scholarly journals Neighborhood Growth Dynamics on the Hamming Plane

10.37236/6400 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Janko Gravner ◽  
David Sivakoff ◽  
Erik Slivken
Keyword(s):  
Young Diagram ◽  
Growth Dynamics ◽  
Vertical Line ◽  
Product Measure ◽  
Two Dimensional ◽  
Young Diagrams ◽  
Entire Plane ◽  
Initial Product ◽  
Entropy Functional ◽  
Hamming Graphs

We initiate the study of general neighborhood growth dynamics on two-dimensional Hamming graphs. The decision to add a point is made by counting the currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram.  We focus on two related extremal quantities. The first is the size of the smallest set that eventually occupies the entire plane. The second is the minimum of an energy-entropy functional that comes from the scaling of the probability of eventual full occupation versus the density of the initial product measure within a rectangle. We demonstrate the existence of this scaling and study these quantities for large Young diagrams.

scholarly journals Search for Young diagrams with large dimensions

2019 ◽  
pp. 33-43
Author(s):  
Vasilii S. Duzhin ◽  
◽  
Anastasia A. Chudnovskaya ◽  
Keyword(s):  
Symmetric Group ◽  
Young Diagram ◽  
Numerical Experiments ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Maximum Dimension ◽  
Asymptotic Combinatorics

Search for Young diagrams with maximum dimensions or, equivalently, search for irreducible representations of the symmetric group $S(n)$ with maximum dimensions is an important problem of asymptotic combinatorics. In this paper, we propose algorithms that transform a Young diagram into another one of the same size but with a larger dimension. As a result of massive numerical experiments, the sequence of $10^6$ Young diagrams with large dimensions was constructed. Furthermore, the proposed algorithms do not change the first 1000 elements of this sequence. This may indicate that most of them have the maximum dimension. It has been found that the dimensions of all Young diagrams of the resulting sequence starting from the 75778th exceed the dimensions of corresponding diagrams of the greedy Plancherel sequence.

scholarly journals Hook lengths in a skew Young diagram

10.37236/1309 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Young Diagram ◽  
Character Degrees ◽  
Young Diagrams ◽  
Inductive Proof

Regev and Vershik (Electronic J. Combinatorics 4 (1997), #R22) have obtained some properties of the set of hook lengths for certain skew Young diagrams, using asymptotic calculations of character degrees. They also conjectured a stronger form of one of their results. We give a simple inductive proof of this conjecture. Very recently, Regev and Zeilberger (Annals of Combinatorics, to appear) have independently proved this conjecture.

Growth dynamics of phase separation for two-dimensional binary fluid

2004 ◽  
Vol 19 (4) ◽  
pp. 37-39
Author(s):  
Yuan Yin-quan ◽  
Lei Shao-ming ◽  
Cai Zi ◽  
Pan Ai-guo
Keyword(s):  
Phase Separation ◽  
Growth Dynamics ◽  
Two Dimensional ◽  
Binary Fluid

Twisted entropy functional for three-manifolds and flows induced on the boundary

2014 ◽  
Vol 11 (04) ◽  
pp. 1450033
Author(s):  
R. Cartas-Fuentevilla
Keyword(s):  
Diffusion Equation ◽  
Oscillatory Behavior ◽  
Two Dimensional ◽  
Manifolds With Boundary ◽  
Orthogonal Groups ◽  
Holonomy Groups ◽  
Entropy Functional ◽  
Chern Simons ◽  
Twisted Flows

Considering a twisted version of the gravitational Chern–Simons action for three-manifolds as a Perelman entropy functional, a generalization of the Cotton flow for the metric, in co-evolution with torsion is developed. In the case of manifolds with boundary, there exists an entropy functional induced on it, and hence metric and torsional flows can be defined. The integrability of these boundary flows is studied in detail. In a particular case, the solutions of the two-dimensional logarithmic diffusion equation determine completely the dynamics of the fields in co-evolution. In this scheme of three-manifolds with evolving boundaries, the orthogonality of the twisted flows to the Yamabe-like flows is established; additionally, the evolution of the holonomy groups of the three-manifold and of its boundary is studied, showing in some cases an oscillatory behavior around orthogonal groups. In this approach, the twisted canonical geometries (TCG) are defined as manifolds with a vanishing contribution of the torsion to the curvature, which will be generated fully by the metric; these geometries evolve to manifolds with curvature generated by both metric and torsion, and cannot smoothly evolve into manifolds with a vanishing torsion.

scholarly journals Permutations Without Long Decreasing Subsequences and Random Matrices

10.37236/929 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Piotr Šniady
Keyword(s):  
Random Matrices ◽  
Young Diagram ◽  
Random Matrix ◽  
Random Permutation ◽  
Fixed Number ◽  
Plancherel Measure ◽  
Young Diagrams ◽  
Gaussian Unitary Ensemble ◽  
Longest Increasing Subsequence ◽  
Unitary Ensemble

We study the shape of the Young diagram $\lambda$ associated via the Robinson–Schensted–Knuth algorithm to a random permutation in $S_n$ such that the length of the longest decreasing subsequence is not bigger than a fixed number $d$; in other words we study the restriction of the Plancherel measure to Young diagrams with at most $d$ rows. We prove that in the limit $n\to\infty$ the rows of $\lambda$ behave like the eigenvalues of a certain random matrix (namely the traceless Gaussian Unitary Ensemble random matrix) with $d$ rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix.

scholarly journals Hole growth dynamics in a two dimensional Leidenfrost droplet

2015 ◽  
Vol 27 (3) ◽  
pp. 031704 ◽  
Author(s):  
Christophe Raufaste ◽  
Franck Celestini ◽  
Alexandre Barzyk ◽  
Thomas Frisch
Keyword(s):  
Growth Dynamics ◽  
Two Dimensional ◽  
Hole Growth

TRANSITIVITY IN TWO-DIMENSIONAL LOCAL LANGUAGES DEFINED BY DOT SYSTEMS

2006 ◽  
Vol 17 (02) ◽  
pp. 435-463 ◽  
Author(s):  
NATAŠA JONOSKA ◽  
JONI BURNETTE PIRNOT
Keyword(s):  
Finite Graph ◽  
Finite State Automaton ◽  
Two Dimensional ◽  
Entire Plane ◽  
Shift Spaces ◽  
Related Language ◽  
Finite State ◽  
Local Languages

The paper investigates two-dimensional recognizable languages that are defined by the so-called "dot systems" that are special subgroups of (ℤ/2ℤ)ℤ2. The dot shapes that provide directional transitivity or mixing for the related language are investigated. It is shown that languages defined by parallelogram shapes fail to be transitive in the direction of a defining vector and hence fail to be mixing, while certain triangular shapes guarantee that the factor language of the associated dot system will be mixing. Dot systems belong to a class of two-dimensional shift spaces that have a factor language such that every admissable block can be extended to a configuration of the entire plane. For this class of shift spaces we introduce a finite graph (i.e., a finite state automaton) that recognizes two-dimensional local languages, then show that certain transitivity properties may be observed from the structure of the finite graph.

scholarly journals Green's functions for the wave, Helmholtz and Poisson equations in a two-dimensional boundless domain

2013 ◽  
Vol 35 (1) ◽  
Author(s):  
Roberto Toscano Couto
Keyword(s):  
Green's Functions ◽  
Fourier Transforms ◽  
Green’S Functions ◽  
Plane Domain ◽  
Integral Representations ◽  
Two Dimensional ◽  
Entire Plane ◽  
Poisson Equations ◽  
New Procedures ◽  
Real Poles

In this work, Green's functions for the two-dimensional wave, Helmholtz and Poisson equations are calculated in the entire plane domain by means of the two-dimensional Fourier transform. New procedures are provided for the evaluation of the improper double integrals related to the inverse Fourier transforms that furnish these Green's functions. The integrals are calculated by using contour integration in the complex plane. The method consists basically in applying the correct prescription for circumventing the real poles of the integrand as well as in using well-known integral representations of some Bessel functions.

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