scholarly journals Primes in explicit short intervals on RH

2016 ◽  
Vol 12 (05) ◽  
pp. 1391-1407 ◽  
Author(s):  
Adrian W. Dudek ◽  
Loïc Grenié ◽  
Giuseppe Molteni
Keyword(s):  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Upper Bounds ◽  
Short Intervals ◽  
The Difference

In this paper, on the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

scholarly journals Prime numbers with a certain extremal type property

2018 ◽  
Vol 17 (1) ◽  
pp. 127-151
Author(s):  
Edward Tutaj
Keyword(s):  
Riemann Hypothesis ◽  
Convex Set ◽  
Counting Function ◽  
Numerical Data ◽  
Prime Numbers ◽  
Affine Function ◽  
Piecewise Affine ◽  
Order Of Magnitude ◽  
Type Property ◽  
Prime Counting Function

Abstract The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points $({e_k},\pi ({e_k}))_1^\infty $ . The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then ${{{e_k} + 1} \over {{e_k}}} = 1$ . The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.

On prime numbers of special kind on short intervals

2006 ◽  
Vol 79 (5-6) ◽  
pp. 848-853
Author(s):  
N. N. Mot’kina
Keyword(s):  
Special Kind ◽  
Prime Numbers ◽  
Short Intervals

Matrix form of interval multivariable gray model and its application

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Sandang Guo ◽  
Yaqian Jing ◽  
Bingjun Li
Keyword(s):  
Three Dimensional ◽  
Original Data ◽  
Upper Bounds ◽  
Matrix Form ◽  
Gray Model ◽  
Interval Numbers ◽  
Content Type ◽  
The Matrix ◽  
Number Sequences ◽  
The Difference

PurposeThe purpose of this paper is to make multivariable gray model to be available for the application on interval gray number sequences directly, the matrix form of interval multivariable gray model (IMGM(1,m,k) model) is constructed to simulate and forecast original interval gray number sequences in this paper.Design/methodology/approachFirstly, the interval gray number is regarded as a three-dimensional column vector, and the parameters of multivariable gray model are expressed in matrix form. Based on the dynamic gray action and optimized background value, the interval multivariable gray model is constructed. Finally, two examples and comparisons are carried out to verify the effectiveness of IMGM(1,m,k) model.FindingsThe model is applied to simulate and predict expert value, foreign direct investment, automobile sales and steel output, respectively. The results show that the proposed model has better simulation and prediction performance than another two models.Practical implicationsDue to the uncertainty information and continuous changing of reality, the interval gray numbers are used to characterize full information of original data. And the IMGM(1,m,k) model not only considers the characteristics of parameters changing with time but also takes into account information on lower, middle and upper bounds of interval gray numbers simultaneously to make better suitable for practical application.Originality/valueThe main contribution of this paper is to propose a new interval multivariable gray model, which considers the interaction between the lower, middle and upper bounds of interval numbers and need not to transform interval gray number sequences into real sequences. According to combining different characteristics of each bound of interval gray numbers, the matrix form of interval multivariable gray model is established to simulate and forecast interval gray numbers. In addition, the model introduces dynamic gray action to reflect the changes of parameters over time. Instead of white equation of classic MGM(1,m), the difference equation is directly used to solve the simulated and predicted values.

Polarization of Separating Invariants

2008 ◽  
Vol 60 (3) ◽  
pp. 556-571 ◽  
Author(s):  
Jan Draisma ◽  
Gregor Kemper ◽  
David Wehlau
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Group Actions ◽  
Upper Bounds ◽  
Weyl’S Theorem ◽  
Finite Group Actions ◽  
Separating Invariants ◽  
The Difference ◽  
Special Case

AbstractWe prove a characteristic free version of Weyl’s theorem on polarization. Our result is an exact analogue ofWeyl’s theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups.

On the optimal control of cancer radiotherapy for non-homogeneous cell populations

1993 ◽  
Vol 25 (01) ◽  
pp. 1-23 ◽  
Author(s):  
L. G. Hanin ◽  
S. T. Rachev ◽  
A. Yu. Yakovlev
Keyword(s):  
Cellular Response ◽  
Optimization Problems ◽  
Complete Solution ◽  
Upper Bounds ◽  
Approximation Procedure ◽  
Probability Metrics ◽  
New Family ◽  
Homogeneous Cell ◽  
The Difference ◽  
Extremal Values

Optimization problems in cancer radiation therapy are considered, with the efficiency functional defined as the difference between expected survival probabilities for normal and neoplastic tissues. Precise upper bounds of the efficiency functional over natural classes of cellular response functions are found. The ‘Lipschitz' upper bound gives rise to a new family of probability metrics. In the framework of the ‘m hit-one target' model of irradiated cell survival the problem of optimal fractionation of the given total dose into n fractions is treated. For m = 1, n arbitrary, and n = 1, 2, m arbitrary, complete solution is obtained. In other cases an approximation procedure is constructed. Stability of extremal values and upper bounds of the efficiency functional with respect to perturbation of radiosensitivity distributions for normal and tumor tissues is demonstrated.

Dimer coverings on the Tower of Hanoi graph

2019 ◽  
Vol 33 (07) ◽  
pp. 1950043
Author(s):  
Wei-Bang Li ◽  
Shu-Chiuan Chang
Keyword(s):  
Diatomic Molecules ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Tower Of Hanoi ◽  
N Stage ◽  
Dimension 2 ◽  
The Difference

We present the number of dimer coverings Nd(n) on the Tower of Hanoi graph THd(n) at n stage with dimension 2 [Formula: see text][Formula: see text]d[Formula: see text][Formula: see text] 5. When the number of vertices v(n) is even, Nd(n) gives the number of close-packed dimers; when the number of vertices is odd, it is impossible to have a close-packed configurations and one of the outmost vertices is allowed to be unoccupied. We define the entropy of absorption of diatomic molecules per vertex as S[Formula: see text][Formula: see text]=[Formula: see text][Formula: see text] Nd(n)/v(n), that can be shown exactly for TH2, while its lower and upper bounds can be derived in terms of the results at a certain n for THd(n) with 3 [Formula: see text][Formula: see text]d[Formula: see text][Formula: see text] 5. We find that the difference between the lower and upper bounds converges rapidly to zero as n increases, such that the value of S[Formula: see text] with d[Formula: see text]=[Formula: see text]3 and 5 can be calculated with at least 100 correct digits.

scholarly journals Square-full numbers in short intervals

1991 ◽  
Vol 110 (1) ◽  
pp. 1-3 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Positive Integer ◽  
Riemann Hypothesis ◽  
Prime Factor ◽  
Short Interval ◽  
Short Intervals ◽  
Image Position

A positive integer n is called square-full if p2|n for every prime factor p of n. Let Q(x) denote the number of square-full integers up to x. It was shown by Bateman and Grosswald [1] thatBateman and Grosswald also remarked that any improvement in the exponent would imply a ‘quasi-Riemann Hypothesis’ of the type for . Thus (1) is essentially as sharp as one can hope for at present. From (1) it follows that, for the number of square-full integers in a short interval, we havewhen and y = o (x½). (It seems more suggestive) to write the interval as (x, x + x½y]) than (x, x + y], since only intervals of length x½ or more can be of relevance here.)

scholarly journals III. The sums of the series of the reciprocals of the prime numbers and of their powers

1882 ◽  
Vol 33 (216-219) ◽  
pp. 4-10 ◽  
Keyword(s):  
Prime Numbers ◽  
Natural Numbers ◽  
The Difference

Euler has shown that it is possible to sum the series of reciprocals of powers of the prime numbers, and he has calculated the values of these sums for the even powers. I thought it of some interest to calculate the sums for the odd powers, and to evaluate a peculiar constant (somewhat analogous to the Eulerian constant,— γ = 0·57721 56649 01532 86060 65) which presents itself, in the series of simple reciprocals of primes, as the difference between the sum of the series and the double logarithmic infinity to the Napierian base ϵ. The summation of these series was shown by Euler to depend upon the Napierian logarithms of the sums of the reciprocals of the powers of the natural numbers.

Index, sub-index and sub-factor of groups with interactions to number theory

2019 ◽  
Vol 19 (06) ◽  
pp. 2050101
Author(s):  
M. H. Hooshmand
Keyword(s):  
Number Theory ◽  
Direct Product ◽  
Prime Numbers ◽  
Additive Number Theory ◽  
Open Problems ◽  
Famous Conjecture ◽  
Future Directions ◽  
Left And Right ◽  
The Difference ◽  
Equivalent Conditions

This paper is the first step of a new topic about groups which has close relations and applications to number theory. Considering the factorization of a group into a direct product of two subsets, and since every subgroup is a left and right factor, we observed that the index conception can be generalized for a class of factors. But, thereafter, we found that every subset [Formula: see text] of a group [Formula: see text] has four related sub-indexes: right, left, upper and lower sub-indexes [Formula: see text], [Formula: see text] which agree with the conception index of subgroups, and all of them are equal if [Formula: see text] is a subgroup or normal sub-semigroup of [Formula: see text]. As a result of the topic, we introduce some equivalent conditions to a famous conjecture for prime numbers (“every even number is the difference of two primes”) that one of them is: the prime numbers set is index stable (i.e. all of its sub-indexes are equal) in integers and [Formula: see text]. Index stable groups (i.e. those whose subsets are all index stable) are a challenging subject of the topic with several results and ideas. Regarding the extension of the theory, we give some methods for evaluation of sub-indexes, by using the left and right differences of subsets. At last, we pose many open problems, questions, a proposal for additive number theory, and show some future directions of researches and projects for the theory.

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