scholarly journals Explicit interval estimates for prime numbers

2021 ◽  
Author(s):  
Michaela Cully-Hugill ◽  
Ethan Lee
Keyword(s):  
Prime Numbers ◽  
Interval Estimates

How to Correct for Chance Agreement in the Estimation of Sensitivity and Specificity of Diagnostic Tests

1994 ◽  
Vol 33 (02) ◽  
pp. 180-186 ◽  
Author(s):  
H. Brenner ◽  
O. Gefeller
Keyword(s):  
Sensitivity And Specificity ◽  
Disease Status ◽  
Disease Prevalence ◽  
Small Sample ◽  
Small Samples ◽  
Interval Estimate ◽  
Traditional Concept ◽  
Interval Estimates ◽  
Major Disadvantage ◽  
Test Result

Abstract:The traditional concept of describing the validity of a diagnostic test neglects the presence of chance agreement between test result and true (disease) status. Sensitivity and specificity, as the fundamental measures of validity, can thus only be considered in conjunction with each other to provide an appropriate basis for the evaluation of the capacity of the test to discriminate truly diseased from truly undiseased subjects. In this paper, chance-corrected analogues of sensitivity and specificity are presented as supplemental measures of validity, which pay attention to the problem of chance agreement and offer the opportunity to be interpreted separately. While recent proposals of chance-correction techniques, suggested by several authors in this context, lead to measures which are dependent on disease prevalence, our method does not share this major disadvantage. We discuss the extension of the conventional ROC-curve approach to chance-corrected measures of sensitivity and specificity. Furthermore, point and asymptotic interval estimates of the parameters of interest are derived under different sampling frameworks for validation studies. The small sample behavior of the estimates is investigated in a simulation study, leading to a logarithmic modification of the interval estimate in order to hold the nominal confidence level for small samples.

A remark on a theorem of the Goldbach-Waring type

2004 ◽  
Vol 41 (3) ◽  
pp. 309-324
Author(s):  
C. Bauer
Keyword(s):  
Prime Numbers

Let pi, 2 ≤ i ≤ 5 be prime numbers. It is proved that all but ≪ x23027/23040+ε even integers N ≤ x can be written as N = p21 + p32 + p43 + p45.

The First 50 Million Prime Numbers

1977 ◽  
Vol 1 (S2) ◽  
pp. 7-19 ◽  
Author(s):  
Don Zagier
Keyword(s):  
Prime Numbers

Density of summable subsequences of a sequence and its applications

2020 ◽  
Vol 70 (3) ◽  
pp. 657-666
Author(s):  
Bingzhe Hou ◽  
Yue Xin ◽  
Aihua Zhang
Keyword(s):  
Prime Numbers ◽  
Linear Operators ◽  
Upper Density ◽  
Asymptotic Densities

AbstractLet x = $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that $\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$xk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if $\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.

Complex group algebras of the direct product of non-isomorphic Suzuki groups

2019 ◽  
Vol 19 (02) ◽  
pp. 2050036
Author(s):  
Morteza Baniasad Azad ◽  
Behrooz Khosravi
Keyword(s):  
Direct Product ◽  
Group Algebra ◽  
Irreducible Character ◽  
Prime Numbers ◽  
Product Formula ◽  
Character Degrees ◽  
Group Algebras ◽  
Complex Group ◽  
Complex Group Algebra ◽  
Suzuki Groups

In this paper, we prove that the direct product [Formula: see text], where [Formula: see text] are distinct numbers, is uniquely determined by its complex group algebra. Particularly, we show that the direct product [Formula: see text], where [Formula: see text]’s are distinct odd prime numbers, is uniquely determined by its order and three irreducible character degrees.

Une Caractérisation des Polynômes Prenant des Valeurs Entières Sur Tous les Nombres Premiers

1996 ◽  
Vol 39 (4) ◽  
pp. 402-407 ◽  
Author(s):  
Jean-Luc Chabert
Keyword(s):  
Prime Numbers

AbstractWe give a characterization of polynomials with rational coefficients which take integral values on the prime numbers: to test a polynomial of degree n, it is enough to consider its values on the integers from 1 to 2n —1.

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