scholarly journals Bounds on Parameters of Minimally Nonlinear Patterns

10.37236/6735 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
P. A. CrowdMath
Keyword(s):  
Upper Bound ◽  
Extremal Function ◽  
Subgraph Isomorphic ◽  
Non Linear ◽  
Ordered Graphs ◽  
P Matrix ◽  
The Extremal Function

Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) \neq O(n)$ but $ex(n, P') = O(n)$ for every proper subpattern $P'$ of $P$. We prove that the ratio between the length and width of any minimally non-linear 0-1 matrix is at most $4$, and that a minimally non-linear 0-1 matrix with $k$ rows has at most $5k-3$ ones. We also obtain an upper bound on the number of minimally non-linear 0-1 matrices with $k$ rows.In addition, we prove corresponding bounds for minimally non-linear ordered graphs. The minimal non-linearity that we investigate for ordered graphs is for the extremal function $ex_{<}(n, G)$, which is the maximum possible number of edges in any ordered graph on $n$ vertices with no ordered subgraph isomorphic to $G$.

scholarly journals Sharper Bounds and Structural Results for Minimally Nonlinear 0-1 Matrices

10.37236/7801 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Jesse Geneson ◽  
Shen-Fu Tsai
Keyword(s):  
Upper Bound ◽  
Extremal Function ◽  
Nonlinear Matrix ◽  
The Extremal Function

The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally nonlinear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every $P'$ that is contained in $P$ but not equal to $P$. Bounds on the number of ones and the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018). In this paper, we improve the upper bound on the number of ones in a minimally nonlinear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a corollary, this improves the upper bound on the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows from $4k-2$ to $4k-4$. We also prove that there are not more than four ones in the top and bottom rows of a minimally nonlinear matrix and that there are not more than six ones in any other row of a minimally nonlinear matrix. Furthermore, we prove that if a minimally nonlinear 0-1 matrix has ones in the same row with exactly $d$ columns between them, then within these columns there are at most $2d-1$ rows above and $2d-1$ rows below with ones.

scholarly journals The Extremal Function and Colin de Verdière Graph Parameter

10.37236/7195 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Rose McCarty
Keyword(s):  
Differential Geometry ◽  
Upper Bound ◽  
Extremal Function ◽  
The Extremal Function ◽  
A Minor ◽  
Monotone Graph

The Colin de Verdière parameter $\mu(G)$ is a minor-monotone graph parameter with connections to differential geometry. We study the conjecture that for every integer $t$, if $G$ is a graph with at least $t$ vertices and $\mu(G) \leq t$, then $|E(G)| \leq t|V(G)|-\binom{t+1}{2}$. We observe a relation to the graph complement conjecture for the Colin de Verdière parameter and prove the conjectured edge upper bound for graphs $G$ such that either $\mu(G) \leq 7$, or $\mu(G) \geq |V(G)|-6$, or the complement of $G$ is chordal, or $G$ is chordal.

scholarly journals Use of Evolutionary Computation to Improve Rock Slope Back Analysis

2020 ◽  
Vol 10 (6) ◽  
pp. 2012
Author(s):  
An-Jui Li ◽  
Abdoulie Fatty ◽  
I-Tung Yang
Keyword(s):  
Finite Element ◽  
Evolutionary Computation ◽  
Upper Bound ◽  
Limit Analysis ◽  
Optimization Problems ◽  
Continuous Functions ◽  
Computational Time ◽  
Irregular Solution ◽  
Non Linear ◽  
Geotechnical Problems

Generally, in geotechnical engineering, back analyses are used to investigate uncertain parameters. Back analyses can be undertaken by considering known conditions, such as failure surfaces, displacements, and structural performances. Many geotechnical problems have irregular solution domains, with the objective function being non-convex, and may not be continuous functions. As such, a complex non-linear optimization function is typically required for most geotechnical problems to attain a better understanding of these uncertainties. Therefore, particle swarm optimization (PSO) and a genetic algorithm (GA) are utilized in this study to facilitate in back analyses mainly based on upper bound finite element limit analysis method. These approaches are part of evolutionary computation, which is appropriate for solving non-linear global optimization problems. By using these techniques with upper-bound finite element limit analysis (UB-FELA), two case studies showed that the results obtained are reasonable and reliable while maintaining a balance between computational time and accuracy.

WELFARE LOSS IN LINEAR PRICE-PREFERENCE PROCUREMENT AUCTIONS

2007 ◽  
Vol 09 (04) ◽  
pp. 719-730
Author(s):  
WINSTON T. H. KOH
Keyword(s):  
Uniform Distribution ◽  
Upper Bound ◽  
Government Procurement ◽  
Welfare Loss ◽  
Procurement Auctions ◽  
Domestic Firms ◽  
Sealed Bid ◽  
Non Linear ◽  
Price Auction

In government procurement auctions, discrimination in favor of one group of participants (e.g. domestic firms, minority bidders) over another group is a common practice. The optimal discriminatory rules for these auctions are typically non-linear and could be administratively complex and costly to implement. In practice, procurement auctions are usually organized as sealed-bid first-price auction with a simple percentage price-preference policy. In this paper, we analyze a model with two bidders that draw their costs from a common uniform distribution, and derive an upper bound to the welfare loss resulting from the use of linear-price preference auctions.

scholarly journals A majorant problem

1992 ◽  
Vol 15 (3) ◽  
pp. 441-447
Author(s):  
Ronen Peretz
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Fourier Coefficients ◽  
Positive Real Part ◽  
Complex Vector ◽  
Positive Real ◽  
Zero Sets ◽  
The Extremal Function

Letf(z)=∑k=0∞akzk,a0≠0be analytic in the unit disc. Any infinite complex vectorθ=(θ0,θ1,θ2,…)such that|θk|=1,k=0,1,2,…, induces a functionfθ(z)=∑k=0∞akθkzkwhich is still analytic in the unit disc.In this paper we study the problem of maximizing thep-means:∫02π|fθ(reiϕ)|pdϕover all possible vectorsθand for values ofrclose to0and for allp<2.It is proved that a maximizing function isf1(z)=−|a0|+∑k=1∞|ak|zkand thatrcould be taken to be any positive number which is smaller than the radius of the largest disc centered at the origin which can be inscribed in the zero sets off1. This problem is originated by a well known majorant problem for Fourier coefficients that was studied by Hardy and Littlewood.One consequence of our paper is that forp<2the extremal function for the Hardy-Littlewood problem should be−|a0|+∑k=1∞|ak|zk.We also give some applications to derive some sharp inequalities for the classes of Schlicht functions and of functions of positive real part.

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