Algebraic methods for the solution of linear functional equations

The equation

$$\sum^{n}_ {i=0} a_{i}f(b_{i}x + (1 - b_{i})y) = 0$$

belongs to the class of linear functional equations. The solutions form a linear space with respect to the usual pointwise operations. According to the classical results of the theory they must be generalized polynomials. New investigations have been started a few years ago. They clarified that the existence of non-trivial solutions depends on the algebraic properties of some related families of parameters. The problem is to find the necessary and sufficient conditions for the existence of non-trivial solutions in terms of these kinds of properties. One of the earliest results is due to Z. Daróczy [1]. It can be considered as the solution of the problem in case of n = 2. We are going to take more steps forward by solving the problem in case of n = 3.

     

Analytically exact spiral scheme for generating uniformly distributed points on the unit sphere

The problem of constructing a set of uniformly-distributed points on the surface of a sphere, also known as the Thomson problem, has a long and interesting history, which dates back to J.J. Thomson in 1904. A particular variant of the Thomson problem that is of great importance to biomedical imaging is that of generating a nearly uniform distribution of points on the sphere via a deterministic scheme. Although the point set generated through the minimization of electrostatic potential is the gold standard, minimizing the electrostatic potential of one thousand points (or charges) or more remains a formidable task. Therefore, a deterministic scheme capable of generating efficiently and accurately a set of uniformly-distributed points on the sphere has an important role to play in many scientific and engineering applications, not the least of which is to serve as an initial solution (with random perturbation) for the electrostatic repulsion scheme. In the work, we will present an analytically exact spiral scheme for generating a highly uniform distribution of points on the unit sphere.     

A unifying theoretical and algorithmic framework for least squares methods of estimation in diffusion tensor imaging

A unifying theoretical and algorithmic framework for diffusion tensor estimation is presented. Theoretical connections among the least squares (LS) methods, (linear least squares (LLS), weighted linear least squares (WLLS), nonlinear least squares (NLS) and their constrained counterparts), are established through their respective objective functions, and higher order derivatives of these objective functions, i.e., Hessian matrices. These theoretical connections provide new insights in designing efficient algorithms for NLS and constrained NLS (CNLS) estimation. Here, we propose novel algorithms of full Newton-type for the NLS and CNLS estimations, which are evaluated with Monte Carlo simulations and compared with the commonly used Levenberg-Marquardt method. The proposed methods have a lower percent of relative error in estimating the trace and lower reduced chi2 value than those of the Levenberg-Marquardt method. These results also demonstrate that the accuracy of an estimate, particularly in a nonlinear estimation problem, is greatly affected by the Hessian matrix. In other words, the accuracy of a nonlinear estimation is algorithm-dependent. Further, this study shows that the noise variance in diffusion weighted signals is orientation dependent when signal-to-noise ratio (SNR) is low (相似文献   

Study of theS-matrix near bound states in the continuum

The behaviour ofS-matrix for potentials generating bound states in continuum in the neighbourhood of the positive bound state energies is studied. It is shown that unlike the case of usual negative energy bound states, theS-matrix does not have a pole at the positive bound state energy but becomes unity at the energy corresponding to bound states in continuum. Calculations ofS-waveS-matrix for a local potential constructed by Stillinger and Herrick and a separable nonlocal potential constructed by the present authors verify these results. Our results indicate that the bound states embedded in continuum constructedvia the von Neumann and Wigner procedure cannot be interpreted as resonances with zero width.     

Sound localization in a new-world frugivorous bat, Artibeus jamaicensis: acuity, use of binaural cues, and relationship to vision

Passive sound-localization acuity and its relationship to vision were determined for the echolocating Jamaican fruit bat (Artibeus jamaicensis). A conditioned avoidance procedure was used in which the animals drank fruit juice from a spout in the presence of sounds from their right, but suppressed their behavior, breaking contact with the spout, whenever a sound came from their left, thereby avoiding a mild shock. The mean minimum audible angle for three bats for a 100-ms noise burst was 10 degrees-marginally superior to the 11.6 degrees threshold for Egyptian fruit bats and the 14 degrees threshold for big brown bats. Jamaican fruit bats were also able to localize both low- and high-frequency pure tones, indicating that they can use both binaural phase- and intensity-difference cues to locus. Indeed, their ability to use the binaural phase cue extends up to 6.3 kHz, the highest frequency so far for a mammal. The width of their field of best vision, defined anatomically as the width of the retinal area containing ganglion-cell densities at least 75% of maximum, is 34 degrees. This value is consistent with the previously established relationship between vision and hearing indicating that, even in echolocating bats, the primary function of passive sound localization is to direct the eyes to sound sources.     

Sub-100 fs SDPF optical soliton compressor for diode laser pulses

We report our studies on a fiber-optic soliton compressor for generation of sub-100 femtosecond (fs) optical pulses out of picosecond (ps) diode laser pulses. The soliton compressor is rather simple and composed of a 15 ～ 20 m-long step-like dispersion profiled fiber (SDPF) in conjunction with a single Er-doped fiber amplifier (EDFA). Careful design of such a SDPF compressor was performed, leading to demonstration of 20-fs class compression performance, and experimental investigation was carried out in detail on the optical pulse propagation in the compression processes. In addition, nonlinear fiber loops were applied to suppression of pulse pedestals, resulting successfully in high quality optical pulses of the 100-fs range.     

The enthalpy of formation of 2II CH

The standard enthalpy of formation, δ_fH^o, of² II CH has been determined at converged levels of ab initio electronic structure theory, including high order coupled cluster and full configuration interaction benchmarks. The atomic Gaussian basis sets employed include the (aug)-cc-p(C)VXZ family with X = 3, 4, 5 and 6. Extrapolations to the complete one-particle basis set and the full configuration interaction limits, where appropriate, have been performed to reduce remaining computational errors. Additional improvements in the enthalpy of formation of ²II CH were achieved by appending the valence-only treatment with core-valence correlation, relativistic effects including spin-orbit correlation, and the diagonal Born-Oppenheimer correction. The recommended values for δfH^o ₀ and δA_f H _o ²⁹⁸ of ²II CH are 592.48^+0.47 _?0.56 kJ mol_?1 and 595.93 ^+0.47 _?0.56 kJ mol_?1, respectively.     

Analytically exact correction scheme for signal extraction from noisy magnitude MR signals

An analytically exact method is proposed to extract the signal intensity and the noise variance simultaneously from noisy magnitude MR signals. This method relies on a fixed point formula of signal-to-noise ratio (SNR) and a correction factor. The correction factor, which is a function of SNR, establishes a fundamental link between the variance of the magnitude MR signal and the variance of the underlying Gaussian noise in the two quadrature channels. A more general but very similar method is developed for parallel signal acquisitions with multiple receiver coils. In the context of MR imaging, the proposed method can be carried out on a pixel-by-pixel basis if the mean and the standard deviation of the magnitude signal are available.