Jacobi's Theorem (other)

	Jacobi's Theorem (other) Jacobi's theorem can refer to: Maximum power theorem, in electrical engineering The result that the determinant of skew-symmetric matrices with odd size vanishes, see skew-symmetric matrix Jacobi's four-square theorem Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer ''n'' can be represented as the sum of four squares. History The theorem was proved in 1834 by Carl Gustav Jakob Jacobi. Theorem Two representatio ..., in number theory Jacobi's theorem (geometry), on concurrent lines associated with any triangle * Jacobi's theorem on the normal indicatrix, in differential geometry * Jacobi's theorem on conjugate points, in differential geometry {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Maximum Power Theorem In electrical engineering, the maximum power transfer theorem states that, to obtain ''maximum'' external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Moritz von Jacobi published the maximum power (transfer) theorem around 1840; it is also referred to as "Jacobi's law". The theorem results in maximum ''power'' transfer from the power source to the load, and not maximum '' efficiency'' of useful power out of total power consumed. If the load resistance is made larger than the source resistance, then efficiency increases (since a higher percentage of the source power is transferred to the load), but the ''magnitude'' of the load power decreases (since the total circuit resistance increases). If the load resistance is made smaller than the source resistance, then efficiency decreases (since most of the power ends up being dissipated in the source). Although the total power ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Skew-symmetric Matrix In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a_ denotes the entry in the i-th row and j-th column, then the skew-symmetric condition is equivalent to Example The matrix :A = \begin 0 & 2 & -45 \\ -2 & 0 & -4 \\ 45 & 4 & 0 \end is skew-symmetric because : -A = \begin 0 & -2 & 45 \\ 2 & 0 & 4 \\ -45 & -4 & 0 \end = A^\textsf . Properties Throughout, we assume that all matrix entries belong to a field \mathbb whose characteristic is not equal to 2. That is, we assume that , where 1 denotes the multiplicative identity and 0 the additive identity of the given field. If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix. * The sum of two skew-symmetric matrices is skew-symmetric. * A scala ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Jacobi's Four-square Theorem Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer ''n'' can be represented as the sum of four squares. History The theorem was proved in 1834 by Carl Gustav Jakob Jacobi. Theorem Two representations are considered different if their terms are in different order or if the integer being squared (not just the square) is different; to illustrate, these are three of the eight different ways to represent 1: : \begin & 1^2 + 0^2 + 0^2 + 0^2 \\ & 0^2 + 1^2 + 0^2 + 0^2 \\ & (-1)^2 + 0^2 + 0^2 + 0^2. \end The number of ways to represent n as the sum of four squares is eight times the sum of the divisors of ''n'' if ''n'' is odd and 24 times the sum of the odd divisors of ''n'' if ''n'' is even (see divisor function), i.e. : r_4(n)=\begin8\sum\limits_m&\textn\text\\2pt24\sum\limits_m&\textn\text. \end Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e. :r_4(n)=8\sum_m. We may also write this as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Jacobi Point In plane geometry, a Jacobi point is a point in the Euclidean plane determined by a triangle ''ABC'' and a triple of angles ''α'', ''β'', and ''γ''. This information is sufficient to determine three points ''X'', ''Y'', and ''Z'' such that ∠''ZAB'' = ∠''YAC'' = ''α'', ∠''XBC'' = ∠''ZBA'' = ''β'', and ∠''YCA'' = ∠''XCB'' = ''γ''. Then, by a theorem of , the lines ''AX, BY,'' and ''CZ'' are concurrent,Glenn T. Vickers, "Reciprocal Jacobi Triangles and the McCay Cubic", ''Forum Geometricorum 15'', 2015, 179–183. http://forumgeom.fau.edu/FG2015volume15/FG201518.pdfGlenn T. Vickers, "The 19 Congruent Jacobi Triangles", ''Forum Geometricorum'' 16, 2016, 339–344. http://forumgeom.fau.edu/FG2016volume16/FG201642.pdf at a point ''N'' called the Jacobi point. The Jacobi point is a generalization of the Fermat point, which is obtained by letting ''α'' = ''β'' = ''γ'' = 60° and triangle ''ABC'' having no angle being greater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]