Some results on linear nabla Riemann-Liouville fractional difference equations

In this paper, we establish some criteria for boundedness, stability properties, and separation of solutions of autonomous nonlinear nabla Riemann-Liouville scalar fractional difference equations. To derive these results, we prove the variation of constants formula for nabla Riemann-Liouville fractional difference equations.     

Constraint incorporation in optimization

Numerical methods for solving constrained optimization problems need to incorporate the constraints in a manner that satisfies essentially competing interests; the incorporation needs to be simple enough that the solution method is tractable, yet complex enough to ensure the validity of the ultimate solution. We introduce a framework for constraint incorporation that identifies a minimal acceptable level of complexity and defines two basic types of constraint incorporation which (with combinations) cover nearly all popular numerical methods for constrained optimization, including trust region methods, penalty methods, barrier methods, penalty-multiplier methods, and sequential quadratic programming methods. The broad application of our framework relies on addition and chain rules for constraint incorporation which we develop here.     

A combinatorial characterization of normalizations of Boolean algebras

In this paper we prove that if a groupoid has exactly distinct n-ary term operations for n=1, 2, 3 and the same number of constant unary term operations for n=0, then it is a normalization of a nontrivial Boolean algebra. This, together with some general facts concerning normalizations of algebras, which we recall, yields a clone characterization of normalizations of nontrivial Boolean algebras: A groupoid (G;·) is clone equivalent to a normalization of a nontrivial Boolean algebra if and only if the value of the free spectrum for (G;·) is for n = 0, 1, 2, 3. In the last section the Minimal Extension Property for the sequence (2, 3) in the class of all groupoids is derived. Received September 15, 2004; accepted in final form October 4, 2005.     

Single machine group scheduling with ordered criteria

The single machine group scheduling problem is considered. Jobs are classified into several groups on the basis of group technology, i.e. jobs of the same group have to be processed jointly. A machine set-up time independent of the group sequence is needed between each two consecutive groups. A schedule specifies the sequence of groups and the sequence of jobs in each group. The quality of a schedule is measured by the criteriaF ₁, ...,F _m ordered by their relative importance. The objective is to minimize the least important criterionF _m subject to the schedule being optimal with respect to the more important criterionF _m–1 which is minimized on the set of schedules minimizing criterionF _m–2 and so on. The most important criterion isF ₁, which is minimized on the set of all feasible schedules. An approach to solve this multicriterion problem in polynomial time is presented if functionsF ₁, ...,F _m have special properties. The total weighted completion time and the total weighted exponential time are the examples of functionsF ₁, ...,F _m–1 and the maximum cost is an example of functionF _m for which our approach can be applied.The research of the authors was partially supported by a KBN Grant No. 3 P 406 003 05, the Fundamental Research Fund of Belarus, Project N 60-242, and the Deutsche Forschungsgemeinschaft, Project Schema, respectively. The paper was completed while the first author was visiting the University of Melbourne.     

2pi ambiguity-free optical distance measurement with subnanometer precision with a novel phase-crossing low-coherence interferometer

We report a highly accurate phase-based technique for measuring arbitrarily long optical distance with subnanometer precision. The method employs a Michelson interferometer with a pair of harmonically related light sources, one cw and the other low coherence. By slightly detuning (~2 nm) the center wavelength of the low-coherence source between scans of the target sample, we can use the phase relationship between the heterodyne signals of the cw and the low-coherence light to measure the separation between reflecting interfaces with subnanometer precision. As this technique is completely free of 2pi ambiguity, an issue that plagues most phase-based techniques, it can be used to measure arbitrarily long optical distances without loss of precision. We demonstrate one application of this technique, the high-precision determination of the differential refractive index.     

Two-dimensional high-resolution NMR spectra in matched B(0) and B(1) field gradients

In a recent publication we presented a method to obtain highly resolved NMR spectra in the presence of an inhomogeneous B(0) field with the help of a matched RF gradient. If RF gradient pulses are combined with "ideal" 90 degrees pulses to form inhomogeneous z rotation pulses, the line broadening caused by the B(0) gradient can be refocused, while the full chemical shift information is maintained. This approach is of potential use for NMR spectroscopy in an inhomogeneous magnetic field produced by an "ex-situ" surface spectrometer. In this contribution, we extend this method toward two-dimensional spectroscopy with high resolution in one or both dimensions. Line narrowing in the indirect dimension can be achieved by two types of nutation echoes, thus leading to depth-sensitive NMR spectra with full chemical shift information. If the nutation echo in the indirect dimension is combined with a stroboscopic acquisition using inhomogeneous z-rotation pulses, highly resolved two-dimensional correlation spectra can be obtained in matched field gradients. Finally, we demonstrate that an INEPT coherence transfer from proton to carbon spins is possible in inhomogeneous B(0) fields. Thus, it is possible to obtain one-dimensional (13)C NMR spectra with increased sensitivity and two-dimensional HETCOR spectra in the presence of B(0) gradients of 0.4 mT/cm. These schemes may be of some value for ex-situ NMR analysis of materials and biological systems.     

Improved measurement of Vub with inclusive semileptonic B decays

We report a new measurement of the Cabibbo-Kobayashi-Maskawa parameter Vub made with a sample of 9.7 x 10(6) BB- events collected with the CLEO II detector. Using heavy quark theory, we combine the observed yield of leptons from semileptonic B decay in the end-point momentum interval 2.2-2.6 GeV/c with recent CLEO II data on B-->X(s)gamma to find Vub = (4.08+/-0.34+/-0.44+/-0.16+/-0.24)x10(-3), where the first two uncertainties are experimental and the last two are from theory.     

Algebraic Properties of the Path Ideal of a Tree

The path ideal (of length t ≥ 2) of a directed graph Γ is the monomial ideal, denoted I _t (Γ), whose generators correspond to the directed paths of length t in Γ. We study some of the algebraic properties of I _t (Γ) when Γ is a tree. We first show that I _t (Γ) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I _t (Γ) is always sequentially Cohen–Macaulay, and the Betti numbers of R/I _t (Γ) do not depend upon the characteristic of the field. We study the case of the line graph in greater detail at the end of the article. We give an exact formula for the projective dimension of these ideals, and in some cases, we compute their arithmetical rank.