Simultaneous Approximation and Algebraic Independence

We establish several new measures of simultaneous algebraic approximations for families of complex numbers related to the classical exponential and elliptic functions. These measures are completely explicit in terms of the degree and height of the algebraic approximations. In some instances, they imply that the field has transcendance degree 2 over . This approach which is ultimately based on the technique of interpolation determinants provides an alternative to Gelfonds transcendence criterion. We also formulate a conjecture about simultaneous algebraic approximation which would yield higher transcendance degrees from these measures.     

Critical points for least-squares problems involving certain analytic functions,with applications to sigmoidal nets

This paper deals with nonlinear least-squares problems involving the fitting to data of parameterized analytic functions. For generic regression data, a general result establishes the countability, and under stronger assumptions finiteness, of the set of functions giving rise to critical points of the quadratic loss function. In the special case of what are usually called single-hidden layer neural networks, which are built upon the standard sigmoidal activation tanh(x) (or equivalently (1 +e ^–x )^–1), a rough upper bound for this cardinality is provided as well.Supported in part by US Air Force Grant AFOSR-94-0293.     

Linear independence of time-frequency translates

The refinement equation plays a key role in wavelet theory and in subdivision schemes in approximation theory. Viewed as an expression of linear dependence among the time-scale translates of , it is natural to ask if there exist similar dependencies among the time-frequency translates of . In other words, what is the effect of replacing the group representation of induced by the affine group with the corresponding representation induced by the Heisenberg group? This paper proves that there are no nonzero solutions to lattice-type generalizations of the refinement equation to the Heisenberg group. Moreover, it is proved that for each arbitrary finite collection , the set of all functions such that is independent is an open, dense subset of . It is conjectured that this set is all of .

     

Coloring finite subsets of uncountable sets

It is consistent for every that and there is a function such that every finite set can be written in at most ways as the union of two distinct monocolored sets. If GCH holds, for every such coloring there is a finite set that can be written at least ways as the union of two sets with the same color.

     

Simultaneous rational approximation to binomial functions

We apply Padé approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers. In particular, we strengthen work of Osgood, Fel´dman and Rickert, proving, for example, that

for (where the latter is an effective constant). Some of the Diophantine consequences of such bounds will be discussed, specifically in the direction of solving simultaneous Pell's equations and norm form equations.

     

Comment on “Analysing flow and heat transfer of a viscoelastic fluid over a semi-infinite horizontal moving flat plate, IJNLM, 43 (2008) 772”

In this report we make corrections to a paper by Rafael Cortell (IJNLM, 43 (2008) 772). In this paper (IJNLM, 43 (2008) 772) the author considered an incorrect equation (by including a wrong term instead of correct term) for his considered flow and heat transfer problem and reported unrealistic results as a consequence. In this communication we have considered the correct equation (replacing the incorrect term by correct one) and solved it analytically. Our analysis shows that the unrealistic behavior as observed by Cortell (IJNLM, 43 (2008) 772) persists no more if one considers the correct term.     

On the limiting Pitman efficiency of some rank tests of independence

The limiting (as the significance level approaches 0) Pitman efficiency of a new “regression-based” rank test of independence to Kendall's tau and Spearman's rho is derived. The result is based on a version of [17], Ann. Statist.4 1003–1011) on coincidence of the limiting Pitman efficiency and the local (as the alternative approaches the hypothesis) approximate Bahadur efficiency. [7], Trans. Amer. Math. Soc.87, 173–186) result is applied to verify the main assumption of Wieand's paper. This approach is shown to be useful in some other situations, also.     

Optimization of pumping mode for double-clad fiber lasers

A theoretical modeling of Yb-doped double-clad fiber lasers under various pumping modes including arbitrary numbers of end-pump and side-pump is introduced. Approximate analytic expressions of distributed lasers along the whole fiber are derived, and their accuracies are investigated. The effect of the pumping mode on the output performance of fiber lasers is discussed. The numerical results show that the approximate analytic solution is in excellent agreement with the exact numerical solution of the rate equations, the output power in the side-pump scheme is lower than that in end-pumping scheme, and more uniform distributions of laser and pump powers can be achieved by adopting the distributed pump mode and optimizing the arrangement of pump powers. However, further flattening the pump distribution by using more pump points can degrade the laser efficiency.     

Update Procedures and the 1‐Consistency of Arithmetic

The 1‐consistency of arithmetic is shown to be equiva ent to the existence of fixed points of a certain type of update procedure, which is implicit in the epsilon‐substitution method.     

Contingency Table Analysis and Inference via Double Index Measures

In this work, we focus on a general family of measures of divergence for estimation and testing with emphasis on conditional independence in cross tabulations. For this purpose, a restricted minimum divergence estimator is used for the estimation of parameters under constraints and a new double index (dual) divergence test statistic is introduced and thoroughly examined. The associated asymptotic theory is provided and the advantages and practical implications are explored via simulation studies.