The Intrinsic Periodic Behaviour of Sequences Related to a Rational Integral

Integers ◽  
2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Periodic Behaviour ◽  
Rational Integral

AbstractFor sequences defined in terms of 2-adic valuations, we exploit the intrinsic periodic behaviour obtained by a double summation. The tool is the Mellin–Perron formula.

The determination of units in totally real cubic fields

1960 ◽  
Vol 56 (4) ◽  
pp. 318-321 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Trial And Error ◽  
Integral Lattice ◽  
Cubic Field ◽  
Cubic Fields ◽  
Trial And Error Method ◽  
Totally Real ◽  
Rational Integral ◽  
Fundamental Units ◽  
Error Method

The determination of a pair of fundamental units in a totally real cubic field involves two operations—finding a pair of independent units (i.e. such that neither is a power of the other) and from these a pair of fundamental units (i.e. a pair ε1; ε2 such that every unit of the field is of the form with rational integral m, n). The first operation may be accomplished by exploring regions of the integral lattice in which two conjugates are small or else by factorizing small primes and comparing different factorizations—a trial-and-error method, but often a quick one. The second operation is accomplished by obtaining inequalities which must be satisfied by a fundamental unit and its conjugates and finding whether or not a unit exists satisfying these inequalities. Recently Billevitch ((1), (2)) has given a method, of the nature of an extension of the first method mentioned above, which involves less work on the second operation but no less on the first.

Augmentation quotients of some nonabelian finite groups

1979 ◽  
Vol 85 (2) ◽  
pp. 261-270 ◽  
Author(s):  
Gerald Losey ◽  
Nora Losey
Keyword(s):  
Group Structure ◽  
Lower Central Series ◽  
Central Series ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Nilpotent Residual ◽  
Periodic Behaviour ◽  
Augmentation Quotient ◽  
Image Position ◽  
A Finite Group

1. LetGbe a group,ZGits integral group ring and Δ = ΔGthe augmentation idealZGBy anaugmentation quotientofGwe mean any one of theZG-moduleswheren, r≥ 1. In recent years there has been a great deal of interest in determining the abelian group structure of the augmentation quotientsQn(G) =Qn,1(G) and(see (1, 2, 7, 8, 9, 12, 13, 14, 15)). Passi(8) has shown that in order to determineQn(G) andPn(G) for finiteGit is sufficient to assume thatGis ap-group. Passi(8, 9) and Singer(13, 14) have obtained information on the structure of these quotients for certain classes of abelianp-groups. However little seems to be known of a quantitative nature for nonabelian groups. In (2) Bachmann and Grünenfelder have proved the following qualitative result: ifGis a finite group then there exist natural numbersn0and π such thatQn(G) ≅Qn+π(G) for alln≥n0; ifGωis the nilpotent residual ofGandG/Gωhas classcthen π divides l.c.m. {1, 2, …,c}. There do not appear to be any examples in the literature of this periodic behaviour forc> 1. One of goals here is to present such examples. These examples will be from the class of finitep-groups in which the lower central series is anNp-series.

PERIODICALLY FORCED HYDROCARBON OXIDATION REACTION IN A CSTR

1996 ◽  
Vol 06 (12a) ◽  
pp. 2321-2341 ◽  
Author(s):  
H. S. SIDHU ◽  
L. K. FORBES ◽  
B. F. GRAY
Keyword(s):  
Period Doubling ◽  
Operating Conditions ◽  
Hydrocarbon Oxidation ◽  
Periodic Modulation ◽  
Chaotic Oscillations ◽  
Industrial Practice ◽  
Dynamical Characteristics ◽  
Input And Output ◽  
Periodic Behaviour ◽  
Continuously Stirred Tank Reactor

In this paper we examine in detail the effects of forcing the thermokinetic or the chain-thermal model of hydrocarbon oxidation (proposed by B. F. Gray and C. H. Yang) in a Continuously Stirred Tank Reactor (CSTR). Here, the reaction has been subjected to periodic modulation of the input and output flows of chemicals. This investigation has uncovered rich non-linear dynamical characteristics including primary resonances, super and sub-harmonic resonances, quasi-periodic solutions and chaotic oscillations. These regions of chaos are normally interrupted by windows of periodic behaviour. The transitions to chaos were mainly found to be of three types: Feigenbaum period-doubling cascade, Ruelle-Takens-Newhouse approach through quasi-periodicity, and intermittency. The presence of these chaotic solutions was confirmed by computing the Lyapunov exponents. The results presented here are of potential benefit to industrial practice, since they show great increases in product selectivity when appropriate operating conditions are chosen in this forcing strategy.

scholarly journals Transition to Periodic Behaviour of Flow Past a Circular Cylinder under the Action of Fluidic Actuation in the Transitional Regime

Energies ◽  
2021 ◽  
Vol 14 (16) ◽  
pp. 5069
Author(s):  
Wasim Sarwar ◽  
Fernando Mellibovsky ◽  
Md. Mahbub Alam ◽  
Farhan Zafar
Keyword(s):  
Circular Cylinder ◽  
Period Doubling ◽  
Underlying Mechanism ◽  
Time Dependent ◽  
Cylinder Wake ◽  
Transitional Regime ◽  
Periodic Behaviour ◽  
Vortex Shedding Frequency ◽  
Fluidic Actuation ◽  
Periodic Dynamics

This study focuses on the numerical investigation of the underlying mechanism of transition from chaotic to periodic dynamics of circular cylinder wake under the action of time-dependent fluidic actuation at the Reynolds number = 2000. The forcing is realized by blowing and suction from the slits located at ±90∘ on the top and bottom surfaces of the cylinder. The inverse period-doubling cascade is the underlying physical mechanism underpinning the wake transition from mild chaos to perfectly periodic dynamics in the spanwise-independent, time-dependent forcing at twice the natural vortex-shedding frequency.

Classes of Positive Definite Unimodular Circulants

1957 ◽  
Vol 9 ◽  
pp. 71-73 ◽  
Author(s):  
Morris Newman ◽  
Olga Taussky
Keyword(s):  
Positive Definite ◽  
Image Position ◽  
Rational Integral

All matrices considered here have rational integral elements. In particular some circulants of this nature are investigated. An n × n circulant is of the formThe following result concerning positive definite unimodular circulants was obtained recently (3 ; 4 ):Let C be a unimodular n × n circulant and assume that C = AA' where A is an n × n matrix and A' its transpose. Then it follows that C = C1C1', where C1 is again a circulant.

scholarly journals Note on Numerical Integration

1928 ◽  
Vol 1 (3) ◽  
pp. 139-148 ◽  
Author(s):  
C. E. Wolff
Keyword(s):  
Numerical Integration ◽  
Integral Function ◽  
Tacit Assumption ◽  
Rational Integral

All the commonly used rules for the approximate quadrature of areas, such as those of Cotes, Simpson, Tchebychef and Gauss, are based on the assumption that y can be expressed as a rational integral function of x with finite coefficients. A tacit assumption is thus made that is not infinite within the range considered, and it is therefore hardly a matter for surprise that the degree of accuracy obtainable by the use of these rules in the case of a curve which touches the end ordinates is very poor.

Periodic behaviour of the X-ray flux from the region near 3U1727-33

Nature ◽  
1976 ◽  
Vol 264 (5584) ◽  
pp. 342-343 ◽  
Author(s):  
N. E. WHITE ◽  
K. O. MASON ◽  
G. BRANDUARDI ◽  
P. W. SANFORD ◽  
L. MARASCHI ◽  
...  
Keyword(s):  
X Ray ◽  
Periodic Behaviour

scholarly journals A Theorem on Rational Integral Symmetric Functions

1929 ◽  
Vol 24 ◽  
pp. i-iii
Author(s):  
John Dougall
Keyword(s):  
Symmetric Functions ◽  
Image Position ◽  
Rational Integral

An identity involving symmetric functions of n letters may in a certain class of cases be extended immediately to a greater number of letters.For example, the theoremmay be writtenand in the latter form it is true for any number of letters.

scholarly journals Transient periodic behaviour related to a saddle-node bifurcation

1987 ◽  
Vol 20 (13) ◽  
pp. 4161-4171 ◽  
Author(s):  
R van Damme ◽  
T P Valkering
Keyword(s):  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Periodic Behaviour
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