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Supergolden Ratio
In mathematics, the supergolden ratio is a geometrical aspect ratio, proportion, given by the unique real polynomial root, solution of the equation Its decimal expansion begins with . The name ''supergolden ratio'' is by analogy with the golden ratio, the positive solution of the equation Definition Three quantities are in the supergolden ratio if \frac =\frac =\frac The ratio is commonly denoted Substituting b=\psi c \, and a=\psi b =\psi^2 c \, in the middle fraction, \psi =\frac. It follows that the supergolden ratio is the unique real solution of the cubic equation \psi^3 -\psi^2 -1 =0. The Minimal polynomial (field theory), minimal polynomial for the reciprocal root is the depressed cubic x^ +x -1, thus the simplest solution with Cubic equation#Cardano's formula, Cardano's formula, \begin w_ &=\left( 1 \pm \frac \sqrt \right) /2 \\ 1 /\psi &=\sqrt[3] +\sqrt[3] \end or, using the Cubic equation#Trigonometric and hyperbolic solutions, hyperbolic sine, : 1 /\p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aspect Ratio
The aspect ratio of a geometry, geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangle is oriented as a "landscape format, landscape". The aspect ratio is most often expressed as two integer numbers separated by a colon (x:y), less commonly as a simple or decimal Fraction (mathematics), fraction. The values x and y do not represent actual widths and heights but, rather, the proportion between width and height. As an example, 8:5, 16:10, 1.6:1, and 1.6 are all ways of representing the same aspect ratio. In objects of more than two dimensions, such as hyperrectangles, the aspect ratio can still be defined as the ratio of the longest side to the shortest side. Applications and uses The term is most commonly used with reference to: * Graphic / image ** Aspect ratio (image), Image aspect ratio ** Display aspect ratio ** Pape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Integer
In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one. Almost integers may be considered interesting when they arise in some context in which they are unexpected. Almost integers relating to the golden ratio and Fibonacci numbers Some examples of almost integers are high powers of the golden ratio \phi=\frac\approx 1.618, for example: : \begin \phi^ & =\frac\approx 3571.00028 \\ pt\phi^ & =2889+1292\sqrt5 \approx 5777.999827 \\ pt\phi^ & =\frac\approx 9349.000107 \end The fact that these powers approach integers is non-coincidental, because the golden ratio is a Pisot–Vijayaraghavan number. The ratios of Fibonacci or Lucas numbers can also make almost integers, for instance: * \frac \approx 1242282009792667284144565908481.999999999999999999999999999999195 * \frac \approx 2010054515457065378082322433761.000000000000000000000000000000497 The above examples can be generalized by the following sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Narayana Pandita (mathematician)
Nārāyaṇa Paṇḍita () (1340–1400) was an Indian mathematician. Plofker writes that his texts were the most significant Sanskrit mathematics treatises after those of Bhaskara II, other than the Kerala school. He wrote the '' Ganita Kaumudi'' (lit. "Moonlight of mathematics") in 1356 about mathematical operations. The work anticipated many developments in combinatorics. Life and Works About his life, the most that is known is that: Narayana Pandit wrote two works, an arithmetical treatise called ''Ganita Kaumudi'' and an algebraic treatise called ''Bijaganita Vatamsa''. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled ''Karmapradipika'' (or ''Karma-Paddhati''). Although the ''Karmapradipika'' contains little original work, it contains seven different methods for squaring numbers, a contribution that is wholly original to the author, as well as contributions to algebra and magic squares. Narayana's other major work ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant-recursive Sequence
In mathematics, an sequence, infinite sequence of numbers s_0, s_1, s_2, s_3, \ldots is called constant-recursive if it satisfies an equation of the form :s_n = c_1 s_ + c_2 s_ + \dots + c_d s_, for all n \ge d, where c_i are constant (mathematics), constants. The equation is called a linear recurrence with constant coefficients, linear recurrence relation. The concept is also known as a linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, or a C-finite sequence. For example, the Fibonacci sequence :0, 1, 1, 2, 3, 5, 8, 13, \ldots, is constant-recursive because it satisfies the linear recurrence F_n = F_ + F_: each number in the sequence is the sum of the previous two. Other examples include the power of two sequence 1, 2, 4, 8, 16, \ldots, where each number is the sum of twice the previous number, and the square number sequence 0, 1, 4, 9, 16, 25, \ldots. All arithmetic progressions, all geometric progressions, and all polynomials are constant-recu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix
Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryotic organism's cells * Matrix (chemical analysis), the non-analyte components of a sample * Matrix (geology), the fine-grained material in which larger objects are embedded * Matrix (composite), the constituent of a composite material * Hair matrix, produces hair * Nail matrix, part of the nail in anatomy Technology * Matrix (mass spectrometry), a compound that promotes the formation of ions * Matrix (numismatics), a tool used in coin manufacturing * Matrix (printing), a mould for casting letters * Matrix (protocol), an open standard for real-time communication * Matrix (record production), or master, a disc used in the production of phonograph records ** Matrix number, of a gramophone record * Diode matrix, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eccentricity (mathematics)
In mathematics, the eccentricity of a Conic section#Eccentricity, conic section is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: * The eccentricity of a circle is 0. * The eccentricity of a non-circular ellipse is between 0 and 1. * The eccentricity of a parabola is 1. * The eccentricity of a hyperbola is greater than 1. * The eccentricity of a pair of Line (geometry), lines is \infty. Two conic sections with the same eccentricity are similarity (geometry), similar. Definitions Any conic section can be defined as the Locus (mathematics), locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the ''eccentricity'', commonly denoted as . The eccentricity can also be defined in terms of the intersection of a plane and a Cone (geometry), double-napped cone associated with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Lambda Function
In mathematics, the modular lambda function λ(τ)\lambda(\tau) is not a modular function (per the Wikipedia definition), but every modular function is a rational function in \lambda(\tau). Some authors use a non-equivalent definition of "modular functions". is a highly symmetric holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve ''X''(2). Over any point τ, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve \mathbb/\langle 1, \tau \rangle, where the map is defined as the quotient by the minus;1involution. The q-expansion, where q = e^ is the nome, is given by: : \lambda(\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \dots. By symmetrizing the lambda function under the can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
J-invariant
In mathematics, Felix Klein's -invariant or function is a modular function of weight zero for the special linear group \operatorname(2,\Z) defined on the upper half-plane of complex numbers. It is the unique such function that is holomorphic away from a simple pole at the cusp such that :j\big(e^\big) = 0, \quad j(i) = 1728 = 12^3. Rational functions of j are modular, and in fact give all modular functions of weight 0. Classically, the j-invariant was studied as a parameterization of elliptic curves over \mathbb, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine). Definition The -invariant can be defined as a function on the upper half-plane \mathcal=\, by :j(\tau) = 1728 \frac = 1728 \frac = 1728 \frac with the third definition implying j(\tau) can be expressed as a cube, also since 1728 = 12^3. The function cannot be continued analytically beyond the upper half-plane due to the natura ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weber Modular Function
In mathematics, the Weber modular functions are a family of three functions ''f'', ''f''1, and ''f''2,''f'', ''f''1 and ''f''2 are not modular functions (per the Wikipedia definition), but every modular function is a rational function in ''f'', ''f''1 and ''f''2. Some authors use a non-equivalent definition of "modular functions". studied by Heinrich Martin Weber. Definition Let q = e^ where ''τ'' is an element of the upper half-plane. Then the Weber functions are :\begin \mathfrak(\tau) &= q^\prod_(1+q^) = \frac = e^\frac,\\ \mathfrak_1(\tau) &= q^\prod_(1-q^) = \frac,\\ \mathfrak_2(\tau) &= \sqrt2\, q^\prod_(1+q^)= \frac. \end These are also the definitions in Duke's paper ''"Continued Fractions and Modular Functions"''.https://www.math.ucla.edu/~wdduke/preprints/bams4.pdf ''Continued Fractions and Modular Functions'', W. Duke, pp 22-23 The function \eta(\tau) is the Dedekind eta function and (e^)^ should be interpreted as e^. The descriptions as \eta quotients immediately imp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Eta Function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory. Definition For any complex number with , let ; then the eta function is defined by, :\eta(\tau) = e^\frac \prod_^\infty \left(1-e^\right) = q^\frac \prod_^\infty \left(1 - q^n\right) . Raising the eta equation to the 24th power and multiplying by gives :\Delta(\tau)=(2\pi)^\eta^(\tau) where is the modular discriminant. The presence of 24 can be understood by connection with other occurrences, such as in the 24-dimensional Leech lattice. The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it. The eta function satisfies the functional equations :\begin \eta(\tau+1) &=e^\frac\eta(\tau),\\ \eta\left(-\frac\right) &= \sqrt\, \eta(\tau).\, \end In the second equation the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often denoted by O_K or \mathcal O_K. Since any integer belongs to K and is an integral element of K, the ring \mathbb is always a subring of O_K. The ring of integers \mathbb is the simplest possible ring of integers. Namely, \mathbb=O_ where \mathbb is the field of rational numbers. And indeed, in algebraic number theory the elements of \mathbb are often called the "rational integers" because of this. The next simplest example is the ring of Gaussian integers \mathbb /math>, consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field \mathbb(i) of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Field
In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 and 1. If d>0, the corresponding quadratic field is called a real quadratic field, and, if d<0, it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a Field extension, subfield of the field of the real numbers. Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important. Ring of integers Discriminant For a nonzero square free integer , the Discriminant of an algebraic number field, discriminant of the quadratic field is |
