. Moreover, we deduce from this that there are alsoalmost serial components, that is regular components C, suchthat any indecomposable XC has a unique regular compositionseries. This composition series coincides with the Auslander–Reitenfiltration of X, given by the maximal chain of irreducible monosending at X. 1991 Mathematics Subject Classification: 16G70,16G20, 16G60, 16E30.     

Small Solutions of Quadratic Diophantine Equations

We consider quadratic diophantine equations of the shape for a polynomial Q(X₁, ..., X_s) Z[X₁, ..., X_s] of degree 2.Let H be an upper bound for the absolute values of the coefficientsof Q, and assume that the homogeneous quadratic part of Q isnon-singular. We prove, for all s 3, the existence of a polynomialbound _s(H) with the following property: if equation (1) hasa solution x Z^s at all, then it has one satisfying For s = 3 and s = 4 no polynomial bounds _s(H) were previouslyknown, and for s 5 we have been able to improve existing boundsquite significantly. 2000 Mathematics Subject Classification11D09, 11E20, 11H06, 11P55.     

A Strengthening of Resolution of Singularities in Characteristic Zero

Let X be a closed subscheme embedded in a scheme W, smooth overa field k of characteristic zero, and let I (X) be the sheafof ideals defining X. Assume that the set of regular pointsof X is dense in X. We prove that there exists a proper, birationalmorphism, : W_r W, obtained as a composition of monoidal transformations,so that if X_r W_r denotes the strict transform of X W then: (1) the morphism : W_r W is an embedded desingularization ofX (as in Hironaka's Theorem); (2) the total transform of I (X) in factors as a product of an invertible sheaf of ideals L supportedon the exceptional locus, and the sheaf of ideals defining thestrict transform of X (that is, . Thus (2) asserts that we can obtain, in a simple manner, theequations defining the desingularization of X. 2000 MathematicalSubject Classification: 14E15.     

Hopf-Cyclic Homology and Relative Cyclic Homology of Hopf-Galois Extensions

Let H be a Hopf algebra and let M_s (H) be the category of allleft H-modules and right H-comodules satisfying appropriatecompatibility relations. An object in M_s (H) will be calleda stable anti-Yetter–Drinfeld module (over H) or a SAYDmodule, for short. To each M M_s (H) we associate, in a functorialway, a cyclic object Z_* (H, M). We show that our constructioncan be used to compute the cyclic homology of the underlyingalgebra structure of H and the relative cyclic homology of H-Galoisextensions. Let K be a Hopf subalgebra of H. For an arbitrary M M_s (K)we define a right H-comodule structure on so that becomes an object in M_s (H). Under some assumptions on K and M we computethe cyclic homology of . As a direct application of this result, we describe the relativecyclic homology of strongly graded algebras. In particular,we calculate the cyclic homology of group algebras and quantumtori. Finally, when H is the enveloping algebra of a Lie algebra g,we construct a spectral sequence that converges to the cyclichomology of H with coefficients in a given SAYD module M. Wealso show that the cyclic homology of almost symmetric algebrasis isomorphic to the cyclic homology of H with coefficientsin a certain SAYD module. 2000 Mathematics Subject Classification16E40 (primary), 16W30 (secondary).