Relations between stable dimension and the preimage counting function on basic sets with overlaps

2009 ◽  
Vol 42 (1) ◽  
pp. 15-27 ◽  
Author(s):  
Eugen Mihailescu ◽  
Mariusz Urbanski
Keyword(s):  
Counting Function ◽  
Stable Dimension ◽  
Basic Sets

On a class of stable conditional measures

2010 ◽  
Vol 31 (5) ◽  
pp. 1499-1515 ◽  
Author(s):  
EUGEN MIHAILESCU
Keyword(s):  
Invariant Measures ◽  
Reversible Systems ◽  
Absolutely Continuous ◽  
Stable Manifolds ◽  
Equilibrium Measures ◽  
Basic Set ◽  
Stable Dimension ◽  
Saddle Type ◽  
Local Stable Manifolds ◽  
Basic Sets

AbstractThe dynamics of endomorphisms (smooth non-invertible maps) presents many differences from that of diffeomorphisms or that of expanding maps; most methods from those cases do not work if the map has a basic set of saddle type with self-intersections. In this paper we study the conditional measures of a certain class of equilibrium measures, corresponding to a measurable partition subordinated to local stable manifolds. We show that these conditional measures are geometric probabilities on the local stable manifolds, thus answering in particular the questions related to the stable pointwise Hausdorff and box dimensions. These stable conditional measures are shown to be absolutely continuous if and only if the respective basic set is a non-invertible repeller. We find also invariant measures of maximal stable dimension, on folded basic sets. Examples are given, too, for such non-reversible systems.

PAL – Propositional Algorithmic Logic

1981 ◽  
Vol 4 (3) ◽  
pp. 675-760
Author(s):  
Grażyna Mirkowska
Keyword(s):  
Data Structures ◽  
Completeness Theorem ◽  
Natural Numbers ◽  
First Order ◽  
Algorithmic Logic ◽  
Axiomatic Systems ◽  
Nondeterministic Program ◽  
Basic Sets

The aim of propositional algorithmic logic is to investigate the properties of program connectives. Complete axiomatic systems for deterministic as well as for nondeterministic interpretations of program variables are presented. They constitute basic sets of tools useful in the practice of proving the properties of program schemes. Propositional theories of data structures, e.g. the arithmetic of natural numbers and stacks, are constructed. This shows that in many aspects PAL is close to first-order algorithmic logic. Tautologies of PAL become tautologies of algorithmic logic after replacing program variables by programs and propositional variables by formulas. Another corollary to the completeness theorem asserts that it is possible to eliminate nondeterministic program variables and replace them by schemes with deterministic atoms.

The Absorption Energy and Equilibrium Structure Analysis of CH4 on Coal Molecules

2013 ◽  
Vol 807-809 ◽  
pp. 2366-2370
Author(s):  
Hua Ping Yang ◽  
Ming Li ◽  
Zheng Xin Yan ◽  
Dong Zhi Yan
Keyword(s):  
Structure Analysis ◽  
Dft Method ◽  
Equilibrium Structure ◽  
Atomic Charges ◽  
Absorption Model ◽  
Absorption Energy ◽  
Molecule Model ◽  
Basic Sets

The coal molecule model and the absorption model of coal and methane were established according to Van der Walls force. The coal molecule model and the absorption model of coal and methane were optimized and obtained absorption energy and equilibrium structure with the DFT method with 6-311G++ basic sets. The effects of coal molecule on the absorption position, absorption energy and absorption distance of CH4 were presented by analyzing the Mulliken atomic charges of coal molecules.

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 $B(z).$

Analysis of Adsorbed Methane Gas Structure and Mullikan Charges

2014 ◽  
Vol 962-965 ◽  
pp. 1118-1122
Author(s):  
Hua Ping Yang ◽  
Ming Li
Keyword(s):  
Adsorption Process ◽  
Adsorption Equilibrium ◽  
Dft Method ◽  
Model Structure ◽  
Dispersion Forces ◽  
Methane Gas ◽  
Adsorbed Methane ◽  
Molecule Model ◽  
Equilibrium Geometries ◽  
Basic Sets

The methane molecule model, the coal molecule model and the absorption model of coal and methane have been thoroughly investigated using the DFT method,Van der Walls force radius as boundary conditions as well. The investigations focuses on optimizing the three types of model structure. Meanwhile, adsorption equilibrium geometries with 6-311G++ basic sets have been established. Through the model analysis of the changes in the structure and Millikan charge from free state to the adsorption states, the results indicate that the adsorption process of methane gas on the coal in the Van der Walls force mainly dispersion forces and induction force, methane molecules are polarized in the adsorption process, the electric dipole moment.

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.

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