scholarly journals Polynomial Bounds for the Grid-Minor Theorem

2016 ◽  
Vol 63 (5) ◽  
pp. 1-65 ◽  
Author(s):  
Chandra Chekuri ◽  
Julia Chuzhoy
Keyword(s):  
Polynomial Bounds ◽  
Grid Minor

Polynomial bounds for probability generating functions

1975 ◽  
Vol 12 (3) ◽  
pp. 507-514 ◽  
Author(s):  
Henry Braun
Keyword(s):  
Lower Bound ◽  
Generating Function ◽  
Generating Functions ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Probability Generating Functions ◽  
Extinction Probabilities ◽  
Polynomial Bounds

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

scholarly journals Proof Theory and Computational Analysis

1997 ◽  
Vol 4 (30) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Proof Theory ◽  
Computational Analysis ◽  
Numerical Data ◽  
Expressive Power ◽  
Rates Of Convergence ◽  
Uniform Bounds ◽  
Survey Paper ◽  
Constructive Proofs ◽  
Polynomial Bounds ◽  
The Relationship

In this survey paper we start with a discussion how functionals of finite type can be used for the proof-theoretic extraction of numerical data (e.g. effective<br />uniform bounds and rates of convergence) from non-constructive proofs in numerical analysis. We focus on the case where the extractability of polynomial bounds is guaranteed.<br />This leads to the concept of hereditarily polynomial bounded analysis (PBA). We indicate the mathematical range of PBA which turns out to be surprisingly large. Finally we discuss the relationship between PBA and so-called feasible analysis<br />FA. It turns out that both frameworks are incomparable. We argue in favor of the thesis that PBA offers the more useful approach for the purpose of extracting mathematically interesting bounds from proofs. In a sequel of appendices to this paper we indicate the expressive power of PBA.

scholarly journals Polynomial bounds for Arakelov invariants of Belyi curves

2014 ◽  
Vol 8 (1) ◽  
pp. 89-140 ◽  
Author(s):  
Ariyan Javanpeykar
Keyword(s):  
Polynomial Bounds

scholarly journals New Refinements and Improvements of Jordan’s Inequality

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 284 ◽  
Author(s):  
Lina Zhang ◽  
Xuesi Ma
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Quadratic Polynomial ◽  
Quartic Polynomial ◽  
Polynomial Bounds ◽  
Jordan’S Inequality

The polynomial bounds of Jordan’s inequality, especially the cubic and quartic polynomial bounds, have been studied and improved in a lot of the literature; however, the linear and quadratic polynomial bounds can not be improved very much. In this paper, new refinements and improvements of Jordan’s inequality are given. We present new lower bounds and upper bounds for strengthened Jordan’s inequality using polynomials of degrees 1 and 2. Our bounds are tighter than the previous results of polynomials of degrees 1 and 2. More importantly, we give new improvements of Jordan’s inequality using polynomials of degree 5, which can achieve much tighter bounds than those previous methods.

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