EXPONENTIAL SUMS OVER PRIMES IN RESIDUE CLASSES

2010 ◽  
Vol 06 (04) ◽  
pp. 905-918 ◽  
Author(s):  
H. MAIER ◽  
A. SANKARANARAYANAN
Keyword(s):  
Exponential Sums ◽  
Complex Numbers ◽  
Exponential Sums Over Primes ◽  
Residue Classes ◽  
Special Cases

We specialize a problem studied by Elliott, the behavior of arbitrary sequences ap of complex numbers on residue classes to prime moduli to the case ap = e(αp). For these special cases, we obtain under certain additional conditions improvements on Elliott's results.

Short cubic exponential sums over primes

2017 ◽  
Vol 296 (1) ◽  
pp. 211-233
Author(s):  
Z. Kh. Rakhmonov ◽  
F. Z. Rakhmonov
Keyword(s):  
Exponential Sums ◽  
Exponential Sums Over Primes

scholarly journals On certain exponential sums over primes

2009 ◽  
Vol 129 (7) ◽  
pp. 1669-1677 ◽  
Author(s):  
H. Maier ◽  
A. Sankaranarayanan
Keyword(s):  
Exponential Sums ◽  
Exponential Sums Over Primes

scholarly journals The generalized criterion of multiaxial random fatigue based on conception proposed by prof. Macha

2019 ◽  
Vol 300 ◽  
pp. 15001
Author(s):  
Tadeusz Łagoda ◽  
Marta Kurek ◽  
Karolina Łagoda
Keyword(s):  
Shear Stress ◽  
Complex Number ◽  
Mathematical Problem ◽  
Equivalent Stress ◽  
Complex Stress ◽  
Complex Numbers ◽  
Pure Bending ◽  
Pure Torsion ◽  
Special Cases ◽  
Random Fatigue

This criterion has been repeatedly verified, analyzed and special cases of this criterion reducing complex stress to equivalent uniaxial were taken into account. Since both normal and shear stress are vectors, we encounter the mathematical problem of adding these vectors, and the question arises how to understand the obtained equivalent stress, because two perpendicular vectors are added with weighting factors. Therefore, in this work it was proposed to adopt a system of complex numbers. Normal stress was defined as the real part and shear stress as imaginary part. As a result, on the basis of the defined complex number and basing on pure bending and pure torsion after transformations, the expression for equivalent stress was identical to the previously proposed criteria defined on the basis of the concept of prof. Macha.

General series involving H-functions

1969 ◽  
Vol 65 (2) ◽  
pp. 461-465
Author(s):  
R. N. Jain
Keyword(s):  
Hypergeometric Function ◽  
Analytic Functions ◽  
General Nature ◽  
Complex Numbers ◽  
Important Class ◽  
Vast Number ◽  
Finite Series ◽  
General Series ◽  
Special Cases ◽  
Image Position

MacRobert (4–7) and Ragab(8) have summed many infinite and finite series of E-functions by expressing the E-functions as Barnes integrals and interchanging the order of summation and integration. Verma (9) has given two general expansions involving E-functions from which, in addition to some new results, all the expansions given by MacRobert and Ragab can be deduced. Proceeding similarly, we have studied general summations involving H-functions. The H-function is the most generalized form of the hypergeometric function. It contains a vast number of well-known analytic functions as special cases and also an important class of symmetrical Fourier kernels of a very general nature. The H-function is defined as (2)where 0 ≤ n ≤ p, 1 ≤ m ≤ q, αj, βj are positive numbers and aj, bj may be complex numbers.

On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

2009 ◽  
Vol 146 (1) ◽  
pp. 1-21 ◽  
Author(s):  
J. BOURGAIN ◽  
M. Z. GARAEV
Keyword(s):  
Prime Order ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Complex Numbers ◽  
Number Of Solutions ◽  
Prime Fields ◽  
Explicit Bounds ◽  
Multiplicative Subgroup ◽  
Image Position

AbstractLet Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the bound for any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds: We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, then Finally we show that if g is a generator of F*p then for any M < p the number of solutions of the equation is less than $M^{3-1/24+o(1)}\Bigl(1+(M^2/p)^{1/24}\Bigr).$. This implies that if p1/2 < M < p, then

scholarly journals Products of a C-Measure and a Locally Integrable Mapping

1957 ◽  
Vol 9 ◽  
pp. 475-486 ◽  
Author(s):  
Marston Morse ◽  
William Transue
Keyword(s):  
Integral Representation ◽  
Topological Space ◽  
Orlicz Spaces ◽  
Complex Numbers ◽  
Bounded Operators ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Special Cases ◽  
Integrable Mapping

Let C be the field of complex numbers and E a locally compact topological space. The authors' theory of C-bimeasures Λ and their Λ-integrals in (1; 2) leads to integral representation of bounded operators from A to B' where A and B are MT-spaces as defined in (3). These MT-spaces include the -spaces and Orlicz spaces as special cases.

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