This paper presents the main results obtained in the field of approximation algorithms in a unified framework. Most of these results have been revisited in order to emphasize two basic tools useful for characterizing approximation classes, that is, combinatorial properties of problems and approximation preserving reducibilities. In particular, after reviewing the most important combinatorial ch...

This paper deals with the Falkner-Skan nonlinear differential equation. An analytic approximate technique, namely, optimal homotopy asymptotic method (OHAM), is employed to propose a procedure to solve a boundary-layer problem. Our method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. The obtained re...

Finding the nearest subspace is a fundamental problem and influential to many applications. In particular, a scalable solution that is fast and accurate for a large problem has a great impact. The existing methods for the problem are, however, useless in a large-scale problem with a large number of subspaces and high dimensionality of the feature space. A cause is that they are designed based o...

A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure is presented that uses linear programming to find a good approximation to the sparse solution on a given refinement level. Then only those parts of the mesh ...

The approximate common divisor problem(ACDP) is to find one or more divisors which is the greatest common divisor of the approximate numbers a and b of two given numbers a0 and b0. Howgrave-Graham[7] has considered the special case of b = b0 and gave a continued fraction approach and a lattice approach to find divisors. Furthermore he raised another lattice approach for ACDP based on Coppersmit...

In this thesis, we present an efficient algorithm for creating and solving hierarchical models of large Markov decision processes (MDPs). As the size of the MDP increases, finding an exact solution becomes intractable, so we expect only to find an approximate solution. We also assume that the hierarchies we create are not necessarily applicable to more than one problem so that we must be able t...

Transient analytical solutions are presented for the overpotential and the voltage for a porous electrode that includes both double-layer charging and a faradaic reaction. In addition, a simplified dynamic model is developed for the same process based on a second order, three-parameter polynomial approximate model. The effects of the parameters such as dimensionless exchange current density, co...

depiction of a solution space with several clusters Figure 2: Left: recursive counting by partitioning the solution space. Right: fragmentation of solution clusters. away the number of clusters in which x1 takes both values, 0 and 1. Thus, #clusters(F) = #clusters(F)|x1=0 + #clusters(F)|x1=1− #clusters(F)|x!=0&x1=1. The tricky part is to compute #clusters(F)|x1=v1 for any v1, as this no longer ...