Mathematics Journals

Mathematics History

2008-03-15T16:58:00.000-07:00

Equations Mathematics

2008-03-15T16:12:00.000-07:00

In the algebra section, we looked at how we can use letters as numbers and how to find their values. There are times, however, when life gets a little harder:

x + y = 25

We cannot say what x and y are in the equation above because there are an infinite number of possibilities:

x = 1, y = 24
x = 21, y = 4 etc.

In general, a single equation with two unknowns (we don't know x and y in this case) cannot be solved. What we need is a second equation involving at least one of the unknown variables.

Simultaneous Equations

Suppose that we were given the following equations:

x + y = 25
3x = 15

We have two different equations with two unknowns - this is enough information for us to solve them both (that is, find the values of x and y). We call these simultaneous equations because we use them at the same time.

In the second equations, we can solve immediately for x (see the algebra pages if you're not happy with this):

3x = 15 so x = 5

Now we put this value of x (called substitution) into the first equation:

5 + y = 25 so y = 20

So the only correct solution for the two equations given at the top of the page is x = 5 and y = 20.

So far so good! Let's look at a harder example:

2a + b = 21
a + b = 15

In this case we cannot just solve for a or b individually, as both variables appear in each equation. Instead we must rearrange an equation to find an unknown in terms of the other:

b = 15 - a
[This is the second equation re-stated]

Now we can substitute this statement for b into the first equation:

2a + b = 21
2a + (15 - a) = 21

This can now be tidied up, giving a correct answer for a:

2a - a = 21 - 15
a = 6

Finally we can use any of the first 3 equations to find b:

b = 15 - 6
b = 9

Testing Simultaneous Equations

Now would be a good time for you to test what you have learned. Work out the value of each variable in the examples below:

2a + b = 21
a + 2b = 24
5p + q = 100
2p - q = -30
m + n = 0
6m - n = 49

Hints: Rearrange one equation to give you one variable. Substitute it, then solve. Be careful with minus signs!

Let's see how you got on:

2a + b = 21
a + 2b = 24
Rearrange one of the equations: b = 21 - 2a
Next substitute in the other: a + 2(21 - 2a) = 24
Solve: a + 42 - 4a = 24
Rearrange: 42 - 24 = 3a
Answer: a = 6
Finally substitute: b = 21 - 2×6
Answer: b = 9
5p + q = 100
2p - q = -30
Rearrange one of the equations: q = 100 - 5p
Next substitute in the other: 2p - (100 - 5p) = -30
Solve: 2p - 100 + 5p = -30
Rearrange: 7p = 70
Answer: p = 10
Finally substitute: q = 100 - 5×10
Answer: q = 50
m + n = 0
6m - n = 49
Rearrange one of the equations: m = -n
Next substitute in the other: 6(-n) - n = 49
Solve: -7n = 49
Answer: n = -7
Finally substitute: m = -(-7)
Answer: m = 7

Quadratic Equations Mathematics

2008-03-15T16:09:00.000-07:00

Brace yourself! These are probably the most complicated equations you'll meet at GCSE maths. That means two things:

You'll have to work a little harder to crack them
You'll get many more marks in exams when you do!

Don't worry why they're called quadratic - it basically means "involving squared powers".

What's a Quadratic?

Quadratic equations take the following form:
ax² + bx + c = 0

Where x is the only variable and a, b and c are just numbers (constants, that may also be zero!)

If a=0 then the equation is not quadratic: bx + c = 0

However, if b=0 then it can be: ax² + c = 0

Whilst if c=0 then it's: ax² + bx = 0

It is all much less confusing with numbers!

Quadratics With Numbers

Normally, of course, equations like ax² + bx + c = 0 are not written with a, b and c: they're usually just numbers.

e.g. 4x² - 3x + 5 = 0

It's your job normally to find the values of x for which the equation works - nightmare!

Let's start with equations of the form: ax² + c = 0

Solving Quadratic Equations

Solving equations like ax² + c = 0 can be quite straightforward.
e.g. x² - 25 = 0

From your work on algebra, you should be able to rearrange the equation to: x² = 25

By taking the square-root of both sides, we end up with:
x = 5

That wasn't too bad, was it? Another solution is x = -5, but we'll look at that another time.

Here's one for you. Find the solution to the equation: x² - 121 = 0.
Once you've worked it out, click here.

Modal Temperatures Mathematics

2008-03-15T16:08:00.000-07:00

Sean asked the following:

We have a record of temperatures and we are asked to find the MODAL TEMPERATURES. Any help would be great.

Thanks for this, Sean! The word MODAL refers to the mode, which is a type of average of a set of numbers.

So, for instance, you could have a set of peak temperatures for a month, and what you're asked for is the mode (or the modal temperature). The mode, as you can see from our tutorial, is the most common of a set of numbers.

There can be more than one mode, so if 20°C was the peak temperature on 5 days, and 22°C was the peak for another 5 days, then you have what's called a bimodal set of numbers. More than 2 modes and it's a multimodal set.

Click on the tutorials link below to learn more about averages. Good luck!

Trigonometry Mathematics

2008-03-15T16:07:00.001-07:00

The word trigonometry is very old. In fact the Ancient Greeks drew all this up. Was it important? Well, for one thing it allowed them to estimate the distance to the Sun!

It all starts with a triangle, specifically a right-angled one:

A right-angle is 90°, rather like the corner of a square. The hypotenuse is the longest side of a right-angle triangle, found always opposite the right angle.

Mathematical Explanation

2008-03-15T16:03:00.000-07:00

"Mathematical Explanation:
Problems and Prospects"
THURSDAY, MARCH 1, 2001 - 4:00 PM; **ROOM 1116, IPST BLDG.

Paolo Mancosu
Department of Philosophy, University of California, Berkeley

In this talk I have three major aims. The first is to introduce the topic of mathematical explanation by listing a number of problems followed by a reflection on the status of research and prospects for further development. The general discussion in the first part motivates the specific contributions to be presented in the remaining two parts of the talk. In the second part I will draw attention to an important tradition in philosophy of mathematics for which explanation is a concern. Here I discuss Mill, Russell, Godel, Lakatos and other philosophers of mathematics on mathematical explanation. The last part of the talk will present a case study of a development in mathematical practice that originates from explanatory concerns, i.e. Alfred Pringsheim's "explanatory" approach to the foundations of complex analysis.

Paolo Mancosu is Associate Professor of Philosophy at U.C. Berkeley. His main interests are in mathematical logic and the history and philosophy of mathematics. He is the author of "Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century" (OUP 1996) and "From Brouwer to Hilbert."(OUP 1998).

If you have questions, contact the CHPS Office at (301) 405-5691 or by e-mail at hp26@umail.umd.edu. Information about colloquia is also available on-line through the CHPSCOL LISTSERV and on the WWW at http://carnap.umd.edu/chps or write to:

Probability and Pascal's Triangle Mathematics

2008-03-15T16:02:00.000-07:00

Date: 28 Feb 1995 14:19:21 -0500
From: Dr. Ken
Subject: Re: Probability and Pascal's Triangle

Hello there!

Well, since I'm not quite sure how much you know about Pascal's Triangle
(for instance, you seem to know that it is connected to the study of
probability), I'll start pretty much from the beginning.

The way I see it, Pascal's Triangle is kind of a collection of neat things
in mathematics.  The way you construct it follows:

                                 1
                               1   1
                             1   2   1
                           1   3   3   1
                         1   4   6   4   1
                       1   5  10   10  5   1
                     1   6  15  20   15  6   1
                   1   7  21  35   35  21  7   1
                                 .
                                 .
                                 .

You start out with the top two rows: 1, and 1 1.  Then to construct each
entry in the next row, you look at the two entries above it (i.e. the one
above it and to the right, and the one above it and to the left).  At the
beginning and the end of each row, when there's only one number above,
put a 1.  You might even think of this rule (for placing the 1's) as included
in the first rule: for instance, to get the first 1 in any line, you add up the
number above and to the left (since there is no number there, pretend it's
zero) and the number above and to the right (1), and get a sum of 1.

When people talk about an entry in Pascal's Triangle, they usually give a
row number and a place in that row, beginning with row zero and place
zero.  For instance, the number 20 appears in row 6, place 3.

That's how you construct Pascal's Triangle.  An interactive version where
you can specify the number of rows you want to see can be found at

 http://mathforum.org/dr.cgi/pascal.cgi  

But Pascal's Triangle is more than just a big triangle of numbers. There
are two huge areas where Pascal's Triangle rears its head, in Algebra
and in Probability/Combinatorics. 

First let's look at the Algebra version.

Let's say you have the polynomial 1+x, and you want to raise it to some
powers, like 1,2,3,4,5,....  If you make a chart of what you get when you
do these power-raisings, you'll get something like this:

(x+1)^0   =                           1
(x+1)^1   =                      1    +    x
(x+1)^2   =                 1    +   2x    +    x^2
(x+1)^3   =             1   +   3x    +   3x^2  +    x^3
(x+1)^4   =         1   +  4x    +   6x^2  +   4x^3  +    x^4
(x+1)^5   =     1   +  5x   +  10x^2  +  10x^3  +   5x^4  +    x^5  .....

If you just look at the coefficients of the polynomials that you get, voila!
Pascal's Triangle!  Because of this connection, the entries in Pascal's
Triangle are called the _binomial_coefficients_.  They are usually written
in parentheses, with one number on top of the other, for instance

20 =  (6)   <--- note: that should be one big set of
     (3)              parentheses, not two small ones.

I don't think it's standard notation, but when I write binomial coefficients
in a text document like this, I usually write them [6:3].

The other main area where Pascal's Triangle shows up is in Probability.
Let's say you have five hats on a rack, and you want to know how many
different ways you can pick two of them to wear.  It doesn't matter
to you which hat is on top, it just matters which two hats you pick.
So this problem amounts to the question "how many different ways can you
choose two objects from a set of five objects?" The answer? It's the
number in the second place in the fifth row, i.e. 10. (Remember that the
first number is in place zero.)

Because of this choosing property, the binomial coefficient [6:3] is
usually read "six choose three."  If you want to find out the probability
of choosing one particular combination of two hats, that probability is 1/10.

There's a pretty simple formula for figuring out the binomial coefficients.
It's
           n!
[n:k] = --------
       k! (n-k)!
                      6 * 5 * 4 * 3 * 2 * 1
For example, [6:3] =  ------------------------  =  20.
                      3 * 2 * 1 * 3 * 2 * 1

That's a basic introduction to Pascal's Triangle.  It certainly also shows
up in lots of other places (for example, the triangular numbers are in
there, if you know what those are), but I think it would be too much for
me to go into those right now.  Thanks for the question!

-Ken "Dr." Math

Mathematics History

2008-03-15T15:57:00.000-07:00

Articles Mathematics

2008-03-15T15:55:00.000-07:00

984, (with S. Galvan) Note di teoria dei modelli dell'aritmetica ("Notes on the theory of models of arithmetic"). I.S.U. Pubblicazioni Università Cattolica, Milano, [57 pp.]

1988a, 40 entries for the area "Logic" in the Dizionario Scientifico Tecnico Garzanti, Garzanti, Milano.[20 pp.]

1988b, (with W. Knorr) Diophantus, in Great lives from history, Salem Press, San Diego, California, pp. 632-637.

1989a, The metaphysics of the calculus: a foundational debate in the Paris Academy of Sciences, 1700-1706, in Historia Mathematica 16, pp. 224-248.

1989b, Nuovi risultati di incompletezza per l'aritmetica di Peano. Indicatori e funzioni velocemente crescenti. ("New incompleteness results for Peano arithmetic. Indicators and rapidly growing functions"). I.S.U. Pubblicazioni Università Cattolica, Milano, [35 pp.]

1990, (with E. Vailati) Detleff Clüver: an early opponent of the infinitesimal calculus, Centaurus, vol. 33, pp. 325-344.

1991a, (with E. Vailati) Torricelli's infinitely long solid and its philosophical reception in the XVIIth century, ISIS, 82, pp. 50-70.

1991b, Generalizing classical and effective model theory in theories of operations and classes, Annals of pure and applied logic, 52, 3, pp. 249-308.

1991c, On the status of proofs by contradiction in the seventeenth century, Synthese, 88, pp. 15-41.

1992a, Aristotelian Logic and Euclidean Mathematics: Seventeenth century developments of the "Quaestio de Certitudine Mathematicarum", Studies in History and Philosophy of Science, 23, 2, pp.241-265.

1992b, Descartes's Géométrie and Revolutions in Mathematics, in Revolutions in Mathematics, ed. D. Gillies, Oxford University Press, pp. 83-116.

1996, (Book) Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, Oxford University Press. [272 pp.]. (Paperback 1999)

1998a, (Book), ed., From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press. [335 pp.]

1998b, Hermann Weyl: predicativity and an intuitionistic excursion, in Mancosu (1998a), pp. 65-85.

1998c, Hilbert and Bernays on Metamathematics, in Mancosu (1998a), pp. 149-88.

1998d, (with W. van Stigt), Intuitionistic Logic, in Mancosu (1998a), pp. 275-285.

1999a, Recent work in the history and philosophy of mathematics from the Renaissance to Berkeley, Metascience, 8, issue 2, pp. 102-124.

1999b, Between Vienna and Berlin: the immediate reception of Gödel’s incompleteness theorems, History and Philosophy of Logic, 20, pp. 33-45.

1999c, Between Russell and Hilbert: Behmann on the foundations of mathematics, The Bulletin of Symbolic Logic, 5, no.3, pp. 303-330.

1999d, Bolzano and Cournot on Mathematical Explanation, Revue d’Histoire des Sciences, 52, pp.429-455.

2000a, On Mathematical Explanation, Growth of Mathematical Knowledge, E. Grosholz and H. Breger eds., Kluwer, pp.103-119

2000b, Four entries (Jakob Bernoulli, Johann Bernoulli, Infinitesimals, Mathematical Infinity) for The Scientific Revolution: An Encyclopedia, W. Applebaum ed., Garland Publishing.

2001, Mathematical Explanation: problems and prospects, Topoi , 20, pp. 97-117.

2002a, On the constructivity of proofs. A debate among Behmann, Bernays, Gödel, and Kaufmann, in Reflections on the foundations of mathematics. Essays in honor of Solomon Feferman, edited by W. Sieg, R. Sommer, and C. Talcott, Association for Symbolic Logic, Lecture Notes in Logic (vol. 15), pp. 346-368.

2002b, Phenomenology and Mathematics: Weyl at a crossroads, in Die Philosophie und die Wissenschaften. Zum Werk Oskar Beckers, Hrsg. von J. Mittelstrass und A. Gethmann-Siefert, Fink-Verlag, München, pp.129-148.

2002c, (with T. Ryckman), Mathematics and Phenomenology. The correspondence between Oskar Becker and Hermann Weyl, Philosophia Mathematica, 10, pp. 130-202.

2003a, (with M. Marion), Wittgenstein’s constructivization of Euler’s proof of the infinity of primes, in The Vienna Circle and Logical Empiricism, ed. by Friedrich Stadler, Kluwer, pp. 171-188.

2003b, The Russellian influence on Hilbert and his school, Synthese, 137, pp. 59-101.

2004, (Essay Review), Gödel’s Collected Works, vols. IV and V, Notre Dame Journal of Formal Logic, 45, no.2, pp. 109-125.

2005a, (Book), co-edited with K. Jørgensen and S. Pedersen, Visualization, Explanation and Reasoning Styles in Mathematics, Springer, pp.x+300.

2005b, Visualization in logic and mathematics, in P. Mancosu, K. Jørgensen and S. Pedersen eds., Visualization, Explanation and Reasoning Styles in Mathematics, Springer, pp. 13-30

2005c, (with J. Hafner), The varieties of mathematical explanation, in P. Mancosu, K. Jørgensen and S. Pedersen eds., Visualization, Explanation and Reasoning Styles in Mathematics, Springer, pp. 215-250

2005d, Das Abenteuer der Vernunft: Oskar Becker and Dietrich Mahnke on the phenomenological foundation of the exact sciences, in Die Philosophie und die Mathematik: Oskar Becker in der mathematischen Grundlagendiskussion, ed. Volker Peckhaus, Wilhelm Fink Verlag: München 2005 (Neuzeit & Gegenwart: Philosophie in Wissenschaft und Gesellschaft), pp. 229-243

2005e, (with T. Ryckman), Geometry, Physics and Phenomenology: the correspondence between O. Becker and H. Weyl, in Die Philosophie und die Mathematik: Oskar Becker in der mathematischen Grundlagendiskussion, ed. Volker Peckhaus, Wilhelm Fink Verlag: München 2005 (Neuzeit & Gegenwart: Philosophie in Wissenschaft und Gesellschaft), pp. 153-228

2005f, Harvard 1940-41: Tarski, Carnap and Quine on a finitistic language of mathematics for science, History and Philosophy of Logic, 26, 2005, 327-357.

2006a, (Encyclopedia entry), Addendum to P. Bernays’ entry for "Hilbert" in Borchert, Donald, ed., Encyclopedia of Philosophy, 2nd edition. Detroit: Macmillan Reference USA.

2006b, Tarski on models and logical consequence, in J. Gray, J. Ferreiros, eds. The Architecture of Modern Mathematics, Oxford University Press, 209-237.

2006c, Acoustics and Optics in the early modern period, in L. Daston and K.Park eds., The Cambridge History of Science, vol . 3: Early Modern Science, Cambridge University Press, 596-631.

2006d, Il programma di Hilbert e i teoremi di incompletezza di Gödel, Rivista di Filosofia Neoscolastica, 98, pp. 489-531.

2007, Descartes and Mathematics, in J. Broughton and J. Carriero, eds., A Companion to Descartes, Blackwell, pp.103-123.

2008, Answers to ‘5 questions’, In V. Hendricks, H. Leitgeb, eds., Philosophy of Mathematics. 5 Questions, Automatic Press/VIP, pp. 193-204.

Forthcoming:

(Article), with C. Badesa and R. Zach, Itineraries in the development of logic and the foundations of mathematics from Russell to Tarski (1900-1936), to appear in Handbook of the History of Logic, L. Haaparanta ed., Oxford University Press [150 pp.] [downloadable at http://www.ucalgary.ca/~zach/papers/history.html]

(Introduction), Introductory note for Paul Bernays “On Hilbert’s thoughts on the foundations of mathematics” (1922) and “On the foundations of arithmetic”(1922), in Paul Bernays: Essays in the Philosophy of Mathematics, vols. I-II, W. Sieg et al. Eds., Open Court, Chicago, [8 pp.]

(Article), Neurath, Tarski and Kokoszynska on the semantic conception of truth, forthcoming in D. Patterson, New Essays on Tarski and Philosophy, Oxford University Press.

(Article) Tarski’s engagement with philosophy, forthcoming in S. Lapointe et al., eds., The Golden Age of Polish Philosophy, Springer

(Book), ed., The Philosophy of Mathematical Practice, forthcoming for Oxford University Press.

(Article), “Mathematical Explanation: Why it Matters”, in P. Mancosu, ed., The Philosophy of Mathematical Practice, forthcoming for Oxford University Press.[15 pp.]

(Article), with Johannes Hafner, “Unification and Explanation: a case study from real algebraic geometry”, in P. Mancosu, ed., The Philosophy of Mathematical Practice, forthcoming for Oxford University Press.[25 pp.]

(Article), “Quine and Tarski on Nominalism”, Oxford Studies in Metaphysics [30 pp.]

(Editorial), Transcription and editorial remarks to Quine’s 1946 lecture “Nominalism”, Oxford Studies in Metaphysics.[10 pp]

Editor, "Interpolations. Essays in honor of William Craig." Special issue of Synthese.(Introduction by P, Mancosu; articles by Craig, Feferman, Demopoulos, M. Friedman, Väänänen, d'Agostino, Renardel de Lavalette, van Benthem)

Updated on Tue Mar 11 08:53:41 -0700 2008 by Paolo Mancosu

Pauline van den Driessche and Harald K. Wimmer, Explicit Polar Decomposition of Companion Matrices, pp. 64-69

2008-03-15T15:13:00.000-07:00

P. van den Driessche and H. K. Wimmer

An explicit formula for the polar decomposition of an n by n nonsingular
companion matrix is derived. The proof involves the largest and smallest
singular values of the companion matrix.

Charles R. Johnson and Brenda K. Kroschel, The Combinatorially Symmetric P-matrix Completion Problem, pp. 59-63

2008-03-15T15:12:00.000-07:00

Charles R. Johnson and Brenda K. Kroschel

An n-by-n real matrix is called a P-matrix if all its principal
minors are positive. The P-matrix completion problem asks which
partial P-matrices have a completion to a P-matrix. Here, we prove
that every partial P-matrix with combinatorially symmetric specified
entries has a P-matrix completion. The general case, in which the
combinatorial symmetry assumption is relaxed, is also discussed.

Rafael Bru, Carmen Coll, and Josep Gelonch, Periodic Coprime Matrix Fraction Decompositions, pp. 44-58

2008-03-15T15:11:00.000-07:00

Rafael Bru, Carmen Coll, and Josep Gelonch

A study is presented of right (left) coprime decompositions of a
collection of N-periodic rational matrices, with some ordered structure.
From a block-ordered right coprime decomposition of a rational matrix of
the given periodic collection, the corresponding block-ordered right
coprime decompositions of the remaining matrices of the collection are
constructed. In addition, those decompositions are N-periodic.

Gerd H. Fricke, Stephen T. Hedetniemi, David P. Jacobs and Vilmar Trevisan, Reducing the Adjacency Matrix of a Tree, pp. 34-43

2008-03-15T15:09:00.000-07:00

Gerd H. Fricke, Stephen T. Hedetniemi, David P. Jacobs and
Vilmar Trevisan


Let T be a tree, A its adjacency matrix, and a be a scalar.
We describe a linear-time algorithm for reducing the matrix
aI + A.  Applications include computing the rank of A,
finding a maximum matching in T, computing the rank and
determinant of the associated neighborhood matrix, and
computing the characteristic polynomial of A.

Vaidyanath Mani and Robert E. Hartwig, Some properties of the q-adic Vandermonde matrix, pp. 18-33

2008-03-15T15:08:00.002-07:00

ELA, Volume 1, pp. 18-33, abstract.

Some Properties of the q-adic Vandermonde Matrix

Vaidyanath Mani and Robert E. Hartwig


The Vandermonde and confluent Vandermonde matrices are of fundamental
significance in matrix theory. A further generalization of the
Vandermonde matrix called the q-adic coefficient matrix was
introduced in [V. Mani and R. E. Hartwig, Lin. Algebra Appl., to appear].
It was demonstrated there that the q-adic coefficient matrix reduces the
Bezout matrix of two polynomials by congruence. This extended the work
of Chen, Fuhrman, and Sansigre among others. In this paper, some
important properties of the $q$-adic  coefficient matrix are studied.
It is shown that the determinant of this matrix is a product of
resultants (like the Vandermonde matrix). The Wronskian-like block
structure of the q-adic coefficient matrix is also explored
using  a modified definition of the partial derivative operator.

Chi-Kwong Li, Nam-Kiu Tsing and Frank Uhlig, Numerical ranges of an operator on an indefinite inner product space, pp. 1-17

2008-03-15T15:08:00.001-07:00

ELA, Volume1, pp. 1-17, abstract.

Numerical Ranges of an Operator on an Indefinite Inner Product Space

Chi-Kwong Li, Nam-Kiu Tsing, and Frank Uhlig

For $n \times n$ complex matrices $A$ and an $n \times n$ Hermitian
matrix $S$, we consider the {\it $S$-numerical range} of $A$ and the
{\it positive $S$-numerical range} of $A$ defined by
\[
W_S(A)=\left\{{\langle Av,v\rangle_S\over \langle
v,v\rangle_S}: v\in\IC^n, \langle v,v\rangle_S\ne 0\right\}
\]
and
\[
W^+_S(A)=\left\{\langle Av,v\rangle_S: v\in\IC^n, \langle v,v\rangle_S
=1\right\},
\]
respectively, where $\langle u,v\rangle_S=v^*Su$. These sets generalize the
classical numerical range, and  they
are closely related to the joint numerical range of three Hermitian
forms and the cone generated by it.
Using some theory of the joint numerical range
we can give a detailed description of $W_S(A)$
and $W_S^+(A)$ for arbitrary Hermitian matrices $S$.
In particular, it is shown that $W_S^+(A)$ is always convex
and $W_S(A)$ is always $p$-convex for all $S$.
Similar results are obtained for the sets
\[
V_S(A) =\left\{{\langle Av,v\rangle \over \langle Sv,v\rangle}:
 v\in\IC^n, \langle Sv,v\rangle \ne 0\right\}, \quad
 V_S^+(A)=\left\{\langle Av,v\rangle:\,v\in\IC^n,\langle Sv,
 v\rangle=1\right\},
\]
where $\langle u,v\rangle=v^*u$.
Furthermore, we characterize those linear operators preserving
$W_S(A)$, $W_S^+(A)$, $V_S(A)$, or $V_S^+(A)$.
Possible generalizations of
our results, including their extensions to bounded linear operators on
an infinite dimensional Hilbert or Krein space, are discussed.

Mathematics Sofwate

2008-03-15T15:04:00.000-07:00

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GAP

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Game matematika yang lucu dan seru buat ade kecil .....

2008-03-15T15:03:00.000-07:00

Aku dapat situs yang nyediain game seru & lucu-lucu buat ade kecil. Game-game ini cocok banget buat guru-guru yang pengen pelajaran matematikanya ga membosankan. Tapi sayangnya semua game yang ada belum bisa di Download. Cuma bisa dimainkan secara online.

Selain game, situs ini juga banyak ngebahas matematika buat anak-anak. So... buat para guru-guru (pengajar) matematika, jangan sungkan-sungkan untuk sering-sering surfing & browsing internet, sapa tau bisa nemuin situs matematika yang kayak gini www.coolmath4kids.com

Mathematicians Find New Solutions To An Ancient Puzzle

2008-03-15T15:00:00.000-07:00

Many people find complex math puzzling, including some mathematicians. Recently, mathematician Daniel J. Madden and retired physicist, Lee W. Jacobi, found solutions to a puzzle that has been around for centuries.

Jacobi and Madden have found a way to generate an infinite number of solutions for a puzzle known as 'Euler's Equation of degree four.'

The equation is part of a branch of mathematics called number theory. Number theory deals with the properties of numbers and the way they relate to each other. It is filled with problems that can be likened to numerical puzzles.

"It's like a puzzle: can you find four fourth powers that add up to another fourth power" Trying to answer that question is difficult because it is highly unlikely that someone would sit down and accidentally stumble upon something like that," said Madden, an associate professor of mathematics at The University of Arizona in Tucson.

Equations are puzzles that need certain solutions "plugged into them" in order to create a statement that obeys the rules of logic.

For example, think of the equation x + 2 = 4. Plugging "3" into the equation doesn't work, but if x = 2, then the equation is correct.

In the mathematical puzzle that Jacobi and Madden worked on, the problem was finding variables that satisfy a Diophantine equation of order four. These equations are so named because they were first studied by the ancient Greek mathematician Diophantus, known as 'the father of algebra.'

In its most simple version, the puzzle they were trying to solve is the equation: (a)(to the fourth power) + (b)(to the fourth power) + (c)(to the fourth power) + (d)(to the fourth power) = (a + b + c + d)(to the fourth power)

That equation, expressed mathematically, is: a4 + b4 +c4 +d4 = (a + b + c + d)4.

Madden and Jacobi found a way to find the numbers to substitute, or plug in, for the a's, b's, c's and d's in the equation. All the solutions they have found so far are very large numbers.

In 1772, Euler, one of the greatest mathematicians of all time, hypothesized that to satisfy equations with higher powers, there would need to be as many variables as that power. For example, a fourth order equation would need four different variables, like the equation above.

Euler's hypothesis was disproved in 1987 by a Harvard graduate student named Noam Elkies. He found a case where only three variables were needed. Elkies solved the equation: (a)(to the fourth power) + (b)(to the fourth power) + (c)(to the fourth power) = e(to the fourth power), which shows only three variables are needed to create a variable that is a fourth power.

Inspired by the accomplishments of the 22-year-old graduate student, Jacobi began working on mathematics as a hobby after he retired from the defense industry in 1989.

Fortunately, this was not the first time he had dealt with Diophantine equations. He was familiar with them because they are commonly used in physics for calculations relating to string theory.

Jacobi started searching for new solutions to the puzzle using methods he found in some number theory texts and academic papers.

He used those resources and Mathematica, a computer program used for mathematical manipulations.

Jacobi initially found a solution for which each of the variables was 200 digits long. This solution was different from the other 88 previously known solutions to this puzzle, so he knew he had found something important.

Jacobi then showed the results to Madden. But Jacobi initially miscopied a variable from his Mathematica computer program, and so the results he showed Madden were incorrect.
"

The solution was wrong, but in an interesting way. It was close enough to make me want to see where the error occurred," Madden said.

When they discovered that the solution was invalid only because of Jacobi's transcription error, they began collaborating to find more solutions.

Madden and Jacobi used elliptic curves to generate new solutions. Each solution contains a seed for creating more solutions, which is much more efficient than previous methods used.

In the past, people found new solutions by using computers to analyze huge amounts of data. That required a lot of computing time and power as the magnitude of the numbers soared.

Now people can generate as many solutions as they wish. There are an infinite number of solutions to this problem, and Madden and Jacobi have found a way to find them all.

"Modern number theory allowed me to see with more clarity the implications of his (Jacobi's) calculations," Madden said.

"It was a nice collaboration," Jacobi said. "I have learned a certain amount of new things about number theory; how to think in terms of number theory, although sometimes I can be stubbornly algebraic."

The article, ""On a4 + b4 +c4 +d4 = (a + b + c + d)4" is published in the March issue of The American Mathematical Monthly.

Source : http://www.sciencedaily.com/releases/2008/03/080314145039.htm

Every Day with Recreational Math

2008-03-15T14:58:00.000-07:00

May be that word is not macth on this time. Couse more people is hate math. But, on this site methematic is a recretional. You can chose puzzle to have fun. Or you may be wanted mathematic game.

math.com is a funny mathematic site. On this site parent, teacher or student cant find all about mathematics.

Game, Article, Worksheet, Quizz, and more.