scholarly journals Distribuion of Leading Digits of Numbers II

2019 ◽  
Vol 14 (1) ◽  
pp. 19-42
Author(s):  
Yukio Ohkubo ◽  
Oto Strauch
Keyword(s):  
Distribution Function ◽  
Real Number ◽  
Prime Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Natural Numbers ◽  
Increasing Functions

AbstractIn this paper, we study the sequence (f (pn))n≥1,where pn is the nth prime number and f is a function of a class of slowly increasing functions including f (x)=logb xr and f (x)=logb(x log x)r,where b ≥ 2 is an integer and r> 0 is a real number. We give upper bounds of the discrepancy D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right) for a distribution function g and a sub-sequence (Ni)i≥1 of the natural numbers. Especially for f (x)= logb xr, we obtain the effective results for an upper bound of D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right).

scholarly journals Quadratic nonresidues below the Burgess bound

2017 ◽  
Vol 13 (03) ◽  
pp. 751-759 ◽  
Author(s):  
William D. Banks ◽  
Victor Z. Guo
Keyword(s):  
Prime Number ◽  
Upper Bound ◽  
Legendre Symbol ◽  
Natural Numbers ◽  
Number Formula

For any odd prime number [Formula: see text], let [Formula: see text] be the Legendre symbol, and let [Formula: see text] be the sequence of positive nonresidues modulo [Formula: see text], i.e. [Formula: see text] for each [Formula: see text]. In 1957, Burgess showed that the upper bound [Formula: see text] holds for any fixed [Formula: see text]. In this paper, we prove that the stronger bound [Formula: see text] holds for all odd primes [Formula: see text] provided that [Formula: see text] where the implied constants are absolute. For fixed [Formula: see text], we also show that there is a number [Formula: see text] such that for all odd primes [Formula: see text], there are [Formula: see text] natural numbers [Formula: see text] with [Formula: see text] provided that [Formula: see text]

scholarly journals Some inequalities arising from analytic summability of functions

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3223-3230
Author(s):  
Sh. Saadat ◽  
M.H. Hooshmand
Keyword(s):  
Functional Equation ◽  
Holomorphic Function ◽  
Upper Bound ◽  
Bernoulli Polynomials ◽  
Upper Bounds ◽  
Bernoulli Numbers ◽  
Trigonometric Functions ◽  
Natural Numbers ◽  
Sums Of Powers ◽  
The Difference

Analytic summability of functions was introduced by the second author in 2016. He utilized Bernoulli numbers and polynomials for a holomorphic function to construct analytic summability. The analytic summand function f? (if exists) satisfies the difference functional equation f?(z) = f (z) + f?(z-1). Moreover, some upper bounds for f? and several inequalities between f and f? were presented by him. In this paper, by using Alzer?s improved upper bound for Bernoulli numbers, we improve those upper bounds and obtain some inequalities and new upper bounds. As some applications of the topic, we obtain several upper bounds for Bernoulli polynomials, sums of powers of natural numbers, (e.g., 1p+2p+3p+...+rp ? 2p! ?p+1 (e?r-1)) and several inequalities for exponential, hyperbolic and trigonometric functions.

Some spectral properties of Aα-matrix

2019 ◽  
Vol 11 (06) ◽  
pp. 1950070
Author(s):  
Shuang Zhang ◽  
Yan Zhu
Keyword(s):  
Real Number ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Spectral Properties ◽  
Upper Bounds ◽  
Largest Eigenvalue ◽  
Number Formula ◽  
Second Largest Eigenvalue ◽  
Spectral Radius Formula

For a real number [Formula: see text], the [Formula: see text]-matrix of a graph [Formula: see text] is defined to be [Formula: see text] where [Formula: see text] and [Formula: see text] are the adjacency matrix and degree diagonal matrix of [Formula: see text], respectively. The [Formula: see text]-spectral radius of [Formula: see text], denoted by [Formula: see text], is the largest eigenvalue of [Formula: see text]. In this paper, we consider the upper bound of the [Formula: see text]-spectral radius [Formula: see text], also we give some upper bounds for the second largest eigenvalue of [Formula: see text]-matrix.

Minimal complementation below uniform upper bounds for the arithmetical degrees

1996 ◽  
Vol 61 (4) ◽  
pp. 1158-1192
Author(s):  
Masahiro Kumabe
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Arithmetical Function ◽  
Natural Numbers

This paper was inspired by Lerman [15] in which he proved various properties of upper bounds for the arithmetical degrees. We discuss the complementation property of upper bounds for the arithmetical degrees. In Lerman [15], it is proved that uniform upper bounds for the arithmetical degrees are jumps of upper bounds for the arithmetical degrees. So any uniform upper bound for the arithmetical degrees is not a minimal upper bound for the arithmetical degrees. Given a uniform upper bound a for the arithmetical degrees, we prove a minimal complementation theorem for the upper bounds for the arithmetical degrees below a. Namely, given such a and b < a which is an upper bound for the arithmetical degrees, there is a minimal upper bound for the arithmetical degrees c such that b ∪ c = a. This answers a question in Lerman [15]. We prove this theorem by different methods depending on whether a has a function which is not dominated by any arithmetical function. We prove two propositions (see §1), of which the theorem is an immediate consequence.Our notation is almost standard. Let A ⊕ B = {2n∣n ∈ A} ∪ {2n + 1∣n + 1∣n ∈ B} for any sets A and B. Let ω be the set of nonnegative natural numbers.

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Abelian Groups ◽  
Class Groups ◽  
Abelian Extensions ◽  
Pointwise Bound ◽  
First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

Upper bounds on the energy dissipation in turbulent precession

1996 ◽  
Vol 321 ◽  
pp. 335-370 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Experimental Data ◽  
Energy Dissipation ◽  
Viscous Dissipation ◽  
Upper Bound ◽  
Dissipation Rate ◽  
Oblate Spheroid ◽  
Upper Bounds ◽  
Precession Rate ◽  
Long Cylinder

Rigorous upper bounds on the viscous dissipation rate are identified for two commonly studied precessing fluid-filled configurations: an oblate spheroid and a long cylinder. The latter represents an interesting new application of the upper-bounding techniques developed by Howard and Busse. A novel ‘background’ method recently introduced by Doering & Constantin is also used to deduce in both instances an upper bound which is independent of the fluid's viscosity and the forcing precession rate. Experimental data provide some evidence that the observed viscous dissipation rate mirrors this behaviour at sufficiently high precessional forcing. Implications are then discussed for the Earth's precessional response.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed

scholarly journals On the Rank of $p$-Schemes

10.37236/3097 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Fateme Raei Barandagh ◽  
Amir Rahnamai Barghi
Keyword(s):  
Lower Bound ◽  
Prime Number ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Let $n>1$ be an integer and $p$ be a prime number. Denote by $\mathfrak{C}_{p^n}$ the class of non-thin association $p$-schemes of degree $p^n$. A sharp upper and lower bounds on the rank of schemes in $\mathfrak{C}_{p^n}$ with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound are constructed. Finally, it is shown that the scheme with minimum rank in $\mathfrak{C}_{p^n}$ is unique up to isomorphism, and it is a fusion of any association $p$-schemes with degree $p^n$.

Models of arithmetic and upper bounds for arithmetic sets

1994 ◽  
Vol 59 (3) ◽  
pp. 977-983 ◽  
Author(s):  
Alistair H. Lachlan ◽  
Robert I. Soare
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Models Of Arithmetic

AbstractWe settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

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