First digit phenomenon and ergodic theory

Robertson, J B; Uppuluri, V R.R.; Rajagopal, A K

doi:10.1016/0022-247X(83)90113-0

Title: First digit phenomenon and ergodic theory

Full Record
Other Related Research

Abstract

For any k belongs to I/sup +/ identical with the set of positive integers, let S(k) denote the set of positive integers beginning with k, i.e., S(k) = )n: k = (n/10/sup r/) for some r belongs to I/sup +/), where (x) denotes the greatest integer less than or equal to x. Consider some finitely additive extension of the density function d(S) = (lim/n ..-->.. infinity) (S intersection (1, N))/N, where S contains or equals I/sup +/ and (S) denotes the cardinality of S. If d(k) = d(S(k)) is defined for all positive integers K, and if the mapping T that maps S into (2S) union (2S + 1) preserves the density d, then the result of Cohen is that d(p) = log/sub 10/ (1 + (1/p)) for p belongs to D. The crucial step here is the specific relation of the transformation T to the multiplication of the elements of S by 2. This motivates us to consider the following problem: Set b = 10 and let a belongs to R/sup +/ be a positive real number. Let theta (x) = ax, where x belongs R/sup +/, and consider the sequence p = (p/sub 1/, p/sub 2/,...) of firstmore »« less

Authors:: Robertson, J B; Uppuluri, V R.R.; Rajagopal, A K

Publication Date:: 1983-09-01

Research Org.:: Univ. of California, Santa Barbara, CA (United States)

OSTI Identifier:: 5936327

DOE Contract Number:: W-7405-ENG-26

Resource Type:: Journal Article

Journal Name:: J. Math. Anal. Appl.; (United States)

Additional Journal Information:: Journal Volume: 95:2

Country of Publication:: United States

Language:: English

Subject:: 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ERGODIC HYPOTHESIS; ENTROPY; MATHEMATICS; RANDOMNESS; TRANSFORMATIONS; HYPOTHESIS; PHYSICAL PROPERTIES; THERMODYNAMIC PROPERTIES; 990200* - Mathematics & Computers

Citation Formats


                    Robertson, J B, Uppuluri, V R.R., and Rajagopal, A K. First digit phenomenon and ergodic theory.  United States: N. p., 1983. 
        Web.  doi:10.1016/0022-247X(83)90113-0.

Copy to clipboard


                    Robertson, J B, Uppuluri, V R.R., & Rajagopal, A K. First digit phenomenon and ergodic theory.  United States.  https://doi.org/10.1016/0022-247X(83)90113-0

Copy to clipboard


                    Robertson, J B, Uppuluri, V R.R., and Rajagopal, A K. 1983.  
        "First digit phenomenon and ergodic theory".  United States.  https://doi.org/10.1016/0022-247X(83)90113-0.

Copy to clipboard


                    
@article{osti_5936327,

  title        = {First digit phenomenon and ergodic theory},

  author       = {Robertson, J B and Uppuluri, V R.R. and Rajagopal, A K},

  abstractNote = {For any k belongs to I/sup +/ identical with the set of positive integers, let S(k) denote the set of positive integers beginning with k, i.e., S(k) = )n: k = (n/10/sup r/) for some r belongs to I/sup +/), where (x) denotes the greatest integer less than or equal to x. Consider some finitely additive extension of the density function d(S) = (lim/n ..-->.. infinity) (S intersection (1, N))/N, where S contains or equals I/sup +/ and (S) denotes the cardinality of S. If d(k) = d(S(k)) is defined for all positive integers K, and if the mapping T that maps S into (2S) union (2S + 1) preserves the density d, then the result of Cohen is that d(p) = log/sub 10/ (1 + (1/p)) for p belongs to D. The crucial step here is the specific relation of the transformation T to the multiplication of the elements of S by 2. This motivates us to consider the following problem: Set b = 10 and let a belongs to R/sup +/ be a positive real number. Let theta (x) = ax, where x belongs R/sup +/, and consider the sequence p = (p/sub 1/, p/sub 2/,...) of first digits of the orbit of theta: )x, theta(x), theta/sup 2/ (x), ...). That is p/sub k/ = (theta /sup k/ (x)/b/sup r/) belongs to D where r = (log/sub 10/ theta/sup k/ (x)). The purpose of this paper is to investigate the dynamical properties of the sequence p. 7 references.},

  doi          = {10.1016/0022-247X(83)90113-0},

  url          = {https://www.osti.gov/biblio/5936327},
  journal      = {J. Math. Anal. Appl.; (United States)},
number       = ,

  volume       = 95:2,

  place        = {United States},

  year         = {1983},

  month        = {9}

}

Copy to clipboard

Journal Article:

https://doi.org/10.1016/0022-247X(83)90113-0

Other availability

Find in Google Scholar

Search WorldCat to find libraries that may hold this journal

Save / Share:

Export Metadata

Save to My Library

Similar records in OSTI.GOV collections:

Properties of a new classification of partitions into at most three parts

Journal Article Stein, P; Everett, C - Adv. Appl. Math.; (United States)

Let n be a positive integer, and let n = x + y + z, x greater than or equal to y greater than or equal to z greater than or equal to 0 be a partition of n into at most three positive parts; throughout the present paper the phase ''partition on n'' will tacitly embody this restriction. The number of such partitions is well known to be p(n, 3) = )(n + 2) (n + 4)/12), where the curly brackets in this formula (but not elsewhere in the paper) indicate nearest integer. Further, let m = x/sup 2/more »« less
https://doi.org/10.1016/0196-8858(80)90011-1
Multiflows and matchings

Conference Ilani, H

Let G be a multigraph, with a set T {improper_subset} V(G) of terminals, and S = (T, U) be a graph (called the scheme), with E(G), {intersection} U = {O}. An integer S-flow in G is a collection of edge-disjoint paths linking pairs of terminals adjacent in S. We deal with the following: Find a maximum S-flow in G. For A {improper_subset} T, put {lambda}(A) := min {l_brace}degX : X {improper_subset} V(G), X{intersection}T = A{r_brace} where deg X is the number of edges between X and {bar X}. Assume the G and S satisfy the following conditions: (1) the degreemore »« less
Vehicle scheduling on a tree with release and handling times

Conference Ibaraki, Toshihide; Karuno, Y; Mochizuki, K; ...

Let T = (V, E, v{sub 0}) be a rooted tree, where V is a set of n vertices, E is a set of edges and v{sub 0} {element_of} V is the root. The travel times d(v{sub i}, v{sub j}) and (v{sub j}, v{sub i}) are associated with each edge (v{sub i}, v{sub j}) {element_of} E, and a job, which is also denoted as v{sub i} is located at each vertex v{sub i}. Each job v{sub i} has release time r(v{sub i}) and handling time h(v{sub i}). The TREE-VSP (Vehicle Scheduling Problem on a tree) asks to find a routingmore »« less
Stability of lexicographical multicommodity flow problem

Conference Davidson, M

We consider the following lexicographical multicommodity network flow problem which generalizes the feasibility problem. Let {theta}{sub 1} = max (x, z) E X(d) {theta} z{sub i} {>=} {theta}f{sub i}, i = 1, ..., M where X(d) is the set of all admissible multicommodity flows x; z E R{sub +}{sup M} is the vector of flow values; parameters d and f are the vectors of edge capacities and flow requirements, respectively. Assuming that the problem is solved and letting z{sub i} = {theta}f{sub i} for all i such that there is no optimal solution (x*, z*) with z* > {theta}{sub 1}f{submore »« less
Structure of shock waves in magnetohydrodynamics

Thesis/Dissertation Hesaaraki, M

The mathematical question of the existence of structure for fast, slow, intermediate, switch-on, and switch-off magnetohydrodynamic shock waves is stated in terms of the following system of ordinary differential equations: ..mu.. dx/sub 1/ = x/sub 1/ - delta x/sub 2/, nu dx/sub 2/ = delta x/sub 1/ + Vx/sub 2/ + epsilon, ..mu../sub 1/ dV = 1/2 x/sub 2//sup 2/ + V - J + p(V, T), k dT = -1/2(x/sub 1//sup 2/ 2 delta x/sub 1/x/sub 2/ + Vx/sub 2//sup 2/) - epsilon x/sub 2/ - 1/2 V/sup 2/ + JV - E + e(V,T), where the symbols ..mu..,more »« less

Similar Records