More Discoveries in Linear Algebra

1976 ◽  
Vol 69 (1) ◽  
pp. 56-58
Author(s):  
Darrell Wellen
Keyword(s):  
Linear Algebra ◽  
Mathematics Teacher ◽  
Recent Issue ◽  
Special Cases

In a recent issue of the Mathematics Teacher, Nicolai (1974) detailed several “discoveries” in linear algebra that were made by him and one of his classes. A look at their ideas yielded several discoveries showing that the relationships described by Nicolai are special cases of more general, interesting situations.

Characterizations of Fully Constrained Poses of Parallel Cable-Driven Robots: A Review

2008 ◽  
Author(s):  
Marc Gouttefarde
Keyword(s):  
Linear Algebra ◽  
Convex Cones ◽  
Mobile Platform ◽  
Polyhedral Cones ◽  
Fundamental Properties ◽  
Special Cases ◽  
Force Closure

The pose of the mobile platform of a parallel cable-driven robot is said to be fully constrained if any wrench can be created at the platform by pulling on it with the cables. A fully constrained pose is also known as a force-closure pose. In this paper, a review of three useful characterizations of a force-closure pose is proposed. These characterizations are stated in the form of theorems for which proofs are presented. Tools from linear algebra allow to derive some of these proofs while the others are more difficult and can hardly be obtained in this manner. Therefore, polyhedral cones, which are special cases of convex cones, are introduced along with some of their well-known fundamental properties. Then, it is shown how the aforementioned difficult proofs can be obtained as direct consequences of these properties.

The Place of Mathematics in General Education

1948 ◽  
Vol 41 (6) ◽  
pp. 274-277
Author(s):  
Edward A. Cameron
Keyword(s):  
General Education ◽  
Mathematics Teacher ◽  
Liberal Education ◽  
Arithmetic Geometry ◽  
National Council ◽  
Recent Issue ◽  
The Subject

The place of mathematics in general education was discussed at least as long ago as some 2500 years, when the Pythagoreans established the quadrivium of arithmetic, geometry, astronomy, and music, subjects which were to be considered the heart of a liberal education for many centuries. That the subject is still being discussed today can be readily verified by consulting almost any recent issue of The Mathematics Teacher. The Eleventh and Fifteenth Yearbooks of the National Council of Teachers of Mathematics contain much valuable information on the subject under discussion, and I heartily recommend them to any teacher of mathematics who has not yet read them.

Why Not Be Sensible About Meaning?

1945 ◽  
Vol 38 (3) ◽  
pp. 99-102
Author(s):  
Harry G. Wheat
Keyword(s):  
Mathematics Teacher ◽  
Elementary Mathematics ◽  
Recent Issue

An article in a recent issue of The Mathematics Teacher* admonishes us to be sensible about the meaning we attempt to teach the pupils in our classes in elementary mathematics. It goes on seriously to question both the sense and the utility of teaching meaning, and still further to suggest that it can't be done anyhow.

Female Mathematics Educators Recommend Books

1984 ◽  
Vol 77 (3) ◽  
pp. 233-234
Author(s):  
Sylvia Lazarnick ◽  
Marny Frantz
Keyword(s):  
Secondary School ◽  
Mathematics Teachers ◽  
Mathematics Teacher ◽  
Secondary Mathematics ◽  
Recent Issue ◽  
Secondary Mathematics Teachers ◽  
School Grades ◽  
Mathematics Educators

A recent issue of the Mathematics Teacher reported the results of an interesting question posed to mathematics educators, “What are the ten most important books for a secondary school (grades 7–12) mathematics teacher to read?” (Leake 1983). As secondary mathematics teachers, we eagerly read the article and at once realized that we needed and wanted to know what books female mathematics educators might recommend. We set out to ask them.

Tips For Beginners: A complete set of elementary rules for testing for divisibility

1963 ◽  
Vol 56 (6) ◽  
pp. 437-442
Author(s):  
Wendell M. Williams
Keyword(s):  
Mathematics Teacher ◽  
Ninth Grade ◽  
Simple Set ◽  
Special Cases ◽  
Complete Set

A number of articles in both The Arithmetic Teacher and The Mathematics Teacher have developed and applied different rules for special cases of divisibility. However, the entire divisibility spectrum can be viewed from one simple set of rules which the average ninth grade pupil can understand and apply.

More on Physical Models for Factoring Polynomials

1974 ◽  
Vol 67 (2) ◽  
pp. 133-138
Author(s):  
Allan A. Gibb
Keyword(s):  
Physical Model ◽  
Mathematics Teacher ◽  
Physical Models ◽  
Recent Issue ◽  
Quadratic Polynomials ◽  
Jerome Bruner ◽  
Factoring Polynomials

In a recent issue of the Mathematics Teacher, James K. Bidwell (1972) details a physical model for factoring quadratic polynomials, which he describes as an extension of a model described by Jerome Bruner (1966). This, in turn, is closely related to Dienes’s (1971) materials.

Linear algebra algorithms as dynamical systems

Acta Numerica ◽  
2008 ◽  
Vol 17 ◽  
pp. 1-86 ◽  
Author(s):  
Moody T. Chu
Keyword(s):  
Dynamical Systems ◽  
Linear Algebra ◽  
Iterative Procedure ◽  
Numerical Algorithms ◽  
Great Effort ◽  
Open Problems ◽  
Special Cases ◽  
Specific Rule ◽  
Discrete Methods ◽  
Realization Process

Any logical procedure that is used to reason or to infer either deductively or inductively, so as to draw conclusions or make decisions, can be called, in a broad sense, a realization process. A realization process usually assumes the recursive form that one state develops into another state by following a certain specific rule. Such an action is generally formalized as a dynamical system. In mathematics, especially for existence questions, a realization process often appears in the form of an iterative procedure or a differential equation. For years researchers have taken great effort to describe, analyse, and modify realization processes for various applications.The thrust in this exposition is to exploit the notion of dynamical systems as a special realization process for problems arising from the field of linear algebra. Several differential equations whose solutions evolve in submanifolds of matrices are cast in fairly general frameworks, of which special cases have been found to afford unified and fundamental insights into the structure and behaviour of existing discrete methods and, now and then, suggest new and improved numerical methods. In some cases, there are remarkable connections between smooth flows and discrete numerical algorithms. In other cases, the flow approach seems advantageous in tackling very difficult open problems. Various aspects of the recent development and application in this direction are discussed in this paper.

scholarly journals THE DISTRIBUTION OF MIXING TIMES IN MARKOV CHAINS

2013 ◽  
Vol 30 (01) ◽  
pp. 1250045 ◽  
Author(s):  
JEFFREY J. HUNTER
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Linear Algebra ◽  
Mixing Time ◽  
Probability Generating Function ◽  
First Passage Times ◽  
Mixing Times ◽  
First Passage ◽  
Special Cases ◽  
Passage Times

The distribution of the "mixing time" or the "time to stationarity" in a discrete time irreducible Markov chain, starting in state i, can be defined as the number of trials to reach a state sampled from the stationary distribution of the Markov chain. Expressions for the probability generating function, and hence the probability distribution of the mixing time, starting in state i, are derived and special cases explored. This extends the results of the author regarding the expected time to mixing [Hunter, JJ (2006). Mixing times with applications to perturbed Markov chains. Linear Algebra and Its Applications, 417, 108–123] and the variance of the times to mixing, [Hunter, JJ (2008). Variances of first passage times in a Markov chain with applications to mixing times. Linear Algebra and Its Applications, 429, 1135–1162]. Some new results for the distribution of the recurrence and the first passage times in a general irreducible three-state Markov chain are also presented.

Performing Operations with Matrices on Spreadsheets

1998 ◽  
Vol 91 (4) ◽  
pp. 338-341
Author(s):  
Dusan Pagon
Keyword(s):  
High School ◽  
Middle School ◽  
Mathematics Education ◽  
Linear Algebra ◽  
Mathematics Teacher ◽  
Middle School Mathematics ◽  
School Mathematics ◽  
High School Mathematics ◽  
Interested Reader ◽  
Basic Concepts

Although created mainly for other purposes, spreadsheets appear to be useful in mathematics education. For instance, Russell (1992) writes about spreadsheet activities in middle school mathematics in his book with the same title. In a Mathematics Teacher article, Hunt (1995) describes how he uses spreadsheets to teach students synthetic substitution, synthetic division, and Newton's method. Our experience shows that the same tool can be useful for performing matru operations and, further on, for introducing students to basic concepts of linear algebra. The interested reader can find additional examples of using spreadsheets in high school mathematics in Sjostrand's (1994) book, which deals with Excel spreadsheets.

Discussion

1926 ◽  
Vol 19 (7) ◽  
pp. 429-432
Author(s):  
John J. Hall
Keyword(s):  
Mathematics Teacher ◽  
Recent Issue ◽  
Negative Numbers ◽  
The Subject

In a recent issue of “The Mathematics Teacher” appeared an article on the subject of the proof or reason of the rule that the product of two negative numbers is a positive number.

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