scholarly journals Logarithmic density of a sequence of integers and density of its ratio set

2003 ◽  
Vol 15 (1) ◽  
pp. 309-318 ◽  
Author(s):  
Ladislav Mišík ◽  
János T. Tóth
Keyword(s):  
Logarithmic Density ◽  
Ratio Set

scholarly journals Sarnak’s conjecture for sequences of almost quadratic word growth

2020 ◽  
pp. 1-56
Author(s):  
REDMOND MCNAMARA
Keyword(s):  
Box Dimension ◽  
Multiplicative Functions ◽  
Logarithmic Density ◽  
Liouville Function ◽  
Sign Patterns

Abstract We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the ( $\kappa -1$ )-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\varepsilon })$ many words of length n where $t = \kappa (\kappa +1)/2$ . We prove a variant of the $1$ -Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension less than $1$ .

A New Concept for Separability Problems in Blind Source Separation

2004 ◽  
Vol 16 (9) ◽  
pp. 1827-1850 ◽  
Author(s):  
Fabian J. Theis
Keyword(s):  
Characteristic Function ◽  
Blind Source Separation ◽  
Random Vector ◽  
Nonlinear Models ◽  
Source Separation ◽  
Direct Proof ◽  
Logarithmic Density ◽  
Know How ◽  
Linear Separability ◽  
Mixing Matrix

The goal of blind source separation (BSS) lies in recovering the original independent sources of a mixed random vector without knowing the mixing structure. A key ingredient for performing BSS successfully is to know the indeterminacies of the problem—that is, to know how the separating model relates to the original mixing model (separability). For linear BSS, Comon (1994) showed using the Darmois-Skitovitch theorem that the linear mixing matrix can be found except for permutation and scaling. In this work, a much simpler, direct proof for linear separability is given. The idea is based on the fact that a random vector is independent if and only if the Hessian of its logarithmic density (resp. characteristic function) is diagonal everywhere. This property is then exploited to propose a new algorithm for performing BSS. Furthermore, first ideas of how to generalize separability results based on Hessian diagonalization to more complicated nonlinear models are studied in the setting of postnonlinear BSS.

scholarly journals SOME REMARKS ON PERMUTATIONS WHICH PRESERVE THE WEIGHTED DENSITY

2013 ◽  
Vol 56 (1) ◽  
pp. 35-45
Author(s):  
Milan Paštéka ◽  
Zuzana Václavíková
Keyword(s):  
Number Theory ◽  
Natural Numbers ◽  
Logarithmic Density ◽  
Weighted Density

ABSTRACT In this paper we study the conditions (1), (2) and (3) for the permutations which preserve the weighted density. These conditions are motivated by the conditions of Lévy group, originated in [Levy, P.: Problèmes concrets d’Analyse Fonctionelle. Gauthier Villars, Paris, 1951], and studied in [Obata, N.: Density of natural numbers and Lévy group, J. Number Theory 30 (1988), 288-297]. In the second part we prove that under some conditions for the sequence of weights, for instance for the logarithmic density, the first two conditions can be launched

Rigidity of measures invariant under the action of a multiplicative semigroup of polynomial growth on 𝕋

2009 ◽  
Vol 30 (1) ◽  
pp. 151-157 ◽  
Author(s):  
MANFRED EINSIEDLER ◽  
ALEXANDER FISH
Keyword(s):  
Probability Measure ◽  
Polynomial Growth ◽  
Borel Probability Measure ◽  
Finite Support ◽  
Multiplicative Semigroup ◽  
Logarithmic Density ◽  
Circle Group ◽  
Borel Probability

AbstractWe prove that if a Borel probability measure on the circle group is invariant under the action of a ‘large’ multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then the measure is either Lebesgue or has finite support.

The Asymptotic Ratio Set and Direct Integral Decompositions of a Von Neumann Algebra

1971 ◽  
Vol 23 (4) ◽  
pp. 598-607 ◽  
Author(s):  
Ole A. Nielsen
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Direct Integral ◽  
Von Neumann ◽  
Ratio Set ◽  
Almost All ◽  
Direct Integral Decomposition ◽  
Integral Decomposition ◽  
Remarkable Progress

The fact that any von Neumann algebra on a separable Hilbert space has an essentially unique direct integral decomposition into factors means that there is a global as well as a local aspect to any partial classification of von Neumann algebras. More precisely, suppose that J is a statement about von Neumann algebras which is either true or false for any given von Neumann algebra. Then a von Neumann algebra is said to satisfy J globally if it satisfies J, and to satsify J locally if almost all the factors appearing in some (and hence in any) central decomposition of it satisfy J . In a recent paper [3], H. Araki and E. J. Woods introduced the notion of the asymptotic ratio set of a factor, and by means of this they made remarkable progress in the classification of factors.

scholarly journals Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

2020 ◽  
Vol 63 (4) ◽  
pp. 837-849 ◽  
Author(s):  
Lucile Devin
Keyword(s):  
Riemann Hypothesis ◽  
Linear Independence ◽  
Classical Case ◽  
Weak Version ◽  
Distribution Of Primes ◽  
Logarithmic Density ◽  
Fixed Integer ◽  
Residue Classes ◽  
Independence Hypothesis ◽  
And Function

AbstractWe generalize current known distribution results on Shanks–Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let $\unicode[STIX]{x1D70B}(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$. For a fixed integer $q$ and distinct invertible congruence classes $a_{0},a_{1},\ldots ,a_{D}$, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real $x$ for which the inequalities $\unicode[STIX]{x1D70B}(x;q,a_{0})>\unicode[STIX]{x1D70B}(x;q,a_{1})>\cdots >\unicode[STIX]{x1D70B}(x;q,a_{D})$ are simultaneously satisfied admits a logarithmic density.

scholarly journals Densities in certain three-way prime number races

2020 ◽  
pp. 1-34
Author(s):  
Jiawei Lin ◽  
Greg Martin
Keyword(s):  
Asymptotic Formula ◽  
Prime Number ◽  
Error Term ◽  
Proof Technique ◽  
Real Numbers ◽  
Logarithmic Density ◽  
Mutual Independence ◽  
Error Terms ◽  
The Relationship

Abstract Let $a_1$ , $a_2$ , and $a_3$ be distinct reduced residues modulo q satisfying the congruences $a_1^2 \equiv a_2^2 \equiv a_3^2 \ (\mathrm{mod}\ q)$ . We conditionally derive an asymptotic formula, with an error term that has a power savings in q, for the logarithmic density of the set of real numbers x for which $\pi (x;q,a_1)> \pi (x;q,a_2) > \pi (x;q,a_3)$ . The relationship among the $a_i$ allows us to normalize the error terms for the $\pi (x;q,a_i)$ in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.

scholarly journals Efficient Gear Ratio Selection of a Single-Speed Drivetrain for Improved Electric Vehicle Energy Consumption

2020 ◽  
Vol 12 (21) ◽  
pp. 9254
Author(s):  
Polychronis Spanoudakis ◽  
Gerasimos Moschopoulos ◽  
Theodoros Stefanoulis ◽  
Nikolaos Sarantinoudis ◽  
Eftichios Papadokokolakis ◽  
...  
Keyword(s):  
Energy Consumption ◽  
Electric Vehicle ◽  
Dynamic Model ◽  
Electric Vehicles ◽  
Gear Ratio ◽  
Test Bed ◽  
Time Data ◽  
Dynamic Simulations ◽  
Ratio Set ◽  
Selection Of

The electric vehicle (EV) market has grown over the last few years and even though electric vehicles do not currently possess a high market segment, it is projected that they will do so by 2030. Currently, the electric vehicle industry is looking to resolve the issue of vehicle range, using higher battery capacities and fast charging. Energy consumption is a key issue which heavily effects charging frequency and infrastructure and, therefore, the widespread use of EVs. Although several factors that influence energy consumption of EVs have been identified, a key technology that can make electric vehicles more energy efficient is drivetrain design and development. Based on electric motors’ high torque capabilities, single-speed transmissions are preferred on many light and urban vehicles. In the context of this paper, a prototype electric vehicle is used as a test bed to evaluate energy consumption related to different gear ratio usage on single-speed transmission. For this purpose, real-time data are recorded from experimental road tests and a dynamic model of the vehicle is created and fine-tuned using dedicated software. Dynamic simulations are performed to compare and evaluate different gear ratio set-ups, providing valuable insights into their effect on energy consumption. The correlation of experimental and simulation data is used for the validation of the dynamic model and the evaluation of the results towards the selection of the optimal gear ratio. Based on the aforementioned data, we provide useful information from numerous experimental and simulation results that can be used to evaluate gear ratio effects on electric vehicles’ energy consumption and, at the same time, help to formulate evolving concepts of smart grid and EV integration.

scholarly journals On the Krieger-Araki-Woods ratio set

1995 ◽  
Vol 47 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Gavin Brown ◽  
Anthony H. Dooley ◽  
Jane Lake
Keyword(s):  
Ratio Set
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