The term "cylinder" has a number of related meanings. In its most general usage, the word "cylinder" refers to a solid bounded by a closed generalized cylinder (a.k.a. cylindrical surface) and two parallel planes (Kern and Bland 1948, p. 32; Harris and Stocker 1998, p. 102). A cylinder of this sort having a polygonal base is therefore a prism (Zwillinger 1995, p. 308). Harris and Stocker (1998, p. 103) use the term "general cylinder" to refer to the solid bounded a closed generalized cylinder.Unfortunately, the term "cylinder" is commonly used not only to refer to the solid bounded by a cylindrical surface, but to the cylindrical surface itself (Zwillinger 1995, p. 311). To make matters worse, according to topologists, a cylindrical surface is not even a true surface, but rather a so-called surface with boundary (Henle 1994, pp. 110 and 129).As if this were..
For vectors and in , the cross product in is defined by(1)(2)where is a right-handed, i.e., positively oriented, orthonormal basis. This can be written in a shorthand notation that takes the form of a determinant(3)where , , and are unit vectors. Here, is always perpendicular to both and , with the orientation determined by the right-hand rule.Special cases involving the unit vectors in three-dimensionalCartesian coordinates are given by(4)(5)(6)The cross product satisfies the general identity(7)Note that is not a usual polar vector, but has slightly different transformation properties and is therefore a so-called pseudovector (Arfken 1985, pp. 22-23). Jeffreys and Jeffreys (1988) use the notation to denote the cross product.The cross product is implemented in the Wolfram Language as Cross[a, b].A mathematical joke asks, "What do you get when you cross a mountain-climber with a mosquito?" The answer is, "Nothing:..
In mathematics, a small positive infinitesimal quantity, usually denoted or , whose limit is usually taken as .The late mathematician P. Erdős also used the term "epsilons" to refer to children (Hoffman 1998, p. 4).
In French and German usage, one milliard equals .American usage does not have a number called the milliard, instead using the term billion to denote . British usage, while formerly using "milliard," has in recent years adopted the American convention (Mish 2003, p. 852). This constitutes a fortunate development for standardization of terminology, albeit a somewhat regrettable development from the point of view that the (former) British convention for representing large numbers is simpler and more logical than the American one.A terrible mathematical joke asks "What American President, with cities in California and Utah named after him, is associated in France and Germany with ?" Answer: Milliard Fillmore (J. vos Post, pers. comm., Apr. 27, 2006).
Bertrand's postulate, also called the Bertrand-Chebyshev theorem or Chebyshev's theorem, states that if , there is always at least one prime between and . Equivalently, if , then there is always at least one prime such that . The conjecture was first made by Bertrand in 1845 (Bertrand 1845; Nagell 1951, p. 67; Havil 2003, p. 25). It was proved in 1850 by Chebyshev (Chebyshev 1854; Havil 2003, p. 25; Derbyshire 2004, p. 124) using non-elementary methods, and is therefore sometimes known as Chebyshev's theorem. The first elementary proof was by Ramanujan, and later improved by a 19-year-old Erdős in 1932.A short verse about Bertrand's postulate states, "Chebyshev said it, but I'll say it again; There's always a prime between and ." While commonly attributed to Erdős or to some other Hungarian mathematician upon Erdős's youthful re-proof the theorem (Hoffman 1998), the quote is actually..
A rational number expressed in the form (in-line notation) or (traditional "display" notation), where is called the numerator and is called the denominator. When written in-line, the slash "/" between numerator and denominator is called a solidus.A mathematical joke states that 4/3 of people don't understand fractions.A proper fraction is a fraction such that , and a reduced fraction is a fraction with common terms canceled out of the numerator and denominator.The Egyptians expressed their fractions as sums (and differences) of unit fractions. Conway and Guy (1996) give a table of Roman notation for fractions, in which multiples of 1/12 (the uncia) were given separate names.The rules for the algebraic combination of fractions are given by(1)(2)(3)(4)Note however that the above results will not necessarily be reducedfractions...
Specifying two adjacent side lengths and of a triangle (with ) and one acute angle opposite does not, in general, uniquely determine a triangle.If , there are two possible triangles satisfying the given conditions (left figure). If , there is one possible triangle (middle figure). If , there are no possible triangles (right figure).Remember: Don't try to prove congruence with the ASS theorem or you will make an ASS out of yourself.An ASS triangle with sides and and excluded angle with has two possible side lengths ,The SSS or SAS theorems can then be used with either choice of to determine the angles and and triangle area .
The mathematical study of the likelihood and probability of events occurring based on known information and inferred by taking a limited number of samples. Statistics plays an extremely important role in many aspects of economics and science, allowing educated guesses to be made with a minimum of expensive or difficult-to-obtain data.A joke told about statistics (or, more precisely, about statisticians), runs as follows. Two statisticians are out hunting when one of them sees a duck. The first takes aim and shoots, but the bullet goes sailing past six inches too high. The second statistician also takes aim and shoots, but this time the bullet goes sailing past six inches too low. The two statisticians then give one another high fives and exclaim, "Got him!" (This joke plays on the fact that the mean of and 6 is 0, so "on average," the two shots hit the duck.)Approximately 73.8474% of extant statistical jokes are maintained..
If is a sentential formula depending on a variable ranging in a set of real numbers, the sentence(1)means(2)An example is the proposition(3)which is true, since the inequality is fulfilled for .The statement can also be rephrased as follows: the terms of the sequence become eventually smaller than 0.0001.There are various mathematical jokes involving "sufficiently large." For example, " for sufficiently large values of 1" and "this feature will ship in version 1.0 for sufficiently large values of 1."
The term "real line" has a number of different meanings in mathematics.Most commonly, "real line" is used to mean real axis, i.e., a line with a fixed scale so that every real number corresponds to a unique point on the line. The generalization of the real line to two dimensions is called the complex plane.The term "real line" is also used to distinguish an ordinary line from a so-called imaginary line which can arise in algebraic geometry.Renteln and Dundes (2005) give the following (bad) mathematical jokes about the real line:Q: What is green and homeomorphic to the open unit interval?A: The real lime.
A point is a 0-dimensional mathematical object which can be specified in -dimensional space using an n-tuple (, , ..., ) consisting of coordinates. In dimensions greater than or equal to two, points are sometimes considered synonymous with vectors and so points in n-dimensional space are sometimes called n-vectors. Although the notion of a point is intuitively rather clear, the mathematical machinery used to deal with points and point-like objects can be surprisingly slippery. This difficulty was encountered by none other than Euclid himself who, in his Elements, gave the vague definition of a point as "that which has no part."The basic geometric structures of higher dimensional geometry--the line, plane, space, and hyperspace--are all built up of infinite numbers of points arranged in particular ways.These facts lead to the mathematical pun, "without geometry, life is pointless."The decimal point in a decimal..
The dihedral group is the symmetry group of an -sided regular polygon for . The group order of is . Dihedral groups are non-Abelian permutation groups for .The th dihedral group is represented in the Wolfram Language as DihedralGroup[n].One group presentation for the dihedral group is .A reducible two-dimensional representation of using real matrices has generators given by and , where is a rotation by radians about an axis passing through the center of a regular -gon and one of its vertices and is a rotation by about the center of the -gon (Arfken 1985, p. 250).Dihedral groups all have the same multiplication table structure. The table for is illustrated above.The cycle index (in variables , ..., ) for the dihedral group is given by(1)where(2)is the cycle index for the cyclic group , means divides , and is the totient function (Harary 1994, p. 184). The cycle indices for the first few are(3)(4)(5)(6)(7)Renteln and Dundes (2005) give..
A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group. Simple groups include the infinite families of alternating groups of degree , cyclic groups of prime order, Lie-type groups, and the 26 sporadic groups.Since all subgroups of an Abelian group are normal and all cyclic groups are Abelian, the only simple cyclic groups are those which have no subgroups other than the trivial subgroup and the improper subgroup consisting of the entire original group. And since cyclic groups of composite order can be written as a group direct product of factor groups, this means that only prime cyclic groups lack nontrivial subgroups. Therefore, the only simple cyclic groups are the prime cyclic groups. Furthermore, these are the only Abelian simple groups.In fact, the classification theorem of finite groups states that such groups can be classified completely..
The group is the unique group of group order 2. is both Abelian and cyclic. Examples include the point groups , , and , the integers modulo 2 under addition (), and the modulo multiplication groups , , and (which are the only modulo multiplication groups isomorphic to ).The group is also trivially simple, and forms the subject for the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern University mathematics department a capella group "The Klein Four."The cycle graph is shown above, and the cycleindex isThe elements satisfy , where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below. 1111The conjugacy classes are and . The only subgroups of are the trivial group and entire group , both of which are trivially normal.The irreducible representation for the group is ...
A finite group is a group having finite group order. Examples of finite groups are the modulo multiplication groups, point groups, cyclic groups, dihedral groups, symmetric groups, alternating groups, and so on.Properties of finite groups are implemented in the Wolfram Language as FiniteGroupData[group, prop].The classification theorem of finite groups states that the finite simple groups can be classified completely into one of five types.A convenient way to visualize groups is using so-called cycle graphs, which show the cycle structure of a given abstract group. For example, cycle graphs of the 5 nonisomorphic groups of order 8 are illustrated above (Shanks 1993, p. 85).Frucht's theorem states that every finite group is the graph automorphism group of a finite undirected graph.The finite (cyclic) group forms the subject for the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern University..
If is any nonempty partially ordered set in which every chain has an upper bound, then has a maximal element. This statement is equivalent to the axiom of choice.Renteln and Dundes (2005) give the following (bad) mathematical jokes about Zorn's lemma:Q: What's sour, yellow, and equivalent to the axiomof choice? A: Zorn's lemon.Q: What is brown, furry, runs to the sea, and is equivalent to the axiomof choice? A: Zorn's lemming.
Zeno's paradoxes are a set of four paradoxes dealingwith counterintuitive aspects of continuous space and time. 1. Dichotomy paradox: Before an object can travel a given distance , it must travel a distance . In order to travel , it must travel , etc. Since this sequence goes on forever, it therefore appears that the distance cannot be traveled. The resolution of the paradox awaited calculus and the proof that infinite geometric series such as can converge, so that the infinite number of "half-steps" needed is balanced by the increasingly short amount of time needed to traverse the distances. 2. Achilles and the tortoise paradox: A fleet-of-foot Achilles is unable to catch a plodding tortoise which has been given a head start, since during the time it takes Achilles to catch up to a given position, the tortoise has moved forward some distance. But this is obviously fallacious since Achilles will clearly pass the tortoise! The resolution..
In number theory (e.g., Ivić 2003), the symbol is commonly used to mean the nested logarithm (also called the repeated logarithm or iterated logarithm) , where is the natural logarithm.Care must be taken based on context as to when the notation denotes the logarithm to base and when it means the -nested natural logarithm.The plots above show , , and in the complex plane.The penchant for formulas and bounds containing a profusion of nested logarithms has led to the following joke. What sound does a drowning analytic number theorist make? A: log log log log... (Havil 2003, p. 115).
Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle (i.e., a one-dimensional closed curve with no intersections that can be embedded in two-dimensional space), the set of all possible positions of the hour and minute hands taken together is topologically equivalent to the surface of a torus (i.e., a two-dimensional a surface that can be embedded in three-dimensional space), and the set of all possible positions of the hour, minute, and second hands taken together are topologically equivalent to a three-dimensional object.The definition of topology leads to the following mathematical..
A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof.Although not absolutely standard, the Greeks distinguished between "problems" (roughly, the construction of various figures) and "theorems" (establishing the properties of said figures; Heath 1956, pp. 252, 262, and 264).According to the Nobel Prize-winning physicist Richard Feynman (1985), any theorem, no matter how difficult to prove in the first place, is viewed as "trivial" by mathematicians once it has been proven. Therefore, there are exactly two types of mathematical objects: trivial ones, and those which have not yet been proven.The late mathematician P. Erdős has often been associated with the observation..
A strange loop is a phenomenon in which, whenever movement is made upwards or downwards through the levels of some hierarchical system, the system unexpectedly arrives back where it started. Hofstadter (1989) uses the strange loop as a paradigm in which to interpret paradoxes in logic (such as Grelling's paradox, the liar's paradox, and Russell's antinomy) and calls a system in which a strange loop appears a tangled hierarchy.Canon 5 from Bach's Musical Offering (sometimes known as Bach's endlessly rising canon) is a musical piece that continues to rise in key, modulating through the entire chromatic scale until it ends in the same key in which it began. This is the first example cited by Hofstadter (1989) as a strange loop.Other examples include the endlessly rising stairs in M. C. Escher 1960 lithograph Ascending and Descending, the endlessly falling waterfall in his 1961 lithograph Waterfall, and the pair of hands drawing each..
A short theorem used in proving a larger theorem. Related concepts are the axiom, porism, postulate, principle, and theorem.The late mathematician P. Erdős has often been associated with the observation that "a mathematician is a machine for converting coffee into theorems" (e.g., Hoffman 1998, p. 7). However, this characterization appears to be due to his friend, Alfred Rényi (MacTutor, Malkevitch). This thought was developed further by Erdős' friend and Hungarian mathematician Paul Turán, who suggested that weak coffee was suitable "only for lemmas" (MacTutor, Malkevitch).
An integral obtained by contour integration. The particular path in the complex plane used to compute the integral is called a contour.As a result of a truly amazing property of holomorphic functions, a closed contour integral can be computed simply by summing the values of the complex residues inside the contour.Watson (1966 p. 20) uses the notation to denote the contour integral of with contour encircling the point once in a counterclockwise direction.Renteln and Dundes (2005) give the following (bad) mathematical joke about contour integrals:Q: What's the value of a contour integral around Western Europe? A: Zero, because all the Poles are in Eastern Europe.
A rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition. A mathematical statement that has been proven is called a theorem.According to Hardy (1999, pp. 15-16), "all physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.... [This opinion], with which I am sure that almost all physicists agree at the bottom of their hearts, is one to which a mathematician ought to have some reply."To prove Hardy's assertion, Feynman is reported to have commented, "A great deal more is known than has been proved" (Derbyshire 2004, p. 291).There is some debate among mathematicians..
The "Foxtrot series" is a mathematical sum that appeared in the June 2, 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007). It arose from a convergence testing problem in a calculus book by Anton, but was inadvertently converted into a summation problem on an alleged final exam by the strip's author:(1)The sum can be done using partial fraction decomposition to obtain(2)(3)(4)(5)(OEIS A127198), where and the last sums have been done in terms of the digamma function and symbolically simplified.
A short mnemonic for remembering the first seven decimal digits of is "How I wish I could calculate pi" (C. Heckman, pers. comm., Feb. 3, 2005). Eight digits are given by "May I have a large container of coffee?" giving 3.1415926 (Gardner 1959; 1966, p. 92; Eves 1990, p. 122, Davis 1993, p. 9). "But I must a while endeavour to reckon right" gives nine correct digits (3.14159265). "May I have a white telephone, or pastel color" (M. Amling, pers. comm., Jul. 31, 2004) also gives nine correct digits.A more substantial mnemonic giving 15 digits (3.14159265358979) is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics," originally due to Sir James Jeans (Gardner 1966, p. 92; Castellanos 1988, p. 152; Eves 1990, p. 122; Davis 1993, p. 9; Blatner 1997, p. 112). A slight extension..
The constant is base of the natural logarithm. is sometimes known as Napier's constant, although its symbol () honors Euler. is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis, and the vertical lines and is 1. In other words,(1)With the possible exception of , is the most important constant in mathematics since it appears in myriad mathematical contexts involving limits and derivatives. The numerical value of is(2)(OEIS A001113). can be defined by the limit(3)(illustrated above), or by the infinite series(4)as first published by Newton (1669; reprinted in Whiteside 1968, p. 225). is given by the unusual limit(5)(Brothers and Knox 1998).Euler (1737; Sandifer 2006) proved that is irrational by proving that has an infinite simple continued fraction (; Nagell 1951), and Liouville proved in 1844 that does not satisfy any quadratic equation with integral coefficients (i.e., if it is..
The derivative of a function represents an infinitesimalchange in the function with respect to one of its variables.The "simple" derivative of a function with respect to a variable is denoted either or(1)often written in-line as . When derivatives are taken with respect to time, they are often denoted using Newton's overdot notation for fluxions,(2)The "d-ism" of Leibniz's eventually won the notation battle against the "dotage" of Newton's fluxion notation (P. Ion, pers. comm., Aug. 18, 2006).When a derivative is taken times, the notation or(3)is used, with(4)etc., the corresponding fluxion notation.When a function depends on more than one variable, a partial derivative(5)can be used to specify the derivative with respect to one or more variables.The derivative of a function with respect to the variable is defined as(6)but may also be calculated more symmetrically as(7)provided the..
A definite integral is an integral(1)with upper and lower limits. If is restricted to lie on the real line, the definite integral is known as a Riemann integral (which is the usual definition encountered in elementary textbooks). However, a general definite integral is taken in the complex plane, resulting in the contour integral(2)with , , and in general being complex numbers and the path of integration from to known as a contour.The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if is the indefinite integral for a continuous function , then(3)This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. Definite integrals may be evaluated in the Wolfram Language using Integrate[f, x, a, b].The question of which definite..
If a circular pizza is divided into 8, 12, 16, ... slices by making cuts at equal angles from an arbitrary point, then the sums of the areas of alternate slices are equal.There is also a second pizza theorem. This one gives the volume of a pizza of thickness and radius :
There are several statistical quantities called means, e.g., harmonic mean, geometric mean, arithmetic-geometric mean, and root-mean-square. When applied to two elements and with , these means satisfy(1)The following table summarizes these means (again applied to two elements and with ), where is a complete elliptic integral of the first kind.meanvalueharmonic meangeometric meanarithmetic-geometric meanarithmetic meanroot-mean-squareThe quantity commonly referred to as "the" mean of a set of values is thearithmetic mean(2)also called the (unweighted) average. Notations for "the" mean of a set of values include macron notation or . The expectation value notation is sometimes also used. The mean of a list of data (i.e., the sample mean) is implemented as Mean[list].In general, a mean is a homogeneous function that has the property that a mean of a set of numbers satisfies(3)The term function centroid is..
The term limit comes about relative to a number of topics from several different branches of mathematics.A sequence of elements in a topological space is said to have limit provided that for each neighborhood of , there exists a natural number so that for all . This very general definition can be specialized in the event that is a metric space, whence one says that a sequence in has limit if for all , there exists a natural number so that(1)for all . In many commonly-encountered scenarios, limits are unique, whereby one says that is the limit of and writes(2)On the other hand, a sequence of elements from an metric space may have several - even infinitely many - different limits provided that is equipped with a topology which fails to be T2. One reads the expression in (1) as "the limit as approaches infinity of is ."The topological notion of convergence can be rewritten to accommodate a wider array of topological spaces by utilizing the language..
An Abelian group is a group for which the elements commute (i.e., for all elements and ). Abelian groups therefore correspond to groups with symmetric multiplication tables.All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator.In the Wolfram Language, the function AbelianGroup[n1, n2, ...] represents the direct product of the cyclic groups of degrees , , ....No general formula is known for giving the number of nonisomorphic finite groups of a given group order. However, the number of nonisomorphic Abelian finite groups of any given group order is given by writing as(1)where the are distinct prime factors, then(2)where is the partition function, which is implemented in the Wolfram Language as FiniteAbelianGroupCount[n]...
Geometry is the study of figures in a space of a given number of dimensions and of a given type. The most common types of geometry are plane geometry (dealing with objects like the point, line, circle, triangle, and polygon), solid geometry (dealing with objects like the line, sphere, and polyhedron), and spherical geometry (dealing with objects like the spherical triangle and spherical polygon). Geometry was part of the quadrivium taught in medieval universities.A mathematical pun notes that without geometry, life is pointless. An old children's joke asks, "What does an acorn say when it grows up?" and answers, "Geometry" ("gee, I'm a tree").Historically, the study of geometry proceeds from a small number of accepted truths (axioms or postulates), then builds up true statements using a systematic and rigorous step-by-step proof. However, there is much more to geometry than this relatively dry textbook..
The integer sequence defined by the recurrence(1)with the initial conditions , , . This recurrence relation is the same as that for the Padovan sequence but with different initial conditions. The first few terms for , 1, ..., are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ... (OEIS A001608).The above cartoon (Amend 2005) shows an unconventional sports application of the Perrin sequence (right panel). (The left two panels instead apply the Fibonacci numbers). is the solution of a third-order linear homogeneous recurrence equation having characteristic equation(2)Denoting the roots of this equation by , , and , with the unique real root, the solution is then(3)Here,(4)is the plastic constant , which is also given by the limit(5)The asymptotic behavior of is(6)The first few primes in this sequence are 2, 3, 2, 5, 5, 7, 17, 29, 277, 367, 853, ... (OEIS A074788), which occur for terms , 3, 4, 5, 6, 7, 10, 12, 20, 21, 24, 34, 38, 75, 122, 166, 236, 355, 356, 930, 1042,..
If is a linear operator on a function space, then is an eigenfunction for and is the associated eigenvalue whenever .Renteln and Dundes (2005) give the following (bad) mathematical joke about eigenfunctions:Q: What do you call a young eigensheep? A: A lamb, duh!
A Hilbert space is a vector space with an inner product such that the norm defined byturns into a complete metric space. If the metric defined by the norm is not complete, then is instead known as an inner product space.Examples of finite-dimensional Hilbert spaces include1. The real numbers with the vector dot product of and . 2. The complex numbers with the vector dot product of and the complex conjugate of . An example of an infinite-dimensional Hilbert space is , the set of all functions such that the integral of over the whole real line is finite. In this case, the inner product isA Hilbert space is always a Banach space, but theconverse need not hold.A (small) joke told in the hallways of MIT ran, "Do you know Hilbert? No? Then what are you doing in his space?" (S. A. Vaughn, pers. comm., Jul. 31, 2005)...
A Banach space is a complete vector space with a norm . Two norms and are called equivalent if they give the same topology, which is equivalent to the existence of constants and such that(1)and(2)hold for all .In the finite-dimensional case, all norms are equivalent. An infinite-dimensional space can have many different norms.A basic example is -dimensional Euclidean space with the Euclidean norm. Usually, the notion of Banach space is only used in the infinite dimensional setting, typically as a vector space of functions. For example, the set of continuous functions on closed interval of the real line with the norm of a function given by(3)is a Banach space, where denotes the supremum.On the other hand, the set of continuous functions on the unit interval with the norm of a function given by(4)is not a Banach space because it is not complete. For instance, the Cauchysequence of functions(5)does not converge to a continuous function.Hilbert..
The word "pole" is used prominently in a number of very different branches of mathematics. Perhaps the most important and widespread usage is to denote a singularity of a complex function. In inversive geometry, the inversion pole is related to inverse points with respect to an inversion circle. The term "pole" is also used to denote the degenerate points and in spherical coordinates, corresponding to the north pole and south pole respectively. "All-poles method" is an alternate term for the maximum entropy method used in deconvolution. In triangle geometry, an orthopole is the point of concurrence certain perpendiculars with respect to a triangle of a given line, and a Simson line pole is similarly defined based on the Simson line of a point with respect to a triangle. In projective geometry, the perspector is sometimes known as the perspective pole.In complex analysis, an analytic function is said to have..
A ring in the mathematical sense is a set together with two binary operators and (commonly interpreted as addition and multiplication, respectively) satisfying the following conditions: 1. Additive associativity: For all , , 2. Additive commutativity: For all , , 3. Additive identity: There exists an element such that for all , , 4. Additive inverse: For every there exists such that , 5. Left and right distributivity: For all , and , 6. Multiplicative associativity: For all , (a ring satisfying this property is sometimes explicitly termed an associative ring). Conditions 1-5 are always required. Though non-associative rings exist, virtually all texts also require condition 6 (Itô 1986, pp. 1369-1372; p. 418; Zwillinger 1995, pp. 141-143; Harris and Stocker 1998; Knuth 1998; Korn and Korn 2000; Bronshtein and Semendyayev 2004).Rings may also satisfy various optional conditions: 7. Multiplicative commutativity:..