When solving l2 optimization problems based on linear filtering with some regularization in signal/image processing such asWiener filtering, the fast Fourier transform (FFT) is often available to reduce its computational complexity. Most of the problems, in which the FFT is used to obtain their solutions, are based on single variable equations. On the other hand, the Karush-Kuhn-Tucker (KKT) sy...

The potential V is a real valued, measurable, locally integrable function. H is then a self-adjoint operator whose domain is an appropriate Sobolev space. Considered as an operator on L2([0,∞)), H is likewise self-adjoint if an appropriate boundary condition at zero, e.g. Dirichlet or Neumann, is specified. The quantum mechanical interpretation is that H0 = − d2 dx2 governs the behavior of a fr...

The edit distance between two words w1, w2 is the minimal number of word operations (letter insertions, deletions, and substitutions) necessary to transform w1 to w2. The edit distance generalizes to languages L1,L2, where the edit distance is the minimal number k such that for every word from L1 there exists a word in L2 with edit distance at most k. We study the edit distance computation prob...

A "basophilic method" for phenolic hydroxyl bridged lanthanide coordination polymers (CPs) was developed. With this method, eleven CPs with the general formula of [Ln(HL1)L1·H(2)O](n) (Ln = Tb (1), Nd (2), Eu (3), Gd (4), La (5), Er (6), Y (7), H(2)L1 = 4-methyl salicylic acid) and [Ln(HL2)L2·2MeOH](n) (Ln = Eu (8), Tb (9), Gd (10), La (11), H(2)L2 = 3-hydroxy-2-naphthoic acid) were synthesized...

A planar Bézier curve whose control points are disks is called a disk Bézier curve. It can be looked as a parametric curve with tolerance in the plane. It is an effective tool to measure or control the error. Based on minimum mean square error, this paper presents an algorithm for optimal multi-degree reduction of disk Bézier curves in L2 norm. First, applying the orthogonal property of Jacobi ...

For each f : [0,∞) → C formally consider its co-Poisson or Müntz transform g(x) = ∑ n≥1 f(nx) − 1 x ∫∞ 0 f(t)dt. For certain f ’s with both f, g ∈ L2(0,∞) it is true that the Riemann hypothesis holds if and only if f is in the L2 closure of the vector space generated by the dilations g(kx), k ∈ N. Such is the case for example when f = χ(0,1] where the above statement reduces to the strong Nyman...

In this work, we consider the numerical solution of a distributed order subdiffusion model, arising in the modeling of ultra-slow diffusion processes. We develop a space semidiscrete scheme based on the Galerkin finite element method, and establish error estimates optimal with respect to data regularity in L2(Ω) and H1(Ω) norms for both smooth and nonsmooth initial data. Further, we propose two...

We study two resonant Hamiltonian systems on the phase space L2(R → C): the quintic one-dimensional continuous resonant equation, and a cubic resonant system that appears as the modified scattering limit of a specific NLS equation. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scale...

For each pair (n, k) with 1 </= k < n, we construct a tight frame (rholambda : lambda in Lambda) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wa...