Search

Standard interpolatory subdivision schemes and their underlying interpolating refinable functions are of interest in CAGD, numerical PDEs, and approximation theory. Generalizing these notions, we introduce and study n s -step interpolatory M -subdivision schemes and their interpolating M -refinable functions with n s ∈ N ∪ { ∞ } and a dilation factor M ∈ N \ { 1 } . We completely characterize C m -convergence and smoothness of n s -step interpolatory subdivision schemes and their interpolating M -refinable functions in terms of their masks. Inspired by n s -step interpolatory stationary subdivision schemes, we further introduce the notion of r-mask quasi-stationary subdivision schemes, and then we characterize their C m -convergence and smoothness properties using only their masks. Moreover, combining n s -step interpolatory subdivision schemes with r-mask quasi-stationary subdivision schemes, we can obtain r n s -step interpolatory subdivision schemes. Examples and construction procedures of convergent n s -step interpolatory M -subdivision schemes are provided to illustrate our results with dilation factors M = 2 , 3 , 4 . In addition, for the dyadic dilation M = 2 and r = 2 , 3 , using r masks with only two-ring stencils, we provide examples of C r -convergent r-step interpolatory r-mask quasi-stationary dyadic subdivision schemes. [ABSTRACT FROM AUTHOR]

When constructing multivariate framelets, it is often unavoidable to work with matrices of multivariate trigonometric polynomials and complicated matrix decomposition problems. These problems become even harder when good properties such as high-order vanishing moments are required on the framelets. In this paper, we establish a new method for constructing multivariate dual framelets with high-order vanishing moments. The underlying scheme of our algorithm is the famous Mixed Extension Principle that allows us to derive the high-pass filters (or framelet generators) from a given pair of refinable filters with high-order linear-phase moments. Our method only involves two steps: (1) directly constructing the first few pairs of high-pass filters by using the linear-phase moment conditions of the refinement filters; (2) solving a system of linear equations to obtain the rest of the high-pass filters. Both are easy to implement for scientific computation, regardless of what dimension or dilation matrix we work with. Apart from high-order vanishing moments, we will see that if the refinement filters take coefficients from some subfield of ℂ that is closed under complex conjugation, so do the high-pass filters. Furthermore, our algorithm gives the upper bounds for the number of high-pass filters in arbitrary dimensions. At the end of the paper, we will give several illustrative examples, from which we can also see that the support sizes of the high-pass filters are comparable with those of the refinement filters. [ABSTRACT FROM AUTHOR]

Laws of physical chemistry such as quantum mechanics, laws of chemical kinetics, and thermodynamics have been employed as a potential tool to generate some converging sum rules. Special emphasis has been given to Rayleigh–Schrödinger perturbation theory and the use of delta function potential to generate a number of sum rules. [ABSTRACT FROM AUTHOR]

Due to properties such as interpolation, smoothness, and spline connections, Hermite subdivision schemes employ fast iterative algorithms for geometrically modeling curves/surfaces in CAGD and for building Hermite wavelets in numerical PDEs. In this paper, we introduce a notion of generalized Hermite (dyadic) subdivision schemes and then we characterize their convergence, smoothness and underlying matrix masks with or without interpolation properties. We also introduce the notion of linear-phase moments for achieving the polynomial-interpolation property. For any given integer m ∈ N , we constructively prove that there always exist convergent smooth generalized Hermite subdivision schemes with linear-phase moments such that their basis vector functions are spline functions in C m (R d) and have linearly independent integer shifts. As by-products, our results resolve convergence, smoothness, and existence of Lagrange, Hermite, or Birkhoff subdivision schemes. Even in dimension one our results significantly generalize and extend many known results on extensively studied univariate Hermite subdivision schemes. To illustrate the theoretical results in this paper, we provide examples of convergent generalized Hermite subdivision schemes with symmetric matrix masks having short support and smooth basis vector functions with or without interpolation property. [ABSTRACT FROM AUTHOR]

Factorization of matrices of Laurent polynomials plays an important role in mathematics and engineering such as wavelet frame construction and filter bank design. Wavelet frames (a.k.a. framelets) are useful in applications such as signal and image processing. Motivated by the recent development of quasi-tight framelets, we study and characterize generalized spectral factorizations with symmetry for 2 × 2 matrices of Laurent polynomials. Applying our result on generalized matrix spectral factorization, we establish a necessary and sufficient condition for the existence of symmetric quasi-tight framelets with two generators. The proofs of all our main results are constructive and therefore, one can use them as construction algorithms. We provide several examples to illustrate our theoretical results on generalized matrix spectral factorization and quasi-tight framelets with symmetry. [ABSTRACT FROM AUTHOR]

Hermite interpolation property is desired in applied and computational mathematics. Hermite and vector subdivision schemes are of interest in CAGD for generating subdivision curves and in computational mathematics for building Hermite wavelets to numerically solve partial differential equations. In contrast to well-studied scalar subdivision schemes, Hermite and vector subdivision schemes employ matrix-valued masks and vector input data, which make their analysis much more complicated and difficult than their scalar counterparts. Under the spectral condition or the spectral chain, analysis of Hermite subdivision schemes through factorization of matrix-valued masks has been extensively studied in the literature and sufficient conditions have been given for the convergence of Hermite subdivision schemes through the contractivity of their derived subdivision schemes. We contribute to the study of Hermite subdivision schemes from a different perspective by investigating vector subdivision operators acting on vector polynomials and by establishing connections among Hermite subdivision schemes, vector cascade algorithms, and refinable vector functions. This approach allows us to characterize and construct all masks for Hermite subdivision schemes, to explain the spectral condition and spectral chain in the literature, to characterize convergence and smoothness of Hermite subdivision schemes using vector cascade algorithms, and to provide simple factorizations of Hermite masks through the normal form of matrix-valued masks such that the Hermite subdivision scheme is convergent if and only if its derived subdivision scheme is contractive. We also constructively prove that there always exist arbitrarily smooth convergent Hermite subdivision schemes, whose basis vector functions are splines and have linearly independent shifts. Several examples of Hermite subdivision schemes with short support and high smoothness are presented to illustrate the results in this paper. [ABSTRACT FROM AUTHOR]

We present a calculation of the masses of the isovector mesons (vector, scalar and pseudoscalar including the established recurrences) using a new method of finite energy QCD sum rules. The method is based on the idea of choosing a suitable integration kernel which minimizes the occurring integral over the cut in the complex energy (squared) plane. We obtain remarkably stable results in a wide range R , where R is the radius of the integration contour. The sum rule predictions agree with the experimental values within the expected accuracy showing that QCD describes single resonances. [ABSTRACT FROM AUTHOR]

In this paper, we give a framework to analyze the convergence and smoothness of Hermite subdivision schemes of any arity. The convergence analysis is based on the connections among Hermite subdivision schemes, vector subdivision schemes and refinable function vectors. We first investigate the relation between sum rules and vector subdivision operators acting on vector polynomials of any arity. To study Hermite subdivision schemes, then we depict all Hermite masks of convergent Hermite subdivision schemes of any arity. Under the normal form of matrix-valued masks, factorizations of Hermite masks of Hermite subdivision schemes with any arity are given. Through a quantity defined by sum rules, we can further estimate the smoothness of Hermite subdivision schemes of any arity. Moreover, we can construct Hermite subdivision schemes with arbitrarily high smoothness. Finally, some numerical examples are shown to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In a passive lossy acoustical system, sum rules derived from passivity explicitly relate the broadband response to the spatial dimension of the system, which provide important design criteria as well. In this work, the theory of Herglotz function is applied systematically to derive sum rules for unidimensional scattering problems relying on passive acoustic treatments which are generally made of rigid, motionless and subwavelength structures saturated by air. The rigid-boundary reflection, soft-boundary reflection and transmission problems are analysed. The derived sum rules are applied for guiding the designs of passive absorbers and mufflers: the required minimum space is directly predicted from the target (i.e. the desired absorption or transmission-loss spectra) without any specific design. Besides, it is possible to break this type of sum rules and fundamental constraints in particular cases. This property, if well used, could result in ultra-compact absorbers working at long wavelength up to infinity. [ABSTRACT FROM AUTHOR]

The collective structure of 106Cd is elucidated by multi-step Coulomb excitation of a 3.849 MeV/A beam of 106Cd on a 1.1 mg/cm2 208Pb target using GRETINA-CHICO2 at ATLAS. Fourteen E2 matrix elements were obtained. The nucleus 106Cd is a prime example of emergent collectivity that possesses a simple structure: it is free of complexity caused by shape coexistence and has a small, but collectively active number of valence nucleons. This work follows in a long and currently active quest to answer the fundamental question of the origin of nuclear collectivity and deformation, notably in the cadmium isotopes. The results are discussed in terms of phenomenological models, the shell model, and Kumar-Cline sums of E2 matrix elements. The 〈02+||E2||21+〉 matrix element is determined for the first time, providing a total, converged measure of the electric quadrupole strength, 〈Q2〉, of the first-excited 21+ level relative to the 01+ ground state, which does not show an increase as expected of harmonic and anharmonic vibrations. Strong evidence for triaxial shapes in weakly collective nuclei is indicated; collective vibrations are excluded. This is contrary to the only other cadmium result of this kind in 114Cd by C. Fahlander et al. (1988) [38], which is complicated by low-lying shape coexistence near midshell.

The Thomas–Reiche–Kuhn (TRK) sum rule is a fundamental consequence of the position–momentum commutation relation for an atomic electron, and it provides an important constraint on the transition matrix elements for an atom. Here, we propose a TRK sum rule for electromagnetic fields which is valid even in the presence of very strong light–matter interactions and/or optical nonlinearities. While the standard TRK sum rule involves dipole matrix moments calculated between atomic energy levels (in the absence of interaction with the field), the sum rule here proposed involves expectation values of field operators calculated between general eigenstates of the interacting light–matter system. This sum rule provides constraints and guidance for the analysis of strongly interacting light–matter systems and can be used to test the validity of approximate effective Hamiltonians often used in quantum optics.

In this paper, we obtain the integrated flavor asymmetry of the sea quarks in the proton, ∫ 0 1 (d ̄ − ū) d x , with the help of the truncated moments approach elaborated in our previous papers. We use the difference between the light sea-quark distributions d ̄ − ū extracted from Drell–Yan (DY) NuSea/E866 measurements of the cross-section ratio σ p d / σ p p and from the recent global analysis of deep inelastic scattering (DIS) F 2 p − n data incorporating the reanalyzed neutron structure function. In our analysis, we also include the most recent DY data from the Fermilab SeaQuest/E906 experiment. [ABSTRACT FROM AUTHOR]

In this paper magnetic dipole moment form factor, G M (2) (Q 2) , describing the radial excitation of decuplet baryons to octet baryons electromagnetic transitions as well as the ratios, R S M = − 1 4 m 2 2 4 m 2 2 Q 2 + (m 2 2 − Q 2 − m 1 2) 2 G C (2) (Q 2) G M (2) (Q 2) and R E M = − G E (2) (Q 2) G M (2) (Q 2) are calculated in the framework of light-cone sum rules. We also estimate the degree of the violation of U-spin symmetry. [ABSTRACT FROM AUTHOR]

Generalized Kramers–Kronig (K–K) type dispersion relations and sum rules are derived in the static limit for the moments of the degenerate four wave mixing susceptibility. The degenerate nonlinear susceptibility is different from a typical use of the conventional K–K dispersion relations, which assume absence of complex poles of a function in the upper half of complex frequency plane, whereas degenerate susceptibility has simultaneous poles in both half planes. In the derivation of the generalized K–K relations the poles and their order are taken into account by utilization of the theorem of residues. The conventional K–K relations can be used to estimate the real and imaginary parts of the second and higher powers of the susceptibility as the effect of the poles is reduced due to a faster convergence of the dispersion relations. The present theory is directly applicable to higher order susceptibilities and can be used in testing of theoretical models describing the degenerate four wave mixing susceptibility in nonlinear optical and terahertz spectroscopy. [ABSTRACT FROM AUTHOR]