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A symbol consisting of three rational numbers that can be used to describe uniform polyhedra based on how a point in a spherical triangle can be selected so as to trace the vertices of regular polygonal faces. For example, the Wythoff symbol for the tetrahedron is . There are four types of Wythoff symbols, , , and , and one exceptional symbol, (which is used for the great dirhombicosidodecahedron).The meaning of the bars may be summarized as follows (Wenninger 1989, p. 10; Messer 2002). Consider a spherical triangle whose angles are , , and . 1. : is a special point within that traces snub polyhedra by even reflections. 2. (or ): is the vertex . 3. (or ): lies on the arc and the bisector of the opposite angle . 4. (or any permutation of the three letters): is the incenter of the triangle . Some special cases in terms of Schläfli symbolsare(1)(2)(3)(4)(5)(6)..

Two lines in two-dimensional Euclidean space are said to be parallel if they do not intersect. In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. Lines in three-space that are not parallel but do not intersect are called skew lines.If lines and are parallel, the notation is used.In a non-Euclidean geometry, the concept of parallelism must be modified from its intuitive meaning. This is accomplished by changing the so-called parallel postulate. While this has counterintuitive results, the geometries so defined are still completely self-consistent.In a triangle , a triangle median bisects all segments parallel to a given side (Honsberger 1995, p. 87).

For a single variate having a distribution with known population mean , the population variance , commonly also written , is defined as(1)where is the population mean and denotes the expectation value of . For a discrete distribution with possible values of , the population variance is therefore(2)whereas for a continuous distribution,it is given by(3)The variance is therefore equal to the second central moment .Note that some care is needed in interpreting as a variance, since the symbol is also commonly used as a parameter related to but not equivalent to the square root of the variance, for example in the log normal distribution, Maxwell distribution, and Rayleigh distribution.If the underlying distribution is not known, then the samplevariance may be computed as(4)where is the sample mean.Note that the sample variance defined above is not an unbiased estimator for the population variance . In order to obtain an unbiased estimator for..

A function related to the divisor function , also sometimes called Ramanujan's tau function. It is defined via the Fourier series of the modular discriminant for , where is the upper half-plane, by(1)(Apostol 1997, p. 20). The tau function is also given by the Cauchyproduct(2)(3)where is the divisor function (Apostol 1997, pp. 24 and 140), , and .The tau function has generating function(4)(5)(6)(7)(8)where is a q-Pochhammer symbol. The first few values are 1, , 252, , 4830, ... (OEIS A000594). The tau function is given by the Wolfram Language function RamanujanTau[n].The series(9)is known as the tau Dirichlet series.Lehmer (1947) conjectured that for all , an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of for which this condition holds.reference3316799Lehmer (1947)214928639999Lehmer..

Symbols used to identify irreducible representations of groups: singly degenerate state which is symmetric with respect to rotation about the principal axis, singly degenerate state which is antisymmetric with respect to rotation about the principal axis, doubly degenerate, triply degenerate, (gerade, symmetric) the sign of the wavefunction does not change on inversion through the center of the atom, (ungerade, antisymmetric) the sign of the wavefunction changes on inversion through the center of the atom, (on or ) the sign of the wavefunction does not change upon rotation about the center of the atom, (on or ) the sign of the wavefunction changes upon rotation about the center of the atom, ' = symmetric with respect to a horizontal symmetry plane , " = antisymmetric with respect to a horizontal symmetry plane . ..

The operation of subtraction, i.e., minus . The operation is denoted . The minus sign "" is also used to denote a negative number, i.e., .

The smallest value of a set, function, etc. The minimum value of a set of elements is denoted or , and is equal to the first element of a sorted (i.e., ordered) version of . For example, given the set , the sorted version is , so the minimum is 1. The maximum and minimum are the simplest order statistics.The minimum value of a variable is commonly denoted (cf. Strang 1988, pp. 286-287 and 301-303) or (Golub and Van Loan 1996, p. 84). In this work, the convention is used.The minimum of a set of elements is implemented in the Wolfram Language as Min[list] and satisfies the identities(1)(2)A continuous function may assume a minimum at a single point or may have minima at a number of points. A global minimum of a function is the smallest value in the entire range of the function, while a local minimum is the smallest value in some local neighborhood.For a function which is continuous at a point , a necessary but not sufficient condition for to have a local..

A large number defined as where the circle notation denotes " in squares," and triangles and squares are expanded in terms of Steinhaus-Moser notation (Steinhaus 1999, pp. 28-29). Here, the typographical error of Steinhaus has been corrected.

Min Max Re Im The sign of a real number, also called sgn or signum, is for a negative number (i.e., one with a minus sign ""), 0 for the number zero, or for a positive number (i.e., one with a plus sign ""). In other words, for real ,(1)For real , this can be written(2)and satisfies(3) for real can also be defined as(4)where is the Heaviside step function.The sign function is implemented in the Wolfram Language for real as Sign[x]. For nonzero complex numbers, Sign[z] returns , where is the complex modulus of . can also be interpreted as an unspecified point on the unit circle in the complex plane (Rich and Jeffrey 1996).

Two geometric figures are said to exhibit geometric congruence (or "be geometrically congruent") iff one can be transformed into the other by an isometry (Coxeter and Greitzer 1967, p. 80). This relationship is written . (Unfortunately, the symbol is also used to denote an isomorphism.)

A multiplicative factor (usually indexed) such as one of the constants in the polynomialIn this polynomial, the monomials are , , ..., , and 1, and the single variable is .

The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon and dodecahedron. It is denoted , or sometimes .The designations "phi" (for the golden ratio conjugate ) and "Phi" (for the larger quantity ) are sometimes also used (Knott), although this usage is not necessarily recommended.The term "golden section" (in German, goldener Schnitt or der goldene Schnitt) seems to first have been used by Martin Ohm in the 1835 2nd edition of his textbook Die Reine Elementar-Mathematik (Livio 2002, p. 6). The first known use of this term in English is in James Sulley's 1875 article on aesthetics in the 9th edition of the Encyclopedia Britannica. The symbol ("phi") was apparently first used by Mark Barr at the beginning of the 20th century in commemoration..

A piecewise function is a function that is defined on a sequence of intervals. Acommon example is the absolute value,(1)Piecewise functions are implemented in the Wolfram Language as Piecewise[val1, cond1, val2, cond2, ...].Additional piecewise functions include the Heaviside step function, rectangle function, and triangle function.Semicolons and commas are sometimes used at the end of either the left or the right column, with particular usage apparently depending on the author. The words "if" and "for" are sometimes used in the right column, as is "otherwise" for the final (default) case.For example, Knuth (1996, pp. 175 and 180) uses the notations(2)(3)(4)both with and without the left-column commas. Similarly, Arfken (1985, pp. 488-489) uses(5)which lacks semicolons but only sometimes lacks right-column commas.In this work, commas and semicolons are not used...

The power tower of order is defined as(1)where is Knuth up-arrow notation (Knuth 1976), which in turn is defined by(2)together with(3)(4)Rucker (1995, p. 74) uses the notation(5)and refers to this operation as "tetration."A power tower can be implemented in the WolframLanguage as PowerTower[a_, k_Integer] := Nest[Power[a, #]&, 1, k]or PowerTower[a_, k_Integer] := Power @@ Table[a, {k}]The following table gives values of for , 2, ... for small .OEIS1A0000271, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...2A0003121, 4, 27, 256, 3125, 46656, ...3A0024881, 16, 7625597484987, ...41, 65536, ...The following table gives for , 2, ... for small .OEIS1A0000121, 1, 1, 1, 1, 1, ...2A0142212, 4, 16, 65536, , ...3A0142223, 27, 7625597484987, ...44, 256, , ...Consider and let be defined as(6)(Galidakis 2004). Then for , is entire with series expansion:(7)Similarly, for , is analytic for in the domain of the principal branch of , with series expansion:(8)For..

The conjugate transpose of an matrix is the matrix defined by(1)where denotes the transpose of the matrix and denotes the conjugate matrix. In all common spaces (i.e., separable Hilbert spaces), the conjugate and transpose operations commute, so(2)The symbol (where the "H" stands for "Hermitian") gives official recognition to the fact that for complex matrices, it is almost always the case that the combined operation of taking the transpose and complex conjugate arises in physical or computation contexts and virtually never the transpose in isolation (Strang 1988, pp. 220-221).The conjugate transpose of a matrix is implemented in the Wolfram Language as ConjugateTranspose[A].The conjugate transpose is also known as the adjoint matrix, adjugate matrix, Hermitian adjoint, or Hermitian transpose (Strang 1988, p. 221). Unfortunately, several different notations are in use as summarized in the..

The largest value of a set, function, etc. The maximum value of a set of elements is denoted or , and is equal to the last element of a sorted (i.e., ordered) version of . For example, given the set , the sorted version is , so the maximum is 5. The maximum and minimum are the simplest order statistics.The maximum value of a variable is commonly denoted (Strang 1988, pp. 286-287 and 301-303) or (Golub and Van Loan 1996, p. 74). In this work, the convention is used.The maximum of a set of elements is implemented in the Wolfram Language as Max[list] and satisfies the identities(1)(2)Definite integrals include(3)(4)A continuous function may assume a maximum at a single point or may have maxima at a number of points. A global maximum of a function is the largest value in the entire range of the function, and a local maximum is the largest value in some local neighborhood.For a function which is continuous at a point , a necessary but not sufficient condition..

A matrix is a concise and useful way of uniquely representing and working with linear transformations. In particular, every linear transformation can be represented by a matrix, and every matrix corresponds to a unique linear transformation. The matrix, and its close relative the determinant, are extremely important concepts in linear algebra, and were first formulated by Sylvester (1851) and Cayley.In his 1851 paper, Sylvester wrote, "For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of lines and columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number , and selecting at will lines and columns, the squares corresponding of th order." Because Sylvester was interested in the determinant formed from the rectangular array of number and not the array itself (Kline..

There are at least two meanings on the word congruent in mathematics. Two geometric figures are said to be congruent if one can be transformed into the other by an isometry (Coxeter and Greitzer 1967, p. 80). This relationship, called geometric congruence, is written . (Unfortunately, the symbol is also used to denote an isomorphism.)A number is said to be congruent to modulo if ( divides ).

Given a series of the formthe notation is used to indicate the coefficient (Sedgewick and Flajolet 1996). This corresponds to the Wolfram Language functions Coefficient[A[z], z, k] and SeriesCoefficient[series, k].

(1)The Roman factorial arises in the definition of the harmonic logarithm and Roman coefficient. It obeys the identities(2)(3)(4)where(5)and(6)

A generalization of the binomial coefficientwhose notation was suggested by Knuth,(1)where is a Roman factorial. The above expression is read "Roman choose ." Whenever the binomial coefficient is defined (i.e., or ), the Roman coefficient agrees with it. However, the Roman coefficients are defined for values for which the binomial coefficients are not, e.g.,(2)(3)where(4)The Roman coefficients also satisfy properties like those of the binomialcoefficient,(5)(6)an analog of Pascal's formula(7)and a curious rotation/reflection law due to Knuth(8)(Roman 1992).

A notation for large numbers due to Steinhaus (1999). In circle notation, is defined as in squares, where numbers written inside squares (and triangles) are interpreted in terms of Steinhaus-Moser notation. The particular number known as the mega is then defined as follows (correcting the typographical error of Steinhaus).

There are a number of functions in mathematics commonly denoted with a Greek letter lambda. Examples of one-variable functions denoted with a lower case lambda include the Carmichael functions, Dirichlet lambda function, elliptic lambda function, and Liouville function. Examples of one-variable functions denoted with an upper case lambda include the Mangoldt function and the lambda function defined by Jahnke and Emden (1945).The triangle function, illustrated above, is commonly denoted .The lambda function defined by Jahnke and Emden (1945) is(1)where is a Bessel function of the first kind and is the gamma function. , and taking gives the special case(2)where is the jinc function.A two-variable lambda function is defined as(3)where is the gamma function (McLachlan et al. 1950, p. 9; Prudnikov et al. 1990, p. 798; Gradshteyn and Ryzhik 2000, p. 1109)...

The symbol used to indicate a root is called a radical, or sometimes a surd. The expression is therefore read " radical ," or "the nth root of ." In the radical symbol, the horizonal line is called the vinculum, the quantity under the vinculum is called the radicand, and the quantity written to the left is called the index.In general, the use of roots is equivalent to the use of fractional exponentsas indicated by the identity(1)a more generalized form of the standard(2)The special case is written and is called the square root of . is called the cube root.Some interesting radical identities are due to Ramanujan, and include the equivalent forms(3)and(4)Another such identity is(5)

The Cauchy principal value of a finite integral of a function about a point with is given by(Henrici 1988, p. 261; Whittaker and Watson 1990, p. 117; Bronshtein and Semendyayev 1997, p. 283). Similarly, the Cauchy principal value of a doubly infinite integral of a function is defined byThe Cauchy principal value is also known as the principal value integral (Henrici 1988, p. 261), finite part (Vladimirov 1971), or partie finie (Vladimirov 1971).The Cauchy principal value of an integral having no nonsimple poles can be computed in the Wolfram Language using Integrate[f, x, a, b, PrincipalValue -> True]. Cauchy principal values of functions with possibly nonsimple poles can be computed numerically using the "CauchyPrincipalValue" method in NIntegrate.Cauchy principal values are important in the theory of generalized functions, where they allow extension of results to .Cauchy principal values..

"Q.E.D." (sometimes written "QED") is an abbreviation for the Latin phrase "quod erat demonstrandum" ("that which was to be demonstrated"), a notation which is often placed at the end of a mathematical proof to indicate its completion. Several symbols are occasionally used as synonyms for Q.E.D. These include a filled square (Unicode U+220E, as used in Mathematics Magazine and American Mathematical Monthly), a filled rectangle (Knuth 1997, pp. 3 and 39), or an empty square .

A quantity is said to be less than if is smaller than , written . If is less than or equal to , the relationship is written . In the Wolfram Language, this is denoted Less[a, b], or a < b.If is much less than , this is written . Statements involving greater than and less than symbols are called inequalities.

A quantity is said to be greater than if is larger than , written . If is greater than or equal to , the relationship is written . In the Wolfram Language, this is denoted Greater[a, b], or a > b.If is much greater than , this is written . Statements involving greater than and less than symbols are called inequalities.

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