Vandermonde matrix

In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row, i.e., an m × n matrix

or

for all indices i and j. (Some authors use the transpose of the above matrix.)

The determinant of a square Vandermonde matrix (where m = n) can be expressed as:

This is called the Vandermonde determinant or Vandermonde polynomial. If all the numbers α_i are distinct, then it is non-zero.

Cramer's rule(Linear Algebra)

In linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution. The solution is expressed in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule in his 1750 Introduction à l'analyse des lignes courbes algébriques (Introduction to the analysis of algebraic curves), although Colin Maclaurin also published the method in his 1748 Treatise of Algebra (and probably knew of the method as early as 1729).


General Case

onsider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows:

where the n by n matrix A has a nonzero determinant, and the vector  is the column vector of the variables.

Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns are given by:

where A_i is the matrix formed by replacing the ith column of A by the column vector b.

The rule holds for systems of equations with coefficients and unknowns in any field, not just in the real numbers. It has recently been shown that Cramer's rule can be implemented in O(n³) time, which is comparable to more common methods of solving systems of linear equations, such as Gaussian elimination.

Hypotrochoid

A hypotrochoid is a curve traced out by a point "attached" to a smaller circle rolling around inside a fixed larger circle. In this example, the hypotrochoid is the red curve that is traced out by the red point 5 units from the center of the black circle of radius 3 as it rolls around inside the blue circle of radius 5. Both hypotrochoids and epitrochoids can be created using the Spirograph drawing toy.

Basel problem-Euler's original proof

The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper On the Number of Primes Less Than a Given Magnitude, in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.

The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series:

The series is approximately equal to 1.644934 . The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be  and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, and it was not until 1741 that he was able to produce a truly rigorous proof.

Law of total variance

In probability theory, the law of total variance, Eve's Law, or variance decomposition formula states that if X and Y are random variables on the same probability space, and the variance of Y is finite, then

In language perhaps better known to statisticians than to probabilists, the two terms are the "unexplained" and the "explained component of the variance" (cf. fraction of variance unexplained, explained variation).

The nomenclature in this article's title parallels the phrase law of total probability. Some writers on probability call this the "conditional variance formula" or use other names.

Note that the conditional expected value E( Y | X ) is a random variable in its own right, whose value depends on the value of X. Notice that the conditional expected value of Y given theevent X = y is a function of y (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!). If we write E( Y | X = y ) = g(y) then the random variable E( Y | X ) is just g(X). Similar comments apply to the conditional variance.

How sunflowers teach math

Eons ago, plants worked out the secret of arranging equal-size seeds in an ever-expanding pattern around a central point so that regardless of the size of the arrangement, the seeds pack evenly. The sunflower is a well-known example of such a “spiral phyllotaxis” pattern:

It’s really magical that this works at all, since the spatial relationship of each seed to its neighbors is unique, changing constantly as the pattern expands outwardly—unlike, say, the cells in a honeycomb, which are all equivalent. I wondered if the same magic could be applied to surfaces that are not flat, like spheres, toruses, or wine glasses. It’s an interesting question from an aesthetic point of view, but also a practical one: the answer has applications in space exploration and modern architecture.

To reproduce the flat sunflower pattern mathematically, you need to know three secrets of the arrangement:

1. Seeds spiral outward from the center, each positioned at a fixed angle relative to its predecessor.
2. The fixed angle is the golden angle, γ = 2π(1 – 1/Φ), where Φ is the golden ratio.
3. The ith seed in the pattern is placed at a distance from the center proportional to the square root of i.                           

The following photos simulate the spiral phyllotaxis pattern on mathematical objects..

Soddy's Hexlet(Generalization of Steiner's chain)

The simplest generalization of a Steiner chain is to allow the given circles to touch or intersect one another. In the former case, this corresponds to a Pappus chain, which has an infinite number of circles.

Soddy's hexlet is a three-dimensional generalization of a Steiner chain of six circles. The centers of the six spheres (the hexlet) travel along the same ellipse as do the centers of the corresponding Steiner chain. The envelope of the hexlet spheres is a Dupin cyclide, the inversion of a torus. The six spheres are not only tangent to the inner and outer sphere, but also to two other spheres, centered above and below the plane of the hexlet centers.

Multiple rings of Steiner chains are another generalization. An ordinary Steiner chain is obtained by inverting an annular chain of tangent circles bounded by two concentric circles. This may be generalized to inverting three or more concentric circles that sandwich annular chains of tangent circles.

Hierarchical Steiner chains are yet another generalization. If the two given circles of an ordinary Steiner chain are nested, i.e., if one lies entirely within the other, then the larger given circle circumscribes the Steiner-chain circles. In a hierarchical Steiner chain, each circle of a Steiner chain is itself the circumscribing given circle of another Steiner chain within it; this process may be repeated indefinitely, forming a fractal.

Steiner chain

In geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles (blue and red in Figure 1), where n is finite and each circle in the chain is tangent to the previous and next circles in the chain. In the usualclosed Steiner chains, the first and last (n^th) circles are also tangent to each other; by contrast, in open Steiner chains, they need not be. The given circles α and β do not intersect, but otherwise are unconstrained; the smaller circle may lie completely inside or outside of the larger circle. In these cases, the centers of Steiner-chain circles lie on an ellipse or a hyperbola, respectively.

Green's Theorem Examples

Green's theorem

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is the two-dimensional special case of the more general Stokes' theorem, and is named after British mathematician George Green.

Let C be a positively oriented, piecewise smooth, simple closed curve in the plane ², and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then

For positive orientation, an arrow pointing in the counterclockwise direction may be drawn in the small circle in the integral symbol.

In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.

Examples of Epsilon-Delta Proof

(ε, δ)-definition of limit

In calculus, the (ε, δ)-definition of limit ("epsilon-delta definition of limit") is a formalization of the notion of limit. It was given by Bernard Bolzano, in 1817, and in a less precise form by Augustin-Louis Cauchy.

Fourier-Motzkin example

Fourier–Motzkin elimination

Fourier–Motzkin elimination, FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can look for both real and integer solutions. Its computational complexity is double-exponential.

Elimination (or ∃-elimination) of variables V from a system of relations (here, linear inequalities) consists in creating another system of the same kind, but without the variables V, such that both systems have the same solutions over the remaining variables.

If one eliminates all variables from a system of linear inequalities, then one obtains a system of constant inequalities, which can be trivially decided to be true or false, such that this system has solutions (is true) if and only if the original system has solutions. As a consequence, elimination of all variables can be used to detect whether a system of inequalities has solutions or not.

Taylor Series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.

It is common practice to approximate a function by using a finite number of terms of its Taylor series. Taylor's theorem gives quantitative estimates on the error in this approximation. Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial. The Taylor series of a function is the limit of that function's Taylor polynomials, provided that the limit exists. A function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an open interval (or a disc in the complex plane) is known as an analytic function.

Fürstenberg's proof of the infinitude of primes

In number theory, Hillel Fürstenberg's proof of the infinitude of primes is a celebrated topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences. Unlike Euclid's classical proof, Fürstenberg's is a proof by contradiction. The proof was published in 1955 in the American Mathematical Monthly while Fürstenberg was still an undergraduate student at Yeshiva University.

Pigeonhole Principle

In mathematics and computer science, the pigeonhole principle states that if n items are put into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. This theorem is exemplified in real-life by truisms like "there must be at least two left gloves or two right gloves in a group of three gloves". It is an example of a counting argument, and despite seeming intuitive it can be used to demonstrate possibly unexpected results; for example, that two people in London have the same number of hairs on their heads.

The first formalization of the idea is believed to have been made by Johann Dirichlet in 1834 under the name Schubfachprinzip ("drawer principle" or "shelf principle"). For this reason it is also commonly called Dirichlet's box principle, Dirichlet's drawer principle or simply "Dirichlet principle"—a name that could also refer to the minimum principle for harmonic functions. The original "drawer" name is still in use in French ("principe des tiroirs"), Italian ("principio dei cassetti") and German ("Schubfachprinzip").

Though the most straightforward application is to finite sets (such as pigeons and boxes), it is also used with infinite sets that cannot be put into one-to-one correspondence. To do so requires the formal statement of the pigeonhole principle, which is "there does not exist an injective function on finite sets whose codomain is smaller than its domain". Advanced mathematical proofs like Siegel's lemma build upon this more general concept.