scholarly journals Some Properties of the Inversion Counting Function

2006 ◽  
Vol 13 (4) ◽  
pp. 783-786
Author(s):  
Neville Robbins
Keyword(s):  
Counting Function ◽  
Positive Residue

Abstract Let 𝑕, 𝑘 be integers such that 0 < 𝑕 < 𝑘 and (𝑕, 𝑘) = 1. If 1 ≤ 𝑖 ≤ 𝑘 – 1, let 𝑟𝑖 be the least positive residue (mod 𝑘) of 𝑕𝑖. Let the permutation For 1 ≤ 𝑖 < 𝑗 ≤ 𝑘 – 1, if 𝑟𝑖 > 𝑟𝑗, this is called an inversion of σ 𝑕, 𝑘. Let 𝐼(𝑕, 𝑘) denote the total number of inversions of σ 𝑕, 𝑘. In this note, we prove several identities concerning 𝐼(𝑕, 𝑘).

LOGARITHMIC DERIVATIVE OF THE BLASCHKE PRODUCT WITH SLOWLY INCREASING COUNTING FUNCTION OF ZEROS

2021 ◽  
Vol 9 (1) ◽  
pp. 164-170
Author(s):  
Y. Gal ◽  
M. Zabolotskyi ◽  
M. Mostova
Keyword(s):  
Blaschke Product ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Counting Function ◽  
Finite System ◽  
Blaschke Products ◽  
Nevanlinna Characteristic ◽  
Number Of Zeros ◽  
Accumulation Points ◽  
Theory Of Value

The Blaschke products form an important subclass of analytic functions on the unit disc with bounded Nevanlinna characteristic and also are meromorphic functions on $\mathbb{C}$ except for the accumulation points of zeros $B(z)$. Asymptotics and estimates of the logarithmic derivative of meromorphic functions play an important role in various fields of mathematics. In particular, such problems in Nevanlinna's theory of value distribution were studied by Goldberg A.A., Korenkov N.E., Hayman W.K., Miles J. and in the analytic theory of differential equations -- by Chyzhykov I.E., Strelitz Sh.I. Let $z_0=1$ be the only boundary point of zeros $(a_n)$ %=1-r_ne^{i\psi_n},$ $-\pi/2+\eta<\psi_n<\pi/2-\eta,$ $r_n\to0+$ as $n\to+\infty,$ of the Blaschke product $B(z);$ $\Gamma_m=\bigcup\limits_{j=1}^{m}\{z:|z|<1,\mathop{\text{arg}}(1-z)=-\theta_j\}=\bigcup\limits_{j=1}^{m}l_{\theta_j},$ $-\pi/2+\eta<\theta_1<\theta_2<\ldots<\theta_m<\pi/2-\eta,$ be a finite system of rays, $0<\eta<1$; $\upsilon(t)$ be continuous on $[0,1)$, $\upsilon(0)=0$, slowly increasing at the point 1 function, that is $\upsilon(t)\sim\upsilon\left({(1+t)}/2\right),$ $t\to1-;$ $n(t,\theta_j;B)$ be a number of zeros $a_n=1-r_ne^{i\theta_j}$ of the product $B(z)$ on the ray $l_{\theta_j}$ such that $1-r_n\leq t,$ $0<t<1.$ We found asymptotics of the logarithmic derivative of $B(z)$ as $z=1-re^{-i\varphi}\to1,$ $-\pi/2<\varphi<\pi/2,$ $\varphi\neq\theta_j,$ under the condition that zeros of $B(z)$ lay on $\Gamma_m$ and $n(t,\theta_j;B)\sim \Delta_j\upsilon(t),$ $t\to1-,$ for all $j=\overline{1,m},$ $0\leq\Delta_j<+\infty.$ We also considered the inverse problem for such $B(z).$

scholarly journals Prime numbers with a certain extremal type property

2018 ◽  
Vol 17 (1) ◽  
pp. 127-151
Author(s):  
Edward Tutaj
Keyword(s):  
Riemann Hypothesis ◽  
Convex Set ◽  
Counting Function ◽  
Numerical Data ◽  
Prime Numbers ◽  
Affine Function ◽  
Piecewise Affine ◽  
Order Of Magnitude ◽  
Type Property ◽  
Prime Counting Function

Abstract The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points $({e_k},\pi ({e_k}))_1^\infty $ . The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then ${{{e_k} + 1} \over {{e_k}}} = 1$ . The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.

Simulation and 3D-Solid Modeling of Skew Bevel Gear

2011 ◽  
Vol 121-126 ◽  
pp. 4685-4689
Author(s):  
Luo Ping Zhang ◽  
Jing Fu ◽  
Bo Yuan Yang
Keyword(s):  
Mathematical Model ◽  
Basic Concept ◽  
Grid Point ◽  
Counting Function ◽  
Tooth Surface ◽  
Solid Modelling ◽  
Kinematics Analysis ◽  
Bevel Gears ◽  
3D Solid ◽  
3D Solid Modeling

In this paper, the basic concept and specific steps of establishing skew bevel gears mathematical model using conjugate meshing principle was introduced. Numerical value counting function of MATLAB was used to collect coordinate of tooth surface grid point, and under the Pro/E environment, data document of coordinate created by MATLAB was imported to establish tooth surface modeling , relying on the powerful function of complicated curve modelling, 3D solid modelling of gear was completed. The results via motion analysis kinematics analysis and interference inspecting under assembling condition show the accuracy of the design ,which supports further design, analysis and manufacture.

scholarly journals Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

2018 ◽  
Vol 2 (4) ◽  
pp. 26 ◽  
Author(s):  
Michel Lapidus ◽  
Hùng Lũ’ ◽  
Machiel Frankenhuijsen
Keyword(s):  
Counting Function ◽  
Minkowski Dimension ◽  
Volume Formula ◽  
Fractal Strings ◽  
Complex Dimensions ◽  
Abscissa Of Convergence ◽  
Tubular Neighborhoods ◽  
Fractal String ◽  
Tube Formulas ◽  
Asymptotic Growth Rate

The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string L p , expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and p-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case.

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