scholarly journals A note on the random walk model arising in double diffusion

1982 ◽  
Vol 24 (2) ◽  
pp. 121-129 ◽  
Author(s):  
L. H. Liyanage ◽  
J. M. Hill ◽  
C. M. Gulati
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Diffusion Model ◽  
General Result ◽  
Generating Functions ◽  
Double Diffusion ◽  
Horizontal Position ◽  
Special Cases ◽  
One Step ◽  
The Right

AbstractThe discrete random walk problem for the unrestricted particle formulated in the double diffusion model given in Hill [2] is solved explicitly. In this model it is assumed that a particle moves along two distinct horizontal paths, say the upper path I and lower path 2. For i = 1, 2, when the particle is in path i, it can move at each jump in one of four possible ways, one step to the right with probability pi, one step to the left with probability qi, remains in the same position with probability ri, or exchanges paths but remains in the same horizontal position with probability si (pi + qi + ri + si = 1). Using generating functions, the probability distribution of the position of an unrestricted particle is derived. Finally some special cases are discussed to illustrate the general result.

The random walk associated by the game of roulette

1981 ◽  
Vol 18 (04) ◽  
pp. 931-936
Author(s):  
James M. Hill ◽  
Chandra M. Gulati
Keyword(s):  
Random Walk ◽  
Power Series ◽  
Generating Functions ◽  
The Right ◽  
Lagrange's Theorem ◽  
Explicit Expressions

The random walk arising in the game of roulette involves an absorbing barrier at the origin; at each step either a unit displacement to the left or a fixed multiple displacement to the right can occur with probabilities q and p respectively. Using generating functions and Lagrange's theorem for the expansion of a function as a power series explicit expressions are deduced for the probabilities of the player's capital at the nth step.

The random walk associated by the game of roulette

1981 ◽  
Vol 18 (4) ◽  
pp. 931-936 ◽  
Author(s):  
James M. Hill ◽  
Chandra M. Gulati
Keyword(s):  
Random Walk ◽  
Power Series ◽  
Generating Functions ◽  
The Right ◽  
Lagrange's Theorem ◽  
Explicit Expressions

The random walk arising in the game of roulette involves an absorbing barrier at the origin; at each step either a unit displacement to the left or a fixed multiple displacement to the right can occur with probabilities q and p respectively. Using generating functions and Lagrange's theorem for the expansion of a function as a power series explicit expressions are deduced for the probabilities of the player's capital at the nth step.

A modified random walk in the presence of partially reflecting barriers

1976 ◽  
Vol 13 (1) ◽  
pp. 169-175 ◽  
Author(s):  
Saroj Dua ◽  
Shobha Khadilkar ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
One Dimensional ◽  
Unit Step ◽  
The One ◽  
The Right ◽  
Reflecting Barriers

The paper deals with the one-dimensional modified random walk in the presence of partially reflecting barriers at a and –b (a, b > 0). The simple one-dimensional random walk on a line is the motion-record of a particle which may extend over (–∞, + ∞) or be restricted to a portion of it by absorbing and/or reflecting barriers. Here we introduce the possibility of a particle staying put along with its moving a unit step to the right or to the left and find the bivariate generating functions of the probabilities of a particle reaching m (0 <m <a) under different conditions.

scholarly journals The speed of a random walk excited by its recent history

2016 ◽  
Vol 48 (1) ◽  
pp. 215-234 ◽  
Author(s):  
Ross G. Pinsky
Keyword(s):  
Random Walk ◽  
Finite Number ◽  
Threshold Level ◽  
Recent History ◽  
Threshold Levels ◽  
One Step ◽  
The Right ◽  
Positive Integers

Abstract Let N and M be positive integers satisfying 1≤ M≤ N, and let 0< p0 < p1 < 1. Define a process {Xn}n=0∞ on ℤ as follows. At each step, the process jumps either one step to the right or one step to the left, according to the following mechanism. For the first N steps, the process behaves like a random walk that jumps to the right with probability p0 and to the left with probability 1-p0. At subsequent steps the jump mechanism is defined as follows: if at least M out of the N most recent jumps were to the right, then the probability of jumping to the right is p1; however, if fewer than M out of the N most recent jumps were to the right then the probability of jumping to the right is p0. We calculate the speed of the process. Then we let N→ ∞ and M/N→ r∈[0,1], and calculate the limiting speed. More generally, we consider the above questions for a random walk with a finite number l of threshold levels, (Mi,pi) i=1l, above the pre-threshold level p0, as well as for one model with l=N such thresholds.

scholarly journals The Maximum of a Symmetric Next Neighbor Walk on the Nonnegative Integers

2014 ◽  
Vol 51 (01) ◽  
pp. 162-173
Author(s):  
Ora E. Percus ◽  
Jerome K. Percus
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Generating Functions ◽  
Two Dimensional ◽  
Symmetric Random Walk ◽  
One Dimensional ◽  
Dimensional Distribution

We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the two-dimensional probability distribution P{S n = x, max1≤j≤n S n = a} of being at position x after n steps, while the maximal location that the walker has achieved during these n steps is a. We also obtain the familiar (marginal) one-dimensional distribution for S n = x, but more importantly that for max1≤j≤n S j = a asymptotically at fixed a 2 / n. We are able to compute and compare the expectations and variances of the two one-dimensional distributions, finding that they have qualitatively similar forms, but differ quantitatively in the anticipated fashion.

A modified random walk in the presence of partially reflecting barriers

1976 ◽  
Vol 13 (01) ◽  
pp. 169-175
Author(s):  
Saroj Dua ◽  
Shobha Khadilkar ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
One Dimensional ◽  
Unit Step ◽  
The One ◽  
The Right ◽  
Reflecting Barriers

The paper deals with the one-dimensional modified random walk in the presence of partially reflecting barriers at a and –b (a, b &gt; 0). The simple one-dimensional random walk on a line is the motion-record of a particle which may extend over (–∞, + ∞) or be restricted to a portion of it by absorbing and/or reflecting barriers. Here we introduce the possibility of a particle staying put along with its moving a unit step to the right or to the left and find the bivariate generating functions of the probabilities of a particle reaching m (0 &lt;m &lt;a) under different conditions.

scholarly journals The Maximum of a Symmetric Next Neighbor Walk on the Nonnegative Integers

2014 ◽  
Vol 51 (1) ◽  
pp. 162-173
Author(s):  
Ora E. Percus ◽  
Jerome K. Percus
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Generating Functions ◽  
Two Dimensional ◽  
Symmetric Random Walk ◽  
One Dimensional ◽  
Dimensional Distribution

We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the two-dimensional probability distribution P{Sn = x, max1≤j≤nSn = a} of being at position x after n steps, while the maximal location that the walker has achieved during these n steps is a. We also obtain the familiar (marginal) one-dimensional distribution for Sn = x, but more importantly that for max1≤j≤nSj = a asymptotically at fixed a2 / n. We are able to compute and compare the expectations and variances of the two one-dimensional distributions, finding that they have qualitatively similar forms, but differ quantitatively in the anticipated fashion.

scholarly journals On Some Features of the Problem of Solvability of Systems of Boolean Equations

2021 ◽  
pp. 18-28
Author(s):  
Vladimir Leontiev ◽  
◽  
Eduard Gordeev ◽  
Keyword(s):  
Generating Functions ◽  
Research Method ◽  
Number Of Solutions ◽  
Right Hand ◽  
Special Cases ◽  
Subsequent Application ◽  
Minimum Number ◽  
Covering Set ◽  
The Right

The purpose of the article is to present new results on combinatorial characteristics of systems of Boolean equations, on which such properties of systems as compatibility, solvability, number of solutions and a number of others depend. The research method is the reduction of applied problems to combinatorial models with the subsequent application of classical methods of combinatorics: the method of generating functions, the method of coefficients, methods for obtaining asymptotics, etc. Obtained result. In this paper, we obtain results concerning the solvability of systems of Boolean equations. The complexity of the problem of “ transformation” of an incompatible system into a joint one is analyzed. An approach to solving the problem of separating the minimum number of joint subsystems from an incompatible system is described and justified. The problem is reduced to the problem of finding the minimum covering set. The system compatibility criterion is obtained. Using the method of coefficients, formulas for finding and estimating the number of solutions for parameterizing the problem on the right-hand sides of equations are derived. The maximum of this number is also investigated depending on the parameter. Formulas for the number of solutions for two special cases are obtained: with a restriction on the number of equations and on the size of the problem parameters

Three-Component [1+1+1] Cyclopropanation with Ruthenium(II)

2018 ◽  
Author(s):  
Tanner C. Jankins ◽  
Robert R. Fayzullin ◽  
Eugene Khaskin
Keyword(s):  
Quaternary Carbon ◽  
Poor Control ◽  
One Step ◽  
Synthetic Routes ◽  
Mechanistic Investigations ◽  
Chiral Centers ◽  
Metal Catalyzed ◽  
The Right ◽  
Substituted Cyclopropanes

We report a one-step, Ru(II)-catalyzed cyclopropanation reaction that is conceptually different from the previously reported protocols that include Corey-Chaykovsky, Simmons-Smith, and metal catalyzed carbene attack on olefins. Under the current protocol, various alcohols are transformed into sulfone substituted cyclopropanes with excellent isolated yields and diastereoselectivities. This new reaction forms highly congested cyclopropane products with three new C–C bonds, three or two new chiral centers and one new quaternary carbon center. 22 examples of isolated substrates are given. Previously reported synthetic routes for similar substrates are all multi-step, linear routes that proceed with overall low yields and poor control of stereochemistry. Experimental mechanistic investigations suggest initial metal-catalyzed dehydrogenation of the alcohol substrate and catalyst independent stepwise attack of two equivalents of sulfone on the aldehyde under basic conditions. While the Ru(II) is only responsible for the initial dehydrogenation step, the rate of aldehyde formation is crucial to maintaining the right balance of intermediates needed to afford the cyclopropane product.

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