Algebras almost commuting with Clifford algebras in a II∞ factor

We are concerned here with certain Banach algebras of operators contained within a fixed II factor N. These algebras may be thought of as noncommutative classifying spaces for the functor Ext _* ^N The basic objects of study are the algebras A _kN (for n=1, 2,...). Here, we are given an essentially unique representation of the complex Clifford algebra C _k N and the elements of A _k are those operators in N which exactly commute with the first k–1 generators of C _k and also commute with the kth generator modulo a symmetric ideal N. Up to isomorphism, these algebras are periodic with period 2.We determine completely the homotopy types of A ₁ ^–1 and A ₂ ^–1 Here, A ₁ ^–1 is homotopy equivalent to the space of (Breuer) Fredholm operators in N, while A ₂ ^–1 is homotopy equivalent to the group K _N ^–1 ={x N^–1¦ x=1+k, k K_N}. We use these results to compute the K-theory of A ₁ and A ₂.For a fixed C ^*-algebra A, we define abelian groups G _k,N(A) of equivalence classes of homomorphisms AA _k. Letting N = M (H) for a II₁ factor M we define similar abelian groups G _k(A, M) where we replace N by L(E) for countably generated right Hilbert M-modules E with (left) actions C _k L(E). Using ideas of Skandalis, we show that G _k,NG_k(A, M) so that the G _k,N are stable half-exact homotopy functors because the G _k(·, M) are such.In general, we show that G _k(A, M)KK ^k(A, M) and so our theory fits neatly into Kasparov KK-theory. We investigate many interesting examples from our point of view.     

Maximal Collections of Intersecting Arithmetic Progressions

Let N _t (k) be the maximum number of k-term arithmetic progressions of real numbers, any two of which have t points in common. We determine N ₂(k) for prime k and all large k, and give upper and lower bounds for N _t (k) when t 3.* Research supported in part by NSF grant DMS-0070618.     

Nonselfadjoint spectral problems for linear pencilsN-λP of ordinary differential operators with λ-linear boundary conditions: Completeness results

We study nonselfadjoint spectral problems for ordinary differential equationsN(y)–P(y)=0 with -linear boundary conditions where the orderp of the differential operatorP is less than the ordern ofN. The present paper addresses the question of the completeness of the eigenfunctions and associated functions in the Sobolev spacesW ₂ ^k (0,1) fork=0,1,...,n. To this end we associate a pencil – of operators acting fromL ₂(0,1) to the larger spaceL ₂x(0,1) ⁿ with the given problem. We establish completeness results for normal problems in certain finite codimensional subspaces ofW ₂ ^k (0,1) which are characterized by means of Jordan chains in 0 of the adjoint of the compact operator = ^–1.Dedicated to Professor Heinz Langer on the occasion of his 60th birthday     

Local Accuracy for Radial Basis Function Interpolation on Finite Uniform Grids

We consider interpolation on a finite uniform grid by means of one of the radial basis functions (RBF) φ(r)=r^γ for γ>0, γ2 or φ(r)=r^γ ln r for γ2 ₊. For each positive integer N, let h=N⁻¹ and let {x_i: i =1, 2, …, (N+1)^d} be the set of vertices of the uniform grid of mesh-size h on the unit d-dimensional cube [0, 1]^d. Given f: [0, 1]^d→ , let s_h be its unique RBF interpolant at the grid vertices: s_h(x_i)=f(x_i), i=1, 2, …, (N+1)^d. For h→0, we show that the uniform norm of the error f−s_h on a compact subset K of the interior of [0, 1]^d enjoys the same rate of convergence to zero as the error of RBF interpolation on the infinite uniform grid h ^d, provided that f is a data function whose partial derivatives in the interior of [0, 1]^d up to a certain order can be extended to Lipschitz functions on [0, 1]^d.     

Nonabelian K-theory: The nilpotent class of K1 and general stability

A functorial filtration GL _n =S^–1L _n S⁰L _n S ⁱ L _n E _n of the general linear group GL _{n, n} 3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S^–1 L _n (A)/S⁰L _n (A) is abelian, that S⁰L _n (A) S¹L _n (A) is a descending central series, and that S ⁱ L _n (A) = E _n(A) whenever i the Bass-Serre dimension of A. In particular, the K-functors k ₁ S ⁱ L _n =S ⁱ L _n /E _n are nilpotent for all i 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism S ⁱ L _n (A)/S ⁱ⁺¹ L _n (A)S ⁱ L _{n+ 1}(A)/S ⁱ⁺¹ L _{n + 1} (A) is injective whenever n i + 3, so that one has stability results without stability conditions, and if A is commutative then S⁰L _n (A) agrees with the special linear group SL _n (A), so that the functor S⁰L _n generalizes the functor SL _n to noncommutative rings. Applying the above to subgroups H of GL _n (A), which are normalized by E _n(A), one obtains that each is contained in a sandwich GL _n (A, ) H E _n(A, ) for a unique two-sided ideal of A and there is a descending S⁰L _n (A)-central series GL _n (A, ) S⁰L _n (A, ) S¹L _n (A, ) S ⁱ L _n (A, ) E _n(A, ) such that S ⁱ L _n (A, )=E _n(A, ) whenever i Bass-Serre dimension of A.Dedicated to Alexander Grothendieck on his sixtieth birthday     

Polynomial projectors preserving homogeneous partial differential equations

A polynomial projector Π of degree d on is said to preserve homogeneous partial differential equations (HPDE) of degree k if for every and every homogeneous polynomial of degree k, q(z)=∑_|α|=ka_αz^α, there holds the implication: q(D)f=0q(D)Π(f)=0. We prove that a polynomial projector Π preserves HPDE of degree if and only if there are analytic functionals with such that Π is represented in the following form

with some , where u_α(z)z^α/α!. Moreover, we give an example of polynomial projectors preserving HPDE of degree k (k1) without preserving HPDE of smaller degree. We also give a characterization of Abel–Gontcharoff projectors as the only Birkhoff polynomial projectors that preserve all HPDE.     

On Some Applications of the Ordinary and Extended Duhamel Products

Let C _A ⁽ⁿ⁾ (D) denote the algebra of all n-times continuously differentiable functions on holomorphic on the unit disk D = {z C : |z| < 1}. We prove that C _A ⁽ⁿ⁾ (D) is a Banach algebra with multiplication the Duhamel product and describe its maximal ideal space. Using the Duhamel product we prove that the extended spectrum of the integration operator , on C _A ⁽ⁿ⁾ (D) is C\{0}. We also use the Duhamel product in calculating the spectral multiplicity of a direct sum of the form A. We also consider the extension of the Duhamel product and describe all invariant subspaces of some weighted shift operators.Original Russian Text Copyright © 2005 Karaev M. T.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 553–566, May–June, 2005.     

Geometry of \mathbb{Z}^d and the Central Limit Theorem for Weakly Dependent Random Fields

We study the asymptotic distribution of where A is a subset of , A _N = A[–N, N] ^d , v(A) = lim _N card(A _N) (2N+1) ^–d (0, 1) and X is a stationary weakly dependent random field. We show that the geometry of A has a relevant influence on the problem. More specifically, S _N(A, X) is asymptotically normal for each X that satisfies certain mixting hypotheses if and only if has a limit F(n; A) as N for each . We also study the class of sets A that satisfy this condition.     

Convex programs with an additional reverse convex constraint

A method is presented for solving a class of global optimization problems of the form (P): minimizef(x), subject toxD,g(x)0, whereD is a closed convex subset ofR ⁿ andf,g are convex finite functionsR ⁿ . Under suitable stability hypotheses, it is shown that a feasible point is optimal if and only if 0=max{g(x):xD,f(x)f( )}. On the basis of this optimality criterion, the problem is reduced to a sequence of subproblemsQ _k ,k=1, 2, ..., each of which consists in maximizing the convex functiong(x) over some polyhedronS _k . The method is similar to the outer approximation method for maximizing a convex function over a compact convex set.     

A property of nonlinear elliptic equations when the right-hand side is a measure

LetA(u)=–diva(x, u, Du) be a Leray-Lions operator defined onW ₀ ^1,p () and be a bounded Radon measure. For anyu SOLA (Solution Obtained as Limit of Approximations) ofA(u)= in ,u=0 on , we prove that the truncationsT _k(u) at heightk satisfyA(T _k(u)) A(u) in the weak * topology of measures whenk + .

---

Résumé SoitA(u)=–diva(x, u, Du) un opérateur de Leray-Lions défini surW ₀ ^1,p () et une mesure de Radon bornée. Pour toutu SOLA (Solution Obtenue comme Limite d'Approximations) deA(u)= dans ,u=0 sur , nous démontrons que les troncaturesT _k(u) à la hauteurk vérifientA(T _k(u)) A(u) dans la topologie faible * des mesures quandk + .

     

On a generalization of a theorem of Erdős and Fuchs

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers, let k≥2 be a fixed integer and denote by r_k(A,n) the number of solutions of a_i1+a_i2++a_ik≤n. Montgomery and Vaughan proved that r₂(A,n)=cn+o(n^1/4) cannot hold for any constant c>0. In this paper, we extend this result to k>2.     

Not all quadrative norms are strongly stable

A norm N on an algebra A is called quadrative if N(x²) ≤ N(x)² for all x A, and strongly stable if N(x^k) ≤ N(x)^k for all x A and all k = 2, 3, 4…. Our main purpose in this note is to show that not all quadrative norms are strongly stable.     

Density of Gabor Frames

A Gabor system is a set of time-frequency shifts S(g, Λ) ={e^{2 π ibx}g(x − a)}_{(a, b) Λ} of a function g L²(R^d). We prove that if a finite union of Gabor systems _{k = 1}^rS(g_k, Λ_k) forms a frame for L²(R^d) then the lower and upper Beurling densities of Λ = _{k = 1}^r Λ_k satisfy D⁻(Λ) ≥ 1 and D ⁺ (Λ) < ∞. This extends recent work of Ramanathan and Steger. Additionally, we prove the conjecture that no collection _{k = 1}^r{g_k(x − a)}_{a Γk} of pure translates can form a frame for L²(R^d).     

The asymptotic properties of the multichannel autoregressive spectral estimates

If we fit a-vector stationary time series using observationsx(1), ...,x(T) with AR models , then the spectral densityf() of {x(t)} can be estimated byf _k ^(T) ()=(2)^– A _k ^(T) (e ^–)^–1 _k ^(T) A _k ^(T) (e ^–i)^–, where are estimates of the variance matrix of(t), the residuals of the best linear prediction. By extending some results for the scalar case, this paper treats the asymptotic properties of the estimates in the multichannel case.     

Best Polynomial Approximations in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> and Widths of Some Classes of Functions

We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions f(x) ∈ L ₂ ^r (r ∈ ℤ₊) and expressions containing moduli of continuity of the kth order ω_k(f(^r), t). Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from L ₂. For the classes (k, r, Ψ_*) defined by ω _k(f ^(r), t) and the majorant , we determine the exact values of different widths in the space L₂.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1458–1466, November, 2004.     

Exponential-cosine operator-valued functional equation in the strong operator topology

LetX be a Banach Space and letB(X) denote the family of bounded linear operators onX. LetR ⁺ = [0, ). A one parameter family of operators {S(t);t R ⁺},S:R ⁺ B(X), is called exponential-cosine operator function ifS(O) =I andS(s +t) – 2S(s)S(t) = (S(2s) – 2S ²(s))S(t –s), for alls, t R ⁺,s t. Let ,fD(A), and ,fD(B). It is shown that for a strongly continuous exponential-cosine operator {S(t)},fD(A ²) implies ₀ ^t (t –u(S(u)fduD(B) and B ₀ ^t (t –u)S(u)fdu =S(t)f –f +tAf – 2A ₀ ^t S(u)fdu + 2A ² ₀ ^t (t –u)S(u)fdu.D(B) is seen to be dense inD(A ²). Some regularity properties ofS(t) have also been obtained.     

Nonvanishing of central Hecke L-values and rank of certain elliptic curves

Let D 7 mod 8 be a positive squarefree integer, and let h_D be the ideal class number of E_D= . Let d1 mod 4 be a squarefree integer relatively prime to D. Then for any integer k0 there is a constant M=M(k), independent of the pair (D,D), such that if (–1)^k=sign (d), (2k+1,h_D)=1, and >(12/)d² (logd+M(k)), then the central L-value L(k+1, _{D, d} ^2k+1 >0. Furthermore, for k1, we can take M(k)=0. Finally, if D=p is a prime, and d>0, then the associated elliptic curve A(p)^d has Mordell–Weil rank 0 (over its definition field) when >(12/)d² log d.     

Fractional Modified Dyadic Integral and Derivative on ℝ<Subscript>+</Subscript>

For functions in the Lebesgue space L(ℝ₊), a modified strong dyadic integral J _α and a modified strong dyadic derivative D ^(α) of fractional order α > 0 are introduced. For a given function f ∈ L(ℝ₊), criteria for the existence of these integrals and derivatives are obtained. A countable set of eigenfunctions for the operators J _α and D ^(α) is indicated. The formulas D ^(α)(J ^α(f)) = f and J _α(D ^(α)(f)) = f are proved for each α > 0 under the condition that . We prove that the linear operator is unbounded, where is the natural domain of J _α. A similar statement for the operator is proved. A modified dyadic derivative d ^(α)(f)(x) and a modified dyadic integral j _α(f)(x) are also defined for a function f ∈ L(ℝ₊) and a given point x ∈ ℝ₊. The formulas d ^(α)(J _α(f))(x) = f(x) and j _α(D ^(α)(f)) = f(x) are shown to be valid at each dyadic Lebesgue point x ∈ ℝ₊ of f.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 64–70, 2005Original Russian Text Copyright © by B. I. GolubovSupported by the Russian Foundation for Basic Research (grant no. 05-01-00206).     

Stability of Jackson Type Network Output

We consider an open Jackson type queueing network N with input epochs sequence I={T _n ⁽⁰⁾,n0}, T ₀ ⁽⁰⁾=0, assume another input ={ _n ⁽⁰⁾} and denote _k =| _k ⁽⁰⁾–T _k ⁽⁰⁾|, ₀=0, _n =max_{1kn k} , n1. Let {T _n } and { _n } be the output points in network N and in modified network, with input , accordingly. We study the long-run stability of the network output, establishing two-sided bounds for output perturbation via input perturbation. In particular, we obtain conditions that imply max _kn |T _k – _k |=o(n ^1/r ) with probability 1 as n for some r>0. This result is also extended to continuous time. We consider successively separate station (service node), tandem and feedforward networks. Then we extend stability analysis to general (feedback) networks and show that in our setting these networks can be reduced to feedforward ones. Similar stability results are also obtained in terms of the number of departures. Application to a tandem network with the overloaded stations is considered.     

The Method of Extremal Metric in Extremal Decomposition Problems with Free Parameters

Let a₁,..., a_n be a system of distinct points on the z-sphere , and let be a system of all non-overlapping simply-connected domains D₁,..., D_n on such that a_k ∈ D_k, k = 1,..., n. Let M (D_k, a_k) be the reduced module of the domain D_k with respect to the point a_k ∈ Dk. In the present paper, we solve some problems concerning the maximum of weighted sums of the reduced modules M (D_k, a_k) in certain families of systems of domains {D_k} described above, where the systems of points {a_k} satisfy prescribed symmetry conditions. In each case, the proof is based on an explicit construction of an admissible metric of the module problem, which is equivalent to the extremal problem under consideration, from known extremal metrics of simpler module problems. Bibliography: 7 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 52–67.