Quantum complexity of the approximation for the classes B(Wp ([0, 1])) and B(Hp ([0, 1]))

We study the approximation of functions from anisotropic Sobolev classes B(W^r_p([0,1]d)) and Hölder-Nikolskii classes B(H^r_p([0,1]d)) in the L_q([0,1]d) norm with q ≤ p in the quantum model of computation. We determine the quantum query complexity of this problem up to logarithmic factors. It shows that the quantum algorithms are significantly better than the classical deterministic or randomized algorithms.     

Best N Term Approximation Spaces for Tensor Product Wavelet Bases

We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor product structure ⊗_q on certain quasi-Banach spaces. We prove that the approximation spaces A^α_q(L₂) and A^α_q(H¹) equal tensor products of Besov spaces B^α_q(L_q), e.g., A^α_q(L₂([0,1]^d)) = B^α_q(L_q([0,1])) ⊗_q · ⊗_q B^α_{q · ·}(L_q([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales of Besov spaces.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm of spectral clusters for compact manifolds with boundary

We use microlocal and paradifferential techniques to obtain L ⁸ norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal L ^q bounds, in the range 2⩽q⩽∞, for L ² - normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp L ^q estimates in higher dimensions for a range of exponents q̅_n⩽q⩽∞. The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS-0099642, and DMS-0354668.     

Space-time means and solutions to a class of nonlinear parabolic equations

Cauchy problem and initial boundary value problem for nonlinear parabolic equation inCB([0,T):L ^p ) orL ^q (0,T; L ^p ) type space are considered. Similar to wave equation and dispersive wave equation, the space-time means for linear parabolic equation are shown and a series of nonlinear estimates for some nonlinear functions are obtained by space-time means. By Banach fixed point principle and usual iterative technique a local mild solution of Cauchy problem or IBV problem is constructed for a class of nonlinear parabolic equations inCB([0,T);L ^p orL ^q (0,T; L ^p ) with ϕ(x)∈L ^r . In critical nonlinear case it is also proved thatT can be taken as infinity provided that ||ϕ(x)||_r is sufficiently small, where (p,q,r) is an admissible triple. Project supported by the National Natural Science Foundation of China (Grant No. 19601005).     

The Characterization of Finite Simple Groups, L3（3^2m-1）（m≥2）, by Their Element Orders

In this paper the following theorem is proved： Every group L3（q） for q = 3^（2m-1）（m≥2） is characterized by its set of element orders.     

The information-based complexity of approximation problem by adaptive Monte Carlo methods

In this paper, we study the complexity of information of approximation problem on the multivariate Sobolev space with bounded mixed derivative MWpr,α(Td), 1 < p < ∞, in the norm of Lq(Td), 1 < q < ∞, by adaptive Monte Carlo methods. Applying the discretization technique and some properties of pseudo-s-scale, we determine the exact asymptotic orders of this problem.     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

The Cauchy problem for nonhomogeneous Navier-Stokes equations inL q([0,T);L p)

In this paper the Cauchy problem for a class of nonhomogeneous Navier-Stokes equations in the infinite cylinderS _T =ℝⁿ x [0,T) is considered. We construct a unique local solution inL ^q([0,T);L ^p (ℝ ⁿ )) for a class of nonhomogeneous Navier-Stokes equations provided that initial data are inL ^r (ℝ ⁿ ), wherer>1 is an exponent determined by the structure of nonlinear terms andp,q are such that 2/q=n(1/r−1/p). Meanwhile under suitable conditions we also obtain thatu(t)≠L ^q([0,∞];L ^p (ℝ ⁿ )) provided that initial data are sufficiently small. This work is supported by the National Natural Sciences Foundation of China and the Foundation of LNM Laboratory of Institute of Mechanics of the Chinese Academy of Sciences.     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

Some extremal properties of multivariate polynomial splines in the metric Lp(?d)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (ℝ^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (ℝ^d in the metric L_p((ℝ^d).     

The generalized hu-meyer formula for random kernels

A theory of Hilbert-space-valued traces and multiple integration is developed for kernels inL ²([0, 1]^p × Θ). The multiple Ogawa and the multiple Stratonovich integrals for such kernels are introduced and sufficient conditions for their existence are obtained. The derivation of the Hu-Meyer formula connecting the multiple Ogawa and the multiple Stratonovich integrals requires the introduction of traces of random kernels. Such a derivation is obtained under appropriate conditions. This research was supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL03-92-G-0008.     

On Hua-Tuan’s conjecture

Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.     

Average B-width and infinite dimensional G-width of some smooth function classes on the line

In this paper, we get the exact values of average σ-B width and infinite dimensional σ-G width of Sobolev class B^r _p(R) in the metric L_p(R) (1≤p≤∞) and obtain the exact (σ∈N) and strong asymptotic (σ>1) results of infinite dimensional σ-G widths of Sobolev-Wiener class W^r _pq (R) in the metric L_q(R) and its dual case W^r _p(R) in the metric L_qp(R) (1≤q≤p≤∞). Supported by the National Natural Science Foundation of China (No. 19671012) and by the Doctoral Programme Foundation of Institution of Higher Education of National Education Committee of China.     

THE GLOBAL APPROXIMATION BY LEFT-BERNSTEIN-DURRMEYER QUASI-INTERPOLANTS IN Lp[0, 1]

In this paper,we will use the 2r-th Ditzian-Totik modulus of smoothness wp^2r(f,t)p to discuss the direct and inverse theorem of approximation by Left-Bernstein-Durrmeyer quasi-interpolants Mn^[2r-1]f for functions of the space Lp[0,1](1≤p≤ ∞)。     

Approximation to the Sobolev classesW q r of functions of several variables by bilinear forms inL p for 2≤q≤p≤∞

We study the approximation of functions of several variables by bilinear forms that are the pairwise products of functions of fewer variables. The order of approximation of Sobolev classesW _q ^r by bilinear forms inL _p for 2≤q≤p≤∞ is found. Translated by N. K. Kulman Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 18–34, July, 1997.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

Quantum integration in Sobolev classes

We study high-dimensional integration in the quantum model of computation. We develop quantum algorithms for integration of functions from Sobolev classes W_p^r([0,1]^d) and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of Novak on integration of functions from Hölder classes.     

Finite 2-generator equilibrated p-groups

Following Blackburn, Deaconescu and Mann, a group G is called an equilibrated group if for any subgroups H,K of G with HK = KH, either H≤NG(K) or K≤NG(H). Continuing their work and based on the classification of metacyclic p-groups given by Newman and Xu, we give a complete classification of 2-generator equilibrated p-groups in this note.     

Weighted estimates for commutators of singular integral operators with non-doubling measures

Letμbe a nonnegative Radon measure on R~d which only satisfiesμ(B(x,r))≤C_0r~n for all x∈R~d,r>0,and some fixed constants C_0>0 and n∈(0,d].In this paper,some weighted weak type estimates with A_(p,(log L)~σ)~ρ(μ) weights are established for the commutators generated by Calder■n-Zygmund singular integral operators with RBMO(μ) functions.     

Moderate deviations for the quenched mean of the super-Brownian motion with random immigration

Moderate deviations for the quenched mean of the super-Brownian motion with random immigration are proved for 3≤d≤6, which fills in the gap between central limit theorem(CLT)and large deviation principle(LDP).