Stability under Small Hilbert–Schmidt Perturbations for <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

This paper studies the tracial stability of C^*-algebras, which is a general property of stability of relations in a Hilbert–Schmidt-type norm defined by a trace on a C^*-algebra. Precise definitions are formulated in terms of tracial ultraproducts. For nuclear C^*-algebras, a characterization of matricial tracial stability in terms of approximation of tracial states by traces of finite-dimensional representations is obtained. For the nonnuclear case, new obstructions and counterexamples are constructed in terms of free entropy theory.     

Dirichlet forms for diffusion in ℝ<Superscript>2</Superscript> and jumps on fractals: The regularity problem

We define a Dirichlet form ɛ describing diffusion in ℝ ^d and jumps in a fractal Γ ⊂ ℝ ^d . The jump measure J is defined as an image of a jump measure j of a process in a non-Archimedean metric space. As the result the jump intensity depends on the hierarchical structure of Γ rather than the geometric distance in ℝ ^d . For a class of fractals in ℝ² we find a condition on the measure j so that the Dirichlet form ɛ is regular. The condition is given in terms of Hausdorff dimension of Γ.     

On the Nitsche conjecture for harmonic mappings in ℝ<Superscript>2</Superscript> and ℝ<Superscript>3</Superscript>

We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. LetA(r, 1)={z:r<|z|<1}. Nitsche's conjecture states that if there exists a univalent harmonic mapping from an annulusA(r, 1), to an annulusA(s, 1), thens is at most 2r/(r ²+1).Lyzzaik's result states thats<t wheret is the length of the Grötzsch's ring domain associated withA(r, 1) (see [5]). Weitsman's result states thats≤1/(1+1/2(r logr)²) (see [8]).Our result for two-dimensional space states thats≤1/(1+1/2 log² r) which improves Weitsman's bound for allr, and Lyzzaik's bound forr close to 1. For three-dimensional space the result states thats≤1/(r?logr).     

Sobolev estimates for the Cauchy–Riemann complex on <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> pseudoconvex domains

Let be a bounded pseudoconvex domain with C ^k boundary, k ≥ 1. In this paper, we will prove that the Cauchy–Riemann operator has a bounded solution operator in the Sobolev space for all .     

Asymptotic unitary equivalence and classification of simple amenable <Emphasis Type="Italic">C</Emphasis><Superscript>∗</Superscript>-algebras

Let C and A be two unital separable amenable simple C ^?-algebras with tracial rank at most one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that ? ₁,? ₂:C→A are two unital monomorphisms. We show that there is a continuous path of unitaries {u _t :t∈[0,∞)} of A such that

$\lim_{t\to\infty}u_t^*\varphi_1(c)u_t=\varphi_2(c)\quad\mbox{for all }c\in C$

if and only if [? ₁]=[? ₂] in \(KK(C,A),\varphi_{1}^{\ddag}=\varphi_{2}^{\ddag},(\varphi_{1})_{T}=(\varphi _{2})_{T}\) and a rotation related map \(\overline{R}_{\varphi_{1},\varphi_{2}}\) associated with ? ₁ and ? ₂ is zero.

Applying this result together with a result of W. Winter, we give a classification theorem for a class \({\mathcal{A}}\) of unital separable simple amenable C ^?-algebras which is strictly larger than the class of separable C ^?-algebras with tracial rank zero or one. Tensor products of two C ^?-algebras in \({\mathcal{A}}\) are again in \({\mathcal{A}}\). Moreover, this class is closed under inductive limits and contains all unital simple ASH-algebras for which the state space of K ₀ is the same as the tracial state space and also some unital simple ASH-algebras whose K ₀-group is ? and whose tracial state spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AH-algebras which are \({\mathcal{Z}}\)-stable are isomorphic to ones with no dimension growth.     

Determination of cusp forms by central values of Rankin–Selberg <Emphasis Type="BoldItalic">L</Emphasis>-functions<Superscript>*</Superscript>

Let f be a fixed holomorphic Hecke eigen cusp form of weight k for \( SL\left( {2,{\mathbb Z}} \right) \), and let \( {\mathcal U} = \left\{ {{u_j}:j \geqslant 1} \right\} \) be an orthonormal basis of Hecke–Maass cusp forms for \( SL\left( {2,{\mathbb Z}} \right) \). We prove an asymptotic formula for the twisted first moment of the Rankin–Selberg L-functions \( L\left( {s,f \otimes {u_j}} \right) \) at \( s = \frac{1}{2} \) as u _j runs over \( {\mathcal U} \). It follows that f is uniquely determined by the central values of the family of Rankin–Selberg L-functions \( \left\{ {L\left( {s,f \otimes {u_j}} \right):{u_j} \in {\mathcal U}} \right\} \).     

Projective Multi-Resolution Analyses for L<Superscript>2</Superscript>(ℝ<Superscript>2</Superscript>)

We define the notion of projective multiresolution analyses, for which, by definition, the initial space corresponds to a finitely generated projective module over the algebra C(Tⁿ) of continuous complex-valued functions on an n-torus. The case of ordinary multi-wavelets is that in which the projective module is actually free. We discuss the properties of projective multiresolution analyses, including the frames which they provide for L²(ⁿ). Then we show how to construct examples for the case of any diagonal 2 × 2 dilation matrix with integer entries, with initial module specified to be any fixed finitely generated projective C(T²)-module. We compute the isomorphism classes of the corresponding wavelet modules.     

A <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Multivariate Clough–Tocher Interpolant

We show that in dimensions four and higher, to insure a smooth interpolant, additional geometric constraints must be imposed on the generalized Clough–Tocher split introduced in Worsey and Farin (Constr. Approx. 3:99–110, [1987]).      

A <Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript>-Regularity Theorem for Nondegenerate CR Mappings

We prove the following regularity result: If , are smooth generic submanifolds and M is minimal, then every C^k-CR-map from M into M which is k-nondegenerate is smooth. As an application, every CR diffeomorphism of k-nondegenerate minimal submanifolds in of class C^k is smooth.     

Weierstrass representation for discrete isotropic surfaces in ℝ<Superscript>2,1</Superscript>, ℝ<Superscript>3,1</Superscript>, and ℝ<Superscript>2,2</Superscript>

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation for hyperbolic surfaces parameterized along isotropic directions in ℝ^2,1, ℝ^3,1, and ℝ^2,2. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a general monotonicity condition and having isotropic edges admits such a representation.     

Hyperbolic Chevalley groups on ℂ<Superscript>2</Superscript>

Let Γ ? U (1, 1) be the subgroup generated by the complex reflections. Suppose that Γ acts discretely on the domain K = {(z ₁, z ₂) ∈ ?² ||z ₁|² ? |z ₂|² < 0} and that the projective group PΓ acts on the unit disk B = {|z ₁/z ₂| < 1} as a Fuchsian group of signature (n ₁, ..., n _s ), s ? 3, n _i ? 2. For such groups, we prove a Chevalley type theorem, i.e., find a necessary and sufficient condition for the quotient space K/Γ to be isomorphic to ?² ? {0}.     

Siegmund Duality for Continuous Time Markov Chains on ℤ<Stack><Subscript>+</Subscript><Superscript>d</Superscript></Stack>

For the continuous time Markov chain with transition function P(t) on Z₊^d, we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P(t). If Q, the q-matrix of P(t), is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of q-matrices, and a sufficient condition under which the Q-function is the Siegmund dual of some Q-function is also given.     

The Primitive Ideals of some Étale Groupoid <Emphasis Type="Italic">C</Emphasis><Superscript>∗</Superscript>-Algebras

We consider the Deaconu–Renault groupoid of an action of a finitely generated free abelian monoid by local homeomorphisms of a locally compact Hausdorff space. We catalogue the primitive ideals of the associated groupoid C ^?-algebra. For a special class of actions we describe the Jacobson topology.     

The <Emphasis Type="Italic">C</Emphasis><Superscript>α</Superscript> regularity of a class of non-homogeneous ultraparabolic equations

We obtain the C ^α regularity for weak solutions of a class of non-homogeneous ultraparabolic equation, with measurable coefficients. The result generalizes our recent C ^α regularity results of homogeneous ultraparabolic equations. This work was supported by National Natural Science Foundation of China (Grant No. 10325104)     

Self-normalized limit theorems for linear processes generated by <Emphasis Type="Italic">ρ</Emphasis>-mixing innovations<Superscript>*</Superscript>

In this paper, we study the asymptotic behavior of the self-normalizer V _n ² for partial sums of linear processes generated by strictly stationary ρ-mixing innovations with infinite variance. Further, by using this we derive self-normalized versions of the CLT, the functional CLT, and the almost sure CLT for partial sums of the processes.     

Total Curvature for <Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript>-Singular Surfaces with Limiting Tangent Bundle

The total curvature of a compact C^∞-immersed surface in Euclidean 3-space ³ can be interpreted as the average number of critical points for a linear ‘height’ function. The Morse inequalities provide an intrinsic topological lower bound for the total curvature and ‘tight’ surfaces, which realize equality, have been an active topic of research. The objective of this paper is to describe the natural notion of total curvature for C^∞-singular surfaces which fail to immerse on C^∞-embedded closed curves, but which have a C^∞-globally defined unit normal (e.g. caustics, or critical images for mappings of 3-manifolds into Euclidean 3-space). For such surfaces total curvature consists of a sum of two-dimensional and one-dimensional integrals, which have various lower bounds. Large sets of LT-surfaces which realize equality are then constructed. As an application, the orthogonal projection of an immersed tight hypersurface in Euclidean 4-space is shown to have LT-tight critical image, and several related inequalities are given. Mathematics Subject Classifications (2000): 57N65, 14P99, 53C21, 53B25, 53B20.     

Stiefel–Whitney invariants for bilinear forms

We examine potential extensions of the Stiefel–Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Witt?s Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)     

Quincunx multiresolution analysis for <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℚ<Stack><Subscript>2</Subscript><Superscript>2</Superscript></Stack>)

With an eye on applications in quantum mechanics and other areas of science, much work has been done to generalize traditional analytic methods to p-adic systems. In 2002 the first paper on p-adic wavelets was published. Since then p-adic wavelet sets, multiresolution analyses, and wavelet frames have all been introduced. However, so far all constructions have involved dilations by p. This paper presents the first construction of a p-adic wavelet system with a more general matrix dilation, laying the foundation for further work in this direction.     

Uniformity seminorms on ℓ<Superscript>∞</Superscript> and applications

A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on ℤ/Nℤ introduced by Gowers in his proof of Szemerédi’s Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg’s proof of Szemerédi’s Theorem) defined by the authors. For each integer k ≥ 1, we define seminorms on ℓ^∞(ℤ) analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences). Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence. As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem.