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.

Finding fundamental units in cubic fields

1982 ◽  
Vol 92 (3) ◽  
pp. 385-389 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Computational Problem ◽  
Cubic Field ◽  
Cubic Fields ◽  
Totally Real ◽  
Fundamental Units

This paper improves a method of Godwin (4) for finding a pair of fundamental units in a totally real cubic field. The determination of such a unit pair is a well known computational problem. There is an old algorithm (circa 1896) of Voronoi which solves this problem, but the algorithm is quite complicated (an account of it is given in the book of Delone and Faddeev ((3), chapter IV, part A)). The method of Godwin is, in principle, much simpler. However, this method also has its drawbacks (more is said about this in Section 4 below). Indeed, when Godwin's student Angell produced his large table (see (1)) of totally real cubic fields some 15 years after (4) appeared, Voronoi's algorithm was used to compute the pairs of fundamental units.

A note on Cusick's theorem on units in totally real cubic fields

1984 ◽  
Vol 95 (1) ◽  
pp. 1-2 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Present Note ◽  
Cubic Field ◽  
Cubic Fields ◽  
Totally Real ◽  
Fundamental Units

Let ε = ε1, with conjugates ε2, ε3, be a unit in a totally real cubic field, and let . Let ε be a unit for which T (ε) is least and let η be a unit, not a power of ε, for which T(η) is least. It was shown by Cusick[l] that ε,η form a pair of fundamental units under certain conditions. The purpose of the present note is to show that these conditions are unnecessary and that ε, η form a pair of fundamental units in all cases.

scholarly journals A numerical algorithm for nth root

2014 ◽  
Vol 7 (1) ◽  
Author(s):  
Natarajan Murugesan ◽  
A.M.S. Ramasamy
Keyword(s):  
Real Number ◽  
Numerical Algorithm ◽  
Positive Real Number ◽  
Direct Method ◽  
Initial Guess ◽  
Trial And Error ◽  
Positive Real ◽  
Trial And Error Method ◽  
Error Method

Presently a direct analytical method is available for the digit-by-digit extraction of the square root of a given positive real number. To calculate the nthroot of a given positive real number one may use trial and error method, iterative method, etc. When one desires to determine the nth root, it is found thatsuch methods are inherent with certain weaknesses like the requirement of an initial guess, a large number of arithmetic operations and several iterativesteps for convergence, etc. There has been no direct method for the determination of the nth root of a given positive real number. This paper focusesattention on developing a numerical algorithm to determine the digit-by-digit extraction of the nth root of a given positive real number up to any desiredaccuracy. Examples are provided to illustrate the algorithm.

The determination of the class-numbers of totally real cubic fields

1961 ◽  
Vol 57 (4) ◽  
pp. 728-730 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Class Number ◽  
Sum Of Squares ◽  
Class Numbers ◽  
Cubic Field ◽  
Cubic Fields ◽  
Totally Real

In a previous paper (2) it was shown how the work of finding the units of a totally real cubic field could be facilitated by consideration of the sum of squares of differences between a number and its conjugates. In the present paper it is shown that the same ideas can be helpful in the calculation of class-numbers, and a list of the fields with class-number greater than unity and discriminant less than 20,000 is given.

CONTEXTUAL SUBTEXT INFORMATION OF A FOREIGN LANGUAGE TEXT: HERMENEUTIC APPROACH

2020 ◽  
Vol 2020 (30) ◽  
pp. 75-87
Author(s):  
Lidiya Derbenyova
Keyword(s):  
Foreign Language ◽  
Optimal Solution ◽  
Decision Making Process ◽  
Trial And Error ◽  
Transmission Of Information ◽  
Hermeneutic Approach ◽  
Trial And Error Method ◽  
Problems Of Translation ◽  
Error Method ◽  
Language Text

The article focuses on the problems of translation in the field of hermeneutics, understood as a methodology in the activity of an interpreter, the doctrine of the interpretation of texts, as a component of the transmission of information in a communicative aspect. The relevance of the study is caused by the special attention of modern linguistics to the under-researched issues of hermeneutics related to the problems of transmission of foreign language text semantics in translation. The process of translation in the aspect of hermeneutics is regarded as the optimum search and decision-making process, which corresponds to a specific set of functional criteria of translation, which can take many divergent forms. The translator carries out a number of specific translation activities: the choice of linguistic means and means of expression in the translation language, replacement and compensation of nonequivalent units. The search for the optimal solution itself is carried out using the “trial and error” method. The translator always acts as an interpreter. Within the boundaries of a individual utterance, it must be mentally reconstructed as conceptual situations, the mentally linguistic actions of the author, which are verbalized in this text.

MXene: a promising photocatalyst for water splitting

2016 ◽  
Vol 4 (29) ◽  
pp. 11446-11452 ◽  
Author(s):  
Zhonglu Guo ◽  
Jian Zhou ◽  
Linggang Zhu ◽  
Zhimei Sun
Keyword(s):  
Water Splitting ◽  
Trial And Error ◽  
Hydrogen Fuel ◽  
Photocatalytic Water Splitting ◽  
Arduous Task ◽  
Trial And Error Method ◽  
Error Method

Identifying suitable photocatalysts for photocatalytic water splitting to produce hydrogen fuelviasunlight is an arduous task by the traditional trial-and-error method.

scholarly journals Measuring the Deformation of a Flat Die by Applying a Laser Beam on a Reflecting Surface

2009 ◽  
Vol 424 ◽  
pp. 197-204 ◽  
Author(s):  
W. Assaad ◽  
H.J.M. Geijselaers ◽  
K.E. Nilsen
Keyword(s):  
Finite Element ◽  
Laser Beam ◽  
Extrusion Process ◽  
Trial And Error ◽  
Element Calculation ◽  
Extrusion Dies ◽  
Cheap Method ◽  
Trial And Error Method ◽  
Reflecting Surface ◽  
Error Method

The design of extrusion dies depends on the experience of the designer. After the die has been manufactured, it is tested during an extrusion process and machined several times until it works properly. The die is designed by a trial and error method which is expensive interms of time consumption and the amount of scrap. Research is going on to replace the trial pressing with finite element simulations that concentrate on material and tool analysis. In order to validate the tool simulations, an experiment is required for measuring the deformation of the die. Measuring the deformation of the die is faced with two main obstacles: high temperature and little free space. To overcome these obstacles a method is tried, which works by applying a laser beam on a reflecting surface. This cheap method is simple, robust and gives good results. This paper describes measuring the deformation of a flat die used to extrude a single U shape profile. In addition, finite element calculation of the die is performed. Finally, a comparison is performed between experimental and numerical results.

Adjustment of a Grazing Incidence Spectrometer for Maximum Resolution

1969 ◽  
Vol 23 (2) ◽  
pp. 128-132 ◽  
Author(s):  
E. Alexander ◽  
B. S. Fraenkel
Keyword(s):  
Grazing Incidence ◽  
Trial And Error ◽  
Routine Method ◽  
Maximum Resolution ◽  
Trial And Error Method ◽  
Error Method

A routine method to adjust a grazing incidence spectrometer for maximum resolution is described. The trial and error method uses as variable, the distance of the slit from the Rowland circle. Examples of resolved doublets are shown.

scholarly journals Directivity Enhancement in Non-Reciprocal Bandpass Filters using an Evolutionary Algorithm

2021 ◽  
Author(s):  
Prantik Dutta ◽  
Arun Gande ◽  
Gopi Ram
Keyword(s):  
Evolutionary Algorithm ◽  
Insertion Loss ◽  
Return Loss ◽  
Bandpass Filter ◽  
Trial And Error ◽  
Swarm Optimization ◽  
Lumped Element ◽  
Trial And Error Method ◽  
Error Method ◽  
Maximum Directivity

In this letter, a non-reciprocal filter with enhanced directivity is analyzed methodically and the filter parameters are optimized using an evolutionary algorithm. The return loss, insertion loss, and isolation characteristics of the filter exhibit a trade-off that makes manual tuning a trial-and-error method. The veracity of the numerical modeling is conformed by designing a 150 MHz lumped element non-reciprocal bandpass filter based on the parameters extracted using an evolutionary algorithm based particle swarm optimization (PSO). The simulated and measured results comply well with the modeling and the results exhibit maximum directivity of 28.2 dB without degradation in insertion loss (1.1 dB) and return loss (16.2 dB) within the passband. The algorithm can be utilized in designing non-reciprocal filters having different center frequencies and bandwidths.

A mathematical model of a brushed DC motor system

2021 ◽  
Vol 2 (2) ◽  
pp. 60-68
Author(s):  
N. N. A. Rahman ◽  
N. M. Yahya
Keyword(s):  
Mathematical Model ◽  
Steady State ◽  
Motor System ◽  
Dc Motor ◽  
Positioning System ◽  
Pd Controller ◽  
Trial And Error ◽  
Trial And Error Method ◽  
Low Torque ◽  
Error Method

Mathematical model has been proposed for some system that involves a brushed DC motor and it is widely used in industry. Brushed DC motor ideals for applications with a low- torque, manage to change pace or speed and it is widely used in many applications such as x-y table positioning system, conveyor systems and other system that required to use the features that brushed DC motor have. Mathematical model of brushed DC motor in order to verify the performance of the DC motor. In this paper, mathematical model of brushed DC motor will be derived from a brushed DC motor circuit that consist of two parts that are electrical and mechanical part. To validate the functionality of mathematical model, the performance of the brushed DC motor without any controller will be compared with the brushed DC motor with the presence of PI-PD controller that will be tuned by trial-and-error method. Performances of both brushed DC motor with and without controller will be compared in terms of transient response which are, rise time, Tr, settling time, Ts, steady state error, ess and lastly percentage overshoot. At the end of the study, the brushed DC motor with PI-PD controller show a better performance compared to the brushed DC motor without any controller.

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