Maximizing the number of unused bins

We analyze the approximation behavior of some of the best-known polynomial-time approximation algorithms for bin-packing under an approximation criterion, called differential ratio, informally the ratio (n - apx(I))/(n - opt(I)), where n is the size of the input list, apx(I) is the size of the solution provided by an approximation algorithm and opt(I) is the size of the optimal one. This measure has originally been introduced by Ausiello, DÁtri and Protasi and more recently revisited, in a more systematic way, by the first and the third authors of the present paper. Under the differential ratio, bin-packing has a natural formulation as the problem of maximizing the number of unused bins. We first show that two basic fit bin-packing algorithms, the first-fit and the best-fit, admit differential approximation ratios 1/2. Next, we show that slightly improved versions of them achieve ratios 2/3. Refining our analysis we show that the famous first-fit-decreasing and best-fit decreasing algorithms achieve differential approximation ratio 3/4. Finally, we show that first-fit-decreasing achieves asymptotic differential approximation ratio 7/9

Self-Similar Factor Approximants

The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents the passage from one approximant to another as the motion realized by a dynamical system with the property of group self-similarity. The derived approximants, because of their form, are named the self-similar factor approximants. These complement the obtained earlier self-similar exponential approximants and self-similar root approximants. The specific feature of the self-similar factor approximants is that their control functions, providing convergence of the computational algorithm, are completely defined from the accuracy-through-order conditions. These approximants contain the Pade approximants as a particular case, and in some limit they can be reduced to the self-similar exponential approximants previously introduced by two of us. It is proved that the self-similar factor approximants are able to reproduce exactly a wide class of functions which include a variety of transcendental functions. For other functions, not pertaining to this exactly reproducible class, the factor approximants provide very accurate approximations, whose accuracy surpasses significantly that of the most accurate Pade approximants. This is illustrated by a number of examples showing the generality and accuracy of the factor approximants even when conventional techniques meet serious difficulties.Comment: 22 pages + 11 ps figure

Uniform approximation of sgn(x) by rational functions with prescribed poles

For $a\in (0,1)$ let $L^k_m(a)$ be the error of the best approximation of the function \sgn(x) on the two symmetric intervals $[-1,-a]\cup[a,1]$ by rational functions with the only possible poles of degree $2k-1$ at the origin and of $2m-1$ at infinity. Then the following limit exists \begin{equation} \lim_{m\to \infty}L^k_m(a)(\frac{1+a}{1-a})^{m-{1/2}} (2m-1)^{k+{1/2}}=\frac 2 \pi(\frac{1-a^2}{2a})^{k+{1/2}} \Gamma(k+\frac 1 2). \end{equation

Computer simulation of the critical behavior of 3D disordered Ising model

The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.Comment: 14 RevTeX pages, 6 figure

Non-leading Logarithms in Principal Value Resummation

We apply the method of principal value resummation of large momentum-dependent radiative corrections to the calculation of the Drell Yan cross section. We sum all next-to-leading logarithms and provide numerical results for the resummed exponent and the corresponding hard scattering function.Comment: 22 pages (LaTeX), 5 figures not included; available upon request from [email protected], ANL-HEP-PR-94-25, ITP-SB-94-3