On Lattice Paths with Several Diagonal Steps

1968 ◽  
Vol 11 (4) ◽  
pp. 537-545 ◽  
Author(s):  
S.G. Mohanty ◽  
B.R. Handa
Keyword(s):  
Positive Integer ◽  
Line Segment ◽  
Lattice Point ◽  
Lattice Points ◽  
Lattice Paths ◽  
Lattice Path ◽  
Vertical Step ◽  
Horizontal Step ◽  
Minimal Lattice

In this note we consider the enumeration of unrestricted and restricted minimal lattice paths from (0, 0) to (m, n), with the following (μ + 2) moves, μ being a positive integer. Let the line segment between two lattice points on which no other lattice point lies be called a step. A lattice path at any stage can have either (1) a vertical step denoted by S0, or (2) a diagonal step parallel to the line x = ty (t = 1,…, μ), denoted by St, or (3) a horizontal step, denoted by Sμ+1.

scholarly journals On Lattice Paths with Diagonal Steps

1964 ◽  
Vol 7 (3) ◽  
pp. 470-472 ◽  
Author(s):  
V.K. Rohatgi
Keyword(s):  
Main Diagonal ◽  
Lattice Paths ◽  
Vertical Step ◽  
Horizontal Step

In [1] L. Moser and W. Zayachkowski considered lattice paths from (0, 0) to (x, y) where the possible moves are of three types: (1) a horizontal step, (2) a vertical step, and (3) a diagonal step. They obtained an expression for the number of paths from (0, 0) to (n, n) lying below the main diagonal except at the terminal points. In this note we extend their results to cover any point (m, n) lying below the main diagonal.

A unified lattice path approach to distributions related to Markov dependent trials

2018 ◽  
Vol 55 (4) ◽  
pp. 523-541
Author(s):  
Babita Goyal ◽  
Kanwar Sen
Keyword(s):  
Probability Distributions ◽  
Lattice Path ◽  
Vertical Step ◽  
Markov Dependent Trials ◽  
Moving Particle ◽  
First Time ◽  
Horizontal Step

For fixed integers n(= 0) and μ, the number of ways in which a moving particle taking a horizontal step with probability p and a vertical step with probability q, touches the line Y = n+μX for the first time, have been counted. The concept has been applied to obtain various probability distributions in independent and Markov dependent trials.

scholarly journals Bounds on the Lattice Point Enumerator via Slices and Projections

2021 ◽  
Author(s):  
Ansgar Freyer ◽  
Martin Henk
Keyword(s):  
Convex Body ◽  
Lattice Point ◽  
Geometric Mean ◽  
Discrete Analogue ◽  
Lattice Points ◽  
General Context ◽  
General Bound

AbstractGardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

scholarly journals A Continuous Analogue of Lattice Path Enumeration

10.37236/8788 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Quang-Nhat Le ◽  
Sinai Robins ◽  
Christophe Vignat ◽  
Tanay Wakhare
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lattice Paths ◽  
Lattice Path ◽  
Continuous Version ◽  
Continuous Analog ◽  
Continuous Analogue ◽  
Lattice Path Enumeration ◽  
Multinomial Coefficients ◽  
General Method

Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.  

An Analysis of the E l /E k /1 Queueing System by Restricted Minimal Lattice Paths

1995 ◽  
Vol 46 (2) ◽  
pp. 245
Author(s):  
Ikuo Arizono ◽  
Hiroshi Ohta ◽  
Stuart J. Deutsch ◽  
Ching-Cheng Wang
Keyword(s):  
Queueing System ◽  
Lattice Paths ◽  
Minimal Lattice

scholarly journals The number of minimal lattice paths restricted by two parallel lines

1983 ◽  
Vol 43 (2-3) ◽  
pp. 249-261 ◽  
Author(s):  
Masako Sato ◽  
Thien Tran Cong
Keyword(s):  
Lattice Paths ◽  
Parallel Lines ◽  
Minimal Lattice

scholarly journals New Proof That the Sum of the Reciprocals of Primes Diverges

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1414
Author(s):  
Vicente Jara-Vera ◽  
Carmen Sánchez-Ávila
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Lattice Points ◽  
Elementary Approach ◽  
Prime Divisors

In this paper, we give a new proof of the divergence of the sum of the reciprocals of primes using the number of distinct prime divisors of positive integer n, and the placement of lattice points on a hyperbola given by n=pr with prime number p. We also offer both a new expression of the average sum of the number of distinct prime divisors, and a new proof of its divergence, which is very intriguing by its elementary approach.

Two classical lattice point problems

1961 ◽  
Vol 57 (4) ◽  
pp. 699-721 ◽  
Author(s):  
K. S. Gangadharan ◽  
A. E. Ingham
Keyword(s):  
Lattice Point ◽  
Error Term ◽  
Divisor Problem ◽  
Lattice Points ◽  
Classical Lattice ◽  
Rectangular Hyperbola ◽  
Number Of Divisors ◽  
Image Position ◽  
Euler's Constant ◽  
Lattice Point Problems

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.

Theoretical Investigation of Alloy Phase Equilibria by Continuous Displacement Cluster Variation Method

2011 ◽  
Vol 172-174 ◽  
pp. 1119-1127
Author(s):  
Tetsuo Mohri
Keyword(s):  
Phase Equilibria ◽  
Lattice Point ◽  
Cluster Variation Method ◽  
Lattice Points ◽  
Variation Method ◽  
Bravais Lattice ◽  
Diffuse Intensity ◽  
Wide Range ◽  
Cluster Variation ◽  
Continuous Displacement

Continuous Displacement Cluster Variation Method is employed to study binary phase equilibria on the two dimensional square lattice with Lennard-Jones type pair potentials. It is confirmed that the transition temperature decreases significantly as compared with the one obtained by conventional Cluster Variation Method. This is ascribed to the distribution of atomic pairs in a wide range of atomic distance, which enables the system to attain the lower free energy. The spatial distribution of atomic species around a Bravais lattice point is visualized. Although the average position of an atom is centred at the Bravais lattice point, the maximum pair probability is not necessarily attained for the pairs located at the neighboring Bravais lattice points. In addition to the real space information, k-space information are calculated in the present study. Among them, the diffuse intensity spectra due to short range ordering and atomic displacement are discussed.

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