Periodic Cyclic Homology of Iwahori–Hecke Algebras

We determine the periodic cyclic homology of the Iwahori–Hecke algebras H _q , for q * not a proper root of unity. (In this paper, by a proper root of unity we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a weakly spectrum preserving morphism of finite type algebras induces an isomorphism in periodic cyclic homology. The concept of a weakly spectrum preserving morphism is defined in this paper, and most of our work is devoted to understanding this class of morphisms. Results of Kazhdan and Lusztig and Lusztig show that, for the indicated values of q, there exists a weakly spectrum preserving morphism _q : H _q J, to a fixed finite type algebra J. This proves that _q induces an isomorphism in periodic cyclic homology and, in particular, that all algebras H _q have the same periodic cyclic homology, for the indicated values of q. The periodic cyclic homology groups of the algebra H ₁ can then be determined directly, using results of Karoubi and Burghelea, because it is the group algebra of an extended affine Weyl group.     

The Trace Conjecture – A Counterexample

In the paper Geometric K-Theory for Lie Groups and Foliations, Baum and Connes conjecture in a remark following Corollary 2 of their famous Isomorphism conjecture that for a finitely generated group with torsion, the trace map tr: K₀(C^*)R maps K₀(C^*) onto the additive subgroup of Q generated by all rational numbers of the form 1/n where n is the order of a finite subgroup of . We construct a counterexample to this conjecture.     

Sharp parameter ranges in the uniform anti-maximum principle for second-order ordinary differential operators

We consider the equation (pu)-qu+wu = f in (0,1) subject to homogenous boundary conditions at x = 0 and x = 1, e.g., u(0) = u(1) = 0. Let ₁ be the first eigenvalue of the corresponding Sturm-Liouville problem. If f 0 but 0 then it is known that there exists > 0 (independent on f) such that for (₁, ₁ + ] any solution u must be negative. This so-called uniform anti-maximum principle (UAMP) goes back to Clément, Peletier [4]. In this paper we establish the sharp values of for which (UAMP) holds. The same phenomenon, including sharp values of , can be shown for the radially symmetric p-Laplacian on balls and annuli in ⁿ provided 1 n < p. The results are illustrated by explicitly computed examples.     

Radii of convexity and close-to-convexity of certain integral representations

Strict upper bounds are determined for ¦s(z)¦, ¦Re s(z)¦, and ¦Im s(z) ¦ in the class of functions s(z)=a _nzⁿ+a_n+1zⁿ⁺¹+... (n¹) regular in ¦z¦<1 and satisfying the condition ¦u (₁) –u (₂) ¦K¦ ₁- ₂¦, where U()=Re s (eⁱ ), K>0, and ₁ and ₂ are arbitrary real numbers. These bounds are used in the determination of radii of convexity and close-to-convexity of certain integral representations.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 581–592, May, 1970.The author wishes to thank L. A. Aksent'ev for his guidance in this work.     

On regular embedding integrally in R3 of metrics of class C4 of negative curvature

On the x₀y plane let there be specified a complete metric of negative curvature K by means of the line element ds²=dx²+B²(x, y) dy², and, in the strip _a={0xa, -4-bounded function B>0,K-²<0 ( and are constants). Then, the metric in strip _a is embedded in R³ by means of a surface of class C³.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 261–266, August, 1973.     

On some extremal problems of approximation theory in the complex plane

In the Hardy Banach spaces H _q , Bergman Banach spaces H_q, and Banach spaces (p, q, ), we determine the exact values of the Kolmogorov, Bernstein, Gelfand, linear, and trigonometric n-widths of classes of functions analytic in the disk |z| < 1 and such that the averaged moduli of continuity of their r-derivatives are majorized by a certain function. For these classes, we also consider the problems of optimal recovery and coding of functions.__________Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1155–1171, September, 2004.     

On Conjectures of Mathai and Borel

Mathai has conjectured that the Cheeger–Gromov invariant ₍₂₎ = ₍₂₎ - is a homotopy invariant of closed manifolds with torsion-free fundamental group. In this paper we prove this statement for closed manifolds M when the rational Borel conjecture is known for = ₁(M), i.e. the assembly map : H _*(B, ) L_*() is an isomorphism. Our discussion evokes the theory of intersection homology and results related to the higher signature problem.     

Bruck-Ryser abstract theorem and symmetric designs

In [3] U. Ott introduced a Bruck-Ryser abstract theorem concerning lattices of an R-module. As one of the applications of this theorem, he obtained a new proof of the Bruck-Ryser theorem for finite projective planes and studied p-curves in a projective plane. In this paper, we apply the Bruck-Ryser abstract theorem in order to give a new proof of the Bruck-Ryser-Chowla theorem for symmetric (v, k, )-design with (v, k, )=1, and to study p-curves in arbitrary symmetric designs.     

On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ), where _i ( _i ) are ordinary moduli of continuity that depend on one variable. In the case where _i ( _i ) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L ₂ for (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ); (ii) in the integral metric L _p (p 1) for (₁, ..., _n ) = c ₁₁ + ... + c _{n n} ; (iii) in the integral metric L _{(2, ..., 2, 2r)} (r = 2, 3, ...) for (₁, ..., _n ) = ₁(₁) + ... + _{n – 1}( _{n – 1}) + c _{n n} .     

On the relation between Fourier and Leont’ev coefficients with respect to smirnov spaces

Yu. Melnik showed that the Leontev coefficients _f () in the Dirichlet series 2}}$$ " align="middle" border="0"> of a function f E ^p (D), 1 < p < , are the Fourier coefficients of some function F L ^p , ([0, 2]) and that the first modulus of continuity of F can be estimated by the first moduli and majorants in f. In the present paper, we extend his results to moduli of arbitrary order.Published in Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 517–526, April, 2004.     

A new example of ‘independence’ and ‘white noise’

Summary We examine the notion of free independence according to Voiculescu. This form of independence is used for defining free white noise or process with stationary and freely independent increments. We prove a general limit theorem giving the combinatorics of infinitely freely divisible states and thus of free white noises with the help of admissible partitions. We realize the free analogues of the Wiener process and of the Poisson process as processes on the full Fock space ofL ² (—).     

On the Uniform Summability of Multiple Walsh-Fourier Series

In this paper we prove that if f C (0, 1 ^N ) and the function f is of bounded partial variation, then the N-dimensional Walsh-Fourier series of the function f is uniformly (C,–) summable (₁ +...+ _N < 1, _i > 0, i = 1,...,N) in the sense of Pringsheim. If ₁ +...+ _N = 1, _i > 0, i = 1,2,...,N, then there exists a continuous function f ₀ of bounded partial variation on [0, 1] ^N such that the Cesàro (C,–) means _m ^– (f₀,Õ) of the N-dimensional Walsh-Fourier series of f ₀ diverge over cubes.     

Strong Laws of Large Numbers by Elementary Tauberian Arguments

For Kolmogorovs strong law of large numbers an alternative short proof is given which weakens Etemadis condition of pairwise independence. The argument uses the known – and elementary – equivalence of (Cesàro) C₁- and C₂-summability for one-sided bounded sequences. Also other strong laws of large numbers are established, in part via Borel summability.     

The Novikov Conjecture for Low Degree Cohomology Classes

We outline a twisted analogue of the Mishchenko–Kasparov approach to prove the Novikov conjecture on the homotopy invariance of the higher signatures. Using our approach, we give a new and simple proof of the homotopy invariance of the higher signatures associated to all cohomology classes of the classifying space that belong to the subring of the cohomology ring of the classifying space that is generated by cohomology classes of degree less than or equal to 2, a result that was first established by Connes and Gromov and Moscovici using other methods. A key new ingredient is the construction of a tautological C* _r (, )-bundle and connection, which can be used to construct a C* _r (, )-index that lies in the Grothendieck group of C* _r (, ), where is a multiplier on the discrete group corresponding to a degree 2 cohomology class. We also utilise a main result of Hilsum and Skandalis to establish our theorem.     

Special Varieties and Classification Theory: An Overview

Lang's conjectures link the geometric, hyperbolic, and arithmetic properties of projective complex varieties of general type. We propose here an extension of these conjectures to arbitrary projective varieties X. This extension rests on the notion of special variety. This class contains manifolds either rationally connected or with vanishing Kodaira dimension. We further construct for any X its core, which is a fibration c _X : XC(X) with general fibre special and orbifold base of general type. This fibration seems to permit us to decompose X according to the dichotomy special vs general type, and not only leads to the above-mentioned extension of Lang's conjectures but also to a simple global view of classification theory.     

Computation of Almost Split Sequences with Applications to Relatively Projective and Prinjective Modules

Let be an Artin algebra, let mod be the category of finitely generated -modules, and let Amod be a contravariantly finite and extension closed subcategory. For an indecomposable and not Ext-projective module CA, we compute the almost split sequence 0ABC0 in A from the almost split sequence 0DTrCEC0 in mod. Since the computation is particularly simple if the minimal right A-approximation of DTrC is indecomposable for all indecomposable and not Ext-projective CA, we manufacture subcategories A with the desired property using orthogonal subcategories. The method of orthogonal subcategories is applied to compute almost split sequences for relatively projective and prinjective modules.     

Taut Distance-Regular Graphs of Odd Diameter

Let denote a bipartite distance-regular graph with diameter D 4, valency k 3, and distinct eigenvalues ₀ > ₁ > ··· > _D. Let M denote the Bose-Mesner algebra of . For 0 i D, let E _i denote the primitive idempotent of M associated with _i . We refer to E ₀ and E _D as the trivial idempotents of M. Let E, F denote primitive idempotents of M. We say the pair E, F is taut whenever (i) E, F are nontrivial, and (ii) the entry-wise product E F is a linear combination of two distinct primitive idempotents of M. We show the pair E, F is taut if and only if there exist real scalars , such that _{i + 1 i + 1} – _{i – 1 i – 1} = _i ( _{i + 1} – _{i – 1}) + _i ( _{i + 1} – _{i – 1}) + (1 i D – 1)where ₀, ₁, ..., _D and ₀, ₁, ..., _D denote the cosine sequences of E, F, respectively. We define to be taut whenever has at least one taut pair of primitive idempotents but is not 2-homogeneous in the sense of Nomura and Curtin. Assume is taut and D is odd, and assume the pair E, F is taut. We show

The role of Fourier modes in extension theorems of Hartogs-Chirka type

We generalize Chirkas theorem on the extension of functions holomorphic in a neighbourhood of (F)(D×D) – where D is the open unit disc and (F) is the graph of a continuous D-valued function F – to the bidisc. We extend holomorphic functions by applying the Kontinuitätssatz to certain continuous families of analytic annuli, which is a procedure suited to configurations not covered by Chirkas theorem.Mathematics Subject classification (2000): 32D15Work supported by NSF Grant DMS-0072237.     

Groups acting simply transitively on the vertices of a building of typeà 2, II: The casesq=2 andq=3

If (P, L) is a projective plane and is a triangle presentation compatible with a point-line correspondence :P L, then gives rise to a group and a thick building of typeà ₂ on the vertices of which acts simply transitively. We find all triangle presentations (up to natural equivalence) compatible with some point-line correspondence :P L, when (P, L) is the projective plane of orderq=2 orq=3. For some, but not all, of these , is isomorphic to the building associated withG=PGL(3,K) whereK is a local field with discrete valuation and residual field of orderq. We identify the for which this is the case, and in these cases, find embeddings of intoG. We also describe the arithmetic nature of these groups.     

Approximation of infinitely differentiable periodic functions by interpolation trigonometric polynomials

We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities on the classes of periodic infinitely differentiable functions C C whose elements can be represented in the form of convolutions with fixed generating kernels. We obtain asymptotic equalities for upper bounds of approximations by interpolation trigonometric polynomials on the classes C _, and C H .Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 495–505, April, 2004.