###
*BASIC* *DEFINITIONS*

"... eshghisharifedu Abstract It is shown that the quadratic graph Q k con sisting of cycles of length k has an valuation a stronger form of the graceful valuation for every positive integer k Fur thermore additional results are obtained from the main theorem of this paper ..."

Abstract
- Add to MetaCart

eshghisharifedu Abstract It is shown that the quadratic graph Q k con sisting of cycles of length k has an valuation a stronger form of the graceful valuation for every positive integer k Fur thermore additional results are obtained from the main theorem of this paper

###
*Basic* *definitions* *Definition*

, 2006

"... A topological space X is a Polish space if it is separable and ..."

###
I. *BASIC* *DEFINITIONS* AND INTERPRETATIONS

"... Abstract — The extended versions of common Laplace and Fourier transforms are given. This is achieved by defining a new function fe(p), p 2 C related to the function to be transformed f (t), t 2 R. Then fe(p) is transformed by an integral whose path is defined on an inclined line on the complex plan ..."

Abstract
- Add to MetaCart

plane. The slope of the path is the parameter of the extended

*definitions*which reduce to common transforms with zero slope. Inverse transforms of the extended versions are also defined. These proposed*definitions*, when applied to filtering in complex ordered fractional Fourier stages, significantly###
1.1. *BASIC* *DEFINITIONS*

"... Abstract. Let G = (L; R; E) be a bipartite graph with color classes L and R and edge set E. A set of two bijections f'1; '2g, '1; '2: L [ R! L [ R, is said to be a 3-biplacement of G if '1(L) = '2(L) = L and E \ ' ..."

Abstract
- Add to MetaCart

Abstract. Let G = (L; R; E) be a bipartite graph with color classes L and R and edge set E. A set of two bijections f'1; '2g, '1; '2: L [ R! L [ R, is said to be a 3-biplacement of G if '1(L) = '2(L) = L and E \ '

###
◮ *Basic* *Definitions* ◮ Classical Motivations

"... ◮ P=polynomial time. Those languages L for which there is an algorithm deciding x ∈ L in time O(|x | c) some fixed c. ◮ E.g. 2-colouring of graphs. ◮ NP=nondeterministic polynomial time. Those languages L for which there is a nondeterministic (guess and check) algorithm deciding x ∈ L in time O(|x | ..."

Abstract
- Add to MetaCart

◮ P=polynomial time. Those languages L for which there is an algorithm deciding x ∈ L in time O(|x | c) some fixed c. ◮ E.g. 2-colouring of graphs. ◮ NP=nondeterministic polynomial time. Those languages L for which there is a nondeterministic (guess and check) algorithm deciding x ∈ L in time O(|x | c) some fixed c. ◮ E.g. 3-colouring of graphs. WHERE DOES PARAMETERIZED COMPLEXITY COME FROM? ◮ A mathematical idealization is to identify “Feasible ” with P. (I won’t even bother looking at the problems with this.) ◮ With this assumption, the theory of NP-hardness is an excellent vehicle for mapping an outer boundary of intractability, for all practical purposes. ◮ Indeed, assuming the reasonable current working assumption that NTM acceptance is Ω(2 n), NP-hardness allows for practical lower bound for exact solution for problems. ◮ A very difficult practical and theoretical problem is “How can we deal with P?”. ◮ More importantly how can we deal with P − FEASIBLE, and map a further boundary of intractability. ◮ Lower bounds in P are really hard to come by. But this theory will allow you establish infeasibility for problems in P, under a reasonable complexity hypothesis. ◮ Also it will indicate to you how to attack the problem if it looks bad. ◮ It is thus both a positive and negative tool kit. I’M DUBIOUS; EXAMPLE? ◮ Below is one application that points at why the completeness theory might interest you. ◮ The great PCP Theorem of Arora et. al. allows us to show that things don’t have PTAS’s on the assumption that P=NP. ◮ Some things actually do have PTAS’s. Lets look at a couple taken from recent major conferences: STOC,

###
1.1 *Basic* *definitions*

"... On the geometry and the deformation of shape represented by a piecewise continuous Bézier ..."

Abstract
- Add to MetaCart

On the geometry and the deformation of shape represented by a piecewise continuous Bézier

###
1.1. *Basic* *definitions*.

, 2007

"... topics and examples of quasigroups. The following lectures then introduce the three main branches of quasigroup representation theory: characters, permutation representations, and modules. 1. Quasigroups ..."

Abstract
- Add to MetaCart

topics and examples of quasigroups. The following lectures then introduce the three main branches of quasigroup representation theory: characters, permutation representations, and modules. 1. Quasigroups