Asymptotic equidistribution of congruence classes with respect to the convolution iterates of a probability vector

2012 ◽  
Vol 82 (10) ◽  
pp. 1849-1852 ◽  
Author(s):  
Gilles Gnacadja
Keyword(s):  
Probability Vector ◽  
Asymptotic Equidistribution ◽  
Congruence Classes

On a Markovian queue with weakly correlated interarrival times

1981 ◽  
Vol 18 (01) ◽  
pp. 190-203 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Time Distribution ◽  
Queueing System ◽  
Interarrival Time ◽  
Probability Vector ◽  
Common Value ◽  
Markovian Queue ◽  
The Stability ◽  
Number Of Customers ◽  
Stationary Probabilities ◽  
Stationary Probability Vector

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated. The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.

scholarly journals Human-Mimetic Estimation of Food Volume from a Single-View RGB Image Using an AI System

Electronics ◽  
2021 ◽  
Vol 10 (13) ◽  
pp. 1556
Author(s):  
Zhengeng Yang ◽  
Hongshan Yu ◽  
Shunxin Cao ◽  
Qi Xu ◽  
Ding Yuan ◽  
...  
Keyword(s):  
Chronic Diseases ◽  
Volume Estimation ◽  
Portion Size ◽  
Inner Product ◽  
Probability Vector ◽  
Internal Reference ◽  
Single View ◽  
Unhealthy Diet ◽  
Simplified Procedure ◽  
Traditional Image

It is well known that many chronic diseases are associated with unhealthy diet. Although improving diet is critical, adopting a healthy diet is difficult despite its benefits being well understood. Technology is needed to allow an assessment of dietary intake accurately and easily in real-world settings so that effective intervention to manage being overweight, obesity, and related chronic diseases can be developed. In recent years, new wearable imaging and computational technologies have emerged. These technologies are capable of performing objective and passive dietary assessments with a much simplified procedure than traditional questionnaires. However, a critical task is required to estimate the portion size (in this case, the food volume) from a digital image. Currently, this task is very challenging because the volumetric information in the two-dimensional images is incomplete, and the estimation involves a great deal of imagination, beyond the capacity of the traditional image processing algorithms. In this work, we present a novel Artificial Intelligent (AI) system to mimic the thinking of dietitians who use a set of common objects as gauges (e.g., a teaspoon, a golf ball, a cup, and so on) to estimate the portion size. Specifically, our human-mimetic system “mentally” gauges the volume of food using a set of internal reference volumes that have been learned previously. At the output, our system produces a vector of probabilities of the food with respect to the internal reference volumes. The estimation is then completed by an “intelligent guess”, implemented by an inner product between the probability vector and the reference volume vector. Our experiments using both virtual and real food datasets have shown accurate volume estimation results.

scholarly journals SUMS OF PRODUCTS OF CONGRUENCE CLASSES AND OF ARITHMETIC PROGRESSIONS

2009 ◽  
Vol 05 (04) ◽  
pp. 625-634
Author(s):  
SERGEI V. KONYAGIN ◽  
MELVYN B. NATHANSON
Keyword(s):  
Arithmetic Progression ◽  
Congruence Class ◽  
Arithmetic Progressions ◽  
Sum Of Products ◽  
Sums Of Products ◽  
Positive Integers ◽  
Congruence Classes

Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a + im : i ∈ N0}. For positive integers a,b,c,d,m the sum of products set Rm(a)Rm(b) + Rm(c)Rm(d) consists of all integers of the form (a+im) · (b+jm)+(c+km)(d+ℓm) for some i,j,k,ℓ ∈ Z. It is proved that if gcd (a,b,c,d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class Rm(ab+cd), and that the sum of products set Pm(a)Pm(b)+Pm(c)Pm eventually coincides with the infinite arithmetic progression Pm(ab+cd).

scholarly journals Lehmer points and visible points on affine varieties over finite fields

2013 ◽  
Vol 156 (2) ◽  
pp. 193-207 ◽  
Author(s):  
KIT-HO MAK ◽  
ALEXANDRU ZAHARESCU
Keyword(s):  
Finite Fields ◽  
Affine Variety ◽  
Asymptotic Results ◽  
Visible Points ◽  
Congruence Classes

AbstractLet V be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on V is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on V are obtained, and the distribution of visible points into different congruence classes is investigated.

scholarly journals Congruence Class Sizes in Finite Sectionally Complemented Lattices

2004 ◽  
Vol 47 (2) ◽  
pp. 191-205 ◽  
Author(s):  
G. Grätzer ◽  
E. T. Schmidt
Keyword(s):  
Congruence Class ◽  
Typical Result ◽  
Complemented Lattices ◽  
The Family ◽  
Complemented Lattice ◽  
Congruence Classes

AbstractThe congruences of a finite sectionally complemented lattice L are not necessarily uniform (any two congruence classes of a congruence are of the same size). To measure how far a congruence Θ of L is from being uniform, we introduce Spec Θ, the spectrum of Θ, the family of cardinalities of the congruence classes of Θ. A typical result of this paper characterizes the spectrum S = (mj | j < n) of a nontrivial congruence Θ with the following two properties:

scholarly journals ACDC: Analysis of Congruent Diversification Classes

2022 ◽  
Author(s):  
Sebastian Hoehna ◽  
Bjoern Tore Kopperud ◽  
Andrew F Magee
Keyword(s):  
Congruence Class ◽  
R Package ◽  
Rate Function ◽  
Diversification Rate ◽  
Extinction Rate ◽  
Diversification Rates ◽  
Common Features ◽  
Alternative Hypotheses ◽  
Rate Functions ◽  
Congruence Classes

Diversification rates inferred from phylogenies are not identifiable. There are infinitely many combinations of speciation and extinction rate functions that have the exact same likelihood score for a given phylogeny, building a congruence class. The specific shape and characteristics of such congruence classes have not yet been studied. Whether speciation and extinction rate functions within a congruence class share common features is also not known. Instead of striving to make the diversification rates identifiable, we can embrace their inherent non-identifiable nature. We use two different approaches to explore a congruence class: (i) testing of specific alternative hypotheses, and (ii) randomly sampling alternative rate function within the congruence class. Our methods are implemented in the open-source R package ACDC (https://github.com/afmagee/ACDC). ACDC provides a flexible approach to explore the congruence class and provides summaries of rate functions within a congruence class. The summaries can highlight common trends, i.e. increasing, flat or decreasing rates. Although there are infinitely many equally likely diversification rate functions, these can share common features. ACDC can be used to assess if diversification rate patterns are robust despite non-identifiability. In our example, we clearly identify three phases of diversification rate changes that are common among all models in the congruence class. Thus, congruence classes are not necessarily a problem for studying historical patterns of biodiversity from phylogenies.

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