3,146 research outputs found
A Penrose polynomial for embedded graphs
We extend the Penrose polynomial, originally defined only for plane graphs,
to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial
of embedded graphs leads to new identities and relations for the Penrose
polynomial which can not be realized within the class of plane graphs. In
particular, by exploiting connections with the transition polynomial and the
ribbon group action, we find a deletion-contraction-type relation for the
Penrose polynomial. We relate the Penrose polynomial of an orientable
checkerboard colourable graph to the circuit partition polynomial of its medial
graph and use this to find new combinatorial interpretations of the Penrose
polynomial. We also show that the Penrose polynomial of a plane graph G can be
expressed as a sum of chromatic polynomials of twisted duals of G. This allows
us to obtain a new reformulation of the Four Colour Theorem
Minimum and maximum against k lies
A neat 1972 result of Pohl asserts that [3n/2]-2 comparisons are sufficient,
and also necessary in the worst case, for finding both the minimum and the
maximum of an n-element totally ordered set. The set is accessed via an oracle
for pairwise comparisons. More recently, the problem has been studied in the
context of the Renyi-Ulam liar games, where the oracle may give up to k false
answers. For large k, an upper bound due to Aigner shows that (k+O(\sqrt{k}))n
comparisons suffice. We improve on this by providing an algorithm with at most
(k+1+C)n+O(k^3) comparisons for some constant C. The known lower bounds are of
the form (k+1+c_k)n-D, for some constant D, where c_0=0.5, c_1=23/32=0.71875,
and c_k=\Omega(2^{-5k/4}) as k goes to infinity.Comment: 11 pages, 3 figure
Shot noise in the chaotic-to-regular crossover regime
We investigate the shot noise for phase-coherent quantum transport in the
chaotic-to-regular crossover regime. Employing the Modular Recursive Green's
Function Method for both ballistic and disordered two-dimensional cavities we
find the Fano factor and the transmission eigenvalue distribution for regular
systems to be surprisingly similar to those for chaotic systems. We argue that
in the case of regular dynamics in the cavity, diffraction at the lead openings
is the dominant source of shot noise. We also explore the onset of the
crossover from quantum to classical transport and develop a quasi-classical
transport model for shot noise suppression which agrees with the numerical
quantum data.Comment: 4 pages, 3 figures, submitted to Phys.Rev.Let
Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform
We enumerate the inequivalent self-dual additive codes over GF(4) of
blocklength n, thereby extending the sequence A090899 in The On-Line
Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a
well-known interpretation as quantum codes. They can also be represented by
graphs, where a simple graph operation generates the orbits of equivalent
codes. We highlight the regularity and structure of some graphs that correspond
to codes with high distance. The codes can also be interpreted as quadratic
Boolean functions, where inequivalence takes on a spectral meaning. In this
context we define PAR_IHN, peak-to-average power ratio with respect to the
{I,H,N}^n transform set. We prove that PAR_IHN of a Boolean function is
equivalent to the the size of the maximum independent set over the associated
orbit of graphs. Finally we propose a construction technique to generate
Boolean functions with low PAR_IHN and algebraic degree higher than 2.Comment: Presented at Sequences and Their Applications, SETA'04, Seoul, South
Korea, October 2004. 17 pages, 10 figure
Imaging geometry through dynamics: the observable representation
For many stochastic processes there is an underlying coordinate space, ,
with the process moving from point to point in or on variables (such as
spin configurations) defined with respect to . There is a matrix of
transition probabilities (whether between points in or between variables
defined on ) and we focus on its ``slow'' eigenvectors, those with
eigenvalues closest to that of the stationary eigenvector. These eigenvectors
are the ``observables,'' and they can be used to recover geometrical features
of
German Everyday Culture and Technology
The project focuses on developing lessons for beginning German classes that combine knowledge and understanding of German-speaking cultures with a focus on daily life, especially that affect students at the High School and College level. This approach reflects the current ACTFL standards for foreign language learning (preparing for the 21st century) and the South Carolina State Standards for Foreign Language. Since technology plays a vital role in student learning, Ms. Patinella will display Power Point presentations, as well as overheads to review documents. Ms. Patinella is able to show DVDs and you tube clips only once approved by the technology individual at her school. This academic year the school is transitioning to incorporate more technology into student learning so that students will have competence in technology
A Multi-media Approach to Building Listening and Speaking Skills in the Context of Cultural Competency
Students at Spartanburg High School recently received Apple MacBooks for use in the classroom. The goal of this project was to come up with short supplementary materials using this new technology to build German listening and speaking skills, as well as cultural competency. The hope was that incorporating technology would hold the interest of the students and provide new opportunities for instruction for the teacher. It was also hoped that the activities created would bring German culture to life for the students. The activities are divided into 4 sections: Nachrichten, Lieder, Jojo sucht das Glück, and Films & Märchen. Each folder contains an information page explaining the contents
A note on the Cops & Robber game on graphs embedded in non-orientable surfaces
The Cops and Robber game is played on undirected finite graphs. A number of
cops and one robber are positioned on vertices and take turns in sliding along
edges. The cops win if they can catch the robber. The minimum number of cops
needed to win on a graph is called its cop number. It is known that the cop
number of a graph embedded on a surface of genus is at most ,
if is orientable (Schroeder 2004), and at most , otherwise
(Nowakowski & Schroeder 1997).
We improve the bounds for non-orientable surfaces by reduction to the
orientable case using covering spaces.
As corollaries, using Schroeder's results, we obtain the following: the
maximum cop number of graphs embeddable in the projective plane is 3; the cop
number of graphs embeddable in the Klein Bottle is at most 4, and an upper
bound is for all other .Comment: 5 pages, 1 figur
The Cop Number of the One-Cop-Moves Game on Planar Graphs
Cops and robbers is a vertex-pursuit game played on graphs. In the classical
cops-and-robbers game, a set of cops and a robber occupy the vertices of the
graph and move alternately along the graph's edges with perfect information
about each other's positions. If a cop eventually occupies the same vertex as
the robber, then the cops win; the robber wins if she can indefinitely evade
capture. Aigner and Frommer established that in every connected planar graph,
three cops are sufficient to capture a single robber. In this paper, we
consider a recently studied variant of the cops-and-robbers game, alternately
called the one-active-cop game, one-cop-moves game or the lazy-cops-and-robbers
game, where at most one cop can move during any round. We show that Aigner and
Frommer's result does not generalise to this game variant by constructing a
connected planar graph on which a robber can indefinitely evade three cops in
the one-cop-moves game. This answers a question recently raised by Sullivan,
Townsend and Werzanski.Comment: 32 page
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