Gauss–Seidel method In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a linear system of equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel, and is similar to the Jacobi method. Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either diagonally dominant, or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823. A publication was not delivered before 1874 by Seidel. Description The Gauss–Seidel method is an iterative technique for solving a square system of n linear equations with unknown x: It is defined by the iteration where the matrix A is decomposed into a lower triangular component , and a strictly upper triangular component U: In more detail, write out A, x and b in their components: and
Stereographic projection 3D illustration of a stereographic projection from the north pole onto a plane below the sphere Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography. In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a stereographic net, shortened to stereonet or Wulff net. History Illustration by Rubens for "Opticorum libri sex philosophis juxta ac mathematicis utiles", by François d'Aiguillon. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians. Definition This section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Properties with E.
Riemann zeta function , which converges when the real part of s is greater than 1. More general representations of ζ(s) for all s are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. This function, as a function of a real argument, was introduced and studied by Leonhard Euler in the first half of the eighteenth century without using complex analysis, which was not available at that time. Bernhard Riemann in his article "On the Number of Primes Less Than a Given Magnitude" published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers. The values of the Riemann zeta function at even positive integers were computed by Euler. Definition Bernhard Riemann's article on the number of primes below a given magnitude. Specific values A058303).
Avogadro constant Previous definitions of chemical quantity involved Avogadro's number, a historical term closely related to the Avogadro constant but defined differently: Avogadro's number was initially defined by Jean Baptiste Perrin as the number of atoms in one gram-molecule of hydrogen. It was later redefined as the number of atoms in 12 grams of the isotope carbon-12 and still later generalized to relate amounts of a substance to their molecular weight. For instance, to a first approximation, 1 gram of hydrogen, which has a mass number of 1 (atomic number 1), has 6.022×1023 hydrogen atoms. Similarly, 12 grams of carbon 12, with the mass number of 12 (atomic number 6), has the same number of carbon atoms, 6.022×1023. Avogadro's number is a dimensionless quantity and has the numerical value of the Avogadro constant given in base units. The Avogadro constant is fundamental to understanding both the makeup of molecules and their interactions and combinations. History [dubious ] Measurement
'Periodic table of shapes' to give a new dimension to math Mathematicians are creating their own version of the periodic table that will provide a vast directory of all the possible shapes in the universe across three, four and five dimensions, linking shapes together in the same way as the periodic table links groups of chemical elements. The three-year project should provide a resource that mathematicians, physicists and other scientists can use for calculations and research in a range of areas, including computer vision, number theory, and theoretical physics. The researchers, from Imperial College London and institutions in Australia, Japan and Russia, are aiming to identify all the shapes across three, four and five dimensions that cannot be divided into other shapes. As these building block shapes are revealed, the mathematicians will work out the equations that describe each shape and through this, they expect to develop a better understanding of the shapes' geometric properties and how different shapes are related to one another.
List of mathematical symbols When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations a different convention may be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and in TeX, as an image. Guide This list is organized by symbol type and is intended to facilitate finding an unfamiliar symbol by its visual appearance. Basic symbols: Symbols widely used in mathematics, roughly through first-year calculus. Basic symbols Symbols based on equality sign Symbols that point left or right Brackets Other non-letter symbols
arXiv.org e-Print archive Warning! High Dimensions Ahead ← Inductio Ex Machina Machine learning is often framed as finding surfaces (usually planes) that separate or fit points in some high dimensional space. Two and three dimensional diagrams are often used as an aid to intuition when thinking about these higher dimensional spaces. The following result (related to me by Marcus Hutter) shows that our low dimensional intuition can easily lead us astray. Consider the diagram in Figure 1. The larger grey spheres have radius 1 and are each touching exactly two other grey spheres. Pythagoras tells us that and so . Figure 1: The blue circle in the middle touches each of the four identical grey circles. We can move to three dimensions and set up an analogue of the two dimensional situation by arranging eight spheres of radius 1 so they are touching exactly three other grey spheres. As in the case with circles, we can place a blue sphere inside the eight grey spheres so that it touches all of them. Figure 2: The blue sphere touches each of the eight identical grey spheres.
Imaginary number An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where a and b are called, respectively, the real part and the imaginary part of the complex number. Imaginary numbers can therefore be thought of as complex numbers whose real part is zero. The name "imaginary number" was coined in the 17th century as a derogatory term, as such numbers were regarded by some as fictitious or useless. The term "imaginary number" now means simply a complex number with a real part equal to 0, that is, a number of the form bi. History An illustration of the complex plane. Although Greek mathematician and engineer Heron of Alexandria is noted as the first to have conceived these numbers, Rafael Bombelli first set down the rules for multiplication of complex numbers in 1572. In 1843 a mathematical physicist, William Rowan Hamilton, extended the idea of an axis of imaginary numbers in the plane to a three-dimensional space of quaternion imaginaries. See also
Complex plane Geometric representation of z and its conjugate z̅ in the complex plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z. The angle φ is the argument of z. In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. Notational conventions In complex analysis the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts, like this: for example: z = 4 + 5i, where x and y are real numbers, and i is the imaginary unit. In the Cartesian plane the point (x, y) can also be represented in polar coordinates as where Here |z| is the absolute value or modulus of the complex number z; θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π; and the last equality (to |z|eiθ) is taken from Euler's formula. Stereographic projections Cutting the plane
Dave's short course in trigonometry Table of Contents Who should take this course? Trigonometry for you Your background How to learn trigonometry Applications of trigonometry Astronomy and geography Engineering and physics Mathematics and its applications What is trigonometry? Trigonometry as computational geometry Angle measurement and tables Background on geometry The Pythagorean theorem An explanation of the Pythagorean theorem Similar triangles Angle measurement The concept of angle Radians and arc length Exercises, hints, and answers About digits of accuracy Chords What is a chord? Ptolemy’s sum and difference formulas Ptolemy’s theorem The sum formula for sines The other sum and difference formulas Summary of trigonometric formulas Formulas for arcs and sectors of circles Formulas for right triangles Formulas for oblique triangles Formulas for areas of triangles Summary of trigonometric identities More important identities Less important identities Truly obscure identities About the Java applet.