On lengths of <Emphasis Type="Italic">H</Emphasis>ℤ-localization towers

In this paper, the H?-length of different groups is studied. By definition, this is the length of the H?-localization tower or the length of the transfinite lower central series of H?-localization. It is proved that, for a free noncyclic group, its H?-length is ≥ ω+2. For a large class of ?[C]-modules M, where C is an infinite cyclic group, it is proved that the H?-length of the semi-direct product M ? C is ≤ ω + 1 and its H?-localization can be described as a central extension of its pro-nilpotent completion. In particular, this class covers modules M, such that M?C is finitely presented and H₂(M ? C) is finite.     

The spectra of composition operators from linear fractional maps acting upon the Dirichlet space

Let φ:D→D be a non-constant linear fractional transformation (necessarily of the form ). Let D denote the Dirichlet space of analytic functions. We determine the spectrum of the composition operator C_?:D→D defined by C_?(f)=f°φ. Eigenfunctions for the operator C_?:H²→H² frequently do not belong to the space D. However, spectral results for the operator C_?:D→D, much like those that have already been demonstrated for the operator C_?:H²→H², are presented in this paper.     

Large deviations of interacting particle systems in infinite volume

We prove a large deviation theorem from the hydrodynamical limit for the empirical measure of Ginzburg-Landau and zero range processes in infinite volume starting from deterministic initial configurations. In the Ginzburg-Landau case the main tool is the study of the evolution of the H_?1 norm and in the zero range case the attractiveness which allows couplings.     

A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C^∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ?C × H for which $\text{D?}{}_{\mathrm{c}}\text{(x) = 0}$; here we write $\text{?}{}_{\mathrm{c}}\text{(x)}$ for $\text{?(c, x)}$. Say ? is of standard type if at all critical points of ?_c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function f_o?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in $\text{R}$ⁿ.     

On some properties of convolution operators in K′1 and S′

Let $\text{H}$′ be either the space $\text{K}$′₁ of distributions of exponential growth or the space $\text{S}$′ of tempered distributions, and let $\text{O}$′_C($\text{H}$′:$\text{H}$′) be the space of convolution operators in $\text{H}$′. In each case $\text{H}$′ is the dual of a space $\text{H}$ of C^∞-functions which are in $\text{O}$′_C($\text{H}$′:$\text{H}$′). We establish necessary and sufficient conditions on the Fourier transform $\text{S}\text{?}$ of ? of S ? $\text{O}$′_C($\text{H}$′:$\text{H}$′) in order that every distribution u? $\text{O}$′_C($\text{H}$′:$\text{H}$′) with S1u?$\text{H}$ be in $\text{H}$. If $\text{H}$′ = $\text{K}$′₁, the condition is equivalent to S×H′¹=H′¹.     

Inviscid limit for the energy-critical complex Ginzburg-Landau equation

In this paper, we consider the limit behavior for the solution of the Cauchy problem of the energy-critical complex Ginzburg-Landau equation in Rn, n?3. In lower dimension case (3?n?6), we show that its solution converges to that of the energy-critical nonlinear Schrödinger equation in , T>0, s=0,1, as a by-product, we get the regularity of solutions in H³ for the nonlinear Schrödinger equation. In higher dimension case (n>6), we get the similar convergent behavior in C(0,T,L²(Rn)). In both cases we obtain the optimal convergent rate.     

Strong summability theorems for H p (Δ d )

We prove that certain means of the (C,α,…,α)-means (α=1/p?1) of the d-dimensional trigonometric Fourier series are uniformly bounded operators from the Hardy space H _p to H _p (1≦p≦2). As a consequence we obtain strong summability theorems concerning (C,α,…,α)-means.     

On spectral theory in discrete programming: References added in proof

In different parts of discrete programming so-called (H,A,L_C)-problems are studied, where one must find an h_o?H (H is a set of permutation), for which L_C(h_o) = min_h?HL_C(h), L_C(h) = c_1h(1) +...+ c_nh(n), C = | c_ij | is an n ×n-matrix over A, and A is a totally ordered commutative semigroup (for example, semigroup of positive real numbers or a finite commutative totally ordered semigroup). We are dealing with the full spectrum of values of the function L_C and not only with the solutions of a (H,A,L_C)-problem. Equivalence theorems for different classes of these problems are proved. Realizability of spectra in some classes of (H,A,L_C)-problems is studied.     

Linear fractional composition operators on H2

If ? is an analytic function mapping the unit diskD into itself, the composition operatorC _? is the operator onH ² given byC _?f=fo?. The structure of the composition operatorC _? is usually complex, even if the function ? is fairly simple. In this paper, we consider composition operators whose symbol ? is a linear fractional transformation mapping the disk into itself. That is, we will assume throughout that $$\varphi \left( z \right) = \frac{{az + b}}{{cz + d}}$$ for some complex numbersa, b, c, d such that ? maps the unit diskD into itself. For this restricted class of examples, we address some of the basic questions of interest to operator theorists, including the computation of the adjoint.     

Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D₄,E₆,E₇,E₈). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G=Z??Z², where ? is 2, 3, 4, and 6, respectively. In this paper, we define a flat deformation H(t,q) of the group algebra C[G] of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra H(t,q) for D₄ is the Cherednik algebra of type C^∨C₁, which was studied by Noumi, Sahi, and Stokman, and controls Askey-Wilson polynomials. We prove that H(t,q) is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on C[G]. We also show that if q is a root of unity, then for generic t the algebra H(t,q) is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra eH(t,q)e provides a quantization of such surfaces. We also discuss connections of H(t,q) with preprojective algebras and Painlevé VI.     

Unitarization of symplectics and stability for causal differential equations in Hilbert space

Let Sp($\text{H}$) be the symplectic group for a complex Hibert space $\text{H}$. Its Lie algebra sp($\text{H}$) contains an open invariant convex cone C₀; each element of C₀ commutes with a unique sympletic complex structure. The Cayley transform $\text{C}$: X∈ sp($\text{H}$)→(I + X)¹∈ Sp($\text{H}$) is analyzed and compared with the exponential mapping. As an application we consider equations of the form $\text{(}\text{d}\text{dt}\text{) S = A(t)S,}\text{where}\text{t → A(t) ?}\text{C}\text{?}{}_0$ is strongly continuous, and show that if ∝_?∞^∞ ∥A(t)∥ dt < 2 and ∝_{? t8}^∞A(t) dt?C₀, the (scattering) operator

, where S_t′(t) is the solution such that S_t′(t′) = I, is in the range of $\text{B}$ restricted to C₀. It follows that S leaves invariant a unique complex structure; in particular, it is conjugate in Sp($\text{H}$) to a unitary operator.     

A generalized weight for linear codes and a Witt-MacWilliams theorem

Let F be a finite field, H a subgroup of $\text{F}{}^1$ of index ν, and α₁,…, α_ν coset representatives. For each n-tuple u = (u₁,…, u_n) ?Fⁿ define W_H(u) = (w₁(u),…, w_ν(u)), where w_m(u) = #{u_i: u_i?α_mH}. An H-monomial map on Fⁿ is an automorphism of Fⁿ whose matrix with respect to the co-ordinate basis is of the form P · D, where P is a permutation matrix and D is a diagonal matrix with non-zero entries from H. Suppose C is an (n, k) code over F (that is, a k-dimensional subspace of Fⁿ) and that ?: C → Fⁿ is an injective homomorphism which preserves W_H in the sense that W_H(?(u)) = W_H(u) for all u ?C. We prove that ? may be extended to an H-monomial map on Fⁿ. This generalization of a theorem of MacWilliams on the (Hamming) equivalence of codes may be considered an analogue of the Witt theorem of metric vector space theory.     

Singular perturbations of abstract wave equations

Given, on the Hilbert space H₀, the self-adjoint operator B and the skew-adjoint operators C₁ and C₂, we consider, on the Hilbert space H?D(B)⊕H₀, the skew-adjoint operator

     

Asymptotics of convex sets in Euclidean and hyperbolic spaces

We study convex sets C of finite (but non-zero) volume in Hn and En. We show that the intersection C_∞ of any such set with the ideal boundary of Hn has Minkowski (and thus Hausdorff) dimension of at most (n−1)/2, and this bound is sharp, at least in some dimensions n. We also show a sharp bound when C_∞ is a smooth submanifold of _∂∞Hn. In the hyperbolic case, we show that for any k?(n−1)/2 there is a bounded section S of C through any prescribed point p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of a convex body in En, and give asymptotic estimates as 1?k?n.     

TheC ℓ Bailey transform and Bailey lemma

TheC _? nonterminating C_? summation theorem is derived by appropriately specializing Gustafson's₆ψ₆ summation theorem for bilateral basic hypergeometric series very well-poised on symplecticC _? groups. From this, the terminating₆?₅ and, hence, terminating₄?₃ summation theorem is obtained. A suitably modified₄?₃ is then used to derive theC _? generalization of the Bailey transform. The transform is then interpreted as a matrix inversion result for two infinite, lower-triangular matrices. This result is used to motivate the definition of theC _? Bailey pair. TheC _? generalization of Bailey's lemma is then proved. This result is inverted, and the concept of the bilateral Bailey chain is discussed. TheC _? Bailey lemma is then used to obtain a connection coefficient result for generalC _? littleq-Jacobi polynomials. All of this work is a natural extension of the unitaryA _?, or equivalentlyU(?+1), case. The classical case, corresponding toA ₁ or equivalentlyU(2), contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur identities. TheC _? nonterminating₆?₅ summation theorem is also used to recover C. Krattenthaler's multivariable summation which he utilized in deriving his refinement of the Bender-Knuth and MacMahon generating functions for certain sets of plane partitions.     

Exponential sets and their geometric motions

Let us call an “exponential set” in a C*-algebraA any set consisting of the exponentialse ^X of all the self-adjoint elementsX of a subspaceH ofA. For example, ifH = A the resulting exponential setG ⁺ consists of all the positive invertible elements ofA, and all other exponential sets are contained in G⁺. An exponential setC ? G⁺ inherits the geometric structure of the space G⁺ when the defining subspaceH has suitable properties. Here we investigate reasonable conditions onH that permit, for example, reduction of the canonical connection of G⁺ toC. As a consequence, in these cases the setC has a rich family of motions that are “rigid” for the geometry of G⁺. In particular we find thatC itself operates on C by the actionL _g a = (g^?1)*ag^? of the groupG of all invertible elements ofA in G⁺, and that the subgroup generated byC is transitive. Similarly, in several cases the productscu withc ε C andu unitary form a closed Lie subgroup ofG that acts onC, withC contained in it. This is the case forH, the space of elements of trace zero, when there is a trace. The conditions onH are all additions to the following basic situation:H is the kernel of a (bounded linear) projection Φ:A → A. For example, ifH is closed under triple brackets [X, [Y, Z]] then parallel transport in G⁺ along geodesics inC through 1 ∈C preserves vectors tangent toC. Similarly, if the symmetric part of [e ^X Ye ^?X ,Z] is inH for allX, Y, Z ∈H ^s thenC is “geodesically convex” in the sense that geodesics tangent toC stay inC. The most interesting cases correspond to a conditional expectation. Two additional conditions produce the groups described in the first paragraph: the case of a Z₂-graded C*-algebra with Φ the projection on the elements of degree 0 (which is automatically a conditional expectation) and the case of a conditional expectation such that the anti-symmetric part ofe ^X Ye ^?Y is in the range of Φ wheneverX, Y are self-adjoint and Φ(X)= Φ(Y) = 0. This is verified for example in the case of central traces.     

Hypercyclicity and unimodular point spectrum

We show that an operator on a separable complex Banach space with sufficiently many eigenvectors associated to eigenvalues of modulus 1 is hypercyclic. We apply this result to construct hypercyclic operators with prescribed K_σ unimodular point spectrum. We show how eigenvectors associated to unimodular eigenvalues can be used to exhibit common hypercyclic vectors for uncountable families of operators, and prove that the family of composition operators C_? on H²(D), where ? is a disk automorphism having +1 as attractive fixed point, has a residual set of common hypercyclic vectors.     

Wavelet-based estimator for the hurst parameters of fractional brownian sheet

It is proposed a class of statistical estimators H =(H_1,…,H_d) for the Hurst parameters H =(H_1,…,H_d) of fractional Brownian field via multi-dimensional wavelet analysis and least squares,which are asymptotically normal.These estimators can be used to detect self-similarity and long-range dependence in multi-dimensional signals,which is important in texture classification and improvement of diffusion tensor imaging(DTI) of nuclear magnetic resonance(NMR).Some fractional Brownian sheets will be simulated and the simulated data are used to validate these estimators.We find that when H_i ≥ 1/2,the estimators are accurate,and when H_i 1/2,there are some bias.     

More on the generalized Macaulay theorem

Let k₁ ? k₂ ? … ? k_n be given positive integers and let F denote the set of vectors (l₁, …, l_n) with integer components satisfying 0 ? l_i ? k_i, i = 1, 2, …, n. If H is a subset of F, let (l)H denote the subset of H consisting of those vectors with component sum l, and let C((l)H) denote the smallest [(l)H] elements of (l)F. The generalized Macaulay theorem due to the author and B. Lindström [3] shows that |Gamma;((C)(l)(H)|, ? |Γ(C((l)H))|, where Γ((l)H) is the setof vectors in F obtainable by subtracting l from a single component of a vector in (l)H. A method is given for computing [Γ(C((l)H)] in this paper. It is analogous to the method for computing |Γ(C(l)H))| in the k₁ = … = k_n = 1 case which has been given independently by Katona [4] and Kruskal [5].