INFINITE SUM OF CONSTANT

2020 ◽  
Vol 7 (6) ◽  
Keyword(s):  
Infinite Sum

Branching processes and functional differential equations determining steady-size distributions in cell populations

1995 ◽  
Vol 32 (01) ◽  
pp. 1-10
Author(s):  
Ziad Taib
Keyword(s):  
Differential Equation ◽  
Branching Process ◽  
Branching Processes ◽  
Functional Differential Equations ◽  
Cell Populations ◽  
Size Distributions ◽  
Multitype Branching Process ◽  
Functional Differential ◽  
Intuitive Interpretation ◽  
Infinite Sum

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.

Numerical integration by systematic sampling

1950 ◽  
Vol 46 (1) ◽  
pp. 111-115 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Numerical Integration ◽  
Bounded Variation ◽  
Mathematical Problem ◽  
Random Variable ◽  
Mean Value ◽  
Lattice Points ◽  
Systematic Sampling ◽  
Image Position ◽  
Infinite Sum

Recent investigations by F. Yates (1) in agricultural statistics suggest a mathematical problem which may be formulated as follows. A function f(x) is known to be of bounded variation and Lebesgue integrable on the range −∞ < x < ∞, and its integral over this range is to be determined. In default of any knowledge of the position of the non-negligible values of the function the best that can be done is to calculate the infinite sumfor some suitable δ and an arbitrary origin t, where s ranges over all possible positive and negative integers including zero. S is evidently of period δ in t and ranges over all its values as t varies from 0 to δ. Previous writers (Aitken (2), p. 45, and Kendall (3)) have examined the resulting errors for fixed t. (They considered only symmetrical functions, and supposed one of the lattice points to be located at the centre.) Here we do not restrict ourselves to symmetrical functions and consider the likely departure of S(t) from J (the required integral) when t is a random variable uniformly distributed in (0, δ). It will be shown that S(t) is distributed about J as mean value, with a variance which will be evaluated as a function of δ, the scale of subdivision.

Exact Frequency Equation of a Linear Structure Carrying Lumped Elements Using the Assumed Modes Method

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
Philip D. Cha ◽  
Siyi Hu
Keyword(s):  
Linear Structure ◽  
Frequency Equation ◽  
Governing Equations ◽  
Linear Structures ◽  
Combined System ◽  
Digamma Function ◽  
Simply Supported ◽  
Lumped Elements ◽  
Finite Sum ◽  
Infinite Sum

Combined systems consisting of linear structures carrying lumped attachments have received considerable attention over the years. In this paper, the assumed modes method is first used to formulate the governing equations of the combined system, and the corresponding generalized eigenvalue problem is then manipulated into a frequency equation. As the number of modes used in the assumed modes method increases, the approximate eigenvalues converge to the exact solutions. Interestingly, under certain conditions, as the number of component modes goes to infinity, the infinite sum term in the frequency equation can be reduced to a finite sum using digamma function. The conditions that must be met in order to reduce an infinite sum to a finite sum are specified, and the closed-form expressions for the infinite sum are derived for certain linear structures. Knowing these expressions allows one to easily formulate the exact frequency equations of various combined systems, including a uniform fixed–fixed or fixed-free rod carrying lumped translational elements, a simply supported beam carrying any combination of lumped translational and torsional attachments, or a cantilever beam carrying lumped translational and/or torsional elements at the beam's tip. The scheme developed in this paper is easy to implement and simple to code. More importantly, numerical experiments show that the eigenvalues obtained using the proposed method match those found by solving a boundary value problem.

Addition of sigmoid-shaped fuzzy intervals using the Dombi operator and infinite sum theorems

2006 ◽  
Vol 157 (7) ◽  
pp. 952-963 ◽  
Author(s):  
József Dombi ◽  
Norbert Győrbíró
Keyword(s):  
Infinite Sum

Closed-Form for Infinite Sum in Bandlimited CDMA

2004 ◽  
Vol 8 (3) ◽  
pp. 138-140 ◽  
Author(s):  
L.A. Jatunov ◽  
V.K. Madisetti
Keyword(s):  
Closed Form ◽  
Infinite Sum

scholarly journals Centered Polygonal Lacunary Sequences

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 943 ◽  
Author(s):  
Keith Sullivan ◽  
Drew Rutherford ◽  
Darin J. Ulness
Keyword(s):  
Rotational Symmetry ◽  
Natural Boundary ◽  
Self Similarity ◽  
Lacunary Sequences ◽  
Special Limit ◽  
Similarity Scaling ◽  
Scaling Features ◽  
Infinite Sum ◽  
Natural Way ◽  
P Cycle

Lacunary functions based on centered polygonal numbers have interesting features which are distinct from general lacunary functions. These features include rotational symmetry of the modulus of the functions and a notion of polished level sets. The behavior and characteristics of the natural boundary for centered polygonal lacunary sequences are discussed. These systems are complicated but, nonetheless, well organized because of their inherent rotational symmetry. This is particularly apparent at the so-called symmetry angles at which the values of the sequence at the natural boundary follow a relatively simple 4 p -cycle. This work examines special limit sequences at the natural boundary of centered polygonal lacunary sequences. These sequences arise by considering the sequence of values along integer fractions of the symmetry angle for centered polygonal lacunary functions. These sequences are referred to here as p-sequences. Several properties of the p-sequences are explored to give insight in the centered polygonal lacunary functions. Fibered spaces can organize these cycles into equivalence classes. This then provides a natural way to approach the infinite sum of the actual lacunary function. It is also seen that the inherent organization of the centered polygonal lacunary sequences gives rise to fractal-like self-similarity scaling features. These features scale in simple ways.

On the serial correlation for number in system in the stationary GI/M/1 bulk arrival and GI/Em/1 queues

1992 ◽  
Vol 29 (2) ◽  
pp. 404-417
Author(s):  
D. A. Stanford ◽  
B. Pagurek
Keyword(s):  
Correlation Coefficient ◽  
Generating Functions ◽  
Serial Correlation ◽  
Correlation Coefficients ◽  
Numerical Examples ◽  
Bulk Arrival ◽  
Serial Correlation Coefficient ◽  
Infinite Sum

The generating functions for the serial covariances for number in system in the stationary GI/M/1 bulk arrival queue with fixed bulk sizes, and the GI/Em/1 queue, are derived. Expressions for the infinite sum of the serial correlation coefficients are also presented, as well as the first serial correlation coefficient in the case of the bulk arrival queue. Several numerical examples are considered.

An Infinite Sum

SIAM Review ◽  
1976 ◽  
Vol 18 (1) ◽  
pp. 117-117 ◽  
Author(s):  
Murray Geller
Keyword(s):  
Infinite Sum

Energetic ion distribution resulting from neutral beam injection in tokamaks

1976 ◽  
Vol 16 (2) ◽  
pp. 149-169 ◽  
Author(s):  
John D. Gaffey
Keyword(s):  
Ion Beam ◽  
Planck Equation ◽  
Magnetized Plasma ◽  
Injection Velocity ◽  
Parallel Electric Field ◽  
Ion Distribution ◽  
Asymptotic Expressions ◽  
Fast Ion ◽  
Infinite Sum ◽  
The Magnetic Field

The Fokker-Planck equation is studied for an energetic ion beam injected into a magnetized plasma consisting of Maxwellian ions and electrons with υthi ≪υb≪ υthe. The time evolution of the fast ion distribution is given in terms of an infinite sum of Legendre polynomials for distributions that are axisymmetric about the magnetic field. The effect of charge exchange is included. The resulting ion distribution is somewhat isotropic for velocities much less than the injection velocity, however, the distribution is sharply peaked in both energy and pitch angle for velocities near the injection velocity. Approximate asymptotic expressions are given for the distribution in the vicinity of the injected beam and for velocities greater than the injection velocity. The effect of a weak parallel electric field is also given.

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