scholarly journals Monotone subsequence via ultrapower

2018 ◽  
Vol 16 (1) ◽  
pp. 149-153
Author(s):  
Piotr Błaszczyk ◽  
Vladimir Kanovei ◽  
Mikhail G. Katz ◽  
Tahl Nowik
Keyword(s):  
Infinite Sequence ◽  
Monotone Subsequence

AbstractAn ultraproduct can be a helpful organizing principle in presenting solutions of problems at many levels, as argued by Terence Tao. We apply it here to the solution of a calculus problem: every infinite sequence has a monotone infinite subsequence, and give other applications.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

scholarly journals Expressiveness and Nash Equilibrium in Iterated Boolean Games

2021 ◽  
Vol 22 (2) ◽  
pp. 1-38
Author(s):  
Julian Gutierrez ◽  
Paul Harrenstein ◽  
Giuseppe Perelli ◽  
Michael Wooldridge
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Infinite Sequence ◽  
Multi Agent Systems ◽  
Temporal Logics ◽  
Agent Systems ◽  
Temporal Properties ◽  
Multi Agent ◽  
Game Theoretic ◽  
Boolean Games

We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model of multi-agent systems [Gutierrez et al. 2013, 2015a]. In such a game, each agent  has a goal  , represented using (a fragment of) Linear Temporal Logic ( ) . The goal  captures agent  ’s preferences, in the sense that the models of  represent system behaviours that would satisfy  . Each player controls a subset of Boolean variables , and at each round in the game, player is at liberty to choose values for variables in any way that she sees fit. Play continues for an infinite sequence of rounds, and so as players act they collectively trace out a model for , which for every player will either satisfy or fail to satisfy their goal. Players are assumed to act strategically, taking into account the goals of other players, in an attempt to bring about computations satisfying their goal. In this setting, we apply the standard game-theoretic concept of (pure) Nash equilibria. The (possibly empty) set of Nash equilibria of an iterated Boolean game can be understood as inducing a set of computations, each computation representing one way the system could evolve if players chose strategies that together constitute a Nash equilibrium. Such a set of equilibrium computations expresses a temporal property—which may or may not be expressible within a particular fragment. The new notion of expressiveness that we formally define and investigate is then as follows: What temporal properties are characterised by the Nash equilibria of games in which agent goals are expressed in specific fragments of  ? We formally define and investigate this notion of expressiveness for a range of fragments. For example, a very natural question is the following: Suppose we have an iterated Boolean game in which every goal is represented using a particular fragment of : is it then always the case that the equilibria of the game can be characterised within ? We show that this is not true in general.

A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables

1976 ◽  
Vol 13 (2) ◽  
pp. 361-364 ◽  
Author(s):  
M. E. Solari ◽  
J. E. A. Dunnage
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Partial Sums ◽  
Dependent Random Variables ◽  
Finite Segment ◽  
Exchangeable Random Variables ◽  
Partial Sum

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

A note on exchangeable events

1979 ◽  
Vol 16 (03) ◽  
pp. 662-664
Author(s):  
Yu-Sheng Hsu
Keyword(s):  
Infinite Sequence ◽  
Finite Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
De Finetti ◽  
Necessary And Sufficient

Kendall [2] gave a thorough discussion of De Finetti constants for a sequence (finite or infinite) of exchangeable events. Galambos [1] and Ridler-Rowe [3] also found some interesting results in this area. In this paper, we intend to give a necessary and sufficient condition for a finite sequence of exchangeable events to be extendable to an infinite sequence of exchangeable events.

scholarly journals On the Existence of Eigenmodes of Linear Quasi-Periodic Differential Equations and their Relation to the MHD Continuum

1982 ◽  
Vol 37 (8) ◽  
pp. 830-839 ◽  
Author(s):  
A. Salat
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Infinite Sequence ◽  
Periodic Coefficient ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Magnetic Surfaces ◽  
Toroidal Geometry

The existence of quasi-periodic eigensolutions of a linear second order ordinary differential equation with quasi-periodic coefficient f{ω1t, ω2t) is investigated numerically and graphically. For sufficiently incommensurate frequencies ω1, ω2, a doubly indexed infinite sequence of eigenvalues and eigenmodes is obtained.The equation considered is a model for the magneto-hydrodynamic “continuum” in general toroidal geometry. The result suggests that continuum modes exist at least on sufficiently ir-rational magnetic surfaces

Noncommutative negative order AKNS equation and its soliton solutions

2018 ◽  
Vol 33 (35) ◽  
pp. 1850209 ◽  
Author(s):  
H. Wajahat A. Riaz ◽  
Mahmood ul Hassan
Keyword(s):  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Curvature Condition ◽  
Integrable Structure ◽  
Zero Curvature ◽  
Negative Order ◽  
Akns Equation

A noncommutative negative order AKNS (NC-AKNS(-1)) equation is studied. To show the integrability of the system, we present explicitly the underlying integrable structure such as Lax pair, zero-curvature condition, an infinite sequence of conserved densities, Darboux transformation (DT) and quasideterminant soliton solutions. Moreover, the NC-AKNS(-1) equation is compared with its commutative counterpart not only on the level of nonlinear evolution equation but also for the explicit solutions.

scholarly journals The Maximal Number of Runs in Standard Sturmian Words

10.37236/2473 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Paweł Baturo ◽  
Marcin Piątkowski ◽  
Wojciech Rytter
Keyword(s):  
Explicit Formula ◽  
Special Class ◽  
Linear Time ◽  
Infinite Sequence ◽  
Integer Sequence ◽  
Sturmian Word ◽  
Compact Representations ◽  
Sturmian Words ◽  
Standard Word

We investigate some repetition problems for a very special class $\mathcal{S}$ of strings called the standard Sturmian words, which  have very compact representations in terms of sequences of integers. Usually the size of this word is exponential with respect to the size of its integer sequence, hence we are dealing with repetition problems in compressed strings. An explicit formula is given for the number $\rho(w)$ of runs in a standard word $w$. We show that $\rho(w)/|w|\le 4/5$ for each $w\in S$, and  there is an infinite sequence of strictly growing words $w_k\in {\mathcal{S}}$ such that $\lim_{k\rightarrow \infty} \frac{\rho(w_k)}{|w_k|} = \frac{4}{5}$. Moreover, we show how to compute the number of runs in a standard Sturmian word in linear time with respect to the size of its compressed representation.

scholarly journals An Erd\H{o}s--Hajnal analogue for permutation classes

2016 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
Vincent Vatter
Keyword(s):  
Monotone Subsequence

Let $\mathcal{C}$ be a permutation class that does not contain all layered permutations or all colayered permutations. We prove that there is a constant $c$ such that every permutation in $\mathcal{C}$ of length $n$ contains a monotone subsequence of length $cn$.

scholarly journals Another Infinite Sequence of Dense Triangle-Free Graphs

10.37236/1381 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Stephan Brandt ◽  
Tomaž Pisanski
Keyword(s):  
Structural Properties ◽  
Chromatic Number ◽  
Infinite Sequence ◽  
Minimum Degree ◽  
Free Graph ◽  
The Core ◽  
Graph Homomorphisms ◽  
Natural Way

The core is the unique homorphically minimal subgraph of a graph. A triangle-free graph with minimum degree $\delta > n/3$ is called dense. It was observed by many authors that dense triangle-free graphs share strong structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms. One infinite sequence of cores of dense maximal triangle-free graphs was known. All graphs in this sequence are 3-colourable. Only two additional cores with chromatic number 4 were known. We show that the additional graphs are the initial terms of a second infinite sequence of cores.

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