scholarly journals On Closure Properties of Irrational and Transcendental Numbers under Addition and Multiplication

2016 ◽  
Vol 13 (3) ◽  
Author(s):  
Shekh Zahid ◽  
Prasanta Ray
Keyword(s):  
Algebraic Numbers ◽  
Undergraduate Mathematics ◽  
Mathematical Methods ◽  
Closure Properties ◽  
New Approach ◽  
Irrational Numbers ◽  
Transcendental Numbers ◽  
Dedekind Cuts ◽  
Mathematics Research

In the article 'There are Truth and Beauty in Undergraduate Mathematics Research’, the author posted a problem regarding the closure properties of irrational and transcendental numbers under addition and multiplication. In this study, we investigate the problem using elementary mathematical methods and provide a new approach to the closure properties of irrational numbers. Further, we also study the closure properties of transcendental numbers. KEYWORDS: Irrational numbers; Transcendental numbers; Dedekind cuts; Algebraic numbers

Developing a Multisemester Interwoven Dynamic Systems Project to Foster Learning and Retention of STEM Material

2004 ◽  
Author(s):  
Peter Avitabile ◽  
Stephen Pennell ◽  
John White
Keyword(s):  
Dynamic Systems ◽  
Basic Material ◽  
Mathematical Methods ◽  
Introductory Courses ◽  
New Approach ◽  
Science Technology ◽  
Educational Career ◽  
And Mathematics ◽  
Relationship Of ◽  
The Relationship

Students generally do not understand how basic STEM (Science, Technology, Engineering and Mathematics) material fits into all of their engineering courses. Basic material is presented in introductory courses but the relationship of the material to subsequent courses is unclear to the student since the practical relevance of the material is not necessarily presented. Students generally hit the “reset button” after each course not realizing the importance of basic STEM material. The capstone experience is supposed to “tie all the pieces together” but this occurs too late in the student’s educational career. A new multisemester interwoven dynamic systems project has been initiated to better integrate the material from differential equations, mathematical methods, laboratory measurements and dynamic systems across several semesters/courses so that the students can better understand the relationship of basic STEM material to an ongoing problem. This paper highlights the overall concept to be addressed by the new approach. The description of the project and modules under development are discussed.

How to characterize provably total functions by local predicativity

1996 ◽  
Vol 61 (1) ◽  
pp. 52-69 ◽  
Author(s):  
Andreas Weiermann
Keyword(s):  
Theoretic Analysis ◽  
Closure Properties ◽  
New Approach

AbstractInspired by Pohlers' proof-theoretic analysis of KPω we give a straightforward non-metamathematical proof of the (well-known) classification of the provably total functions of PA, PA + TI(⊰ ↾) (where it is assumed that the well-ordering ⊰ has some reasonable closure properties) and KPω. Our method relies on a new approach to subrecursion due to Buchholz, Cichon and the author.

New Books

1932 ◽  
Vol 25 (4) ◽  
pp. 238-241
Keyword(s):  
High School ◽  
Elementary Mathematics ◽  
School Teacher ◽  
High School Teacher ◽  
Complex Numbers ◽  
Algebraic Numbers ◽  
Historical Information ◽  
Transcendental Numbers

It is a trite and rather patronizing statement that a certain book should be read by every high school teacher. If it were to be used at all, however, it might be used in connection with Professor Bell's charming little work, which came out a few months ago. It is a combination of historical information and material to show the nature of the various branches of elementary mathematics, with some excursions into such fields as complex numbers, transformations, groups, algebraic numbers, transcendental numbers, and the infinite in mathematics. It begins with a treatment of the purposes of mathematics, and ends with a reference to some of the prominent theories of the last quarter of a century. It allows the reader to find out without undue difficulty the nature and bases of the postulates of mathematics, the development of the underlying rules of the science, the significance of invariants and projections, and the nature of geometry as considered by mathematics of the present day.

scholarly journals Multi-Criteria Analysis and Prediction of Network Incidents Using Monitoring System

2017 ◽  
Vol 1 (1) ◽  
pp. 29 ◽  
Author(s):  
Lukas Macura ◽  
Miroslav Voznak
Keyword(s):  
Monitoring System ◽  
Network Flow ◽  
Flow Analysis ◽  
Mathematical Methods ◽  
New Approach ◽  
Creative Commons ◽  
Data Flows ◽  
Data Correlations ◽  
Computationally Intensive ◽  
Network Technologies

Today, network technologies can handle throughputs up to 100Gbps, transporting 200 million packets per second on a single link. Such high bandwidths impact network flow analysis and as a result require significantly more powerful hardware. Methods used today concentrate mainly on analyzes of data flows and patterns. It is nearly impossible to actively look for anomalies in network packets and flows for a small amount of change of monitoring patterns could result in big increases in potentially false positive incidents. This paper focuses on multi-criteria analyzes of systems generated data in order to predict incidents. We prove that systems generated monitoring data are an appropriate source to analyze and enable for much more focused and less computationally intensive monitoring operations. By using appropriate mathematical methods to analyze stored data it is possible to obtain useful information. During our work, some interesting anomalies in networks were found by utilizing simple data correlations using monitoring system Zabbix. We concluded that it is possible to declare that deeper analysis is possible due to Zabbix monitoring system and its features like Open-Source core, documented API and SQL backend for data. The result of this work is a new approach to the analysis containing algorithms which allow to identify significant items in monitoring system. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

scholarly journals SIMULTANEOUS APPROXIMATION BY CONJUGATE ALGEBRAIC NUMBERS IN FIELDS OF TRANSCENDENCE DEGREE ONE

2005 ◽  
Vol 01 (03) ◽  
pp. 357-382 ◽  
Author(s):  
DAMIEN ROY
Keyword(s):  
Simultaneous Approximation ◽  
Transcendence Degree ◽  
Algebraic Numbers ◽  
Bounded Degree ◽  
Root Of Unity ◽  
Transcendental Numbers ◽  
Algebraic Difference ◽  
Conjugate Algebraic Numbers ◽  
The Given ◽  
Real Complex

We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers ξ1, …, ξt by conjugate algebraic numbers of bounded degree over ℚ, provided that the given transcendental numbers ξ1, …, ξt generate over ℚ a field of transcendence degree one. We provide sharper estimates for example when ξ1, …, ξt form an arithmetic progression with non-zero algebraic difference, or a geometric progression with non-zero algebraic ratio different from a root of unity. In this case, we also obtain by duality a version of Gel'fond's transcendence criterion expressed in terms of polynomials of bounded degree taking small values at ξ1, …, ξt.

scholarly journals Electronic ceramic structure within the Voronoi cells model and microstructure fractals contacts surfaces new frontier applications

2013 ◽  
Vol 45 (2) ◽  
pp. 223-232 ◽  
Author(s):  
V.V. Mitic ◽  
V. Paunovic ◽  
S. Jankovic ◽  
V. Pavlovic ◽  
I. Antolovic ◽  
...  
Keyword(s):  
Mathematical Statistic ◽  
Mathematical Statistics ◽  
Fractal Nature ◽  
Mathematical Methods ◽  
Material Microstructure ◽  
Ceramic Structure ◽  
New Approach ◽  
Voronoi Cells ◽  
Real Model ◽  
New Frontier

In this study, in order to establish grain shapes of sintered ceramics, new approach on correlation between microstructure and doped BaTiO3 -ceramics properties based, on Voronoi model and mathematical statistics calculations on fractal geometry, has been developed. BaTiO3-ceramics doped with Yb2O3 (from 0.1 to 1.0wt% of Yb) were prepared by using conventional solid state procedure and were sintered from 1320?C to 1380?C for four hours. The microstructure of sintered specimens was investigated by Scanning electron microscope JEOL-SEM-5300. For better and deeper characterization and understanding of the ceramics material microstructure, the methods which include the fractal nature structure, and also Voronoi model and mathematical statistics calculations, are applied. In our research the Voronoi is one specific interface between fractal structure nature and different stochastically contact surfaces, defined by statistical mathematical methods. Also, the Voronoi model practically provided possibility to control the ceramics microstructure fractal nature. Mathematical statistic methods enabled establishing the real model for the prognosis based on correlation: synthesis-structures-properties.

scholarly journals Algebraic values of sines and cosines and their arguments

2021 ◽  
Vol 61 ◽  
pp. 21-28
Author(s):  
Edmundas Mazėtis ◽  
Grigorijus Melničenko
Keyword(s):  
Rational Numbers ◽  
Trigonometric Functions ◽  
Algebraic Numbers ◽  
Transcendental Numbers

The article introduces the reader to some amazing properties of trigonometric functions. It turns out that if the values of the arguments of the functions sin x, cos x, tg x and ctg x, expressed in radians, are algebraic numbers, then the values of these functions are transcendental numbers. Hence, it follows that the values of all angles of the pseudo-Heronian triangle, including the values of all angles of the Pythagoras or Heron triangle, expressed in radians, are transcendental numbers. If the arguments of functions sin x and cos x, expressed in radians, are equal to x = r 2 \pi, where r are rational numbers, then the values of the functions are algebraic numbers. It should be noted that in this case the argument x = r 2\pi  is transcendental and, if expressed in degrees, becomes a rational.

scholarly journals Algebraic Numbers as Product of Powers of Transcendental Numbers

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 887 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Crucial Role ◽  
Symmetric Functions ◽  
Transcendental Number ◽  
Algebraic Numbers ◽  
Elementary Symmetric Functions ◽  
Transcendental Numbers ◽  
Definition Of

The elementary symmetric functions play a crucial role in the study of zeros of non-zero polynomials in C [ x ] , and the problem of finding zeros in Q [ x ] leads to the definition of algebraic and transcendental numbers. Recently, Marques studied the set of algebraic numbers in the form P ( T ) Q ( T ) . In this paper, we generalize this result by showing the existence of algebraic numbers which can be written in the form P 1 ( T ) Q 1 ( T ) ⋯ P n ( T ) Q n ( T ) for some transcendental number T, where P 1 , … , P n , Q 1 , … , Q n are prescribed, non-constant polynomials in Q [ x ] (under weak conditions). More generally, our result generalizes results on the arithmetic nature of z w when z and w are transcendental.

scholarly journals Forecast of seismic resistance of structures by the results of numerical simulation of deformation properties of soil bases

2021 ◽  
Vol 303 ◽  
pp. 01015
Author(s):  
Mikhail Sokolov ◽  
Sergey Prostov ◽  
Andrey Pokatilov
Keyword(s):  
Numerical Simulation ◽  
Seismic Intensity ◽  
Seismic Resistance ◽  
Mathematical Methods ◽  
New Approach ◽  
Geological Surveys ◽  
Deformation Properties ◽  
Before And After ◽  
Real Objects ◽  
Properties Of Soil

The object of the research is the forecast of seismic resistance when strengthening the soil foundations of structures. The purpose of the work is to numerically estimate the total increment of seismic intensity during artificial transformation and strengthening of foundation soils based on the results of geomechanical modeling. The study is based on classical mathematical methods for modeling soil foundations in a flat nonlinear setting. A new approach to determining the total increment of seismic intensity is presented, based on determining the ratio of the values of subsidence of the foundations of buildings and structures before and after soil strengthening. The paper presents the results of predicting changes in seismic resistance for real objects, obtained from the data of engineering and geological surveys and numerical computer models. It was found that due to the transformation of soil foundations, seismic resistance can decrease by more than 0.5 points. This technique can be used both to adjust the scoring for individual objects and to clarify the boundaries of seismic zones on OCP maps.

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