A note on trigonometric sums in several variables

1986 ◽  
Vol 99 (2) ◽  
pp. 189-193 ◽  
Author(s):  
R. W. K. Odoni
Keyword(s):  
Random Walk ◽  
Broad Class ◽  
Upper Bounds ◽  
Total Degree ◽  
Trigonometric Sums ◽  
Absolute Value ◽  
Independent Variables ◽  
Several Variables ◽  
The Absolute ◽  
Image Position

In [3, 4] we showed how the use of a random-walk analogue can be made to yield non-trivial information about the behaviour of certain trigonometric sums in one variable. Our aim here is to show how our method can be adapted to yield similar results for a broad class of trigonometric sums in several variables. Letbe a polynomial in v independent variables with integral coefficients. We choose integers n ≥ 0, d ≥ 1 and p ≥ 2 with p prime, and assume that f(x) has total degree ≤ d + 1. We shall consider the problem of obtaining non-trivial upper bounds for the absolute value of sums of the typewhere P = {1, 2, …, p} and f is non-constant.

scholarly journals Pointwise characteristic factors for Wiener–Wintner double recurrence theorem

2015 ◽  
Vol 36 (4) ◽  
pp. 1037-1066 ◽  
Author(s):  
IDRIS ASSANI ◽  
DAVID DUNCAN ◽  
RYO MOORE
Keyword(s):  
Full Measure ◽  
Orthogonal Complement ◽  
Ergodic System ◽  
Absolute Value ◽  
Recurrence Theorem ◽  
The Absolute ◽  
Image Position ◽  
Null Set

In this paper we extend Bourgain’s double recurrence result to the Wiener–Wintner averages. Let $(X,{\mathcal{F}},{\it\mu},T)$ be a standard ergodic system. We will show that for any $f_{1},f_{2}\in L^{\infty }(X)$, the double recurrence Wiener–Wintner average $$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)e^{2{\it\pi}int}\end{eqnarray}$$ converges off a single null set of $X$ independent of $t$ as $N\rightarrow \infty$. Furthermore, we will show a uniform Wiener–Wintner double recurrence result: if either $f_{1}$ or $f_{2}$ belongs to the orthogonal complement of the Conze–Lesigne factor, then there exists a set of full measure such that the supremum on $t$ of the absolute value of the averages above converges to $0$.

scholarly journals ON THE ABSOLUTE VALUE OF THE NEUTRINO MASS

2011 ◽  
Vol 26 (21) ◽  
pp. 1555-1559 ◽  
Author(s):  
DRAGAN SLAVKOV HAJDUKOVIC
Keyword(s):  
Neutrino Mass ◽  
Neutrino Oscillations ◽  
Upper Bounds ◽  
Neutrino Mixing ◽  
Absolute Value ◽  
The Absolute ◽  
Good Agreement

The neutrino oscillations probabilities depend on mass squared differences; in the case of three-neutrino mixing, there are two independent differences, which have been measured experimentally. In order to calculate the absolute masses of neutrinos, we have conjectured a third relation, in the form of a sum of squared masses. The calculated masses look plausible and are in good agreement with the upper bounds coming from astrophysics.

scholarly journals ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART II

2012 ◽  
Vol 85 (2) ◽  
pp. 202-216 ◽  
Author(s):  
BARBARA PRZEBIERACZ
Keyword(s):  
Functional Equation ◽  
Abelian Group ◽  
Functional Equations ◽  
Additive Functions ◽  
Absolute Value ◽  
Real Functions ◽  
The Absolute ◽  
Image Position

AbstractWe investigate the Pexider-type functional equation where f, g, h are real functions defined on an abelian group G. We solve this equation under the assumptions G=ℝ and f is continuous.

A rational invariant for certain infinite discrete groups

1993 ◽  
Vol 113 (3) ◽  
pp. 473-478
Author(s):  
F. E. A. Johnson
Keyword(s):  
Euler Characteristic ◽  
Discrete Groups ◽  
Fundamental Groups ◽  
Intersection Form ◽  
Absolute Value ◽  
Rational Invariant ◽  
The Absolute ◽  
Image Position ◽  
Aspherical Manifolds

We introduce a rational-valued invariant which is capable of distinguishing between the commensurability classes of certain discrete groups, namely, the fundamental groups of smooth closed orientable aspherical manifolds of dimensional 4k(k ≥ 1) whose Euler characteristic χ(Λ) is non-zero. The invariant in question is the quotientwhere Sign (Λ) is the absolute value of the signature of the intersection formand [Λ] is a generator of H4k(Λ; ℝ).

scholarly journals ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART I

2012 ◽  
Vol 85 (2) ◽  
pp. 191-201 ◽  
Author(s):  
BARBARA PRZEBIERACZ
Keyword(s):  
Functional Equation ◽  
Abelian Group ◽  
Functional Equations ◽  
Additive Functions ◽  
Absolute Value ◽  
Real Functions ◽  
The Absolute ◽  
Image Position

AbstractWe investigate the Pexider-type functional equation where f,g,h are real functions defined on an abelian group G.

scholarly journals Inequalities for two specific classes of functions using Chebyshev functional

Filomat ◽  
2011 ◽  
Vol 25 (4) ◽  
pp. 153-163
Author(s):  
Mohammad Masjed-Jamei
Keyword(s):  
Lower Bounds ◽  
Special Functions ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Absolute Value ◽  
Lp Spaces ◽  
Suitable Tool ◽  
Chebyshev Functional ◽  
The Absolute

In this paper, we introduce two specific classes of functions in Lp-spaces that can generate new and known inequalities in the literature. By using some recent results related to the Chebyshev functional, we then obtain upper bounds for the absolute value of the two introduced functions and consider three particular examples. One of these examples is a suitable tool for finding upper and lower bounds of some incomplete special functions such as incomplete gamma and beta functions.

scholarly journals Quantitative Equidistribution for Certain Quadruples in Quasi-Random Groups: Erratum

2015 ◽  
Vol 25 (3) ◽  
pp. 484-485 ◽  
Author(s):  
TIM AUSTIN
Keyword(s):  
First Line ◽  
Absolute Value ◽  
Formula Group ◽  
The Absolute ◽  
Image Position ◽  
The Right ◽  
Random Groups ◽  
Complex Valued

In my recent paper [1] there is a mistake in the proof of Corollary 3. The first line of the displayed equation in that proof asserts that \[\int_G|\langle u,\pi^g v\rangle_V|^2\,\rm{d} g = \int_G\langle u\otimes u,(\pi^g\otimes \pi^g)(v\otimes v)\rangle_{V\otimes V}\, \rm{d} g.\] However, since the paper uses complex-valued representations, the integrand on the right here may not retain the absolute value of that on the left. Without this equality, the proof of Corollary 3 can no longer be reduced to an application of Lemma 2. However, it can be proved directly from Schur Orthogonality along very similar lines to the proof of Lemma 2.

scholarly journals On the growth of the cyclotomic polynomial in the interval (0, 1)

1957 ◽  
Vol 3 (2) ◽  
pp. 102-104 ◽  
Author(s):  
P. Erdös
Keyword(s):  
Positive Constant ◽  
Cyclotomic Polynomial ◽  
Absolute Value ◽  
The Absolute ◽  
Image Position

Letbe the rath cyclotomic polynomial, and denote by An the absolute value of the largest coefficient of Fn(x).Schur proved thatand Emma Lehmer [5] showed that An>cn1/3 for infinitely many n; in fact she proved that n can be chosen as the product of three distinct primes. I proved [3] that there exists a positive constant q such that, for infinitely many nand Bateman [1] proved very simply that, for every ∈>0 and all n>no(∈),

Quantitative χn analysis of HREM images with applications to planar defects

1996 ◽  
Vol 54 ◽  
pp. 98-99
Author(s):  
H. Zhang ◽  
L. D. Marks
Keyword(s):  
Experimental Data ◽  
Cross Correlation ◽  
Planar Defects ◽  
Absolute Value ◽  
Background Levels ◽  
Correlation Analyses ◽  
The Absolute ◽  
Image Position ◽  
R Factors ◽  
Calculated Image

A number of different methods have been suggested in the literature for using HREM in a quantitative fashion, including R factors and cross-correlation analyses. The problem with many of these is that it is difficult to realistically gauge the errors involved when they are applied to real systems. For instance, R-factors defined by:(1)(where n=1 or 2 and Ic is the calculated image, Ie the experimental data) assume a signal independent error. Furthermore, the absolute value of R is strongly dependent upon background levels which is misleading.

Trigonometric sums of Heilbronn's type

1985 ◽  
Vol 98 (3) ◽  
pp. 389-396 ◽  
Author(s):  
R. W. K. Odoni
Keyword(s):  
Taylor Expansion ◽  
Prime Power ◽  
Chinese Remainder Theorem ◽  
General Procedure ◽  
Additive Number Theory ◽  
Trigonometric Sums ◽  
Trigonometric Sum ◽  
Absolute Value ◽  
Image Position ◽  
Trivial Estimate

In problems of additive number theory one frequently needs to obtain a non-trivial estimate for the absolute value of a trigonometric sum of the typewhere f(X) ε ℤ [X] and 1 ≤ m ε ℤ. The general procedure is first to reduce the estimation to the case where m is a prime power, by means of the Chinese Remainder Theorem. The case m = pr (p prime) can often be reduced to that of a lower power of p, by a substitution of the type x = u + vps (where 0 ≤ u < ps and 0 ≤ v < pr-s), followed by the use of a p-adic Taylor expansion f(u+psv) = f(u) +psvf′(u) +…. Frequently this gives T(f, pr) = 0 when r ≥ 2, or at least allows one to reduce to the case m = p. In the latter case an appeal to Weil's estimateusually gives a good estimate for (0.1), at least if deg f = o(√p).

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