scholarly journals On some mean value results involving djζ(1/2 + it)dj

2003 ◽  
Vol 127 (28) ◽  
pp. 17-29
Author(s):  
Aleksandar Ivic
Keyword(s):  
Mean Value ◽  
Fourth Moment ◽  
Mean Square ◽  
Error Terms ◽  
The Mean

Several problems involving E(T) and E2(T), the error terms in the mean square and mean fourth moment formula for |?(1/2 + it)|, are discussed. In particular it is proved that ?0T? E(t)E2(T)dt?T7/4(logT)7/2loglogT. .

scholarly journals On a sum analogous to Dedekind sum and its mean square value formula

2002 ◽  
Vol 32 (1) ◽  
pp. 47-55 ◽  
Author(s):  
Zhang Wenpeng
Keyword(s):  
Asymptotic Property ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Square ◽  
Dedekind Sum ◽  
The Mean

The main purpose of this paper is using the mean value theorem of DirichletL-functions to study the asymptotic property of a sum analogous to Dedekind sum, and give an interesting mean square value formula.

scholarly journals One Kind Special Gauss Sums and their Mean Square Values

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jianhong Zhao ◽  
Jiejie Gao
Keyword(s):  
Mean Value ◽  
Gauss Sums ◽  
Mean Square ◽  
Analytic Methods ◽  
The Mean

In this paper, we introduce one kind special Gauss sums; then, using the elementary and analytic methods to study the mean value properties of these kind sums, we obtain several exact calculating formulae for them.

scholarly journals On the mean square of the Riemann zeta function and the divisor problem

2008 ◽  
Vol 83 (97) ◽  
pp. 71-86
Author(s):  
Yifan Yang
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Integral Transforms ◽  
Divisor Problem ◽  
Mean Square ◽  
The Riemann Zeta Function ◽  
Dirichlet Divisor Problem ◽  
Error Terms ◽  
Riemann Zeta ◽  
The Mean

Let ?(T) and E(T) be the error terms in the classical Dirichlet divisor problem and in the asymptotic formula for the mean square of the Riemann zeta function in the critical strip, respectively. We show that ?(T) and E(T) are asymptotic integral transforms of each other. We then use this integral representation of ?(T) to give a new proof of a result of M. Jutila.

scholarly journals Failure Probability Analysis of Interlayer Sliding Belts of Gravity Dams Under Seismic Load

2014 ◽  
Vol 8 (1) ◽  
pp. 85-91 ◽  
Author(s):  
Qiang Xu ◽  
Jian-Yun Chen ◽  
Chunfeng Zhao ◽  
Jing Li ◽  
Hongyuan Yue
Keyword(s):  
Failure Probability ◽  
Mean Value ◽  
Seismic Load ◽  
Mean Square ◽  
Mean Square Deviation ◽  
Concrete Gravity Dam ◽  
Gravity Dams ◽  
The Mean ◽  
Improved Model ◽  
Pseudo Excitation

In this paper, an improved model is presented for analysis of failure probability of the interlayer sliding belts under seismic loads. Firstly, using the theory of the Markov chain, the relation between failure probabilities of specified interlayer sliding belts and elements in this interlayer sliding belt is deduced. Then, the failure function is proposed according to the destructive characteristic of concrete, the pseudo excitation method is utilized in order to obtain the probability distribution of element stresses in specified interlayer sliding belts, and the improved response surface method based on weighted regression is used to calculate the failure probability of elements in specified interlayer sliding belts. Finally, an algorithm is established to calculate the failure probability of the specified interlayer sliding belts. In this paper, the mean value and variance of the tensile strength of elements are changed when interlayer sliding belt is developed. The numerical results show that the conditional failure probability in specified interlayer sliding belts at the head of the dam tends to decrease. However, the tendency of conditional failure probability in the other specified interlayer sliding belts is complicated. And the interlayer sliding belt at head of the concrete gravity dam is the most dangerous. In addition, the tendencies of the mean value and mean square deviation of stresses in different specified interlayer sliding belts are similar. The mean value of stress in different specified interlayer sliding belts tends to decrease but the tendency of mean square deviation changes from decrease to increase. The range of the mean value and the mean square deviation of stress in specified interlayer sliding belts at the heel of the dam is the greatest of all.

scholarly journals Meteorological Temperature and Humidity Prediction from Fourier-Statistical Analysis of Hourly Data

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Alejandro Castañeda-Miranda ◽  
M. de Icaza-Herrera ◽  
Víctor M. Castaño
Keyword(s):  
Angular Position ◽  
Geographical Location ◽  
Mean Value ◽  
Mean Square ◽  
Mean Square Deviation ◽  
Hourly Data ◽  
Humidity Measurements ◽  
The Mean ◽  
Length Of The Day ◽  
New Variables

The temperature readings for all the 365 days and the 24 hours may be fitted through a 3 × 3 matrix (the so-called T-matrix). The mean square deviation between this fit and the actual meteorological measurements is smaller than three degrees Celsius. Four entries of this (nonsymmetric) matrix may be fixed by other means, leaving only five independent components. However, the same method applied to the humidity measurements produces a larger mean square deviation. A strong stochastical connection is found between the T-temperature matrix and the U-humidity matrix. The computer program, in C, may be used to adjust a (2M + 1) × (2m + 1) matrix simply by changing the arguments at the command line and has been tested with m and M ranging from zero to 11 (eleven) (more than 24 readings per day are necessary for larger values of m). The physical meaning of these constants is given only in the case m = M = 1. Our results have also been connected to fundamental cosmological properties: Earth’s orbit, the ecliptic angle, and the latitude of Querétaro (or whatever geographical location is chosen). A separate program calculates the angular position of the Sun as measured in the sky of Querétaro, to determine the length of the day or the mean value of the solar cosine. This work introduces several new variables which happen to be stochastically connected.

The mean-square value of the divisor function

2016 ◽  
Vol 100 (548) ◽  
pp. 203-212
Author(s):  
Peter Shiu
Keyword(s):  
Counting Function ◽  
Mean Value ◽  
Mean Square ◽  
Divisor Function ◽  
Dirichlet’S Theorem ◽  
Fixed Power ◽  
The Mean ◽  
Dirichlet's Theorem ◽  
Image Position ◽  
Prime Counting Function

The behaviour of the divisor function d (n) is rather tricky. For a prime p, we have d(p) = 2, but if n is the product of the first k primes then, by Chebyshev's estimate for the prime counting function [1, Theorem 414], we have so thatfor such n then, d (n) is ‘unusually large’ — it can exceed any fixed power of log n, for example.In [2] Jameson gives, amongst other things, a derivation of Dirichlet's theorem, which shows that the mean-value of the divisor function in an interval containing n is log n. However, the result is somewhat deceptive because, for most n, the value of d (n) is substantially smaller than log n.

The investigations of pulsar integral radio luminosities

1996 ◽  
Vol 160 ◽  
pp. 59-60
Author(s):  
Igor’ F. Malov ◽  
Oleg I. Malov ◽  
Valerij M. Malofeev
Keyword(s):  
Mean Value ◽  
Main Mechanism ◽  
Square Root ◽  
Mean Square ◽  
Pulsar Period ◽  
Light Cylinder ◽  
Curvature Radiation ◽  
The Mean ◽  
The Difference ◽  
Integral Radio

We have calculated accurate integral radio luminositiesLfor 232 pulsars (Malov et al., 1994) using new average spectra of these objects. Histogram ofL-distribution is characterized by the mean value < logL>= 28.45 and by the mean-square-root deviationS= 1.0. We have analysed also data for short-periodic pulsars (P < 0.1 s) and long-periodic ones (P > 1 s) separately.The main goal of such separation was to test the hypothesis on two types of pulsars (Malov, 1987): i) for the first group of objects radiation is emitted from the neighbourhood of the light cylinder (r=rLC=cP/2π, P is the pulsar period), ii) for the second one emission is generated at distancesr≪rLC. In the second case the main mechanism of radiation is curvature radiation. For the first group of pulsars the radiation is connected with the cyclotron mechanism. The difference between two basic mechanisms and the locations of the emission generation regions must cause some differences in the observable features for these two classes of pulsars.

Numerical simulation of the flow and heat transfer around a cylinder with a pulsating approaching flow at a low Reynolds number

2001 ◽  
Vol 215 (1) ◽  
pp. 105-119 ◽  
Author(s):  
G Papadakis ◽  
G Bergeles
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Vortex Shedding ◽  
Mean Value ◽  
Velocity Pulsation ◽  
Mean Square ◽  
Front Part ◽  
Lower Side ◽  
Flow And Heat Transfer ◽  
The Mean

Two-dimensional numerical simulations of flow and heat transfer around a cylinder at a Reynolds number Re= 100 have been performed in order to investigate the effect of imposed inlet velocity pulsation on the heat transfer and flow fields. First the code is validated against existing results from the literature and then several external frequencies are examined. The numerical results confirm the existence of a vortex shedding lock-on regime where the wake behaves in a very ordered manner (completely periodic). Outside the lock-on region the flow is quasiperiodic. The length and centre of the mean recirculating zone downstream of the cylinder are also affected by the external pulsation. Regarding heat transfer, the results indicate that by imposing an external velocity pulsation, the root mean square (r.m.s.) of the local Nusselt number Nu increases, but the mean value increases only in the area downstream of the separation point. The mechanism responsible for this is identified: hot fluid is engulfed by stronger vortices (compared with the steady approaching flow case) shed from the upper and lower side of the cylinder and returned close to the downstream stagnation point. This mechanism also explains the observed variation in Nu with time. In the front part of the cylinder, the Nu varies almost sinusoidally and closely follows the imposed external velocity pulsation. The results indicate also that there is a range of external frequencies where the time and spatially averaged Nu number is maximized.

scholarly journals On some mean value results for the zeta-function and a divisor problem

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2315-2327
Author(s):  
Aleksandar Ivic
Keyword(s):  
Zeta Function ◽  
Error Term ◽  
Mean Value ◽  
Divisor Problem ◽  
Mean Square ◽  
Asymptotic Formulae ◽  
Dirichlet Divisor Problem ◽  
The Mean

Let ?(x) denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be ?*(x) = -?(x) + 2?(2x)-1/2?(4x). We show that ?T+H,T ?*(t/2?)|?(1/2+it)|2dt<< HT1/6log7/2 T (T2/3+? ? H = H(T) ? T), ?T,0 ?(t)|?(1/2+it)|2dt << T9/8(log T)5/2, and obtain asymptotic formulae for ?T,0 (?*(t/2?))2|?( 1/2+it)|2 dt, ?T0 (?*(t/2?))3|?(1/+it)|2 dt. The importance of the ?*-function comes from the fact that it is the analogue of E(T), the error term in the mean square formula for |?(1/2+it)|2. We also show, if E*(T) = E(T)-2??*(T/(2?)), ?T0 E*(t)Ej(t)|?(1/2+it)|2 dt << j,? T7/6+j/4+? (j=1,2,3).

The mean value of the fluctuations in pressure and pressure gradient in a turbulent fluid

1938 ◽  
Vol 34 (4) ◽  
pp. 534-539
Author(s):  
A. E. Green
Keyword(s):  
Unit Volume ◽  
Isotropic Turbulence ◽  
Fluid Motion ◽  
Mean Value ◽  
Mean Square ◽  
Turbulent Fluid ◽  
Dissipation Of Energy ◽  
The Mean ◽  
Rate Of Dissipation ◽  
Image Position

1. Taylor has shown (1) that two characteristic lengthsλ and λη may be defined for turbulent fluid motion. The length λ, which is connected with the dissipation of energy, is, for isotropic turbulence, given bywhere is the mean rate of dissipation of energy per unit volume and represents the mean square value of any component of velocity. The length λη can be defined in terms of thuswhere For isotropic turbulence Taylor assumed thatwhere B is a constant. Since the turbulence is isotropic,and so, from (1), (2), (3) and (4) we have

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