scholarly journals A Survey of Forbidden Configuration Results

10.37236/2379 ◽  
2013 ◽  
Vol 1000 ◽  
Author(s):  
Richard Anstee
Keyword(s):  
Fundamental Result ◽  
Asymptotic Results ◽  
Column Permutation ◽  
The Matrix ◽  
Simple Matrix ◽  
Forbidden Configuration

Let $F$ be a $k\times \ell$ (0,1)-matrix. We say a (0,1)-matrix $A$ has $F$ as a configuration if there is a submatrix of $A$ which is a row and column permutation of $F$. In the language of sets, a configuration is a trace and in the language of hypergraphs a configuration is a subhypergraph.Let $F$ be a given $k\times \ell$ (0,1)-matrix. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. The matrix $F$ need not be simple. We define $\hbox{forb}(m,F)$ as the maximum number of columns of any simple $m$-rowed matrix $A$ which do not contain $F$ as a configuration. Thus if $A$ is an $m\times n$ simple matrix which has no submatrix which is a row and column permutation of $F$ then $n\le\hbox{forb}(m,F)$. Or alternatively if $A$ is an $m\times (\hbox{forb}(m,F)+1)$ simple matrix then $A$ has a submatrix which is a row and column permutation of $F$. We call $F$ a forbidden configuration. The fundamental result is due to Sauer, Perles and Shelah, Vapnik and Chervonenkis. For $K_k$ denoting the $k\times 2^k$ submatrix of all (0,1)-columns on $k$ rows, then $\hbox{forb}(m,K_k)=\binom{m}{k-1}+\binom{m}{k-2}+\cdots \binom{m}{0}$. We seek asymptotic results for $\hbox{forb}(m,F)$ for a fixed $F$ and as $m$ tends to infinity . A conjecture of Anstee and Sali predicts the asymptotically best constructions from which to derive the asymptotics of $\hbox{forb}(m,F)$. The conjecture has helped guide the research and has been verified for $k\times \ell$ $F$ with $k=1,2,3$ and for simple $F$ with $k=4$ as well as other cases including $\ell=1,2$. We also seek exact values for $\hbox{forb}(m,F)$. 

scholarly journals Color critical hypergraphs and forbidden configurations

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Richard Anstee ◽  
Balin Fleming ◽  
Zoltán Füredi ◽  
Attila Sali
Keyword(s):  
Linear Algebra ◽  
Upper Bound ◽  
Sufficient Condition ◽  
Column Permutation ◽  
International Audience ◽  
Simple Matrix ◽  
Forbidden Configuration ◽  
Forbidden Configurations

International audience The present paper connects sharpenings of Sauer's bound on forbidden configurations with color critical hypergraphs. We define a matrix to be \emphsimple if it is a $(0,1)-matrix$ with no repeated columns. Let $F$ be $a k× l (0,1)-matrix$ (the forbidden configuration). Assume $A$ is an $m× n$ simple matrix which has no submatrix which is a row and column permutation of $F$. We define $forb(m,F)$ as the best possible upper bound on n, for such a matrix $A$, which depends on m and $F$. It is known that $forb(m,F)=O(m^k)$ for any $F$, and Sauer's bond states that $forb(m,F)=O(m^k-1)$ fore simple $F$. We give sufficient condition for non-simple $F$ to have the same bound using linear algebra methods to prove a generalization of a result of Lovász on color critical hypergraphs.

Small forbidden configurations V: Exact bounds for 4 × 2 cases

2011 ◽  
Vol 48 (1) ◽  
pp. 1-22
Author(s):  
Richard Anstee ◽  
Farzin Barekat ◽  
Attila Sali
Keyword(s):  
Column Permutation ◽  
Single Column ◽  
Simple Matrix ◽  
Forbidden Configuration ◽  
Forbidden Configurations ◽  
Exact Bounds

The present paper continues the work begun by Anstee, Ferguson, Griggs, Kamoosi and Sali on small forbidden configurations. We define a matrix to besimpleif it is a (0, 1)-matrix with no repeated columns. LetFbe ak× (0, 1)-matrix (the forbidden configuration). AssumeAis anm×nsimple matrix which has no submatrix which is a row and column permutation ofF. We define forb (m, F) as the largestn, which would depend onmandF, so that such anAexists.DefineFabcdas the (a+b+c+d) × 2 matrix consisting ofarows of [11],brows of [10],crows of [01] anddrows of [00]. With the exception ofF2110, we compute forb (m; Fabcd) for all 4 × 2Fabcd. A number of cases follow easily from previous results and general observations. A number follow by clever inductions based on a single column such as forb (m; F1111) = 4m− 4 and forb (m; F1210) = forb (m; F1201) = forb (m; F0310) = (2m)+m+ 2 (proofs are different). A different idea proves forb (m; F0220) = (2m) + 2m− 1 with the forbidden configuration being related to a result of Kleitman. Our results suggest that determining forb (m; F2110) is heavily related to designs and we offer some constructions of matrices avoidingF2110using existing designs.

scholarly journals Small Forbidden Configurations III

10.37236/997 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
R. P. Anstee ◽  
N. Kamoosi
Keyword(s):  
Column Permutation ◽  
Simple Matrix ◽  
Forbidden Configuration ◽  
Forbidden Configurations ◽  
Q Matrix

The present paper continues the work begun by Anstee, Ferguson, Griggs and Sali on small forbidden configurations. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. Let $F$ be a $k\times l$ (0,1)-matrix (the forbidden configuration). Assume $A$ is an $m\times n$ simple matrix which has no submatrix which is a row and column permutation of $F$. We define ${\hbox{forb}}(m,F)$ as the largest $n$, which would depend on $m$ and $F$, so that such an $A$ exists. 'Small' refers to the size of $k$ and in this paper $k=2$. For $p\le q$, we set $F_{pq}$ to be the $2\times (p+q)$ matrix with $p$ $\bigl[{1\atop0}\bigr]$'s and $q$ $\bigl[{0\atop1}\bigr]$'s. We give new exact values: ${\hbox{forb}}(m,F_{0,4})=\lfloor {5m\over2}\rfloor +2$, ${\hbox{forb}}(m,F_{1,4})=\lfloor {11m\over4}\rfloor +1$, ${\hbox{forb}}(m,F_{1,5})=\lfloor {15m\over4}\rfloor +1$, ${\hbox{forb}}(m,F_{2,4})=\lfloor {10m\over3}-{4\over3}\rfloor$ and ${\hbox{forb}}(m,F_{2,5})=4m$ (For ${\hbox{forb}}(m,F_{1,4})$, ${\hbox{forb}}(m,F_{1,5})$ we obtain equality only for certain classes modulo 4). In addition we provide a surprising construction which shows ${\hbox{forb}}(m,F_{pq})\ge \bigl({p+q\over2}+O(1)\bigr)m$.

Properties of a Dielectric Elastomer Actuator Modified by Dispersion of Functionalised Carbon Nanotubes

2012 ◽  
Vol 79 ◽  
pp. 41-46 ◽  
Author(s):  
Fabia Galantini ◽  
Sabrina Bianchi ◽  
Valter Castelvetro ◽  
Irene Anguillesi ◽  
Giuseppe Gallone
Keyword(s):  
Carbon Nanotubes ◽  
Dielectric Constant ◽  
Mechanical Performance ◽  
Broad Class ◽  
Dielectric Elastomer ◽  
Low Dielectric ◽  
Dielectric Elastomer Actuators ◽  
Electromechanical Transduction ◽  
The Matrix ◽  
Simple Matrix

Among the broad class of electro-active polymers, dielectric elastomer actuators represent a rapidly growing technology for electromechanical transduction. In order to further develop this applied science, the high driving voltages currently needed must be reduced. For this purpose, one of the most promising and adopted approach is to increase the dielectric constant while maintaining both low dielectric losses and high mechanical compliance. In this work, a dielectric elastomer was prepared by dispersing functionalised carbon nanotubes into a polyurethane matrix and the effects of filler dispersion into the matrix were studied in terms of dielectric, mechanical and electro-mechanical performance. An interesting increment of the dielectric constant was observed throughout the collected spectrum while the loss factor remained almost unchanged with respect to the simple matrix, indicating that conductive percolation paths did not arise in such a system. Consequences of the chemical functionalisation of carbon nanotubes with respect to the use of unmodified filler were also studied and discussed along with rising benefits and drawbacks for the whole composite material.

Stationary waiting-time distributions in the GI/PH/1 queue

1981 ◽  
Vol 18 (4) ◽  
pp. 901-912 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Waiting Time ◽  
Embedded Markov Chain ◽  
Asymptotic Results ◽  
Waiting Time Distributions ◽  
Structured Systems ◽  
Queue Discipline ◽  
The Matrix ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Stationary Waiting Time

It is known that the stable GI/PH/1 queue has an embedded Markov chain whose invariant probability vector is matrix-geometric with a rate matrix R. In terms of the matrix R, the stationary waiting-time distributions at arrivals, at an arbitrary time point and until the customer's departure may be evaluated by solving finite, highly structured systems of linear differential equations with constant coefficients. Asymptotic results, useful in truncating the computations, are also obtained. The queue discipline is first-come, first-served.

Matrix Analysis of Shell Structures with Flexible Frames

1958 ◽  
Vol 9 (4) ◽  
pp. 361-394 ◽  
Author(s):  
J. S. Przemieniecki
Keyword(s):  
Thermal Loading ◽  
Arbitrary Cross Section ◽  
Shell Structures ◽  
Matrix Analysis ◽  
Internal Load ◽  
Flexibility Matrix ◽  
The Matrix ◽  
Simple Matrix ◽  
Curved Panels ◽  
And Diffusion

SummaryA simple matrix method is presented for the deflection and stress analysis of cylindrical shell structures of arbitrary cross section stiffened by flexible frames. The method is an extension to fuselage structures of the Matrix Force Method developed by Argyris, in which the internal load system in the structure consists of two parts:—(a) synthetic load distribution, represented by the matrix b0, satisfying the external and internal equations of equilibrium, and(b)self-equilibrating load systems, represented by the matrix b1, which are introduced to satisfy compatibility conditions. The magnitudes of these self-equilibrating load systems are determined from the generalised compatibility equations formulated using the flexibility matrix f for the un-assembled elements of the structure. The self-equilibrating systems are non-orthogonal, but are arranged in such a way that the mixing between one system and another is kept to a minimum and, consequently, the resulting compatibility equations are well-conditioned. The three basic matrices, b0, b1;and f, are compiled using only very simple formulae. The matrices b0and b1depend on the geometry of the structure, while the flexibility matrix f is a function of geometry and elastic properties. The present analysis is applied to cut-out problems in fuselage structures. It can also be used for problems involving thermal loading and diffusion of loads in curved panels stiffened by flexible frames.

scholarly journals Designing matrix organizations that work: Lessons from the P&G case

2009 ◽  
Vol 2 (1) ◽  
pp. 179 ◽  
Author(s):  
Ronald Jean Degen
Keyword(s):  
Structure Design ◽  
Matrix Structure ◽  
Systems Planning ◽  
The Us ◽  
The Matrix ◽  
Simple Matrix ◽  
Proper Power ◽  
Matrix Organization ◽  
The 1960S ◽  
Information Systems Planning

The matrix organization concept emerged from the US aerospace industry in the 1960s and was adopted by many companies in the early 1970s. In the late 1970s and early 1980s many companies were experiencing trouble with its operation and many argued like Peters & Waterman in their bestseller In search of excellence in 1982 (p. 306) that the matrix was too complex to work properly. Galbraith (2009, p. 10-14) explains that the reason for the problems were that the matrix in these organizations was wrongly adopted, hastily installed, and inappropriately implemented. He explains that adopting a matrix structure requires a collaborative organization form, proper power, and accountability distribution, complementing changes to the information systems, planning and budgeting process, the performance evaluation and bonus system, and so on. The purpose of this paper is to illustrate why companies adopted the matrix, what problems they had, the solutions for these problems based on Galbraith (2009) and other authors like Davis & Lawrence (1977), and the state of the art of matrix structure design today like the P&G front-back hybrid matrix organization. To illustrate the historical evolution of organization structure to the simple matrix and then to more complex matrix organizations we used the P&G case (Piskorski & Spadini 2007).

Influence of Matrix Circuit Switching Device Junction Capacitance on Piezoelectric Actuator Drive Performance

2009 ◽  
Vol 3 (3) ◽  
pp. 313-318 ◽  
Author(s):  
Katsushi Furutani ◽  
◽  
Taizo Makino
Keyword(s):  
Current Pulse ◽  
Piezoelectric Actuator ◽  
Degrees Of Freedom ◽  
Field Effect Transistors ◽  
Piezoelectric Actuators ◽  
Circuit Switching ◽  
Switching Device ◽  
The Matrix ◽  
Simple Matrix ◽  
Drive Circuit

Piezoelectric actuators are widely used as fine-motion actuators for positioning devices. Current pulse driving reduces displacement hysteresis, and the drive circuit provides current pulses considered constant charge pulses to the piezoelectric actuator. The circuit for devices with multiple degrees of freedom using multiple piezoelectric actuators should be simple. Matrix circuits are used to drive liquid crystal displays to reduce the number of drive-circuit control signals and components. A group of 2 × 2 piezoelectric actuators was driven alternately using a 4-switch matrix circuit, and two field effect transistors were used as a fast matrix circuit switch. Piezoelectric actuator drive performance was studied in the case of using the current pulse circuit and the matrix circuit. When the matrix circuit drove actuators performing as capacitive loads, switching device junction capacitors adversely affected drive performance.

scholarly journals NONPERTURBATIVE DOUBLE SCALING LIMITS

2003 ◽  
Vol 18 (04) ◽  
pp. 577-591 ◽  
Author(s):  
FRANK FERRARI
Keyword(s):  
Gauge Theory ◽  
Path Integrals ◽  
Partition Functions ◽  
Scaling Limits ◽  
Random Surfaces ◽  
Critical Metrics ◽  
Fundamental Limitations ◽  
The Matrix ◽  
Classic Approach ◽  
Simple Matrix

Recently, the author has proposed a generalization of the matrix and vector models approach to the theory of random surfaces and polymers. The idea is to replace the simple matrix or vector (path)-integrals by gauge theory or nonlinear σ model (path)-integrals. We explain how this solves one of the most fundamental limitations of the classic approach: we automatically obtain nonperturbative definitions in non-Borel summable cases. This is exemplified in the simplest possible examples involving O (N) symmetric nonlinear σ models with N-dimensional target spaces, for which we construct (multi)critical metrics. The nonperturbative definitions of the double scaled, manifestly positive, partition functions rely on remarkable identities involving (path)-integrals.

A Matrix-Inversion Method for Gamma-Source Mapping From Gamma-Count Data

2013 ◽  
Author(s):  
Ian Adsley ◽  
Richard K. Bull ◽  
Claire Burgess
Keyword(s):  
Matrix Inversion ◽  
Source Distribution ◽  
Inversion Method ◽  
Practical Applications ◽  
Highly Active ◽  
Co60 Source ◽  
The Matrix ◽  
Simple Matrix ◽  
Inversion Procedure ◽  
Matrix Inversion Method

In a previous paper (1) it was proposed that a simple matrix inversion method could be used to extract source distributions from gamma-count maps, using simple models to calculate the response matrix. The method was tested using numerically generated count maps. In the present work a 100 kBq Co60 source has been placed on a gridded surface and the count rate measured using a NaI scintillation detector. The resulting map of gamma counts was used as input to the matrix inversion procedure and the source position recovered. A multisource array was simulated by superposition of several single-source count maps and the source distribution was again recovered using matrix inversion. The measurements were performed for several detector heights. The effects of uncertainties in source-detector distances on the matrix-inversion method are also examined. The results from this work give confidence in the application of the method to practical applications, such as the segregation of highly active objects amongst fuel-element debris.

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