Construction of nonseparable dual -wavelet frames in

Suppose and are two pairs of dual M-wavelet frames and N-wavelet frames in and , respectively, where M and N are s₁×s₁ and s₂×s₂ dilation matrices with a₁(|det(M)|-1) and (2a₂+1)(|det(N)|-1). Moreover, their mask symbols both satisfy mixed extension principle (MEP). Based on the mask symbols, a family of nonseparable dual Ω-wavelet frames in are constructed, where s=s₁+s₂, and with Θ and M^-1Θ both being integer matrices. Then a convolution scheme for improving regularity of wavelet frames is given. From the nonseparable dual Ω-wavelet frames, nonseparable Ω-wavelet frames with high regularity can be constructed easily. We give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in . From the dual Ω-wavelet frames, nonseparable dual Ω-wavelet frames with symmetry can be obtained easily. In the end, two examples are given.     

Instance-optimality in probability with an -minimization decoder

Let Φ(ω), ωΩ, be a family of n×N random matrices whose entries _i,j are independent realizations of a symmetric, real random variable η with expectation and variance . Such matrices are used in compressed sensing to encode a vector by y=Φx. The information y holds about x is extracted by using a decoder . The most prominent decoder is the ℓ₁-minimization decoder Δ which gives for a given the element which has minimal ℓ₁-norm among all with Φz=y. This paper is interested in properties of the random family Φ(ω) which guarantee that the vector will with high probability approximate x in to an accuracy comparable with the best k-term error of approximation in for the range kan/log₂(N/n). This means that for the above range of k, for each signal , the vector satisfies

with high probability on the draw of Φ. Here, Σ_k consists of all vectors with at most k nonzero coordinates. The first result of this type was proved by Wojtaszczyk [P. Wojtaszczyk, Stability and instance optimality for Gaussian measurements in compressed sensing, Found. Comput. Math., in press] who showed this property when η is a normalized Gaussian random variable. We extend this property to more general random variables, including the particular case where η is the Bernoulli random variable which takes the values with equal probability. The proofs of our results use geometric mapping properties of such random matrices some of which were recently obtained in [A. Litvak, A. Pajor, M. Rudelson, N. Tomczak-Jaegermann, Smallest singular value of random matrices and geometry of random polytopes, Adv. Math. 195 (2005) 491–523].     

Double point self-transverse immersions of

A self-transverse immersion of a smooth manifold M^8k in has a double point self-intersection set which is the image of an immersion of a smooth four-dimensional manifold, cobordent to P⁴, P²×P², P⁴+P²×P² or a boundary. We will prove that for any value of k>1 the double point self-intersection set is a boundary. If k=1, then there exists an immersion of P²×P²×P²×P² in with double point manifold boundary and odd number of triple points. In particular any immersion of oriented manifold in this dimension has double point manifold cobordant to a boundary.     

Erdős–Ko–Rado theorems for permutations and set partitions

Let Sym([n]) denote the collection of all permutations of [n]={1,…,n}. Suppose is a family of permutations such that any two of its elements (when written in its cycle decomposition) have at least t cycles in common. We prove that for sufficiently large n, with equality if and only if is the stabilizer of t fixed points. Similarly, let denote the collection of all set partitions of [n] and suppose is a family of set partitions such that any two of its elements have at least t blocks in common. It is proved that, for sufficiently large n, with equality if and only if consists of all set partitions with t fixed singletons, where B_n is the nth Bell number.     

Widths of weighted Sobolev classes on the ball

We study the Kolmogorov n-widths and the linear n-widths of weighted Sobolev classes on the unit ball B^d in L_q,μ, where L_q,μ, 1≤q≤∞, denotes the weighted L_q space of functions on B^d with respect to weight . Optimal asymptotic orders of and as n→∞ are obtained for all 1≤p,q≤∞ and μ≥0.     

Asymptotics of the orthogonal polynomials for the Szegő class with a polynomial weight

Let p be a trigonometric polynomial, non-negative on the unit circle . We say that a measure σ on belongs to the polynomial Szegő class, if , σ_s is singular, and

For the associated orthogonal polynomials {_n}, we obtain pointwise asymptotics inside the unit disc . Then we show that these asymptotics hold in L²-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.     

Quadratic and cubic invariants of unipotent affine automorphisms

Let K be an arbitrary field of characteristic zero, P_n:=K[x₁,…,x_n] be a polynomial algebra, and , for n2. Let σ^′Aut_K(P_n) be given by

It is proved that the algebra of invariants, , is a polynomial algebra in n−1 variables which is generated by quadratic and cubic (free) generators that are given explicitly.Let σAut_K(P_n) be given by

It is well known that the algebra of invariants, , is finitely generated (theorem of Weitzenböck [R. Weitzenböck, Über die invarianten Gruppen, Acta Math. 58 (1932) 453–494]), has transcendence degree n−1, and that one can give an explicit transcendence basis in which the elements have degrees 1,2,3,…,n−1. However, it is an old open problem to find explicit generators for F_n. We find an explicit vector space basis for the quadratic invariants, and prove that the algebra of invariants is a polynomial algebra over in n−2 variables which is generated by quadratic and cubic (free) generators that are given explicitly.The coefficients of these quadratic and cubic invariants throw light on the ‘unpredictable combinatorics’ of invariants of affine automorphisms and of SL₂-invariants.     

Spaces of holomorphic functions in regular domains

Let Ω be a regular domain in the complex plane , . Let be the linear space over of the holomorphic functions f in Ω such that f⁽ⁿ⁾ is bounded in Ω and is continuously extendible to the closure of Ω, n=0,1,2,… . We endow , in a natural manner, with a structure of Fréchet space and we obtain dense subspaces F of , with good topological linear properties, also satisfying that each function f of F, distinct from zero, does not extend holomorphically outside Ω.     

On linear combinations of two tripotent, idempotent, and involutive matrices

Let A=c₁A₁+c₂A₂, wherec₁, c₂ are nonzero complex numbers and (A₁,A₂) is a pair of two n×n nonzero matrices. We consider the problem of characterizing all situations where a linear combination of the form A=c₁A₁+c₂A₂ is (i) a tripotent or an involutive matrix when are commuting involutive or commuting tripotent matrices, respectively, (ii) an idempotent matrix when are involutive matrices, and (iii) an involutive matrix when are involutive or idempotent matrices.     

Branching rules for unitary groups and spectra of invariant differential operators on complex Grassmannians

In this paper, we prove a combinatorial rule describing the restriction of any irreducible representation of U(n+m) to the subgroup U(n)×U(m). We also derive similar rules for the reductions from SU(n+m) to S(U(n)×U(m)), and from SU(n+m) to SU(n)×SU(m). As applications of these representation-theoretic results, we compute the spectra of the Bochner–Laplacian on powers of the determinant bundle over the complex Grassmannian . The spectrum of the Dirac operator acting on the spin Grassmannian is also partially computed. A further application is given by the determination of the spectrum of the Hodge–Laplacian acting on the space of smooth functions on the unit determinant bundle over .     

The Van der Waerden conjecture for mixed discriminants

We prove that the mixed discriminant of doubly stochastic n-tuples of semidefinite hermitian n×n matrices is bounded below by and that this bound is uniquely attained at the n-tuple . This result settles a conjecture posed by R. Bapat in 1989. We consider various generalizations and applications of this result.     

Regular matrices and their strong preservers over semirings

An m×n matrix A over a semiring is called regular if there is an n×m matrix G over such that AGA=A. We study the problem of characterizing those linear operators T on the matrices over a semiring such that T(X) is regular if and only if X is. Complete characterizations are obtained for many semirings including the Boolean algebra, the nonnegative reals, the nonnegative integers and the fuzzy scalars.     

Hamilton cycles in random lifts of graphs

An n-lift of a graph K is a graph with vertex set V(K)×[n], and for each edge (i,j)E(K) there is a perfect matching between {i}×[n] and {j}×[n]. If these matchings are chosen independently and uniformly at random then we say that we have a random n-lift. We show that there are constants h₁,h₂ such that if h≥h₁ then a random n-lift of the complete graph K_h is hamiltonian and if h≥h₂ then a random n-lift of the complete bipartite graph K_h,h is hamiltonian .     

Vector refinement equations with infinitely supported masks

In this paper we investigate the L₂-solutions of vector refinement equations with exponentially decaying masks and a general dilation matrix. A vector refinement equation with a general dilation matrix and exponentially decaying masks is of the form

where the vector of functions φ=(φ₁,…,φ_r)^T is in is an exponentially decaying sequence of r×r matrices called refinement mask and M is an s×s integer matrix such that lim_n→∞M^-n=0. Associated with the mask a and dilation matrix M is a linear operator Q_a on given by

The iterative scheme is called vector subdivision scheme or vector cascade algorithm. The purpose of this paper is to provide a necessary and sufficient condition to guarantee the sequence to converge in L₂-norm. As an application, we also characterize biorthogonal multiple refinable functions, which extends some main results in [B. Han, R.Q. Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, Appl. Comput. Harmon. Anal., to appear] and [R.Q. Jia, Convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets, Advances in Wavelet (Hong Kong, 1997), Springer, Singapore, 1998, pp. 199–227] to the general setting.     

Monotone Jacobi parameters and non-Szegő weights

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for a_n≡1, b_n=−Cn^−β (), one has on (−2,2), and near x=2, where

     

Varieties of modules for

Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety , that is the set of pairs of (n×n)-matrices (A,B) such that A²=B²=[A,B]=0, is equidimensional. can be identified with the ‘variety of n-dimensional modules’ for , or equivalently, for k[X,Y]/(X²,Y²). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic >2. We also prove that if e²=0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of as a direct sum of indecomposable components of varieties of -modules.     

Splitting methods for SU(N) loop approximation

The problem of finding the correct asymptotic rate of approximation by polynomial loops in dependence of the smoothness of the elements of a loop group seems not well-understood in general. For matrix Lie groups such as , it can be viewed as a problem of nonlinearly constrained trigonometric approximation. Motivated by applications to optical FIR filter design and control, we present some initial results for the case of -loops, N≥2. In particular, using representations via the exponential map and first order splitting methods, we prove that the best approximation of an -loop belonging to a Hölder–Zygmund class , α>1/2, by a polynomial -loop of degree ≤n is of the order O(n^−α/(1+α)) as n→∞. Although this approximation rate is not considered final, to our knowledge it is the first general, nontrivial result of this type.     

Zeros of Bernoulli-type functions and best approximations

The zero sets of (D+a)ⁿg(t) with in the (t,a)-plane are investigated for and .The results are used to determine entire interpolations to functions , which give representations for the best approximation and best one-sided approximation from the class of functions of exponential type η>0 to .     

On wavelets related to the Walsh series

For any integers p,n≥2 necessary and sufficient conditions are given for scaling filters with pⁿ many terms to generate a p-multiresolution analysis in . A method for constructing orthogonal compactly supported p-wavelets on is described. Also, an adaptive p-wavelet approximation in is considered.     

A note on the distance between two consecutive zeros of m-orthogonal polynomials for a generalized Jacobi weight

Let , with

-1=x_0n<x_1n<<x_nn<x_n+1,n=1