scholarly journals Impulse Propagation in Compositions and Words

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Margaret Archibald ◽  
Aubrey Blecher ◽  
Charlotte Brennan ◽  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Generating Function ◽  
Finite Alphabet ◽  
Impulse Propagation ◽  
The Right

We consider compositions of n represented as bargraphs and subject these to repeated impulses which start from the left at the top level and destroy horizontally connected parts. This is repeated while moving to the right first and then downwards to the next row and the statistic of interest is the number of impulses needed to annihilate the whole composition. We achieve this by conceptualizing a generating function that tracks compositions as well as the number of impulses used. This conceptualization is repeated for words (over a finite alphabet) represented by bargraphs.

scholarly journals Changes of electrophysiological parameters in patients with atrial flutter

Medicina ◽  
2007 ◽  
Vol 43 (8) ◽  
pp. 614 ◽  
Author(s):  
Diana Žaliaduonytė-Pekšienė ◽  
Tomas Kazakevičius ◽  
Vytautas Zabiela ◽  
Vytautas Šileikis ◽  
Remigijus Vaičiulis ◽  
...  
Keyword(s):  
Response Time ◽  
Atrial Flutter ◽  
Right Atrium ◽  
Propagation Speed ◽  
Control Group ◽  
Type I ◽  
Stimulus Response ◽  
Refractory Periods ◽  
Impulse Propagation ◽  
The Right

Objectives. The aim of the study was to study some anatomic and electrophysiological features of the right atrium, related to the presence of atrial flutter. Materials and methods. A total 23 patients with type I atrial flutter and 22 patients without atrial flutter were studied. Right atrium size was assessed using echocardiography before intracardiac examination and radiofrequency ablation. Results. Effective refractory periods of coronary sinus, high right atrium, low right atrium were different comparing with the control group (P<0.05). A stimulus–response time between high right atrium and low right atrium positions in anterograde and retrograde ways, an impulse propagation speed along the lateral wall of the right atrium were statistically different comparing both groups (P<0.05). There was a significant correlation among effective refractory periods measured in different sites of the right atrium (r²=0.64, 0.44, 0.44, respectively). All measured effective refractory periods also correlated with stimulus–response time in anterograde way (P<0.05) and impulse propagation speed (P<0.05). Right atrium dimensions were significantly larger in atrial flutter group. There was no correlation between the right atrium dimensions and measured electrophysiological parameters in both groups.Conclusions. The presence of atrial flutter associates with diffuse alterations of the right atrium, but not the focal or single changes of refractoriness.

A note on the equilibrium M/G/1 queue length

1988 ◽  
Vol 25 (1) ◽  
pp. 228-231 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Infinite Divisibility ◽  
Queue Length Distribution ◽  
Service Distribution ◽  
Bulk Arrival ◽  
Finite Sum ◽  
The Right

This note concerns the distribution of the equilibrium M/G/1 queue length. A representation for the probability generating function is given which allows for an explicit finite sum representation of the associated probabilities. The radius of convergence of the probability generating function and an asymptotic formula for the right tail of the distribution also follow from this representation, as well as infinite divisibility of the queue-length distribution when the service distribution is infinitely divisible. Extension of these results to the bulk arrival case is straightforward.

scholarly journals Counting Finite Languages by Total Word Length

Integers ◽  
2011 ◽  
Vol 11 (6) ◽  
Author(s):  
Stefan Gerhold
Keyword(s):  
Explicit Expression ◽  
Generating Function ◽  
Word Length ◽  
Finite Alphabet ◽  
Alphabet Size ◽  
Total Length ◽  
Large Alphabet ◽  
Finite Language

AbstractWe investigate the number of sets of words that can be formed from a finite alphabet, counted by the total length of the words in the set. An explicit expression for the counting sequence is derived from the generating function, and asymptotics for large alphabet size and large total word length are discussed. Moreover, we derive a Gaussian limit law for the number of words in a random finite language.

scholarly journals A Note on Naturally Embedded Ternary Trees

10.37236/629 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Markus Kuba
Keyword(s):  
Generating Function ◽  
Counting Problem ◽  
Unique Path ◽  
External Node ◽  
Enumeration Problems ◽  
Ternary Tree ◽  
The Right

In this note we consider ternary trees naturally embedded in the plane in a deterministic way. The root has position zero, or in other words label zero, and the three children of a node with position $j\in\mathbb{Z}$ have positions $j-1$, $j$, and $j+1$. We derive the generating function of embedded ternary trees where all internal nodes have labels less than or equal to $j$, with $j\in\mathbb{N}$. Furthermore, we study the generating function of the number of ternary trees of size $n$ with a given number of internal nodes with label $j$. Moreover, we discuss generalizations of this counting problem to several labels at the same time. We also study a refinement of the depth of the external node of rank $s$, with $0\le s\le 2n$, by keeping track of the left, center, and right steps on the unique path from the root to the external node. The $2n+1$ external nodes of a ternary tree are ranked from the left to the right according to an inorder traversal of the tree. Finally, we discuss generalizations of the considered enumeration problems to embedded $d$-ary trees.

A note on the equilibrium M/G/1 queue length

1988 ◽  
Vol 25 (01) ◽  
pp. 228-231 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Infinite Divisibility ◽  
Queue Length Distribution ◽  
Service Distribution ◽  
Bulk Arrival ◽  
Finite Sum ◽  
The Right

This note concerns the distribution of the equilibrium M/G/1 queue length. A representation for the probability generating function is given which allows for an explicit finite sum representation of the associated probabilities. The radius of convergence of the probability generating function and an asymptotic formula for the right tail of the distribution also follow from this representation, as well as infinite divisibility of the queue-length distribution when the service distribution is infinitely divisible. Extension of these results to the bulk arrival case is straightforward.

A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes

1978 ◽  
Vol 15 (2) ◽  
pp. 235-242 ◽  
Author(s):  
Martin I. Goldstein
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Uniform Limit ◽  
Second Moment ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
The Right

Let Z(t) ··· (Z1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v.It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1]k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant.This result is then used to give a new proof of the exponential limit law.

A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes

1978 ◽  
Vol 15 (02) ◽  
pp. 235-242
Author(s):  
Martin I. Goldstein
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Uniform Limit ◽  
Second Moment ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
The Right

Let Z(t) ··· (Z 1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v. It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1] k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant. This result is then used to give a new proof of the exponential limit law.

Sequence Enumeration and the de Bruijn-Van Aardenne Ehrenfest-Smith-Tutte Theorem

1979 ◽  
Vol 31 (3) ◽  
pp. 488-495 ◽  
Author(s):  
D. M. Jackson ◽  
I. P. Goulden
Keyword(s):  
Linear System ◽  
Generating Function ◽  
Power Series Expansion ◽  
Finite Alphabet ◽  
System Of Equations ◽  
Implicit Functions ◽  
Linear System Of Equations ◽  
Lagrange Theorem ◽  
Matrix Identity ◽  
De Bruijn

The de Bruijn—van Aardenne Ehrenfest— Smith—Tutte theorem [1] is a theorem which connects the number of Eulerian dicircuits in a directed graph with the number of rooted spanning arborescences. In this paper we obtain a proof of this theorem by considering sequences over a finite alphabet, and we show that the theorem emerges from the generating function for a certain type of sequence. The generating function for the set of sequences is obtained as the solution of a linear system of equations in Section 2. The power series expansion for the solution of this system is obtained by means of the multivariate form of the Lagrange theorem for implicit functions, and is given in Section 3, together with a restatement of the theorem as a matrix identity.

scholarly journals A Fourier-Neumann series and its application to the reduction of triple cosine series

1973 ◽  
Vol 14 (2) ◽  
pp. 198-201 ◽  
Author(s):  
C. J. Tranter ◽  
J. C. Cooke
Keyword(s):  
Generating Function ◽  
Neumann Series ◽  
Definite Integral ◽  
Cosine Series ◽  
Jacobi Expansion ◽  
Image Position ◽  
The Right

The Jacobi expansionis well known and easily obtained from the generating function of the Besselcoefficients. The sum of the series on the right of equation (1) when sin (n+½)x is replaced by cos (n+½)x cannot be found in this way but it can be expressed in terms of a definite integral as shown below. The result so obtained is useful in reducing certain triple cosine series to dual series and so simplifying the solution given by one of us for such series in an earlier paper [1].

scholarly journals Strings with Maximally Many Distinct Subsequences and Substrings

10.37236/1761 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Abraham Flaxman ◽  
Aram W. Harrow ◽  
Gregory B. Sorkin
Keyword(s):  
Generating Function ◽  
Probabilistic Method ◽  
Finite Alphabet ◽  
Asymptotic Estimates ◽  
Extremal Combinatorics ◽  
De Bruijn

A natural problem in extremal combinatorics is to maximize the number of distinct subsequences for any length-$n$ string over a finite alphabet $\Sigma$; this value grows exponentially, but slower than $2^n$. We use the probabilistic method to determine the maximizing string, which is a cyclically repeating string. The number of distinct subsequences is exactly enumerated by a generating function, from which we also derive asymptotic estimates. For the alphabet $\Sigma=\{1,2\}$, $\,(1,2,1,2,\dots)$ has the maximum number of distinct subsequences, namely ${\rm Fib}(n+3)-1 \sim \left((1+\sqrt5)/2\right)^{n+3} \! / \sqrt{5}$. We also consider the same problem with substrings in lieu of subsequences. Here, we show that an appropriately truncated de Bruijn word attains the maximum. For both problems, we compare the performance of random strings with that of the optimal ones.

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