scholarly journals The integer part of nonlinear forms with prime variables

2021 ◽  
Vol 7 (1) ◽  
pp. 1147-1154
Author(s):  
Weiping Li ◽  
◽  
Guohua Chen ◽  
Keyword(s):  
Integer Part

<abstract><p>In this paper, we discuss problems that integer part of nonlinear forms with prime variables represent primes infinitely. We prove that under suitable conditions there exist infinitely many primes $ p_j, p $ such that $ [\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^k] = p $ and $ [\lambda_1p_1^3+\cdots+\lambda_4p_4^3+\lambda_5p_5^k] = p $ with $ k\geq 2 $ and $ k\geq 3 $ respectively, which improve the author's earlier results.</p></abstract>

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

Some Tauberian conditions for Cesàro summability method

2012 ◽  
Vol 62 (2) ◽  
Author(s):  
İbrahi̇m Çanak ◽  
Ümi̇t Totur
Keyword(s):  
Finite Number ◽  
Oscillatory Behavior ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Integer Part ◽  
Real Numbers ◽  
General Control ◽  
Integer Order ◽  
Tauberian Conditions ◽  
General Control Modulo

AbstractLet u = (u n) be a sequence of real numbers whose generator sequence is Cesàro summable to a finite number. We prove that (u n) is slowly oscillating if the sequence of Cesàro means of (ω n(m−1)(u)) is increasing and the following two conditions are hold: $$\begin{gathered} \left( {\lambda - 1} \right)\mathop {\lim \sup }\limits_n \left( {\frac{1} {{\left[ {\lambda n} \right] - n}}\sum\limits_{k = n + 1}^{\left[ {\lambda n} \right]} {\left( {\omega _k^{\left( m \right)} \left( u \right)} \right)^q } } \right)^{\frac{1} {q}} = o\left( 1 \right), \lambda \to 1^ + , q > 1, \hfill \\ \left( {1 - \lambda } \right)\mathop {\lim \sup }\limits_n \left( {\frac{1} {{n - \left[ {\lambda n} \right]}}\sum\limits_{k = \left[ {\lambda n} \right] + 1}^n {\left( {\omega _k^{\left( m \right)} \left( u \right)} \right)^q } } \right)^{\frac{1} {q}} = o\left( 1 \right), \lambda \to 1^ - , q > 1, \hfill \\ \end{gathered}$$ where (ω n(m) (u)) is the general control modulo of the oscillatory behavior of integer order m ≥ 1 of a sequence (u n) defined in [DİK, F.: Tauberian theorems for convergence and subsequential convergence with moderately oscillatory behavior, Math. Morav. 5, (2001), 19–56] and [λn] denotes the integer part of λn.

The integer part of qα + β

2014 ◽  
pp. 127-146
Author(s):  
Jonathan Borwein ◽  
Alf van der Poorten ◽  
Jeffrey Shallit ◽  
Wadim Zudilin
Keyword(s):  
Integer Part

scholarly journals A result for semi-regular continued fractions

1969 ◽  
Vol 10 (1-2) ◽  
pp. 145-154 ◽  
Author(s):  
P. E. Blanksby
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Two Dimensional ◽  
Integer Part ◽  
Image Position

If Φ is a real number with |Φ| ≧ 1, then a semiregular continuet fraction development of Φ is denoted by where the ai are integers such that |ai| ≧ 2. The expansions arise geo-. metrically by considering the sequence of divided cells of two-dimensional grids (see [1]), and are described by the following algorithm: for all n ≧ 0, taking Φ = Φ.0 Hence where in this case the square brackets are used to signify the integer-part function. It follows that each irrational Φ has uncountably many such expansions, none of which has a constantly equal to 2 (or -2) for large n.

scholarly journals Integer part polynomial correlation sequences

2016 ◽  
Vol 38 (4) ◽  
pp. 1525-1542 ◽  
Author(s):  
ANDREAS KOUTSOGIANNIS
Keyword(s):  
Error Term ◽  
Uniform Density ◽  
Integer Part ◽  
Ergodic Averages ◽  
Multiple Correlation ◽  
Transference Principle ◽  
The Mean ◽  
Correlation Sequences ◽  
Measure Preserving Actions ◽  
Invent Math

Following an approach presented by Frantzikinakis [Multiple correlation sequences and nilsequences. Invent. Math. 202(2) (2015), 875–892], we prove that any multiple correlation sequence defined by invertible measure preserving actions of commuting transformations with integer part polynomial iterates is the sum of a nilsequence and an error term, which is small in uniform density. As an intermediate result, we show that multiple ergodic averages with iterates given by the integer part of real-valued polynomials converge in the mean. Also, we show that under certain assumptions the limit is zero. A transference principle, communicated to us by M. Wierdl, plays an important role in our arguments by allowing us to deduce results for $\mathbb{Z}$-actions from results for flows.

scholarly journals Gap Problems for Integer Part and Fractional Part Sequences

1995 ◽  
Vol 50 (1) ◽  
pp. 66-86 ◽  
Author(s):  
A.S. Fraenkel ◽  
R. Holzman
Keyword(s):  
Fractional Part ◽  
Integer Part

scholarly journals Steganographic Scheme based on Message-Cover matching

2018 ◽  
Vol 8 (5) ◽  
pp. 3594
Author(s):  
Youssef Taouil ◽  
El Bachir Ameur
Keyword(s):  
Wavelet Transform ◽  
Discrete Wavelet Transform ◽  
Secret Message ◽  
Discrete Wavelet ◽  
Large Set ◽  
Integer Part ◽  
Least Significant Bit ◽  
Secret Data ◽  
Trade Off ◽  
Good Trade

Steganography is one of the techniques that enter into the field of information   security, it is the art of dissimulating data into digital files in an imperceptible way that does not arise the suspicion. In this paper, a steganographic method based on the Faber-Schauder discrete wavelet transform is proposed. The embedding of the secret data is performed in Least Significant Bit (LSB) of the integer part of the wavelet coefficients. The secret message is decomposed into pairs of bits, then each pair is transformed into another pair based on a permutation that allows to obtain the most matches possible between the message and the LSB of the coefficients. To assess the performance of the proposed method, experiments were carried out on a large set of images, and a comparison to prior works is accomplished. Results show a good level of imperceptibility and a good trade-off imperceptibility-capacity compared to literature.

scholarly journals The Division Algorithm in Complex Bases

1996 ◽  
Vol 39 (1) ◽  
pp. 47-54 ◽  
Author(s):  
William J. Gilbert
Keyword(s):  
Iterated Function System ◽  
Complex Numbers ◽  
Integer Part ◽  
Division Algorithm ◽  
Fractal Boundary ◽  
Gaussian Integer ◽  
Gaussian Integers ◽  
Linear Maps ◽  
Iterated Function ◽  
Long Division

AbstractComplex numbers can be represented in positional notation using certain Gaussian integers as bases and digit sets. We describe a long division algorithm to divide one Gaussian integer by another, so that the quotient is a periodic expansion in such a complex base. To divide by the Gaussian integer w in the complex base b, using a digit set D, the remainder must be in the set wT(b,D) ∩ ℤ[i], where T(b,D) is the set of complex numbers with zero integer part in the base. The set T(b,D) tiles the plane, and can be described geometrically as the attractor of an iterated function system of linear maps. It usually has a fractal boundary. The remainder set can be determined algebraically from the cycles in a certain directed graph.

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