On Metric Dimension of Functigraphs

The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=f(u)\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2 \le \dim(C(G, f)) \le 2n-3$, if $G$ is a connected graph of order $n \ge 3$ and $f$ is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.Comment: 10 pages, 7 figure

Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph admits a $2$-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph $G$ with $|E(G)| \equiv 0 \pmod 2$ has a $2$-edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we give a detailed insight into the conjectures of Ban-Linial and Wormald and provide evidence of a strong relation of both of them with Ando's conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above mentioned conjectures. Moreover, we prove Ban-Linial's conjecture for cubic cycle permutation graphs. As a by-product of studying $2$-edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.Comment: 33 pages; submitted for publicatio

Videoconferencing via satellite. Opening Congress to the people: Technical report

The feasibility of using satellite videoconferencing as a mechanism for informed dialogue between Congressmen and constituents to strengthen the legislative process was evaluated. Satellite videoconferencing was defined as a two-way interactive television with the TV signals transmitted by satellite. With videoconferencing, one or more Congressmen in Washington, D. C. can see, hear and talk with groups of citizens at distant locations around the country. Simultaneously, the citizens can see, hear and talk with the Congressmen

Cheminement des carrières de direction dans la fonction publique au Canada

Cet article présente un résumé d'une étude entreprise auprès des hauts fonctionnaires dans la fonction publique du Canada. Elle porte sur le cheminement des carrières et touche aussi à la mobilité, à l'occupation et aux sources de recrutement.Who are the most capable executives in the Federal Public Service ? What are their experience characteristics ? How mobile have they been ? What are their academic backgrounds ? How does age relate to seniority and earning power ?These and related questions must be answered if manpower at upper government levels is to be managed more effectively.A fourteen-month study recently completed by the Personnel Consulting Division, Bureau of Management Consulting Services, Ottawa, has made a major contribution towards providing these answers.In the Public Service of Canada, at the time the study was made in July, 1965, there were some 625 executives with salaries ranging from $16,000 to$30,000. These are the men and women who fill the top three or four levels in each department or agency. Officially, they are known as Senior Officers 1, 2, 3. and Deputy Ministers (Deputy Heads).Data on 591 of these executives was obtained from files and various other sources, coded and fed into a Bendix G-20 computer along with some 40 questions, the answers to which provided the following information :The typical executive is 51 years old and probably has a B.A. in Social Management Sciences (Arts, History, Political Science, Commerce, Sociology, etc.). While he could have graduated from any university in any province, the chances are greatest that he obtained his degree from the University of Toronto. He has had military service, reads a second language and joined the Public Service at a junior managerial level at 34 years of age, after having worked outside the Public Service for 11 years.At the time of his appointment to the executive level, he was 46 years of age and had 23 years of working experience. As of July 1, 1967, he had 28 years working experience and was earning $21,000 a year.BASIC PERSONAL DATAIn considering the personal data in more detail, it was found that ages ranged from 30 to 69, with an average age of 51.2. The average ages for the four levels (Senior Officer 1, 2, 3, and Deputy Minister — Deputy Head), were 50.3 52.6, 51.7 and 54.9 years respectively. The similarity of the averages implies career progression problems and also highlights the serious executive replacement problem the government faces in the next 10-15 years.A relatively high level of education was found among the government's top executives. 81$14,000 plus). Close to 20% entered directly at the executive level.SALARYAs of July 1, 1967, the average executive in the Public Service of Canada was earning $20,927 a year.It should be noted that this is a total figure because Public Servants do not receive bonusses or stock options, and there is not profit to be « profit-shared ».For the three career path categories they were :Public Service Only$21,297Private-Public 20,861Public-Private-Public 20,862Subsequent to this study, the government released proposed salary scales for the executive levels ranging from a maximum of $23,500 for Senior Executive Officers 1 up to a maximum of$40,000 for certain Deputy Ministers.APPOINTMENT TO EXECUTIVE LEVELThe average age at time of appointment to the Senior Officer 1 level was 45.6, to the SO 2 level 48, SO 3 level 47 and DM-DH 52. The similarity of these figures is interesting, since one would expect an executive to progress from the lowest to highest level. Also, based on today's standards, the ages would appear to be rather high.TRENDS IN AGE AT APPOINTMENTIt is generally believed that there has been a trend to appoint younger officers to the executive levels, but the study casts some doubt on the belief, certainly as far as government is concerned. However, more recent data obtained since the completion of the study indicates a change in the trend with the average ages of Senior Officers 1 appointed in 1966, 1967 and 1968, decreasing from 48 to 46 to 43 years respectively.EXPERIENCEThe average government executive has 28 years experience, almost two-thirds of which will have been spent in the Public Service. He will have spent about 15% in the category we called « Business and Self-Employed » and 10% in military service.However, there were indications that the more time executives spent in industry the less capable they were of adjusting to the political and Public Service atmosphere. Similarly it was found that extended career military service tended to reduce the rate of progress.RECRUITMENTPrior to July, 1967, about 76% of all appointments to the executive level were made from within the individuals present department and about 9% were from other departments. Therefore, some 85% of all appointments were from within the Public Service. Of the remaining 15%, some 10% came from Industry and 5% were recruited from Provincial Governments (2.4%), Universities (1.5%) and the Armed Forces (.3%).MOBILITYWithin the Public Service, a great interest has recently developed in the movement, or lack of movement, of management level personnel from one department to another. Traditionally, Public Servants have tended to remain in one department, rising to senior levels within what many claim to be a narrow occupational field.From the findings it appeared that moving from one department to another, within the Public Service, tended to improve the possibilities of achieving more rapid progress.A further analysis of movement after reaching the executive level showed that slightly more than 12% of all executives changed departments at least once after reaching this level and 1.3% had changed twice. This figure is changing rapidly as over 30% of the appointments made in 1968, at the executive level, resulted in movement between departments

Speed of synchronization in complex networks of neural oscillators Analytic results based on Random Matrix Theory

We analyze the dynamics of networks of spiking neural oscillators. First, we present an exact linear stability theory of the synchronous state for networks of arbitrary connectivity. For general neuron rise functions, stability is determined by multiple operators, for which standard analysis is not suitable. We describe a general non-standard solution to the multi-operator problem. Subsequently, we derive a class of rise functions for which all stability operators become degenerate and standard eigenvalue analysis becomes a suitable tool. Interestingly, this class is found to consist of networks of leaky integrate and fire neurons. For random networks of inhibitory integrate-and-fire neurons, we then develop an analytical approach, based on the theory of random matrices, to precisely determine the eigenvalue distribution. This yields the asymptotic relaxation time for perturbations to the synchronous state which provides the characteristic time scale on which neurons can coordinate their activity in such networks. For networks with finite in-degree, i.e. finite number of presynaptic inputs per neuron, we find a speed limit to coordinating spiking activity: Even with arbitrarily strong interaction strengths neurons cannot synchronize faster than at a certain maximal speed determined by the typical in-degree.Comment: 17 pages, 12 figures, submitted to Chao

Depleted pyrochlore antiferromagnets

I consider the class of "depleted pyrochlore" lattices of corner-sharing triangles, made by removing spins from a pyrochlore lattice such that every tetrahedron loses exactly one. Previously known examples are the "hyperkagome" and "kagome staircase". I give criteria in terms of loops for whether a given depleted lattice can order analogous to the kagome \sqrt{3} \times \sqrt{three} state, and also show how the pseudo-dipolar correlations (due to local constraints) generalize to even the random depleted case.Comment: 6pp IOP latex, 1 figure; Proc. "Highly Frustrated Magnetism 2008", Sept 2008, Braunschwei

Operational Improvements From the In-Trail Procedure in the North Atlantic Organized Track System

This paper explains the computerized batch processing experiment examining the operational impacts of the introduction of Automatic Dependent Surveillance-Broadcast (ADS-B) equipment and the In-Trail Procedure (ITP) to the North Atlantic Organized Track System (NATOTS). This experiment was conducted using the Traffic Manager (TMX), a desktop simulation capable of simulating airspace environments and aircraft operations. ADS-B equipment can enable the use of new ground and airborne procedures, such as the ITP. The ITP is among the first of these new procedures, which will make use of improved situation awareness in the local surrounding airspace of ADS-B equipped aircraft to enable more efficient oceanic flight level changes. The data collected were analyzed with respect to multiple operationally relevant parameters including fuel burn, request approval rates, and the distribution of fuel savings. This experiment showed that through the use of ADS-B or ADS-B and the ITP that operational improvements and benefits could be achieved

A Comparison between the Zero Forcing Number and the Strong Metric Dimension of Graphs

The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)-S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. The \emph{strong metric dimension}, $sdim(G)$, of a graph $G$ is the minimum among cardinalities of all strong resolving sets: $W \subseteq V(G)$ is a \emph{strong resolving set} of $G$ if for any $u, v \in V(G)$, there exists an $x \in W$ such that either $u$ lies on an $x-v$ geodesic or $v$ lies on an $x-u$ geodesic. In this paper, we prove that $Z(G) \le sdim(G)+3r(G)$ for a connected graph $G$, where $r(G)$ is the cycle rank of $G$. Further, we prove the sharp bound $Z(G) \leq sdim(G)$ when $G$ is a tree or a unicyclic graph, and we characterize trees $T$ attaining $Z(T)=sdim(T)$. It is easy to see that $sdim(T+e)-sdim(T)$ can be arbitrarily large for a tree $T$; we prove that $sdim(T+e) \ge sdim(T)-2$ and show that the bound is sharp.Comment: 8 pages, 5 figure

On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields

In this paper we discuss off-shell representations of N-extended supersymmetry in one dimension, ie, N-extended supersymmetric quantum mechanics, and following earlier work on the subject codify them in terms of certain graphs, called Adinkras. This framework provides a method of generating all Adinkras with the same topology, and so also all the corresponding irreducible supersymmetric multiplets. We develop some graph theoretic techniques to understand these diagrams in terms of a relatively small amount of information, namely, at what heights various vertices of the graph should be "hung". We then show how Adinkras that are the graphs of N-dimensional cubes can be obtained as the Adinkra for superfields satisfying constraints that involve superderivatives. This dramatically widens the range of supermultiplets that can be described using the superspace formalism and organizes them. Other topologies for Adinkras are possible, and we show that it is reasonable that these are also the result of constraining superfields using superderivatives. The family of Adinkras with an N-cubical topology, and so also the sequence of corresponding irreducible supersymmetric multiplets, are arranged in a cyclical sequence called the main sequence. We produce the N=1 and N=2 main sequences in detail, and indicate some aspects of the situation for higher N.Comment: LaTeX, 58 pages, 52 illustrations in color; minor typos correcte