scholarly journals Asymptotics of the Average Height of $2$–Watermelons with a Wall

10.37236/982 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Markus Fulmek
Keyword(s):  
Average Height ◽  
Exact Enumeration ◽  
Standard Methods ◽  
Dyck Paths ◽  
Classical Work ◽  
De Bruijn ◽  
Enumeration Formula ◽  
Special Case

We generalize the classical work of de Bruijn, Knuth and Rice (giving the asymptotics of the average height of Dyck paths of length $n$) to the case of $p$–watermelons with a wall (i.e., to a certain family of $p$ nonintersecting Dyck paths; simple Dyck paths being the special case $p=1$.) An exact enumeration formula for the average height is easily obtained by standard methods and well–known results. However, straightforwardly computing the asymptotics turns out to be quite complicated. Therefore, we work out the details only for the simple case $p=2$.

scholarly journals Nearest $$\varOmega $$-stable matrix via Riemannian optimization

2021 ◽  
Author(s):  
Vanni Noferini ◽  
Federico Poloni
Keyword(s):  
Optimization Problem ◽  
Small Scale ◽  
Frobenius Norm ◽  
Unitary Matrices ◽  
Riemannian Optimization ◽  
Important Special Case ◽  
Closed Set ◽  
Standard Methods ◽  
Special Case ◽  
Stable Matrix

AbstractWe study the problem of finding the nearest $$\varOmega $$ Ω -stable matrix to a certain matrix A, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set $$\varOmega $$ Ω . Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.

scholarly journals Ascent Sequences Avoiding Pairs of Patterns

10.37236/4479 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Andrew M. Baxter ◽  
Lara K. Pudwell
Keyword(s):  
Pattern Avoidance ◽  
Exact Enumeration ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
Generating Trees ◽  
Precise Bound ◽  
Ascent Sequences ◽  
Pattern Avoiding Permutations

Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and Steingrímsson. In this paper, we consider ascent sequences of length $n$ avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the Erdős-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound.

scholarly journals A De Bruijn–Erdős Theorem for Chordal Graphs

10.37236/3527 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Laurent Beaudou ◽  
Adrian Bondy ◽  
Xiaomin Chen ◽  
Ehsan Chiniforooshan ◽  
Maria Chudnovsky ◽  
...  
Keyword(s):  
Metric Spaces ◽  
Chordal Graphs ◽  
Combinatorial Theorem ◽  
De Bruijn ◽  
Special Case

A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chávtal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces induced by connected chordal graphs.

scholarly journals The height of watermelons with wall - Extended Abstract

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Thomas Feierl
Keyword(s):  
Average Height ◽  
Height Distribution ◽  
Famous Result ◽  
Plane Trees ◽  
International Audience ◽  
De Bruijn

International audience We derive asymptotics for the moments of the height distribution of watermelons with $p$ branches with wall. This generalises a famous result by de Bruijn, Knuth and Rice on the average height of planted plane trees, and a result by Fulmek on the average height of watermelons with two branches.

scholarly journals Lower Bounds, and Exact Enumeration in Particular Cases, for the Probability of Existence of a Universal Cycle or a Universal Word for a Set of Words

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 778
Author(s):  
Herman Z. Q. Chen ◽  
Sergey Kitaev ◽  
Brian Y. Sun
Keyword(s):  
Lower Bounds ◽  
Exact Enumeration ◽  
Letter Alphabet ◽  
De Bruijn Sequences ◽  
De Bruijn

A universal cycle, or u-cycle, for a given set of words is a circular word that contains each word from the set exactly once as a contiguous subword. The celebrated de Bruijn sequences are a particular case of such a u-cycle, where a set in question is the set A n of all words of length n over a k-letter alphabet A. A universal word, or u-word, is a linear, i.e., non-circular, version of the notion of a u-cycle, and it is defined similarly. Removing some words in A n may, or may not, result in a set of words for which u-cycle, or u-word, exists. The goal of this paper is to study the probability of existence of the universal objects in such a situation. We give lower bounds for the probability in general cases, and also derive explicit answers for the case of removing up to two words in A n , or the case when k = 2 and n ≤ 4 .

The use in additive number theory of numbers without large prime factors

1993 ◽  
Vol 345 (1676) ◽  
pp. 363-376
Keyword(s):  
Historical Perspective ◽  
Exponential Sums ◽  
Upper Bounds ◽  
Mean Value ◽  
Additive Number Theory ◽  
The Past ◽  
Classical Work ◽  
Prime Factors ◽  
Special Case ◽  
Large Natural Number

In the past few years considerable progress has been made with regard to the known upper bounds for G ( k ) in Waring’s problem, that is, the smallest s such that every sufficiently large natural number is the sum of at most 8 k th powers of natural numbers. This has come about through the development of techniques using properties of numbers having only relatively small prime factors. In this article an account of these developments is given, and they are illustrated initially in a historical perspective through the special case of cubes. In particular the connection with the classical work of Davenport on smaller values of k is demonstrated. It is apparent that the fundamental ideas and the underlying mean value theorems and estimates for exponential sums have numerous applications and a brief account is given of some of them.

scholarly journals Double-Dimer Pairings and Skew Young Diagrams

10.37236/617 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Richard W. Kenyon ◽  
David B. Wilson
Keyword(s):  
Bipartite Graph ◽  
Perfect Matching ◽  
Outer Face ◽  
Young Diagrams ◽  
Grid Spacing ◽  
Dimer Model ◽  
Dyck Paths ◽  
Interesting Special Case ◽  
Special Case

We study the number of tilings of skew Young diagrams by ribbon tiles shaped like Dyck paths, in which the tiles are "vertically decreasing". We use these quantities to compute pairing probabilities in the double-dimer model: Given a planar bipartite graph $G$ with special vertices, called nodes, on the outer face, the double-dimer model is formed by the superposition of a uniformly random dimer configuration (perfect matching) of $G$ together with a random dimer configuration of the graph formed from $G$ by deleting the nodes. The double-dimer configuration consists of loops, doubled edges, and chains that start and end at the boundary nodes. We are interested in how the chains connect the nodes. An interesting special case is when the graph is $\varepsilon(\mathbb Z\times\mathbb N)$ and the nodes are at evenly spaced locations on the boundary $\mathbb R$ as the grid spacing $\varepsilon\to0$.

scholarly journals Extending the parking space

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Andrew Berget ◽  
Brendon Rhoades
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Algebraic Combinatorics ◽  
Parking Space ◽  
Parking Functions ◽  
Dyck Paths ◽  
International Audience ◽  
Special Case

International audience The action of the symmetric group $S_n$ on the set $\mathrm{Park}_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $\mathrm{Park}_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module $V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $\mathrm{Park}_n$. We describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization. L’action du groupe symétrique $S_n$ sur l’ensemble $\mathrm{Park}_n$ des fonctions de stationnement de longueur $n$ a reçu beaucoup d’attention dans la combinatoire algébrique. Nous démontrons que l’action de $S_n$ sur $\mathrm{Park}_n$ s’étend à une action de $S_{n+1}$. Plus précisément, nous construisons un gradué $S_{n+1}$-module $V_n$ telles que la restriction de $S_n$ est isomorphe à $\mathrm{Park}_n$. Nous décrivons la $S_n$-Frobenius caractères des modules $V_n$ à tous les degrés et décrivent le $S_{n+1}$-Frobenius caractères de $V_n$ en degrés extrêmes. Nous donnons une généralisation bivariée $V_n^{(\ell, m)}$ de notre module $V_n$ dont la représentation théorie est régie par une généralisation bivariée des chemins de Dyck. Une généralisation Fuss de nos résultats est un cas particulier de cette généralisation bivariée.

Extreme self-sacrifice beyond fusion: Moral expansiveness and the special case of allyship

2018 ◽  
Vol 41 ◽  
Author(s):  
Daniel Crimston ◽  
Matthew J. Hornsey
Keyword(s):  
General Theory ◽  
Biological Markers ◽  
Natural Environment ◽  
Relevant Dimension ◽  
Special Case

AbstractAs a general theory of extreme self-sacrifice, Whitehouse's article misses one relevant dimension: people's willingness to fight and die in support of entities not bound by biological markers or ancestral kinship (allyship). We discuss research on moral expansiveness, which highlights individuals’ capacity to self-sacrifice for targets that lie outside traditional in-group markers, including racial out-groups, animals, and the natural environment.

3-D reconstruction of molecular shape from microcrystal thin sections

1983 ◽  
Vol 41 ◽  
pp. 748-749
Author(s):  
S. Cusack ◽  
J.-C. Jésior
Keyword(s):  
Three Dimensional ◽  
Molecular Shape ◽  
Biological Molecules ◽  
Fourier Space ◽  
Single Particles ◽  
Thin Sections ◽  
Standard Methods ◽  
X Ray ◽  
X Ray Crystallography ◽  
Dimensional Reconstruction

Three-dimensional reconstruction techniques using electron microscopy have been principally developed for application to 2-D arrays (i.e. monolayers) of biological molecules and symmetrical single particles (e.g. helical viruses). However many biological molecules that crystallise form multilayered microcrystals which are unsuitable for study by either the standard methods of 3-D reconstruction or, because of their size, by X-ray crystallography. The grid sectioning technique enables a number of different projections of such microcrystals to be obtained in well defined directions (e.g. parallel to crystal axes) and poses the problem of how best these projections can be used to reconstruct the packing and shape of the molecules forming the microcrystal.Given sufficient projections there may be enough information to do a crystallographic reconstruction in Fourier space. We however have considered the situation where only a limited number of projections are available, as for example in the case of catalase platelets where three orthogonal and two diagonal projections have been obtained (Fig. 1).

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