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Wikipedia:Division by infinity#0
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In mathematics, division by infinity is division where the divisor (denominator) is ∞. In ordinary arithmetic, this does not have a well-defined meaning, since ∞ is a mathematical concept that does not correspond to a specific number, and moreover, there is no nonzero real number that, when added to itself an infinite number of times, gives a finite number, unless you address the concept of indeterminate forms. However, "dividing by ∞" can be given meaning as an informal way of expressing the limit of dividing a number by larger and larger divisors.: 201–204 Using mathematical structures that go beyond the real numbers, it is possible to define numbers that have infinite magnitude yet can still be manipulated in ways much like ordinary arithmetic. For example, on the extended real number line, dividing any real number by infinity yields zero, while in the surreal number system, dividing 1 by the infinite number ω {\displaystyle \omega } yields the infinitesimal number ϵ {\displaystyle \epsilon } .: 12 In floating-point arithmetic, any finite number divided by ± ∞ {\displaystyle \pm \infty } is equal to positive or negative zero if the numerator is finite. Otherwise, the result is NaN. The challenges of providing a rigorous meaning of "division by infinity" are analogous to those of defining division by zero. == Use in technology == As infinity is difficult to deal with for most calculators and computers, many do not have a formal way of computing division by infinity. Calculators such as the TI-84 and most household calculators do not have an infinity button so it is impossible to type into the calculator 'x divided by infinity' so instead users can type a large number such as "1e99" ( 1 × 10 99 {\displaystyle 1\times 10^{99}} ) or "-1e99". By typing in some number divided by a sufficiently large number the output will be 0. In some cases this fails as there is either an overflow error or if the numerator is also a sufficiently large number then the output may be 1 or a real number. In the Wolfram language, dividing an integer by infinity will result in the result 0. Also, in some calculators such as the TI-Nspire, 1 divided by infinity can be evaluated as 0. == Use in calculus == === Integration === In calculus, taking the integral of a function is defined finding the area under a curve. This can be done simply by breaking up this area into rectangular sections and taking the sum of these sections. These are called Riemann sums. As the sections get narrower, the Riemann sum becomes an increasingly accurate approximation of the true area. Taking the limit of these Riemann sums, in which the sections can be heuristically regarded as "infinitely thin", gives the definite integral of the function over the prescribed interval. Conceptually this results in dividing the interval by infinity to result in infinitely small pieces.: 255–259 On a different note when taking an integral where one of the boundaries is infinity this is defined as an improper integral. To determine this one would take the limit as a variable a approaches infinity substituting a in for the infinity sign. This would then allow the integral to be evaluated and then the limit would be taken. In many cases evaluating this would result in a term being divided by infinity. In this case in order to evaluate the integral one would assume this to be zero. This allows for the integral to be assumed to converge meaning a finite answer can be determined from the integral using this assumption. === L'Hôpital's rule === When given a ratio between two functions, the limit of this ratio can be evaluated by computing the limit of each function separately. Where the limit of the function in the denominator is infinity, and the numerator does not allow the ratio to be well determined, the limit of the ratio is said to be of indeterminate form. An example of this is: ∞ ∞ {\displaystyle {\frac {\infty }{\infty }}} Using L'Hôpital's rule to evaluate limits of fractions where the denominator tends towards infinity can produce results other than 0. If lim x → c f ′ ( x ) g ′ ( x ) {\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}} then lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}} So if lim x → c | f ( x ) | = lim x → c | g ( x ) | = ∞ , {\displaystyle \lim _{x\to c}|f(x)|=\lim _{x\to c}|g(x)|=\infty ,} then lim x → c f ( x ) g ( x ) = L {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=L} This means that, when using limits to give meaning to division by infinity, the result of "dividing by infinity" does not always equal 0. == References == == External links == How to Divide by Zero by Bill Shillito
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Wikipedia:Division by zero#0
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In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case. Using fraction notation, the general example can be written as a 0 {\displaystyle {\tfrac {a}{0}}} , where a {\displaystyle a} is the dividend (numerator). The usual definition of the quotient in elementary arithmetic is the number which yields the dividend when multiplied by the divisor. That is, c = a b {\displaystyle c={\tfrac {a}{b}}} is equivalent to c ⋅ b = a . {\displaystyle c\cdot b=a.} By this definition, the quotient q = a 0 {\displaystyle q={\tfrac {a}{0}}} is nonsensical, as the product q ⋅ 0 {\displaystyle q\cdot 0} is always 0 {\displaystyle 0} rather than some other number a . {\displaystyle a.} Following the ordinary rules of elementary algebra while allowing division by zero can create a mathematical fallacy, a subtle mistake leading to absurd results. To prevent this, the arithmetic of real numbers and more general numerical structures called fields leaves division by zero undefined, and situations where division by zero might occur must be treated with care. Since any number multiplied by zero is zero, the expression 0 0 {\displaystyle {\tfrac {0}{0}}} is also undefined. Calculus studies the behavior of functions in the limit as their input tends to some value. When a real function can be expressed as a fraction whose denominator tends to zero, the output of the function becomes arbitrarily large, and is said to "tend to infinity", a type of mathematical singularity. For example, the reciprocal function, f ( x ) = 1 x , {\displaystyle f(x)={\tfrac {1}{x}},} tends to infinity as x {\displaystyle x} tends to 0. {\displaystyle 0.} When both the numerator and the denominator tend to zero at the same input, the expression is said to take an indeterminate form, as the resulting limit depends on the specific functions forming the fraction and cannot be determined from their separate limits. As an alternative to the common convention of working with fields such as the real numbers and leaving division by zero undefined, it is possible to define the result of division by zero in other ways, resulting in different number systems. For example, the quotient a 0 {\displaystyle {\tfrac {a}{0}}} can be defined to equal zero; it can be defined to equal a new explicit point at infinity, sometimes denoted by the infinity symbol ∞ {\displaystyle \infty } ; or it can be defined to result in signed infinity, with positive or negative sign depending on the sign of the dividend. In these number systems division by zero is no longer a special exception per se, but the point or points at infinity involve their own new types of exceptional behavior. In computing, an error may result from an attempt to divide by zero. Depending on the context and the type of number involved, dividing by zero may evaluate to positive or negative infinity, return a special not-a-number value, or crash the program, among other possibilities. == Elementary arithmetic == === The meaning of division === The division N / D = Q {\displaystyle N/D=Q} can be conceptually interpreted in several ways. In quotitive division, the dividend N {\displaystyle N} is imagined to be split up into parts of size D {\displaystyle D} (the divisor), and the quotient Q {\displaystyle Q} is the number of resulting parts. For example, imagine ten slices of bread are to be made into sandwiches, each requiring two slices of bread. A total of five sandwiches can be made ( 10 2 = 5 {\displaystyle {\tfrac {10}{2}}=5} ). Now imagine instead that zero slices of bread are required per sandwich (perhaps a lettuce wrap). Arbitrarily many such sandwiches can be made from ten slices of bread, as the bread is irrelevant. The quotitive concept of division lends itself to calculation by repeated subtraction: dividing entails counting how many times the divisor can be subtracted before the dividend runs out. Because no finite number of subtractions of zero will ever exhaust a non-zero dividend, calculating division by zero in this way never terminates. Such an interminable division-by-zero algorithm is physically exhibited by some mechanical calculators. In partitive division, the dividend N {\displaystyle N} is imagined to be split into D {\displaystyle D} parts, and the quotient Q {\displaystyle Q} is the resulting size of each part. For example, imagine ten cookies are to be divided among two friends. Each friend will receive five cookies ( 10 2 = 5 {\displaystyle {\tfrac {10}{2}}=5} ). Now imagine instead that the ten cookies are to be divided among zero friends. How many cookies will each friend receive? Since there are no friends, this is an absurdity. In another interpretation, the quotient Q {\displaystyle Q} represents the ratio N : D . {\displaystyle N:D.} For example, a cake recipe might call for ten cups of flour and two cups of sugar, a ratio of 10 : 2 {\displaystyle 10:2} or, proportionally, 5 : 1. {\displaystyle 5:1.} To scale this recipe to larger or smaller quantities of cake, a ratio of flour to sugar proportional to 5 : 1 {\displaystyle 5:1} could be maintained, for instance one cup of flour and one-fifth cup of sugar, or fifty cups of flour and ten cups of sugar. Now imagine a sugar-free cake recipe calls for ten cups of flour and zero cups of sugar. The ratio 10 : 0 , {\displaystyle 10:0,} or proportionally 1 : 0 , {\displaystyle 1:0,} is perfectly sensible: it just means that the cake has no sugar. However, the question "How many parts flour for each part sugar?" still has no meaningful numerical answer. A geometrical appearance of the division-as-ratio interpretation is the slope of a straight line in the Cartesian plane. The slope is defined to be the "rise" (change in vertical coordinate) divided by the "run" (change in horizontal coordinate) along the line. When this is written using the symmetrical ratio notation, a horizontal line has slope 0 : 1 {\displaystyle 0:1} and a vertical line has slope 1 : 0. {\displaystyle 1:0.} However, if the slope is taken to be a single real number then a horizontal line has slope 0 1 = 0 {\displaystyle {\tfrac {0}{1}}=0} while a vertical line has an undefined slope, since in real-number arithmetic the quotient 1 0 {\displaystyle {\tfrac {1}{0}}} is undefined. The real-valued slope y x {\displaystyle {\tfrac {y}{x}}} of a line through the origin is the vertical coordinate of the intersection between the line and a vertical line at horizontal coordinate 1 , {\displaystyle 1,} dashed black in the figure. The vertical red and dashed black lines are parallel, so they have no intersection in the plane. Sometimes they are said to intersect at a point at infinity, and the ratio 1 : 0 {\displaystyle 1:0} is represented by a new number ∞ {\displaystyle \infty } ; see § Projectively extended real line below. Vertical lines are sometimes said to have an "infinitely steep" slope. === Inverse of multiplication === Division is the inverse of multiplication, meaning that multiplying and then dividing by the same non-zero quantity, or vice versa, leaves an original quantity unchanged; for example ( 5 × 3 ) / 3 = {\displaystyle (5\times 3)/3={}} ( 5 / 3 ) × 3 = 5 {\displaystyle (5/3)\times 3=5} . Thus a division problem such as 6 3 = ? {\displaystyle {\tfrac {6}{3}}={?}} can be solved by rewriting it as an equivalent equation involving multiplication, ? × 3 = 6 , {\displaystyle {?}\times 3=6,} where ? {\displaystyle {?}} represents the same unknown quantity, and then finding the value for which the statement is true; in this case the unknown quantity is 2 , {\displaystyle 2,} because 2 × 3 = 6 , {\displaystyle 2\times 3=6,} so therefore 6 3 = 2. {\displaystyle {\tfrac {6}{3}}=2.} An analogous problem involving division by zero, 6 0 = ? , {\displaystyle {\tfrac {6}{0}}={?},} requires determining an unknown quantity satisfying ? × 0 = 6. {\displaystyle {?}\times 0=6.} However, any number multiplied by zero is zero rather than six, so there exists no number which can substitute for ? {\displaystyle {?}} to make a true statement. When the problem is changed to 0 0 = ? , {\displaystyle {\tfrac {0}{0}}={?},} the equivalent multiplicative statement is ? × 0 = 0 {\displaystyle {?}\times 0=0} ; in this case any value can be substituted for the unknown quantity to yield a true statement, so there is no single number which can be assigned as the quotient 0 0 . {\displaystyle {\tfrac {0}{0}}.} Because of these difficulties, quotients where the divisor is zero are traditionally taken to be undefined, and division by zero is not allowed. === Fallacies === A compelling reason for not allowing division by zero is that allowing it leads to fallacies. When working with numbers, it is easy to identify an illegal division by zero. For example: From 0 × 1 = 0 {\displaystyle 0\times 1=0} and 0 × 2 = 0 {\displaystyle 0\times 2=0} one gets 0 × 1 = 0 × 2. {\displaystyle 0\times 1=0\times 2.} Cancelling 0 from both sides yields 1 = 2 {\displaystyle 1=2} , a false statement. The fallacy here arises from the assumption that it is legitimate to cancel 0 like any other number, whereas, in fact, doing so is a form of division by 0. Using algebra, it is possible to disguise a division by zero to obtain an invalid proof. For example: This is essentially the same fallacious computation as the previous numerical version, but the division by zero was obfuscated because we wrote 0 as x − 1. == Early attempts == The Brāhmasphuṭasiddhānta of Brahmagupta (c. 598–668) is the earliest text to treat zero as a number in its own right and to define operations involving zero. According to Brahmagupta, A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero. In 830, Mahāvīra unsuccessfully tried to correct the mistake Brahmagupta made in his book Ganita Sara Samgraha: "A number remains unchanged when divided by zero." Bhāskara II's Līlāvatī (12th century) proposed that division by zero results in an infinite quantity, A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to a 0 {\textstyle {\tfrac {a}{0}}} is contained in Anglo-Irish philosopher George Berkeley's criticism of infinitesimal calculus in 1734 in The Analyst ("ghosts of departed quantities"). == Calculus == Calculus studies the behavior of functions using the concept of a limit, the value to which a function's output tends as its input tends to some specific value. The notation lim x → c f ( x ) = L {\textstyle \lim _{x\to c}f(x)=L} means that the value of the function f {\displaystyle f} can be made arbitrarily close to L {\displaystyle L} by choosing x {\displaystyle x} sufficiently close to c . {\displaystyle c.} In the case where the limit of the real function f {\displaystyle f} increases without bound as x {\displaystyle x} tends to c , {\displaystyle c,} the function is not defined at x , {\displaystyle x,} a type of mathematical singularity. Instead, the function is said to "tend to infinity", denoted lim x → c f ( x ) = ∞ , {\textstyle \lim _{x\to c}f(x)=\infty ,} and its graph has the line x = c {\displaystyle x=c} as a vertical asymptote. While such a function is not formally defined for x = c , {\displaystyle x=c,} and the infinity symbol ∞ {\displaystyle \infty } in this case does not represent any specific real number, such limits are informally said to "equal infinity". If the value of the function decreases without bound, the function is said to "tend to negative infinity", − ∞ . {\displaystyle -\infty .} In some cases a function tends to two different values when x {\displaystyle x} tends to c {\displaystyle c} from above ( x → c + {\displaystyle x\to c^{+}} ) and below ( x → c − {\displaystyle x\to c^{-}} ); such a function has two distinct one-sided limits. A basic example of an infinite singularity is the reciprocal function, f ( x ) = 1 / x , {\displaystyle f(x)=1/x,} which tends to positive or negative infinity as x {\displaystyle x} tends to 0 {\displaystyle 0} : lim x → 0 + 1 x = + ∞ , lim x → 0 − 1 x = − ∞ . {\displaystyle \lim _{x\to 0^{+}}{\frac {1}{x}}=+\infty ,\qquad \lim _{x\to 0^{-}}{\frac {1}{x}}=-\infty .} In most cases, the limit of a quotient of functions is equal to the quotient of the limits of each function separately, lim x → c f ( x ) g ( x ) = lim x → c f ( x ) lim x → c g ( x ) . {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {\displaystyle \lim _{x\to c}f(x)}{\displaystyle \lim _{x\to c}g(x)}}.} However, when a function is constructed by dividing two functions whose separate limits are both equal to 0 , {\displaystyle 0,} then the limit of the result cannot be determined from the separate limits, so is said to take an indeterminate form, informally written 0 0 . {\displaystyle {\tfrac {0}{0}}.} (Another indeterminate form, ∞ ∞ , {\displaystyle {\tfrac {\infty }{\infty }},} results from dividing two functions whose limits both tend to infinity.) Such a limit may equal any real value, may tend to infinity, or may not converge at all, depending on the particular functions. For example, in lim x → 1 x 2 − 1 x − 1 , {\displaystyle \lim _{x\to 1}{\dfrac {x^{2}-1}{x-1}},} the separate limits of the numerator and denominator are 0 {\displaystyle 0} , so we have the indeterminate form 0 0 {\displaystyle {\tfrac {0}{0}}} , but simplifying the quotient first shows that the limit exists: lim x → 1 x 2 − 1 x − 1 = lim x → 1 ( x − 1 ) ( x + 1 ) x − 1 = lim x → 1 ( x + 1 ) = 2. {\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=\lim _{x\to 1}{\frac {(x-1)(x+1)}{x-1}}=\lim _{x\to 1}(x+1)=2.} == Alternative number systems == === Extended real line === The affinely extended real numbers are obtained from the real numbers R {\displaystyle \mathbb {R} } by adding two new numbers + ∞ {\displaystyle +\infty } and − ∞ , {\displaystyle -\infty ,} read as "positive infinity" and "negative infinity" respectively, and representing points at infinity. With the addition of ± ∞ , {\displaystyle \pm \infty ,} the concept of a "limit at infinity" can be made to work like a finite limit. When dealing with both positive and negative extended real numbers, the expression 1 / 0 {\displaystyle 1/0} is usually left undefined. However, in contexts where only non-negative values are considered, it is often convenient to define 1 / 0 = + ∞ {\displaystyle 1/0=+\infty } . === Projectively extended real line === The set R ∪ { ∞ } {\displaystyle \mathbb {R} \cup \{\infty \}} is the projectively extended real line, which is a one-point compactification of the real line. Here ∞ {\displaystyle \infty } means an unsigned infinity or point at infinity, an infinite quantity that is neither positive nor negative. This quantity satisfies − ∞ = ∞ {\displaystyle -\infty =\infty } , which is necessary in this context. In this structure, a 0 = ∞ {\displaystyle {\frac {a}{0}}=\infty } can be defined for nonzero a, and a ∞ = 0 {\displaystyle {\frac {a}{\infty }}=0} when a is not ∞ {\displaystyle \infty } . It is the natural way to view the range of the tangent function and cotangent functions of trigonometry: tan(x) approaches the single point at infinity as x approaches either +π/2 or −π/2 from either direction. This definition leads to many interesting results. However, the resulting algebraic structure is not a field, and should not be expected to behave like one. For example, ∞ + ∞ {\displaystyle \infty +\infty } is undefined in this extension of the real line. === Riemann sphere === The subject of complex analysis applies the concepts of calculus in the complex numbers. Of major importance in this subject is the extended complex numbers C ∪ { ∞ } , {\displaystyle \mathbb {C} \cup \{\infty \},} the set of complex numbers with a single additional number appended, usually denoted by the infinity symbol ∞ {\displaystyle \infty } and representing a point at infinity, which is defined to be contained in every exterior domain, making those its topological neighborhoods. This can intuitively be thought of as wrapping up the infinite edges of the complex plane and pinning them together at the single point ∞ , {\displaystyle \infty ,} a one-point compactification, making the extended complex numbers topologically equivalent to a sphere. This equivalence can be extended to a metrical equivalence by mapping each complex number to a point on the sphere via inverse stereographic projection, with the resulting spherical distance applied as a new definition of distance between complex numbers; and in general the geometry of the sphere can be studied using complex arithmetic, and conversely complex arithmetic can be interpreted in terms of spherical geometry. As a consequence, the set of extended complex numbers is often called the Riemann sphere. The set is usually denoted by the symbol for the complex numbers decorated by an asterisk, overline, tilde, or circumflex, for example C ^ = C ∪ { ∞ } . {\displaystyle {\hat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}.} In the extended complex numbers, for any nonzero complex number z , {\displaystyle z,} ordinary complex arithmetic is extended by the additional rules z 0 = ∞ , {\displaystyle {\tfrac {z}{0}}=\infty ,} z ∞ = 0 , {\displaystyle {\tfrac {z}{\infty }}=0,} ∞ + 0 = ∞ , {\displaystyle \infty +0=\infty ,} ∞ + z = ∞ , {\displaystyle \infty +z=\infty ,} ∞ ⋅ z = ∞ . {\displaystyle \infty \cdot z=\infty .} However, 0 0 {\displaystyle {\tfrac {0}{0}}} , ∞ ∞ {\displaystyle {\tfrac {\infty }{\infty }}} , and 0 ⋅ ∞ {\displaystyle 0\cdot \infty } are left undefined. == Higher mathematics == The four basic operations – addition, subtraction, multiplication and division – as applied to whole numbers (positive integers), with some restrictions, in elementary arithmetic are used as a framework to support the extension of the realm of numbers to which they apply. For instance, to make it possible to subtract any whole number from another, the realm of numbers must be expanded to the entire set of integers in order to incorporate the negative integers. Similarly, to support division of any integer by any other, the realm of numbers must expand to the rational numbers. During this gradual expansion of the number system, care is taken to ensure that the "extended operations", when applied to the older numbers, do not produce different results. Loosely speaking, since division by zero has no meaning (is undefined) in the whole number setting, this remains true as the setting expands to the real or even complex numbers. As the realm of numbers to which these operations can be applied expands there are also changes in how the operations are viewed. For instance, in the realm of integers, subtraction is no longer considered a basic operation since it can be replaced by addition of signed numbers. Similarly, when the realm of numbers expands to include the rational numbers, division is replaced by multiplication by certain rational numbers. In keeping with this change of viewpoint, the question, "Why can't we divide by zero?", becomes "Why can't a rational number have a zero denominator?". Answering this revised question precisely requires close examination of the definition of rational numbers. In the modern approach to constructing the field of real numbers, the rational numbers appear as an intermediate step in the development that is founded on set theory. First, the natural numbers (including zero) are established on an axiomatic basis such as Peano's axiom system and then this is expanded to the ring of integers. The next step is to define the rational numbers keeping in mind that this must be done using only the sets and operations that have already been established, namely, addition, multiplication and the integers. Starting with the set of ordered pairs of integers, {(a, b)} with b ≠ 0, define a binary relation on this set by (a, b) ≃ (c, d) if and only if ad = bc. This relation is shown to be an equivalence relation and its equivalence classes are then defined to be the rational numbers. It is in the formal proof that this relation is an equivalence relation that the requirement that the second coordinate is not zero is needed (for verifying transitivity). Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures. === Non-standard analysis === In the hyperreal numbers, division by zero is still impossible, but division by non-zero infinitesimals is possible. The same holds true in the surreal numbers. === Distribution theory === In distribution theory one can extend the function 1 x {\textstyle {\frac {1}{x}}} to a distribution on the whole space of real numbers (in effect by using Cauchy principal values). It does not, however, make sense to ask for a "value" of this distribution at x = 0; a sophisticated answer refers to the singular support of the distribution. === Linear algebra === In matrix algebra, square or rectangular blocks of numbers are manipulated as though they were numbers themselves: matrices can be added and multiplied, and in some cases, a version of division also exists. Dividing by a matrix means, more precisely, multiplying by its inverse. Not all matrices have inverses. For example, a matrix containing only zeros is not invertible. One can define a pseudo-division, by setting a/b = ab+, in which b+ represents the pseudoinverse of b. It can be proven that if b−1 exists, then b+ = b−1. If b equals 0, then b+ = 0. === Abstract algebra === In abstract algebra, the integers, the rational numbers, the real numbers, and the complex numbers can be abstracted to more general algebraic structures, such as a commutative ring, which is a mathematical structure where addition, subtraction, and multiplication behave as they do in the more familiar number systems, but division may not be defined. Adjoining a multiplicative inverses to a commutative ring is called localization. However, the localization of every commutative ring at zero is the trivial ring, where 0 = 1 {\displaystyle 0=1} , so nontrivial commutative rings do not have inverses at zero, and thus division by zero is undefined for nontrivial commutative rings. Nevertheless, any number system that forms a commutative ring can be extended to a structure called a wheel in which division by zero is always possible. However, the resulting mathematical structure is no longer a commutative ring, as multiplication no longer distributes over addition. Furthermore, in a wheel, division of an element by itself no longer results in the multiplicative identity element 1 {\displaystyle 1} , and if the original system was an integral domain, the multiplication in the wheel no longer results in a cancellative semigroup. The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as rings and fields. In a field, every nonzero element is invertible under multiplication; as above, division poses problems only when attempting to divide by zero. This is likewise true in a skew field (which for this reason is called a division ring). However, in other rings, division by nonzero elements may also pose problems. For example, the ring Z/6Z of integers mod 6. The meaning of the expression 2 2 {\textstyle {\frac {2}{2}}} should be the solution x of the equation 2 x = 2 {\displaystyle 2x=2} . But in the ring Z/6Z, 2 is a zero divisor. This equation has two distinct solutions, x = 1 and x = 4, so the expression 2 2 {\textstyle {\frac {2}{2}}} is undefined. In field theory, the expression a b {\textstyle {\frac {a}{b}}} is only shorthand for the formal expression ab−1, where b−1 is the multiplicative inverse of b. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when b is zero. Modern texts, that define fields as a special type of ring, include the axiom 0 ≠ 1 for fields (or its equivalent) so that the zero ring is excluded from being a field. In the zero ring, division by zero is possible, which shows that the other field axioms are not sufficient to exclude division by zero in a field. == Computer arithmetic == === Floating-point arithmetic === In computing, most numerical calculations are done with floating-point arithmetic, which since the 1980s has been standardized by the IEEE 754 specification. In IEEE floating-point arithmetic, numbers are represented using a sign (positive or negative), a fixed-precision significand and an integer exponent. Numbers whose exponent is too large to represent instead "overflow" to positive or negative infinity (+∞ or −∞), while numbers whose exponent is too small to represent instead "underflow" to positive or negative zero (+0 or −0). A NaN (not a number) value represents undefined results. In IEEE arithmetic, division of 0/0 or ∞/∞ results in NaN, but otherwise division always produces a well-defined result. Dividing any non-zero number by positive zero (+0) results in an infinity of the same sign as the dividend. Dividing any non-zero number by negative zero (−0) results in an infinity of the opposite sign as the dividend. This definition preserves the sign of the result in case of arithmetic underflow. For example, using single-precision IEEE arithmetic, if x = −2−149, then x/2 underflows to −0, and dividing 1 by this result produces 1/(x/2) = −∞. The exact result −2150 is too large to represent as a single-precision number, so an infinity of the same sign is used instead to indicate overflow. === Integer arithmetic === Integer division by zero is usually handled differently from floating point since there is no integer representation for the result. CPUs differ in behavior: for instance x86 processors trigger a hardware exception, while PowerPC processors silently generate an incorrect result for the division and continue, and ARM processors can either cause a hardware exception or return zero. Because of this inconsistency between platforms, the C and C++ programming languages consider the result of dividing by zero undefined behavior. In typical higher-level programming languages, such as Python, an exception is raised for attempted division by zero, which can be handled in another part of the program. === In proof assistants === Many proof assistants, such as Rocq (previously known as Coq) and Lean, define 1/0 = 0. This is due to the requirement that all functions are total. Such a definition does not create contradictions, as further manipulations (such as cancelling out) still require that the divisor is non-zero. == Historical accidents == On 21 September 1997, a division by zero error in the "Remote Data Base Manager" aboard USS Yorktown (CG-48) brought down all the machines on the network, causing the ship's propulsion system to fail. == See also == Zero divisor Zero to the power of zero L'Hôpital's rule == Notes == == Sources == Bunch, Bryan (1982), Mathematical Fallacies and Paradoxes, New York: Van Nostrand Reinhold, ISBN 0-442-24905-5 (Dover reprint 1997) Cheng, Eugenia (2023), Is Math(s) Real? How Simple Questions Lead Us to Mathematics' Deepest Truths, Basic Books, ISBN 978-1-541-60182-6 Klein, Felix (1925), Elementary Mathematics from an Advanced Standpoint / Arithmetic, Algebra, Analysis, translated by Hedrick, E. R.; Noble, C. A. (3rd ed.), Dover Hamilton, A. G. (1982), Numbers, Sets, and Axioms, Cambridge University Press, ISBN 978-0521287616 Henkin, Leon; Smith, Norman; Varineau, Verne J.; Walsh, Michael J. (2012), Retracing Elementary Mathematics, Literary Licensing LLC, ISBN 978-1258291488 Schumacher, Carol (1996), Chapter Zero : Fundamental Notions of Abstract Mathematics, Addison-Wesley, ISBN 978-0-201-82653-1 Zazkis, Rina; Liljedahl, Peter (2009), "Stories that explain", Teaching Mathematics as Storytelling, Sense Publishers, pp. 51–65, doi:10.1163/9789087907358_008, ISBN 978-90-8790-734-1 == Further reading == Northrop, Eugene P. (1944), Riddles in Mathematics: A Book of Paradoxes, New York: D. Van Nostrand, Ch. 5 "Thou Shalt Not Divide By Zero", pp. 77–96 Seife, Charles (2000), Zero: The Biography of a Dangerous Idea, New York: Penguin, ISBN 0-14-029647-6 Suppes, Patrick (1957), Introduction to Logic, Princeton: D. Van Nostrand, §8.5 "The Problem of Division by Zero" and §8.7 "Five Approaches to Division by Zero" (Dover reprint, 1999) Tarski, Alfred (1941), Introduction to Logic and to the Methodology of Deductive Sciences, Oxford University Press, §53 "Definitions whose definiendum contains the identity sign"
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Wikipedia:Divsha Amirà#0
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Divsha Amirà (Hebrew: דיבשה אמירה; 1899 – 9 April 1966) was an Israeli mathematician and educator. == Biography == Amirà was born in Brańsk, Russian Empire to Rivka (née Garbuz) and Aharon Itin. She immigrated to Israel with her family in 1906. Her father was one of the founders of Ahuzat Bayit (today Tel Aviv), a founder of the Tel Aviv Great Synagogue, and the owner of the first publishing house in Jaffa. She graduated in the second class of the Herzliya Gymnasium in 1914. Amirà studied at the University of Göttingen and obtained her doctorate from the University of Geneva in 1924 under the guidance of Herman Müntz. Her doctoral thesis, published in 1925, provided a projective synthesis of Euclidean geometry. == Pedagogic career == After leaving Geneva, Amirà worked at Gymnasia Rehavia in Jerusalem, and taught several courses on geometry at the Einstein Institute of Mathematics. She later taught at the Levinsky College of Education and Beit-Hakerem High School, where her students included such future mathematicians as Ernst G. Straus. == Published works == Amirà published an introductory school textbook on geometry in 1938, following the axiomatic approach of Hilbert's Grundlagen der Geometrie. She published a more advanced textbook on the same topic in 1963. == See also == Education in Israel Women in Israel == References == Divsha Amirà at the Mathematics Genealogy Project
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Wikipedia:Dixmier conjecture#0
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In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism. Tsuchimoto in 2005, and independently Belov-Kanel and Kontsevich in 2007, showed that the Dixmier conjecture is stably equivalent to the Jacobian conjecture. == References ==
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Wikipedia:Dixon's identity#0
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In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating a hypergeometric sum. These identities famously follow from the MacMahon Master theorem, and can now be routinely proved by computer algorithms (Ekhad 1990). == Statements == The original identity, from (Dixon 1891), is ∑ k = − a a ( − 1 ) k ( 2 a k + a ) 3 = ( 3 a ) ! ( a ! ) 3 . {\displaystyle \sum _{k=-a}^{a}(-1)^{k}{2a \choose k+a}^{3}={\frac {(3a)!}{(a!)^{3}}}.} A generalization, also sometimes called Dixon's identity, is ∑ k ∈ Z ( − 1 ) k ( a + b a + k ) ( b + c b + k ) ( c + a c + k ) = ( a + b + c ) ! a ! b ! c ! {\displaystyle \sum _{k\in \mathbb {Z} }(-1)^{k}{a+b \choose a+k}{b+c \choose b+k}{c+a \choose c+k}={\frac {(a+b+c)!}{a!b!c!}}} where a, b, and c are non-negative integers (Wilf 1994, p. 156). The sum on the left can be written as the terminating well-poised hypergeometric series ( b + c b − a ) ( c + a c − a ) 3 F 2 ( − 2 a , − a − b , − a − c ; 1 + b − a , 1 + c − a ; 1 ) {\displaystyle {b+c \choose b-a}{c+a \choose c-a}{}_{3}F_{2}(-2a,-a-b,-a-c;1+b-a,1+c-a;1)} and the identity follows as a limiting case (as a tends to an integer) of Dixon's theorem evaluating a well-poised 3F2 generalized hypergeometric series at 1, from (Dixon 1902): 3 F 2 ( a , b , c ; 1 + a − b , 1 + a − c ; 1 ) = Γ ( 1 + a / 2 ) Γ ( 1 + a / 2 − b − c ) Γ ( 1 + a − b ) Γ ( 1 + a − c ) Γ ( 1 + a ) Γ ( 1 + a − b − c ) Γ ( 1 + a / 2 − b ) Γ ( 1 + a / 2 − c ) . {\displaystyle \;_{3}F_{2}(a,b,c;1+a-b,1+a-c;1)={\frac {\Gamma (1+a/2)\Gamma (1+a/2-b-c)\Gamma (1+a-b)\Gamma (1+a-c)}{\Gamma (1+a)\Gamma (1+a-b-c)\Gamma (1+a/2-b)\Gamma (1+a/2-c)}}.} This holds for Re(1 + 1⁄2a − b − c) > 0. As c tends to −∞ it reduces to Kummer's formula for the hypergeometric function 2F1 at −1. Dixon's theorem can be deduced from the evaluation of the Selberg integral. == q-analogues == A q-analogue of Dixon's formula for the basic hypergeometric series in terms of the q-Pochhammer symbol is given by 4 φ 3 [ a − q a 1 / 2 b c − a 1 / 2 a q / b a q / c ; q , q a 1 / 2 / b c ] = ( a q , a q / b c , q a 1 / 2 / b , q a 1 / 2 / c ; q ) ∞ ( a q / b , a q / c , q a 1 / 2 , q a 1 / 2 / b c ; q ) ∞ {\displaystyle \;_{4}\varphi _{3}\left[{\begin{matrix}a&-qa^{1/2}&b&c\\&-a^{1/2}&aq/b&aq/c\end{matrix}};q,qa^{1/2}/bc\right]={\frac {(aq,aq/bc,qa^{1/2}/b,qa^{1/2}/c;q)_{\infty }}{(aq/b,aq/c,qa^{1/2},qa^{1/2}/bc;q)_{\infty }}}} where |qa1/2/bc| < 1. == References == Dixon, A.C. (1891), "On the sum of the cubes of the coefficients in a certain expansion by the binomial theorem", Messenger of Mathematics, 20: 79–80, JFM 22.0258.01 Dixon, A.C. (1902). "Summation of a certain series". Proc. London Math. Soc. 35 (1): 284–291. doi:10.1112/plms/s1-35.1.284. JFM 34.0490.02. Gessel, Ira; Stanton, Dennis (1985). "Short proofs of Saalschütz's and Dixon's theorems". Journal of Combinatorial Theory, Series A. 38 (1): 87–90. doi:10.1016/0097-3165(85)90026-3. ISSN 1096-0899. MR 0773560. Zbl 0559.05008. Ekhad, Shalosh B. (1990), "A very short proof of Dixon's theorem", Journal of Combinatorial Theory, Series A, 54 (1): 141–142, doi:10.1016/0097-3165(90)90014-N, ISSN 1096-0899, MR 1051787, Zbl 0707.05007 Ward, James (1991). "100 years of Dixon's identity". Irish Mathematical Society Bulletin. 0027 (27): 46–54. doi:10.33232/BIMS.0027.46.54. ISSN 0791-5578. MR 1185413. Zbl 0795.01009. Mikic, Jovan (2016). "A proof of Dixon's identity". J. Int. Seq. 19: #16.5.3. Wilf, Herbert S. (1994), Generatingfunctionology (2nd ed.), Boston, MA: Academic Press, ISBN 0-12-751956-4, Zbl 0831.05001
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Wikipedia:Djairo Guedes de Figueiredo#0
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Djairo Guedes de Figueiredo (academic signature: D. G. De Figueiredo, born on 2 April 1934, in Limoeiro do Norte) is a Brazilian mathematician noted for his research on differential equations, elliptic operators, and calculus of variations. He is considered the greatest analyst from Brazil. He was the president of the Brazilian Mathematical Society from 1977 to 1979. Figueiredo is a well-known figure among mathematicians in analysis and differential equations and among Brazilian students in physics, engineering and mathematics. He has received many Brazilian national and international prizes, both for his research in pure mathematics and also for his popular mathematics textbooks (about analysis and differential equations) and expository writing papers. In 1995 he received the National Order of Scientific Merit and in 2004 the title of "Doctor Honoris Causa" from the Federal University of Paraíba. In 2009, he became a member of the National Academy of Science of Buenos Aires. In 2011, he became the first Brazilian to receive a gold medal from the Telesio-Galilei Academy of Sciences "for his great contribution to Mathematics, especially to the theory of elliptical partial differential equations". He was a Ph.D. student of Louis Nirenberg at New York University, and is currently a titular professor at UNICAMP, a position he began in 1988. He is a recipient of Brazil's National Order of Scientific Merit in mathematics (1995). Since 1969 he has been a member of the Brazilian Academy of Sciences. == Selected papers == D. G. de Figueiredo, P. L. Lions, R. D. Nussbaum. "A priori estimates and existence of positive solutions of semilinear elliptic equations", Journal de Mathématiques Pures et Appliquées, 61, 1982, pp. 41–63. D. G. de Figueiredo, P. L. Felmer . "On superquadratic elliptic systems", Transactions of the American Mathematical Society, v. 343, n. 1, 1994, pp. 99–116. P. Clément, D. G. de Figueiredo . "Positive solutions of semilinear elliptic systems", Communications in Partial Differential Equations, v. 17, n. 5–6., 1992, pp. 923–940. The book Selected Papers of Djairo Guedes Figueiredo has been published by Springer, as part of the collection Selected Works of Outstanding Brazilian Mathematicians (Google Preview). == Books == Análise I (1975, in Portuguese) Análise de Fourier e Equações Diferenciais Parciais (1977, in Portuguese) Equações Diferenciais Aplicadas (1979, in Portuguese) Números Irracionais e Transcendentes (1974, in Portuguese) Equações Elípticas não Lineares (1977, in Portuguese) Lectures on the Ekeland Variational Principle with Applications and Detours (Springer Verlag, 1989) == References ==
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Wikipedia:Dmitrii Menshov#0
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Dmitrii Yevgenyevich Menshov (also spelled Men'shov, Menchoff, Menšov, Menchov; Russian: Дми́трий Евгéньевич Меньшóв; 18 April 1892 – 25 November 1988) was a Soviet and Russian mathematician known for his contributions to the theory of trigonometric series. == Biography == Dmitrii Menshov studied languages as a schoolboy, but from the age of 13 he began to show great interest in mathematics and physics. In 1911, he completed high school with a gold medal. After a semester at the Moscow Engineering School, he enrolled at Moscow State University in 1912 and became a student of Nikolai Luzin. In 1916, Menshov completed his dissertation on the topic of trigonometric series. He became a docent of Moscow State University in 1918. Soon after, he moved to Nizhny Novgorod where he was appointed a professor of the Ivanovsky Pedagogical Institute. After a few years, he returned to Moscow in 1922 and began to teach at Moscow State University. In 1935, Menshov became a full professor of Moscow State University and was awarded the title of Doctor of Physical and Mathematical Sciences. He gave lectures at Moscow State University and also Moscow State Pedagogical University in numerical analysis, complex functions, and differential equations to undergraduate and graduate students. In this position, he taught and influenced an entire generation of young up-and-coming Russian mathematicians and physicists, including such renowned scientists as his student Sergey Stechkin. He received the Stalin Prize in 1951 and was elected to the position of corresponding member of the Russian Academy of Sciences in 1953. His construction of a Fourier series with non-zero coefficients which converges to zero almost everywhere gave rise to the theory of Menshov sets. He proved the Rademacher–Menchov theorem, the Looman–Menchoff theorem, and the Lusin–Menchoff theorem. Menshov was an Invited Speaker of the ICM in 1928 in Bologna and in 1958 in Edinburgh. == References == Vinogradova, I. A.; Vladimirov, V. S.; Gonchar, A. A.; et al. (1989), "Dmitrii Evgen'evich Men'shov", Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 44 (5): 149–151, doi:10.1070/RM1989v044n05ABEH002213, ISSN 0042-1316, MR 1040273 == External links == O'Connor, John J.; Robertson, Edmund F., "Dmitrii Menshov", MacTutor History of Mathematics Archive, University of St Andrews Dmitrii Menshov at the Mathematics Genealogy Project Mini Biography – from the Russian Academy of Sciences (including picture)
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Wikipedia:Dmitry Chelkak#0
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Dmitry Sergeevich Chelkak (Дмитрий Сергеевич Челкак; born January 1979 in Leningrad) is a Russian American mathematician. Chelkak graduated from Saint Petersburg State University in 1995 with a diploma in 2000 and received his doctorate in 2003 from the Steklov Institute in Saint Petersburg. In 2000 he was with an Euler scholarship in Heidelberg and later in Potsdam. He is a senior researcher at the Steklov Institute in Saint Petersburg and was also a lecturer at the Saint Petersburg State University from 2004 to 2010 and at the Chebyshev Laboratory from 2010 to 2014. He was from 2014 to 2015 at ETH Zurich and from 2015 to 2016 a visiting professor in Geneva. His research deals with conformal invariance of two-dimensional lattice models at criticality, specifically the Ising models of statistical mechanics, in which he showed universality and conformal invariance at criticality with the Fields medalist Stanislav Smirnov. Chelkak also does research on spectral theory, especially inverse spectral problems of one-dimensional differential operators. In 1995 he received the gold medal at the International Mathematical Olympiad. In 2004 he was awarded the "Young Mathematician" Prize of the St. Petersburg Mathematical Society. In 2008 he received the Pierre Deligne Prize in Moscow. In 2014 he received the Salem Prize. In 2018 was an invited speaker at the International Congress of Mathematicians in Rio de Janeiro with talk Planar Ising model at criticality: state-of-the-art and perspectives. == Selected publications == Chelkak, D.; Kargaev, P.; Korotyaev, E. (2004). "Inverse Problem for Harmonic Oscillator Perturbed by Potential, Characterization". Communications in Mathematical Physics. 249 (1): 133–196. Bibcode:2004CMaPh.249..133C. doi:10.1007/s00220-004-1105-8. S2CID 119850806. Chelkak, D.; Korotyaev, E. (2006). "Spectral estimates for Schrödinger operators with periodic matrix potentials on the real line". International Mathematics Research Notices. 2006: 60314. doi:10.1155/IMRN/2006/60314. S2CID 17678384.{{cite journal}}: CS1 maint: unflagged free DOI (link) Chelkak, D.; Korotyaev, E. (2009). "Weyl–Titchmarsh functions of vector-valued Sturm–Liouville operators on the unit interval". Journal of Functional Analysis. 257 (5): 1546–1588. arXiv:0808.2547. doi:10.1016/j.jfa.2009.05.010. S2CID 16767606. Chelkak, D.; Smirnov, S. (2011). "Discrete complex analysis on isoradial graphs". Advances in Mathematics. 228 (3): 1590–1630. arXiv:0810.2188. doi:10.1016/j.aim.2011.06.025. S2CID 15161035. Chelkak, D.; Smirnov, S. (2012). "Universality in the 2D Ising model and conformal invariance of fermionic observables". Inventiones Mathematicae. 189 (3): 515–580. arXiv:0910.2045. Bibcode:2012InMat.189..515C. doi:10.1007/s00222-011-0371-2. S2CID 54789807. Chelkak, D.; Cimasoni, D.; Kassel, A. (2017). "Revisiting the combinatorics of the 2D Ising model". Annales de l'Institut Henri Poincaré D. 4 (3): 309–385. arXiv:1507.08242. Bibcode:2017AIHPD...4..309C. doi:10.4171/AIHPD/42. S2CID 116918297. == References == == External links == mathnet.ru "Dmitry Chelkak — 2D Ising model; combinatorics, CFT/CLE description at criticality and beyond". YouTube. IHÉS. 18 May 2017. "Planar Ising model at criticality: State-of-the-art and perspectives — Dmitry Chelkak". YouTube. Rio ICM2018. 2 October 2018. Dmitry Chelkak - Planar Ising model: from combinatorics to CFT and s-embeddings, Lectures 1–4, U. of Virginia Integrable Probability Summer School "Lecture 1". YouTube. 29 May 2019. "Lecture 2". YouTube. 29 May 2019. "Lecture 3". YouTube. 29 May 2019. "Lecture 4". YouTube. 31 May 2019.
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Wikipedia:Dmitry Dolgopyat#0
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Dmitry Dolgopyat is a Russian-American mathematician specializing in dynamical systems, a field that studies the time evolution of natural and abstract systems. An internationally acclaimed lecturer, he holds the position of Distinguished University Professor at the University of Maryland, and is a foreign member of the Academia Europaea. == Education == Dmitry Dolgopyat graduated from Moscow State School 57 mathematical class in 1989. From 1989 to 1994, he was an undergraduate student at Moscow State University. From 1994 to 1997, Dolgopyat was enrolled at Princeton University, where he earned a PhD under the guidance of Yakov Sinai. == Career == From September 1999 to June 2003, Dmitry Dolgopyat served as an assistant professor at Penn State University. Dolgopyat joined the University of Maryland as an associate professor from September 2002 to June 2006. During this period, he also spent a year at the Institute for Advanced Study (IAS) in Princeton (2002-2003). He briefly returned to Penn State University as a professor from September 2006 to June 2007 before settling at the University of Maryland as a professor in September 2007, a position he holds to the present day. Additionally, Dolgopyat spent a year at the University of Toronto and the Fields Institute from 2010 to 2011. He has also served on the editorial boards of the Journal of Modern Dynamics, Nonlinearity, Ergodic Theory and Dynamical Systems, Annales Henri Poincaré, and the Journal of the American Mathematical Society, as well as on the Prize Committee of the International Bolyai Prize Committee. == Recognition == Dolgopyat was named a Distinguished University Professor in 2022 - the highest academic honor bestowed by the University of Maryland. In 2020, he was elected a foreign member of the Academia Europaea - a pan-European Academy of Humanities, Letters, Law, and Sciences. In 2009, he was awarded the Michael Brin Prize in Dynamical Systems for his fundamental contributions to the theory of hyperbolic dynamics. In 2009, Dmitry Dolgopyat received the Annales Henri Poincaré Prize for the article Unbounded Orbits for Semicircular Outer Billiard coauthored with Bassam Fayad. Earlier academic honors included the Sloan Fellowship (Fall 2000 - Spring 2002) and the Miller Fellowship (Fall 1997 - Spring 1999). == Guest Lectures and Academic Visits == Dolgopyat was an invited speaker at the 2003 International Congress on Mathematical Physics in Lisbon, at the 2006 International Congress of Mathematicians in Madrid, plenary speaker at the 2006 Canadian Mathematical Society Winter Meeting in Toronto, 2009 International Workshop Dynamics Beyond Uniform Hyperbolicity at the Beijing International Center for Mathematical Research, and at the 2012 International Congress in Mathematical Physics in Aalborg, Denmark. delivered mini-courses at the International Centre for Theoretical Physics in Trieste, Italy, and at the Centro di Ricerca Matematica Ennio De Giorgi in Pisa Italy. Other longer mini-courses included DANCE (Dynamics, Attractors, Nonlinearity, Chaos & Stability) Winter School in Murcia, Spain, Program on Hyperbolic Dynamics in Vienna, School on Dynamics & Complexity at the University of the Republic (Uruguay), Special trimester on dynamical systems in Pisa, Dynamics Beyond Uniform Hyperbolicity in Evanston, with shorter visits to Manchester University, United Kingdom, Centro de Investigacion en Matematicas, Guanajuato, Mexico, Hebrew University, Jerusalem, Israel, Newton Institute, Cambridge, United Kingdom, Instituto de Matematica Pura e Aplicada, Rio de Janeiro, Brazil, Northwestern University, Evanston, Illinois, Scuola Normale, Pisa, Ecole Polytechnique, ETH, Zurich, IAS, Princeton, CalTech, Research Institute for Mathematical Sciences, Kyoto, Institut Henri Poincaré, Paris, Erwin Schroedinger Institute, Vienna, Centre International de Rencontres Mathématiques, Luminy, France, University of Toronto/ Fields Institute, University of Bristol, EPFL, Lausanne, Switzerland, Weizmann institute, Israel, Institute for Computational and Experimental Research in Mathematics, Providence, RI. == References ==
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Wikipedia:Dmitry Faddeev#0
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Dmitry Konstantinovich Faddeev (Russian: Дми́трий Константи́нович Фадде́ев, IPA: [ˈdmʲitrʲɪj kənstɐnʲˈtʲinəvʲɪtɕ fɐˈdʲe(j)ɪf]; 30 June 1907 – 20 October 1989) was a Soviet mathematician. == Biography == Dmitry was born June 30, 1907, about 200 kilometers southwest of Moscow on his father's estate. His father Konstantin Tikhonovich Faddeev was an engineer while his mother was a doctor and appreciator of music who instilled the love for music in Dmitry. Friends found his piano playing entertaining. In 1928 he graduated from Petrograd State University, as it was then called. His teachers included Ivan Matveyevich Vinogradov and Boris Nicolaevich Delone. In 1930 he married Vera Nicolaevna Zamyatina (Faddeeva). They had three children, including the mathematical physicist Ludvig Faddeev. Dmitry Faddeev's students included Mark Bashmakov (ru), Zenon Borevich, Lyudmyla Nazarova, Andrei Roiter, Alexander Skopin, and Anatoly Yakovlev (ru). == Contributions == D. K. Faddeev and V. N. Faddeeva co-authored Numerical Methods in Linear Algebra in 1960, followed by an enlarged edition in 1963. For instance, they developed an idea of Urbain Leverrier to produce an algorithm to find the resolvent matrix ( A − s I ) − 1 {\displaystyle (A-sI)^{-1}} of a given matrix A. By iteration, the method computed the adjugate matrix and characteristic polynomial for A. Dmitry was committed to mathematics education and aware of the need for graded sets of mathematical exercises. With Iliya Samuilovich Sominskii he wrote Problems in Higher Algebra. He was one of the founders of the Russian Mathematical Olympiads. He was one of the founders of the a Physics-Mathematics secondary school later named after him. == See also == Faddeev–LeVerrier algorithm == References == Aleksandrov, A. D.; Bashmakov, M. I.; Borevich, Z. I.; Kublanovskaya, V. N.; Nikulin, M. S.; Skopin, A. I.; Yakovlev, A. V. (1989). "Dmitrii Konstantinovich Faddeev (on his eightieth birthday)". Russian Mathematical Surveys. 44 (3). Translated by Lofthouse, A. Russian Academy of Sciences: 223–231. doi:10.1070/RM1989v044n03ABEH002126. ISSN 0042-1316. S2CID 250913337 – via Saint Petersburg Mathematical Society. Borevich, Z. I.; Linnik, Yu. V.; Skopin, A. I. (1968). "Дмитрий Константинович Фаддеев (к шестидесятилетию со дня рождения)" [Dmitrii Konstantinovich Faddeev (on his sixtieth birthday)]. Russian Mathematical Surveys (in Russian). 23 (3). Russian Academy of Sciences: 169–175. doi:10.1070/RM1968v023n03ABEH003777. ISSN 0042-1316. S2CID 250895230. == External links == O'Connor, John J.; Robertson, Edmund F., "Dmitry Faddeev", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Dmitry Grave#0
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Nikolai Grigorievich Chebotaryov (often spelled Chebotarov or Chebotarev; Russian: Никола́й Григо́рьевич Чеботарёв; Ukrainian: Мико́ла Григо́рович Чеботарьо́в; 15 June [O.S. 3 June] 1894 – 2 July 1947) was a Soviet mathematician. He is best known for the Chebotaryov density theorem. He was a student of Dmitry Grave. Chebotaryov worked on the algebra of polynomials, in particular examining the distribution of the zeros. He also studied Galois theory and wrote a textbook on the subject titled Basic Galois Theory. His ideas were used by Emil Artin to prove the Artin reciprocity law. He worked with his student Anatoly Dorodnov on a generalization of the quadrature of the lune, and proved the conjecture now known as the Chebotarev theorem on roots of unity. == Early life == Nikolai Chebotaryov was born on 15 June 1894 in Kamianets-Podilskyi, Russian Empire (now in Ukraine). He entered the department of physics and mathematics at Kiev University in 1912. In 1928, he became a professor at Kazan University, remaining there for the rest of his life. He died on 2 July 1947. He was an atheist. On 14 May 2010, a memorial plaque for Nikolai Chebotaryov was unveiled on the main administration building of I.I. Mechnikov Odessa National University. == References ==
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Wikipedia:Dmitry Ioffe#0
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Dmitry (Dima) Ioffe (April 5, 1963 - October 1, 2020) was an Israeli mathematician, specializing in probability theory. == Biography == Dmitry Ioffe obtained his diploma from the Moscow Mining Institute in 1985 and his PhD in mathematics in 1991 from the Technion, under the supervision of Ross Pinsky. He then spent a post-doc at the University of California, Davis and the Courant Institute. He was an assistant professor at Northwestern University (1993-1995) and a researcher at the Weierstrass Institute of Analysis and Stochastics (WIAS) in Berlin (1995-1997), before returning to the Technion, where he spent the rest of his life as professor. From 2014, he was the incumbent of the Alexander Goldberg chair in management sciences. == Scientific work == Ioffe made fundamental contributions to several areas of statistical mechanics, including random interface models, interacting particle systems, polymers in random environments, random perturbations of dynamical systems, metastability and homogenization. In particular, he extended the Dobrushin-Kotecky-Shlosman two dimensional Wulff construction to the full range of subcritical temperatures and developed with Bodineau and Velenik a robust analytic alternative that worked also in higher dimension. With collaborators, he developed the Ornstein-Zernike theory (at temperatures above criticality) and introduced a diamond representation for a range of models including self-avoiding walks, Bernoulli percolation, Ising ferromagnets and polymers. He also made important contributions to the analysis of quantum spin systems and metastability. == Personal life == Ioffe's family applied for permission to leave the USSR for Israel in 1976 but was refused and hence he became refusenik. It was only in 1987, following a hunger strike by his father, mathematician Alexander Ioffe, that Dmitry and his family were allowed to emigrate. == Prizes and honors == Ioffe got the Prix de l’Institut Henri Poincare (2005) and a Humboldt research award (2011). == Selected publications == Ioffe, Dmitry (1995). "Exact large deviation bounds up to Tc for the Ising model in two dimensions". Probability Theory and Related Fields. 102 (3): 313–330. doi:10.1007/BF01192464. S2CID 1332517. Ioffe, Dmitry; Schonmann, Roberto (1998). "Dobrushin-Kotecky-Shlosman theorem up to the critical temperature". Communications in Mathematical Physics. 199 (1): 117–167. Bibcode:1998CMaPh.199..117I. doi:10.1007/s002200050497. S2CID 11896158. Bodineau, Thierry; Ioffe, Dmitry; Velenik, Yvan (2001). "Winterbottom construction for finite range ferromagnetic models: an L1-approach". Journal of Statistical Physics. 105: 93–131. arXiv:math/0101174. doi:10.1023/A:1012277926007. Campanino, Massimo; Ioffe, Dmitry; Velenik, Yvan (2003). "Ornstein-Zernike theory for finite range Ising models above Tc". Probability Theory and Related Fields. 125 (3): 305–349. arXiv:math/0111274. doi:10.1007/s00440-002-0229-z. S2CID 18054507. Ioffe, Dmitry (2009). "Stochastic geometry of classical and quantum Ising models". Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics. Vol. 2144. pp. 87–127. arXiv:math-ph/0610025. doi:10.1007/978-3-540-92796-9_2. ISBN 978-3-540-92795-2. S2CID 119670695. Ioffe, Dmitry; Velenik, Yvan (2012). "Crossing random walks and stretched polymers at weak disorder". The Annals of Probability. 40 (2): 209–235. arXiv:1002.4289. doi:10.1214/10-AOP625. S2CID 54771817. Bianchi, Alessandra; Bovier, Anton; Ioffe, Dmitry (2012). "Pointwise estimates and exponential laws in metastable systems via coupling methods". The Annals of Probability. 40: 339–371. arXiv:0909.1242. doi:10.1214/10-AOP622. S2CID 14138474. Ioffe, Dmitry (2015). "Multidimensional Random Polymers: A Renewal Approach". Random Walks, Random Fields, and Disordered Systems. Lecture Notes in Mathematics. Vol. 11. pp. 339–369. arXiv:1412.0229. doi:10.1007/978-3-319-19339-7_4. ISBN 978-3-319-19338-0. S2CID 117484180. == References == == External links == Dmitry Ioffe, Low temperature interfaces and level lines in the critical prewetting regime, CIRM Dmitry Ioffe, Technion. Dmitry Ioffe, Minerva foundation Obituary. Obituary: Medallion lecturer Dmitry Ioffe, Institute of Mathematical Statistics. Obituary to Dmitry Ioffe, Bonn University.
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Wikipedia:Dmitry Kramkov#0
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Dmitry Olegovich Kramkov (Russian: Дмитрий Олегович Крамков) is a Russian mathematician at Carnegie Mellon University. His research field are statistics and financial mathematics. Kramkov obtained his doctorate from Steklov Institute of Mathematics in 1992, under supervision of Albert Shiryaev. In 1996 he was awarded an EMS Prize for his work in filtered statistical experiments. Kramkov's optional decomposition theorem is named after him. == References ==
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Wikipedia:Dmitry Matveyevich Smirnov#0
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Dmitry Matveyevich Smirnov (Russian: Дмитрий Матвеевич Смирнов; 27 October 1919 in Shilovo, Seredskii District, Ivanovo Oblast, Soviet Union – 14 April 2005) was a Soviet mathematician working in group theory and Jónsson–Tarski algebras. == References == == Bibliography == "Dmitrii Matveevich Smirnov (on his fiftieth birthday)" (PDF), Algebra i Logika, 8 (4): 493–496, 1969 Volʹbot, A. D.; Goncharov, S. S.; Gorbuno, V. A. (1989), "Dmitriĭ Matveevich Smirnov (on the occasion of his seventieth birthday)", Algebra i Logika, 28 (5): 491–492, MR 1087567 Dmitry Matveyevich Smirnov at the Mathematics Genealogy Project
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Wikipedia:Dmitry Sychugov#0
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Dmitry Sychugov (Russian: Дми́трий Ю́рьевич Сычу́гов) (born 1955) is a Russian mathematician, Dr.Sc., Professor, a professor at the Faculty of Computer Science at the Moscow State University. He graduated from the faculty MSU CMC (1976). Has been working at Moscow State University since 1980. He defended the thesis «Mathematical modeling of plasma confinement processes in toroidal traps» for the degree of Doctor of Physical and Mathematical Sciences (2013). He is the author of five books and more than 80 scientific articles. Area of scientific interests: mathematical modeling, computational physics of plasma. == References == == Bibliography == Grigoriev, Evgeny (2010). Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory. Moscow: Publishing house of Moscow University. pp. 205–206. ISBN 978-5-211-05838-5. == External links == MSU CMC(in Russian) Scientific works of Dmitry Sychugov Scientific works of Dmitry Sychugov(in English)
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Wikipedia:Domain (mathematical analysis)#0
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In mathematical analysis, a domain or region is a non-empty, connected, and open set in a topological space. In particular, it is any non-empty connected open subset of the real coordinate space Rn or the complex coordinate space Cn. A connected open subset of coordinate space is frequently used for the domain of a function. The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the term domain, some use the term region, some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as non-empty connected open subset. == Conventions == One common convention is to define a domain as a connected open set but a region as the union of a domain with none, some, or all of its limit points. A closed region or closed domain is the union of a domain and all of its limit points. Various degrees of smoothness of the boundary of the domain are required for various properties of functions defined on the domain to hold, such as integral theorems (Green's theorem, Stokes theorem), properties of Sobolev spaces, and to define measures on the boundary and spaces of traces (generalized functions defined on the boundary). Commonly considered types of domains are domains with continuous boundary, Lipschitz boundary, C1 boundary, and so forth. A bounded domain is a domain that is bounded, i.e., contained in some ball. Bounded region is defined similarly. An exterior domain or external domain is a domain whose complement is bounded; sometimes smoothness conditions are imposed on its boundary. In complex analysis, a complex domain (or simply domain) is any connected open subset of the complex plane C. For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function. In the study of several complex variables, the definition of a domain is extended to include any connected open subset of Cn. In Euclidean spaces, one-, two-, and three-dimensional regions are curves, surfaces, and solids, whose extent are called, respectively, length, area, and volume. == Historical notes == Definition. An open set is connected if it cannot be expressed as the sum of two open sets. An open connected set is called a domain. German: Eine offene Punktmenge heißt zusammenhängend, wenn man sie nicht als Summe von zwei offenen Punktmengen darstellen kann. Eine offene zusammenhängende Punktmenge heißt ein Gebiet. According to Hans Hahn, the concept of a domain as an open connected set was introduced by Constantin Carathéodory in his famous book (Carathéodory 1918). In this definition, Carathéodory considers obviously non-empty disjoint sets. Hahn also remarks that the word "Gebiet" ("Domain") was occasionally previously used as a synonym of open set. The rough concept is older. In the 19th and early 20th century, the terms domain and region were often used informally (sometimes interchangeably) without explicit definition. However, the term "domain" was occasionally used to identify closely related but slightly different concepts. For example, in his influential monographs on elliptic partial differential equations, Carlo Miranda uses the term "region" to identify an open connected set, and reserves the term "domain" to identify an internally connected, perfect set, each point of which is an accumulation point of interior points, following his former master Mauro Picone: according to this convention, if a set A is a region then its closure A is a domain. == See also == Analytic polyhedron – Subset of complex n-space bounded by analytic functions Caccioppoli set – Region with boundary of finite measure Hermitian symmetric space#Classical domains – Manifold with inversion symmetry Interval (mathematics) – All numbers between two given numbers Lipschitz domain Whitehead's point-free geometry – Geometric theory based on regions == Notes == == References == Ahlfors, Lars (1953). Complex Analysis. McGraw-Hill. Bremermann, H. J. (1956). "Complex Convexity". Transactions of the American Mathematical Society. 82 (1): 17–51. doi:10.1090/S0002-9947-1956-0079100-2. JSTOR 1992976. Carathéodory, Constantin (1918). Vorlesungen über reelle Funktionen [Lectures on real functions] (in German). B. G. Teubner. JFM 46.0376.12. MR 0225940. Reprinted 1968 (Chelsea). Carathéodory, Constantin (1964) [1954]. Theory of Functions of a Complex Variable, vol. I (2nd ed.). Chelsea. English translation of Carathéodory, Constantin (1950). Functionentheorie I (in German). Birkhäuser. Carrier, George; Krook, Max; Pearson, Carl (1966). Functions of a Complex Variable: Theory and Technique. McGraw-Hill. Churchill, Ruel (1948). Introduction to Complex Variables and Applications (1st ed.). McGraw-Hill.Churchill, Ruel (1960). Complex Variables and Applications (2nd ed.). McGraw-Hill. ISBN 9780070108530. {{cite book}}: ISBN / Date incompatibility (help) Dieudonné, Jean (1960). Foundations of Modern Analysis. Academic Press. Eves, Howard (1966). Functions of a Complex Variable. Prindle, Weber & Schmidt. p. 105. Forsyth, Andrew (1893). Theory of Functions of a Complex Variable. Cambridge. JFM 25.0652.01. Fuchs, Boris; Shabat, Boris (1964). Functions of a complex variable and some of their applications, vol. 1. Pergamon. English translation of Фукс, Борис; Шабат, Борис (1949). Функции комплексного переменного и некоторые их приложения (PDF) (in Russian). Физматгиз. Goursat, Édouard (1905). Cours d'analyse mathématique, tome 2 [A course in mathematical analysis, vol. 2] (in French). Gauthier-Villars. Hahn, Hans (1921). Theorie der reellen Funktionen. Erster Band [Theory of Real Functions, vol. I] (in German). Springer. JFM 48.0261.09. Krantz, Steven; Parks, Harold (1999). The Geometry of Domains in Space. Birkhäuser. Kreyszig, Erwin (1972) [1962]. Advanced Engineering Mathematics (3rd ed.). Wiley. ISBN 9780471507284. Kwok, Yue-Kuen (2002). Applied Complex Variables for Scientists and Engineers. Cambridge. Miranda, Carlo (1955). Equazioni alle derivate parziali di tipo ellittico (in Italian). Springer. MR 0087853. Zbl 0065.08503. Translated as Miranda, Carlo (1970). Partial Differential Equations of Elliptic Type. Translated by Motteler, Zane C. (2nd ed.). Springer. MR 0284700. Zbl 0198.14101. Picone, Mauro (1923). "Parte Prima – La Derivazione" (PDF). Lezioni di analisi infinitesimale, vol. I [Lessons in infinitesimal analysis] (in Italian). Circolo matematico di Catania. JFM 49.0172.07. Rudin, Walter (1974) [1966]. Real and Complex Analysis (2nd ed.). McGraw-Hill. ISBN 9780070542334. Solomentsev, Evgeny (2001) [1994], "Domain", Encyclopedia of Mathematics, EMS Press Sveshnikov, Aleksei; Tikhonov, Andrey (1978). The Theory Of Functions Of A Complex Variable. Mir. English translation of Свешников, Алексей; Ти́хонов, Андре́й (1967). Теория функций комплексной переменной (in Russian). Наука. Townsend, Edgar (1915). Functions of a Complex Variable. Holt. Whittaker, Edmund (1902). A Course Of Modern Analysis (1st ed.). Cambridge. JFM 33.0390.01. Whittaker, Edmund; Watson, George (1915). A Course Of Modern Analysis (2nd ed.). Cambridge.
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Wikipedia:Domain of a function#0
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In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname {dom} (f)} or dom f {\displaystyle \operatorname {dom} f} , where f is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be". More precisely, given a function f : X → Y {\displaystyle f\colon X\to Y} , the domain of f is X. In modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that X and Y are both sets of real numbers, the function f can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the x-axis of the graph, as the projection of the graph of the function onto the x-axis. For a function f : X → Y {\displaystyle f\colon X\to Y} , the set Y is called the codomain: the set to which all outputs must belong. The set of specific outputs the function assigns to elements of X is called its range or image. The image of f is a subset of Y, shown as the yellow oval in the accompanying diagram. Any function can be restricted to a subset of its domain. The restriction of f : X → Y {\displaystyle f\colon X\to Y} to A {\displaystyle A} , where A ⊆ X {\displaystyle A\subseteq X} , is written as f | A : A → Y {\displaystyle \left.f\right|_{A}\colon A\to Y} . == Natural domain == If a real function f is given by a formula, it may be not defined for some values of the variable. In this case, it is a partial function, and the set of real numbers on which the formula can be evaluated to a real number is called the natural domain or domain of definition of f. In many contexts, a partial function is called simply a function, and its natural domain is called simply its domain. === Examples === The function f {\displaystyle f} defined by f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} cannot be evaluated at 0. Therefore, the natural domain of f {\displaystyle f} is the set of real numbers excluding 0, which can be denoted by R ∖ { 0 } {\displaystyle \mathbb {R} \setminus \{0\}} or { x ∈ R : x ≠ 0 } {\displaystyle \{x\in \mathbb {R} :x\neq 0\}} . The piecewise function f {\displaystyle f} defined by f ( x ) = { 1 / x x ≠ 0 0 x = 0 , {\displaystyle f(x)={\begin{cases}1/x&x\not =0\\0&x=0\end{cases}},} has as its natural domain the set R {\displaystyle \mathbb {R} } of real numbers. The square root function f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} has as its natural domain the set of non-negative real numbers, which can be denoted by R ≥ 0 {\displaystyle \mathbb {R} _{\geq 0}} , the interval [ 0 , ∞ ) {\displaystyle [0,\infty )} , or { x ∈ R : x ≥ 0 } {\displaystyle \{x\in \mathbb {R} :x\geq 0\}} . The tangent function, denoted tan {\displaystyle \tan } , has as its natural domain the set of all real numbers which are not of the form π 2 + k π {\displaystyle {\tfrac {\pi }{2}}+k\pi } for some integer k {\displaystyle k} , which can be written as R ∖ { π 2 + k π : k ∈ Z } {\displaystyle \mathbb {R} \setminus \{{\tfrac {\pi }{2}}+k\pi :k\in \mathbb {Z} \}} . == Other uses == The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis, a domain is a non-empty connected open subset of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} or the complex coordinate space C n . {\displaystyle \mathbb {C} ^{n}.} Sometimes such a domain is used as the domain of a function, although functions may be defined on more general sets. The two concepts are sometimes conflated as in, for example, the study of partial differential equations: in that case, a domain is the open connected subset of R n {\displaystyle \mathbb {R} ^{n}} where a problem is posed, making it both an analysis-style domain and also the domain of the unknown function(s) sought. == Set theoretical notions == For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form f: X → Y. == See also == Argument of a function Attribute domain Bijection, injection and surjection Codomain Domain decomposition Effective domain Endofunction Image (mathematics) Lipschitz domain Naive set theory Range of a function Support (mathematics) == Notes == == References == Bourbaki, Nicolas (1970). Théorie des ensembles. Éléments de mathématique. Springer. ISBN 9783540340348. Eccles, Peter J. (11 December 1997). An Introduction to Mathematical Reasoning: Numbers, Sets and Functions. Cambridge University Press. ISBN 978-0-521-59718-0. Mac Lane, Saunders (25 September 1998). Categories for the Working Mathematician. Springer Science & Business Media. ISBN 978-0-387-98403-2. Scott, Dana S.; Jech, Thomas J. (31 December 1971). Axiomatic Set Theory, Part 1. American Mathematical Soc. ISBN 978-0-8218-0245-8. Sharma, A. K. (2010). Introduction To Set Theory. Discovery Publishing House. ISBN 978-81-7141-877-0. Stewart, Ian; Tall, David (1977). The Foundations of Mathematics. Oxford University Press. ISBN 978-0-19-853165-4.
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Wikipedia:Domain theory#0
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Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and is closely related to topology. == Motivation and intuition == The primary motivation for the study of domains, which was initiated by Dana Scott in the late 1960s, was the search for a denotational semantics of the lambda calculus. In this formalism, one considers "functions" specified by certain terms in the language. In a purely syntactic way, one can go from simple functions to functions that take other functions as their input arguments. Using again just the syntactic transformations available in this formalism, one can obtain so-called fixed-point combinators (the best-known of which is the Y combinator); these, by definition, have the property that f(Y(f)) = Y(f) for all functions f. To formulate such a denotational semantics, one might first try to construct a model for the lambda calculus, in which a genuine (total) function is associated with each lambda term. Such a model would formalize a link between the lambda calculus as a purely syntactic system and the lambda calculus as a notational system for manipulating concrete mathematical functions. The combinator calculus is such a model. However, the elements of the combinator calculus are functions from functions to functions; in order for the elements of a model of the lambda calculus to be of arbitrary domain and range, they could not be true functions, only partial functions. Scott got around this difficulty by formalizing a notion of "partial" or "incomplete" information to represent computations that have not yet returned a result. This was modeled by considering, for each domain of computation (e.g. the natural numbers), an additional element that represents an undefined output, i.e. the "result" of a computation that never ends. In addition, the domain of computation is equipped with an ordering relation, in which the "undefined result" is the least element. The important step to finding a model for the lambda calculus is to consider only those functions (on such a partially ordered set) that are guaranteed to have least fixed points. The set of these functions, together with an appropriate ordering, is again a "domain" in the sense of the theory. But the restriction to a subset of all available functions has another great benefit: it is possible to obtain domains that contain their own function spaces, i.e. one gets functions that can be applied to themselves. Beside these desirable properties, domain theory also allows for an appealing intuitive interpretation. As mentioned above, the domains of computation are always partially ordered. This ordering represents a hierarchy of information or knowledge. The higher an element is within the order, the more specific it is and the more information it contains. Lower elements represent incomplete knowledge or intermediate results. Computation then is modeled by applying monotone functions repeatedly on elements of the domain in order to refine a result. Reaching a fixed point is equivalent to finishing a calculation. Domains provide a superior setting for these ideas since fixed points of monotone functions can be guaranteed to exist and, under additional restrictions, can be approximated from below. == A guide to the formal definitions == In this section, the central concepts and definitions of domain theory will be introduced. The above intuition of domains being information orderings will be emphasized to motivate the mathematical formalization of the theory. The precise formal definitions are to be found in the dedicated articles for each concept. A list of general order-theoretic definitions, which include domain theoretic notions as well can be found in the order theory glossary. The most important concepts of domain theory will nonetheless be introduced below. === Directed sets as converging specifications === As mentioned before, domain theory deals with partially ordered sets to model a domain of computation. The goal is to interpret the elements of such an order as pieces of information or (partial) results of a computation, where elements that are higher in the order extend the information of the elements below them in a consistent way. From this simple intuition it is already clear that domains often do not have a greatest element, since this would mean that there is an element that contains the information of all other elements—a rather uninteresting situation. A concept that plays an important role in the theory is that of a directed subset of a domain; a directed subset is a non-empty subset of the order in which any two elements have an upper bound that is an element of this subset. In view of our intuition about domains, this means that any two pieces of information within the directed subset are consistently extended by some other element in the subset. Hence we can view directed subsets as consistent specifications, i.e. as sets of partial results in which no two elements are contradictory. This interpretation can be compared with the notion of a convergent sequence in analysis, where each element is more specific than the preceding one. Indeed, in the theory of metric spaces, sequences play a role that is in many aspects analogous to the role of directed sets in domain theory. Now, as in the case of sequences, we are interested in the limit of a directed set. According to what was said above, this would be an element that is the most general piece of information that extends the information of all elements of the directed set, i.e. the unique element that contains exactly the information that was present in the directed set, and nothing more. In the formalization of order theory, this is just the least upper bound of the directed set. As in the case of the limit of a sequence, the least upper bound of a directed set does not always exist. Naturally, one has a special interest in those domains of computations in which all consistent specifications converge, i.e. in orders in which all directed sets have a least upper bound. This property defines the class of directed-complete partial orders, or dcpo for short. Indeed, most considerations of domain theory do only consider orders that are at least directed complete. From the underlying idea of partially specified results as representing incomplete knowledge, one derives another desirable property: the existence of a least element. Such an element models that state of no information—the place where most computations start. It also can be regarded as the output of a computation that does not return any result at all. === Computations and domains === Now that we have some basic formal descriptions of what a domain of computation should be, we can turn to the computations themselves. Clearly, these have to be functions, taking inputs from some computational domain and returning outputs in some (possibly different) domain. However, one would also expect that the output of a function will contain more information when the information content of the input is increased. Formally, this means that we want a function to be monotonic. When dealing with dcpos, one might also want computations to be compatible with the formation of limits of a directed set. Formally, this means that, for some function f, the image f(D) of a directed set D (i.e. the set of the images of each element of D) is again directed and has as a least upper bound the image of the least upper bound of D. One could also say that f preserves directed suprema. Also note that, by considering directed sets of two elements, such a function also has to be monotonic. These properties give rise to the notion of a Scott-continuous function. Since this often is not ambiguous one also may speak of continuous functions. === Approximation and finiteness === Domain theory is a purely qualitative approach to modeling the structure of information states. One can say that something contains more information, but the amount of additional information is not specified. Yet, there are some situations in which one wants to speak about elements that are in a sense much simpler (or much more incomplete) than a given state of information. For example, in the natural subset-inclusion ordering on some powerset, any infinite element (i.e. set) is much more "informative" than any of its finite subsets. If one wants to model such a relationship, one may first want to consider the induced strict order < of a domain with order ≤. However, while this is a useful notion in the case of total orders, it does not tell us much in the case of partially ordered sets. Considering again inclusion-orders of sets, a set is already strictly smaller than another, possibly infinite, set if it contains just one less element. One would, however, hardly agree that this captures the notion of being "much simpler". === Way-below relation === A more elaborate approach leads to the definition of the so-called order of approximation, which is more suggestively also called the way-below relation. An element x is way below an element y, if, for every directed set D with supremum such that y ⊑ sup D {\displaystyle y\sqsubseteq \sup D} , there is some element d in D such that x ⊑ d {\displaystyle x\sqsubseteq d} . Then one also says that x approximates y and writes x ≪ y {\displaystyle x\ll y} . This does imply that x ⊑ y {\displaystyle x\sqsubseteq y} , since the singleton set {y} is directed. For an example, in an ordering of sets, an infinite set is way above any of its finite subsets. On the other hand, consider the directed set (in fact, the chain) of finite sets { 0 } , { 0 , 1 } , { 0 , 1 , 2 } , … {\displaystyle \{0\},\{0,1\},\{0,1,2\},\ldots } Since the supremum of this chain is the set of all natural numbers N, this shows that no infinite set is way below N. However, being way below some element is a relative notion and does not reveal much about an element alone. For example, one would like to characterize finite sets in an order-theoretic way, but even infinite sets can be way below some other set. The special property of these finite elements x is that they are way below themselves, i.e. x ≪ x {\displaystyle x\ll x} . An element with this property is also called compact. Yet, such elements do not have to be "finite" nor "compact" in any other mathematical usage of the terms. The notation is nonetheless motivated by certain parallels to the respective notions in set theory and topology. The compact elements of a domain have the important special property that they cannot be obtained as a limit of a directed set in which they did not already occur. Many other important results about the way-below relation support the claim that this definition is appropriate to capture many important aspects of a domain. === Bases of domains === The previous thoughts raise another question: is it possible to guarantee that all elements of a domain can be obtained as a limit of much simpler elements? This is quite relevant in practice, since we cannot compute infinite objects but we may still hope to approximate them arbitrarily closely. More generally, we would like to restrict to a certain subset of elements as being sufficient for getting all other elements as least upper bounds. Hence, one defines a base of a poset P as being a subset B of P, such that, for each x in P, the set of elements in B that are way below x contains a directed set with supremum x. The poset P is a continuous poset if it has some base. Especially, P itself is a base in this situation. In many applications, one restricts to continuous (d)cpos as a main object of study. Finally, an even stronger restriction on a partially ordered set is given by requiring the existence of a base of finite elements. Such a poset is called algebraic. From the viewpoint of denotational semantics, algebraic posets are particularly well-behaved, since they allow for the approximation of all elements even when restricting to finite ones. As remarked before, not every finite element is "finite" in a classical sense and it may well be that the finite elements constitute an uncountable set. In some cases, however, the base for a poset is countable. In this case, one speaks of an ω-continuous poset. Accordingly, if the countable base consists entirely of finite elements, we obtain an order that is ω-algebraic. === Special types of domains === A simple special case of a domain is known as an elementary or flat domain. This consists of a set of incomparable elements, such as the integers, along with a single "bottom" element considered smaller than all other elements. One can obtain a number of other interesting special classes of ordered structures that could be suitable as "domains". We already mentioned continuous posets and algebraic posets. More special versions of both are continuous and algebraic cpos. Adding even further completeness properties one obtains continuous lattices and algebraic lattices, which are just complete lattices with the respective properties. For the algebraic case, one finds broader classes of posets that are still worth studying: historically, the Scott domains were the first structures to be studied in domain theory. Still wider classes of domains are constituted by SFP-domains, L-domains, and bifinite domains. All of these classes of orders can be cast into various categories of dcpos, using functions that are monotone, Scott-continuous, or even more specialized as morphisms. Finally, note that the term domain itself is not exact and thus is only used as an abbreviation when a formal definition has been given before or when the details are irrelevant. == Important results == A poset D is a dcpo if and only if each chain in D has a supremum. (The 'if' direction relies on the axiom of choice.) If f is a continuous function on a domain D then it has a least fixed point, given as the least upper bound of all finite iterations of f on the least element ⊥: fix ( f ) = ⨆ n ∈ N f n ( ⊥ ) {\displaystyle \operatorname {fix} (f)=\bigsqcup _{n\in \mathbb {N} }f^{n}(\bot )} . This is the Kleene fixed-point theorem. The ⊔ {\displaystyle \sqcup } symbol is the directed join. == Generalizations == A continuity space is a generalization of metric spaces and posets that can be used to unify the notions of metric spaces and domains. == See also == Category theory Denotational semantics Scott domain Scott information system Type theory == Further reading == G. Gierz; K. H. Hofmann; K. Keimel; J. D. Lawson; M. Mislove; D. S. Scott (2003). "Continuous Lattices and Domains". Encyclopedia of Mathematics and its Applications. Vol. 93. Cambridge University Press. ISBN 0-521-80338-1. Samson Abramsky, Achim Jung (1994). "Domain theory" (PDF). In S. Abramsky; D. M. Gabbay; T. S. E. Maibaum (eds.). Handbook of Logic in Computer Science. Vol. III. Oxford University Press. pp. 1–168. ISBN 0-19-853762-X. Retrieved 2007-10-13. Alex Simpson (2001–2002). "Part III: Topological Spaces from a Computational Perspective". Mathematical Structures for Semantics. Archived from the original on 2005-04-27. Retrieved 2007-10-13. D. S. Scott (1975). "Data types as lattices". In Müller, G.H.; Oberschelp, A.; Potthoff, K. (eds.). ISILC Logic Conference. Lecture Notes in Mathematics. Vol. 499. Springer-Verlag. pp. 579–651. doi:10.1007/BFb0079432. ISBN 978-3-540-07534-9. Scott, Dana (1976). "Data Types as Lattices". SIAM Journal on Computing. 5 (3): 522–587. doi:10.1137/0205037. Carl A. Gunter (1992). Semantics of Programming Languages. MIT Press. ISBN 9780262570954. B. A. Davey; H. A. Priestley (2002). Introduction to Lattices and Order (2nd ed.). Cambridge University Press. ISBN 0-521-78451-4. Carl Hewitt; Henry Baker (August 1977). "Actors and Continuous Functionals" (PDF). Proceedings of IFIP Working Conference on Formal Description of Programming Concepts. Archived (PDF) from the original on April 12, 2019. V. Stoltenberg-Hansen; I. Lindstrom; E. R. Griffor (1994). Mathematical Theory of Domains. Cambridge University Press. ISBN 0-521-38344-7. == External links == Introduction to Domain Theory by Graham Hutton, University of Nottingham
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Wikipedia:Dominique Perrin#0
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Dominique Pierre Perrin (b. 1946) is a French mathematician and theoretical computer scientist known for his contributions to coding theory and to combinatorics on words. He is a professor of the University of Marne-la-Vallée and currently serves as the President of ESIEE Paris. == Biography == Perrin earned his PhD from Paris 7 University in 1975. In his early career, he was a CNRS researcher (1970–1977) and taught at the University of Chile (1972–1973). Later, he worked as a professor at the University of Rouen (1977–1983), Paris 7 University (1983–1993), and École Polytechnique (1982–2002). Since 1993, Perrin is a professor at the University of Marne-la-Vallée, and since 2004, he is the President of ESIEE Paris. Perrin is a member of Academia Europaea since 1989. == Scientific contributions == Perrin has been a member of the Lothaire group of mathematicians that developed the foundations of combinatorics on words. He has co-authored three scientific monographs: "Theory of Codes" (1985), "Codes and Automata" (2009), and "Infinite Words" (2004), as well as the three Lothaire books. Perrin has published around 50 research articles in formal language theory. == References == == External links == Dominique Perrin at DBLP Bibliography Server Dominique Perrin at the Mathematics Genealogy Project
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Wikipedia:Dominique Picard#0
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Dominique Brigitte Picard (born March 3, 1952) is a French mathematician who works as a professor in the Laboratoire de Probabilités et Modèles Aléatoires of Paris Diderot University. Her research concerns the statistical applications of wavelets. == Education == Picard's doctoral advisor was Didier Dacunha-Castelle. == Recognition == She was an invited speaker at the International Congress of Mathematicians in 2006, in the section on probability and statistics. At the congress, she spoke on her work with Gérard Kerkyacharian on "Estimation in inverse problems and second-generation wavelets". She was elected to the National Academy of Sciences as an International Member in 2023. == Selected publications == With Valentine Genon-Catalot, Picard is the author of a book on asymptotic theory in statistics, Elements De Statistique Asymptotique (Springer, 1993). With Wolfgang Härdle, Gerard Kerkyacharian, and Alexander Tsybakov, she is the author of Wavelets, Approximation, and Statistical Applications (Springer, Lecture Notes in Statistics, 1998). She is also the coauthor of a highly-cited paper in the Journal of the Royal Statistical Society (1995) surveying the wavelet-shrinkage method for nonparametric curve estimation. == References == == External links == Home page Dominique Picard publications indexed by Google Scholar
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Wikipedia:Domninus of Larissa#0
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Domninus of Larissa (Greek: Δομνῖνος; c. 420 – c. 480) was an ancient Hellenistic Syrian mathematician. == Life == Domninus of Larissa, Syria was, simultaneously with Proclus, a pupil of Syrianus. Domninus is said to have corrupted the doctrines of Plato by mixing up with them his private opinions. This called forth a treatise from Proclus, intended as a statement of the genuine principles of Platonism. Marinus writes about a rivalry between Domninus and Proclus about how Plato's work should be interpreted, [Syrianus] offered to discourse to them on either the Orphic theories or the oracles; but Domninus wanted Orphism, Proclus the oracles, and they had not agreed when Syrianus died... The Athenian academy eventually choose Proclus' interpretation over Domninus' and Proclus would later become the head of the academy. After Proclus' promotion, Domninus left Athens and returned to Larissa. It is said that once when Domninus was ill and coughing up blood, he took to eating copious amounts of pork, despite the fact that he was Jewish, because a physician prescribed it as a treatment. He is also said to have taught Asclepiodotus, until Asclepiodotus became so argumentative that Domninus no longer admitted him into his company. == Works == Domninus is remembered for authoring a Manual of Introductory Arithmetic (Greek: Ἐγχειρίδιον ἀριθμητικῆς εἰσαγωγῆς), which was edited by Boissonade and had two articles by Tannery written about it. The Manual of Introductory Arithmetic was a concise and well arranged overview of the theory of numbers. It covered numbers, proportions and means. It is important since it is a reaction against Nicomachus' Introductio arithmetica and a return to the doctrine of Euclid. Domninus is also believed to have authored a tract entitled how a ratio can be taken out of a ratio (Greek: Πῶς ἔστι λόγον ἐκ λόγου ἀφελεῖν), which studies the manipulation of ratios into other forms. Bulmer-Thomas believe that it was written, at least in part, by Domninus, but Heath casts some doubt on the authorship by stating that if it wasn't written by Domninus then it at least comes from the same period as him. Domninus may have also written a work entitled Elements of Arithmetic as referred to near the end of his Manual of Introductory Arithmetic, although whether or not he ever wrote this book is unknown. == See also == Heliodorus of Larissa == Citations and footnotes == == References == Heath, Thomas Little (1981). A History of Greek Mathematics, Volume II. Dover publications. ISBN 0-486-24074-6. Ivor Bulmer-Thomas, Biography in Dictionary of Scientific Biography (New York 1970–1990). Peter Brown, The Manual of Domninus in Harvard Review of Philosophy (2000) == External links == O'Connor, John J.; Robertson, Edmund F., "Domninus of Larissa", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Dona Strauss#0
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Dona Anschel Papert Strauss (born April 1934) is a South African mathematician working in topology and functional analysis. Her doctoral thesis was one of the initial sources of pointless topology. She has also been active in the political left, lost one of her faculty positions over her protests of the Vietnam War, and became a founder of European Women in Mathematics. Mathematician Neil Hindman, with whom Strauss wrote a book on the Stone–Čech compactification of topological semigroups, has stated the following as advice for other mathematicians: "Find someone who is smarter than you are and get them to put your name on their papers", writing that for him, that someone was Dona Strauss. == Education and career == Strauss is originally from South Africa, the descendant of Jewish immigrants from Eastern Europe. Her father was a physicist at the University of Cape Town. She grew up in the Eastern Cape, and earned a master's degree in mathematics at the University of Cape Town. She completed her Ph.D. at the University of Cambridge in 1958. Her dissertation, Lattices of Functions, Measures, and Open Sets, was supervised by Frank Smithies. After completing her doctorate, she took a faculty position at the University of London. Following her husband's dream of living on a farm in Vermont, she moved to Dartmouth College in 1966. By 1972, she was working at the University of Hull and circa 2008 she became a professor at the University of Leeds. After retiring, she has been listed by Leeds as an honorary visiting fellow. == Activism == In South Africa, Strauss developed a strong antipathy to racial discrimination from a combination of being a Jew at the time of the Holocaust and her own observations of South African society. At the University of Cape Town, she became a member of the Non-European Unity Movement. After completing her degree, she left the country in protest over apartheid; her parents also left South Africa, after her father's retirement, for Israel. In the 1950s, she regularly published editorial works in Socialist Review, and in the 1960s she was active in Solidarity (UK). As an assistant professor at Dartmouth College in 1969, Strauss took part in a student anti-war protest that occupied Parkhurst Hall, the building that housed the college administration. In response, Dartmouth announced that Strauss and another faculty protester would not have their contracts renewed, and that they would be suspended from the faculty and "denied all rights and privileges of membership on the Dartmouth faculty", the first time in the college's history that it had taken this step. In 1986, Strauss became one of the five founders of European Women in Mathematics, together with Bodil Branner, Caroline Series, Gudrun Kalmbach, and Marie-Françoise Roy. == Books == Strauss is the co-author of: Algebra in the Stone-Čech compactification: Theory and applications (with Neil Hindman, De Gruyter Expositions in Mathematics 27, Walter de Gruyter & Co., 1998; 2nd ed., 2012) Banach algebras on semigroups and on their compactifications (with H. Garth Dales and Anthony T.-M. Lau, Memoirs of the American Mathematical Society 205, 2010) Banach spaces of continuous functions as dual spaces (with H. Garth Dales, Frederick K. Dashiell Jr., and Anthony T.-M. Lau, CMS Books in Mathematics, Springer, 2016) == Recognition == In 2009 the University of Cambridge hosted a meeting, "Algebra and Analysis around the Stone-Cech Compactification", in honour of Strauss's 75th birthday. == Personal life == Strauss married (as the first of his four wives) Seymour Papert. Papert was also South African, and became a co-author and fellow student of Frank Smithies with Strauss at Cambridge. She met her second husband, Edmond Strauss, at the University of London. She is a strong amateur chess player, and was director of the Brighton and Hove Progressive Synagogue for 2014–2015. == References ==
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Wikipedia:Donald A. Dawson#0
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Donald Andrew Dawson (born June 4, 1937) is a Canadian mathematician, specializing in probability. == Education and career == Dawson received in 1958 his bachelor's degree and in 1959 his master's degree from McGill University and in 1963 his PhD from MIT under Henry McKean with thesis Constructions of Diffusions with Specified Mean Hitting Times and Hitting Probabilities. In 1962/63 he was an engineer in the aerospace department of Raytheon. At McGill University he became in 1963 an assistant professor and in 1967 an associate professor. At Carleton University he became in 1970 an associate professor and in 1971 a professor, working in this position until 1996. From 1996 to 2000 Dawson was the director of the Fields Institute and during these years also an adjunct professor at the University of Toronto. From 2000 to 2010 he was an adjunct professor at McGill University. == Research == Dawson works on stochastic processes, measure-valued processes, and hierarchical stochastic systems with applications in information systems, genetics, evolutionary biology, and economics. He has written 8 monographs and over 150 refereed publications. In 1994 he was an invited speaker at the International Congress of Mathematicians in Zürich with lecture Interaction and hierarchy in measure-valued processes. From 2003 to 2005 he was the president of the Bernoulli Society. == Honors and awards == Fellow of the International Statistical Institute 1975 Fellow of the Institute of Mathematical Statistics 1977 Killam Senior Research Scholar 1977–1979 Fellow of the Royal Society of Canada, Academy of Sciences 1987 Gold Medal of the Statistical Society of Canada 1991 Jeffery–Williams Prize 1994 Max Planck Research Award for International Cooperation 1996–2000 CRM-Fields-PIMS Prize 2004 Honorary Member of the Statistical Society of Canada 2004 Fellow of the Royal Society 2010 Fellow of the American Mathematical Society 2012 Inaugural fellow of the Canadian Mathematical Society, 2018 == Selected works == with Edwin A. Perkins: Measure-valued processes and Renormalization of Branching Particle Systems, in R. Carmona, B. Rozovskii Stochastic Partial Differential Equations: Six Perspectives, American Mathematical Society Mathematical Surveys and Monographs, vol. 64, 1999, pp. 45–106. with J. T. Cox, A. Greven: Mutually catalytic super branching random walks: large finite systems and renormalization analysis, American Mathematical Society 2004 as editor: Measure-valued processes, stochastic partial differential equations, and interacting systems, American Mathematical Society 1994 with Edwin Perkins: Historical processes, American Mathematical Society 1991 with J. Gärtner: Large deviations, free energy functional and quasi-potential for a mean field model of interacting diffusions, American Mathematical Society 1989 == References == == External links == Homepage
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Wikipedia:Donald Bentley#0
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Donald Lyon Bentley is an American statistician and mathematician. A doctoral student of biostatistician Rupert Griel Miller at the Stanford University School of Humanities and Sciences, Bentley graduated with a Doctor of Philosophy in Applied Mathematics and Statistics in 1962. He then taught at the Mathematics and Statistics Department of Pomona College in Claremont, California from 1964 to 2001, becoming Lingurn H. Burkhead Professor of Mathematics, an endowed chair, before retiring to become a professor emeritus. He was also president of the Southern California Chapter of the American Statistical Association from 1987 to 1988, and was named a Fellow of the American Statistical Association in 1990. Bentley is known locally for his role in creating Pomona College's tradition of revering the number 47. It began in the summer of 1964, when two students, Laurie Mets and Bruce Elgin, conducted a research project seeking to find out whether the number occurs more often in nature than would be expected by chance. They documented various 47 sightings, and Bentley produced a false mathematical proof that 47 was equal to all other integers. The number became a meme among the class, which spread once the academic year began and snowballed over time. Many Pomona alumni have since deliberately inserted 47 references into their work. In the early 2010s, the college's clock tower would chime on the 47th minute of the hour. == References ==
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Wikipedia:Donald Kingsbury#0
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Donald MacDonald Kingsbury (born 12 February 1929, in San Francisco) is an American–Canadian science fiction author. Kingsbury taught mathematics at McGill University, Montreal, from 1956 until his retirement in 1986. == Bibliography == === Books === Courtship Rite. New York : Simon and Schuster, July 1982. ISBN 0-671-44033-0. (Nominated for Hugo for Best Novel in 1983) (Compton Crook Award winner) (Prometheus Award Hall of Fame 2016 winner) Published in UK as Geta. The Moon Goddess and the Son. New York : Baen Books, December 1986. ISBN 0-671-55958-3. (Short version nominated for Hugo Award for Best Novella in 1980) Psychohistorical Crisis. New York : Tor Books, December 2001. ISBN 0-7653-4195-6. (Winner, 2002 Prometheus Award) The Finger Pointing Solward has been awaited ever since the publication of Courtship Rite. Kingsbury has never finished the story, noting as far back as September 1982 that he was still "polishing" it (see interview with Robert J. Sawyer) and as recently as his self-supplied Readercon biography in July 2006. Artist Donato Giancola placed a copy of the intended cover on his gallery page: this cover was used in 2016 for the Bradley P. Beaulieu collection In the Stars I'll Find You. In 1994, an excerpt was published as "The Cauldron". === Short fiction === "The Ghost Town", Astounding Science Fiction, June 1952. "Shipwright", Analog, April 1978. "To Bring in the Steel", Analog, July 1978. "The Moon Goddess and the Son", Analog, December 1979. "The Survivor", Man-Kzin Wars IV, September 1991. "The Heroic Myth of Lieutenant Nora Argamentine", Man-Kzin Wars VI, July 1994. "The Cauldron", Northern Stars: The Anthology of Canadian Science Fiction, September 1994. "Historical Crisis", Far Futures, December 1995. == References == == External links == Donald Kingsbury Donald Kingsbury interviewed by Robert J. Sawyer Donald Kingsbury at the Internet Speculative Fiction Database
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Wikipedia:Donald S. Passman#0
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Donald Steven Passman (born March 28, 1940, in New York City) is an American mathematician, specializing in ring theory, group theory, and Lie algebra theory. == Biography == After attending the Bronx High School of Science, Passman matriculated at the Polytechnic Institute of Brooklyn, where he graduated with B.S. in 1960. He then became a graduate student in mathematics at Harvard University, where he graduated with M.A. in 1961 and Ph.D. in 1964. His doctoral dissertation was written under the supervision of Richard Brauer. Passman was an assistant professor from 1964 to 1966 at the University of California, Los Angeles (U.C.L.A.) and from 1966 to 1969 at Yale University. At the University of Wisconsin–Madison, he was from 1969 to 1971 an associate professor, from 1971 to 1995 a full professor, and from 1995 to 2011 the Richard Brauer Professor of Mathematics. In 2011 he retired as professor emeritus. Passman has written 7 books and more than 180 research publications. He has given over 70 invited addresses, not only in North America but also in Europe, Brazil, Israel, and Turkey. He has been an editor for several mathematical journals, including the International Journal of Mathematics, Game Theory and Algebra (1991–2013), Beiträge zur Algebra und Geometrie (1993–2013), Algebras and Representation Theory (2001–2011), and the Journal of Algebra and its Applications (2001–2016). His research interests include finite and infinite groups, noncommutative ring theory, group rings and enveloping algebras of Lie algebras. In 1963 Passman married Marjorie Mednick. They have two children, Barbara and Jonathan, and five grandchildren, Samuel, Rebecca, Abraham, Jordan and Eve. == Awards and honors == 1977 — Lester R. Ford Award for 1976 article What is a group ring? 1989 — Plenary speaker, Canadian Math Society summer meeting, Windsor Ontario, June 1989 2000 — Deborah and Franklin Haimo Award for Distinguished College or University Teaching of Mathematics, Mathematical Association of America 2005 — Conference, held June 10–12 2005 at the University of Wisconsin–Madison, in honor of Donald S. Passman 2012 — elected a 2013 Fellow of the American Mathematical Society == Books == Permutation Groups, Benjamin, New York, (1968), pbk edition, Dover, Mineola, (2012). Infinite Group Rings, Marcel Dekker, New York, (1971). The Algebraic Structure of Group Rings, Wiley-Interscience, New York (1977), [Krieger, Malabar, (1985)], pbk edition, Dover, Mineola, (2011). Group rings, Crossed Products and Galois Theory, CBMS Conference Notes, AMS, Providence, 1986. Infinite Crossed Products, Academic Press, Boston, (1989), pbk edition, Dover, Mineola, (2013). A Course in Ring Theory, Wadsworth, Pacific Grove, (1991), pbk edition, Chelsea-AMS, Providence, (2004). Lectures on Linear Algebra, World Scientific, Singapore, (2022), [1] == References == == External links == "Donald S. Passman, Abstracts & PDFs".
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Wikipedia:Donald Solitar#0
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Donald Solitar (September 5, 1932 in Brooklyn, New York, United States – April 28, 2008 in Toronto, Canada) was an American and Canadian mathematician, known for his work in combinatorial group theory. The Baumslag–Solitar groups are named after him and Gilbert Baumslag, after their joint 1962 paper on these groups. == Life == Solitar competed on the mathematics team of Brooklyn Technical High School with his future co-author Abe Karrass, one year ahead of him in school. He graduated from Brooklyn College in 1953 (with the assistance of tutoring from Karrass, who went to New York University) and went to Princeton University for graduate study in mathematics. However, his intended mentor there, Emil Artin, was no longer interested in group theory, so he left with a master's degree and earned his doctorate from New York University instead, in 1958, under the supervision of Wilhelm Magnus. After finishing his studies, he joined the faculty of Adelphi University in 1959, and Karrass soon joined him there as a doctoral student, earning a Ph.D. under Solitar's supervision in 1961; this was the first Ph.D. awarded at Adelphi. Karrass remained on the faculty with Solitar, where they founded a summer institute for high school mathematics teachers. Solitar moved to Polytechnic University in 1967, and then (as department chair) to York University in 1968, along with Karrass. Solitar married J. Francien Hageman, a Dutch woman, in 1976. He died of a heart attack on April 28, 2008. == Selected publications == Books Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (1966), Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, MR 0207802. Research articles Baumslag, Gilbert; Solitar, Donald (1962), "Some two-generator one-relator non-Hopfian groups", Bulletin of the American Mathematical Society, 68 (3): 199–201, doi:10.1090/s0002-9904-1962-10745-9, MR 0142635. Karrass, A.; Solitar, D. (1970), "The subgroups of a free product of two groups with an amalgamated subgroup", Transactions of the American Mathematical Society, 150 (1): 227–255, doi:10.2307/1995492, JSTOR 1995492, MR 0260879. Karrass, A.; Pietrowski, A.; Solitar, D. (1973), "Finite and infinite cyclic extensions of free groups", Journal of the Australian Mathematical Society, 16 (4): 458–466, doi:10.1017/s1446788700015445, MR 0349850. == Awards and honors == Solitar became a Fellow of the Royal Society of Canada in 1982. == References ==
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Wikipedia:Donato Acciaioli#0
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Donato Acciaioli (15 March 1428 – 28 August 1478) was an Italian scholar and statesman. He was known for his learning, especially in Greek and mathematics, and for his services to his native state, the Republic of Florence. == Biography == He was born in Florence, Italy. He was educated under the patronage or guidance of Jacopo Piccolomini-Ammannati (1422–1479), who subsequently was named cardinal. He also putatively gained his knowledge of the classics from Lionardo and Carlo Marsuppini (1399–1453) and from the refugee scholar from Byzantium, Giovanni Argiropolo. Having previously been entrusted with several important embassies, in 1473 he became Gonfalonier of Florence, one of the nine citizens selected by drawing lots every two months, who formed the government. He died at Milan in 1478, when on his way to Paris to ask the aid of Louis XI on behalf of the Florentines against Pope Sixtus IV. His body was taken back to Florence and buried in the church of the Carthusian order at the public expense, and his daughters were endowed by his fellow-citizens, since he had little in terms of wealth. He wrote Latin translations of some of Plutarch's Lives (Florence, 1478); Commentaries on Aristotle's Ethics, Politics, Physics, and De anima; the lives of Hannibal, Scipio and Charlemagne as well as the biography of the grand seneschal of the Kingdom of Naples, Niccolò Acciaioli by Matteo Palmieri. In the work on Aristotle he had the cooperation of his master John Argyropulus. == See also == Zanobi Acciaioli, Librarian of the Vatican, of the same family == References ==
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Wikipedia:Donna DeEtte Elbert#0
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Donna DeEtte Elbert (27 January 1928 – 15 January 2019) was an American mathematician and scientist. == Early life and education == Born 27 January 1928 in Williams Bay, Wisconsin to William Lawrence Elbert and Sue Melicent Hatch, Donna DeEtte Elbert was the second of three siblings. She attended Williams Bay Elementary School and Williams Bay High School, graduating in 1945. When Elbert accepted her position to work for astrophysicist Subrahmanyan Chandrasekhar in 1948, she did not have any official education in advanced mathematics. It was only after Elbert’s employer Chandrasekhar encouraged her to enroll in advanced mathematics courses at the University of Wisconsin–Madison did she formally obtain education in advanced mathematics such as calculus. Despite her now-formal start in college mathematics, she pursued and graduated with a Bachelors of Fine Arts degree from School of the Art Institute of Chicago in 1974. She also received six weeks of education from Parsons School of Design in New York City during the summer of 1956, during which Chandrasekhar worked at Los Alamos. == Science career == At 20 years of age and without college education, Elbert began working as a human "computer" for Subrahmanyan Chandrasekhar at the Yerkes Observatory in Wisconsin in the autumn of 1948. She later also worked both at Yerkes and the University of Chicago. Although she originally intended to work under Chandrasekhar only long enough so that she could afford attending design college, she continued to work for the astrophysicist for over the next thirty years. Elbert’s first major set of contributions to Chandrasekhar’s research, which resulted in her explicit name recognition, was computing solutions to sophisticated differential equations to numerically and algebraically solve for variables in relation to Heisenberg’s theory of turbulence. Although she did not gain co-authorship for her mathematical work, Chandrasekhar did give her his thanks in the paper’s closing remarks: "In conclusion, I wish to record my indebtedness to Miss Donna Elbert for valuable assistance with the various numerical integrations involved in the preparation of this paper". After continuing to provide Chandrasekhar with mathematical assistance, he encouraged Elbert, who had no prior official education in advanced mathematics, to study advanced mathematics courses at the University of Wisconsin, Madison. Elbert achieved co-authorship of 18 papers with Chandrasekhar with her work in analyzing turbulence, magnetohydrodynamics, polarization of the sunlit sky, rotating flows, convection, and other topics as she progressed into a more central role in Chardrasekhar’s group’s research, promoting herself beyond her original role as a computer. Despite this step-up, Elbert still conducted much of Chandrasekhar’s numerical work, often producing solutions that were more further simplified compared to Chandrasekhar’s. Elbert also authored her own paper, “Bessel and Related Functions Which Occur in Hydromagnetics," published in The Astrophysical Journal in 1957. Elbert continued to conduct research with Chandrasekhar until 1979. == Elbert range == During Elbert’s research with Chandrasekhar on the book later published under the title Hydrodynamic and Hydromagnetic Stability, Elbert noted a range of values in the hydrodynamic and hydromagnetic marginal stability curves which result in local minima surrounded by extreme changes. Despite Elbert’s key insight and extensive work on the book, she was not given co-authorship of the work by Chandrasekhar. Instead, Chandrasekhar thanked Elbert in only a single footnote. Researcher and scholar Susanne Horn of the Coventry University (UK) and postdoctoral associate Jonathan Aurnou of UCLA (USA) now build on Elbert’s key insight about a range of body’s specific values on hydrodynamic and hydromagnetic marginal stability curves that lead to unusually strong magnetic fields in their publication The Elbert Range of magnetostrophic convection I. Linear Theory. This specific range of values is now known as the Elbert range. Horn and Arnou reveal that studying the Elbert Range can yield crucial insight into research on objects such as stars and exoplanets. In the case of exoplanets, objects that fall within the Elbert Range, such as earth, have substantially strong magnetic fields that can deflect harmful radiation, increasing the probability that life similar to that we know exists on that exoplanet. === Very Briefly: The Elbert Range === Bodies that have fluid and conductive interiors, such as Earth with its molten liquid metallic core, can create their own magnetic fields due to the movement of charge within their conductive cores. The fluid’s motion depends largely on two factors: 1) the combination of the body’s rotational velocity and size, which affects the Coriolis Force on the fluids, and 2) convection of the fluid caused by differences in temperature in different sections of the fluid. For bodies that lie in the Elbert Range, the strength of the motion of the conductive fluid caused by the Coriolis Effect and convection are approximately equal, causing the fluid to flow more uniformly in an orderly manner. This uniformity of flow allows for the generation of strong magnetic fields around the body. On the other hand, most bodies do not lie in the Elbert Range, and their conductive cores (if they have one) do not flow in an orderly fashion. Disparate contributions of fluid flow caused by the Coriolis Effect and convection cause disruptive flow patterns, resulting in only weak magnetic fields. Horn and Arnou expanded on Elbert’s work regarding the Elbert Range with modern computational and analytical tools. == Personal life == Despite working long hours for Chandrasekhar, Elbert and her family still remained in close touch with her community in Williams Bay. She served as the treasurer of Walworth County Historical Society for 15 years, and her father owned a local barbershop from 1929 to 1970. She picked up many hobbies, including art, piano, and genealogy, because of which she joined the Mayflower Society and Daughters of the Revolution. At one point she took piano lessons with Chandrasekhar, though he quit after their teacher progressed through the basics too fast for Chandrasekhar’s liking. On vacations, Elbert would read books recommended to her by her boss and discuss them with him upon her return. Elbert died on 15 January 2019 at the age of 90 at Aurora Lakeland Medical Center in Wisconsin due to a brief illness. == References ==
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Wikipedia:Dorette Pronk#0
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Dorothea Ariette (Dorette) Pronk (born 1968) is a Dutch and Canadian mathematician specializing in category theory and categorical approaches to differentiation. She is a professor of mathematics at Dalhousie University. As well as for her research, she is also known for her work promoting mathematics competitions in Canada. == Education and career == Pronk is originally from Rotterdam. She became a mathematics student at Utrecht University in The Netherlands, where she earned a master's degree in 1991 and completed her Ph.D. in 1995. Her doctoral dissertation, Groupoid Representations for Sheaves on Orbifolds, was jointly promoted by Dirk van Dalen and Ieke Moerdijk. She took a faculty position at Dalhousie University in 2000, after previously being a postdoctoral researcher there. She is a professor of mathematics at Dalhousie. Pronk began working in mathematics competitions as an observer for the Canadian team at the 1998 and 1999 International Mathematical Olympiads. She chaired Canada's committee on the IMO from 2014 to 2015, the mathematical competitions committee of the Canadian Mathematical Society beginning in 2016, and Canada's committee on the European Girls' Mathematical Olympiad since 2018. She has also served as a leader of Canada's mathematics team. At Dalhousie, she has organized a mathematics challenge club and Math circles focused both on Nova Scotia students and on First Nations students. == Recognition == Pronk was the 2023 recipient of the Graham Wright Award for Distinguished Service of the Canadian Mathematical Society. In the same year she was named as a Fellow of the Canadian Mathematical Society. == Personal life == Pronk was raised in a strict denomination of Reformed (Calvinist) Christianity in the Netherlands. After resisting the call because of her strict background, she has been participating in messianic dance through All Nations Christian Reformed Church in Halifax, Nova Scotia since 2002. == References == == External links == Dorette Pronk publications indexed by Google Scholar Dorette Pronk in nLab
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Wikipedia:Dorina Mitrea#0
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Dorina Irena-Rita Mitrea (born April 30, 1965) is a Romanian-American mathematician known for her work in harmonic analysis, partial differential equations, and the theory of distributions, and in mathematics education. She is a professor of mathematics and chair of the mathematics department at Baylor University. == Education and career == Mitrea earned a master's degree in 1987 from the University of Bucharest. Her thesis, Riemann’s Theorem for Simply Connected Riemann Surfaces, was supervised by Cabiria Andreian Cazacu. She completed her doctorate in 1996 from the University of Minnesota. Her dissertation, Layer Potential Operators and Boundary Value Problems for Differential Forms on Lipschitz Domains, was supervised by Eugene Barry Fabes. Mitrea joined the University of Missouri mathematics faculty in 1996, and became M. & R. Houchins Distinguished Professor of Mathematics at the University of Missouri in 2016. She moved to Baylor as professor and chair in 2019. == Books == Mitrea is the author of: Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds (with Marius Mitrea and Michael E. Taylor, Memoirs of the American Mathematical Society, 2001) Calculus Connections: Mathematics for Middle School Teachers (with Asma Harcharras, Pearson Prentice Hall, 2007) Distributions, Partial Differential Equations, and Harmonic Analysis (Universitext, Springer, 2013; 2nd ed., 2018) Groupoid Metrization Theory: With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis (with Irina Mitrea, Marius Mitrea, and Sylvie Monniaux, Birkhäuser, 2013) The Hodge-Laplacian: Boundary Value Problems on Riemannian Manifolds (with Irina Mitrea, Marius Mitrea, and Michael E. Taylor, De Gruyter, 2016) L p {\displaystyle L^{p}} -Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets (with Steve Hofmann, Marius Mitrea, and Andrew J. Morris, Memoirs of the American Mathematical Society, 2017) Singular Integral Operators, Quantitative Flatness, and Boundary Problems (with Juan José Marín, José María Martell, Irina Mitrea, and Marius Mitrea, Progress in Mathematics, 344, Birkhäuser, 2022) Geometric Harmonic Analysis I: A Sharp Divergence Theorem with Nontangential Pointwise Traces (with Irina Mitrea and Marius Mitrea, Developments in Mathematics 72, Springer, 2022. ISBN 978-3031059490) Geometric Harmonic Analysis II: Function Spaces Measuring Size and Smoothness on Rough Sets (with Irina Mitrea and Marius Mitrea, Developments in Mathematics 73, Springer, 2022. ISBN 978-3031137174) Geometric Harmonic Analysis III: Integral Representations, Calderón-Zygmund Theory, Fatou Theorems, and Applications to Scattering (with Irina Mitrea and Marius Mitrea, Developments in Mathematics 74, Springer, 2023. ISBN 978-3-031-22737-0, doi:10.1007/978-3-031-22735-6) Geometric Harmonic Analysis IV: Boundary Layer Potentials on Uniformly Rectifiable Domains, and Applications to Complex Analysis (with Irina Mitrea and Marius Mitrea, Developments in Mathematics 75, Springer, 2023. ISBN 978-3031291814) Geometric Harmonic Analysis V: Fredholm Theory and Finer Estimates for Integral Operators, with Applications to Boundary Problems (with Irina Mitrea and Marius Mitrea, Developments in Mathematics 76, Springer, 2023.ISBN 978-3031315602) == Recognition == Mitrea was elected as a Fellow of the American Mathematical Society in the 2024 class of fellows. == Personal life == She is married to Marius Mitrea. Her husband is also a mathematician, and moved with Mitrea from Missouri to Baylor. == References == == External links == Dorina Mitrea publications indexed by Google Scholar
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Wikipedia:Dorothy Geneva Styles#0
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Dorothy Geneva Styles (December 13, 1922 - February 12, 1984) was an American composer, mathematician, organist, and poet. Styles was born in El Dorado, Arkansas, to Minnie A. Shelnut and Alfred Alexander Styles. She demonstrated musical talent as a child, performing on WEXL radio at age 10, and giving music lessons as a teenager. Styles married Dennis Glenn Van Eck in 1941 and divorced him in 1945. She graduated from the Detroit Institute of Musical Arts and received a B.Mus. from the University of Detroit Mercy in 1945, a B.S. from Columbia University in 1954, and an M.A. from the University of Michigan in 1970. Styles taught and also worked as an organist at Hazel Park Baptist Tabernacle in Hazel Park, Michigan, and as a choir director at St. Timothy’s Evangelical Lutheran Church in Wayne, Michigan. Her publications included: == Prose == A Prime Number Theorem An Extension of the Idea of Countability as Applied to Real Numbers Centaur Projections of the Natural Harmonic Series: Some Implications Sea Chanty Young Verses for the Early Old == Vocal == “I Sing a Song” “Japanese Raindrops” (music by Bernadette Daria Bohdanowycz; words by Styles) “Love Song” “Lullaby” “Mother, Tell Me” “Mrs. Santa Claus Loves Mr. Santa Claus” “Pledge of Allegiance to the Flag” == References ==
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Wikipedia:Dorte Olesen#0
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Dorte Marianne Olesen (born 1948) is a Danish mathematician. In 1988 at Roskilde University, she became the first Danish woman to be appointed a full professor of mathematics. She has also played a leading role in the development of education and research networks, both in Denmark and at the European level. == Early life, education, and family == Born on 8 January 1948 in Hillerød, she was the daughter of the medical specialist and academic Knud Henning Olesen (1920–2007) and the physician Irene Mariane Pedersen (1919–2004). After matriculating from Sortedam Gymnasium in Copenhagen, following in her parents¨footsteps she began to read medicine at Copenhagen University, hoping to become a biophysicist. As this was not possible, she studied mathematics instead, graduating in 1973 and receiving the university's gold medal for a dissertation on operator algebra. She went on to Odense University where she received a Lic.Scient (equivalent to a PhD) in mathematics in 1975. She also went on study trips to Philadelphia (1971–72) and Marseille (1974 & 1979) and was a guest professor at Berkeley (1984–85). When she was 23, Olesen married one of her assistant teachers at Copenhagen University, Gert Kjærgaard Pedersen (1940–2004), who became a prominent mathematics professor. Together they had three children, Just (born 1976), Oluf (1980) and Cecilie (1984). == Career == Olesen returned to Copenhagen University as a senior scholar in 1975, becoming a lecturer at the university's mathematical institute in 1980. In 1988, she was appointed Professor of Mathematics at Roskilde University, the first Danish woman to become a full professor in the field. In 1989, she was appointed executive director of UNI-C, a Danish government department devoted to promoting the use of information technology in research and education. She managed the development of UNI-C until 2011, establishing the Danish NREN research network, developing computer services and experimenting with online support for education. In parallel, she also promoted the use of the internet for education, including its use in elementary schools, and supported the development of e-business for Denmark. At the European level, in 1992 Olesen became a member of the European Commission's High Performance Computing and Networking Advisory Committee. From 2001 to 2005, she served on the commission's Expert Group on ICT in Education while acting as president of TERENA, the Trans-European Research and Education Networking Association, from 2003 to 2009. From 2010 to 2011 she participated in the commission's High Level Expert Group on the Future of GÉANT. In 2013, she was a member of the Expert Group on the Implementation of the ERA Communications. Finally, from 2014 to 2017, she served on the Board of Directors of GÉANT during the merger of DANTE and TERENA. == Awards == In 1992, Olesen was awarded the Order of the Dannebrog and in 2000, was honoured as a Knight of the First Class. She received the Tagea Brandts Rejselegat in 1987. == References ==
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Wikipedia:Dottie number#0
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In mathematics, the Dottie number or the cosine constant is a constant that is the unique real root of the equation cos x = x {\displaystyle \cos x=x} , where the argument of cos {\displaystyle \cos } is in radians. The decimal expansion of the Dottie number is given by: D = 0.739085133215160641655312087673... (sequence A003957 in the OEIS). Since cos ( x ) − x {\displaystyle \cos(x)-x} is decreasing and its derivative is non-zero at cos ( x ) − x = 0 {\displaystyle \cos(x)-x=0} , it only crosses zero at one point. This implies that the equation cos ( x ) = x {\displaystyle \cos(x)=x} has only one real solution. It is the single real-valued fixed point of the cosine function and is a nontrivial example of a universal attracting fixed point. It is also a transcendental number because of the Lindemann–Weierstrass theorem. The generalised case cos z = z {\displaystyle \cos z=z} for a complex variable z {\displaystyle z} has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points. == History == The constant appeared in publications as early as 1860s. Norair Arakelian used lowercase ayb (ա) from the Armenian alphabet to denote the constant. The constant name was coined by Samuel R. Kaplan in 2007. It originates from a professor of French named Dottie who observed the number by repeatedly pressing the cosine button on her calculator. The Dottie number, for which an exact series expansion can be obtained using the Faà di Bruno formula, has interesting connections with the Kepler and Bertrand's circle problems. == Identities == The Dottie number appears in the closed form expression of some integrals: ∫ 0 ∞ ln ( 4 ( x + sinh x ) 2 + π 2 4 ( x − sinh x ) 2 + π 2 ) d x = π 2 − 2 π D {\displaystyle \int _{0}^{\infty }\ln \left({\frac {4\left(x+\sinh x\right)^{2}+\pi ^{2}}{4(x-\sinh x)^{2}+\pi ^{2}}}\right)\mathrm {d} x=\pi ^{2}-2\pi D} ∫ 0 ∞ 3 π 2 + 4 ( x − sinh x ) 2 ( 3 π 2 + 4 ( x − sinh x ) 2 ) 2 + 16 π 2 ( x − sinh x ) 2 d x = 1 8 + 8 1 − D 2 {\displaystyle \int _{0}^{\infty }{\frac {3\pi ^{2}+4(x-\sinh x)^{2}}{(3\pi ^{2}+4(x-\sinh x)^{2})^{2}+16\pi ^{2}(x-\sinh x)^{2}}}\,\mathrm {d} x={\frac {1}{8+8{\sqrt {1-D^{2}}}}}} Using the Taylor series of the inverse of f ( x ) = cos ( x ) − x {\displaystyle f(x)=\cos(x)-x} at π 2 {\textstyle {\frac {\pi }{2}}} (or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series: D = π 2 + ∑ n o d d a n π n {\displaystyle D={\frac {\pi }{2}}+\sum _{n\,\mathrm {odd} }a_{n}\pi ^{n}} where each a n {\displaystyle a_{n}} is a rational number defined for odd n as a n = 1 n ! 2 n lim m → π 2 ∂ n − 1 ∂ m n − 1 ( cos m m − π / 2 − 1 ) − n = − 1 4 , − 1 768 , − 1 61440 , − 43 165150720 , … {\displaystyle {\begin{aligned}a_{n}&={\frac {1}{n!2^{n}}}\lim _{m\to {\frac {\pi }{2}}}{\frac {\partial ^{n-1}}{\partial m^{n-1}}}{\left({\frac {\cos m}{m-\pi /2}}-1\right)^{-n}}\\&=-{\frac {1}{4}},-{\frac {1}{768}},-{\frac {1}{61440}},-{\frac {43}{165150720}},\ldots \end{aligned}}} The Dottie number can also be expressed as: D = 1 − ( 2 I 1 2 − 1 ( 1 2 , 3 2 ) − 1 ) 2 , {\displaystyle D={\sqrt {1-\left(2I_{\frac {1}{2}}^{-1}\left({\frac {1}{2}},{\frac {3}{2}}\right)-1\right)^{2}}},} where I − 1 {\displaystyle I^{-1}} is the inverse of the regularized beta function. This value can be obtained using Kepler's equation, along with other equivalent closed forms. In Microsoft Excel and LibreOffice Calc spreadsheets, the Dottie number can be expressed in closed form as SQRT(1-(2*BETA.INV(1/2,1/2,3/2)-1)^2). In the Mathematica computer algebra system, the Dottie number is Sqrt[1 - (2 InverseBetaRegularized[1/2, 1/2, 3/2] - 1)^2]. Another closed form representation: D = − tanh ( 2 arctanh ( 1 3 InvT ( 1 4 , 3 ) ) ) = − 2 3 InvT ( 1 4 , 3 ) InvT 2 ( 1 4 , 3 ) + 3 , {\displaystyle D=-\tanh \left(2{\text{ arctanh}}\left({\frac {1}{\sqrt {3}}}\operatorname {InvT} \left({\frac {1}{4}},3\right)\right)\right)=-{\frac {2{\sqrt {3}}{\operatorname {InvT} \left({\frac {1}{4}},3\right)}}{\operatorname {InvT} ^{2}\left({\frac {1}{4}},3\right)+3}},} where InvT {\displaystyle \operatorname {InvT} } is the inverse survival function of Student's t-distribution. In Microsoft Excel and LibreOffice Calc, due to the specifics of the realization of `TINV` function, this can be expressed as formulas 2 *SQRT(3)* TINV(1/2, 3)/(TINV(1/2, 3)^2+3) and TANH(2*ATANH(1/SQRT(3) * TINV(1/2,3))). == Notes == == References == == External links == Miller, T. H. (Feb 1890). "On the numerical values of the roots of the equation cosx = x". Proceedings of the Edinburgh Mathematical Society. 9: 80–83. doi:10.1017/S0013091500030868. Salov, Valerii (2012). "Inevitable Dottie Number. Iterals of cosine and sine". arXiv:1212.1027. Azarian, Mohammad K. (2008). "ON THE FIXED POINTS OF A FUNCTION AND THE FIXED POINTS OF ITS COMPOSITE FUNCTIONS" (PDF). International Journal of Pure and Applied Mathematics.
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Wikipedia:Douady rabbit#0
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A Douady rabbit is a fractal derived from the Julia set of the function f c ( z ) = z 2 + c {\textstyle f_{c}(z)=z^{2}+c} , when parameter c {\displaystyle c} is near the center of one of the period three bulbs of the Mandelbrot set for a complex quadratic map. It is named after French mathematician Adrien Douady. == Background == The Douady rabbit is generated by iterating the Mandelbrot set map z n + 1 = z n 2 + c {\displaystyle z_{n+1}=z_{n}^{2}+c} on the complex plane, where parameter c {\displaystyle c} is fixed to lie in one of the two period three bulb off the main cardioid and z {\displaystyle z} ranging over the plane. The resulting image can be colored by corresponding each pixel with a starting value z 0 {\displaystyle z_{0}} and calculating the amount of iterations required before the value of z n {\displaystyle z_{n}} escapes a bounded region, after which it will diverge toward infinity. It can also be described using the logistic map form of the complex quadratic map, specifically z n + 1 = M z n := γ z n ( 1 − z n ) . {\displaystyle z_{n+1}={\mathcal {M}}z_{n}:=\gamma z_{n}\left(1-z_{n}\right).} which is equivalent to w n + 1 = w n 2 + c {\displaystyle w_{n+1}=w_{n}^{2}+c} . Irrespective of the specific iteration used, the filled Julia set associated with a given value of γ {\displaystyle \gamma } (or μ {\displaystyle \mu } ) consists of all starting points z 0 {\displaystyle z_{0}} (or w 0 {\displaystyle w_{0}} ) for which the iteration remains bounded. Then, the Mandelbrot set consists of those values of γ {\displaystyle \gamma } (or μ {\displaystyle \mu } ) for which the associated filled Julia set is connected. The Mandelbrot set can be viewed with respect to either γ {\displaystyle \gamma } or μ {\displaystyle \mu } . Noting that μ {\displaystyle \mu } is invariant under the substitution γ → 2 − γ {\displaystyle \gamma \to 2-\gamma } , the Mandelbrot set with respect to γ {\displaystyle \gamma } has additional horizontal symmetry. Since z {\displaystyle z} and w {\displaystyle w} are affine transformations of one another, or more specifically a similarity transformation, consisting of only scaling, rotation and translation, the filled Julia sets look similar for either form of the iteration given above. == Detailed description == You can also describe the Douady rabbit utilising the Mandelbrot set with respect to γ {\displaystyle \gamma } as shown in the graph above. In this figure, the Mandelbrot set superficially appears as two back-to-back unit disks with sprouts or buds, such as the sprouts at the one- and five-o'clock positions on the right disk or the sprouts at the seven- and eleven-o'clock positions on the left disk. When γ {\displaystyle \gamma } is within one of these four sprouts, the associated filled Julia set in the mapping plane is said to be a Douady rabbit. For these values of γ {\displaystyle \gamma } , it can be shown that M {\displaystyle {\mathcal {M}}} has z = 0 {\displaystyle z=0} and one other point as unstable (repelling) fixed points, and z = ∞ {\displaystyle z=\infty } as an attracting fixed point. Moreover, the map M 3 {\displaystyle {\mathcal {M}}^{3}} has three attracting fixed points. A Douady rabbit consists of the three attracting fixed points z 1 {\displaystyle z_{1}} , z 2 {\displaystyle z_{2}} , and z 3 {\displaystyle z_{3}} and their basins of attraction. For example, Figure 4 shows the Douady rabbit in the z {\displaystyle z} plane when γ = γ D = 2.55268 − 0.959456 i {\displaystyle \gamma =\gamma _{D}=2.55268-0.959456i} , a point in the five-o'clock sprout of the right disk. For this value of γ {\displaystyle \gamma } , the map M {\displaystyle {\mathcal {M}}} has the repelling fixed points z = 0 {\displaystyle z=0} and z = .656747 − .129015 i {\displaystyle z=.656747-.129015i} . The three attracting fixed points of M 3 {\displaystyle {\mathcal {M}}^{3}} (also called period-three fixed points) have the locations z 1 = 0.499997032420304 − ( 1.221880225696050 × 10 − 6 ) i ( r e d ) , z 2 = 0.638169999974373 − ( 0.239864000011495 ) i ( g r e e n ) , z 3 = 0.799901291393262 − ( 0.107547238170383 ) i ( y e l l o w ) . {\displaystyle {\begin{aligned}z_{1}&=0.499997032420304-(1.221880225696050\times 10^{-6})i{\;}{\;}{\mathrm {(red)} },\\z_{2}&=0.638169999974373-(0.239864000011495)i{\;}{\;}{\mathrm {(green)} },\\z_{3}&=0.799901291393262-(0.107547238170383)i{\;}{\;}{\mathrm {(yellow)} }.\end{aligned}}} The red, green, and yellow points lie in the basins B ( z 1 ) {\displaystyle B(z_{1})} , B ( z 2 ) {\displaystyle B(z_{2})} , and B ( z 3 ) {\displaystyle B(z_{3})} of M 3 {\displaystyle {\mathcal {M}}^{3}} , respectively. The white points lie in the basin B ( ∞ ) {\displaystyle B(\infty )} of M {\displaystyle {\mathcal {M}}} . The action of M {\displaystyle {\mathcal {M}}} on these fixed points is given by the relations M z 1 = z 2 {\displaystyle {\mathcal {M}}z_{1}=z_{2}} , M z 2 = z 3 {\displaystyle {\mathcal {M}}z_{2}=z_{3}} , and M z 3 = z 1 {\displaystyle {\mathcal {M}}z_{3}=z_{1}} . Corresponding to these relations there are the results M B ( z 1 ) = B ( z 2 ) o r M r e d ⊆ g r e e n , M B ( z 2 ) = B ( z 3 ) o r M g r e e n ⊆ y e l l o w , M B ( z 3 ) = B ( z 1 ) o r M y e l l o w ⊆ r e d . {\displaystyle {\begin{aligned}{\mathcal {M}}B(z_{1})&=B(z_{2}){\;}{\mathrm {or} }{\;}{\mathcal {M}}{\;}{\mathrm {red} }\subseteq {\mathrm {green} },\\{\mathcal {M}}B(z_{2})&=B(z_{3}){\;}{\mathrm {or} }{\;}{\mathcal {M}}{\;}{\mathrm {green} }\subseteq {\mathrm {yellow} },\\{\mathcal {M}}B(z_{3})&=B(z_{1}){\;}{\mathrm {or} }{\;}{\mathcal {M}}{\;}{\mathrm {yellow} }\subseteq {\mathrm {red} }.\end{aligned}}} As a second example, Figure 5 shows a Douady rabbit when γ = 2 − γ D = − .55268 + .959456 i {\displaystyle \gamma =2-\gamma _{D}=-.55268+.959456i} , a point in the eleven-o'clock sprout on the left disk ( μ {\displaystyle \mu } is invariant under this transformation). This rabbit is more symmetrical in the plane. The period-three fixed points then are located at z 1 = 0.500003730675024 + ( 6.968273875812428 × 10 − 6 ) i ( r e d ) , z 2 = − 0.138169999969259 + ( 0.239864000061970 ) i ( g r e e n ) , z 3 = − 0.238618870661709 − ( 0.264884797354373 ) i ( y e l l o w ) . {\displaystyle {\begin{aligned}z_{1}&=0.500003730675024+(6.968273875812428\times 10^{-6})i{\;}{\;}({\mathrm {red} }),\\z_{2}&=-0.138169999969259+(0.239864000061970)i{\;}{\;}({\mathrm {green} }),\\z_{3}&=-0.238618870661709-(0.264884797354373)i{\;}{\;}({\mathrm {yellow} }).\end{aligned}}} The repelling fixed points of M {\displaystyle {\mathcal {M}}} itself are located at z = 0 {\displaystyle z=0} and z = 1.450795 + 0.7825835 i {\displaystyle z=1.450795+0.7825835i} . The three major lobes on the left, which contain the period-three fixed points z 1 {\displaystyle z_{1}} , z 2 {\displaystyle z_{2}} , and z 3 {\displaystyle z_{3}} , meet at the fixed point z = 0 {\displaystyle z=0} , and their counterparts on the right meet at the point z = 1 {\displaystyle z=1} . It can be shown that the effect of M {\displaystyle {\mathcal {M}}} on points near the origin consists of a counterclockwise rotation about the origin of arg ( γ ) {\displaystyle \arg(\gamma )} , or very nearly 120 ∘ {\displaystyle 120^{\circ }} , followed by scaling (dilation) by a factor of | γ | = 1.1072538 {\displaystyle |\gamma |=1.1072538} . == Variants == A twisted rabbit is the composition of a rabbit polynomial with n {\displaystyle n} powers of Dehn twists about its ears. The corabbit is the symmetrical image of the rabbit. Here parameter c ≈ − 0.1226 − 0.7449 i {\displaystyle c\approx -0.1226-0.7449i} . It is one of 2 other polynomials inducing the same permutation of their post-critical set are the rabbit. === 3D === The Julia set has no direct analog in three dimensions. === 4D === A quaternion Julia set with parameters c = − 0.123 + 0.745 i {\displaystyle c=-0.123+0.745i} and a cross-section in the x y {\displaystyle xy} plane. The Douady rabbit is visible in the cross-section. === Embedded === A small embedded homeomorphic copy of rabbit in the center of a Julia set === Fat === The fat rabbit or chubby rabbit has c at the root of the 1/3-limb of the Mandelbrot set. It has a parabolic fixed point with 3 petals. === n-th eared === In general, the rabbit for the p e r i o d − ( n + 1 ) {\displaystyle period-(n+1)} th bulb of the main cardioid will have n {\displaystyle n} ears For example, a period four bulb rabbit has three ears. === Perturbed === Perturbed rabbit Perturbed rabbit == Twisted rabbit problem == In the early 1980s, Hubbard posed the so-called twisted rabbit problem, a polynomial classification problem. The goal is to determine Thurston equivalence types of functions of complex numbers that usually are not given by a formula (these are called topological polynomials): given a topological quadratic whose branch point is periodic with period three, determining which quadratic polynomial it is Thurston equivalent to determining the equivalence class of twisted rabbits, i.e. composite of the rabbit polynomial with nth powers of Dehn twists about its ears. The problem was originally solved by Laurent Bartholdi and Volodymyr Nekrashevych using iterated monodromic groups. The generalization of the problem to the case where the number of post-critical points is arbitrarily large has been solved as well. == Gallery == == See also == Dragon curve Herman ring Siegel disc == References == == External links == Weisstein, Eric W. "Douady Rabbit Fractal". MathWorld. Dragt, A. "Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics". Adrien Douady: La dynamique du lapin (1996) - video on the YouTube This article incorporates material from Douady Rabbit on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Wikipedia:Doug Stinson#0
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Douglas Robert Stinson (born 1956 in Guelph, Ontario) is a Canadian mathematician and cryptographer, currently a Professor Emeritus at the University of Waterloo. Stinson received his B.Math from the University of Waterloo in 1978, his M.Sc. from Ohio State University in 1980, and his Ph.D. from the University of Waterloo in 1981. He was at the University of Manitoba from 1981 to 1989, and the University of Nebraska-Lincoln from 1990 to 1998. In 2011 he was named as a Fellow of the Royal Society of Canada. Stinson is the author of over 300 research publications as well as the mathematics-based cryptography textbook Cryptography: Theory and Practice (ISBN 9781584885085). == Selected publications == Stinson, Doug R. (Nov 1997). "On Some Methods for Unconditionally Secure Key Distribution and Broadcast Encryption". Designs, Codes and Cryptography. 12 (3): 215–243. doi:10.1023/A:1008268610932. S2CID 14778421. == See also == List of University of Waterloo people == References == == External links == Doug Stinson's home page
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Wikipedia:Douglas Northcott#0
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Douglas Geoffrey Northcott, FRS (31 December 1916 – 8 April 2005) was a British mathematician who worked on ideal theory. == Early life and career == Northcott was born Douglas Geoffrey Robertson in Kensington on 31 December 1916 to Clara Freda (née Behl) (1894–1958) and her first husband Geoffrey Douglas Spence Robertson (1894–1978). His mother remarried in 1919 to Arthur Hugh Kynaston Northcott (1887–1952). In 1935, he legally adopted his step-father's surname. He was educated in London, then at Christ's Hospital and St John's College, Cambridge, where he started research under the supervision of G.H. Hardy. His work was interrupted by active service during World War II. Captured at Singapore, he survived his time as a prisoner of war in Japan, and returned to Cambridge at the end of the war. Back at Cambridge, he published his dissertation "Abstract Tauberian theorems with applications to power series and Hilbert series ". He then turned to algebra under the influence of Emil Artin, whom he had met while visiting Princeton University. He became a Research Fellow of St John's College in 1948. In 1949, he proved an important result in the theory of heights, namely that there are only finitely many algebraic numbers of bounded degree and bounded height. In analogy to this result, a set of algebraic numbers is said to satisfy the Northcott property if there are only finitely many elements of bounded height. In 1952, he moved to the Town Trust Chair of Pure Mathematics at Sheffield University. He remained at Sheffield until his retirement in 1982, also serving as Head of Department and Dean of Pure Science. In 1954, Douglas Northcott and David Rees introduced in a joint paper the Northcott-Rees theory of reductions and integral closures, which has subsequently been influential in commutative algebra. == Awards == Northcott was awarded the London Mathematical Society Junior Berwick Prize in 1953 and served as LMS Vice-President during 1968-69. He was elected Fellow of the Royal Society in 1961. == Family life == In 1949, at Cambridge, Northcott married Rose Hilda Austin (1917-1992), with two daughters, Anne Patricia (born 1950) and Pamela Rose (1952-1992). == Publications == Northcott, D. G. Multilinear algebra. Cambridge University Press, Cambridge, 1984. ISBN 0-521-26269-0 Northcott, D. G. A first course of homological algebra. Reprint of 1973 edition. Cambridge University Press, Cambridge-New York, 1980. ISBN 0-521-29976-4 Northcott, D. G. Affine sets and affine groups. London Mathematical Society Lecture Note Series, 39. Cambridge University Press, Cambridge-New York, 1980. ISBN 0-521-22909-X Northcott, D. G. Finite free resolutions. Cambridge Tracts in Mathematics, No. 71. Cambridge University Press, Cambridge-New York-Melbourne, 1976. Northcott, D. G. Lessons on rings, modules and multiplicities. Cambridge University Press, London 1968 Northcott, D. G. An introduction to homological algebra. Cambridge University Press, New York 1960 Northcott, D. G. Ideal theory. Cambridge Tracts in Mathematics and Mathematical Physics, No. 42. Cambridge, at the University Press, 1953. == References ==
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Wikipedia:Douglas Quadling#0
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Douglas Arthur Quadling (1926–2015) was an English mathematician, school master and educationalist who was one of the four drivers behind the School Mathematics Project (SMP) in the 1960s and 70s. == Life == Quadling was educated at the City of London School. In 1939 the school was moved out of London, at the start of World War II, with most of the pupils attending Marlborough College though not accommodated there. Quadling had use of the College library at weekends, was influenced by Gordon Nobbs, one of the masters, and decided on a teaching career. In 1943 he won a scholarship to Emmanuel College, Cambridge. Graduating as a wrangler, with a two-year Part II in the Mathematical Tripos, Quadling worked briefly at the end of the war at Fort Halstead for the Ministry of Supply. At this period, based near Orpington, he met and influenced the young Michael Saward, who found him "a pleasant if somewhat owlish young man", while canvassing support for the Crusaders. Quadling taught at Mill Hill School from 1946 to 1952. He was at Marlborough College from 1952 to 1967, becoming Head of Mathematics and a housemaster (C2). He was then a tutor at the Cambridge Institute of Education, from 1968 to 1985. As a novice mathematics teacher in the late 1940s, Quadling joined the Mathematical Association, serving as its President in 1980–1. He spoke at a mathematics education conference in Ghana in 1968. He took over from Edwin A. Maxwell as editor of the Mathematical Gazette in 1971; his successor in 1980 was Victor Bryant. In 1983 he was awarded the OBE for services to mathematical education. He married Ruth Starte of the Cambridge Institute of Education. == Creation of School Mathematics Project == The School Mathematics Project, which changed the course of mathematics teaching in Britain, arose from a meeting between Quadling and three others, Martyn Cundy of Sherborne School, Tom Jones of Winchester College and Professor Bryan Thwaites of University of Southampton, in September 1961. Cundy, like Quadling, was involved with the Mathematical Association. By 1963 the compilation of new SMP mathematics syllabuses had been given to Cundy, Jones, Quadling and T. D. Morris of Charterhouse School. From July 1964 three examination boards offered an SMP syllabus for the General Certificate of Education. When the A-level syllabus was constructed, Cundy and Quadling wrote with John Durran, Laurence Ellis, Colin Goldsmith, Tim Lewis and others. == Views == In public life, Quadling was known for lamenting the state of mathematics education, advocating the need for university courses which were more practical and scientific, in contrast to, say, the exacting Mathematical Tripos at the University of Cambridge. It was at school level, however, that he had greatest influence, through the SMP. == Selected publications == Quadling was head-hunted as a textbook writer at the Mathematical Association conference in 1955, by the authors C. V. Durell and Alan Robson (Marlborough College), and A. V. Ready of George Bell & Sons. There resulted his books on mechanics with A. R. D. Ramsay, also of Marlborough. His textbooks are noted for their mathematical rigour and thoroughness, and their attention to practical application. Textbooks D.A. Quadling (1 December 1955), Mathematical Analysis, Oxford University Press, ISBN 0-19-832518-5 {{citation}}: ISBN / Date incompatibility (help) D.A. Quadling; A.R.D. Ramsay (1971) [1959 1st ed.], Elementary Mechanics, vol. 1 (Metric, 2nd ed.), Bell and Hyman, ISBN 0-7135-1690-9 D.A. Quadling; A.R.D. Ramsay (1971) [1957 1st ed.], Elementary Mechanics, vol. 2 (Metric, 2nd ed.), Bell and Hyman, ISBN 0-7135-1700-X Douglas Quadling (2004), Mechanics 1 for OCR, Cambridge University Press, ISBN 0-521-54900-0 Douglas Quadling (2004), Mechanics 2 for OCR, Cambridge University Press, ISBN 0-521-54901-9 Douglas Quadling (2005), Mechanics 3 and 4 for OCR, Cambridge University Press, ISBN 0-521-54902-7 == References ==
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Wikipedia:Dov Tamari#0
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In mathematics, a Tamari lattice, introduced by Dov Tamari (1962), is a partially ordered set in which the elements consist of different ways of grouping a sequence of objects into pairs using parentheses; for instance, for a sequence of four objects abcd, the five possible groupings are ((ab)c)d, (ab)(cd), (a(bc))d, a((bc)d), and a(b(cd)). Each grouping describes a different order in which the objects may be combined by a binary operation; in the Tamari lattice, one grouping is ordered before another if the second grouping may be obtained from the first by only rightward applications of the associative law (xy)z = x(yz). For instance, applying this law with x = a, y = bc, and z = d gives the expansion (a(bc))d = a((bc)d), so in the ordering of the Tamari lattice (a(bc))d ≤ a((bc)d). In this partial order, any two groupings g1 and g2 have a greatest common predecessor, the meet g1 ∧ g2, and a least common successor, the join g1 ∨ g2. Thus, the Tamari lattice has the structure of a lattice. The Hasse diagram of this lattice is isomorphic to the graph of vertices and edges of an associahedron. The number of elements in a Tamari lattice for a sequence of n + 1 objects is the nth Catalan number Cn. The Tamari lattice can also be described in several other equivalent ways: It is the poset of sequences of n integers a1, ..., an, ordered coordinatewise, such that i ≤ ai ≤ n and if i ≤ j ≤ ai then aj ≤ ai (Huang & Tamari 1972). It is the poset of binary trees with n leaves, ordered by tree rotation operations. It is the poset of ordered forests, in which one forest is earlier than another in the partial order if, for every j, the jth node in a preorder traversal of the first forest has at least as many descendants as the jth node in a preorder traversal of the second forest (Knuth 2005). It is the poset of triangulations of a convex n-gon, ordered by flip operations that substitute one diagonal of the polygon for another. == Notation == The Tamari lattice of the groupings of n+1 objects is called Tn. The corresponding associahedron is called Kn+1. In The Art of Computer Programming T4 is called the Tamari lattice of order 4 and its Hasse diagram K5 the associahedron of order 4. == References == Chapoton, F. (2005), "Sur le nombre d'intervalles dans les treillis de Tamari", Séminaire Lotharingien de Combinatoire (in French), 55 (55): 2368, arXiv:math/0602368, Bibcode:2006math......2368C, MR 2264942. Csar, Sebastian A.; Sengupta, Rik; Suksompong, Warut (2014), "On a Subposet of the Tamari Lattice", Order, 31 (3): 337–363, arXiv:1108.5690, doi:10.1007/s11083-013-9305-5, MR 3265974. Early, Edward (2004), "Chain lengths in the Tamari lattice", Annals of Combinatorics, 8 (1): 37–43, doi:10.1007/s00026-004-0203-9, MR 2061375. Friedman, Haya; Tamari, Dov (1967), "Problèmes d'associativité: Une structure de treillis finis induite par une loi demi-associative", Journal of Combinatorial Theory (in French), 2 (3): 215–242, doi:10.1016/S0021-9800(67)80024-3, MR 0238984. Geyer, Winfried (1994), "On Tamari lattices", Discrete Mathematics, 133 (1–3): 99–122, doi:10.1016/0012-365X(94)90019-1, MR 1298967. Huang, Samuel; Tamari, Dov (1972), "Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law", Journal of Combinatorial Theory, Series A, 13: 7–13, doi:10.1016/0097-3165(72)90003-9, MR 0306064. Knuth, Donald E. (2005), "Draft of Section 7.2.1.6: Generating All Trees", The Art of Computer Programming, vol. IV, p. 34. Tamari, Dov (1962), "The algebra of bracketings and their enumeration", Nieuw Archief voor Wiskunde, Series 3, 10: 131–146, MR 0146227.
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Wikipedia:Drigganita#0
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Drigganita (दृग्गणित; IAST: dṛggaṇita, from dṛk-gaṇita, "sight-calculation"), also called the Drik system, is a system of astronomical computations followed by several traditional astronomers, astrologers and almanac makers in India. In this system the computations are performed using certain basic constants derived from observations of astronomical phenomena. The almanacs computed using the methods of Drigganita are referred to as Drigganita Panchangas. The Drigganita system is in contrast to the method followed by some other almanac makers who use the values given in the ancient astronomical treatise known by the name Surya Siddhanta. The almanacs computed using this treatise are known as Sydhantic Panchangas. They are also known as Vakya Panchangas. In the history of astronomy in India, two different Drigganita systems have been introduced at two different points of time and at two different geographical locations. The first system was introduced by the Kerala astronomer-mathematicians Parameshvara (1380-1460) and Damodara in the fifteenth century. Incidentally, Drigganita is also the title of a book authored by Parameshvara through which the Drik system was promulgated. In the nineteenth century, a second Drigganita system was introduced by Chinthamani Ragoonatha Chary (1822 – 5 February 1880) an Indian astronomer attached to the then Madras Observatory. == Drigganita of Paramesvara == The Drigganita system propounded by Parameshvara was a revision of the Parahita system introduced by Haridatta in the year 683 CE. No new methodology was introduced as part of the Drigganita system. Instead, new multipliers and divisors were given for the computation of the Kali days and for the calculation of the mean positions of the planets. Revised values are given for the positions of planets at zero Kali. Also the values of the sines of arc of anomaly (manda-jya) and of commutation (sighra-jya) are revised and are given for intervals of 6 degrees. A large number of manuals have been composed describing the Drik system. Since the results obtained using the Drigganita system are more accurate, the astronomers and astrologers use the system for casting horoscopes, for conducting astrological queries and for the computations of eclipses. However, the older parahita system continues to be used for fixing auspicious times for rituals and ceremonies. == Drigganita of Ragoonatha Chary == Chintamani Ragoonathachary, a native astronomer took the initiative to modify and publish a new almanac and thereby introduced a change in the calendrical system followed in the Tamil region. It was clear during the middle of the nineteenth century that the traditional calendars were way off the mark. Not only were there errors in the position of stars, the old system predicted eclipses when there would be none. As the traditional almanac was seen to be quite inaccurate, Chatre and Khetkar in Bombay, Venkatakrishna Raya and Ragoonathachari in Madras proposed Drigganitha Panchang to replace the traditional Panchang computations based on the Vakya Panchang. Ragoonathachari had to face the criticism of the traditionalists who argued against such improvements and criticized him for his scientific zeal. Ragoonathachary’s Drigganitha Panchang not only provided the traditional five calendarical elements but also provided concordance with English months and dates. Therefore, this Panchang was of more practical utility; native officials working in government establishments or those dealing with government found it handy. At the end of a lot of, often acrimonious, Drig vs. Vakya debate, a meeting was called at Sankara Mutt at Kumbakonam and the meeting arrived at the conclusion that the Drig system needs to be followed and a new almanac to be prepared on those lines. Accordingly, the head of the Mutt, issued a srimugam (message of blessing) in 1877 affirming support for the Drig system. From then on a Drig almanac began to be published under the auspices of Kanchi Sankaracharya Matt in the name of 'Sri Kanchi Math almanac'. == See also == Parahita Tirugaṇita-pañcāṅgam == References ==
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Wikipedia:Dror Bar-Natan#0
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Dror Bar-Natan (Hebrew: דרוֹר בָר-נָתָן; born January 30, 1966) is a professor at the University of Toronto Department of Mathematics, Canada. His main research interests include knot theory, finite type invariants, and Khovanov homology. == Education == Bar-Natan earned his B.Sc. in mathematics at Tel Aviv University in 1984. After performing his military service as a teacher, he went to study at Princeton University in 1987. He obtained his Ph.D. in mathematics from Princeton in 1991, under the direction of physicist Edward Witten. == Professorship == After holding a Benjamin Peirce Assistant Professorship at Harvard University for four years from 1991 to 1995, he returned to Israel, and became Associate Professor at the Hebrew University of Jerusalem. He moved to the University of Toronto in 2002, and was promoted to Full Professor in 2006. == Personal life == Bar-Natan holds US, Israeli, and Canadian citizenship, and currently resides in Canada. Bar-Natan originally refused to take the Canadian citizenship oath because it would require him to swear allegiance to royalty. He later decided to become a citizen but publicly announced his intention to renounce the oath immediately after becoming a citizen, which he did so in front of the presiding judge at his citizenship ceremony on November 30, 2015. From his former marriage to mathematician Yael Karshon he has two sons, Assaf and Itai. == Research == In 1999, Bar-Natan collaborated on a paper with the goal of mathematically refuting claims made in The Bible Code by Michael Drosnin that hidden messages could be deciphered from within the Bible. In particular, the paper demonstrated that practically any "code" could be found within the Bible, thereby debunking Drosnin's "discovery" of specific codes. This work is outside the main scope of his academic interests, although he is known for it because of the popularity of The Bible Code. Academically, Bar-Natan has made significant contributions to the formalization of Khovanov homology. Bar-Natan was a member of the Editorial Board for the journal Compositio Mathematica for 10 years, until 2010. == Selected publication == Bar-Natan, Dror (1995). "On the Vassiliev knot invariants". Topology. 34 (2): 423–472. doi:10.1016/0040-9383(95)93237-2. MR 1318886. == References == == External links == "Bar-Natan", The Knot Atlas. Citizenship disavowal website maintained by Bar-Natan
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Wikipedia:Dual basis#0
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In linear algebra, given a vector space V {\displaystyle V} with a basis B {\displaystyle B} of vectors indexed by an index set I {\displaystyle I} (the cardinality of I {\displaystyle I} is the dimension of V {\displaystyle V} ), the dual set of B {\displaystyle B} is a set B ∗ {\displaystyle B^{*}} of vectors in the dual space V ∗ {\displaystyle V^{*}} with the same index set I {\displaystyle I} such that B {\displaystyle B} and B ∗ {\displaystyle B^{*}} form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V ∗ {\displaystyle V^{*}} . If it does span V ∗ {\displaystyle V^{*}} , then B ∗ {\displaystyle B^{*}} is called the dual basis or reciprocal basis for the basis B {\displaystyle B} . Denoting the indexed vector sets as B = { v i } i ∈ I {\displaystyle B=\{v_{i}\}_{i\in I}} and B ∗ = { v i } i ∈ I {\displaystyle B^{*}=\{v^{i}\}_{i\in I}} , being biorthogonal means that the elements pair to have an inner product equal to 1 if the indexes are equal, and equal to 0 otherwise. Symbolically, evaluating a dual vector in V ∗ {\displaystyle V^{*}} on a vector in the original space V {\displaystyle V} : v i ⋅ v j = δ j i = { 1 if i = j 0 if i ≠ j , {\displaystyle v^{i}\cdot v_{j}=\delta _{j}^{i}={\begin{cases}1&{\text{if }}i=j\\0&{\text{if }}i\neq j{\text{,}}\end{cases}}} where δ j i {\displaystyle \delta _{j}^{i}} is the Kronecker delta symbol. == Introduction == To perform operations with a vector, we must have a straightforward method of calculating its components. In a Cartesian frame the necessary operation is the dot product of the vector and the base vector. For example, x = x 1 i 1 + x 2 i 2 + x 3 i 3 {\displaystyle \mathbf {x} =x^{1}\mathbf {i} _{1}+x^{2}\mathbf {i} _{2}+x^{3}\mathbf {i} _{3}} where { i 1 , i 2 , i 3 } {\displaystyle \{\mathbf {i} _{1},\mathbf {i} _{2},\mathbf {i} _{3}\}} is the basis in a Cartesian frame. The components of x {\displaystyle \mathbf {x} } can be found by x k = x ⋅ i k . {\displaystyle x^{k}=\mathbf {x} \cdot \mathbf {i} _{k}.} However, in a non-Cartesian frame, we do not necessarily have e i ⋅ e j = 0 {\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=0} for all i ≠ j {\displaystyle i\neq j} . However, it is always possible to find vectors e i {\displaystyle \mathbf {e} ^{i}} in the dual space such that x i = e i ( x ) ( i = 1 , 2 , 3 ) . {\displaystyle x^{i}=\mathbf {e} ^{i}(\mathbf {x} )\qquad (i=1,2,3).} The equality holds when the e i {\displaystyle \mathbf {e} ^{i}} s are the dual basis of e i {\displaystyle \mathbf {e} _{i}} s. Notice the difference in position of the index i {\displaystyle i} . == Existence and uniqueness == The dual set always exists and gives an injection from V into V∗, namely the mapping that sends vi to vi. This says, in particular, that the dual space has dimension greater or equal to that of V. However, the dual set of an infinite-dimensional V does not span its dual space V∗. For example, consider the map w in V∗ from V into the underlying scalars F given by w(vi) = 1 for all i. This map is clearly nonzero on all vi. If w were a finite linear combination of the dual basis vectors vi, say w = ∑ i ∈ K α i v i {\textstyle w=\sum _{i\in K}\alpha _{i}v^{i}} for a finite subset K of I, then for any j not in K, w ( v j ) = ( ∑ i ∈ K α i v i ) ( v j ) = 0 {\textstyle w(v_{j})=\left(\sum _{i\in K}\alpha _{i}v^{i}\right)\left(v_{j}\right)=0} , contradicting the definition of w. So, this w does not lie in the span of the dual set. The dual of an infinite-dimensional space has greater dimension (this being a greater infinite cardinality) than the original space has, and thus these cannot have a basis with the same indexing set. However, a dual set of vectors exists, which defines a subspace of the dual isomorphic to the original space. Further, for topological vector spaces, a continuous dual space can be defined, in which case a dual basis may exist. === Finite-dimensional vector spaces === In the case of finite-dimensional vector spaces, the dual set is always a dual basis and it is unique. These bases are denoted by B = { e 1 , … , e n } {\displaystyle B=\{e_{1},\dots ,e_{n}\}} and B ∗ = { e 1 , … , e n } {\displaystyle B^{*}=\{e^{1},\dots ,e^{n}\}} . If one denotes the evaluation of a covector on a vector as a pairing, the biorthogonality condition becomes: ⟨ e i , e j ⟩ = δ j i . {\displaystyle \left\langle e^{i},e_{j}\right\rangle =\delta _{j}^{i}.} The association of a dual basis with a basis gives a map from the space of bases of V to the space of bases of V∗, and this is also an isomorphism. For topological fields such as the real numbers, the space of duals is a topological space, and this gives a homeomorphism between the Stiefel manifolds of bases of these spaces. == A categorical and algebraic construction of the dual space == Another way to introduce the dual space of a vector space (module) is by introducing it in a categorical sense. To do this, let A {\displaystyle A} be a module defined over the ring R {\displaystyle R} (that is, A {\displaystyle A} is an object in the category R - M o d {\displaystyle R{\text{-}}\mathbf {Mod} } ). Then we define the dual space of A {\displaystyle A} , denoted A ∗ {\displaystyle A^{\ast }} , to be Hom R ( A , R ) {\displaystyle {\text{Hom}}_{R}(A,R)} , the module formed of all R {\displaystyle R} -linear module homomorphisms from A {\displaystyle A} into R {\displaystyle R} . Note then that we may define a dual to the dual, referred to as the double dual of A {\displaystyle A} , written as A ∗ ∗ {\displaystyle A^{\ast \ast }} , and defined as Hom R ( A ∗ , R ) {\displaystyle {\text{Hom}}_{R}(A^{\ast },R)} . To formally construct a basis for the dual space, we shall now restrict our view to the case where F {\displaystyle F} is a finite-dimensional free (left) R {\displaystyle R} -module, where R {\displaystyle R} is a ring with unity. Then, we assume that the set X {\displaystyle X} is a basis for F {\displaystyle F} . From here, we define the Kronecker Delta function δ x y {\displaystyle \delta _{xy}} over the basis X {\displaystyle X} by δ x y = 1 {\displaystyle \delta _{xy}=1} if x = y {\displaystyle x=y} and δ x y = 0 {\displaystyle \delta _{xy}=0} if x ≠ y {\displaystyle x\neq y} . Then the set S = { f x : F → R | f x ( y ) = δ x y } {\displaystyle S=\lbrace f_{x}:F\to R\;|\;f_{x}(y)=\delta _{xy}\rbrace } describes a linearly independent set with each f x ∈ Hom R ( F , R ) {\displaystyle f_{x}\in {\text{Hom}}_{R}(F,R)} . Since F {\displaystyle F} is finite-dimensional, the basis X {\displaystyle X} is of finite cardinality. Then, the set S {\displaystyle S} is a basis to F ∗ {\displaystyle F^{\ast }} and F ∗ {\displaystyle F^{\ast }} is a free (right) R {\displaystyle R} -module. == Examples == For example, the standard basis vectors of R 2 {\displaystyle \mathbb {R} ^{2}} (the Cartesian plane) are { e 1 , e 2 } = { ( 1 0 ) , ( 0 1 ) } {\displaystyle \left\{\mathbf {e} _{1},\mathbf {e} _{2}\right\}=\left\{{\begin{pmatrix}1\\0\end{pmatrix}},{\begin{pmatrix}0\\1\end{pmatrix}}\right\}} and the standard basis vectors of its dual space ( R 2 ) ∗ {\displaystyle (\mathbb {R} ^{2})^{*}} are { e 1 , e 2 } = { ( 1 0 ) , ( 0 1 ) } . {\displaystyle \left\{\mathbf {e} ^{1},\mathbf {e} ^{2}\right\}=\left\{{\begin{pmatrix}1&0\end{pmatrix}},{\begin{pmatrix}0&1\end{pmatrix}}\right\}{\text{.}}} In 3-dimensional Euclidean space, for a given basis { e 1 , e 2 , e 3 } {\displaystyle \{\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\}} , the biorthogonal (dual) basis { e 1 , e 2 , e 3 } {\displaystyle \{\mathbf {e} ^{1},\mathbf {e} ^{2},\mathbf {e} ^{3}\}} can be found by formulas below: e 1 = ( e 2 × e 3 V ) T , e 2 = ( e 3 × e 1 V ) T , e 3 = ( e 1 × e 2 V ) T . {\displaystyle \mathbf {e} ^{1}=\left({\frac {\mathbf {e} _{2}\times \mathbf {e} _{3}}{V}}\right)^{\mathsf {T}},\ \mathbf {e} ^{2}=\left({\frac {\mathbf {e} _{3}\times \mathbf {e} _{1}}{V}}\right)^{\mathsf {T}},\ \mathbf {e} ^{3}=\left({\frac {\mathbf {e} _{1}\times \mathbf {e} _{2}}{V}}\right)^{\mathsf {T}}.} where T denotes the transpose and V = ( e 1 ; e 2 ; e 3 ) = e 1 ⋅ ( e 2 × e 3 ) = e 2 ⋅ ( e 3 × e 1 ) = e 3 ⋅ ( e 1 × e 2 ) {\displaystyle V\,=\,\left(\mathbf {e} _{1};\mathbf {e} _{2};\mathbf {e} _{3}\right)\,=\,\mathbf {e} _{1}\cdot (\mathbf {e} _{2}\times \mathbf {e} _{3})\,=\,\mathbf {e} _{2}\cdot (\mathbf {e} _{3}\times \mathbf {e} _{1})\,=\,\mathbf {e} _{3}\cdot (\mathbf {e} _{1}\times \mathbf {e} _{2})} is the volume of the parallelepiped formed by the basis vectors e 1 , e 2 {\displaystyle \mathbf {e} _{1},\,\mathbf {e} _{2}} and e 3 . {\displaystyle \mathbf {e} _{3}.} In general the dual basis of a basis in a finite-dimensional vector space can be readily computed as follows: given the basis f 1 , … , f n {\displaystyle f_{1},\ldots ,f_{n}} and corresponding dual basis f 1 , … , f n {\displaystyle f^{1},\ldots ,f^{n}} we can build matrices F = [ f 1 ⋯ f n ] G = [ f 1 ⋯ f n ] {\displaystyle {\begin{aligned}F&={\begin{bmatrix}f_{1}&\cdots &f_{n}\end{bmatrix}}\\G&={\begin{bmatrix}f^{1}&\cdots &f^{n}\end{bmatrix}}\end{aligned}}} Then the defining property of the dual basis states that G T F = I {\displaystyle G^{\mathsf {T}}F=I} Hence the matrix for the dual basis G {\displaystyle G} can be computed as G = ( F − 1 ) T {\displaystyle G=\left(F^{-1}\right)^{\mathsf {T}}} == See also == Reciprocal lattice Miller index Zone axis == Notes == == References == Lebedev, Leonid P.; Cloud, Michael J.; Eremeyev, Victor A. (2010). Tensor Analysis With Applications to Mechanics. World Scientific. ISBN 978-981431312-4. "Finding the Dual Basis". Stack Exchange. May 27, 2012.
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Wikipedia:Dual basis in a field extension#0
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In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite field extension L/K, by using the field trace. This requires the property that the field trace TrL/K provides a non-degenerate quadratic form over K. This can be guaranteed if the extension is separable; it is automatically true if K is a perfect field, and hence in the cases where K is finite, or of characteristic zero. A dual basis () is not a concrete basis like the polynomial basis or the normal basis; rather it provides a way of using a second basis for computations. Consider two bases for elements in a finite field, GF(pm): B 1 = α 0 , α 1 , … , α m − 1 {\displaystyle B_{1}={\alpha _{0},\alpha _{1},\ldots ,\alpha _{m-1}}} and B 2 = γ 0 , γ 1 , … , γ m − 1 {\displaystyle B_{2}={\gamma _{0},\gamma _{1},\ldots ,\gamma _{m-1}}} then B2 can be considered a dual basis of B1 provided Tr ( α i ⋅ γ j ) = { 0 , if i ≠ j 1 , otherwise . {\displaystyle \operatorname {Tr} (\alpha _{i}\cdot \gamma _{j})={\begin{cases}0,&\operatorname {if} \ i\neq j\\1,&\operatorname {otherwise} .\end{cases}}} Here the trace of a value in GF(pm) can be calculated as follows: Tr ( β ) = ∑ i = 0 m − 1 β p i {\displaystyle \operatorname {Tr} (\beta )=\sum _{i=0}^{m-1}\beta ^{p^{i}}} Using a dual basis can provide a way to easily communicate between devices that use different bases, rather than having to explicitly convert between bases using the change of bases formula. Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with multiplication by the multiplicative identity (usually 1). == References == Lidl, Rudolf; Niederreiter, Harald (1994). Introduction to finite fields and their applications. Cambridge: Cambridge University Press. doi:10.1017/cbo9781139172769. ISBN 9781139172769., Definition 2.30, p. 54.
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Wikipedia:Dual graph#0
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In the mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs (graphs that are already embedded in the plane) rather than planar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph. Historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and is in turn generalized combinatorially by the concept of a dual matroid. Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces. These notions of dual graphs should not be confused with a different notion, the edge-to-vertex dual or line graph of a graph. The term dual is used because the property of being a dual graph is symmetric, meaning that if H is a dual of a connected graph G, then G is a dual of H. When discussing the dual of a graph G, the graph G itself may be referred to as the "primal graph". Many other graph properties and structures may be translated into other natural properties and structures of the dual. For instance, cycles are dual to cuts, spanning trees are dual to the complements of spanning trees, and simple graphs (without parallel edges or self-loops) are dual to 3-edge-connected graphs. Graph duality can help explain the structure of mazes and of drainage basins. Dual graphs have also been applied in computer vision, computational geometry, mesh generation, and the design of integrated circuits. == Examples == === Cycles and dipoles === The unique planar embedding of a cycle graph divides the plane into only two regions, the inside and outside of the cycle, by the Jordan curve theorem. However, in an n-cycle, these two regions are separated from each other by n different edges. Therefore, the dual graph of the n-cycle is a multigraph with two vertices (dual to the regions), connected to each other by n dual edges. Such a graph is called a multiple edge, linkage, or sometimes a dipole graph. Conversely, the dual to an n-edge dipole graph is an n-cycle. === Dual polyhedra === According to Steinitz's theorem, every polyhedral graph (the graph formed by the vertices and edges of a three-dimensional convex polyhedron) must be planar and 3-vertex-connected, and every 3-vertex-connected planar graph comes from a convex polyhedron in this way. Every three-dimensional convex polyhedron has a dual polyhedron; the dual polyhedron has a vertex for every face of the original polyhedron, with two dual vertices adjacent whenever the corresponding two faces share an edge. Whenever two polyhedra are dual, their graphs are also dual. For instance the Platonic solids come in dual pairs, with the octahedron dual to the cube, the dodecahedron dual to the icosahedron, and the tetrahedron dual to itself. Polyhedron duality can also be extended to duality of higher dimensional polytopes, but this extension of geometric duality does not have clear connections to graph-theoretic duality. === Self-dual graphs === A plane graph is said to be self-dual if it is isomorphic to its dual graph. The wheel graphs provide an infinite family of self-dual graphs coming from self-dual polyhedra (the pyramids). However, there also exist self-dual graphs that are not polyhedral, such as the one shown. Servatius & Christopher (1992) describe two operations, adhesion and explosion, that can be used to construct a self-dual graph containing a given planar graph; for instance, the self-dual graph shown can be constructed as the adhesion of a tetrahedron with its dual. It follows from Euler's formula that every self-dual graph with n vertices has exactly 2n − 2 edges. Every simple self-dual planar graph contains at least four vertices of degree three, and every self-dual embedding has at least four triangular faces. == Properties == Many natural and important concepts in graph theory correspond to other equally natural but different concepts in the dual graph. Because the dual of the dual of a connected plane graph is isomorphic to the primal graph, each of these pairings is bidirectional: if concept X in a planar graph corresponds to concept Y in the dual graph, then concept Y in a planar graph corresponds to concept X in the dual. === Simple graphs versus multigraphs === The dual of a simple graph need not be simple: it may have self-loops (an edge with both endpoints at the same vertex) or multiple edges connecting the same two vertices, as was already evident in the example of dipole multigraphs being dual to cycle graphs. As a special case of the cut-cycle duality discussed below, the bridges of a planar graph G are in one-to-one correspondence with the self-loops of the dual graph. For the same reason, a pair of parallel edges in a dual multigraph (that is, a length-2 cycle) corresponds to a 2-edge cutset in the primal graph (a pair of edges whose deletion disconnects the graph). Therefore, a planar graph is simple if and only if its dual has no 1- or 2-edge cutsets; that is, if it is 3-edge-connected. The simple planar graphs whose duals are simple are exactly the 3-edge-connected simple planar graphs. This class of graphs includes, but is not the same as, the class of 3-vertex-connected simple planar graphs. For instance, the figure showing a self-dual graph is 3-edge-connected (and therefore its dual is simple) but is not 3-vertex-connected. === Uniqueness === Because the dual graph depends on a particular embedding, the dual graph of a planar graph is not unique, in the sense that the same planar graph can have non-isomorphic dual graphs. In the picture, the blue graphs are isomorphic but their dual red graphs are not. The upper red dual has a vertex with degree 6 (corresponding to the outer face of the blue graph) while in the lower red graph all degrees are less than 6. Hassler Whitney showed that if the graph is 3-connected then the embedding, and thus the dual graph, is unique. By Steinitz's theorem, these graphs are exactly the polyhedral graphs, the graphs of convex polyhedra. A planar graph is 3-vertex-connected if and only if its dual graph is 3-vertex-connected. Moreover, a planar biconnected graph has a unique embedding, and therefore also a unique dual, if and only if it is a subdivision of a 3-vertex-connected planar graph (a graph formed from a 3-vertex-connected planar graph by replacing some of its edges by paths). For some planar graphs that are not 3-vertex-connected, such as the complete bipartite graph K2,4, the embedding is not unique, but all embeddings are isomorphic. When this happens, correspondingly, all dual graphs are isomorphic. Because different embeddings may lead to different dual graphs, testing whether one graph is a dual of another (without already knowing their embeddings) is a nontrivial algorithmic problem. For biconnected graphs, it can be solved in polynomial time by using the SPQR trees of the graphs to construct a canonical form for the equivalence relation of having a shared mutual dual. For instance, the two red graphs in the illustration are equivalent according to this relation. However, for planar graphs that are not biconnected, this relation is not an equivalence relation and the problem of testing mutual duality is NP-complete. === Cuts and cycles === A cutset in an arbitrary connected graph is a subset of edges defined from a partition of the vertices into two subsets, by including an edge in the subset when it has one endpoint on each side of the partition. Removing the edges of a cutset necessarily splits the graph into at least two connected components. A minimal cutset (also called a bond) is a cutset with the property that every proper subset of the cutset is not itself a cut. A minimal cutset of a connected graph necessarily separates its graph into exactly two components, and consists of the set of edges that have one endpoint in each component. A simple cycle is a connected subgraph in which each vertex of the cycle is incident to exactly two edges of the cycle. In a connected planar graph G, every simple cycle of G corresponds to a minimal cutset in the dual of G, and vice versa. This can be seen as a form of the Jordan curve theorem: each simple cycle separates the faces of G into the faces in the interior of the cycle and the faces of the exterior of the cycle, and the duals of the cycle edges are exactly the edges that cross from the interior to the exterior. The girth of any planar graph (the size of its smallest cycle) equals the edge connectivity of its dual graph (the size of its smallest cutset). This duality extends from individual cutsets and cycles to vector spaces defined from them. The cycle space of a graph is defined as the family of all subgraphs that have even degree at each vertex; it can be viewed as a vector space over the two-element finite field, with the symmetric difference of two sets of edges acting as the vector addition operation in the vector space. Similarly, the cut space of a graph is defined as the family of all cutsets, with vector addition defined in the same way. Then the cycle space of any planar graph and the cut space of its dual graph are isomorphic as vector spaces. Thus, the rank of a planar graph (the dimension of its cut space) equals the cyclotomic number of its dual (the dimension of its cycle space) and vice versa. A cycle basis of a graph is a set of simple cycles that form a basis of the cycle space (every even-degree subgraph can be formed in exactly one way as a symmetric difference of some of these cycles). For edge-weighted planar graphs (with sufficiently general weights that no two cycles have the same weight) the minimum-weight cycle basis of the graph is dual to the Gomory–Hu tree of the dual graph, a collection of nested cuts that together include a minimum cut separating each pair of vertices in the graph. Each cycle in the minimum weight cycle basis has a set of edges that are dual to the edges of one of the cuts in the Gomory–Hu tree. When cycle weights may be tied, the minimum-weight cycle basis may not be unique, but in this case it is still true that the Gomory–Hu tree of the dual graph corresponds to one of the minimum weight cycle bases of the graph. In directed planar graphs, simple directed cycles are dual to directed cuts (partitions of the vertices into two subsets such that all edges go in one direction, from one subset to the other). Strongly oriented planar graphs (graphs whose underlying undirected graph is connected, and in which every edge belongs to a cycle) are dual to directed acyclic graphs in which no edge belongs to a cycle. To put this another way, the strong orientations of a connected planar graph (assignments of directions to the edges of the graph that result in a strongly connected graph) are dual to acyclic orientations (assignments of directions that produce a directed acyclic graph). In the same way, dijoins (sets of edges that include an edge from each directed cut) are dual to feedback arc sets (sets of edges that include an edge from each cycle). === Spanning trees === A spanning tree may be defined as a set of edges that, together with all of the vertices of the graph, forms a connected and acyclic subgraph. But, by cut-cycle duality, if a set S of edges in a planar graph G is acyclic (has no cycles), then the set of edges dual to S has no cuts, from which it follows that the complementary set of dual edges (the duals of the edges that are not in S) forms a connected subgraph. Symmetrically, if S is connected, then the edges dual to the complement of S form an acyclic subgraph. Therefore, when S has both properties – it is connected and acyclic – the same is true for the complementary set in the dual graph. That is, each spanning tree of G is complementary to a spanning tree of the dual graph, and vice versa. Thus, the edges of any planar graph and its dual can together be partitioned (in multiple different ways) into two spanning trees, one in the primal and one in the dual, that together extend to all the vertices and faces of the graph but never cross each other. In particular, the minimum spanning tree of G is complementary to the maximum spanning tree of the dual graph. However, this does not work for shortest path trees, even approximately: there exist planar graphs such that, for every pair of a spanning tree in the graph and a complementary spanning tree in the dual graph, at least one of the two trees has distances that are significantly longer than the distances in its graph. An example of this type of decomposition into interdigitating trees can be seen in some simple types of mazes, with a single entrance and no disconnected components of its walls. In this case both the maze walls and the space between the walls take the form of a mathematical tree. If the free space of the maze is partitioned into simple cells (such as the squares of a grid) then this system of cells can be viewed as an embedding of a planar graph, in which the tree structure of the walls forms a spanning tree of the graph and the tree structure of the free space forms a spanning tree of the dual graph. Similar pairs of interdigitating trees can also be seen in the tree-shaped pattern of streams and rivers within a drainage basin and the dual tree-shaped pattern of ridgelines separating the streams. This partition of the edges and their duals into two trees leads to a simple proof of Euler’s formula V − E + F = 2 for planar graphs with V vertices, E edges, and F faces. Any spanning tree and its complementary dual spanning tree partition the edges into two subsets of V − 1 and F − 1 edges respectively, and adding the sizes of the two subsets gives the equation E = (V − 1) + (F − 1) which may be rearranged to form Euler's formula. According to Duncan Sommerville, this proof of Euler's formula is due to K. G. C. Von Staudt’s Geometrie der Lage (Nürnberg, 1847). In nonplanar surface embeddings the set of dual edges complementary to a spanning tree is not a dual spanning tree. Instead this set of edges is the union of a dual spanning tree with a small set of extra edges whose number is determined by the genus of the surface on which the graph is embedded. The extra edges, in combination with paths in the spanning trees, can be used to generate the fundamental group of the surface. === Additional properties === Any counting formula involving vertices and faces that is valid for all planar graphs may be transformed by planar duality into an equivalent formula in which the roles of the vertices and faces have been swapped. Euler's formula, which is self-dual, is one example. Another given by Harary involves the handshaking lemma, according to which the sum of the degrees of the vertices of any graph equals twice the number of edges. In its dual form, this lemma states that in a plane graph, the sum of the numbers of sides of the faces of the graph equals twice the number of edges. The medial graph of a plane graph is isomorphic to the medial graph of its dual. Two planar graphs can have isomorphic medial graphs only if they are dual to each other. A planar graph with four or more vertices is maximal (no more edges can be added while preserving planarity) if and only if its dual graph is both 3-vertex-connected and 3-regular. A connected planar graph is Eulerian (has even degree at every vertex) if and only if its dual graph is bipartite. A Hamiltonian cycle in a planar graph G corresponds to a partition of the vertices of the dual graph into two subsets (the interior and exterior of the cycle) whose induced subgraphs are both trees. In particular, Barnette's conjecture on the Hamiltonicity of cubic bipartite polyhedral graphs is equivalent to the conjecture that every Eulerian maximal planar graph can be partitioned into two induced trees. If a planar graph G has Tutte polynomial TG(x,y), then the Tutte polynomial of its dual graph is obtained by swapping x and y. For this reason, if some particular value of the Tutte polynomial provides information about certain types of structures in G, then swapping the arguments to the Tutte polynomial will give the corresponding information for the dual structures. For instance, the number of strong orientations is TG(0,2) and the number of acyclic orientations is TG(2,0). For bridgeless planar graphs, graph colorings with k colors correspond to nowhere-zero flows modulo k on the dual graph. For instance, the four color theorem (the existence of a 4-coloring for every planar graph) can be expressed equivalently as stating that the dual of every bridgeless planar graph has a nowhere-zero 4-flow. The number of k-colorings is counted (up to an easily computed factor) by the Tutte polynomial value TG(1 − k,0) and dually the number of nowhere-zero k-flows is counted by TG(0,1 − k). An st-planar graph is a connected planar graph together with a bipolar orientation of that graph, an orientation that makes it acyclic with a single source and a single sink, both of which are required to be on the same face as each other. Such a graph may be made into a strongly connected graph by adding one more edge, from the sink back to the source, through the outer face. The dual of this augmented planar graph is itself the augmentation of another st-planar graph. == Variations == === Directed graphs === In a directed plane graph, the dual graph may be made directed as well, by orienting each dual edge by a 90° clockwise turn from the corresponding primal edge. Strictly speaking, this construction is not a duality of directed planar graphs, because starting from a graph G and taking the dual twice does not return to G itself, but instead constructs a graph isomorphic to the transpose graph of G, the graph formed from G by reversing all of its edges. Taking the dual four times returns to the original graph. === Weak dual === The weak dual of a plane graph is the subgraph of the dual graph whose vertices correspond to the bounded faces of the primal graph. A plane graph is outerplanar if and only if its weak dual is a forest. For any plane graph G, let G+ be the plane multigraph formed by adding a single new vertex v in the unbounded face of G, and connecting v to each vertex of the outer face (multiple times, if a vertex appears multiple times on the boundary of the outer face); then, G is the weak dual of the (plane) dual of G+. === Infinite graphs and tessellations === The concept of duality applies as well to infinite graphs embedded in the plane as it does to finite graphs. However, care is needed to avoid topological complications such as points of the plane that are neither part of an open region disjoint from the graph nor part of an edge or vertex of the graph. When all faces are bounded regions surrounded by a cycle of the graph, an infinite planar graph embedding can also be viewed as a tessellation of the plane, a covering of the plane by closed disks (the tiles of the tessellation) whose interiors (the faces of the embedding) are disjoint open disks. Planar duality gives rise to the notion of a dual tessellation, a tessellation formed by placing a vertex at the center of each tile and connecting the centers of adjacent tiles. The concept of a dual tessellation can also be applied to partitions of the plane into finitely many regions. It is closely related to but not quite the same as planar graph duality in this case. For instance, the Voronoi diagram of a finite set of point sites is a partition of the plane into polygons within which one site is closer than any other. The sites on the convex hull of the input give rise to unbounded Voronoi polygons, two of whose sides are infinite rays rather than finite line segments. The dual of this diagram is the Delaunay triangulation of the input, a planar graph that connects two sites by an edge whenever there exists a circle that contains those two sites and no other sites. The edges of the convex hull of the input are also edges of the Delaunay triangulation, but they correspond to rays rather than line segments of the Voronoi diagram. This duality between Voronoi diagrams and Delaunay triangulations can be turned into a duality between finite graphs in either of two ways: by adding an artificial vertex at infinity to the Voronoi diagram, to serve as the other endpoint for all of its rays, or by treating the bounded part of the Voronoi diagram as the weak dual of the Delaunay triangulation. Although the Voronoi diagram and Delaunay triangulation are dual, their embedding in the plane may have additional crossings beyond the crossings of dual pairs of edges. Each vertex of the Delaunay triangle is positioned within its corresponding face of the Voronoi diagram. Each vertex of the Voronoi diagram is positioned at the circumcenter of the corresponding triangle of the Delaunay triangulation, but this point may lie outside its triangle. === Nonplanar embeddings === The concept of duality can be extended to graph embeddings on two-dimensional manifolds other than the plane. The definition is the same: there is a dual vertex for each connected component of the complement of the graph in the manifold, and a dual edge for each graph edge connecting the two dual vertices on either side of the edge. In most applications of this concept, it is restricted to embeddings with the property that each face is a topological disk; this constraint generalizes the requirement for planar graphs that the graph be connected. With this constraint, the dual of any surface-embedded graph has a natural embedding on the same surface, such that the dual of the dual is isomorphic to and isomorphically embedded to the original graph. For instance, the complete graph K7 is a toroidal graph: it is not planar but can be embedded in a torus, with each face of the embedding being a triangle. This embedding has the Heawood graph as its dual graph. The same concept works equally well for non-orientable surfaces. For instance, K6 can be embedded in the projective plane with ten triangular faces as the hemi-icosahedron, whose dual is the Petersen graph embedded as the hemi-dodecahedron. Even planar graphs may have nonplanar embeddings, with duals derived from those embeddings that differ from their planar duals. For instance, the four Petrie polygons of a cube (hexagons formed by removing two opposite vertices of the cube) form the hexagonal faces of an embedding of the cube in a torus. The dual graph of this embedding has four vertices forming a complete graph K4 with doubled edges. In the torus embedding of this dual graph, the six edges incident to each vertex, in cyclic order around that vertex, cycle twice through the three other vertices. In contrast to the situation in the plane, this embedding of the cube and its dual is not unique; the cube graph has several other torus embeddings, with different duals. Many of the equivalences between primal and dual graph properties of planar graphs fail to generalize to nonplanar duals, or require additional care in their generalization. Another operation on surface-embedded graphs is the Petrie dual, which uses the Petrie polygons of the embedding as the faces of a new embedding. Unlike the usual dual graph, it has the same vertices as the original graph, but generally lies on a different surface. Surface duality and Petrie duality are two of the six Wilson operations, and together generate the group of these operations. === Matroids and algebraic duals === An algebraic dual of a connected graph G is a graph G* such that G and G* have the same set of edges, any cycle of G is a cut of G*, and any cut of G is a cycle of G*. Every planar graph has an algebraic dual, which is in general not unique (any dual defined by a plane embedding will do). The converse is actually true, as settled by Hassler Whitney in Whitney's planarity criterion: A connected graph G is planar if and only if it has an algebraic dual. The same fact can be expressed in the theory of matroids. If M is the graphic matroid of a graph G, then a graph G* is an algebraic dual of G if and only if the graphic matroid of G* is the dual matroid of M. Then Whitney's planarity criterion can be rephrased as stating that the dual matroid of a graphic matroid M is itself a graphic matroid if and only if the underlying graph G of M is planar. If G is planar, the dual matroid is the graphic matroid of the dual graph of G. In particular, all dual graphs, for all the different planar embeddings of G, have isomorphic graphic matroids. For nonplanar surface embeddings, unlike planar duals, the dual graph is not generally an algebraic dual of the primal graph. And for a non-planar graph G, the dual matroid of the graphic matroid of G is not itself a graphic matroid. However, it is still a matroid whose circuits correspond to the cuts in G, and in this sense can be thought of as a combinatorially generalized algebraic dual of G. The duality between Eulerian and bipartite planar graphs can be extended to binary matroids (which include the graphic matroids derived from planar graphs): a binary matroid is Eulerian if and only if its dual matroid is bipartite. The two dual concepts of girth and edge connectivity are unified in matroid theory by matroid girth: the girth of the graphic matroid of a planar graph is the same as the graph's girth, and the girth of the dual matroid (the graphic matroid of the dual graph) is the edge connectivity of the graph. == Applications == Along with its use in graph theory, the duality of planar graphs has applications in several other areas of mathematical and computational study. In geographic information systems, flow networks (such as the networks showing how water flows in a system of streams and rivers) are dual to cellular networks describing drainage divides. This duality can be explained by modeling the flow network as a spanning tree on a grid graph of an appropriate scale, and modeling the drainage divide as the complementary spanning tree of ridgelines on the dual grid graph. In computer vision, digital images are partitioned into small square pixels, each of which has its own color. The dual graph of this subdivision into squares has a vertex per pixel and an edge between pairs of pixels that share an edge; it is useful for applications including clustering of pixels into connected regions of similar colors. In computational geometry, the duality between Voronoi diagrams and Delaunay triangulations implies that any algorithm for constructing a Voronoi diagram can be immediately converted into an algorithm for the Delaunay triangulation, and vice versa. The same duality can also be used in finite element mesh generation. Lloyd's algorithm, a method based on Voronoi diagrams for moving a set of points on a surface to more evenly spaced positions, is commonly used as a way to smooth a finite element mesh described by the dual Delaunay triangulation. This method improves the mesh by making its triangles more uniformly sized and shaped. In the synthesis of CMOS circuits, the function to be synthesized is represented as a formula in Boolean algebra. Then this formula is translated into two series–parallel multigraphs. These graphs can be interpreted as circuit diagrams in which the edges of the graphs represent transistors, gated by the inputs to the function. One circuit computes the function itself, and the other computes its complement. One of the two circuits is derived by converting the conjunctions and disjunctions of the formula into series and parallel compositions of graphs, respectively. The other circuit reverses this construction, converting the conjunctions and disjunctions of the formula into parallel and series compositions of graphs. These two circuits, augmented by an additional edge connecting the input of each circuit to its output, are planar dual graphs. == History == The duality of convex polyhedra was recognized by Johannes Kepler in his 1619 book Harmonices Mundi. Recognizable planar dual graphs, outside the context of polyhedra, appeared as early as 1725, in Pierre Varignon's posthumously published work, Nouvelle Méchanique ou Statique. This was even before Leonhard Euler's 1736 work on the Seven Bridges of Königsberg that is often taken to be the first work on graph theory. Varignon analyzed the forces on static systems of struts by drawing a graph dual to the struts, with edge lengths proportional to the forces on the struts; this dual graph is a type of Cremona diagram. In connection with the four color theorem, the dual graphs of maps (subdivisions of the plane into regions) were mentioned by Alfred Kempe in 1879, and extended to maps on non-planar surfaces by Lothar Heffter in 1891. Duality as an operation on abstract planar graphs was introduced by Hassler Whitney in 1931. == Notes == == External links == Weisstein, Eric W., "Dual graph", MathWorld
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Wikipedia:Dual norm#0
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In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space. == Definition == Let X {\displaystyle X} be a normed vector space with norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} and let X ∗ {\displaystyle X^{*}} denote its continuous dual space. The dual norm of a continuous linear functional f {\displaystyle f} belonging to X ∗ {\displaystyle X^{*}} is the non-negative real number defined by any of the following equivalent formulas: ‖ f ‖ = sup { | f ( x ) | : ‖ x ‖ ≤ 1 and x ∈ X } = sup { | f ( x ) | : ‖ x ‖ < 1 and x ∈ X } = inf { c ∈ [ 0 , ∞ ) : | f ( x ) | ≤ c ‖ x ‖ for all x ∈ X } = sup { | f ( x ) | : ‖ x ‖ = 1 or 0 and x ∈ X } = sup { | f ( x ) | : ‖ x ‖ = 1 and x ∈ X } this equality holds if and only if X ≠ { 0 } = sup { | f ( x ) | ‖ x ‖ : x ≠ 0 and x ∈ X } this equality holds if and only if X ≠ { 0 } {\displaystyle {\begin{alignedat}{5}\|f\|&=\sup &&\{\,|f(x)|&&~:~\|x\|\leq 1~&&~{\text{ and }}~&&x\in X\}\\&=\sup &&\{\,|f(x)|&&~:~\|x\|<1~&&~{\text{ and }}~&&x\in X\}\\&=\inf &&\{\,c\in [0,\infty )&&~:~|f(x)|\leq c\|x\|~&&~{\text{ for all }}~&&x\in X\}\\&=\sup &&\{\,|f(x)|&&~:~\|x\|=1{\text{ or }}0~&&~{\text{ and }}~&&x\in X\}\\&=\sup &&\{\,|f(x)|&&~:~\|x\|=1~&&~{\text{ and }}~&&x\in X\}\;\;\;{\text{ this equality holds if and only if }}X\neq \{0\}\\&=\sup &&{\bigg \{}\,{\frac {|f(x)|}{\|x\|}}~&&~:~x\neq 0&&~{\text{ and }}~&&x\in X{\bigg \}}\;\;\;{\text{ this equality holds if and only if }}X\neq \{0\}\\\end{alignedat}}} where sup {\displaystyle \sup } and inf {\displaystyle \inf } denote the supremum and infimum, respectively. The constant 0 {\displaystyle 0} map is the origin of the vector space X ∗ {\displaystyle X^{*}} and it always has norm ‖ 0 ‖ = 0. {\displaystyle \|0\|=0.} If X = { 0 } {\displaystyle X=\{0\}} then the only linear functional on X {\displaystyle X} is the constant 0 {\displaystyle 0} map and moreover, the sets in the last two rows will both be empty and consequently, their supremums will equal sup ∅ = − ∞ {\displaystyle \sup \varnothing =-\infty } instead of the correct value of 0. {\displaystyle 0.} Importantly, a linear function f {\displaystyle f} is not, in general, guaranteed to achieve its norm ‖ f ‖ = sup { | f ( x ) | : ‖ x ‖ ≤ 1 , x ∈ X } {\displaystyle \|f\|=\sup\{|f(x)|:\|x\|\leq 1,x\in X\}} on the closed unit ball { x ∈ X : ‖ x ‖ ≤ 1 } , {\displaystyle \{x\in X:\|x\|\leq 1\},} meaning that there might not exist any vector u ∈ X {\displaystyle u\in X} of norm ‖ u ‖ ≤ 1 {\displaystyle \|u\|\leq 1} such that ‖ f ‖ = | f u | {\displaystyle \|f\|=|fu|} (if such a vector does exist and if f ≠ 0 , {\displaystyle f\neq 0,} then u {\displaystyle u} would necessarily have unit norm ‖ u ‖ = 1 {\displaystyle \|u\|=1} ). R.C. James proved James's theorem in 1964, which states that a Banach space X {\displaystyle X} is reflexive if and only if every bounded linear function f ∈ X ∗ {\displaystyle f\in X^{*}} achieves its norm on the closed unit ball. It follows, in particular, that every non-reflexive Banach space has some bounded linear functional that does not achieve its norm on the closed unit ball. However, the Bishop–Phelps theorem guarantees that the set of bounded linear functionals that achieve their norm on the unit sphere of a Banach space is a norm-dense subset of the continuous dual space. The map f ↦ ‖ f ‖ {\displaystyle f\mapsto \|f\|} defines a norm on X ∗ . {\displaystyle X^{*}.} (See Theorems 1 and 2 below.) The dual norm is a special case of the operator norm defined for each (bounded) linear map between normed vector spaces. Since the ground field of X {\displaystyle X} ( R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } ) is complete, X ∗ {\displaystyle X^{*}} is a Banach space. The topology on X ∗ {\displaystyle X^{*}} induced by ‖ ⋅ ‖ {\displaystyle \|\cdot \|} turns out to be stronger than the weak-* topology on X ∗ . {\displaystyle X^{*}.} == The double dual of a normed linear space == The double dual (or second dual) X ∗ ∗ {\displaystyle X^{**}} of X {\displaystyle X} is the dual of the normed vector space X ∗ {\displaystyle X^{*}} . There is a natural map φ : X → X ∗ ∗ {\displaystyle \varphi :X\to X^{**}} . Indeed, for each w ∗ {\displaystyle w^{*}} in X ∗ {\displaystyle X^{*}} define φ ( v ) ( w ∗ ) := w ∗ ( v ) . {\displaystyle \varphi (v)(w^{*}):=w^{*}(v).} The map φ {\displaystyle \varphi } is linear, injective, and distance preserving. In particular, if X {\displaystyle X} is complete (i.e. a Banach space), then φ {\displaystyle \varphi } is an isometry onto a closed subspace of X ∗ ∗ {\displaystyle X^{**}} . In general, the map φ {\displaystyle \varphi } is not surjective. For example, if X {\displaystyle X} is the Banach space L ∞ {\displaystyle L^{\infty }} consisting of bounded functions on the real line with the supremum norm, then the map φ {\displaystyle \varphi } is not surjective. (See L p {\displaystyle L^{p}} space). If φ {\displaystyle \varphi } is surjective, then X {\displaystyle X} is said to be a reflexive Banach space. If 1 < p < ∞ , {\displaystyle 1<p<\infty ,} then the space L p {\displaystyle L^{p}} is a reflexive Banach space. == Examples == === Dual norm for matrices === The Frobenius norm defined by ‖ A ‖ F = ∑ i = 1 m ∑ j = 1 n | a i j | 2 = trace ( A ∗ A ) = ∑ i = 1 min { m , n } σ i 2 {\displaystyle \|A\|_{\text{F}}={\sqrt {\sum _{i=1}^{m}\sum _{j=1}^{n}\left|a_{ij}\right|^{2}}}={\sqrt {\operatorname {trace} (A^{*}A)}}={\sqrt {\sum _{i=1}^{\min\{m,n\}}\sigma _{i}^{2}}}} is self-dual, i.e., its dual norm is ‖ ⋅ ‖ F ′ = ‖ ⋅ ‖ F . {\displaystyle \|\cdot \|'_{\text{F}}=\|\cdot \|_{\text{F}}.} The spectral norm, a special case of the induced norm when p = 2 {\displaystyle p=2} , is defined by the maximum singular values of a matrix, that is, ‖ A ‖ 2 = σ max ( A ) , {\displaystyle \|A\|_{2}=\sigma _{\max }(A),} has the nuclear norm as its dual norm, which is defined by ‖ B ‖ 2 ′ = ∑ i σ i ( B ) , {\displaystyle \|B\|'_{2}=\sum _{i}\sigma _{i}(B),} for any matrix B {\displaystyle B} where σ i ( B ) {\displaystyle \sigma _{i}(B)} denote the singular values. If p , q ∈ [ 1 , ∞ ] {\displaystyle p,q\in [1,\infty ]} the Schatten ℓ p {\displaystyle \ell ^{p}} -norm on matrices is dual to the Schatten ℓ q {\displaystyle \ell ^{q}} -norm. === Finite-dimensional spaces === Let ‖ ⋅ ‖ {\displaystyle \|\cdot \|} be a norm on R n . {\displaystyle \mathbb {R} ^{n}.} The associated dual norm, denoted ‖ ⋅ ‖ ∗ , {\displaystyle \|\cdot \|_{*},} is defined as ‖ z ‖ ∗ = sup { z ⊺ x : ‖ x ‖ ≤ 1 } . {\displaystyle \|z\|_{*}=\sup\{z^{\intercal }x:\|x\|\leq 1\}.} (This can be shown to be a norm.) The dual norm can be interpreted as the operator norm of z ⊺ , {\displaystyle z^{\intercal },} interpreted as a 1 × n {\displaystyle 1\times n} matrix, with the norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} on R n {\displaystyle \mathbb {R} ^{n}} , and the absolute value on R {\displaystyle \mathbb {R} } : ‖ z ‖ ∗ = sup { | z ⊺ x | : ‖ x ‖ ≤ 1 } . {\displaystyle \|z\|_{*}=\sup\{|z^{\intercal }x|:\|x\|\leq 1\}.} From the definition of dual norm we have the inequality z ⊺ x = ‖ x ‖ ( z ⊺ x ‖ x ‖ ) ≤ ‖ x ‖ ‖ z ‖ ∗ {\displaystyle z^{\intercal }x=\|x\|\left(z^{\intercal }{\frac {x}{\|x\|}}\right)\leq \|x\|\|z\|_{*}} which holds for all x {\displaystyle x} and z . {\displaystyle z.} The dual of the dual norm is the original norm: we have ‖ x ‖ ∗ ∗ = ‖ x ‖ {\displaystyle \|x\|_{**}=\|x\|} for all x . {\displaystyle x.} (This need not hold in infinite-dimensional vector spaces.) The dual of the Euclidean norm is the Euclidean norm, since sup { z ⊺ x : ‖ x ‖ 2 ≤ 1 } = ‖ z ‖ 2 . {\displaystyle \sup\{z^{\intercal }x:\|x\|_{2}\leq 1\}=\|z\|_{2}.} (This follows from the Cauchy–Schwarz inequality; for nonzero z , {\displaystyle z,} the value of x {\displaystyle x} that maximises z ⊺ x {\displaystyle z^{\intercal }x} over ‖ x ‖ 2 ≤ 1 {\displaystyle \|x\|_{2}\leq 1} is z ‖ z ‖ 2 . {\displaystyle {\tfrac {z}{\|z\|_{2}}}.} ) The dual of the ℓ ∞ {\displaystyle \ell ^{\infty }} -norm is the ℓ 1 {\displaystyle \ell ^{1}} -norm: sup { z ⊺ x : ‖ x ‖ ∞ ≤ 1 } = ∑ i = 1 n | z i | = ‖ z ‖ 1 , {\displaystyle \sup\{z^{\intercal }x:\|x\|_{\infty }\leq 1\}=\sum _{i=1}^{n}|z_{i}|=\|z\|_{1},} and the dual of the ℓ 1 {\displaystyle \ell ^{1}} -norm is the ℓ ∞ {\displaystyle \ell ^{\infty }} -norm. More generally, Hölder's inequality shows that the dual of the ℓ p {\displaystyle \ell ^{p}} -norm is the ℓ q {\displaystyle \ell ^{q}} -norm, where q {\displaystyle q} satisfies 1 p + 1 q = 1 , {\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1,} that is, q = p p − 1 . {\displaystyle q={\tfrac {p}{p-1}}.} As another example, consider the ℓ 2 {\displaystyle \ell ^{2}} - or spectral norm on R m × n {\displaystyle \mathbb {R} ^{m\times n}} . The associated dual norm is ‖ Z ‖ 2 ∗ = sup { t r ( Z ⊺ X ) : ‖ X ‖ 2 ≤ 1 } , {\displaystyle \|Z\|_{2*}=\sup\{\mathbf {tr} (Z^{\intercal }X):\|X\|_{2}\leq 1\},} which turns out to be the sum of the singular values, ‖ Z ‖ 2 ∗ = σ 1 ( Z ) + ⋯ + σ r ( Z ) = t r ( Z ⊺ Z ) , {\displaystyle \|Z\|_{2*}=\sigma _{1}(Z)+\cdots +\sigma _{r}(Z)=\mathbf {tr} ({\sqrt {Z^{\intercal }Z}}),} where r = r a n k Z . {\displaystyle r=\mathbf {rank} Z.} This norm is sometimes called the nuclear norm. === Lp and ℓp spaces === For p ∈ [ 1 , ∞ ] , {\displaystyle p\in [1,\infty ],} p-norm (also called ℓ p {\displaystyle \ell _{p}} -norm) of vector x = ( x n ) n {\displaystyle \mathbf {x} =(x_{n})_{n}} is ‖ x ‖ p := ( ∑ i = 1 n | x i | p ) 1 / p . {\displaystyle \|\mathbf {x} \|_{p}~:=~\left(\sum _{i=1}^{n}\left|x_{i}\right|^{p}\right)^{1/p}.} If p , q ∈ [ 1 , ∞ ] {\displaystyle p,q\in [1,\infty ]} satisfy 1 / p + 1 / q = 1 {\displaystyle 1/p+1/q=1} then the ℓ p {\displaystyle \ell ^{p}} and ℓ q {\displaystyle \ell ^{q}} norms are dual to each other and the same is true of the L p {\displaystyle L^{p}} and L q {\displaystyle L^{q}} norms, where ( X , Σ , μ ) , {\displaystyle (X,\Sigma ,\mu ),} is some measure space. In particular the Euclidean norm is self-dual since p = q = 2. {\displaystyle p=q=2.} For x T Q x {\displaystyle {\sqrt {x^{\mathrm {T} }Qx}}} , the dual norm is y T Q − 1 y {\displaystyle {\sqrt {y^{\mathrm {T} }Q^{-1}y}}} with Q {\displaystyle Q} positive definite. For p = 2 , {\displaystyle p=2,} the ‖ ⋅ ‖ 2 {\displaystyle \|\,\cdot \,\|_{2}} -norm is even induced by a canonical inner product ⟨ ⋅ , ⋅ ⟩ , {\displaystyle \langle \,\cdot ,\,\cdot \rangle ,} meaning that ‖ x ‖ 2 = ⟨ x , x ⟩ {\displaystyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}} for all vectors x . {\displaystyle \mathbf {x} .} This inner product can expressed in terms of the norm by using the polarization identity. On ℓ 2 , {\displaystyle \ell ^{2},} this is the Euclidean inner product defined by ⟨ ( x n ) n , ( y n ) n ⟩ ℓ 2 = ∑ n x n y n ¯ {\displaystyle \langle \left(x_{n}\right)_{n},\left(y_{n}\right)_{n}\rangle _{\ell ^{2}}~=~\sum _{n}x_{n}{\overline {y_{n}}}} while for the space L 2 ( X , μ ) {\displaystyle L^{2}(X,\mu )} associated with a measure space ( X , Σ , μ ) , {\displaystyle (X,\Sigma ,\mu ),} which consists of all square-integrable functions, this inner product is ⟨ f , g ⟩ L 2 = ∫ X f ( x ) g ( x ) ¯ d x . {\displaystyle \langle f,g\rangle _{L^{2}}=\int _{X}f(x){\overline {g(x)}}\,\mathrm {d} x.} The norms of the continuous dual spaces of ℓ 2 {\displaystyle \ell ^{2}} and ℓ 2 {\displaystyle \ell ^{2}} satisfy the polarization identity, and so these dual norms can be used to define inner products. With this inner product, this dual space is also a Hilbert space. == Properties == Given normed vector spaces X {\displaystyle X} and Y , {\displaystyle Y,} let L ( X , Y ) {\displaystyle L(X,Y)} be the collection of all bounded linear mappings (or operators) of X {\displaystyle X} into Y . {\displaystyle Y.} Then L ( X , Y ) {\displaystyle L(X,Y)} can be given a canonical norm. When Y {\displaystyle Y} is a scalar field (i.e. Y = C {\displaystyle Y=\mathbb {C} } or Y = R {\displaystyle Y=\mathbb {R} } ) so that L ( X , Y ) {\displaystyle L(X,Y)} is the dual space X ∗ {\displaystyle X^{*}} of X . {\displaystyle X.} As usual, let d ( x , y ) := ‖ x − y ‖ {\displaystyle d(x,y):=\|x-y\|} denote the canonical metric induced by the norm on X , {\displaystyle X,} and denote the distance from a point x {\displaystyle x} to the subset S ⊆ X {\displaystyle S\subseteq X} by d ( x , S ) := inf s ∈ S d ( x , s ) = inf s ∈ S ‖ x − s ‖ . {\displaystyle d(x,S)~:=~\inf _{s\in S}d(x,s)~=~\inf _{s\in S}\|x-s\|.} If f {\displaystyle f} is a bounded linear functional on a normed space X , {\displaystyle X,} then for every vector x ∈ X , {\displaystyle x\in X,} | f ( x ) | = ‖ f ‖ d ( x , ker f ) , {\displaystyle |f(x)|=\|f\|\,d(x,\ker f),} where ker f = { k ∈ X : f ( k ) = 0 } {\displaystyle \ker f=\{k\in X:f(k)=0\}} denotes the kernel of f . {\displaystyle f.} == See also == Convex conjugate – Generalization of the Legendre transformation Hölder's inequality – Inequality between integrals in Lp spaces Lp space – Function spaces generalizing finite-dimensional p norm spaces Operator norm – Measure of the "size" of linear operators Polarization identity – Formula relating the norm and the inner product in a inner product space == Notes == == References == Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer. ISBN 9783540326960. Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge University Press. ISBN 9780521833783. Diestel, Joe (1984). Sequences and series in Banach spaces. New York: Springer-Verlag. ISBN 0-387-90859-5. OCLC 9556781. Hashimoto, Kazuo; Nakamura, Gen; Oharu, Shinnosuke (1986-01-01). "Riesz's lemma and orthogonality in normed spaces" (PDF). Hiroshima Mathematical Journal. 16 (2). Hiroshima University - Department of Mathematics. doi:10.32917/hmj/1206130429. ISSN 0018-2079. Kolmogorov, A.N.; Fomin, S.V. (1957). Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces. Rochester: Graylock Press. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. == External links == Notes on the proximal mapping by Lieven Vandenberge
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Wikipedia:Dual number#0
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In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form a + bε, where a and b are real numbers, and ε is a symbol taken to satisfy ε 2 = 0 {\displaystyle \varepsilon ^{2}=0} with ε ≠ 0 {\displaystyle \varepsilon \neq 0} . Dual numbers can be added component-wise, and multiplied by the formula ( a + b ε ) ( c + d ε ) = a c + ( a d + b c ) ε , {\displaystyle (a+b\varepsilon )(c+d\varepsilon )=ac+(ad+bc)\varepsilon ,} which follows from the property ε2 = 0 and the fact that multiplication is a bilinear operation. The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements. == History == Dual numbers were introduced in 1873 by William Clifford, and were used at the beginning of the twentieth century by the German mathematician Eduard Study, who used them to represent the dual angle which measures the relative position of two skew lines in space. Study defined a dual angle as θ + dε, where θ is the angle between the directions of two lines in three-dimensional space and d is a distance between them. The n-dimensional generalization, the Grassmann number, was introduced by Hermann Grassmann in the late 19th century. == Modern definition == In modern algebra, the algebra of dual numbers is often defined as the quotient of a polynomial ring over the real numbers ( R ) {\displaystyle (\mathbb {R} )} by the principal ideal generated by the square of the indeterminate, that is R [ X ] / ⟨ X 2 ⟩ . {\displaystyle \mathbb {R} [X]/\left\langle X^{2}\right\rangle .} It may also be defined as the exterior algebra of a one-dimensional vector space with ε {\displaystyle \varepsilon } as its basis element. == Division == Division of dual numbers is defined when the real part of the denominator is non-zero. The division process is analogous to complex division in that the denominator is multiplied by its conjugate in order to cancel the non-real parts. Therefore, to evaluate an expression of the form a + b ε c + d ε {\displaystyle {\frac {a+b\varepsilon }{c+d\varepsilon }}} we multiply the numerator and denominator by the conjugate of the denominator: a + b ε c + d ε = ( a + b ε ) ( c − d ε ) ( c + d ε ) ( c − d ε ) = a c − a d ε + b c ε − b d ε 2 c 2 + c d ε − c d ε − d 2 ε 2 = a c − a d ε + b c ε − 0 c 2 − 0 = a c + ε ( b c − a d ) c 2 = a c + b c − a d c 2 ε {\displaystyle {\begin{aligned}{\frac {a+b\varepsilon }{c+d\varepsilon }}&={\frac {(a+b\varepsilon )(c-d\varepsilon )}{(c+d\varepsilon )(c-d\varepsilon )}}\\[5pt]&={\frac {ac-ad\varepsilon +bc\varepsilon -bd\varepsilon ^{2}}{c^{2}+cd\varepsilon -cd\varepsilon -d^{2}\varepsilon ^{2}}}\\[5pt]&={\frac {ac-ad\varepsilon +bc\varepsilon -0}{c^{2}-0}}\\[5pt]&={\frac {ac+\varepsilon (bc-ad)}{c^{2}}}\\[5pt]&={\frac {a}{c}}+{\frac {bc-ad}{c^{2}}}\varepsilon \end{aligned}}} which is defined when c is non-zero. If, on the other hand, c is zero while d is not, then the equation a + b ε = ( x + y ε ) d ε = x d ε + 0 {\displaystyle {a+b\varepsilon =(x+y\varepsilon )d\varepsilon }={xd\varepsilon +0}} has no solution if a is nonzero is otherwise solved by any dual number of the form b/d + yε. This means that the non-real part of the "quotient" is arbitrary and division is therefore not defined for purely nonreal dual numbers. Indeed, they are (trivially) zero divisors and clearly form an ideal of the associative algebra (and thus ring) of the dual numbers. == Matrix representation == The dual number a + b ε {\displaystyle a+b\varepsilon } can be represented by the square matrix ( a b 0 a ) {\displaystyle {\begin{pmatrix}a&b\\0&a\end{pmatrix}}} . In this representation the matrix ( 0 1 0 0 ) {\displaystyle {\begin{pmatrix}0&1\\0&0\end{pmatrix}}} squares to the zero matrix, corresponding to the dual number ε {\displaystyle \varepsilon } . Generally, if ε {\displaystyle \varepsilon } is a nilpotent matrix, then B = {x I + y ε {\displaystyle \varepsilon } : x, y real} is a subalgebra isomorphic to the algebra of dual numbers. In the case of 2x2 real matrices M(2,R), ε {\displaystyle \varepsilon } can be taken as any matrix of the form ( a b c − a ) {\displaystyle {\begin{pmatrix}a&b\\c&-a\end{pmatrix}}} with p = a2 + bc = 0. The dual numbers are one of three isomorphism classes of real 2-algebras in M(2,R). When p > 0 the subalgebra B is isomorphic to split-complex numbers, and when p < 0, B is isomorphic to the complex plane. == Automatic differentiation == One application of dual numbers is automatic differentiation. Any polynomial P ( x ) = p 0 + p 1 x + p 2 x 2 + ⋯ + p n x n {\displaystyle P(x)=p_{0}+p_{1}x+p_{2}x^{2}+\cdots +p_{n}x^{n}} with real coefficients can be extended to a function of a dual-number-valued argument, P ( a + b ε ) = p 0 + p 1 ( a + b ε ) + ⋯ + p n ( a + b ε ) n = p 0 + p 1 a + p 2 a 2 + ⋯ + p n a n + p 1 b ε + 2 p 2 a b ε + ⋯ + n p n a n − 1 b ε = P ( a ) + b P ′ ( a ) ε , {\displaystyle {\begin{aligned}P(a+b\varepsilon )&=p_{0}+p_{1}(a+b\varepsilon )+\cdots +p_{n}(a+b\varepsilon )^{n}\\[2mu]&=p_{0}+p_{1}a+p_{2}a^{2}+\cdots +p_{n}a^{n}+p_{1}b\varepsilon +2p_{2}ab\varepsilon +\cdots +np_{n}a^{n-1}b\varepsilon \\[5mu]&=P(a)+bP'(a)\varepsilon ,\end{aligned}}} where P ′ {\displaystyle P'} is the derivative of P . {\displaystyle P.} More generally, any (analytic) real function can be extended to the dual numbers via its Taylor series: f ( a + b ε ) = ∑ n = 0 ∞ f ( n ) ( a ) b n ε n n ! = f ( a ) + b f ′ ( a ) ε , {\displaystyle f(a+b\varepsilon )=\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)b^{n}\varepsilon ^{n}}{n!}}=f(a)+bf'(a)\varepsilon ,} since all terms involving ε2 or greater powers are trivially 0 by the definition of ε. By computing compositions of these functions over the dual numbers and examining the coefficient of ε in the result we find we have automatically computed the derivative of the composition. A similar method works for polynomials of n variables, using the exterior algebra of an n-dimensional vector space. == Geometry == The "unit circle" of dual numbers consists of those with a = ±1 since these satisfy zz* = 1 where z* = a − bε. However, note that e b ε = ∑ n = 0 ∞ ( b ε ) n n ! = 1 + b ε , {\displaystyle e^{b\varepsilon }=\sum _{n=0}^{\infty }{\frac {\left(b\varepsilon \right)^{n}}{n!}}=1+b\varepsilon ,} so the exponential map applied to the ε-axis covers only half the "circle". Let z = a + bε. If a ≠ 0 and m = b/a, then z = a(1 + mε) is the polar decomposition of the dual number z, and the slope m is its angular part. The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since (1 + pε)(1 + qε) = 1 + (p + q)ε. In absolute space and time the Galilean transformation ( t ′ , x ′ ) = ( t , x ) ( 1 v 0 1 ) , {\displaystyle \left(t',x'\right)=(t,x){\begin{pmatrix}1&v\\0&1\end{pmatrix}}\,,} that is t ′ = t , x ′ = v t + x , {\displaystyle t'=t,\quad x'=vt+x,} relates the resting coordinates system to a moving frame of reference of velocity v. With dual numbers t + xε representing events along one space dimension and time, the same transformation is effected with multiplication by 1 + vε. === Cycles === Given two dual numbers p and q, they determine the set of z such that the difference in slopes ("Galilean angle") between the lines from z to p and q is constant. This set is a cycle in the dual number plane; since the equation setting the difference in slopes of the lines to a constant is a quadratic equation in the real part of z, a cycle is a parabola. The "cyclic rotation" of the dual number plane occurs as a motion of its projective line. According to Isaak Yaglom,: 92–93 the cycle Z = {z : y = αx2} is invariant under the composition of the shear x 1 = x , y 1 = v x + y {\displaystyle x_{1}=x,\quad y_{1}=vx+y} with the translation x ′ = x 1 = v 2 a , y ′ = y 1 + v 2 4 a . {\displaystyle x'=x_{1}={\frac {v}{2a}},\quad y'=y_{1}+{\frac {v^{2}}{4a}}.} == Applications in mechanics == Dual numbers find applications in mechanics, notably for kinematic synthesis. For example, the dual numbers make it possible to transform the input/output equations of a four-bar spherical linkage, which includes only rotoid joints, into a four-bar spatial mechanism (rotoid, rotoid, rotoid, cylindrical). The dualized angles are made of a primitive part, the angles, and a dual part, which has units of length. See screw theory for more. == Algebraic geometry == In modern algebraic geometry, the dual numbers over a field k {\displaystyle k} (by which we mean the ring k [ ε ] / ( ε 2 ) {\displaystyle k[\varepsilon ]/(\varepsilon ^{2})} ) may be used to define the tangent vectors to the points of a k {\displaystyle k} -scheme. Since the field k {\displaystyle k} can be chosen intrinsically, it is possible to speak simply of the tangent vectors to a scheme. This allows notions from differential geometry to be imported into algebraic geometry. In detail: The ring of dual numbers may be thought of as the ring of functions on the "first-order neighborhood of a point" – namely, the k {\displaystyle k} -scheme Spec ( k [ ε ] / ( ε 2 ) ) {\displaystyle \operatorname {Spec} (k[\varepsilon ]/(\varepsilon ^{2}))} . Then, given a k {\displaystyle k} -scheme X {\displaystyle X} , k {\displaystyle k} -points of the scheme are in 1-1 correspondence with maps Spec k → X {\displaystyle \operatorname {Spec} k\to X} , while tangent vectors are in 1-1 correspondence with maps Spec ( k [ ε ] / ( ε 2 ) ) → X {\displaystyle \operatorname {Spec} (k[\varepsilon ]/(\varepsilon ^{2}))\to X} . The field k {\displaystyle k} above can be chosen intrinsically to be a residue field. To wit: Given a point x {\displaystyle x} on a scheme S {\displaystyle S} , consider the stalk S x {\displaystyle S_{x}} . Observe that S x {\displaystyle S_{x}} is a local ring with a unique maximal ideal, which is denoted m x {\displaystyle {\mathfrak {m}}_{x}} . Then simply let k = S x / m x {\displaystyle k=S_{x}/{\mathfrak {m}}_{x}} . == Generalizations == This construction can be carried out more generally: for a commutative ring R one can define the dual numbers over R as the quotient of the polynomial ring R[X] by the ideal (X2): the image of X then has square equal to zero and corresponds to the element ε from above. === Arbitrary module of elements of zero square === There is a more general construction of the dual numbers. Given a commutative ring R {\displaystyle R} and a module M {\displaystyle M} , there is a ring R [ M ] {\displaystyle R[M]} called the ring of dual numbers which has the following structures: It is the R {\displaystyle R} -module R ⊕ M {\displaystyle R\oplus M} with the multiplication defined by ( r , i ) ⋅ ( r ′ , i ′ ) = ( r r ′ , r i ′ + r ′ i ) {\displaystyle (r,i)\cdot \left(r',i'\right)=\left(rr',ri'+r'i\right)} for r , r ′ ∈ R {\displaystyle r,r'\in R} and i , i ′ ∈ I . {\displaystyle i,i'\in I.} The algebra of dual numbers is the special case where M = R {\displaystyle M=R} and ε = ( 0 , 1 ) . {\displaystyle \varepsilon =(0,1).} == Superspace == Dual numbers find applications in physics, where they constitute one of the simplest non-trivial examples of a superspace. Equivalently, they are supernumbers with just one generator; supernumbers generalize the concept to n distinct generators ε, each anti-commuting, possibly taking n to infinity. Superspace generalizes supernumbers slightly, by allowing multiple commuting dimensions. The motivation for introducing dual numbers into physics follows from the Pauli exclusion principle for fermions. The direction along ε is termed the "fermionic" direction, and the real component is termed the "bosonic" direction. The fermionic direction earns this name from the fact that fermions obey the Pauli exclusion principle: under the exchange of coordinates, the quantum mechanical wave function changes sign, and thus vanishes if two coordinates are brought together; this physical idea is captured by the algebraic relation ε2 = 0. == Projective line == The idea of a projective line over dual numbers was advanced by Grünwald and Corrado Segre. Just as the Riemann sphere needs a north pole point at infinity to close up the complex projective line, so a line at infinity succeeds in closing up the plane of dual numbers to a cylinder.: 149–153 Suppose D is the ring of dual numbers x + yε and U is the subset with x ≠ 0. Then U is the group of units of D. Let B = {(a, b) ∈ D × D : a ∈ U or b ∈ U}. A relation is defined on B as follows: (a, b) ~ (c, d) when there is a u in U such that ua = c and ub = d. This relation is in fact an equivalence relation. The points of the projective line over D are equivalence classes in B under this relation: P(D) = B/~. They are represented with projective coordinates [a, b]. Consider the embedding D → P(D) by z → [z, 1]. Then points [1, n], for n2 = 0, are in P(D) but are not the image of any point under the embedding. P(D) is mapped onto a cylinder by projection: Take a cylinder tangent to the double number plane on the line {yε : y ∈ R}, ε2 = 0. Now take the opposite line on the cylinder for the axis of a pencil of planes. The planes intersecting the dual number plane and cylinder provide a correspondence of points between these surfaces. The plane parallel to the dual number plane corresponds to points [1, n], n2 = 0 in the projective line over dual numbers. == See also == Smooth infinitesimal analysis Perturbation theory Infinitesimal Screw theory Dual-complex number Laguerre transformations Grassmann number Automatic differentiation == References == === Further reading ===
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Wikipedia:Dual space#0
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In mathematics, any vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V , {\displaystyle V,} together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for dual include polarer Raum [Hahn 1927], espace conjugué, adjoint space [Alaoglu 1940], and transponierter Raum [Schauder 1930] and [Banach 1932]. The term dual is due to Bourbaki 1938. == Algebraic dual space == Given any vector space V {\displaystyle V} over a field F {\displaystyle F} , the (algebraic) dual space V ∗ {\displaystyle V^{*}} (alternatively denoted by V ∨ {\displaystyle V^{\lor }} or V ′ {\displaystyle V'} ) is defined as the set of all linear maps φ : V → F {\displaystyle \varphi :V\to F} (linear functionals). Since linear maps are vector space homomorphisms, the dual space may be denoted hom ( V , F ) {\displaystyle \hom(V,F)} . The dual space V ∗ {\displaystyle V^{*}} itself becomes a vector space over F {\displaystyle F} when equipped with an addition and scalar multiplication satisfying: ( φ + ψ ) ( x ) = φ ( x ) + ψ ( x ) ( a φ ) ( x ) = a ( φ ( x ) ) {\displaystyle {\begin{aligned}(\varphi +\psi )(x)&=\varphi (x)+\psi (x)\\(a\varphi )(x)&=a\left(\varphi (x)\right)\end{aligned}}} for all φ , ψ ∈ V ∗ {\displaystyle \varphi ,\psi \in V^{*}} , x ∈ V {\displaystyle x\in V} , and a ∈ F {\displaystyle a\in F} . Elements of the algebraic dual space V ∗ {\displaystyle V^{*}} are sometimes called covectors, one-forms, or linear forms. The pairing of a functional φ {\displaystyle \varphi } in the dual space V ∗ {\displaystyle V^{*}} and an element x {\displaystyle x} of V {\displaystyle V} is sometimes denoted by a bracket: φ ( x ) = [ x , φ ] {\displaystyle \varphi (x)=[x,\varphi ]} or φ ( x ) = ⟨ x , φ ⟩ {\displaystyle \varphi (x)=\langle x,\varphi \rangle } . This pairing defines a nondegenerate bilinear mapping ⟨ ⋅ , ⋅ ⟩ : V × V ∗ → F {\displaystyle \langle \cdot ,\cdot \rangle :V\times V^{*}\to F} called the natural pairing. === Finite-dimensional case === If V {\displaystyle V} is finite-dimensional, then V ∗ {\displaystyle V^{*}} has the same dimension as V {\displaystyle V} . Given a basis { e 1 , … , e n } {\displaystyle \{\mathbf {e} _{1},\dots ,\mathbf {e} _{n}\}} in V {\displaystyle V} , it is possible to construct a specific basis in V ∗ {\displaystyle V^{*}} , called the dual basis. This dual basis is a set { e 1 , … , e n } {\displaystyle \{\mathbf {e} ^{1},\dots ,\mathbf {e} ^{n}\}} of linear functionals on V {\displaystyle V} , defined by the relation e i ( c 1 e 1 + ⋯ + c n e n ) = c i , i = 1 , … , n {\displaystyle \mathbf {e} ^{i}(c^{1}\mathbf {e} _{1}+\cdots +c^{n}\mathbf {e} _{n})=c^{i},\quad i=1,\ldots ,n} for any choice of coefficients c i ∈ F {\displaystyle c^{i}\in F} . In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations e i ( e j ) = δ j i {\displaystyle \mathbf {e} ^{i}(\mathbf {e} _{j})=\delta _{j}^{i}} where δ j i {\displaystyle \delta _{j}^{i}} is the Kronecker delta symbol. This property is referred to as the bi-orthogonality property. For example, if V {\displaystyle V} is R 2 {\displaystyle \mathbb {R} ^{2}} , let its basis be chosen as { e 1 = ( 1 / 2 , 1 / 2 ) , e 2 = ( 0 , 1 ) } {\displaystyle \{\mathbf {e} _{1}=(1/2,1/2),\mathbf {e} _{2}=(0,1)\}} . The basis vectors are not orthogonal to each other. Then, e 1 {\displaystyle \mathbf {e} ^{1}} and e 2 {\displaystyle \mathbf {e} ^{2}} are one-forms (functions that map a vector to a scalar) such that e 1 ( e 1 ) = 1 {\displaystyle \mathbf {e} ^{1}(\mathbf {e} _{1})=1} , e 1 ( e 2 ) = 0 {\displaystyle \mathbf {e} ^{1}(\mathbf {e} _{2})=0} , e 2 ( e 1 ) = 0 {\displaystyle \mathbf {e} ^{2}(\mathbf {e} _{1})=0} , and e 2 ( e 2 ) = 1 {\displaystyle \mathbf {e} ^{2}(\mathbf {e} _{2})=1} . (Note: The superscript here is the index, not an exponent.) This system of equations can be expressed using matrix notation as [ e 11 e 12 e 21 e 22 ] [ e 11 e 21 e 12 e 22 ] = [ 1 0 0 1 ] . {\displaystyle {\begin{bmatrix}e^{11}&e^{12}\\e^{21}&e^{22}\end{bmatrix}}{\begin{bmatrix}e_{11}&e_{21}\\e_{12}&e_{22}\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}.} Solving for the unknown values in the first matrix shows the dual basis to be { e 1 = ( 2 , 0 ) , e 2 = ( − 1 , 1 ) } {\displaystyle \{\mathbf {e} ^{1}=(2,0),\mathbf {e} ^{2}=(-1,1)\}} . Because e 1 {\displaystyle \mathbf {e} ^{1}} and e 2 {\displaystyle \mathbf {e} ^{2}} are functionals, they can be rewritten as e 1 ( x , y ) = 2 x {\displaystyle \mathbf {e} ^{1}(x,y)=2x} and e 2 ( x , y ) = − x + y {\displaystyle \mathbf {e} ^{2}(x,y)=-x+y} . In general, when V {\displaystyle V} is R n {\displaystyle \mathbb {R} ^{n}} , if E = [ e 1 | ⋯ | e n ] {\displaystyle E=[\mathbf {e} _{1}|\cdots |\mathbf {e} _{n}]} is a matrix whose columns are the basis vectors and E ^ = [ e 1 | ⋯ | e n ] {\displaystyle {\hat {E}}=[\mathbf {e} ^{1}|\cdots |\mathbf {e} ^{n}]} is a matrix whose columns are the dual basis vectors, then E ^ T ⋅ E = I n , {\displaystyle {\hat {E}}^{\textrm {T}}\cdot E=I_{n},} where I n {\displaystyle I_{n}} is the identity matrix of order n {\displaystyle n} . The biorthogonality property of these two basis sets allows any point x ∈ V {\displaystyle \mathbf {x} \in V} to be represented as x = ∑ i ⟨ x , e i ⟩ e i = ∑ i ⟨ x , e i ⟩ e i , {\displaystyle \mathbf {x} =\sum _{i}\langle \mathbf {x} ,\mathbf {e} ^{i}\rangle \mathbf {e} _{i}=\sum _{i}\langle \mathbf {x} ,\mathbf {e} _{i}\rangle \mathbf {e} ^{i},} even when the basis vectors are not orthogonal to each other. Strictly speaking, the above statement only makes sense once the inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } and the corresponding duality pairing are introduced, as described below in § Bilinear products and dual spaces. In particular, R n {\displaystyle \mathbb {R} ^{n}} can be interpreted as the space of columns of n {\displaystyle n} real numbers, its dual space is typically written as the space of rows of n {\displaystyle n} real numbers. Such a row acts on R n {\displaystyle \mathbb {R} ^{n}} as a linear functional by ordinary matrix multiplication. This is because a functional maps every n {\displaystyle n} -vector x {\displaystyle x} into a real number y {\displaystyle y} . Then, seeing this functional as a matrix M {\displaystyle M} , and x {\displaystyle x} as an n × 1 {\displaystyle n\times 1} matrix, and y {\displaystyle y} a 1 × 1 {\displaystyle 1\times 1} matrix (trivially, a real number) respectively, if M x = y {\displaystyle Mx=y} then, by dimension reasons, M {\displaystyle M} must be a 1 × n {\displaystyle 1\times n} matrix; that is, M {\displaystyle M} must be a row vector. If V {\displaystyle V} consists of the space of geometrical vectors in the plane, then the level curves of an element of V ∗ {\displaystyle V^{*}} form a family of parallel lines in V {\displaystyle V} , because the range is 1-dimensional, so that every point in the range is a multiple of any one nonzero element. So an element of V ∗ {\displaystyle V^{*}} can be intuitively thought of as a particular family of parallel lines covering the plane. To compute the value of a functional on a given vector, it suffices to determine which of the lines the vector lies on. Informally, this "counts" how many lines the vector crosses. More generally, if V {\displaystyle V} is a vector space of any dimension, then the level sets of a linear functional in V ∗ {\displaystyle V^{*}} are parallel hyperplanes in V {\displaystyle V} , and the action of a linear functional on a vector can be visualized in terms of these hyperplanes. === Infinite-dimensional case === If V {\displaystyle V} is not finite-dimensional but has a basis e α {\displaystyle \mathbf {e} _{\alpha }} indexed by an infinite set A {\displaystyle A} , then the same construction as in the finite-dimensional case yields linearly independent elements e α {\displaystyle \mathbf {e} ^{\alpha }} ( α ∈ A {\displaystyle \alpha \in A} ) of the dual space, but they will not form a basis. For instance, consider the space R ∞ {\displaystyle \mathbb {R} ^{\infty }} , whose elements are those sequences of real numbers that contain only finitely many non-zero entries, which has a basis indexed by the natural numbers N {\displaystyle \mathbb {N} } . For i ∈ N {\displaystyle i\in \mathbb {N} } , e i {\displaystyle \mathbf {e} _{i}} is the sequence consisting of all zeroes except in the i {\displaystyle i} -th position, which is 1. The dual space of R ∞ {\displaystyle \mathbb {R} ^{\infty }} is (isomorphic to) R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} , the space of all sequences of real numbers: each real sequence ( a n ) {\displaystyle (a_{n})} defines a function where the element ( x n ) {\displaystyle (x_{n})} of R ∞ {\displaystyle \mathbb {R} ^{\infty }} is sent to the number ∑ n a n x n , {\displaystyle \sum _{n}a_{n}x_{n},} which is a finite sum because there are only finitely many nonzero x n {\displaystyle x_{n}} . The dimension of R ∞ {\displaystyle \mathbb {R} ^{\infty }} is countably infinite, whereas R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} does not have a countable basis. This observation generalizes to any infinite-dimensional vector space V {\displaystyle V} over any field F {\displaystyle F} : a choice of basis { e α : α ∈ A } {\displaystyle \{\mathbf {e} _{\alpha }:\alpha \in A\}} identifies V {\displaystyle V} with the space ( F A ) 0 {\displaystyle (F^{A})_{0}} of functions f : A → F {\displaystyle f:A\to F} such that f α = f ( α ) {\displaystyle f_{\alpha }=f(\alpha )} is nonzero for only finitely many α ∈ A {\displaystyle \alpha \in A} , where such a function f {\displaystyle f} is identified with the vector ∑ α ∈ A f α e α {\displaystyle \sum _{\alpha \in A}f_{\alpha }\mathbf {e} _{\alpha }} in V {\displaystyle V} (the sum is finite by the assumption on f {\displaystyle f} , and any v ∈ V {\displaystyle v\in V} may be written uniquely in this way by the definition of the basis). The dual space of V {\displaystyle V} may then be identified with the space F A {\displaystyle F^{A}} of all functions from A {\displaystyle A} to F {\displaystyle F} : a linear functional T {\displaystyle T} on V {\displaystyle V} is uniquely determined by the values θ α = T ( e α ) {\displaystyle \theta _{\alpha }=T(\mathbf {e} _{\alpha })} it takes on the basis of V {\displaystyle V} , and any function θ : A → F {\displaystyle \theta :A\to F} (with θ ( α ) = θ α {\displaystyle \theta (\alpha )=\theta _{\alpha }} ) defines a linear functional T {\displaystyle T} on V {\displaystyle V} by T ( ∑ α ∈ A f α e α ) = ∑ α ∈ A f α T ( e α ) = ∑ α ∈ A f α θ α . {\displaystyle T\left(\sum _{\alpha \in A}f_{\alpha }\mathbf {e} _{\alpha }\right)=\sum _{\alpha \in A}f_{\alpha }T(e_{\alpha })=\sum _{\alpha \in A}f_{\alpha }\theta _{\alpha }.} Again, the sum is finite because f α {\displaystyle f_{\alpha }} is nonzero for only finitely many α {\displaystyle \alpha } . The set ( F A ) 0 {\displaystyle (F^{A})_{0}} may be identified (essentially by definition) with the direct sum of infinitely many copies of F {\displaystyle F} (viewed as a 1-dimensional vector space over itself) indexed by A {\displaystyle A} , i.e. there are linear isomorphisms V ≅ ( F A ) 0 ≅ ⨁ α ∈ A F . {\displaystyle V\cong (F^{A})_{0}\cong \bigoplus _{\alpha \in A}F.} On the other hand, F A {\displaystyle F^{A}} is (again by definition), the direct product of infinitely many copies of F {\displaystyle F} indexed by A {\displaystyle A} , and so the identification V ∗ ≅ ( ⨁ α ∈ A F ) ∗ ≅ ∏ α ∈ A F ∗ ≅ ∏ α ∈ A F ≅ F A {\displaystyle V^{*}\cong \left(\bigoplus _{\alpha \in A}F\right)^{*}\cong \prod _{\alpha \in A}F^{*}\cong \prod _{\alpha \in A}F\cong F^{A}} is a special case of a general result relating direct sums (of modules) to direct products. If a vector space is not finite-dimensional, then its (algebraic) dual space is always of larger dimension (as a cardinal number) than the original vector space. This is in contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if the latter is infinite-dimensional. The proof of this inequality between dimensions results from the following. If V {\displaystyle V} is an infinite-dimensional F {\displaystyle F} -vector space, the arithmetical properties of cardinal numbers implies that d i m ( V ) = | A | < | F | | A | = | V ∗ | = m a x ( | d i m ( V ∗ ) | , | F | ) , {\displaystyle \mathrm {dim} (V)=|A|<|F|^{|A|}=|V^{\ast }|=\mathrm {max} (|\mathrm {dim} (V^{\ast })|,|F|),} where cardinalities are denoted as absolute values. For proving that d i m ( V ) < d i m ( V ∗ ) , {\displaystyle \mathrm {dim} (V)<\mathrm {dim} (V^{*}),} it suffices to prove that | F | ≤ | d i m ( V ∗ ) | , {\displaystyle |F|\leq |\mathrm {dim} (V^{\ast })|,} which can be done with an argument similar to Cantor's diagonal argument. The exact dimension of the dual is given by the Erdős–Kaplansky theorem. === Bilinear products and dual spaces === If V is finite-dimensional, then V is isomorphic to V∗. But there is in general no natural isomorphism between these two spaces. Any bilinear form ⟨·,·⟩ on V gives a mapping of V into its dual space via v ↦ ⟨ v , ⋅ ⟩ {\displaystyle v\mapsto \langle v,\cdot \rangle } where the right hand side is defined as the functional on V taking each w ∈ V to ⟨v, w⟩. In other words, the bilinear form determines a linear mapping Φ ⟨ ⋅ , ⋅ ⟩ : V → V ∗ {\displaystyle \Phi _{\langle \cdot ,\cdot \rangle }:V\to V^{*}} defined by [ Φ ⟨ ⋅ , ⋅ ⟩ ( v ) , w ] = ⟨ v , w ⟩ . {\displaystyle \left[\Phi _{\langle \cdot ,\cdot \rangle }(v),w\right]=\langle v,w\rangle .} If the bilinear form is nondegenerate, then this is an isomorphism onto a subspace of V∗. If V is finite-dimensional, then this is an isomorphism onto all of V∗. Conversely, any isomorphism Φ {\displaystyle \Phi } from V to a subspace of V∗ (resp., all of V∗ if V is finite dimensional) defines a unique nondegenerate bilinear form ⟨ ⋅ , ⋅ ⟩ Φ {\displaystyle \langle \cdot ,\cdot \rangle _{\Phi }} on V by ⟨ v , w ⟩ Φ = ( Φ ( v ) ) ( w ) = [ Φ ( v ) , w ] . {\displaystyle \langle v,w\rangle _{\Phi }=(\Phi (v))(w)=[\Phi (v),w].\,} Thus there is a one-to-one correspondence between isomorphisms of V to a subspace of (resp., all of) V∗ and nondegenerate bilinear forms on V. If the vector space V is over the complex field, then sometimes it is more natural to consider sesquilinear forms instead of bilinear forms. In that case, a given sesquilinear form ⟨·,·⟩ determines an isomorphism of V with the complex conjugate of the dual space Φ ⟨ ⋅ , ⋅ ⟩ : V → V ∗ ¯ . {\displaystyle \Phi _{\langle \cdot ,\cdot \rangle }:V\to {\overline {V^{*}}}.} The conjugate of the dual space V ∗ ¯ {\displaystyle {\overline {V^{*}}}} can be identified with the set of all additive complex-valued functionals f : V → C such that f ( α v ) = α ¯ f ( v ) . {\displaystyle f(\alpha v)={\overline {\alpha }}f(v).} === Injection into the double-dual === There is a natural homomorphism Ψ {\displaystyle \Psi } from V {\displaystyle V} into the double dual V ∗ ∗ = hom ( V ∗ , F ) {\displaystyle V^{**}=\hom(V^{*},F)} , defined by ( Ψ ( v ) ) ( φ ) = φ ( v ) {\displaystyle (\Psi (v))(\varphi )=\varphi (v)} for all v ∈ V , φ ∈ V ∗ {\displaystyle v\in V,\varphi \in V^{*}} . In other words, if e v v : V ∗ → F {\displaystyle \mathrm {ev} _{v}:V^{*}\to F} is the evaluation map defined by φ ↦ φ ( v ) {\displaystyle \varphi \mapsto \varphi (v)} , then Ψ : V → V ∗ ∗ {\displaystyle \Psi :V\to V^{**}} is defined as the map v ↦ e v v {\displaystyle v\mapsto \mathrm {ev} _{v}} . This map Ψ {\displaystyle \Psi } is always injective; and it is always an isomorphism if V {\displaystyle V} is finite-dimensional. Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a natural isomorphism. Infinite-dimensional Hilbert spaces are not isomorphic to their algebraic double duals, but instead to their continuous double duals. === Transpose of a linear map === If f : V → W is a linear map, then the transpose (or dual) f∗ : W∗ → V∗ is defined by f ∗ ( φ ) = φ ∘ f {\displaystyle f^{*}(\varphi )=\varphi \circ f\,} for every φ ∈ W ∗ {\displaystyle \varphi \in W^{*}} . The resulting functional f ∗ ( φ ) {\displaystyle f^{*}(\varphi )} in V ∗ {\displaystyle V^{*}} is called the pullback of φ {\displaystyle \varphi } along f {\displaystyle f} . The following identity holds for all φ ∈ W ∗ {\displaystyle \varphi \in W^{*}} and v ∈ V {\displaystyle v\in V} : [ f ∗ ( φ ) , v ] = [ φ , f ( v ) ] , {\displaystyle [f^{*}(\varphi ),\,v]=[\varphi ,\,f(v)],} where the bracket [·,·] on the left is the natural pairing of V with its dual space, and that on the right is the natural pairing of W with its dual. This identity characterizes the transpose, and is formally similar to the definition of the adjoint. The assignment f ↦ f∗ produces an injective linear map between the space of linear operators from V to W and the space of linear operators from W∗ to V∗; this homomorphism is an isomorphism if and only if W is finite-dimensional. If V = W then the space of linear maps is actually an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that (fg)∗ = g∗f∗. In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself. It is possible to identify (f∗)∗ with f using the natural injection into the double dual. If the linear map f is represented by the matrix A with respect to two bases of V and W, then f∗ is represented by the transpose matrix AT with respect to the dual bases of W∗ and V∗, hence the name. Alternatively, as f is represented by A acting on the left on column vectors, f∗ is represented by the same matrix acting on the right on row vectors. These points of view are related by the canonical inner product on Rn, which identifies the space of column vectors with the dual space of row vectors. === Quotient spaces and annihilators === Let S {\displaystyle S} be a subset of V {\displaystyle V} . The annihilator of S {\displaystyle S} in V ∗ {\displaystyle V^{*}} , denoted here S 0 {\displaystyle S^{0}} , is the collection of linear functionals f ∈ V ∗ {\displaystyle f\in V^{*}} such that [ f , s ] = 0 {\displaystyle [f,s]=0} for all s ∈ S {\displaystyle s\in S} . That is, S 0 {\displaystyle S^{0}} consists of all linear functionals f : V → F {\displaystyle f:V\to F} such that the restriction to S {\displaystyle S} vanishes: f | S = 0 {\displaystyle f|_{S}=0} . Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) the orthogonal complement. The annihilator of a subset is itself a vector space. The annihilator of the zero vector is the whole dual space: { 0 } 0 = V ∗ {\displaystyle \{0\}^{0}=V^{*}} , and the annihilator of the whole space is just the zero covector: V 0 = { 0 } ⊆ V ∗ {\displaystyle V^{0}=\{0\}\subseteq V^{*}} . Furthermore, the assignment of an annihilator to a subset of V {\displaystyle V} reverses inclusions, so that if { 0 } ⊆ S ⊆ T ⊆ V {\displaystyle \{0\}\subseteq S\subseteq T\subseteq V} , then { 0 } ⊆ T 0 ⊆ S 0 ⊆ V ∗ . {\displaystyle \{0\}\subseteq T^{0}\subseteq S^{0}\subseteq V^{*}.} If A {\displaystyle A} and B {\displaystyle B} are two subsets of V {\displaystyle V} then A 0 + B 0 ⊆ ( A ∩ B ) 0 . {\displaystyle A^{0}+B^{0}\subseteq (A\cap B)^{0}.} If ( A i ) i ∈ I {\displaystyle (A_{i})_{i\in I}} is any family of subsets of V {\displaystyle V} indexed by i {\displaystyle i} belonging to some index set I {\displaystyle I} , then ( ⋃ i ∈ I A i ) 0 = ⋂ i ∈ I A i 0 . {\displaystyle \left(\bigcup _{i\in I}A_{i}\right)^{0}=\bigcap _{i\in I}A_{i}^{0}.} In particular if A {\displaystyle A} and B {\displaystyle B} are subspaces of V {\displaystyle V} then ( A + B ) 0 = A 0 ∩ B 0 {\displaystyle (A+B)^{0}=A^{0}\cap B^{0}} and ( A ∩ B ) 0 = A 0 + B 0 . {\displaystyle (A\cap B)^{0}=A^{0}+B^{0}.} If V {\displaystyle V} is finite-dimensional and W {\displaystyle W} is a vector subspace, then W 00 = W {\displaystyle W^{00}=W} after identifying W {\displaystyle W} with its image in the second dual space under the double duality isomorphism V ≈ V ∗ ∗ {\displaystyle V\approx V^{**}} . In particular, forming the annihilator is a Galois connection on the lattice of subsets of a finite-dimensional vector space. If W {\displaystyle W} is a subspace of V {\displaystyle V} then the quotient space V / W {\displaystyle V/W} is a vector space in its own right, and so has a dual. By the first isomorphism theorem, a functional f : V → F {\displaystyle f:V\to F} factors through V / W {\displaystyle V/W} if and only if W {\displaystyle W} is in the kernel of f {\displaystyle f} . There is thus an isomorphism ( V / W ) ∗ ≅ W 0 . {\displaystyle (V/W)^{*}\cong W^{0}.} As a particular consequence, if V {\displaystyle V} is a direct sum of two subspaces A {\displaystyle A} and B {\displaystyle B} , then V ∗ {\displaystyle V^{*}} is a direct sum of A 0 {\displaystyle A^{0}} and B 0 {\displaystyle B^{0}} . === Dimensional analysis === The dual space is analogous to a "negative"-dimensional space. Most simply, since a vector v ∈ V {\displaystyle v\in V} can be paired with a covector φ ∈ V ∗ {\displaystyle \varphi \in V^{*}} by the natural pairing ⟨ x , φ ⟩ := φ ( x ) ∈ F {\displaystyle \langle x,\varphi \rangle :=\varphi (x)\in F} to obtain a scalar, a covector can "cancel" the dimension of a vector, similar to reducing a fraction. Thus while the direct sum V ⊕ V ∗ {\displaystyle V\oplus V^{*}} is a 2 n {\displaystyle 2n} -dimensional space (if V {\displaystyle V} is n {\displaystyle n} -dimensional), V ∗ {\displaystyle V^{*}} behaves as an ( − n ) {\displaystyle (-n)} -dimensional space, in the sense that its dimensions can be canceled against the dimensions of V {\displaystyle V} . This is formalized by tensor contraction. This arises in physics via dimensional analysis, where the dual space has inverse units. Under the natural pairing, these units cancel, and the resulting scalar value is dimensionless, as expected. For example, in (continuous) Fourier analysis, or more broadly time–frequency analysis: given a one-dimensional vector space with a unit of time t {\displaystyle t} , the dual space has units of frequency: occurrences per unit of time (units of 1 / t {\displaystyle 1/t} ). For example, if time is measured in seconds, the corresponding dual unit is the inverse second: over the course of 3 seconds, an event that occurs 2 times per second occurs a total of 6 times, corresponding to 3 s ⋅ 2 s − 1 = 6 {\displaystyle 3s\cdot 2s^{-1}=6} . Similarly, if the primal space measures length, the dual space measures inverse length. == Continuous dual space == When dealing with topological vector spaces, the continuous linear functionals from the space into the base field F = C {\displaystyle \mathbb {F} =\mathbb {C} } (or R {\displaystyle \mathbb {R} } ) are particularly important. This gives rise to the notion of the "continuous dual space" or "topological dual" which is a linear subspace of the algebraic dual space V ∗ {\displaystyle V^{*}} , denoted by V ′ {\displaystyle V'} . For any finite-dimensional normed vector space or topological vector space, such as Euclidean n-space, the continuous dual and the algebraic dual coincide. This is however false for any infinite-dimensional normed space, as shown by the example of discontinuous linear maps. Nevertheless, in the theory of topological vector spaces the terms "continuous dual space" and "topological dual space" are often replaced by "dual space". For a topological vector space V {\displaystyle V} its continuous dual space, or topological dual space, or just dual space (in the sense of the theory of topological vector spaces) V ′ {\displaystyle V'} is defined as the space of all continuous linear functionals φ : V → F {\displaystyle \varphi :V\to {\mathbb {F} }} . Important examples for continuous dual spaces are the space of compactly supported test functions D {\displaystyle {\mathcal {D}}} and its dual D ′ , {\displaystyle {\mathcal {D}}',} the space of arbitrary distributions (generalized functions); the space of arbitrary test functions E {\displaystyle {\mathcal {E}}} and its dual E ′ , {\displaystyle {\mathcal {E}}',} the space of compactly supported distributions; and the space of rapidly decreasing test functions S , {\displaystyle {\mathcal {S}},} the Schwartz space, and its dual S ′ , {\displaystyle {\mathcal {S}}',} the space of tempered distributions (slowly growing distributions) in the theory of generalized functions. === Properties === If X is a Hausdorff topological vector space (TVS), then the continuous dual space of X is identical to the continuous dual space of the completion of X. === Topologies on the dual === There is a standard construction for introducing a topology on the continuous dual V ′ {\displaystyle V'} of a topological vector space V {\displaystyle V} . Fix a collection A {\displaystyle {\mathcal {A}}} of bounded subsets of V {\displaystyle V} . This gives the topology on V {\displaystyle V} of uniform convergence on sets from A , {\displaystyle {\mathcal {A}},} or what is the same thing, the topology generated by seminorms of the form ‖ φ ‖ A = sup x ∈ A | φ ( x ) | , {\displaystyle \|\varphi \|_{A}=\sup _{x\in A}|\varphi (x)|,} where φ {\displaystyle \varphi } is a continuous linear functional on V {\displaystyle V} , and A {\displaystyle A} runs over the class A . {\displaystyle {\mathcal {A}}.} This means that a net of functionals φ i {\displaystyle \varphi _{i}} tends to a functional φ {\displaystyle \varphi } in V ′ {\displaystyle V'} if and only if for all A ∈ A ‖ φ i − φ ‖ A = sup x ∈ A | φ i ( x ) − φ ( x ) | ⟶ i → ∞ 0. {\displaystyle {\text{ for all }}A\in {\mathcal {A}}\qquad \|\varphi _{i}-\varphi \|_{A}=\sup _{x\in A}|\varphi _{i}(x)-\varphi (x)|{\underset {i\to \infty }{\longrightarrow }}0.} Usually (but not necessarily) the class A {\displaystyle {\mathcal {A}}} is supposed to satisfy the following conditions: Each point x {\displaystyle x} of V {\displaystyle V} belongs to some set A ∈ A {\displaystyle A\in {\mathcal {A}}} : for all x ∈ V there exists some A ∈ A such that x ∈ A . {\displaystyle {\text{ for all }}x\in V\quad {\text{ there exists some }}A\in {\mathcal {A}}\quad {\text{ such that }}x\in A.} Each two sets A ∈ A {\displaystyle A\in {\mathcal {A}}} and B ∈ A {\displaystyle B\in {\mathcal {A}}} are contained in some set C ∈ A {\displaystyle C\in {\mathcal {A}}} : for all A , B ∈ A there exists some C ∈ A such that A ∪ B ⊆ C . {\displaystyle {\text{ for all }}A,B\in {\mathcal {A}}\quad {\text{ there exists some }}C\in {\mathcal {A}}\quad {\text{ such that }}A\cup B\subseteq C.} A {\displaystyle {\mathcal {A}}} is closed under the operation of multiplication by scalars: for all A ∈ A and all λ ∈ F such that λ ⋅ A ∈ A . {\displaystyle {\text{ for all }}A\in {\mathcal {A}}\quad {\text{ and all }}\lambda \in {\mathbb {F} }\quad {\text{ such that }}\lambda \cdot A\in {\mathcal {A}}.} If these requirements are fulfilled then the corresponding topology on V ′ {\displaystyle V'} is Hausdorff and the sets U A = { φ ∈ V ′ : ‖ φ ‖ A < 1 } , for A ∈ A {\displaystyle U_{A}~=~\left\{\varphi \in V'~:~\quad \|\varphi \|_{A}<1\right\},\qquad {\text{ for }}A\in {\mathcal {A}}} form its local base. Here are the three most important special cases. The strong topology on V ′ {\displaystyle V'} is the topology of uniform convergence on bounded subsets in V {\displaystyle V} (so here A {\displaystyle {\mathcal {A}}} can be chosen as the class of all bounded subsets in V {\displaystyle V} ). If V {\displaystyle V} is a normed vector space (for example, a Banach space or a Hilbert space) then the strong topology on V ′ {\displaystyle V'} is normed (in fact a Banach space if the field of scalars is complete), with the norm ‖ φ ‖ = sup ‖ x ‖ ≤ 1 | φ ( x ) | . {\displaystyle \|\varphi \|=\sup _{\|x\|\leq 1}|\varphi (x)|.} The stereotype topology on V ′ {\displaystyle V'} is the topology of uniform convergence on totally bounded sets in V {\displaystyle V} (so here A {\displaystyle {\mathcal {A}}} can be chosen as the class of all totally bounded subsets in V {\displaystyle V} ). The weak topology on V ′ {\displaystyle V'} is the topology of uniform convergence on finite subsets in V {\displaystyle V} (so here A {\displaystyle {\mathcal {A}}} can be chosen as the class of all finite subsets in V {\displaystyle V} ). Each of these three choices of topology on V ′ {\displaystyle V'} leads to a variant of reflexivity property for topological vector spaces: If V ′ {\displaystyle V'} is endowed with the strong topology, then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just called reflexive. If V ′ {\displaystyle V'} is endowed with the stereotype dual topology, then the corresponding reflexivity is presented in the theory of stereotype spaces: the spaces reflexive in this sense are called stereotype. If V ′ {\displaystyle V'} is endowed with the weak topology, then the corresponding reflexivity is presented in the theory of dual pairs: the spaces reflexive in this sense are arbitrary (Hausdorff) locally convex spaces with the weak topology. === Examples === Let 1 < p < ∞ be a real number and consider the Banach space ℓ p of all sequences a = (an) for which ‖ a ‖ p = ( ∑ n = 0 ∞ | a n | p ) 1 p < ∞ . {\displaystyle \|\mathbf {a} \|_{p}=\left(\sum _{n=0}^{\infty }|a_{n}|^{p}\right)^{\frac {1}{p}}<\infty .} Define the number q by 1/p + 1/q = 1. Then the continuous dual of ℓ p is naturally identified with ℓ q: given an element φ ∈ ( ℓ p ) ′ {\displaystyle \varphi \in (\ell ^{p})'} , the corresponding element of ℓ q is the sequence ( φ ( e n ) ) {\displaystyle (\varphi (\mathbf {e} _{n}))} where e n {\displaystyle \mathbf {e} _{n}} denotes the sequence whose n-th term is 1 and all others are zero. Conversely, given an element a = (an) ∈ ℓ q, the corresponding continuous linear functional φ {\displaystyle \varphi } on ℓ p is defined by φ ( b ) = ∑ n a n b n {\displaystyle \varphi (\mathbf {b} )=\sum _{n}a_{n}b_{n}} for all b = (bn) ∈ ℓ p (see Hölder's inequality). In a similar manner, the continuous dual of ℓ 1 is naturally identified with ℓ ∞ (the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces c (consisting of all convergent sequences, with the supremum norm) and c0 (the sequences converging to zero) are both naturally identified with ℓ 1. By the Riesz representation theorem, the continuous dual of a Hilbert space is again a Hilbert space which is anti-isomorphic to the original space. This gives rise to the bra–ket notation used by physicists in the mathematical formulation of quantum mechanics. By the Riesz–Markov–Kakutani representation theorem, the continuous dual of certain spaces of continuous functions can be described using measures. === Transpose of a continuous linear map === If T : V → W is a continuous linear map between two topological vector spaces, then the (continuous) transpose T′ : W′ → V′ is defined by the same formula as before: T ′ ( φ ) = φ ∘ T , φ ∈ W ′ . {\displaystyle T'(\varphi )=\varphi \circ T,\quad \varphi \in W'.} The resulting functional T′(φ) is in V′. The assignment T → T′ produces a linear map between the space of continuous linear maps from V to W and the space of linear maps from W′ to V′. When T and U are composable continuous linear maps, then ( U ∘ T ) ′ = T ′ ∘ U ′ . {\displaystyle (U\circ T)'=T'\circ U'.} When V and W are normed spaces, the norm of the transpose in L(W′, V′) is equal to that of T in L(V, W). Several properties of transposition depend upon the Hahn–Banach theorem. For example, the bounded linear map T has dense range if and only if the transpose T′ is injective. When T is a compact linear map between two Banach spaces V and W, then the transpose T′ is compact. This can be proved using the Arzelà–Ascoli theorem. When V is a Hilbert space, there is an antilinear isomorphism iV from V onto its continuous dual V′. For every bounded linear map T on V, the transpose and the adjoint operators are linked by i V ∘ T ∗ = T ′ ∘ i V . {\displaystyle i_{V}\circ T^{*}=T'\circ i_{V}.} When T is a continuous linear map between two topological vector spaces V and W, then the transpose T′ is continuous when W′ and V′ are equipped with "compatible" topologies: for example, when for X = V and X = W, both duals X′ have the strong topology β(X′, X) of uniform convergence on bounded sets of X, or both have the weak-∗ topology σ(X′, X) of pointwise convergence on X. The transpose T′ is continuous from β(W′, W) to β(V′, V), or from σ(W′, W) to σ(V′, V). === Annihilators === Assume that W is a closed linear subspace of a normed space V, and consider the annihilator of W in V′, W ⊥ = { φ ∈ V ′ : W ⊆ ker φ } . {\displaystyle W^{\perp }=\{\varphi \in V':W\subseteq \ker \varphi \}.} Then, the dual of the quotient V / W can be identified with W⊥, and the dual of W can be identified with the quotient V′ / W⊥. Indeed, let P denote the canonical surjection from V onto the quotient V / W ; then, the transpose P′ is an isometric isomorphism from (V / W )′ into V′, with range equal to W⊥. If j denotes the injection map from W into V, then the kernel of the transpose j′ is the annihilator of W: ker ( j ′ ) = W ⊥ {\displaystyle \ker(j')=W^{\perp }} and it follows from the Hahn–Banach theorem that j′ induces an isometric isomorphism V′ / W⊥ → W′. === Further properties === If the dual of a normed space V is separable, then so is the space V itself. The converse is not true: for example, the space ℓ 1 is separable, but its dual ℓ ∞ is not. === Double dual === In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator Ψ : V → V′′ from a normed space V into its continuous double dual V′′, defined by Ψ ( x ) ( φ ) = φ ( x ) , x ∈ V , φ ∈ V ′ . {\displaystyle \Psi (x)(\varphi )=\varphi (x),\quad x\in V,\ \varphi \in V'.} As a consequence of the Hahn–Banach theorem, this map is in fact an isometry, meaning ‖ Ψ(x) ‖ = ‖ x ‖ for all x ∈ V. Normed spaces for which the map Ψ is a bijection are called reflexive. When V is a topological vector space then Ψ(x) can still be defined by the same formula, for every x ∈ V, however several difficulties arise. First, when V is not locally convex, the continuous dual may be equal to { 0 } and the map Ψ trivial. However, if V is Hausdorff and locally convex, the map Ψ is injective from V to the algebraic dual V′∗ of the continuous dual, again as a consequence of the Hahn–Banach theorem. Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual V′, so that the continuous double dual V′′ is not uniquely defined as a set. Saying that Ψ maps from V to V′′, or in other words, that Ψ(x) is continuous on V′ for every x ∈ V, is a reasonable minimal requirement on the topology of V′, namely that the evaluation mappings φ ∈ V ′ ↦ φ ( x ) , x ∈ V , {\displaystyle \varphi \in V'\mapsto \varphi (x),\quad x\in V,} be continuous for the chosen topology on V′. Further, there is still a choice of a topology on V′′, and continuity of Ψ depends upon this choice. As a consequence, defining reflexivity in this framework is more involved than in the normed case. == See also == Covariance and contravariance of vectors Dual module Dual norm Duality (mathematics) Duality (projective geometry) Pontryagin duality Reciprocal lattice – dual space basis, in crystallography == Notes == == References == == Bibliography == Axler, Sheldon Jay (2015). Linear Algebra Done Right (3rd ed.). Springer. ISBN 978-3-319-11079-0. Bourbaki, Nicolas (1989). Elements of mathematics, Algebra I. Springer-Verlag. ISBN 3-540-64243-9. Bourbaki, Nicolas (2003). Elements of mathematics, Topological vector spaces. Springer-Verlag. Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces (2nd ed.). Springer. ISBN 0-387-90093-4. Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. ISBN 978-0-8218-4419-9. Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001 Tu, Loring W. (2011). An Introduction to Manifolds (2nd ed.). Springer. ISBN 978-1-4419-7400-6. Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3rd ed.). AMS Chelsea Publishing. ISBN 0-8218-1646-2.. Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973). Gravitation. W. H. Freeman. ISBN 0-7167-0344-0. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Rudin, Walter (1973). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259. Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. Robertson, A.P.; Robertson, W. (1964). Topological vector spaces. Cambridge University Press. Schaefer, Helmut H. (1966). Topological vector spaces. New York: The Macmillan Company. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. == External links == Weisstein, Eric W. "Dual Vector Space". MathWorld.
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Wikipedia:Dukagjin Pupovci#0
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Dukagjin Pupovci (born 5 June 1964) is a Kosovo-Albanian professor, education expert and a critic of the education system in Kosovo. He has contributed to the development of important policy documents and has co-authored numerous studies and articles in the field of education and research in Kosovo. == Career == Pupovci holds a Doctor of Philosophy degree in mathematics and was a lecturer at numerous universities, including the Universiteti i Prishtinës in Pristina. He was the executive director of the Kosovo Education Center from 2000 to 2021. He was Deputy Minister of Education in the Government of Kosovo from April 2021 to December 2022. == References ==
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Wikipedia:Dustin Clausen#0
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Dustin Clausen is an American-Canadian mathematician known for his contributions to algebraic K-theory and the development of condensed mathematics, in collaboration with Peter Scholze. His research interests include the intersections of number theory and homotopy theory. == Early life and education == Dustin Clausen completed his undergraduate studies at Harvard University. While at Harvard he spent a semester studying at the Math in Moscow program. He received his PhD in 2013 from the Massachusetts Institute of Technology (MIT), where he was supervised by Jacob Lurie. His doctoral thesis was titled "Arithmetic Duality in Algebraic K-Theory." == Academic career == After earning his PhD, Clausen spent five years as a postdoctoral researcher at the University of Copenhagen. He then moved to Bonn, Germany, where he first served as a postdoctoral researcher at the University of Bonn and subsequently as the head of a research group at the Max Planck Institute for Mathematics. In 2020, Clausen returned to the University of Copenhagen as an associate professor. Since 2023, he has held a position as a permanent professor at the Institut des Hautes Études Scientifiques (IHÉS). == Research and contributions == Clausen's research has focused on algebraic K-theory and its connections to number theory and homotopy theory. Along with Peter Scholze, he has developed the concept of condensed mathematics, which aims to provide a framework for topological algebraic structures. == Awards and honors == NSF Graduate Research Fellowship (2008) David Mumford Prize (2008) Hoopes Prize (2008) Hartmann Foundation's Diploma Prize (2022) == Personal life == Dustin Clausen is the grandson of mathematician John T. Tate (1925–2019), and the great grandson of Emil Artin. == References == == External links == Dustin Clausen in Mathematics Genealogy Project
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Wikipedia:Dwight Barkley#0
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Dwight Barkley (born 7 January 1959) is a professor of mathematics at the University of Warwick. == Education and career == Barkley obtained his PhD in physics from the University of Texas at Austin in 1988. He then spent one year at Caltech working with Philip Saffman followed by three years at Princeton University where he worked with Yannís Keverkidis and Steven Orszag. In 1992 he was awarded both NSF and NATO postdoctoral fellowships. In 1994 he joined the faculty at the University of Warwick. == Research == Barkley studies waves in excitable media such as the Belousov–Zhabotinsky reaction, heart tissue, and neurons. He is the author of the Barkley Model of excitable media and discoverer of the role of Euclidean symmetry in spiral-wave dynamics. In 1997, Laurette Tuckerman and Dwight Barkley coined the term "bifurcation analysis for time steppers" for techniques involving the modification of time-stepping computer codes to perform the tasks of bifurcation analysis. He has applied this approach in several areas of fluid dynamics, in particular to stability analysis of the cylinder wake and of the backward-facing step. Barkley also works on the transition to turbulence in shear flows, including the formation of turbulent-laminar bands and the critical point for pipe flow. Exploiting an analogy with the transition between excitable and bistable media, Barkley derived a model for pipe flow which captures most features of transition to turbulence, in particular the behavior of turbulent regions called puffs and slugs. He is also known for deriving an equation to estimate how long it will be until a child in a car asks the question "are we there yet?" == Awards == In 2005 he was awarded the J. D. Crawford Prize for outstanding research in nonlinear science, "for his development of high quality, robust and efficient numerical algorithms for pattern formation phenomena in spatially extended dynamical systems". In 2008 he was elected Fellow of the American Physical Society "for combining computation and dynamical systems analyses to obtain remarkable insights into hydrodynamic instabilities and patterns in diverse systems, including flow past a cylinder, channel flow, laminar-turbulent bands, and thermal convection." That same year he was also elected fellow of the Institute of Mathematics and Its Applications. In 2009-2010 he was a Royal Society–Leverhulme Trust Senior Research Fellow. In 2016 he was elected Fellow of the Society for Industrial and Applied Mathematics "for innovative combinations of analysis and computation to obtain fundamental insights into complex dynamics of spatially extended systems." In 2024, he was named a Fluids Mechanics Fellow of Euromech "for his profound contributions to transition to turbulence, nonlinear dynamics, pattern formation, hydrodynamic instabilities, and the Euler singularity through combination of large-scale computing with insightful dynamical systems analysis and modelling". == Selected publications == Barkley, Dwight; Kness, Mark; Tuckerman, Laurette S. (1990), "Spiral-wave dynamics in a simple model of excitable media: the transition from simple to compound rotation", Physical Review A, Third Series, 42 (4): 2489–2491, Bibcode:1990PhRvA..42.2489B, doi:10.1103/PhysRevA.42.2489, MR 1068482, PMID 9904313. Barkley, Dwight (1991), "A model for fast computer simulation of waves in excitable media", Physica D: Nonlinear Phenomena, 49 (1–2): 61–70, Bibcode:1991PhyD...49...61B, doi:10.1016/0167-2789(91)90194-E. Barkley, Dwight (January 1994), "Euclidean symmetry and the dynamics of rotating spiral waves", Physical Review Letters, 72 (1): 164–167, Bibcode:1994PhRvL..72..164B, doi:10.1103/physrevlett.72.164, PMID 10055592. Barkley, Dwight; Henderson, Ronald D. (September 1996), "Three-dimensional Floquet stability analysis of the wake of a circular cylinder", Journal of Fluid Mechanics, 322: 215–241, Bibcode:1996JFM...322..215B, doi:10.1017/s0022112096002777, S2CID 53610776. Avila, K.; Moxey, D.; de Lozar, A.; Avila, M.; Barkley, D.; Hof, B. (July 2011), "The onset of turbulence in pipe flow", Science, 333 (6039): 192–196, Bibcode:2011Sci...333..192A, doi:10.1126/science.1203223, PMID 21737736, S2CID 22560587. Barkley, D. (2016), "Theoretical perspective on the route to turbulence in a pipe" (PDF), Journal of Fluid Mechanics, 803: P1, Bibcode:2016JFM...803P...1B, doi:10.1017/jfm.2016.465 == References == == External links == Google scholar profile Archived 20 April 2016 at the Wayback Machine Dwight Barkley at the Mathematics Genealogy Project
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Wikipedia:Dwight Duffus#0
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Dwight Albert Duffus is a Canadian-American mathematician, the Goodrich C. White Professor of Mathematics & Computer Science at Emory University and editor-in-chief of the journal Order. Duffus did his undergraduate studies at the University of Regina, graduating in 1974; he received his Ph.D. in 1978 from the University of Calgary under the supervision of Ivan Rival. In 1986 Duffus received Emory University's Emory Williams Teaching Award, its highest award for teaching excellence. He served as chair of the Mathematics & Computer Science Department at Emory for many years, beginning in 1991. == References ==
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Wikipedia:Dyadic derivative#0
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In mathematical analysis, the dyadic derivative is a concept that extends the notion of classical differentiation to functions defined on the dyadic group or the dyadic field. Unlike classical differentiation, which is based on the limit of difference quotients, dyadic differentiation is defined using dyadic (binary) addition and reflects the discontinuous nature of Walsh functions. == Definition == === Pointwise dyadic derivative === For a function f {\displaystyle f} defined on [0,1), the first pointwise dyadic derivative of f {\displaystyle f} at a point x {\displaystyle x} is defined as: f [ 1 ] ( x ) = lim m → ∞ ∑ j = 0 m − 1 2 j − 1 [ f ( x ) − f ( x ⊕ 2 − j − 1 ) ] {\displaystyle f^{[1]}(x)=\lim _{m\to \infty }\sum _{j=0}^{m-1}2^{j-1}[f(x)-f(x\oplus 2^{-j-1})]} if this limit exists. Here, ⊕ {\displaystyle \oplus } denotes the dyadic addition operation, which is defined using the dyadic (binary) representation of numbers. That is, if x = ∑ j = 0 ∞ x j 2 − j − 1 {\displaystyle x=\sum _{j=0}^{\infty }x_{j}2^{-j-1}} and y = ∑ j = 0 ∞ y j 2 − j − 1 {\displaystyle y=\sum _{j=0}^{\infty }y_{j}2^{-j-1}} with x j , y j ∈ { 0 , 1 } {\displaystyle x_{j},y_{j}\in \{0,1\}} , then x ⊕ y = ∑ j = 0 ∞ ( x j ⊕ y j ) 2 − j − 1 {\displaystyle x\oplus y=\sum _{j=0}^{\infty }(x_{j}\oplus y_{j})2^{-j-1}} , where x j ⊕ y j = ( x j + y j ) ( mod 2 ) {\displaystyle x_{j}\oplus y_{j}=(x_{j}+y_{j}){\pmod {2}}} . Higher-order dyadic derivatives are defined recursively: f [ r ] ( x ) = ( f [ r − 1 ] ) [ 1 ] ( x ) {\displaystyle f^{[r]}(x)=(f^{[r-1]})^{[1]}(x)} for r ∈ N {\displaystyle r\in \mathbb {N} } . === Strong dyadic derivative === The strong dyadic derivative is defined in the context of function spaces. Let X ( 0 , 1 ) {\displaystyle X(0,1)} denote one of the function spaces L p ( 0 , 1 ) {\displaystyle L^{p}(0,1)} for 1 ≤ p ≤ ∞ {\displaystyle 1\leq p\leq \infty } (Lp space); L ∞ ( 0 , 1 ) {\displaystyle L^{\infty }(0,1)} (L∞ space); or C ⊕ [ 0 , 1 ] {\displaystyle C^{\oplus }[0,1]} (the space of dyadically continuous functions). If f ∈ X ( 0 , 1 ) {\displaystyle f\in X(0,1)} and there exists g ∈ X ( 0 , 1 ) {\displaystyle g\in X(0,1)} such that lim m → ∞ ‖ ∑ j = 0 m − 1 2 j − 1 [ f ( ⋅ ) − f ( ⋅ ⊕ 2 − j − 1 ) ] − g ( ⋅ ) ‖ X = 0 {\displaystyle \lim _{m\to \infty }\left\|\sum _{j=0}^{m-1}2^{j-1}[f(\cdot )-f(\cdot \oplus 2^{-j-1})]-g(\cdot )\right\|_{X}=0} , then g {\displaystyle g} is called the first strong dyadic derivative of f {\displaystyle f} , denoted by g = D [ 1 ] f {\displaystyle g=D^{[1]}f} . Higher-order derivatives can be defined recursively similar to pointwise dyadic derivatives. == Properties == Similar to the classic derivative in calculus, the dyadic derivative possesses several properties. === Linearity === The dyadic derivative is a linear operator. If functions f {\displaystyle f} and g {\displaystyle g} are dyadically differentiable and α , β {\displaystyle \alpha ,\beta } are constants, then α f + β g {\displaystyle \alpha f+\beta g} is dyadically differentiable: ( α f + β g ) [ 1 ] = α f [ 1 ] + β g [ 1 ] {\displaystyle (\alpha f+\beta g)^{[1]}=\alpha f^{[1]}+\beta g^{[1]}} . === Closure === The dyadic differentiation operator is closed; that is, if f {\displaystyle f} is in the domain of the operator, then its dyadic derivative also belongs to the same function space. === Inverse operator === There exists a dyadic integration operator that serves as an inverse to the dyadic differentiation operator, analogous to the fundamental theorem of calculus. === Relationship to the Walsh-Fourier transform === For functions f {\displaystyle f} where D [ 1 ] f ∈ L 1 ( G ) {\displaystyle D^{[1]}f\in L_{1}(G)} exists, the Walsh-Fourier transform satisfies: [ D [ 1 ] f ] ∧ ( χ ) = | χ | f ^ ( χ ) {\displaystyle [D^{[1]}f]^{\wedge }(\chi )=|\chi |{\hat {f}}(\chi )} for all characters χ {\displaystyle \chi } , where | χ | {\displaystyle |\chi |} represents the norm of the character. === Eigenfunctions === The Walsh functions ψ k {\displaystyle \psi _{k}} are eigenfunctions of the dyadic differentiation operator with corresponding eigenvalues related to their index: D [ 1 ] ψ k = ψ k [ 1 ] = k ψ k {\displaystyle D^{[1]}\psi _{k}=\psi _{k}^{[1]}=k\psi _{k}} and D [ r ] ψ k = ψ k [ r ] = k r ψ k {\displaystyle D^{[r]}\psi _{k}=\psi _{k}^{[r]}=k^{r}\psi _{k}} . This eigenfunction property makes Walsh functions naturally suited for analysis involving dyadic derivatives, similar to how complex exponentials e i k x {\displaystyle e^{ikx}} are eigenfunctions of classical differentiation. === Characterization of differentiable functions === Thanks to a generalization of a result of Butzer and Wagner, Theorem (Skvorcov—Wade). Let f {\displaystyle f} be continuous on [ 0 , 1 ) {\displaystyle [0,1)} , and let f [ 1 ] {\displaystyle f^{[1]}} exist for all but countably many points x ∈ ( 0 , 1 ) {\displaystyle x\in (0,1)} . Then f {\displaystyle f} is constant. This result implies that it is more interesting to consider functions that are not continuous over the entire interval. A generalization of the above result shows that: Theorem. A bounded function defined on [ 0 , 1 ) {\displaystyle [0,1)} with a countable set of discontinuities (exclusively of jump discontinuities) that have at most a finite number of cluster points is pointwise dyadically differentiable except on a countable set if and only if it is a piecewise constant function. == Examples == == History == The dyadic derivative was introduced by mathematician James Edmund Gibbs in the context of Walsh functions and further developed by Paul Butzer and Heinz-Joseph Wagner. Further contributions came from C. W. Onneweer, who extended the concept to fractional differentiation and p-adic fields. In 1979, Onneweer provided alternative definitions to the dyadic derivatives. == See also == Walsh function Haar wavelet Harmonic analysis Walsh transform == References ==
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Wikipedia:Dyson conjecture#0
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In mathematics, the Dyson conjecture (Freeman Dyson 1962) is a conjecture about the constant term of certain Laurent polynomials, proved independently in 1962 by Wilson and Gunson. Andrews generalized it to the q-Dyson conjecture, proved by Zeilberger and Bressoud and sometimes called the Zeilberger–Bressoud theorem. Macdonald generalized it further to more general root systems with the Macdonald constant term conjecture, proved by Cherednik. == Dyson conjecture == The Dyson conjecture states that the Laurent polynomial ∏ 1 ≤ i ≠ j ≤ n ( 1 − t i / t j ) a i {\displaystyle \prod _{1\leq i\neq j\leq n}(1-t_{i}/t_{j})^{a_{i}}} has constant term ( a 1 + a 2 + ⋯ + a n ) ! a 1 ! a 2 ! ⋯ a n ! . {\displaystyle {\frac {(a_{1}+a_{2}+\cdots +a_{n})!}{a_{1}!a_{2}!\cdots a_{n}!}}.} The conjecture was first proved independently by Wilson (1962) and Gunson (1962). Good (1970) later found a short proof, by observing that the Laurent polynomials, and therefore their constant terms, satisfy the recursion relations F ( a 1 , … , a n ) = ∑ i = 1 n F ( a 1 , … , a i − 1 , … , a n ) . {\displaystyle F(a_{1},\dots ,a_{n})=\sum _{i=1}^{n}F(a_{1},\dots ,a_{i}-1,\dots ,a_{n}).} The case n = 3 of Dyson's conjecture follows from the Dixon identity. Sills & Zeilberger (2006) and (Sills 2006) used a computer to find expressions for non-constant coefficients of Dyson's Laurent polynomial. == Dyson integral == When all the values ai are equal to β/2, the constant term in Dyson's conjecture is the value of Dyson's integral 1 ( 2 π ) n ∫ 0 2 π ⋯ ∫ 0 2 π ∏ 1 ≤ j < k ≤ n | e i θ j − e i θ k | β d θ 1 ⋯ d θ n . {\displaystyle {\frac {1}{(2\pi )^{n}}}\int _{0}^{2\pi }\cdots \int _{0}^{2\pi }\prod _{1\leq j<k\leq n}|e^{i\theta _{j}}-e^{i\theta _{k}}|^{\beta }\,d\theta _{1}\cdots d\theta _{n}.} Dyson's integral is a special case of Selberg's integral after a change of variable and has value Γ ( 1 + β n / 2 ) Γ ( 1 + β / 2 ) n {\displaystyle {\frac {\Gamma (1+\beta n/2)}{\Gamma (1+\beta /2)^{n}}}} which gives another proof of Dyson's conjecture in this special case. == q-Dyson conjecture == Andrews (1975) found a q-analog of Dyson's conjecture, stating that the constant term of ∏ 1 ≤ i < j ≤ n ( x i x j ; q ) a i ( q x j x i ; q ) a j {\displaystyle \prod _{1\leq i<j\leq n}\left({\frac {x_{i}}{x_{j}}};q\right)_{a_{i}}\left({\frac {qx_{j}}{x_{i}}};q\right)_{a_{j}}} is ( q ; q ) a 1 + ⋯ + a n ( q ; q ) a 1 ⋯ ( q ; q ) a n . {\displaystyle {\frac {(q;q)_{a_{1}+\cdots +a_{n}}}{(q;q)_{a_{1}}\cdots (q;q)_{a_{n}}}}.} Here (a;q)n is the q-Pochhammer symbol. This conjecture reduces to Dyson's conjecture for q = 1, and was proved by Zeilberger & Bressoud (1985), using a combinatorial approach inspired by previous work of Ira Gessel and Dominique Foata. A shorter proof, using formal Laurent series, was given in 2004 by Ira Gessel and Guoce Xin, and an even shorter proof, using a quantitative form, due to Karasev and Petrov, and independently to Lason, of Noga Alon's Combinatorial Nullstellensatz, was given in 2012 by Gyula Karolyi and Zoltan Lorant Nagy. The latter method was extended, in 2013, by Shalosh B. Ekhad and Doron Zeilberger to derive explicit expressions of any specific coefficient, not just the constant term; see http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qdyson.html, for detailed references. == Macdonald conjectures == Macdonald (1982) extended the conjecture to arbitrary finite or affine root systems, with Dyson's original conjecture corresponding to the case of the An−1 root system and Andrews's conjecture corresponding to the affine An−1 root system. Macdonald reformulated these conjectures as conjectures about the norms of Macdonald polynomials. Macdonald's conjectures were proved by (Cherednik 1995) using doubly affine Hecke algebras. Macdonald's form of Dyson's conjecture for root systems of type BC is closely related to Selberg's integral. == References == Andrews, George E. (1975), "Problems and prospects for basic hypergeometric functions", Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), Boston, MA: Academic Press, pp. 191–224, MR 0399528 Cherednik, I. (1995), "Double Affine Hecke Algebras and Macdonald's Conjectures", The Annals of Mathematics, 141 (1): 191–216, doi:10.2307/2118632, JSTOR 2118632 Dyson, Freeman J. (1962), "Statistical theory of the energy levels of complex systems. I", Journal of Mathematical Physics, 3 (1): 140–156, Bibcode:1962JMP.....3..140D, doi:10.1063/1.1703773, ISSN 0022-2488, MR 0143556 Good, I. J. (1970), "Short proof of a conjecture by Dyson", Journal of Mathematical Physics, 11 (6): 1884, Bibcode:1970JMP....11.1884G, doi:10.1063/1.1665339, ISSN 0022-2488, MR 0258644 Gunson, J. (1962), "Proof of a conjecture by Dyson in the statistical theory of energy levels", Journal of Mathematical Physics, 3 (4): 752–753, Bibcode:1962JMP.....3..752G, doi:10.1063/1.1724277, ISSN 0022-2488, MR 0148401 Macdonald, I. G. (1982), "Some conjectures for root systems", SIAM Journal on Mathematical Analysis, 13 (6): 988–1007, doi:10.1137/0513070, ISSN 0036-1410, MR 0674768 Sills, Andrew V. (2006), "Disturbing the Dyson conjecture, in a generally GOOD way", Journal of Combinatorial Theory, Series A, 113 (7): 1368–1380, arXiv:1812.05557, doi:10.1016/j.jcta.2005.12.005, ISSN 1096-0899, MR 2259066, S2CID 1565705 Sills, Andrew V.; Zeilberger, Doron (2006), "Disturbing the Dyson conjecture (in a GOOD way)", Experimental Mathematics, 15 (2): 187–191, arXiv:1812.04490, doi:10.1080/10586458.2006.10128959, ISSN 1058-6458, MR 2253005, S2CID 14594152 Wilson, Kenneth G. (1962), "Proof of a conjecture by Dyson", Journal of Mathematical Physics, 3 (5): 1040–1043, Bibcode:1962JMP.....3.1040W, doi:10.1063/1.1724291, ISSN 0022-2488, MR 0144627 Zeilberger, Doron; Bressoud, David M. (1985), "A proof of Andrews' q-Dyson conjecture", Discrete Mathematics, 54 (2): 201–224, doi:10.1016/0012-365X(85)90081-0, ISSN 0012-365X, MR 0791661
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Wikipedia:Déborah Oliveros#0
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Déborah Oliveros Braniff is a Mexican mathematician whose research interests include discrete geometry, combinatorics, and convex geometry, including the geometry of bodies of constant width and related topics. == Education and career == After earning an undergraduate degree in mathematics from the National Autonomous University of Mexico (UNAM) in 1992, and earning a master's degree in 1994 under the mentorship of Mónica Clapp, Oliveros continued at UNAM for graduate study in mathematics, with doctoral research on an unsolved question of Stanislaw Ulam concerning the buoyancy of floating convex bodies. Her 1997 dissertation on the topic, Los volantines : sistemas dinamicos asociados al problema de la flotacion de los cuerpos, was jointly supervised by Luis Montejano and Javier Bracho. She became a professor at UNAM in 1996, but left in 1999 for postdoctoral research at the University of Calgary in Canada. She became a professor there from 2001 to 2005, when she returned to a professorship at UNAM. She became one of the founders of the branch of the UNAM Institute of Mathematics at the UNAM Juriquilla campus, and directed the institute for 2015–2016. She also holds an affiliation with the Faculty of Engineering of the Autonomous University of Queretaro. == Book == Oliveros is a coauthor with Horst Martini and Luis Montejano of the book Bodies of Constant Width: An Introduction to Convex Geometry with Applications (Birkhäuser, 2019). == Recognition == UNAM gave Oliveros the "Reconocimiento Sor Juana Inés de la Cruz" award in 2014. She is a member of the Mexican Academy of Sciences. == References == == External links == Home page Déborah Oliveros publications indexed by Google Scholar
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Wikipedia:E-dense semigroup#0
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In abstract algebra, an E-dense semigroup (also called an E-inversive semigroup) is a semigroup in which every element a has at least one weak inverse x, meaning that xax = x. The notion of weak inverse is (as the name suggests) weaker than the notion of inverse used in a regular semigroup (which requires that axa=a). The above definition of an E-inversive semigroup S is equivalent with any of the following: for every element a ∈ S there exists another element b ∈ S such that ab is an idempotent. for every element a ∈ S there exists another element c ∈ S such that ca is an idempotent. This explains the name of the notion as the set of idempotents of a semigroup S is typically denoted by E(S). The concept of E-inversive semigroup was introduced by Gabriel Thierrin in 1955. Some authors use E-dense to refer only to E-inversive semigroups in which the idempotents commute. More generally, a subsemigroup T of S is said dense in S if, for all x ∈ S, there exists y ∈ S such that both xy ∈ T and yx ∈ T. A semigroup with zero is said to be an E*-dense semigroup if every element other than the zero has at least one non-zero weak inverse. Semigroups in this class have also been called 0-inversive semigroups. == Examples == Any regular semigroup is E-dense (but not vice versa). Any eventually regular semigroup is E-dense. Any periodic semigroup (and in particular, any finite semigroup) is E-dense. == See also == Dense set E-semigroup == References == == Further reading == Mitsch, H. "Introduction to E-inversive semigroups." Semigroups (Braga, 1999), 114–135. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. ISBN 9810243928
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Wikipedia:E-semigroup#0
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In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): x ⋅ y, or simply xy, denotes the result of applying the semigroup operation to the ordered pair (x, y). Associativity is formally expressed as that (x ⋅ y) ⋅ z = x ⋅ (y ⋅ z) for all x, y and z in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. As in the case of groups or magmas, the semigroup operation need not be commutative, so x ⋅ y is not necessarily equal to y ⋅ x; a well-known example of an operation that is associative but non-commutative is matrix multiplication. If the semigroup operation is commutative, then the semigroup is called a commutative semigroup or (less often than in the analogous case of groups) it may be called an abelian semigroup. A monoid is an algebraic structure intermediate between semigroups and groups, and is a semigroup having an identity element, thus obeying all but one of the axioms of a group: existence of inverses is not required of a monoid. A natural example is strings with concatenation as the binary operation, and the empty string as the identity element. Restricting to non-empty strings gives an example of a semigroup that is not a monoid. Positive integers with addition form a commutative semigroup that is not a monoid, whereas the non-negative integers do form a monoid. A semigroup without an identity element can be easily turned into a monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory. Semigroups should not be confused with quasigroups, which are generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroups preserve from groups the notion of division. Division in semigroups (or in monoids) is not possible in general. The formal study of semigroups began in the early 20th century. Early results include a Cayley theorem for semigroups realizing any semigroup as a transformation semigroup, in which arbitrary functions replace the role of bijections in group theory. A deep result in the classification of finite semigroups is Krohn–Rhodes theory, analogous to the Jordan–Hölder decomposition for finite groups. Some other techniques for studying semigroups, like Green's relations, do not resemble anything in group theory. The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata via the syntactic monoid. In probability theory, semigroups are associated with Markov processes. In other areas of applied mathematics, semigroups are fundamental models for linear time-invariant systems. In partial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time. There are numerous special classes of semigroups, semigroups with additional properties, which appear in particular applications. Some of these classes are even closer to groups by exhibiting some additional but not all properties of a group. Of these we mention: regular semigroups, orthodox semigroups, semigroups with involution, inverse semigroups and cancellative semigroups. There are also interesting classes of semigroups that do not contain any groups except the trivial group; examples of the latter kind are bands and their commutative subclass – semilattices, which are also ordered algebraic structures. == Definition == A semigroup is a set S together with a binary operation ⋅ (that is, a function ⋅ : S × S → S) that satisfies the associative property: For all a, b, c ∈ S, the equation (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) holds. More succinctly, a semigroup is an associative magma. == Examples of semigroups == Empty semigroup: the empty set forms a semigroup with the empty function as the binary operation. Semigroup with one element: there is essentially only one (specifically, only one up to isomorphism), the singleton {a} with operation a · a = a. Semigroup with two elements: there are five that are essentially different. A null semigroup on any nonempty set with a chosen zero, or a left/right zero semigroup on any set. The "flip-flop" monoid: a semigroup with three elements representing the three operations on a switch – set, reset, and do nothing. The set of positive integers with addition. (With 0 included, this becomes a monoid.) The set of integers with minimum or maximum. (With positive/negative infinity included, this becomes a monoid.) Square nonnegative matrices of a given size with matrix multiplication. Any ideal of a ring with the multiplication of the ring. The set of all finite strings over a fixed alphabet Σ with concatenation of strings as the semigroup operation – the so-called "free semigroup over Σ". With the empty string included, this semigroup becomes the free monoid over Σ. A probability distribution F together with all convolution powers of F, with convolution as the operation. This is called a convolution semigroup. Transformation semigroups and monoids. The set of continuous functions from a topological space to itself with composition of functions forms a monoid with the identity function acting as the identity. More generally, the endomorphisms of any object of a category form a monoid under composition. The product of faces of an arrangement of hyperplanes. == Basic concepts == === Identity and zero === A left identity of a semigroup S (or more generally, magma) is an element e such that for all x in S, e ⋅ x = x. Similarly, a right identity is an element f such that for all x in S, x ⋅ f = x. Left and right identities are both called one-sided identities. A semigroup may have one or more left identities but no right identity, and vice versa. A two-sided identity (or just identity) is an element that is both a left and right identity. Semigroups with a two-sided identity are called monoids. A semigroup may have at most one two-sided identity. If a semigroup has a two-sided identity, then the two-sided identity is the only one-sided identity in the semigroup. If a semigroup has both a left identity and a right identity, then it has a two-sided identity (which is therefore the unique one-sided identity). A semigroup S without identity may be embedded in a monoid formed by adjoining an element e ∉ S to S and defining e ⋅ s = s ⋅ e = s for all s ∈ S ∪ {e}. The notation S1 denotes a monoid obtained from S by adjoining an identity if necessary (S1 = S for a monoid). Similarly, every magma has at most one absorbing element, which in semigroup theory is called a zero. Analogous to the above construction, for every semigroup S, one can define S0, a semigroup with 0 that embeds S. === Subsemigroups and ideals === The semigroup operation induces an operation on the collection of its subsets: given subsets A and B of a semigroup S, their product A · B, written commonly as AB, is the set { ab | a ∈ A and b ∈ B }. (This notion is defined identically as it is for groups.) In terms of this operation, a subset A is called a subsemigroup if AA is a subset of A, a right ideal if AS is a subset of A, and a left ideal if SA is a subset of A. If A is both a left ideal and a right ideal then it is called an ideal (or a two-sided ideal). If S is a semigroup, then the intersection of any collection of subsemigroups of S is also a subsemigroup of S. So the subsemigroups of S form a complete lattice. An example of a semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of a commutative semigroup, when it exists, is a group. Green's relations, a set of five equivalence relations that characterise the elements in terms of the principal ideals they generate, are important tools for analysing the ideals of a semigroup and related notions of structure. The subset with the property that every element commutes with any other element of the semigroup is called the center of the semigroup. The center of a semigroup is actually a subsemigroup. === Homomorphisms and congruences === A semigroup homomorphism is a function that preserves semigroup structure. A function f : S → T between two semigroups is a homomorphism if the equation f(ab) = f(a)f(b). holds for all elements a, b in S, i.e. the result is the same when performing the semigroup operation after or before applying the map f. A semigroup homomorphism between monoids preserves identity if it is a monoid homomorphism. But there are semigroup homomorphisms that are not monoid homomorphisms, e.g. the canonical embedding of a semigroup S without identity into S1. Conditions characterizing monoid homomorphisms are discussed further. Let f : S0 → S1 be a semigroup homomorphism. The image of f is also a semigroup. If S0 is a monoid with an identity element e0, then f(e0) is the identity element in the image of f. If S1 is also a monoid with an identity element e1 and e1 belongs to the image of f, then f(e0) = e1, i.e. f is a monoid homomorphism. Particularly, if f is surjective, then it is a monoid homomorphism. Two semigroups S and T are said to be isomorphic if there exists a bijective semigroup homomorphism f : S → T. Isomorphic semigroups have the same structure. A semigroup congruence ~ is an equivalence relation that is compatible with the semigroup operation. That is, a subset ~ ⊆ S × S that is an equivalence relation and x ~ y and u ~ v implies xu ~ yv for every x, y, u, v in S. Like any equivalence relation, a semigroup congruence ~ induces congruence classes [a]~ = {x ∈ S | x ~ a} and the semigroup operation induces a binary operation ∘ on the congruence classes: [u]~ ∘ [v]~ = [uv]~ Because ~ is a congruence, the set of all congruence classes of ~ forms a semigroup with ∘, called the quotient semigroup or factor semigroup, and denoted S / ~. The mapping x ↦ [x]~ is a semigroup homomorphism, called the quotient map, canonical surjection or projection; if S is a monoid then quotient semigroup is a monoid with identity [1]~. Conversely, the kernel of any semigroup homomorphism is a semigroup congruence. These results are nothing more than a particularization of the first isomorphism theorem in universal algebra. Congruence classes and factor monoids are the objects of study in string rewriting systems. A nuclear congruence on S is one that is the kernel of an endomorphism of S. A semigroup S satisfies the maximal condition on congruences if any family of congruences on S, ordered by inclusion, has a maximal element. By Zorn's lemma, this is equivalent to saying that the ascending chain condition holds: there is no infinite strictly ascending chain of congruences on S. Every ideal I of a semigroup induces a factor semigroup, the Rees factor semigroup, via the congruence ρ defined by x ρ y if either x = y, or both x and y are in I. === Quotients and divisions === The following notions introduce the idea that a semigroup is contained in another one. A semigroup T is a quotient of a semigroup S if there is a surjective semigroup morphism from S to T. For example, (Z/2Z, +) is a quotient of (Z/4Z, +), using the morphism consisting of taking the remainder modulo 2 of an integer. A semigroup T divides a semigroup S, denoted T ≼ S if T is a quotient of a subsemigroup S. In particular, subsemigroups of S divides T, while it is not necessarily the case that there are a quotient of S. Both of those relations are transitive. == Structure of semigroups == For any subset A of S there is a smallest subsemigroup T of S that contains A, and we say that A generates T. A single element x of S generates the subsemigroup { xn | n ∈ Z+ }. If this is finite, then x is said to be of finite order, otherwise it is of infinite order. A semigroup is said to be periodic if all of its elements are of finite order. A semigroup generated by a single element is said to be monogenic (or cyclic). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive integers with the operation of addition. If it is finite and nonempty, then it must contain at least one idempotent. It follows that every nonempty periodic semigroup has at least one idempotent. A subsemigroup that is also a group is called a subgroup. There is a close relationship between the subgroups of a semigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of the subgroup. For each idempotent e of the semigroup there is a unique maximal subgroup containing e. Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups. Here the term maximal subgroup differs from its standard use in group theory. More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and has a minimal ideal and at least one idempotent. The number of finite semigroups of a given size (greater than 1) is (obviously) larger than the number of groups of the same size. For example, of the sixteen possible "multiplication tables" for a set of two elements {a, b}, eight form semigroups whereas only four of these are monoids and only two form groups. For more on the structure of finite semigroups, see Krohn–Rhodes theory. == Special classes of semigroups == A monoid is a semigroup with an identity element. A group is a monoid in which every element has an inverse element. A subsemigroup is a subset of a semigroup that is closed under the semigroup operation. A cancellative semigroup is one having the cancellation property: a · b = a · c implies b = c and similarly for b · a = c · a. Every group is a cancellative semigroup, and every finite cancellative semigroup is a group. A band is a semigroup whose operation is idempotent. A semilattice is a semigroup whose operation is idempotent and commutative. 0-simple semigroups. Transformation semigroups: any finite semigroup S can be represented by transformations of a (state-) set Q of at most |S| + 1 states. Each element x of S then maps Q into itself x : Q → Q and sequence xy is defined by q(xy) = (qx)y for each q in Q. Sequencing clearly is an associative operation, here equivalent to function composition. This representation is basic for any automaton or finite-state machine (FSM). The bicyclic semigroup is in fact a monoid, which can be described as the free semigroup on two generators p and q, under the relation pq = 1. C0-semigroups. Regular semigroups. Every element x has at least one inverse y that satisfies xyx = x and yxy = y; the elements x and y are sometimes called "mutually inverse". Inverse semigroups are regular semigroups where every element has exactly one inverse. Alternatively, a regular semigroup is inverse if and only if any two idempotents commute. Affine semigroup: a semigroup that is isomorphic to a finitely-generated subsemigroup of Zd. These semigroups have applications to commutative algebra. == Structure theorem for commutative semigroups == There is a structure theorem for commutative semigroups in terms of semilattices. A semilattice (or more precisely a meet-semilattice) (L, ≤) is a partially ordered set where every pair of elements a, b ∈ L has a greatest lower bound, denoted a ∧ b. The operation ∧ makes L into a semigroup that satisfies the additional idempotence law a ∧ a = a. Given a homomorphism f : S → L from an arbitrary semigroup to a semilattice, each inverse image Sa = f−1{a} is a (possibly empty) semigroup. Moreover, S becomes graded by L, in the sense that SaSb ⊆ Sa∧b. If f is onto, the semilattice L is isomorphic to the quotient of S by the equivalence relation ~ such that x ~ y if and only if f(x) = f(y). This equivalence relation is a semigroup congruence, as defined above. Whenever we take the quotient of a commutative semigroup by a congruence, we get another commutative semigroup. The structure theorem says that for any commutative semigroup S, there is a finest congruence ~ such that the quotient of S by this equivalence relation is a semilattice. Denoting this semilattice by L, we get a homomorphism f from S onto L. As mentioned, S becomes graded by this semilattice. Furthermore, the components Sa are all Archimedean semigroups. An Archimedean semigroup is one where given any pair of elements x, y , there exists an element z and n > 0 such that xn = yz. The Archimedean property follows immediately from the ordering in the semilattice L, since with this ordering we have f(x) ≤ f(y) if and only if xn = yz for some z and n > 0. == Group of fractions == The group of fractions or group completion of a semigroup S is the group G = G(S) generated by the elements of S as generators and all equations xy = z that hold true in S as relations. There is an obvious semigroup homomorphism j : S → G(S) that sends each element of S to the corresponding generator. This has a universal property for morphisms from S to a group: given any group H and any semigroup homomorphism k : S → H, there exists a unique group homomorphism f : G → H with k = fj. We may think of G as the "most general" group that contains a homomorphic image of S. An important question is to characterize those semigroups for which this map is an embedding. This need not always be the case: for example, take S to be the semigroup of subsets of some set X with set-theoretic intersection as the binary operation (this is an example of a semilattice). Since A.A = A holds for all elements of S, this must be true for all generators of G(S) as well, which is therefore the trivial group. It is clearly necessary for embeddability that S have the cancellation property. When S is commutative this condition is also sufficient and the Grothendieck group of the semigroup provides a construction of the group of fractions. The problem for non-commutative semigroups can be traced to the first substantial paper on semigroups. Anatoly Maltsev gave necessary and sufficient conditions for embeddability in 1937. == Semigroup methods in partial differential equations == Semigroup theory can be used to study some problems in the field of partial differential equations. Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as an ordinary differential equation on a function space. For example, consider the following initial/boundary value problem for the heat equation on the spatial interval (0, 1) ⊂ R and times t ≥ 0: { ∂ t u ( t , x ) = ∂ x 2 u ( t , x ) , x ∈ ( 0 , 1 ) , t > 0 ; u ( t , x ) = 0 , x ∈ { 0 , 1 } , t > 0 ; u ( t , x ) = u 0 ( x ) , x ∈ ( 0 , 1 ) , t = 0. {\displaystyle {\begin{cases}\partial _{t}u(t,x)=\partial _{x}^{2}u(t,x),&x\in (0,1),t>0;\\u(t,x)=0,&x\in \{0,1\},t>0;\\u(t,x)=u_{0}(x),&x\in (0,1),t=0.\end{cases}}} Let X = L2((0, 1) R) be the Lp space of square-integrable real-valued functions with domain the interval (0, 1) and let A be the second-derivative operator with domain D ( A ) = { u ∈ H 2 ( ( 0 , 1 ) ; R ) | u ( 0 ) = u ( 1 ) = 0 } , {\displaystyle D(A)={\big \{}u\in H^{2}((0,1);\mathbf {R} ){\big |}u(0)=u(1)=0{\big \}},} where H 2 {\displaystyle H^{2}} is a Sobolev space. Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space X: { u ˙ ( t ) = A u ( t ) ; u ( 0 ) = u 0 . {\displaystyle {\begin{cases}{\dot {u}}(t)=Au(t);\\u(0)=u_{0}.\end{cases}}} On an heuristic level, the solution to this problem "ought" to be u ( t ) = exp ( t A ) u 0 . {\displaystyle u(t)=\exp(tA)u_{0}.} However, for a rigorous treatment, a meaning must be given to the exponential of tA. As a function of t, exp(tA) is a semigroup of operators from X to itself, taking the initial state u0 at time t = 0 to the state u(t) = exp(tA)u0 at time t. The operator A is said to be the infinitesimal generator of the semigroup. == History == The study of semigroups trailed behind that of other algebraic structures with more complex axioms such as groups or rings. A number of sources attribute the first use of the term (in French) to J.-A. de Séguier in Élements de la Théorie des Groupes Abstraits (Elements of the Theory of Abstract Groups) in 1904. The term is used in English in 1908 in Harold Hinton's Theory of Groups of Finite Order. Anton Sushkevich obtained the first non-trivial results about semigroups. His 1928 paper "Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit" ("On finite groups without the rule of unique invertibility") determined the structure of finite simple semigroups and showed that the minimal ideal (or Green's relations J-class) of a finite semigroup is simple. From that point on, the foundations of semigroup theory were further laid by David Rees, James Alexander Green, Evgenii Sergeevich Lyapin, Alfred H. Clifford and Gordon Preston. The latter two published a two-volume monograph on semigroup theory in 1961 and 1967 respectively. In 1970, a new periodical called Semigroup Forum (currently published by Springer Verlag) became one of the few mathematical journals devoted entirely to semigroup theory. The representation theory of semigroups was developed in 1963 by Boris Schein using binary relations on a set A and composition of relations for the semigroup product. At an algebraic conference in 1972 Schein surveyed the literature on BA, the semigroup of relations on A. In 1997 Schein and Ralph McKenzie proved that every semigroup is isomorphic to a transitive semigroup of binary relations. In recent years researchers in the field have become more specialized with dedicated monographs appearing on important classes of semigroups, like inverse semigroups, as well as monographs focusing on applications in algebraic automata theory, particularly for finite automata, and also in functional analysis. == Generalizations == If the associativity axiom of a semigroup is dropped, the result is a magma, which is nothing more than a set M equipped with a binary operation that is closed M × M → M. Generalizing in a different direction, an n-ary semigroup (also n-semigroup, polyadic semigroup or multiary semigroup) is a generalization of a semigroup to a set G with a n-ary operation instead of a binary operation. The associative law is generalized as follows: ternary associativity is (abc)de = a(bcd)e = ab(cde), i.e. the string abcde with any three adjacent elements bracketed. n-ary associativity is a string of length n + (n − 1) with any n adjacent elements bracketed. A 2-ary semigroup is just a semigroup. Further axioms lead to an n-ary group. A third generalization is the semigroupoid, in which the requirement that the binary relation be total is lifted. As categories generalize monoids in the same way, a semigroupoid behaves much like a category but lacks identities. Infinitary generalizations of commutative semigroups have sometimes been considered by various authors. == See also == Absorbing element Biordered set Compact semigroup Empty semigroup Generalized inverse Identity element Light's associativity test Principal factor Quantum dynamical semigroup Semigroup ring Weak inverse == Notes == == Citations == == References ==
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Wikipedia:E. G. Glagoleva#0
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Elena Georgievna Glagoleva (Russian: Елена Георгиевна Глаголева, 8 April 1926 – 20 July 2015) was a Soviet and Russian mathematician and mathematics educator who organized a correspondence school for the mathematics in the Soviet Union based at Moscow State University, and as part of the project coauthored two mathematics textbooks with Israel Gelfand. She is the author of: Метод координат (with I. M. Gelfand and A. A. Kirillov, 1964); translated into English by Richard A. Silverman as The Coordinate Method (Pocket Mathematical Library, Gordon & Breach, 1969), and by Leslie Cohn and David Sookne as The Method of Coordinates (Library of School Mathematics, MIT Press, 1967; Dover, 2002) Функции и графики (with I. M. Gelfand and E. E. Schnol, 1965); translated into English as Functions and Graphs by Richard A. Silverman (Pocket Mathematical Library, Gordon & Breach, 1969) and by Thomas Walsh and Randell Magee (MIT Press, 1969; Birkhäuser, 1990; Dover, 2002); translated into German by Reinhard Hoffmann as Funktionen und ihre graphische Darstellung (Teubner, 1971) Электричество в живых организмах (Electricity in Living Organisms, with M. B. Berkinblit, 1988) == References ==
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Wikipedia:E. T. Whittaker#0
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Sir Edmund Taylor Whittaker (24 October 1873 – 24 March 1956) was a British mathematician, physicist, and historian of science. Whittaker was a leading mathematical scholar of the early 20th century who contributed widely to applied mathematics and was renowned for his research in mathematical physics and numerical analysis, including the theory of special functions, along with his contributions to astronomy, celestial mechanics, the history of physics, and digital signal processing. Among the most influential publications in Whittaker's bibliography, he authored several popular reference works in mathematics, physics, and the history of science, including A Course of Modern Analysis (better known as Whittaker and Watson), Analytical Dynamics of Particles and Rigid Bodies, and A History of the Theories of Aether and Electricity. Whittaker is also remembered for his role in the relativity priority dispute, as he credited Henri Poincaré and Hendrik Lorentz with developing special relativity in the second volume of his History, a dispute which has lasted several decades, though scientific consensus has remained with Einstein. Whittaker served as the Royal Astronomer of Ireland early in his career, a position he held from 1906 through 1912, before moving on to the chair of mathematics at the University of Edinburgh for the next three decades and, towards the end of his career, received the Copley Medal and was knighted. The School of Mathematics of the University of Edinburgh holds The Whittaker Colloquium, a yearly lecture, in his honour and the Edinburgh Mathematical Society promotes an outstanding young Scottish mathematician once every four years with the Sir Edmund Whittaker Memorial Prize, also given in his honour. == Life == === Early life and education === Edmund Taylor Whittaker was born in Southport, in Lancashire, the son of Selina Septima (née Taylor) and John Whittaker. He was described as an "extremely delicate child", necessitating his mother to home school him until he was 11 years old, when he was sent off to Manchester Grammar School. Ernest Barker, a classmate of Whittaker's at the Grammar School with whom he shared the office of prefect, later recalled his personality: "He had a gay, lively, bubbling spirit: he was ready for every prank: he survives in my memory as a natural actor; and I think he could also, on occasion, produce a merry poem." While at the school, Whittaker studied on the "classical side", devoting three-fifths of his time to Latin and Greek. Whittaker struggled with the poetry and drama which was required by the upper school, and expressed gratitude for being allowed to leave these studies behind and specialise in mathematics. In December 1891 Whittaker received an entrance scholarship to Trinity College, Cambridge. After completing his education at the Manchester Grammar School he went on to study mathematics and physics there from 1892 to 1895. He entered Trinity College as a minor scholar in October 1892 to study mathematics. Whittaker was the pupil of Andrew Russell Forsyth and George Howard Darwin while at Trinity College and received tutoring throughout his first two years. With an interest more in applied than pure mathematics, Whittaker won the Sheepshanks Astronomical Exhibition in 1894 as an undergraduate. He graduated as Second Wrangler in the Cambridge Tripos examination in 1895. The Senior Wrangler that year was Thomas John I'Anson Bromwich and Whittaker tied John Hilton Grace for second, all three along with three other participants, including Bertram Hopkinson, went on to be elected Fellows of the Royal Society. He also received the Tyson Medal for Mathematics and Astronomy in 1896. === Career === Whittaker was a fellow of Trinity College, Cambridge from 1896 to 1906 when he was appointed Andrews Professor of Astronomy at Trinity College Dublin and Royal Astronomer of Ireland. He held these posts until 1912, when he was appointed chair of mathematics at the University of Edinburgh, a role he went on to hold for just over a third of a century. Throughout his career, he wrote papers on automorphic functions and special functions in pure mathematics as well as on electromagnetism, general relativity, numerical analysis and astronomy in applied mathematics and physics, and was also interested in topics in biography, history, philosophy and theology. He also made several important innovations in Edinburgh that had a large impact on mathematical education and societies there. ==== Trinity College, Cambridge ==== In 1896, Whittaker was elected as a Fellow of Trinity College, Cambridge, and remained at Cambridge as a teacher until 1906. In 1897, Whittaker was awarded the Smith Prize for his work on the paper "On the connexion of algebraic functions with automorphic functions", published in 1888. In 1902, Whittaker found a general solution to Laplace's equation, which received popular news coverage as a "remarkable discovery", though the mathematician Horace Lamb noted that it did not offer any new features. He also wrote several celebrated books in his early career, publishing A Course of Modern Analysis in 1902 and following it up with A Treatise on the Analytical Dynamics of Particles and Rigid Bodies just two years later in 1904. In September of that year, Whittaker was forced to sell six silver forks at an auction to pay back taxes which he had previously refused to pay due to the Education Act 1902 requiring citizens to pay taxes to fund local Christian schools, such as the Roman Catholic Church and the Church of England. Prior to being compelled by a magistrate to repay the tax burden, Whittaker was one of several activists who engaged in passive resistance by refusing to pay the taxes. In 1905, Whittaker was elected as a fellow of the Royal Society in recognition of his achievements. ==== Trinity College Dublin ==== In 1906, Whittaker was appointed Andrews Professor of Astronomy at Trinity College Dublin, which came with the title Royal Astronomer of Ireland. He succeeded Charles Jasper Joly at the post and was appointed upon recommendation from the astronomer Robert Stawell Ball. Ball's recommendation, which was published in a collection of his letters in 1915, stated that Whittaker was the only person he knew who could "properly succeed Joly" and that the role would "suit him in every way". He then describes Whittaker as "modest" and "charming" and as "a man who has infinite capacity for making things go". Ball said Whittaker was a world-leading expert in theoretical astronomy and that, in relation to Whittaker's discovery of a general solution to Laplace's equation, notes that he "has already made one discovery which the greatest mathematician of the last two centuries would be proud to have placed to his credit". The Royal Astronomers acted as directors for the Dunsink Observatory, which used outdated astronomy equipment; it was understood that the primary responsibility of the role was to teach mathematical physics at Trinity College. During this time, the relative leisure of his post allowed him to complete the reading required to write his third major book A History of the Theories of Aether and Electricity, from the age of Descartes to the close of the nineteenth century. Also during this time, he wrote the book The Theory of Optical Instruments, published six astronomy papers, and published collected astronomical observations. ==== University of Edinburgh ==== Whittaker became Professor of Mathematics at the University of Edinburgh in January 1912, where he remained for the rest of his career. The role was left vacant by the death of his predecessor, George Chrystal in 1911. He was elected as a Fellow of the Royal Society of Edinburgh in 1912, after being nominated by Cargill Gilston Knott, Ralph Allan Sampson, James Gordon MacGregor and Sir William Turner. He served as Secretary to the Society from 1916 to 1922, the Vice President from 1925 to 1928 and from 1937 to 1939, and was President of the Society from 1939 to 1944, through the war years. Whittaker began holding "research lectures" in mathematics at the university, typically given twice a week. He was said to be a great lecturer by one of his previous attendees, who stated that his "clear diction, his felicity of language and his enthusiasm could not fail to evoke a response" and that he was very good with illustrations. Freeman Dyson commented on Whittaker's lecture style by saying that students were "warmed, not only by the physical presence of a big crowd packed together, but by the mental vigour and enthusiasm of the old man". Whittaker's efforts helped transform the Edinburgh Mathematical Society from a teachers society to an academic research society and was a major driving force in introducing computational mathematics education to the UK and America. Shortly after coming to Edinburgh, Whittaker established the Edinburgh Mathematical Laboratory, one of the UK's first mathematical laboratories. The laboratory was the first attempt of a systematic treatment of numerical analysis in Great Britain and friends of Whittaker have said he believes it his most notable contribution to the education of mathematics. Subjects taught at the laboratory included interpolation, the method of least squares, systems of linear equations, determinants, roots of transcendental equations, practical Fourier analysis, definite integrals, and numerical solution of differential equations. The laboratory program was so successful, it resulted in many requests for an extra summer course to allow others to attend who previously were unable, ultimately leading to the establishment of a colloquium through the Edinburgh Mathematical Society. In 1913, Whittaker established the Edinburgh Mathematical Society Colloquium and the first was held over five days in August of that year. The textbook The calculus of observations was compiled from courses given at the Laboratory over a ten-year period; the book was well received and ultimately went through four editions. === Fellowships and academic positions === Outside of the Royal Astronomer of Ireland and his roles in the Royal Society of Edinburgh, Whittaker held several notable academic posts, including president of the Mathematical Association from 1920 through 1921, president of the Mathematical and Physical Section (Section A) of the British Science Association in 1927, and was president of the London Mathematical Society from 1828 through 1829. Whittaker also held the Gunning Victoria Jubilee Prize Lectureship for "his service to mathematics" with the Royal Society of Edinburgh from 1924 through 1928. He was elected either Honorary Fellow or Foreign Member in a number of academic organisations, including the Accademia dei Lincei in 1922, the Societa Reale di Napoli in 1936, the American Philosophical Society in 1944, the Académie royale de Belgique in 1946, the Faculty of Actuaries in 1918, the Benares Mathematical Society in 1920, the Indian Mathematical Society in 1924, and the Mathematical Association in 1935. In 1956, he was elected as a corresponding member of the Geometry section of the French Academy of Sciences a few days before his death. Whittaker was also awarded honorary doctorates from several universities, including two LLDs from the University of St Andrews in 1926 and the University of California in 1934, an ScD from the Trinity College Dublin in 1906, and two D.Sc.s from the National University of Ireland in 1939 and University of Manchester in 1944. === Later life === Whittaker published many works on philosophy and theism in the last years of his career and during his retirement in addition to his work on the second edition of A History of the Theories of Aether and Electricity. He released two books on Christianity and published several books and papers on the philosophy of Arthur Eddington. ==== Christianity ==== Whittaker was a Christian and became a convert to the Roman Catholic Church in 1930. In relation to that, Pope Pius XI awarded him with the Pro Ecclesia et Pontifice in 1935 and appointed him to the Pontifical Academy of Sciences in 1936. He was a member of the academy from 1936 onward and served as Honorary President of the Newman Society from 1943 to 1945. Whittaker published two book-length works on the topic of Christianity, including The beginning and end of the world and Space and spirit. The first of which was the result of the 1942 Riddell Memorial Lectures at Durham while the second is based on his 1946 Donnellan Lecture at Trinity College Dublin. It has been remarked by physics historian Helge Kragh, that in these books, Whittaker was "the only physical scientist of the first rank" who defended the strong entropic creation argument, which holds that as entropy always increases, the Universe must have started at a point of minimum entropy, which they argue implies the existence of a god. Whittaker published several articles which draw connections between science, philosophy and theism between 1947 and 1952 in the BBC magazine The Listener, one of which Religion and the nature of the universe was republished in American Vogue, making him "a rare, if not unique, example of a man whose published work not only crossed disciplinary boundaries, but was published everywhere from Nature to Vogue." ==== Retirement ==== Whittaker retired from his position as chair of the mathematics department at the University of Edinburgh in September 1946, a role he held for over 33 years. He was awarded emeritus professor status at the university which he retained until his death. In retirement, Whittaker worked tirelessly on the second edition of his A History of the Theories of Aether and Electricity, releasing The Classical Theories just a few years later. He also continued publishing works in philosophy and theism. James Robert McConnell noted that Whittaker's research in the connection between physics and philosophy spanned nearly forty publications written over his last 15 years. During the three years prior to the publication of second volume of his History, Whittaker had already determined that he was going to give priority for the discovery of special relativity to Henri Poincaré and Hendrik Lorentz in the new book. Max Born, a friend of Whittaker's, wrote a letter to Einstein in September 1953 explaining that he had done all he could over the previous three years to convince Whittaker to change his mind about Einstein's role, but Whittaker was resolved in the idea and, according to Born, he "cherished" and "loved to talk" about it. Born told Einstein that Whittaker insists that all the important features were developed by Poincaré while Lorentz "quite plainly had the physical interpretation", which annoyed Born as Whittaker was a "great authority in the English speaking countries" and he was worried that Whittaker's view would influence others. ==== Death ==== Whittaker died at his home, 48 George Square, Edinburgh, on 24 March 1956. He was buried at Mount Vernon Cemetery in Edinburgh, with "mathematical precision at a depth of 6 ft. 6 inches", according to the cemetery register. His entry in the Biographical Memoirs of Fellows of the Royal Society was written by George Frederick James Temple in November 1956. He received published obituaries from Alexander Aitken, Herbert Dingle, Gerald James Whitrow, and William Hunter McCrea, among others. His house was owned by the University of Edinburgh and was demolished in the 1960s to expand the campus and now holds the William Robertson Building. === Personal life === In 1901, while at Cambridge, he married Mary Ferguson Macnaghten Boyd, the daughter of a Presbyterian minister and granddaughter of Thomas Jamieson Boyd. They had five children, two daughters and three sons including the mathematician John Macnaghten Whittaker (1905–1984). His elder daughter, Beatrice, married Edward Taylor Copson, who would later become Professor of Mathematics at the University of St Andrews. George Frederick James Temple noted that Whittaker's home in Edinburgh was "a great centre of social and intellectual activity where liberal hospitality was dispensed to students of all ages", and went on to note that Whittaker had a happy home life and was well loved by his family. Whittaker kept a piano in his home which he did not know how to play, but enjoyed listening to friends play when they would come to visit. Whittaker was also known to take a personal interest in his students and would invite them to social gatherings at his house. He also continued to keep track of his Honours students over the years. His home was also the location of many unofficial interviews that would have a large impact on a student's future career. After his death, William Hunter McCrea described Whittaker as having a "quick wit" with an "ever-present sense of humour" and being "the most unselfish of men with a delicate sense of what would give help or pleasure to others". He notes that Whittaker had a "vast number of friends" and that he "never missed an opportunity to do or say something on behalf of any one of them". == Legacy == In addition to his textbooks and other works, several of which remain in print, Whittaker is remembered for his research in automorphic functions, numerical analysis, harmonic analysis, and general relativity. He has several theorems and functions named in his honour. In June 1958, two years after his death, an entire issue of the Proceedings of the Edinburgh Mathematical Society was dedicated to his life and works. The volume included an article by Robert Alexander Rankin on Whittaker's work on automorphic functions, an article on Whittaker's work on numerical analysis by Alexander Aitken, his work on Harmonic functions was covered in an article by Temple, John Lighton Synge wrote about his contributions to the theory of relativity, and James Robert McConnell wrote about Whittaker's philosophy. Among others, Whittaker coined the terms cardinal function and Mathieu function. The School of Mathematics of the University of Edinburgh holds the annual Whittaker Colloquium in his honour. Funded by a donation from his family in 1958, the Edinburgh Mathematical Society promotes an outstanding young Scottish mathematician once every four years with the Sir Edmund Whittaker Memorial Prize, also given in his honour. === Namesakes and notable research === Whittaker is the eponym of the Whittaker function or Whittaker integral, in the theory of confluent hypergeometric functions. This makes him also the eponym of the Whittaker model in the local theory of automorphic representations. He published also on algebraic functions, though they were typically limited to special cases. Whittaker had a lifelong interest in automorphic functions and he published three papers on the topic throughout his career. Among other contributions, he found the general expression for the Bessel functions as integrals involving Legendre functions. Whittaker also made contributions to the theory of partial differential equations, harmonic functions and other special functions of mathematical physics, including finding a general solution to Laplace's equation that became a standard part of potential theory. Whittaker developed a general solution of the Laplace equation in three dimensions and the solution of the wave equation. === Notable works === Whittaker wrote three scientific treatises which were highly influential, A Course of Modern Analysis, Analytical Dynamics of Particles and Rigid Bodies, and The Calculus of Observations. In 1956, Gerald James Whitrow stated that two of them not only were required reading for British mathematicians, but were regarded as fundamental components of their personal libraries. Despite the success of these books and his other researchers and their influence in mathematics and physics, the second edition of Whittaker's A History of the Theories of Aether and Electricity has been called his "magnum opus". In reference to the title's popularity, William Hunter McCrea predicted that future readers would have a hard time acknowledging it was the result of just "a few years at both ends of a career of the highest distinction in other pursuits." Whittaker also wrote The theory of optical instruments during his time as Royal Astronomer of Ireland as well as several books on philosophy and theism. Whittaker's bibliography in the Biographical Memoirs of Fellows of the Royal Society includes 11 books and monographs, 56 mathematics and physics articles, 35 philosophy and history articles, and 21 biographical articles, excluding popular and semi-popular articles published in magazines such as Scientific American. In the bibliography compiled by McCrea in 1957, there are 13 books and monographs and the same journal articles, also excluding popular articles. Among other topics, Whittaker wrote a total of ten papers on electromagnetism and general relativity. ==== Whittaker & Watson ==== Whittaker was the original author of the classic textbook A Course of Modern Analysis, first published in 1902. There were three more editions of the book all in collaboration with George Neville Watson, resulting in the famous colloquial name Whittaker & Watson. The work is subtitled an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions and is a classic textbook in mathematical analysis, remaining in print continuously since its release over a hundred years ago. It covered topics previously unavailable in English, such as complex analysis, mathematical analysis, and the Special functions used in mathematical physics. George Frederick James Temple noted that it was unmatched in these aspects "for many years". The book was an edited set of lecture notes from the Cambridge Tripos courses Whittaker taught and contained results from mathematicians such as Augustin-Louis Cauchy and Karl Weierstrass which were relatively unknown to English speaking countries. A. C. Aitken noted the books have been widely influential in the study of special functions and their associated differential equations as well as in the study of functions of complex variables. ==== Analytical Dynamics of Particles and Rigid Bodies ==== Whittaker's second major work, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies was first published in 1904, and quickly became a classic textbook in mathematical physics and analytical dynamics, a branch of classical mechanics. It has remained in print for most of its lifetime, over more than a hundred years, and has been said to have "remarkable longevity". The book represented the forefront of development at the time of publication, where many reviewers noted it contained material otherwise non-existent in the English language. The book was a landmark textbook, providing the first systematic treatment in English for the theory of Hamiltonian dynamics, which played a fundamental role in the development of quantum mechanics. A. C. Aitken called the book "epoch making in a very precise sense", noting that just before the development of the theory of relativity, the book provided a detailed summary of classical dynamics and the progress that had been made in Lagrangian mechanics and Hamiltonian mechanics, including work from Henri Poincaré and Tullio Levi-Civita. The book has received many recommendations, including from Victor Lenzen in 1952, nearly 50 years after its initial publication, who said the book was still the "best exposition of the subject on the highest possible level". It was noted in a 2014 article covering the book's development, published in the Archive for History of Exact Sciences, that the book was used for more than just a historical book, where it was pointed out that of the 114 books and papers that cited the book between 2000 and 2012, "only three are of a historical nature". In that same period, the book was said to be "highly recommended to advanced readers" in the 2006 engineering textbook Principles of Engineering Mechanics. ==== A History of the Theories of Aether and Electricity ==== In 1910, Whittaker wrote A History of the Theories of Aether and Electricity, which gave a detailed account of the aether theories from René Descartes to Hendrik Lorentz and Albert Einstein, including the contributions of Hermann Minkowski. The book was well received and established Whittaker as a respected historian of science. A second, revised and extended edition was later released. The first volume, subtitled the classical theories, was published in 1951 and served as a revised and updated edition of the first book. The second volume, published in 1953, extended this work covering the years 1900–1926. Notwithstanding a notorious controversy on Whittaker's views on the history of special relativity, covered in volume two of the second edition, the books are considered authoritative references on the history of classical electromagnetism and are considered classic books in the history of physics. Due to the book's role in the relativity priority dispute, however, the second volume is cited far less than the first volume and first edition, except in connection with the controversy. === Relativity priority dispute === Whittaker is also remembered for his role in the relativity priority dispute, a historical controversy over credit for the development of special relativity. In a chapter named "The Relativity Theory of Poincaré and Lorentz" in the second volume of the second edition of A History of the Theories of Aether and Electricity, Whittaker credited Henri Poincaré and Hendrik Lorentz for developing the theory; he attributed relatively little importance to Einstein's special relativity paper, saying it "set forth the relativity theory of Poincaré and Lorentz with some amplifications, and which attracted much attention". Max Born, a friend of Whittaker's, wrote to Einstein expressing his concern about the book's publication and wrote a rebuttal in his 1956 book. The controversy was also mentioned in one of Whittaker's obituaries by Gerald James Whitrow, who said that he had written Whittaker a letter explaining how the latter's views "did not do justice to the originality of Einstein's philosophy", but remarked that he understood why Whittaker felt the need to correct the popular misconception that Einstein's contribution was unique. Max Born's rebuttal, published in his 1956 book, also argues that while the contributions of Lorentz and Poincaré should not be overlooked, it was the postulates and philosophy of Einstein's theory that "distinguishes Einstein’s work from his predecessors and gives us the right to speak of Einstein’s theory of relativity, in spite of Whittaker’s different opinion". Though the dispute has lasted decades, most scholars have rejected Whittaker's arguments and scientific consensus has continued to hold that special relativity was Einstein's development. == Philosophy == Whittaker's views on philosophy was analysed by James Robert McConnell for the Whittaker Memorial Volume of the Proceedings of the Edinburgh Mathematical Society. McConnell noted that Whittaker's research into the connections between physics and philosophy were spread across approximately forty publications. Whittaker's worldview was classified as neo-Cartesianism in the volume, a philosophy described as being "founded on the principle that the search for a universal science should be modelled on the procedure of physicomathematicians." McConnell notes several of Whittaker's original contributions to René Descartes' philosophical system, but goes on to sum up the work by saying that while he admired Whittaker's attempt at the problem, he was not satisfied with the many transitions between mathematics, aesthetics, ethics. He stated that "If the transitions from mathematics to moral values are not firmly established, Whittaker's attempt does not succeed in remedying the defects of Descartes' solution." Whittaker published work in several other areas of philosophy, including research on Eddington's principle, a conjecture by Arthur Eddington that all quantitative propositions in physics can be derived from qualitative assertions. In addition to publishing Eddington's Fundamental Theory, Whittaker wrote two other books pertaining to Eddington's philosophy. Whittaker also wrote at length about the impacts of then-recent discoveries in astronomy on religion and theology, determinism and free will, and natural theology. In the conclusion of his article, McConnell sums up Whittaker's philosophic works as appearing as though it came from "that of the scholarly Christian layman". On metaphysics, he goes on to note that there are few scholars who are competent in both physics and metaphysics and states that future work could benefit and draw inspiration from Whittaker's research in the area. == Awards and honours == In 1931, Whittaker received the Sylvester Medal from the Royal Society for "his original contributions to both pure and applied mathematics". He then received the De Morgan Medal from the London Mathematical Society in 1935, an award given once every three years for outstanding contributions to mathematics. He received several honours in his 70s, including being knighted in 1945 by King George VI, and in 1954, receiving the Royal Society's Copley Medal, its highest award, "for his distinguished contributions to both pure and applied mathematics and to theoretical physics". In the opening remarks of the 1954 address of President Edgar Adrian to the Royal Society, Adrian announces Whittaker as that years Copley medallist saying he is probably the most well-known British mathematician at the time, due to "his numerous, varied and important contributions" as well as the offices he had held. Noting contributions to nearly all fields of applied mathematics and then-recent contributions to pure mathematics, relativity, electromagnetism, and quantum mechanics, Adrian goes on to say that the "astonishing quantity and quality of his work is probably unparalleled in modern mathematics and it is most appropriate that the Royal Society should confer on Whittaker its most distinguished award." Whittaker also gave several distinguished lectures, some of which formed the base of books he would later write. He held the Rouse Ball lectureship at Trinity College, Cambridge in 1926, the Bruce-Preller lectureship of the Royal Society of Edinburgh in 1931, and the Selby lectureship at the University of Cardiff in 1933. He also held the Hitchcock professorship at the University of California in 1934, the Riddell lectureship at the University at Durham (Newcastle) in 1942, the Guthrie lectureship of the Royal Physical Society of Edinburgh in 1943, and the Donnellan lectureship at the Trinity College Dublin in 1946. He gave the Tarner Lecture at Trinity College, Cambridge in 1947 and held the Larmor lectureship of the Royal Irish Academy and the Herbert Spencer lectureship at the University of Oxford, both in 1948. == See also == List of fellows of the Royal Society elected in 1905 List of Cambridge mathematicians List of mathematicians born in the 19th century List of theoretical physicists == References == == Bibliography == == Further reading == Watson, G. Alistair (1 November 2009). "The history and development of numerical analysis in Scotland: a personal perspective" (PDF). The Birth of Numerical Analysis. World Scientific. pp. 161–177. doi:10.1142/9789812836267_0010. ISBN 978-981-283-625-0. Retrieved 26 October 2020. Rankin, R. A. (June 1983). "The first hundred years (1883–1983)" (PDF). Proceedings of the Edinburgh Mathematical Society. 26 (2): 135–150. doi:10.1017/S0013091500016849. ISSN 1464-3839. Butzer, P. L.; Ferreira, P. J. S. G.; Higgins, J. R.; Saitoh, S.; Schmeisser, G.; Stens, R. L. (1 April 2011). "Interpolation and Sampling: E.T. Whittaker, K. Ogura and Their Followers". Journal of Fourier Analysis and Applications. 17 (2): 320–354. Bibcode:2011JFAA...17..320B. doi:10.1007/s00041-010-9131-8. ISSN 1531-5851. S2CID 122954185. == External links == O'Connor, John J.; Robertson, Edmund F. (October 2003), "Edmund Taylor Whittaker", MacTutor History of Mathematics Archive, University of St Andrews "Whittaker and the Aether". MathPages.com. "Sir Edmund Taylor Whittaker | British mathematician". Encyclopedia Britannica. Retrieved 27 October 2020. "Whittaker, Edmund Taylor". encyclopedia.com. Retrieved 27 October 2020.
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Wikipedia:Early Algebra#0
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Early Algebra is an approach to early mathematics teaching and learning. It is about teaching traditional topics in more profound ways. It is also an area of research in mathematics education. Traditionally, algebra instruction has been postponed until adolescence. However, data of early algebra researchers shows ways to teach algebraic thinking much earlier. The National Council of Teachers of Mathematics (NCTM) integrates algebra into its Principles and Standards starting from Kindergarten. One of the major goals of early algebra is generalizing number and set ideas. It moves from particular numbers to patterns in numbers. This includes generalizing arithmetic operations as functions, as well as engaging children in noticing and beginning to formalize properties of numbers and operations such as the commutative property, identities, and inverses. Students historically have had a very difficult time adjusting to algebra for a number of reasons. Researchers have found that by working with students on such ideas as developing rules for the use of letters to stand in for numbers and the true meaning of the equals symbol (it is a balance point, and does not mean "put the answer next"), children are much better prepared for formal algebra instruction. Teacher professional development in this area consists of presenting common student misconceptions and then developing lessons to move students out of faulty ways of thinking and into correct generalizations. The use of true, false, and open number sentences can go a long way toward getting students thinking about the properties of number and operations and the meaning of the equals sign. Research areas in early algebra include use of representations, such as symbols, graphs and tables; cognitive development of students; viewing arithmetic as a part of algebraic conceptual fields == Notes == == References == Blanton, M. L. Algebra and the Elementary Classroom: Transforming Thinking, Transforming Practice. (Heinemann, 2008). J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the Early Grades. (Lawrence Erlbaum and Associates, 2007). Schliemann, A.D., Carraher, D.W., & Brizuela, B. Bringing Out the Algebraic Character of Arithmetic: From Children's Ideas to Classroom Practice. (Lawrence Erlbaum Associates, 2007). Carraher, D., Schliemann, A.D., Brizuela, B., & Earnest, D. (2006). Arithmetic and Algebra in early Mathematics Education. Journal for Research in Mathematics Education, Vol 37.[1] National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. (Author, 2000) == External links == Tufts/TERC Early Algebra Project
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Wikipedia:Earthquake map#0
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In hyperbolic geometry, an earthquake map is a method of changing one hyperbolic manifold into another, introduced by William Thurston (1986). == Earthquake maps == Given a simple closed geodesic on an oriented hyperbolic surface and a real number t, one can cut the manifold along the geodesic, slide the edges a distance t to the left, and glue them back. This gives a new hyperbolic surface, and the (possibly discontinuous) map between them is an example of a left earthquake. More generally one can do the same construction with a finite number of disjoint simple geodesics, each with a real number attached to it. The result is called a simple earthquake. An earthquake is roughly a sort of limit of simple earthquakes, where one has an infinite number of geodesics, and instead of attaching a positive real number to each geodesic one puts a measure on them. A geodesic lamination of a hyperbolic surface is a closed subset with a foliation by geodesics. A left earthquake E consists of a map between copies of the hyperbolic plane with geodesic laminations, that is an isometry from each stratum of the foliation to a stratum. Moreover, if A and B are two strata then E−1AEB is a hyperbolic transformation whose axis separates A and B and which translates to the left, where EA is the isometry of the whole plane that restricts to E on A, and likewise for B. == Earthquake theorem == Thurston's earthquake theorem states that for any two points x, y of a Teichmüller space there is a unique left earthquake from x to y. It was proved by William Thurston in a course in Princeton in 1976–1977, but at the time he did not publish it, and the first published statement and proof was given by Kerckhoff (1983), who used it to solve the Nielsen realization problem. == References == Kerckhoff, Steven P. (1983), "The Nielsen realization problem", Annals of Mathematics, Second Series, 117 (2): 235–265, CiteSeerX 10.1.1.353.3593, doi:10.2307/2007076, ISSN 0003-486X, JSTOR 2007076, MR 0690845 Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Paris: Société Mathématique de France, 1979, ISBN 978-99920-1-230-7, MR 0568308 Thurston, William P. (1986), "Earthquakes in two-dimensional hyperbolic geometry", in D.B.A. Epstein (ed.), Low dimensional topology and Kleinian groups, Cambridge University Press, ISBN 978-0-521-33905-6
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Wikipedia:Eben Matlis#0
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Eben Matlis (August 28, 1923 - March 27, 2015) was a mathematician known for his contributions to the theory of rings and modules, especially for his work with injective modules over commutative Noetherian rings, and his introduction of Matlis duality. Matlis earned his Ph.D. at the University of Chicago in 1958, with Irving Kaplansky as advisor. He is an emeritus professor at Northwestern University and was a member of the Institute for Advanced Study from August 1962 to June 1963. == Selected works == Matlis, Eben (1958), "Injective modules over Noetherian rings", Pacific Journal of Mathematics, 8 (3): 511–528, doi:10.2140/pjm.1958.8.511, ISSN 0030-8730, MR 0099360 Matlis, Eben (1972), Torsion-free modules, University of Chicago Press, MR 0344237 Matlis, Eben (1973), 1-dimensional Cohen-Macaulay rings, Lecture Notes in Mathematics, Vol. 327, Berlin, New York: Springer-Verlag, MR 0357391 == References == == External links == Eben Matlis at the Mathematics Genealogy Project
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Wikipedia:Eberhard Becker#0
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Eberhard Becker (born July 23, 1943 in Stavenhagen) is a German mathematician whose career was spent at the University of Dortmund. A very active researcher in algebra, he later became rector of the university there. During his term as rector, it was renamed the Technical University of Dortmund. == Education and career == Becker received his Ph.D. at the University of Hamburg in 1972 with the dissertation, "Contributions to the theory of semi-simple quadratic algebra" under advisor Hel Braun. He completed his habilitation at the University of Cologne in 1976. In 1979 Becker was appointed to the mathematics department at the University of Dortmund. His research included work in the algebraic theory of quadratic forms and real algebraic geometry. His collaborators included Manfred Knebusch and Alex F. T. W. Rosenberg. He supervised over a dozen doctoral students, including Markus Schweighofer, Susanne Pumplün, and Thorsten Wörmann. In the mid 1980s, Becker proved that the expression (1 + t 2 {\displaystyle t^{2}} )/(2 + t 2 {\displaystyle t^{2}} ) was a sum of 4th powers, 6th powers, 8th powers and so on. He offered a bottle of champaign to anyone who could find explicit representations of this form. Bruce Reznick came up with a solution in 1994. After working as institute director, dean and member of the Senate at the University of Dortmund, Becker became rector there on April 30, 2002 following the resignation of Hans-Jürgen Klein. During his term in office the Senate decided, at his request, to change the name from “University of Dortmund” to “Technical University of Dortmund”. This decision was not without controversy, particularly in the humanities departments, but was confirmed at the crucial Senate meeting by two-thirds of the vote. On September 1, 2008 he was replaced by Ursula Gather. On the occasion of his retirement on November 7, 2008, the faculty organized a celebratory colloquium for him. To mark his 80th birthday, a conference on Quadratic Forms and Real Algebra was held in October 2023 at the University of Dortmund. == References == == External links == Eberhard Becker at the Mathematics Genealogy Project Papers by Eberhard Becker 1994 - 2005
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Wikipedia:Eckhard Meinrenken#0
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Eckhard Meinrenken is a German-Canadian mathematician working in differential geometry and mathematical physics. He is a professor at University of Toronto. == Education and career == Meinrenken studied Physics at Albert-Ludwigs-Universität Freiburg, where he obtained a Diplom in 1990 and a PhD in 1994, with a thesis entitled Vielfachheitsformeln für die Quantisierung von Phasenräumen (Multiplicity formulas for the quantization of phase spaces), under the supervision of Hartmann Römer. He was a postdoc at Massachusetts Institute of Technology from 1995 to 1997, and then he joined University of Toronto Department of Mathematics in 1998 as assistant professor. In 2000 he become Associated Professor and since 2004 he is Full Professor at the same university. Meinrenken was awarded in 2001 an André Aisenstadt Prize, in 2003 a McLean Award and in 2007 a NSERC Steacie Memorial Fellowship. In 2002 he was invited speaker at the International Congress of Mathematicians in Beijing and in 2008 he was elected Fellow of the Royal Society of Canada. == Research == Meinrenken's research interests lie in the fields of differential geometry and mathematical physics. In particular, he works on symplectic geometry, Lie theory and Poisson geometry. Among his most important contributions, in 1998 he proved, together with Reyer Sjamaar the conjecture "quantisation commutes with reduction", originally formulated in 1982 by Guillemin and Sternberg. In the same year, together with Anton Alekseev and Anton Malkin, he introduced Lie group-valued moment maps in symplectic geometry. Meinrenken is author of more than 50 research papers in peer-reviewed journals, as well as a monograph on Clifford algebras. He has supervised 9 PhD students as of 2021. == References ==
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Wikipedia:Eckhard Platen#0
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Eckhard Platen is a German/Australian mathematician, financial economist, academic, and author. He is an emeritus Professor of Quantitative Finance at the University of Technology Sydney. Platen is most known for his research on numerical methods for stochastic differential equations and their application in finance along with the generalization of the classical mathematical finance theory by his benchmark approach. He has authored and co-authored research papers and five books including Numerical Solution of Stochastic Differential Equations, A Benchmark Approach to Quantitative Finance and Functionals of Multi-dimensional Diffusions with Applications to Finance. He is the recipient of the 1992 Best Paper Award in Mathematical Finance, was named Honorary Professor at the University of Cape Town from 2014 to 2019 and at the Australian National University from 2015 to 2020, and is a Fellow of the Australian Mathematical Society. == Education == Platen earned an MSc in Mathematics in 1972 and a PhD in Probability Theory in 1975 from the Technical University Dresden, followed by a DSc in Science at the Academy of Sciences, Berlin in 1985. == Career == Platen began his academic career in 1975 as a Research Fellow at the Weierstrass Institute at the Academy of Sciences Berlin, holding the position of Head of the Sector Stochastics from 1987 to 1990. Later, in 1991, he assumed the role of Senior Fellow at the Institute of Advanced Studies at the Australian National University in Canberra, serving as the Founding Head of the Centre for Financial Mathematics from 1994 to 1997. In 1997, he took on a joint appointment between the School of Finance and Economics and the School of Mathematical Sciences, and as the chair in Quantitative Finance at the University of Technology Sydney. He remained a Research Director of the Quantitative Finance Research Centre at the University of Technology Sydney from 1998 until 2021 and has held the position of emeritus Professor of Quantitative Finance since 2021. Platen founded the Quantitative Methods in Finance annual conference series in 1993, where he served as chair for 25 years. Later, he became President of the Bachelier Finance Society from 2014 to 2015 and has been a Director of the Scientific Association of Mathematical Finance since 2021. == Research == Platen has contributed to the field of mathematics and financial economics by studying numerical methods and quantitative finance and proposing the benchmark approach for finance, insurance and economics. == Works == Platen has authored and co-authored five books on numerical methods and quantitative finance. Earlier, he focused on the numerical solution of stochastic differential equations, writing three books on the topic including Numerical Solution of Stochastic Differential Equations with Peter Kloeden, Numerical Solution of SDE Through Computer Experiments with Kloeden and Henri Schurz, and Numerical Solution of Stochastic Differential Equations with Jumps in Finance with Nicola Bruti-Liberati. About the first book, Francesco Gianfelici remarked, "...the need for proper SDE methodologies in a numerical context is increasingly pressing and provides the motivation and the starting point of this excellent book written by Kloeden and Platen." Later, Platen published monographs on his benchmark approach, namely A Benchmark Approach to Quantitative Finance with David Heath. In a review for Quantitative Finance, Wolfgang Runggaldier commented "The book thus presents itself as a comprehensive treatment of Quantitative Finance and distinguishes itself from analogous treatments by using a novel approach, namely the benchmark approach." He also co-wrote the book Functionals of Multi-dimensional Diffusions with Applications to Finance with Jan Baldeaux, which explored the systemic derivation of explicit formulas for functionals of diffusions. === Numerical solution of stochastic differential equations === Platen's work on stochastic differential equations has focused on a general theory for their numerical solution. He contended that the availability of a stochastic analogue to the deterministic Taylor formula would be essential for a numerical theory for stochastic differential equations. Together with Wagner, he discovered the stochastic Taylor formula, and then developed systematically a theory for the efficient numerical solution of stochastic differential equations. With various co-authors, he made seminal contributions on numerical stability, and stochastic delay equations. === Benchmark approach === Platen studied finance theory and proposed the benchmark approach as a novel, very general modelling method. Since the 1990s, he has focused on financial market modelling, derivative pricing, insurance and long-term risk management, while critiquing the existing classical mathematical finance theory. He concluded that the best performing portfolio of a market, which can be found by maximizing its expected growth rate, should form the centrepiece of a more general finance theory that he called the "benchmark approach", which offered a broader modelling framework with new relevant phenomena, and he presented its systematic formulation first in his books and later in research. Using Li symmetry group methods and entropy maximization, he identified conservation laws in finance, in the sense of Noether's Theorem, and discovered the typical least disturbed financial market dynamics. == Awards and honors == 2014 – Honorary Professor, University of Cape Town 2015 – Honorary Professor, Australian National University == Bibliography == === Books === Numerical Solution of Stochastic Differential Equations (1992) ISBN 978-3540540625 Numerical Solution of SDE Through Computer Experiments (1994) ISBN 978-3540570745 Numerical Solution of Stochastic Differential Equations with Jumps in Finance (2010) ISBN 978-3642120572 A Benchmark Approach to Quantitative Finance (2006) ISBN 978-3540262121 Functionals of Multi-dimensional Diffusions with Applications to Finance (2013) ISBN 978-3319007465 === Selected articles === Platen, E. & Wagner W. (1982) On a Taylor formula for a class of Ito processes. Probability and Mathematical Statistics, 3 (1), 37–51. Hofmann, N., Platen, E. & Schweizer, M. (1992). Option pricing under incompleteness and stochastic volatility. Mathematical Finance, 2 (3), 153–187. Milstein, G.N., Platen, E. & Schurz, H. (1998). Balanced implicit methods for stiff stochastic systems. SIAM Journal on Numerical Analysis, 35 (3), 1010–1019. Platen, E. & Schweizer, M. (1998). On feedback effects from hedging derivatives. Mathematical Finance, 8 (1), 67–84. Platen, E. (1999). An introduction to numerical methods for stochastic differential equations. Acta Numerica, 8, 197–246. Küchler, U. & Platen, E. (2000). Strong discrete time approximation of stochastic differential equations with time delay. Mathematics and Computers in Simulation, 54, 189–205. Platen, E. (2002). Arbitrage in continuous complete markets. Advances in Applied Probability, 33 (2), 540–558. Craddock, M. & Platen, E. (2004). Symmetry group methods for fundamental solutions. Journal of Differential Equations, 207 (2), 285–302. Platen, E. (2006). A benchmark approach to finance. Mathematical Finance, 16 (1), 131–151. Filipovic, D. & Platen, E. (2009). Consistent market extensions under the benchmark approach. Mathematical Finance, 19 (1), 41–52. Du, K. & Platen, E. (2016). Benchmarked risk minimization. Mathematical Finance. doi: 10.1111/mafi.12065 Baldeaux, J. & Ignatieva, K. & Platen, E. (2017). Detecting money market bubbles. Journal of Banking & Finance. 87, 369–379. Fergusson, K. & Platen, E. (2023). Less-expensive long-term annuities linked to mortality, cash and equity. Annals of Actuarial Science. 17, 170–207. == References ==
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Wikipedia:Eddy Campbell#0
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Eddy Campbell is a Canadian mathematician, university professor, and university administrator. He served as the president of the University of New Brunswick from 2009 - 2019. H. E. A. (Eddy) Campbell earned two degrees in mathematics from Memorial University of Newfoundland, and completed his doctorate at the University of Toronto. He did post-doctoral work at the University of Western Ontario. In 1983, he joined the Department of Mathematics and Statistics at Queen's University in Kingston, Ontario, eventually rising to head of that department. His main research interest is the invariant theory of finite groups. Campbell then served as Associate Dean of the Faculty of Arts and Science at Queen's. He served as president of the Canadian Mathematical Society from 2004 to 2006. Campbell returned to his original alma mater in May 2004, to become a Vice-president, Academic at Memorial. Upon the resignation of President Alex Meisen, he then stepped into the role of president (pro tempore) and vice-chancellor on January 1, 2008. Campbell applied for the permanent position with wide support within the MUN community, and was shortlisted by the Search Committee. However, in late July 2008, Minister of Education Joan Burke, who was not part of the selection process, stated after an interview with two shortlisted candidates that neither was acceptable to her. This was widely criticized as political interference in the autonomy of the University. Campbell immediately withdrew his name from consideration, and the Chair of Memorial's Board of Trustees, Gil Dalton, also withdrew as leader of the search committee in September 2008. Campbell remained in his position until September 2009, when he accepted the position of President of the University of New Brunswick. In January 2014, he took part in negotiations to end the first strike of academic staff in UNB's 225-plus years history. He later approved compensation to UNB students. The labour dispute was resolved by arbitration. In 2018 the Canadian Mathematical Society listed him in their inaugural class of fellows. == References ==
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Wikipedia:Edgar Krahn#0
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Edgar Krahn (1 October [O.S. 19 September] 1894 – 6 March 1961) was an Estonian mathematician. Krahn was born in Sootaga (now Laiuse, Jõgeva County), Governorate of Livonia, as a member of the Baltic German minority. He died in Rockville, Maryland, United States. Krahn studied at the University of Tartu and the University of Göttingen. He graduated at Tartu in 1918, received his doctoral degree at Göttingen in 1926, with Richard Courant as his advisor, and his habilitation took place at Tartu in 1928. He is co-author of the Rayleigh–Faber–Krahn inequality. Krahn worked in Estonia, Germany, the United Kingdom, and the United States in the following areas of pure and applied mathematics: Differential geometry Differential equations Bausparmathematik, which is distantly related to insurance mathematics Probability theory Gas dynamics Elasticity theory == See also == List of Baltic German scientists == References ==
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Wikipedia:Edge and vertex spaces#0
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In the mathematical discipline of graph theory, the edge space and vertex space of an undirected graph are vector spaces defined in terms of the edge and vertex sets, respectively. These vector spaces make it possible to use techniques of linear algebra in studying the graph. == Definition == Let G := ( V , E ) {\displaystyle G:=(V,E)} be a finite undirected graph. The vertex space V ( G ) {\displaystyle {\mathcal {V}}(G)} of G is the vector space over the finite field of two elements Z / 2 Z := { 0 , 1 } {\displaystyle \mathbb {Z} /2\mathbb {Z} :=\lbrace 0,1\rbrace } of all functions V → Z / 2 Z {\displaystyle V\rightarrow \mathbb {Z} /2\mathbb {Z} } . Every element of V ( G ) {\displaystyle {\mathcal {V}}(G)} naturally corresponds the subset of V which assigns a 1 to its vertices. Also every subset of V is uniquely represented in V ( G ) {\displaystyle {\mathcal {V}}(G)} by its characteristic function. The edge space E ( G ) {\displaystyle {\mathcal {E}}(G)} is the Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } -vector space freely generated by the edge set E. The dimension of the vertex space is thus the number of vertices of the graph, while the dimension of the edge space is the number of edges. These definitions can be made more explicit. For example, we can describe the edge space as follows: elements are subsets of E {\displaystyle E} , that is, as a set E ( G ) {\displaystyle {\mathcal {E}}(G)} is the power set of E vector addition is defined as the symmetric difference: P + Q := P △ Q P , Q ∈ E ( G ) {\displaystyle P+Q:=P\triangle Q\qquad P,Q\in {\mathcal {E}}(G)} scalar multiplication is defined by: 0 ⋅ P := ∅ P ∈ E ( G ) {\displaystyle 0\cdot P:=\emptyset \qquad P\in {\mathcal {E}}(G)} 1 ⋅ P := P P ∈ E ( G ) {\displaystyle 1\cdot P:=P\qquad P\in {\mathcal {E}}(G)} The singleton subsets of E form a basis for E ( G ) {\displaystyle {\mathcal {E}}(G)} . One can also think of V ( G ) {\displaystyle {\mathcal {V}}(G)} as the power set of V made into a vector space with similar vector addition and scalar multiplication as defined for E ( G ) {\displaystyle {\mathcal {E}}(G)} . == Properties == The incidence matrix H {\displaystyle H} for a graph G {\displaystyle G} defines one possible linear transformation H : E ( G ) → V ( G ) {\displaystyle H:{\mathcal {E}}(G)\to {\mathcal {V}}(G)} between the edge space and the vertex space of G {\displaystyle G} . The incidence matrix of G {\displaystyle G} , as a linear transformation, maps each edge to its two incident vertices. Let v u {\displaystyle vu} be the edge between v {\displaystyle v} and u {\displaystyle u} then H ( v u ) = v + u {\displaystyle H(vu)=v+u} The cycle space and the cut space are subspaces of the edge space. == References == Diestel, Reinhard (2005), Graph Theory (3rd ed.), Springer, ISBN 3-540-26182-6 (the electronic 3rd edition is freely available on author's site). == See also == Cycle space Cut space
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Wikipedia:Edge-transitive graph#0
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In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2. In other words, a graph is edge-transitive if its automorphism group acts transitively on its edges. == Examples and properties == The number of connected simple edge-transitive graphs on n vertices is 1, 1, 2, 3, 4, 6, 5, 8, 9, 13, 7, 19, 10, 16, 25, 26, 12, 28 ... (sequence A095424 in the OEIS) Edge-transitive graphs include all symmetric graphs, such as the vertices and edges of the cube. Symmetric graphs are also vertex-transitive (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. Every connected edge-transitive graph that is not vertex-transitive must be bipartite, (and hence can be colored with only two colors), and either semi-symmetric or biregular. Examples of edge but not vertex transitive graphs include the complete bipartite graphs K m , n {\displaystyle K_{m,n}} where m ≠ n, which includes the star graphs K 1 , n {\displaystyle K_{1,n}} . For graphs on n vertices, there are (n-1)/2 such graphs for odd n and (n-2) for even n. Additional edge transitive graphs which are not symmetric can be formed as subgraphs of these complete bi-partite graphs in certain cases. Subgraphs of complete bipartite graphs Km,n exist when m and n share a factor greater than 2. When the greatest common factor is 2, subgraphs exist when 2n/m is even or if m=4 and n is an odd multiple of 6. So edge transitive subgraphs exist for K3,6, K4,6 and K5,10 but not K4,10. An alternative construction for some edge transitive graphs is to add vertices to the midpoints of edges of a symmetric graph with v vertices and e edges, creating a bipartite graph with e vertices of order 2, and v of order 2e/v. An edge-transitive graph that is also regular, but still not vertex-transitive, is called semi-symmetric. The Gray graph, a cubic graph on 54 vertices, is an example of a regular graph which is edge-transitive but not vertex-transitive. The Folkman graph, a quartic graph on 20 vertices is the smallest such graph. The vertex connectivity of an edge-transitive graph always equals its minimum degree. == See also == Edge-transitive (in geometry) == References == == External links == Weisstein, Eric W. "Edge-transitive graph". MathWorld.
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Wikipedia:Edinburgh Mathematical Society#0
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The Edinburgh Mathematical Society is a mathematical society for academics in Scotland. == History == The Society was founded in 1883 by a group of Edinburgh school teachers and academics, on the initiative of Alexander Yule Fraser FRSE and Andrew Jeffrey Gunion Barclay FRSE, both maths teachers at George Watson's College, and Cargill Gilston Knott, the assistant of Peter Guthrie Tait, professor of physics at the University of Edinburgh. The first president, elected at first meeting on 2 February 1883, was J.S. Mackay, the head mathematics master at the Edinburgh Academy. The Society was founded at a time when mathematics societies were being created around the world, but it was unusual in being founded by school teachers rather than university lecturers. This was because, due to the very small number of mathematical academic positions in Scotland at the time, many skilled mathematics graduates chose to become schoolteachers instead. The fifty five founding members contained teachers, ministers and students, as well as a number of academics from the University of Cambridge. The proportion of teachers remained high compared to other mathematical societies, and by 1926 university members made up only one-third of the total members. However, the dominance of teachers in the numbers of the society declined towards the 1930s, and between 1930 and 1935 no papers were presented in the Proceedings by teachers. This was due to an increase in the number of academic positions available and the new requirement for teachers to undergo an additional year of vocational training. The Edinburgh Mathematical Society is now mainly for academics. == Activity == The Society organises and funds meetings and other research events throughout Scotland. There are normally eight meetings a year, at which talks are presented by mathematicians. Every four years it awards the Sir Edmund Whittaker Memorial Prize to an outstanding mathematician with a Scottish connection. The Society is a corporate member of the European Mathematical Society, and in 2008 it became a member of the Council for the Mathematical Sciences. == Journals == The society releases an academic journal, the Proceedings of the Edinburgh Mathematical Society, published by Cambridge University Press (ISSN 0013–0915.) The Proceedings were first published in 1884, and are issued three times a year, covering a range of pure and applied mathematics subjects. Between 1909 and 1961, the Society also published the Edinburgh Mathematical Notes, on the suggestion of George Alexander Gibson, a professor at the University of Glasgow, who wished to remove the more elementary or pedagogical articles from the Proceedings. == See also == List of Mathematical Societies == References == == External links == Official website
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Wikipedia:Edison Farah#0
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Edison Farah (Capivari, April 14, 1915 - São Paulo, April 14, 2006) was a Brazilian mathematician, professor at the University of São Paulo. He was a founding member of the Mathematical Society of São Paulo, founded in 1945, and a full member of the Academy of Sciences of the State of São Paulo. == Biography == Edison was born in the city of Capivari, in 1915. He was the eleventh child among the 12 children of José Ignácio Farah, a Lebanese immigrant from Baalbek, and Eduarda Llamas, a Spanish immigrant from Iznájar, in the province of Córdoba. The couple met on the ship that brought them to Brazil, which left Lebanon with a stopover in Spain. The name Edison, which was at odds with the Lebanese and Spanish names of their brothers, was in honor of the American inventor Thomas Edison. Initially, the couple settled in the city of Capivari, then in Rio das Pedras and in Piracicaba. José Ignácio was a merchant and even had a store in Capivari, a cotton warehouse. Edison and his brothers used to play climbing the beams of the store's shed, where they would fall on the piles of stored cotton. It was while still a child that his taste for science began to blossom. He also made his own musical instruments, such as a bamboo flute. Music would be one of his lifelong interests. The 1929 crisis shook the family business and so the older sons had to look for jobs to help with the household income. His brother Nacif graduated first in pharmacy at the Odontology and Pharmacy School of Ribeirão Preto and was a great incentive for Edison's career. After finishing his studies in Piracicaba, graduating in teaching, Edison began teaching mathematics, physics and music, as early as 1937, at the O Piracicabano Educational Institute. Edison was also a violinist with the Piracicaba Symphony Orchestra, having composed pieces for piano, violin and string quartet. == Career == Among the teachers Edison had in high school was Professor Francisco Mariano da Costa, who taught mathematics and, noticing Edison's aptitude for mathematics, encouraged him to study mathematics in São Paulo. He participated in the admission contest of the University of São Paulo and obtained a degree in mathematics in 1941. In 1942 he moved to the São Paulo capital, where he became assistant of the chair of Mathematical Analysis and then, in 1945, assistant of the chair of Superior Analysis, under the direction of Professor André Weil. In 1945 he married Yvone Farah, a German descendant, with whom he had three sons: Cláudio, Flavio (both architects) and Sergio (mechanical engineer). In 1950, Edison obtained his doctorate in mathematics with the thesis "Sobre a Medida de Lebesgue", under the guidance of Omar Catunda. In 1954 he was elevated to the title of full professor in Higher Analysis, where he published the paper "Algumas proposições equivalentes ao Axioma da Escolha", a pioneering work in Brazil in the area of the Axiomatic Set Theory. He taught at the Pontifical Catholic University of São Paulo (PUC) from 1942 to 1954. From 1970, when the Institute of Mathematics and Statistics was created, Edison worked there for another ten years. He taught several specialized courses at the Institute of Mathematical Research and at the School of Philosophy, Sciences and Letters. His areas of expertise and interest were Set Theory, General Topology, Measurement and Integration Theory, and Functional Analysis. He published about 20 articles in renowned journals in the field of mathematics and published three reference books. He advised several students at the University of São Paulo and at other universities in the country. One of his most notable mentors was the mathematician Newton da Costa. He played a fundamental role in the formation of groups dedicated to Logic and the Foundations of Mathematics at USP and the University of Campinas. He retired from the University of São Paulo in 1980, and taught for a few more years at PUC, in São Paulo. == Death == Edison died in the São Paulo capital on April 14, 2006, at the age of 91. == See also == University of São Paulo Pontifical Catholic University of São Paulo State University of Campinas Newton da Costa Chaim Samuel Hönig Lebesgue measure == References == == Bibliography == Magalhães, Luiz Eduardo (2015). Humanistas e Cientistas do Brasil. São Paulo: Edusp. ISBN 9788531415296. Nobre, Sergio; Berato, Fábio; Saraiva, Luis (2011). Anais/Actas do 6o Encontro Luso-Brasileiro de História da Matemática. Natal: Sociedade Brasileira de História da Matemática. ISBN 978-85-89097-67-3. Trivizoli, Lucieli M. (2008). Sociedade de matemática de São Paulo: um estudo histórico-institucional (PDF) (Thesis). Universidade Estadual Paulista (UNESP).
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Wikipedia:Edith de Leeuw#0
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Edith Desiree de Leeuw (born April 12, 1962) is a Dutch psychologist, statistician, research methodologist, and professor in survey methodology and survey quality, at the University of Utrecht. She is known for her work in the field of survey research. == Biography == Born in Amsterdam, De Leeuw attended the Lely Lyceum in Amsterdam. She obtained her BA in psychology in 1975 at the University of Amsterdam, where in 1982 she also obtained her MA in psychology. In 1992 she obtained her PhD in the social and cultural sciences at the VU University Amsterdam under Hans van der Zouwen and Don Mellenbergh with the thesis, entitled "Data quality in mail, telephone and face to face surveys." De Leeuw started her academic career as Assistant Coordinator at the SISWO institute, Research Institute for Social and Economic Sciences in 1981. In 1983–84 she was assistant professor of psychology at the University of Utrecht. At the University of Amsterdam she started as assistant professor of psychology in 1983, and associate professor of education in 1985. From 1988 to 1991 she was research fellow for the Netherlands Organisation for Scientific Research, and from 1991 to 1995 Senior Research Fellow at the VU University Amsterdam. In 1999 she moved back to the University of Utrecht as Senior lecturer Methods & Statistics, and was appointed full professor in survey methodology and survey quality in 2009. In 1987 De Leeuw had received a Fulbright scholarship to study at the Social and Economic Sciences Research Center of Washington State University. She was also visiting scholar at University of California, Los Angeles, research fellow at the Inter Universities Joint Institute for Psychometrics and Socio Metrics (IOPS) in the Netherlands, and a visiting fellow at the International University of Surrey. She is associate editor of the Journal of Official Statistics since 2000. She came into prominence as assistant to Wim T. Schippers in the National Science Quiz, where she participated from 1994 to 2002. == Recognition == De Leeuw won the outstanding service award of the European Survey Research Association in 2017. == Selected publications == de Leeuw, Edith Desiree. Data Quality in Mail, Telephone and Face to Face Surveys. TT Publikaties, 1992. de Leeuw, Edith Desirée, Joop J. Hox and Don A. Dillman, eds. International handbook of survey methodology. Taylor & Francis, 2008. Articles, a selection De Leeuw, Edith D., and Johannes Van der Zouwen. "Data quality in telephone and face to face surveys: a comparative meta-analysis." Telephone survey methodology (1988): 283–299. Hox, Joop J., and Edith D. De Leeuw. "A comparison of nonresponse in mail, telephone, and face-to-face surveys." Quality and Quantity 28.4 (1994): 329–344. Deeg, Dorly JH, et al. "Attrition in the Longitudinal Aging Study Amsterdam: The effect of differential inclusion in side studies." Journal of clinical epidemiology 55.4 (2002): 319–328. De Leeuw, Edith D., and W. de Heer. "Trends in household survey nonresponse: A longitudinal and international comparison." in: Survey Nonresponse, Groves, R.M. et al. (eds.), (2002): 41–54. De Leeuw, Edith D. "To mix or not to mix data collection modes in surveys." Journal of Official Statistics. Stockholm – 21.2 (2005): 233. == References == == External links == prof. dr. Edith de Leeuw at University of Utrecht Edith de Leeuw homepage
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Wikipedia:Edmonds matrix#0
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In graph theory, the Edmonds matrix A {\displaystyle A} of a balanced bipartite graph G = ( U , V , E ) {\displaystyle G=(U,V,E)} with sets of vertices U = { u 1 , u 2 , … , u n } {\displaystyle U=\{u_{1},u_{2},\dots ,u_{n}\}} and V = { v 1 , v 2 , … , v n } {\displaystyle V=\{v_{1},v_{2},\dots ,v_{n}\}} is defined by A i j = { x i j ( u i , v j ) ∈ E 0 ( u i , v j ) ∉ E {\displaystyle A_{ij}=\left\{{\begin{array}{ll}x_{ij}&(u_{i},v_{j})\in E\\0&(u_{i},v_{j})\notin E\end{array}}\right.} where the xij are indeterminates. One application of the Edmonds matrix of a bipartite graph is that the graph admits a perfect matching if and only if the polynomial det(A) in the xij is not identically zero. Furthermore, the number of perfect matchings is equal to the number of monomials in the polynomial det(A), and is also equal to the permanent of A {\displaystyle A} . In addition, the rank of A {\displaystyle A} is equal to the maximum matching size of G {\displaystyle G} . The Edmonds matrix is named after Jack Edmonds. The Tutte matrix is a generalisation to non-bipartite graphs. == References == R. Motwani, P. Raghavan (1995). Randomized Algorithms. Cambridge University Press. p. 167. ISBN 9780521474658. Allen B. Tucker (2004). Computer Science Handbook. CRC Press. p. 12.19. ISBN 1-58488-360-X.
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Wikipedia:Edmund Alfred Cornish#0
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Edmund Alfred Cornish DSc, FAA (7 January 1909 – 31 January 1973) was one of Australia's eminent mathematicians and statisticians. He was appointed an (inaugural) Foundation Fellow of the Australian Academy of Science (in 1954). He worked as the Officer-in-charge in Mathematical Statistics section of Commonwealth Scientific and Industrial Research Organisation in Adelaide from 1941. In 1954 he became the Chief of this division and served till his death in 1973. == References == == External links == Cornish, Edmund Alfred (1909–1973) – Lectures on Mathematical Statistics, 1945–1946 in Adelaide University Library
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Wikipedia:Edmund Hlawka#0
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Edmund Hlawka (5 November 1916, Bruck an der Mur, Styria – 19 February 2009) was an Austrian mathematician. He was a leading number theorist. Hlawka did most of his work at the Vienna University of Technology. He was also a visiting professor at Princeton University and the Sorbonne. Hlawka died on 19 February 2009 in Vienna. == Education and career == Hlawka studied at the University of Vienna from 1934 to 1938, when he gained his doctorate under Nikolaus Hofreiter. Among his PhD students were Rainer Burkard, later to become president of the Austrian Society for Operations Research, graph theorist Gert Sabidussi, Cole Prize winner Wolfgang M. Schmidt, Walter Knödel who became one of the first German computer science professors, and Hermann Maurer, also a computer scientist. Through these and other students, Hlawka has nearly 1500 academic descendants. Hlawka was awarded the Decoration for Services to the Republic of Austria in 2007. == Honours and awards == Decoration for Science and Art (Austria, 1963) City of Vienna Prize for the Humanities (1969) Decoration for Services to the Republic of Austria, Grand Decoration of Honour in Gold with Star (2007); Grand Decoration of Honour in Gold (1987) Wilhelm Exner Medal (1982). Joseph Johann Ritter von Prechtl Medal (1989) Erwin Schrödinger Prize == See also == Minkowski–Hlawka theorem Koksma–Hlawka inequality 10763 Hlawka, an asteroid named after Edmund Hlawka == References == Schmidt, Wolfgang M. (2009). "Edmund Hlawka (1916–2009)". Acta Arithmetica. 139 (4): 303–320. Bibcode:2009AcAri.139..303S. doi:10.4064/aa139-4-1.
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Wikipedia:Edmund Husserl#0
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Edmund Gustav Albrecht Husserl ( HUUSS-url, US also HUUSS-ər-əl; Austrian German: [ˈɛdmʊnd ˈhʊsɐl]; 8 April 1859 – 27 April 1938) was an Austrian-German philosopher and mathematician who established the school of phenomenology. In his early work, he elaborated critiques of historicism and of psychologism in logic based on analyses of intentionality. In his mature work, he sought to develop a systematic foundational science based on the so-called phenomenological reduction. Arguing that transcendental consciousness sets the limits of all possible knowledge, Husserl redefined phenomenology as a transcendental-idealist philosophy. Husserl's thought profoundly influenced 20th-century philosophy, and he remains a notable figure in contemporary philosophy and beyond. Husserl studied mathematics, taught by Karl Weierstrass and Leo Königsberger, and philosophy taught by Franz Brentano and Carl Stumpf. He taught philosophy as a Privatdozent at Halle from 1887, then as professor, first at Göttingen from 1901, then at Freiburg from 1916 until he retired in 1928, after which he remained highly productive. In 1933, under racial laws of the Nazi Party, Husserl was banned from using the library of the University of Freiburg due to his Jewish family background and months later resigned from the Deutsche Akademie. Following an illness, he died in Freiburg in 1938. == Life and career == === Youth and education === Husserl was born in 1859 in Proßnitz in the Margraviate of Moravia in the Austrian Empire (today Prostějov in the Czech Republic). He was born into a Jewish family, the second of four children. His father was a milliner. His childhood was spent in Prostějov, where he attended the secular primary school. Then Husserl traveled to Vienna to study at the Realgymnasium there, followed next by the Staatsgymnasium in Olmütz. At the University of Leipzig from 1876 to 1878, Husserl studied mathematics, physics, and astronomy. At Leipzig, he was inspired by philosophy lectures given by Wilhelm Wundt, one of the founders of modern psychology. Then he moved to the Frederick William University of Berlin (the present-day Humboldt University of Berlin) in 1878, where he continued his study of mathematics under Leopold Kronecker and Karl Weierstrass. In Berlin, he found a mentor in Tomáš Garrigue Masaryk (then a former philosophy student of Franz Brentano and later the first president of Czechoslovakia) and attended Friedrich Paulsen's philosophy lectures. In 1881, he left for the University of Vienna to complete his mathematics studies under the supervision of Leo Königsberger (a former student of Weierstrass). At Vienna in 1883, he obtained his PhD with the work Beiträge zur Variationsrechnung (Contributions to the Calculus of variations). Evidently, as a result of his becoming familiar with the New Testament during his twenties, Husserl asked to be baptized into the Lutheran Church in 1886. Husserl's father, Adolf, had died in 1884. Herbert Spiegelberg writes, "While outward religious practice never entered his life any more than it did that of most academic scholars of the time, his mind remained open for the religious phenomenon as for any other genuine experience." At times Husserl saw his goal as one of moral "renewal". Although a steadfast proponent of a radical and rational autonomy in all things, Husserl could also speak "about his vocation and even about his mission under God's will to find new ways for philosophy and science," observes Spiegelberg. Following his PhD in mathematics, Husserl returned to Berlin to work as the assistant to Karl Weierstrass. Yet already Husserl had felt the desire to pursue philosophy. Then Weierstrass became very ill. Husserl became free to return to Vienna, where, after serving a short military duty, he devoted his attention to philosophy. In 1884, at the University of Vienna he attended the lectures of Franz Brentano on philosophy and philosophical psychology. Brentano introduced him to the writings of Bernard Bolzano, Hermann Lotze, J. Stuart Mill, and David Hume. Husserl was so impressed by Brentano that he decided to dedicate his life to philosophy; indeed, Franz Brentano is often credited as being his most important influence, e.g., with regard to intentionality. Following academic advice, two years later in 1886 Husserl followed Carl Stumpf, a former student of Brentano, to the University of Halle, seeking to obtain his habilitation which would qualify him to teach at the university level. There, under Stumpf's supervision, he wrote his habilitation thesis, Über den Begriff der Zahl (On the Concept of Number), in 1887, which would serve later as the basis for his first important work, Philosophie der Arithmetik (1891). In 1887, Husserl married Malvine Steinschneider, a union that would last over fifty years. In 1892, their daughter Elizabeth was born, in 1893 their son Gerhart, and in 1894 their son Wolfgang. Elizabeth would marry in 1922, and Gerhart in 1923; Wolfgang, however, became a casualty of the First World War. Gerhart would become a philosopher of law, contributing to the subject of comparative law, teaching in the United States and after the war in Austria. === Professor of philosophy === Following his marriage, Husserl began his long teaching career in philosophy. He started in 1887 as a Privatdozent at the University of Halle. In 1891, he published his Philosophie der Arithmetik. Psychologische und logische Untersuchungen which, drawing on his prior studies in mathematics and philosophy, proposed a psychological context as the basis of mathematics. It drew the adverse notice of Gottlob Frege, who criticized its psychologism. In 1901, Husserl with his family moved to the University of Göttingen, where he taught as extraordinarius professor. Just prior to this, a major work of his, Logische Untersuchungen (Halle, 1900–1901), was published. Volume One contains seasoned reflections on "pure logic" in which he carefully refutes "psychologism". This work was well received and became the subject of a seminar given by Wilhelm Dilthey; Husserl in 1905 traveled to Berlin to visit Dilthey. Two years later, in Italy, he paid a visit to Franz Brentano, his inspiring old teacher, and to the mathematician Constantin Carathéodory. Kant and Descartes were also now influencing his thought. In 1910, he became joint editor of the journal Logos. During this period, Husserl had delivered lectures on internal time consciousness, which several decades later his former students Edith Stein and Martin Heidegger edited for publication. In 1912, in Freiburg, the journal Jahrbuch für Philosophie und Phänomenologische Forschung ("Yearbook for Philosophy and Phenomenological Research") was founded by Husserl and his school, which published articles of their phenomenological movement from 1913 to 1930. His important work Ideen was published in its first issue (Vol. 1, Issue 1, 1913). Before beginning Ideen, Husserl's thought had reached the stage where "each subject is 'presented' to itself, and to each all others are 'presentiated' (Vergegenwärtigung), not as parts of nature but as pure consciousness". Ideen advanced his transition to a "transcendental interpretation" of phenomenology, a view later criticized by, among others, Jean-Paul Sartre. In Ideen Paul Ricœur sees the development of Husserl's thought as leading "from the psychological cogito to the transcendental cogito". As phenomenology further evolves, it leads (when viewed from another vantage point in Husserl's 'labyrinth') to "transcendental subjectivity". Also in Ideen Husserl explicitly elaborates the phenomenological and eidetic reductions. Ivan Ilyin and Karl Jaspers visited Husserl at Göttingen. In October 1914, both his sons were sent to fight on the Western Front of World War I, and the following year, one of them, Wolfgang Husserl, was badly injured. On 8 March 1916, on the battlefield of Verdun, Wolfgang was killed in action. The next year, his other son, Gerhart Husserl was wounded in the war but survived. His own mother, Julia, died. In November 1917, one of his outstanding students and later a noted philosophy professor in his own right, Adolf Reinach, was killed in the war while serving in Flanders. Husserl had transferred in 1916 to the University of Freiburg (in Freiburg im Breisgau) where he continued bringing his work in philosophy to fruition, now as a full professor. Edith Stein served as his personal assistant during his first few years in Freiburg, followed later by Martin Heidegger from 1920 to 1923. The mathematician Hermann Weyl began corresponding with him in 1918. Husserl gave four lectures on the phenomenological method at University College London in 1922. The University of Berlin in 1923 called on him to relocate there, but he declined the offer. In 1926, Heidegger dedicated his book Sein und Zeit (Being and Time) to him "in grateful respect and friendship." Husserl remained in his professorship at Freiburg until he requested retirement, teaching his last class on 25 July 1928. A Festschrift to celebrate his seventieth birthday was presented to him on 8 April 1929. Despite retirement, Husserl gave several notable lectures. The first, at Paris in 1929, led to Méditations cartésiennes (Paris 1931). Husserl here reviews the phenomenological epoché (or phenomenological reduction), presented earlier in his pivotal Ideen (1913), in terms of a further reduction of experience to what he calls a 'sphere of ownness.' From within this sphere, which Husserl enacts to show the impossibility of solipsism, the transcendental ego finds itself always already paired with the lived body of another ego, another monad. This 'a priori' interconnection of bodies, given in perception, is what founds the interconnection of consciousnesses known as transcendental intersubjectivity, which Husserl would go on to describe at length in volumes of unpublished writings. There has been a debate over whether or not Husserl's description of ownness and its movement into intersubjectivity is sufficient to reject the charge of solipsism, to which Descartes, for example, was subject. One argument against Husserl's description works this way: instead of infinity and the Deity being the ego's gateway to the Other, as in Descartes, Husserl's ego in the Cartesian Meditations itself becomes transcendent. It remains, however, alone (unconnected). Only the ego's grasp "by analogy" of the Other (e.g., by conjectural reciprocity) allows the possibility for an 'objective' intersubjectivity, and hence for community. In 1933, the racial laws of the new National Socialist German Workers Party were enacted. On 6 April Husserl was banned from using the library at the University of Freiburg, or any other academic library; the following week, after a public outcry, he was reinstated. Yet his colleague Heidegger was elected Rector of the university on 21–22 April, and joined the Nazi Party. By contrast, in July Husserl resigned from the Deutsche Akademie. Later, Husserl lectured at Prague in 1935 and Vienna in 1936, which resulted in a very differently styled work that, while innovative, is no less problematic: Die Krisis (Belgrade 1936). Husserl describes here the cultural crisis gripping Europe, then approaches a philosophy of history, discussing Galileo, Descartes, several British philosophers, and Kant. The apolitical Husserl before had specifically avoided such historical discussions, pointedly preferring to go directly to an investigation of consciousness. Merleau-Ponty and others question whether Husserl here does not undercut his own position, in that Husserl had attacked in principle historicism, while specifically designing his phenomenology to be rigorous enough to transcend the limits of history. On the contrary, Husserl may be indicating here that historical traditions are merely features given to the pure ego's intuition, like any other. A longer section follows on the "lifeworld" [Lebenswelt], one not observed by the objective logic of science, but a world seen through subjective experience. Yet a problem arises similar to that dealing with 'history' above, a chicken-and-egg problem. Does the lifeworld contextualize and thus compromise the gaze of the pure ego, or does the phenomenological method nonetheless raise the ego up transcendent? These last writings presented the fruits of his professional life. Since his university retirement, Husserl had "worked at a tremendous pace, producing several major works." After suffering a fall in the autumn of 1937, the philosopher became ill with pleurisy. Edmund Husserl died in Freiburg on 27 April 1938, having just turned 79. His wife, Malvine, survived him. Eugen Fink, his research assistant, delivered his eulogy. Gerhard Ritter was the only Freiburg faculty member to attend the funeral, as an anti-Nazi protest. === Heidegger and the Nazi era === Husserl was rumoured to have been denied the use of the library at Freiburg as a result of the anti-Jewish legislation of April 1933. Relatedly, among other disabilities, Husserl was unable to publish his works in Nazi Germany [see above footnote to Die Krisis (1936)]. It was also rumoured that his former pupil Martin Heidegger informed Husserl that he was discharged, but it was actually the previous rector. Apparently, Husserl and Heidegger had moved apart during the 1920s, which became clearer after 1928 when Husserl retired and Heidegger succeeded to his university chair. In the summer of 1929 Husserl had studied carefully selected writings of Heidegger, coming to the conclusion that on several of their key positions they differed: e.g., Heidegger substituted Dasein ["Being-there"] for the pure ego, thus transforming phenomenology into an anthropology, a type of psychologism strongly disfavored by Husserl. Such observations of Heidegger, along with a critique of Max Scheler, were put into a lecture Husserl gave to various Kant Societies in Frankfurt, Berlin, and Halle during 1931 entitled Phänomenologie und Anthropologie. In the wartime 1941 edition of Heidegger's primary work, Being and Time (Sein und Zeit, first published in 1927), the original dedication to Husserl was removed. This was not due to a negation of the relationship between the two philosophers, however, but rather was the result of a suggested censorship by Heidegger's publisher who feared that the book might otherwise be banned by the Nazi regime. The dedication can still be found in a footnote on page 38, thanking Husserl for his guidance and generosity. Husserl had died three years earlier. In post-war editions of Sein und Zeit the dedication to Husserl is restored. The complex, troubled, and sundered philosophical relationship between Husserl and Heidegger has been widely discussed. On 4 May 1933, Professor Edmund Husserl addressed the recent regime change in Germany and its consequences:The future alone will judge which was the true Germany in 1933, and who were the true Germans—those who subscribe to the more or less materialistic-mythical racial prejudices of the day, or those Germans pure in heart and mind, heirs to the great Germans of the past whose tradition they revere and perpetuate.After his death, Husserl's manuscripts, amounting to approximately 40,000 pages of "Gabelsberger" stenography and his complete research library, were in 1939 smuggled to the Catholic University of Leuven in Belgium by the Franciscan priest Herman Van Breda. There they were deposited at Leuven to form the Husserl-Archives of the Higher Institute of Philosophy. Much of the material in his research manuscripts has since been published in the Husserliana critical edition series. == Development of his thought == === Several early themes === In his first works, Husserl combined mathematics, psychology, and philosophy with the goal of providing a sound foundation for mathematics. He analyzed the psychological process needed to obtain the concept of number and then built up a theory on this analysis. He used methods and concepts taken from his teachers. From Weierstrass, he derived the idea of generating the concept of number by counting a certain collection of objects. From Brentano and Stumpf, he took the distinction between proper and improper presenting.: 159 In an example, Husserl explained this in the following way: if someone is standing in front of a house, they have a proper, direct presentation of that house, but if they are looking for it and ask for directions, then these directions (e.g. the house on the corner of this and that street) are an indirect, improper presentation. In other words, the person can have a proper presentation of an object if it is actually present, and an improper (or symbolic, as Husserl also calls it) one if they only can indicate that object through signs, symbols, etc. Husserl's Logical Investigations (1900–1901) is considered the starting point for the formal theory of wholes and their parts known as mereology. Another important element that Husserl took over from Brentano was intentionality, the notion that the main characteristic of consciousness is that it is always intentional. While often simplistically summarised as "aboutness" or the relationship between mental acts and the external world, Brentano defined it as the main characteristic of mental phenomena, by which they could be distinguished from physical phenomena. Every mental phenomenon, every psychological act, has a content, is directed at an object (the intentional object). Every belief, desire, etc. has an object that it is about: the believed, the wanted. Brentano used the expression "intentional inexistence" to indicate the status of the objects of thought in the mind. The property of being intentional, of having an intentional object, was the key feature to distinguish mental phenomena and physical phenomena, because physical phenomena lack intentionality altogether. === The elaboration of phenomenology === Some years after the 1900–1901 publication of his main work, the Logische Untersuchungen (Logical Investigations), Husserl made some key conceptual elaborations which led him to assert that to study the structure of consciousness, one would have to distinguish between the act of consciousness and the phenomena at which it is directed (the objects as intended). Knowledge of essences would only be possible by "bracketing" all assumptions about the existence of an external world. This procedure he called "epoché". These new concepts prompted the publication of the Ideen (Ideas) in 1913, in which they were at first incorporated, and a plan for a second edition of the Logische Untersuchungen. From the Ideen onward, Husserl concentrated on the ideal, essential structures of consciousness. The metaphysical problem of establishing the reality of what people perceive, as distinct from the perceiving subject, was of little interest to Husserl in spite of his being a transcendental idealist. Husserl proposed that the world of objects—and of ways in which people direct themselves toward and perceive those objects—is normally conceived of in what he called the "natural attitude", which is characterized by a belief that objects exist distinct from the perceiving subject and exhibit properties that people see as emanating from them (this attitude is also called physicalist objectivism). Husserl proposed a radical new phenomenological way of looking at objects by examining how people, in their many ways of being intentionally directed toward them, actually "constitute" them (to be distinguished from materially creating objects or objects merely being figments of the imagination); in the Phenomenological standpoint, the object ceases to be something simply "external" and ceases to be seen as providing indicators about what it is, and becomes a grouping of perceptual and functional aspects that imply one another under the idea of a particular object or "type". The notion of objects as real is not expelled by phenomenology, but "bracketed" as a way in which people regard objects—instead of a feature that inheres in an object's essence, founded in the relation between the object and the perceiver. To better understand the world of appearances and objects, phenomenology attempts to identify the invariant features of how objects are perceived and pushes attributions of reality into their role as an attribution about the things people perceive (or an assumption underlying how people perceive objects). The major dividing line in Husserl's thought is the turn to transcendental idealism. In a later period, Husserl began to wrestle with the complicated issues of intersubjectivity, specifically, how communication about an object can be assumed to refer to the same ideal entity (Cartesian Meditations, Meditation V). Husserl tries new methods of bringing his readers to understand the importance of phenomenology to scientific inquiry (and specifically to psychology) and what it means to "bracket" the natural attitude. The Crisis of the European Sciences is Husserl's unfinished work that deals most directly with these issues. In it, Husserl for the first time attempts a historical overview of the development of Western philosophy and science, emphasizing the challenges presented by their increasingly one-sided empirical and naturalistic orientation. Husserl declares that mental and spiritual reality possess their own reality independent of any physical basis, and that a science of the mind ('Geisteswissenschaft') must be established on as scientific a foundation as the natural sciences have managed: "It is my conviction that intentional phenomenology has for the first time made spirit as spirit the field of systematic scientific experience, thus effecting a total transformation of the task of knowledge." == Husserl's thought == Husserl's thought is revolutionary in several ways, most notably in the distinction between "natural" and "phenomenological" modes of understanding. In the former, sense-perception in correspondence with the material realm constitutes the known reality, and understanding is premised on the accuracy of the perception and the objective knowability of what is called the "real world". Phenomenological understanding strives to be rigorously "presuppositionless" by means of what Husserl calls "phenomenological reduction". This reduction is not conditioned but rather transcendental: in Husserl's terms, pure consciousness of absolute Being. In Husserl's work, consciousness of any given thing calls for discerning its meaning as an "intentional object". Such an object does not simply strike the senses, to be interpreted or misinterpreted by mental reason; it has already been selected and grasped, grasping being an etymological connotation, of percipere, the root of "perceive". === Meaning and object === From Logical Investigations (1900/1901) to Experience and Judgment (published in 1939), Husserl expressed clearly the difference between meaning and object. He identified several different kinds of names. For example, there are names that have the role of properties that uniquely identify an object. Each of these names expresses a meaning and designates the same object. Examples of this are "the victor in Jena" and "the loser in Waterloo", or "the equilateral triangle" and "the equiangular triangle"; in both cases, both names express different meanings, but designate the same object. There are names which have no meaning, but have the role of designating an object: "Aristotle", "Socrates", and so on. Finally, there are names which designate a variety of objects. These are called "universal names"; their meaning is a "concept" and refers to a series of objects (the extension of the concept). The way people know sensible objects is called "sensible intuition". Husserl also identifies a series of "formal words" which are necessary to form sentences and have no sensible correlates. Examples of formal words are "a", "the", "more than", "over", "under", "two", "group", and so on. Every sentence must contain formal words to designate what Husserl calls "formal categories". There are two kinds of categories: meaning categories and formal-ontological categories. Meaning categories relate judgments; they include forms of conjunction, disjunction, forms of plural, among others. Formal-ontological categories relate objects and include notions such as set, cardinal number, ordinal number, part and whole, relation, and so on. The way people know these categories is through a faculty of understanding called "categorial intuition". Through sensible intuition, consciousness constitutes what Husserl calls a "situation of affairs" (Sachlage). It is a passive constitution where objects themselves are presented. To this situation of affairs, through categorial intuition, people are able to constitute a "state of affairs" (Sachverhalt). One situation of affairs, through objective acts of consciousness (acts of constituting categorially) can serve as the basis for constituting multiple states of affairs. For example, suppose a and b are two sensible objects in a certain situation of affairs. It can be used as the basis to say, "a<b" and "b>a", two judgments which designate the same state of affairs. For Husserl, a sentence has a proposition or judgment as its meaning, and refers to a state of affairs which has a situation of affairs as a reference base.: 35 === Formal and regional ontology === Husserl sees ontology as a science of essences. Sciences of essences are contrasted with factual sciences: the former are knowable a priori and provide the foundation for the later, which are knowable a posteriori. Ontology as a science of essences is not interested in actual facts, but in the essences themselves, whether they have instances or not. Husserl distinguishes between formal ontology, which investigates the essence of objectivity in general, and regional ontologies, which study regional essences that are shared by all entities belonging to the region. Regions correspond to the highest genera of concrete entities: material nature, personal consciousness, and interpersonal spirit. Husserl's method for studying ontology and sciences of essence in general is called eidetic variation. It involves imagining an object of the kind under investigation and varying its features. The changed feature is inessential to this kind if the object can survive its change, otherwise it belongs to the kind's essence. For example, a triangle remains a triangle if one of its sides is extended, but it ceases to be a triangle if a fourth side is added. Regional ontology involves applying this method to the essences corresponding to the highest genera. === Philosophy of logic and mathematics === Husserl believed that truth-in-itself has as ontological correlate being-in-itself, just as meaning categories have formal-ontological categories as correlates. Logic is a formal theory of judgment, that studies the formal a priori relations among judgments using meaning categories. Mathematics, on the other hand, is formal ontology; it studies all the possible forms of being (of objects). Hence, for both logic and mathematics, the different formal categories are the objects of study, not the sensible objects themselves. The problem with the psychological approach to mathematics and logic is that it fails to account for the fact that this approach is about formal categories, and not simply about abstractions from sensibility alone. The reason why sensible objects are not dealt with in mathematics is because of another faculty of understanding called "categorial abstraction." Through this faculty, people are able to get rid of sensible components of judgments, and just focus on formal categories themselves. Thanks to "eidetic reduction" (or "essential intuition"), people are able to grasp the possibility, impossibility, necessity, and contingency among concepts and among formal categories. Categorial intuition, along with categorial abstraction and eidetic reduction, are the basis for logical and mathematical knowledge. Husserl criticized the logicians of his day for not focusing on the relation between subjective processes that offer objective knowledge of pure logic. All subjective activities of consciousness need an ideal correlate, and objective logic (constituted noematically), as it is constituted by consciousness, needs a noetic correlate (the subjective activities of consciousness). Husserl stated that logic has three strata, each further away from consciousness and psychology than those that precede it. The first stratum is what Husserl called a "morphology of meanings" concerning a priori ways to relate judgments to make them meaningful. In this stratum, people elaborate a "pure grammar" or a logical syntax, and he would call its rules "laws to prevent non-sense", which would be similar to what logic calls today "formation rules". Mathematics, as logic's ontological correlate, also has a similar stratum, a "morphology of formal-ontological categories". The second stratum would be called by Husserl "logic of consequence" or the "logic of non-contradiction" which explores all possible forms of true judgments. He includes here syllogistic classic logic, propositional logic, and that of predicates. This is a semantic stratum, and the rules of this stratum would be the "laws to avoid counter-sense" or "laws to prevent contradiction". They are very similar to today's logic "transformation rules". Mathematics also has a similar stratum which is based, among others, on the pure theory of pluralities and a pure theory of numbers. They provide a science of the conditions of possibility of any theory whatsoever. Husserl also talked about what he called "logic of truth," which consists of the formal laws of possible truth and its modalities, and precedes the third logical stratum. The third stratum is metalogical, what he called a "theory of all possible forms of theories." It explores all possible theories in an a priori fashion, rather than the possibility of theory in general. Theories of possible relations between pure forms of theories could be established; these logical relations could, in turn, be investigated using deduction. The logician is free to see the extension of this deductive, theoretical sphere of pure logic. The ontological correlate to the third stratum is the "theory of manifolds". In formal ontology, it is a free investigation where a mathematician can assign several meanings to several symbols, and all their possible valid deductions in a general and indeterminate manner. It is, properly speaking, the most universal mathematics of all. Through the positing of certain indeterminate objects (formal-ontological categories) as well as any combination of mathematical axioms, mathematicians can explore the apodeictic connections between them, as long as consistency is preserved. According to Husserl, this view of logic and mathematics accounted for the objectivity of a series of mathematical developments of his time, such as n-dimensional manifolds (both Euclidean and non-Euclidean), Hermann Grassmann's theory of extensions, William Rowan Hamilton's Hamiltonians, Sophus Lie's theory of transformation groups, and Cantor's set theory. Jacob Klein was one student of Husserl who pursued this line of inquiry, seeking to "desedimentize" mathematics and the mathematical sciences. == Husserl and psychologism == === Philosophy of arithmetic and Frege === After obtaining his PhD in mathematics, Husserl began analyzing the foundations of mathematics from a psychological point of view. In his On the Concept of Number (1887) and in his Philosophy of Arithmetic (1891), Husserl sought, by employing Brentano's descriptive psychology, to define the natural numbers in a way that advanced the methods and techniques of Karl Weierstrass, Richard Dedekind, Georg Cantor, Gottlob Frege, and other contemporary mathematicians. Later, in the first volume of his Logical Investigations, the Prolegomena of Pure Logic, Husserl, while attacking the psychologistic point of view in logic and mathematics, also appears to reject much of his early work, although the forms of psychologism analysed and refuted in the Prolegomena did not apply directly to his Philosophy of Arithmetic. Some scholars question whether Frege's negative review of the Philosophy of Arithmetic helped turn Husserl towards modern Platonism, but he had already discovered the work of Bernard Bolzano independently around 1890/91. In his Logical Investigations, Husserl explicitly mentioned Bolzano, G. W. Leibniz and Hermann Lotze as inspirations for his newer position. Husserl's review of Ernst Schröder, published before Frege's landmark 1892 article, clearly distinguishes sense from reference; thus, Husserl's notions of noema and object also arose independently. Likewise, in his criticism of Frege in the Philosophy of Arithmetic, Husserl remarks on the distinction between the content and the extension of a concept. Moreover, the distinction between the subjective mental act, namely the content of a concept, and the (external) object, was developed independently by Brentano and his school, and may have surfaced as early as Brentano's 1870s lectures on logic. Scholars such as J. N. Mohanty, Claire Ortiz Hill, and Guillermo E. Rosado Haddock, among others, have argued that Husserl's so-called change from psychologism to Platonism came about independently of Frege's review.: 253–262 For example, the review falsely accuses Husserl of subjectivizing everything, so that no objectivity is possible, and falsely attributes to him a notion of abstraction whereby objects disappear until all that remains are numbers as mere ghosts. Contrary to what Frege states, in Husserl's Philosophy of Arithmetic, there are already two different kinds of representations: subjective and objective. Moreover, objectivity is clearly defined in that work. Frege's attack seems to be directed at certain foundational doctrines then current in Weierstrass's Berlin School, of which Husserl and Cantor cannot be said to be orthodox representatives. Furthermore, various sources indicate that Husserl changed his mind about psychologism as early as 1890, a year before he published the Philosophy of Arithmetic. Husserl stated that by the time he published that book, he had already changed his mind—that he had doubts about psychologism from the very outset. He attributed this change of mind to his reading of Leibniz, Bolzano, Lotze, and David Hume. Husserl makes no mention of Frege as a decisive factor in this change. In his Logical Investigations, Husserl mentions Frege only twice, once in a footnote to point out that he had retracted three pages of his criticism of Frege's The Foundations of Arithmetic, and again to question Frege's use of the word Bedeutung to designate "reference" rather than "meaning" (sense). In a letter dated 24 May 1891, Frege thanked Husserl for sending him a copy of the Philosophy of Arithmetic and Husserl's review of Ernst Schröder's Vorlesungen über die Algebra der Logik. In the same letter, Frege used the review of Schröder's book to analyze Husserl's notion of the sense of reference of concept words. Hence Frege recognized, as early as 1891, that Husserl distinguished between sense and reference. Consequently, Frege and Husserl independently elaborated a theory of sense and reference before 1891. Commentators argue that Husserl's notion of noema has nothing to do with Frege's notion of sense, because noemata are necessarily fused with noeses, which are the conscious activities of consciousness. Noemata have three different levels: The substratum, which is never presented to consciousness, and is the support of all the properties of the object; The noematic senses, which are the different ways the objects are presented to us; The modalities of being (possible, doubtful, existent, non-existent, absurd, and so on). Consequently, in intentional activities, even non-existent objects can be constituted, and form part of the whole noema. Frege, however, did not conceive of objects as forming parts of senses: If a proper name denotes a non-existent object, it does not have a reference, hence, concepts with no objects have no truth value in arguments. Moreover, Husserl did not maintain that predicates of sentences designate concepts. According to Frege, the reference of a sentence is a truth value; for Husserl, it is a "state of affairs." Frege's notion of "sense" is unrelated to Husserl's noema, while the latter's notions of "meaning" and "object" differ from those of Frege. In detail, Husserl's conception of logic and mathematics differs from that of Frege, who held that arithmetic could be derived from logic. For Husserl, this is not the case: mathematics (with the exception of geometry) is the ontological correlate of logic, and while both fields are related, neither one is strictly reducible to the other. === Husserl's criticism of psychologism === Reacting against authors such as John Stuart Mill, Christoph von Sigwart and his own former teacher Brentano, Husserl criticised their psychologism in mathematics and logic, i.e. their conception of these abstract and a priori sciences as having an essentially empirical foundation and a prescriptive or descriptive nature. According to psychologism, logic would not be an autonomous discipline, but a branch of psychology, either proposing a prescriptive and practical "art" of correct judgement (as Brentano and some of his more orthodox students did) or a description of the factual processes of human thought. Husserl pointed out that the failure of anti-psychologists to defeat psychologism was a result of being unable to distinguish between the foundational, theoretical side of logic, and the applied, practical side. Pure logic does not deal at all with "thoughts" or "judgings" as mental episodes but with a priori laws and conditions for any theory and any judgments whatsoever, conceived as propositions in themselves. "Here 'Judgement' has the same meaning as 'proposition', understood, not as a grammatical, but as an ideal unity of meaning. This is the case with all the distinctions of acts or forms of judgement, which provide the foundations for the laws of pure logic. Categorial, hypothetical, disjunctive, existential judgements, and however else we may call them, in pure logic are not names for classes of judgements, but for ideal forms of propositions." Since "truth-in-itself" has "being-in-itself" as ontological correlate, and since psychologists reduce truth (and hence logic) to empirical psychology, the inevitable consequence is scepticism. Psychologists have not been successful either in showing how induction or psychological processes can justify the absolute certainty of logical principles, such as the principles of identity and non-contradiction. It is therefore futile to base certain logical laws and principles on uncertain processes of the mind. This confusion made by psychologism (and related disciplines such as biologism and anthropologism) can be due to three specific prejudices: The first prejudice is the supposition that logic is somehow normative in nature. Husserl argues that logic is theoretical, i.e., that logic itself proposes a priori laws which are themselves the basis of the normative side of logic. Since mathematics is related to logic, he cites an example from mathematics: a formula like "(a + b)(a – b) = a² – b²" does not offer any insight into how to think mathematically. It just expresses a truth. A proposition that says: "The product of the sum and the difference of a and b should give the difference of the squares of a and b" does express a normative proposition, but this normative statement is based on the theoretical statement "(a + b)(a – b) = a² – b²". For psychologists, the acts of judging, reasoning, deriving, and so on, are all psychological processes. Therefore, it is the role of psychology to provide the foundation of these processes. Husserl states that this effort made by psychologists is a "metábasis eis állo génos" (Ancient Greek: μετάβασις εἰς ἄλλο γένος, lit. 'a transgression to another field').: 344 It is a metábasis because psychology cannot provide any foundations for a priori laws which themselves are the basis for all correct thought. Psychologists have the problem of confusing intentional activities with the object of these activities. It is important to distinguish between the act of judging and the judgment itself, the act of counting and the number itself, and so on. Counting five objects is undeniably a psychological process, but the number 5 is not. Judgments can be true or not true. Psychologists argue that judgments are true because they become "evidently" true to us.: 261 This evidence, a psychological process that "guarantees" truth, is indeed a psychological process. Husserl responds by saying that truth itself, as well as logical laws, always remain valid regardless of psychological "evidence" that they are true. No psychological process can explain the a priori objectivity of these logical truths. From this criticism to psychologism, the distinction between psychological acts and their intentional objects, and the difference between the normative side of logic and the theoretical side, derives from a Platonist conception of logic. This means that logical and mathematical laws should be regarded as being independent of the human mind, and also as an autonomy of meanings. It is essentially the difference between the real (everything subject to time) and the ideal or irreal (everything that is atemporal), such as logical truths, mathematical entities, mathematical truths, and meanings in general. == Influence == David Carr commented on Husserl following in his 1970 dissertation at Yale: "It is well known that Husserl was always disappointed at the tendency of his students to go their own way, to embark upon fundamental revisions of phenomenology rather than engage in the communal task" as originally intended by the radical new science. Notwithstanding, he did attract philosophers to phenomenology. Martin Heidegger is the best known of Husserl's students, the one whom Husserl chose as his successor at Freiburg. Heidegger's magnum opus Being and Time was dedicated to Husserl. They shared their thoughts and worked alongside each other for over a decade at the University of Freiburg, Heidegger being Husserl's assistant during 1920–1923. Heidegger's early work followed his teacher, but with time he began to develop new insights distinctively variant. Husserl became increasingly critical of Heidegger's work, especially in 1929, and included pointed criticism of Heidegger in lectures he gave during 1931. Heidegger, while acknowledging his debt to Husserl, followed a political position offensive and harmful to Husserl after the Nazis came to power in 1933, Husserl being of Jewish origin and Heidegger infamously being then a Nazi proponent. Academic discussion of Husserl and Heidegger is extensive. At Göttingen in 1913, Adolf Reinach "was now Husserl's right hand. He was above all the mediator between Husserl and the students, for he understood extremely well how to deal with other persons, whereas Husserl was pretty much helpless in this respect." He was an original editor of Husserl's new journal, Jahrbuch; one of his works (giving a phenomenological analysis of the law of obligations) appeared in its first issue. Reinach was widely admired and a remarkable teacher. Husserl, in his 1917 obituary, wrote, "He wanted to draw only from the deepest sources, he wanted to produce only work of enduring value. And through his wise restraint, he succeeded in this." Edith Stein was Husserl's student at Göttingen and Freiburg while she wrote her doctoral thesis The Empathy Problem as it Developed Historically and Considered Phenomenologically (1916). She then became his assistant at Freiburg in 1916–18. She later adapted her phenomenology to the modern school of modern Thomism. Ludwig Landgrebe became assistant to Husserl in 1923. From 1939, he collaborated with Eugen Fink at the Husserl-Archives in Leuven. In 1954, he became the leader of the Husserl-Archives. Landgrebe is known as one of Husserl's closest associates, but also for his independent views relating to history, religion, and politics as seen from the viewpoints of existentialist philosophy and metaphysics. Eugen Fink was a close associate of Husserl during the 1920s and 1930s. He wrote the Sixth Cartesian Meditation which Husserl said was the truest expression and continuation of his own work. Fink delivered the eulogy for Husserl in 1938. Roman Ingarden, an early student of Husserl at Freiburg, corresponded with Husserl into the mid-1930s. Ingarden did not accept, however, the later transcendental idealism of Husserl which he thought would lead to relativism. Ingarden has written his work in German and Polish. In his Spór o istnienie świata (Ger.: "Der Streit um die Existenz der Welt", Eng.: "Dispute about existence of the world"), he created his own realistic position, which also helped to spread phenomenology in Poland. Max Scheler met Husserl in Halle in 1901 and found in his phenomenology a methodological breakthrough for his own philosophy. Scheler, who was at Göttingen when Husserl taught there, was one of the original few editors of the journal Jahrbuch für Philosophie und Phänomenologische Forschung (1913). Scheler's work Formalism in Ethics and Nonformal Ethics of Value appeared in the new journal (1913 and 1916) and drew acclaim. The personal relationship between the two men, however, became strained, due to Scheler's legal troubles, and Scheler returned to Munich. Although Scheler later criticised Husserl's idealistic logical approach and proposed instead a "phenomenology of love", he states that he remained "deeply indebted" to Husserl throughout his work. Nicolai Hartmann was once thought to be at the center of phenomenology, but perhaps no longer. In 1921, the prestige of Hartmann, the neo-Kantian, who was Professor of Philosophy at the University of Marburg, was added to the Movement; he "publicly declared his solidarity with the actual work of die Phänomenologie." Yet Hartmann's connections were with Max Scheler and the Munich circle; Husserl himself evidently did not consider him a phenomenologist. His philosophy, however, is said to include an innovative use of the method. Emmanuel Levinas in 1929 gave a presentation at one of Husserl's last seminars in Freiburg. Also that year, he wrote on Husserl's Ideen (1913) a long review published by a French journal. With Gabrielle Peiffer, Levinas translated into French Husserl's Méditations cartésiennes (1931). He was at first impressed with Heidegger and began a book on him, but broke off the project when Heidegger became involved with the Nazis. After the war he wrote on Jewish spirituality; most of his family had been murdered by the Nazis in Lithuania. Levinas then began to write works that would become widely known and admired. Alfred Schutz's Phenomenology of the Social World seeks to rigorously ground Max Weber's interpretive sociology in Husserl's phenomenology. Husserl was impressed by this work and asked Schutz to be his assistant. Jean-Paul Sartre was also largely influenced by Husserl, although he later came to disagree with key points in his analyses. Sartre rejected Husserl's transcendental interpretations begun in his Ideen (1913) and instead followed Heidegger's ontology. Maurice Merleau-Ponty's Phenomenology of Perception is influenced by Edmund Husserl's work on perception, intersubjectivity, intentionality, space, and temporality, including Husserl's theory of retention and protention. Merleau-Ponty's description of 'motor intentionality' and sexuality, for example, retains the important structure of the noetic/noematic correlation of Ideen I, yet further concretizes what it means for Husserl when consciousness particularizes itself into modes of intuition. Merleau-Ponty's most clearly Husserlian work is, perhaps, "the Philosopher and His Shadow." Depending on the interpretation of Husserl's accounts of eidetic intuition, given in Husserl's Phenomenological Psychology and Experience and Judgment, it may be that Merleau-Ponty did not accept the "eidetic reduction" nor the "pure essence" said to result. Merleau-Ponty was the first student to study at the Husserl-archives in Leuven. Gabriel Marcel explicitly rejected existentialism, due to Sartre, but not phenomenology, which has enjoyed a wide following among French Catholics. He appreciated Husserl, Scheler, and (but with apprehension) Heidegger. His expressions like "ontology of sensability" when referring to the body, indicate influence by phenomenological thought. Kurt Gödel is known to have read Cartesian Meditations. He expressed very strong appreciation for Husserl's work, especially with regard to "bracketing" or "epoché". Hermann Weyl's interest in intuitionistic logic and impredicativity appears to have resulted from his reading of Husserl. He was introduced to Husserl's work through his wife, Helene Joseph, herself a student of Husserl at Göttingen. Colin Wilson has used Husserl's ideas extensively in developing his "New Existentialism," particularly in regards to his "intentionality of consciousness," which he mentions in a number of his books. Rudolf Carnap was also influenced by Husserl, not only concerning Husserl's notion of essential insight that Carnap used in his Der Raum, but also his notion of "formation rules" and "transformation rules" is founded on Husserl's philosophy of logic. Karol Wojtyla, who would later become Pope John Paul II, was influenced by Husserl. Phenomenology appears in his major work, The Acting Person (1969). Originally published in Polish, it was translated by Andrzej Potocki and edited by Anna-Teresa Tymieniecka in the Analecta Husserliana. The Acting Person combines phenomenological work with Thomistic ethics. Paul Ricœur has translated many works of Husserl into French and has also written many of his own studies of the philosopher. Among other works, Ricœur employed phenomenology in his Freud and Philosophy (1965). Jacques Derrida wrote several critical studies of Husserl early in his academic career. These included his dissertation, The Problem of Genesis in Husserl's Philosophy, and also his introduction to The Origin of Geometry. Derrida continued to make reference to Husserl in works such as Of Grammatology. Stanisław Leśniewski and Kazimierz Ajdukiewicz were inspired by Husserl's formal analysis of language. Accordingly, they employed phenomenology in the development of categorial grammar. José Ortega y Gasset visited Husserl at Freiburg in 1934. He credited phenomenology for having 'liberated him' from a narrow neo-Kantian thought. While perhaps not a phenomenologist himself, he introduced the philosophy to Iberia and Latin America. Wilfrid Sellars, an influential figure in the so-called "Pittsburgh School" (Robert Brandom, John McDowell) had been a student of Marvin Farber, a pupil of Husserl, and was influenced by phenomenology through him: Marvin Farber led me through my first careful reading of the Critique of Pure Reason and introduced me to Husserl. His combination of utter respect for the structure of Husserl's thought with the equally firm conviction that this structure could be given a naturalistic interpretation was undoubtedly a key influence on my own subsequent philosophical strategy. In his 1942 essay The Myth of Sisyphus, absurdist philosopher Albert Camus acknowledges Husserl as a previous philosopher who described and attempted to deal with the feeling of the absurd, but claims he committed "philosophical suicide" by elevating reason and ultimately arriving at ubiquitous Platonic forms and an abstract god. Hans Blumenberg received his habilitation in 1950, with a dissertation on ontological distance, an inquiry into the crisis of Husserl's phenomenology. Roger Scruton, despite some disagreements with Husserl, drew upon his work in Sexual Desire (1986).: 3–4 The influence of the Husserlian phenomenological tradition in the 21st century extends beyond the confines of the European and North American legacies. It has already started to impact (indirectly) scholarship in Eastern and Oriental thought, including research on the impetus of philosophical thinking in the history of ideas in Islam. == Bibliography == === In German === 1887. Über den Begriff der Zahl. Psychologische Analysen (On the Concept of Number; habilitation thesis) 1891. Philosophie der Arithmetik. Psychologische und logische Untersuchungen (Philosophy of Arithmetic) 1900. Logische Untersuchungen. Erster Teil: Prolegomena zur reinen Logik (Logical Investigations, Vol. 1: Prolegomena to Pure Logic) 1901. Logische Untersuchungen. Zweiter Teil: Untersuchungen zur Phänomenologie und Theorie der Erkenntnis (Logical Investigations, Vol. 2) 1911. Philosophie als strenge Wissenschaft (included in Phenomenology and the Crisis of Philosophy: Philosophy as Rigorous Science and Philosophy and the Crisis of European Man) 1913. Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Erstes Buch: Allgemeine Einführung in die reine Phänomenologie (Ideas: General Introduction to Pure Phenomenology) 1923–24. Erste Philosophie. Zweiter Teil: Theorie der phänomenologischen Reduktion (First Philosophy, Vol. 2: Phenomenological Reductions) 1925. Erste Philosophie. Erster Teil: Kritische Ideengeschichte (First Philosophy, Vol. 1: Critical History of Ideas) 1928. Vorlesungen zur Phänomenologie des inneren Zeitbewusstseins (Lectures on the Phenomenology of the Consciousness of Internal Time) 1929. Formale und transzendentale Logik. Versuch einer Kritik der logischen Vernunft (Formal and Transcendental Logic) 1930. Nachwort zu meinen "Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie" (Postscript to my "Ideas") 1936. Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie: Eine Einleitung in die phänomenologische Philosophie (The Crisis of European Sciences and Transcendental Phenomenology: An Introduction to Phenomenological Philosophy) 1939. Erfahrung und Urteil. Untersuchungen zur Genealogie der Logik. (Experience and Judgment) 1950. Cartesianische Meditationen (translation of Méditations cartésiennes (Cartesian Meditations, 1931)) 1952. Ideen II: Phänomenologische Untersuchungen zur Konstitution (Ideas II: Studies in the Phenomenology of Constitution) 1952. Ideen III: Die Phänomenologie und die Fundamente der Wissenschaften (Ideas III: Phenomenology and the Foundations of the Sciences) 1973. Zur Phänomenologie der Intersubjektivität (On the Phenomenology of Intersubjectivity) === In English === Philosophy of Arithmetic, Willard, Dallas, trans., 2003 [1891]. Dordrecht: Kluwer. Logical Investigations, 1973 [1900, 2nd revised edition 1913], Findlay, J. N., trans. London: Routledge. "Philosophy as Rigorous Science", translated in Quentin Lauer, S.J., editor, 1965 [1910] Phenomenology and the Crisis of Philosophy. New York: Harper & Row. Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy – First Book: General Introduction to a Pure Phenomenology, 1982 [1913]. Kersten, F., trans. The Hague: Nijhoff. Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy – Second Book: Studies in the Phenomenology of Constitution, 1989. R. Rojcewicz and A. Schuwer, translators. Dordrecht: Kluwer. Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy – Third Book: Phenomenology and the Foundations of the Sciences, 1980, Klein, T. E., and Pohl, W. E., translators. Dordrecht: Kluwer. On the Phenomenology of the Consciousness of Internal Time (1893–1917), 1990 [1928]. Brough, J.B., trans. Dordrecht: Kluwer. Cartesian Meditations, 1960 [1931]. Cairns, D., trans. Dordrecht: Kluwer. Formal and Transcendental Logic, 1969 [1929], Cairns, D., trans. The Hague: Nijhoff. Experience and Judgement, 1973 [1939], Churchill, J. S., and Ameriks, K., translators. London: Routledge. The Crisis of European Sciences and Transcendental Phenomenology, 1970 [1936/54], Carr, D., trans. Evanston: Northwestern University Press. "Universal Teleology". Telos 4 (Fall 1969). New York: Telos Press. === Anthologies === Willard, Dallas, trans., 1994. Early Writings in the Philosophy of Logic and Mathematics. Dordrecht: Kluwer. Welton, Donn, ed., 1999. The Essential Husserl. Bloomington: Indiana University Press. == See also == Early phenomenology Experimental phenomenology List of phenomenologists == Notes == == Citations == == Further reading == Adorno, Theodor W., 2013. Against Epistemology. Cambridge: Polity Press. ISBN 978-0745665382 Bernet, Rudolf, et al., 1993. Introduction to Husserlian Phenomenology. Evanston: Northwestern University Press. ISBN 0-8101-1030-X Derrida, Jacques, 1954 (French), 2003 (English). The Problem of Genesis in Husserl's Philosophy. Chicago & London: University of Chicago Press. --------, 1962 (French), 1976 (English). Introduction to Husserl's The Origin of Geometry. Includes Derrida's translation of Appendix III of Husserl's 1936 The Crisis of European Sciences and Transcendental Phenomenology. --------, 1967 (French), 1973 (English). Speech and Phenomena (La Voix et le Phénomène), and other Essays on Husserl's Theory of Signs. ISBN 0-8101-0397-4 Fink, Eugen 1995, Sixth Cartesian meditation. The Idea of a Transcendental Theory of Method with textual notations by Edmund Husserl. Translated with an introduction by Ronald Bruzina, Bloomington: Indiana University Press. Hill, C. O., 1991. Word and Object in Husserl, Frege, and Russell: The Roots of Twentieth-Century Philosophy. Ohio Univ. Press. Hopkins, Burt C., (2011). The Philosophy of Husserl. Durham: Acumen. Levinas, Emmanuel, 1963 (French), 1973 (English). The Theory of Intuition in Husserl's Phenomenology. Evanston: Northwestern University Press. Köchler, Hans, 1982. Edmund Husserl's Theory of Meaning. The Hague: Martinus Nijhoff. --------, 1982. Husserl and Frege. Bloomington: Indiana University Press. Moran, D. and Cohen, J., 2012, The Husserl Dictionary. London, Continuum Press. Natanson, Maurice, 1973. Edmund Husserl: Philosopher of Infinite Tasks. Evanston: Northwestern University Press. ISBN 0-8101-0425-3 Ortiz Hill, Claire; da Silva, Jairo Jose, eds. (1997). The Road Not Taken: On Husserl's Philosophy of Logic and Mathematics. College Publications. Ricœur, Paul, 1967. Husserl: An Analysis of His Phenomenology. Evanston: Northwestern University Press. Rollinger, R. D., 2008. Austrian Phenomenology: Brentano, Husserl, Meinong, and Others on Mind and Language. Frankfurt am Main: Ontos-Verlag. ISBN 978-3-86838-005-7 Sokolowski, Robert. Introduction to Phenomenology. New York: Cambridge University Press, 1999. ISBN 978-0-521-66792-0 Smith, B.; Smith, D. W., eds. (1995), The Cambridge Companion to Husserl, Cambridge: Cambridge University Press, ISBN 0-521-43616-8 Smith, David Woodruff, 2007. Husserl. London: Routledge. Zahavi, Dan, 2003. Husserl's Phenomenology. Stanford: Stanford University Press. ISBN 0-8047-4546-3 == External links == === Husserl archives === Husserl-Archives Leuven, the main Husserl-Archive in Leuven, International Centre for Phenomenological Research. Husserliana: Edmund Husserl Gesammelte Werke, the ongoing critical edition of Husserl's works. Husserliana: Materialien, edition for lectures and shorter works. Edmund Husserl Collected Works, English translation of Husserl's works. Husserl-Archives at the University of Cologne. Husserl-Archives Freiburg. Archives Husserl de Paris, at the École normale supérieure, Paris. Works by or about Edmund Husserl at the Internet Archive === Other links === Beyer, Christian. "Edmund Husserl". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Smith, David Woodruff. "Phenomenology". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Papers on Edmund Husserl by Barry Smith English translation of "Vienna Lecture" (1935): "Philosophy and the Crisis of European Humanity" The Husserl Page by Bob Sandmeyer. Includes a number of online texts in German and English. Husserl.net, open content project. "Edmund Husserl: Formal Ontology and Transcendental Logic." Resource guide on Husserl's logic and formal ontology, with annotated bibliography. The Husserl Circle. Cartesian Meditations in Internet Archive Ideas, Part I in Internet Archive Edmund Husserl on the Open Commons of Phenomenology. Complete bibliography and links to all German texts, including Husserliana vols. I–XXVIII
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Wikipedia:Edriss Titi#0
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Edriss Saleh Titi (Arabic: إدريس صالح تيتي, Hebrew: אדריס סאלח תיתי; born 22 March 1957 in Acre, Israel) is an Arab-Israeli mathematician. He is Professor of Nonlinear Mathematical Science at the University of Cambridge. He also holds the Arthur Owen Professorship of Mathematics at Texas A&M University, and serves as Professor of Computer Science and Applied Mathematics at the Weizmann Institute of Science and Professor Emeritus at the University of California, Irvine. == Selected works == Cao, Chongsheng; Titi, Edriss S. (2007). "Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics". Annals of Mathematics. 166. Princeton University Press: 245–267. arXiv:math/0503028. doi:10.4007/annals.2007.166.245. S2CID 12690494. Foias, Ciprian; Holm, Darryl D.; Titi, Edriss S. (2002). "The three dimensional viscous Camassa–Holm equations, and their relation to the Navier–Stokes equations and turbulence theory". Journal of Dynamics and Differential Equations. 14 (14): 1–35. arXiv:nlin/0103039. doi:10.1023/A:1012984210582. S2CID 16616840. Foias, Ciprian; Holm, Darryl D.; Titi, Edriss S. (2001). "The Navier–Stokes-alpha model of fluid turbulence". Physica D: Nonlinear Phenomena. 152. North-Holland: 505–519. arXiv:nlin/0103037. Bibcode:2001PhyD..152..505F. doi:10.1016/S0167-2789(01)00191-9. S2CID 9174030. Titi, Edriss S. (1990). "On approximate inertial manifolds to the Navier-Stokes equations". Journal of Mathematical Analysis and Applications. 149 (2). Academic Press: 540–557. doi:10.1016/0022-247X(90)90061-J. == References == == External links == Edriss Titi at the Mathematics Genealogy Project
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Wikipedia:Eduard Feireisl#0
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Eduard Feireisl (born 16 December 1957 in Kladno) is a Czech mathematician. After studying from 1973 to 1977 at secondary school in Nové Strašecí, Feireisl studied mathematics at Charles University in Prague from 1977 and graduated there in 1982. He received his doctorate in 1986 from the Institute of Mathematics of the Czechoslovak Academy of Sciences with thesis Critical points of non-differentiable functionals: existence of solutions to problems of mathematical elasticity theory under the supervision of Vladimir Lovicar. During the 1980s he worked as an assistant professor at the Department of Mathematics of the Faculty of Mechanical Engineering Czech Technical University in Prague (CTU). He studied at the Institute of Mathematics of the Czechoslovak Academy of Sciences (as a member since 1988) and habilitated there in 1999. He became a lecturer at the Charles University in 2009 and was appointed there to a full professorship in 2011. Feireisl spent in 1989 half a year in Oxford, in 1993/94 a sabbatical year at the Complutense University of Madrid, and in 1998 and in 1999 half a year at the University of Franche-Comté in Besançon. He was also as visiting scholar for 12 months from 2001 to 2013 at Henri Poincaré University in Nancy and for 3 months in 2000 at Ohio State University. He was in 2004/05 at the TU Munich, from 2008 to 2010 a visiting professor at the Central European University in Budapest, and in 2012 at the Erwin Schrödinger Institute in Vienna. For 2018 to 2021 he was appointed an Einstein Visiting Fellow at TU Berlin. His research deals with partial differential equations, infinite dimensional dynamical systems, and mathematical problems of hydrodynamics. He received in 2004 and 2009 the Prize of the Academy of Sciences of the Czech Republic, in 2015 the Neuron Award, and in 2017 the gold medal of Charles University, as well as the Bernard Bolzano Honorary Medal from the Czech Academy of Sciences. In 2012, he chaired the scientific committee of the European Congress of Mathematicians in Krakow. He was an invited speaker in 2002 at the International Congress of Mathematicians in Beijing, and at the conference Dynamics, Equations and Applications in Kraków in 2019. In 2018 he was a member of the Fields Medal Selection Committee. In 2013 he received an Advanced Grant from the European Research Council (ERC) for the study of mathematical modeling of gas movement and heat exchange. == Selected publications == === Articles === Feireisl, Eduard (1998). "Finite energy travelling waves for nonlinear damped wave equations". Quarterly of Applied Mathematics. 56 (1): 55–70. doi:10.1090/qam/1604876. ISSN 0033-569X. Feireisl, Eduard; Petzeltová, Hana (2002). "Global existence for a quasi-linear evolution equation with a non-convex energy". Transactions of the American Mathematical Society. 354 (4): 1421–1435. doi:10.1090/S0002-9947-01-02950-6. ISSN 0002-9947. Asymptotic analysis of the full Navier–Stokes–Fourier system: From compressible to incompressible fluid flows, Russian Mathematical Surveys, vol. 62, 2007, pp. 511–533 doi:10.1070/RM2007v062n03ABEH004416 Dynamical systems approach to models in fluid mechanics, Russian Mathematical Surveys, vol. 69, 2014, pp. 331–357 doi:10.1070/RM2014v069n02ABEH004890 === Books === Dynamics of viscous compressible fluids, Oxford UP 2004 as editor with Constantine Dafermos: Handbook of differential equations: Evolutionary equations, Elsevier 2004 with Dalibor Pražák: Asymptotic behavior of dynamical systems in fluid mechanics, American Institute of Mathematical Sciences 2010 with Trygve G. Karper, Milan Pokorný: Mathematical Theory of Compressible Viscous Fluids: Analysis and Numerics, Birkhäuser 2016 with John M. Ball, Felix Otto: Mathematical thermodynamics of complex fluids : Cetraro, Italy 2015, Lecture notes in mathematics 2200, Springer 2017 with Antonín Novotný: Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser 2017 with Dominic Breit, Martina Hofmanová: Stochastically forced compressible fluid flows, De Gruyter 2018 == References ==
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Wikipedia:Eduard Čech#0
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Eduard Čech (Czech: [ˈɛduart ˈtʃɛx]; 29 June 1893 – 15 March 1960) was a Czech mathematician. His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification (in topology) and the notion of Čech cohomology. He was the first to publish a proof of Tychonoff's theorem in 1937. == Biography == He was born in Stračov, then in Bohemia, Austria-Hungary, now in the Czech Republic. His father was Čeněk Čech, a policeman, and his mother was Anna Kleplová. After attending the gymnasium in Hradec Králové, Čech was admitted to the Philosophy Faculty of Charles University of Prague in 1912. In 1915 he was drafted into the Austro-Hungarian Army and participated in World War I, after which he completed his undergraduate degree in 1918. He received his doctoral degree in 1920 at Charles University; his thesis, titled On Curves and Plane Elements of the Third Order, was written under the direction of Karel Petr. In 1921–1922 he collaborated with Guido Fubini in Turin, Italy. He taught at Masaryk University in Brno and at Charles University. Ivo Babuška, Vlastimil Dlab, Zdeněk Frolík, Věra Trnková, and Petr Vopěnka have been doctoral students of Čech. He attended the First International Topological Conference held in Moscow 4–10 September 1935. He made two presentations there: "Accessibility and homology" and "Betti groups with different coefficient groups". He died in Prague in 1960. == Publications == Čech, E. (1935), "Les groupes de Betti d'un complexe infini", Fundamenta Mathematicae, 25 (1): 33–44, doi:10.4064/fm-25-1-33-44, hdl:10338.dmlcz/501039 Čech, E. (1936), "Multiplications on a complex", Annals of Mathematics, 37 (3): 681–697, doi:10.2307/1968483, hdl:10338.dmlcz/501047, JSTOR 1968483 Čech, E. (1937), "On bicompact spaces", Annals of Mathematics, 38 (4): 823–844, doi:10.2307/1968839, hdl:10338.dmlcz/100459, JSTOR 1968839 == See also == Čech closure operator Čech cohomology Čech nerve Stone–Čech compactification Tychonoff's theorem == References == == External links == List of publications from Czech Digital Mathematics Library
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Wikipedia:Eduardo D. Sontag#0
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Eduardo Daniel Sontag (born April 16, 1951, in Buenos Aires, Argentina) is an Argentine-American mathematician, and distinguished university professor at Northeastern University, who works in the fields control theory, dynamical systems, systems molecular biology, cancer and immunology, theoretical computer science, neural networks, and computational biology. == Biography == Sontag received his Licenciado degree from the mathematics department at the University of Buenos Aires in 1972, and his Ph.D. in Mathematics under Rudolf Kálmán at the Center for Mathematical Systems Theory at the University of Florida in 1976. From 1977 to 2017, he was with the department of mathematics at Rutgers, The State University of New Jersey, where he was a Distinguished Professor of Mathematics as well as a Member of the Graduate Faculty of the Department of Computer Science and the Graduate Faculty of the Department of Electrical and Computer Engineering, and a Member of the Rutgers Cancer Institute of NJ. In addition, Dr. Sontag served as the head of the undergraduate Biomathematics Interdisciplinary Major, director of the Center for Quantitative Biology, and director of graduate studies of the Institute for Quantitative Biomedicine. In January 2018, Dr. Sontag was appointed as a University Distinguished Professor in the Department of Electrical and Computer Engineering and the Department of BioEngineering at Northeastern University, where he is also an affiliate member of the Department of Mathematics and the Department of Chemical Engineering. Since 2006, he has been a research affiliate at the Laboratory for Information and Decision Systems, MIT, and since 2018 he has been a member of the faculty in the Program in Therapeutic Science, Laboratory for Systems Pharmacology at Harvard Medical School. Eduardo Sontag has authored over five hundred research papers and monographs and book chapters in the above areas with about 60,000 citations and an h-index of 104. He is in the editorial board of several journals, including: IET Proceedings Systems Biology, Synthetic and Systems Biology International Journal of Biological Sciences, and Journal of Computer and Systems Sciences, and is a former board member of SIAM Review, IEEE Transactions on Automatic Control, Systems and Control Letters, Dynamics and Control, Neurocomputing, Neural Networks, Neural Computing Surveys, Control-Theory and Advanced Technology, Nonlinear Analysis: Hybrid Systems, and Control, Optimization and the Calculus of Variations. In addition, he is a co-founder and co-Managing Editor of Mathematics of Control, Signals, and Systems. Sontag was married to Frances David-Sontag, who died in 2017. His daughter Laura Kleiman is founder and CEO at Reboot Rx, and his son David Sontag leads the MIT Clinical Machine Learning Group. == Work == His work in control theory led to the introduction of the concept of input-to-state stability (ISS), a stability theory notion for nonlinear systems, and control-Lyapunov functions. Many of the subsequent results were proved in collaboration with his student Yuan Wang and with David Angeli. In systems biology, Sontag introduced together with David Angeli the concept of input/output monotone system. In theory of computation, he proved the first results on computational complexity in nonlinear controllability, and introduced together with his student Hava Siegelmann a new approach to analog computation and super-Turing computing. == Awards and honors == Sontag became an Institute of Electrical and Electronics Engineers (IEEE) Fellow in 1993. He was awarded the Reid Prize in Mathematics in 2001, the 2002 Hendrik W. Bode Lecture Prize from the IEEE, the 2002 Board of Trustees Award for Excellence in Research from Rutgers University, the 2005 Teacher/Scholar Award from Rutgers University, and the 2011 IEEE Control Systems Award. In 2022, he was awarded the Richard E. Bellman Control Heritage Award, which is the highest recognition in control theory and engineering in the United States. He was honored “for pioneering contributions to stability analysis and nonlinear control, and for advancing the control theoretic foundations of systems biology.” In 2011 he became a fellow of the Society for Industrial and Applied Mathematics, in 2012 a fellow of the American Mathematical Society, and in 2014 a fellow of the International Federation of Automatic Control. Sontag was elected a Member of the American Academy of Arts and Sciences in April 2024. Sontag was elected into the US National Academy of Sciences in April 2025. == Publications == Sontag is co-author of several hundred research papers, as well as three books: 1972, Topics in Artificial Intelligence (in Spanish, Buenos Aires: Prolam, 1972) 1979, Polynomial Response Maps (Berlin: Springer, 1979). 1998, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd Edition (Texts in Applied Mathematics, Volume 6, Second Edition, New York: Springer, 1998) == Selected Public Research Rankings == Research.com top 100 US electrical engineers. Research.com top 100 US mathematicians. Most-cited author in: IEEE Transactions on Automatic Control in 1981, 1996, 1997; Systems and Control Letters 1989, 1991, 1995, 1998, and lifetime of journal; SIAM Journal on Control and Optimization 1983, 1986; Theoretical Computer Science 1994; as well as many other journal/years. Elsevier/Stanford list of top 0.5% among 2% top scientists worldwide. MathScinet list of three most-cited applied mathematicians who got PhD in 1976. == References == == External links == Link to Eduardo Sontag's Homepage Eduardo D. Sontag at the Mathematics Genealogy Project
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Wikipedia:Eduardo Héctor Zarantonello#0
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Eduardo Héctor Zarantonello (1918–2010) was an Argentine mathematician working on analysis. His doctorate was awarded by the Universidad Nacional de La Plata. == References == Tirao, J. (2011), "Eduardo H. Zarantonello (1918–2010)", Revista de la Unión Matemática Argentina, 52 (1): i–vi, ISSN 0041-6932, MR 2816203 Eduardo Héctor Zarantonello at the Mathematics Genealogy Project
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Wikipedia:Edward Barbeau#0
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Edward Barbeau is a Canadian mathematician and a Canadian Mathematical Educator. He is a Professor Emeritus at the University of Toronto Department of Mathematics. == Awards == Fellowship of the Ontario Institute for Studies in Education. David Hilbert Award from the World Federation of National Mathematics Competitions. Adrien Pouliot Award from the Canadian Mathematical Society. Inaugural fellow of the Canadian Mathematical Society, 2018 == References == == External links == Edward Barbeau at the Mathematics Genealogy Project Edward J. Barbeau archival papers held at the University of Toronto Archives and Records Management Services
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Wikipedia:Edward Bierstone#0
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Edward Bierstone (born (1946-12-21)December 21, 1946) is a Canadian mathematician at the University of Toronto who specializes in singularity theory, analytic geometry, and differential analysis. == Education and career == He received his B.Sc. from the University of Toronto and his Ph.D. at Brandeis University in 1972. He was a visiting scholar at the Institute for Advanced Study in the summer of 1973. He served as the Director of the Fields Institute from 2009 to 2013. == Recognition == Bierstone was elected a member of the Royal Society of Canada in 1992 and, together with Pierre Milman, received the Jefferey Williams Prize in 2005. In 2012 he became a fellow of the American Mathematical Society. In 2018 the Canadian Mathematical Society listed him in their inaugural class of fellows. == Notes == == External links == Edward Bierstone at the Mathematics Genealogy Project personal webpage
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Wikipedia:Edward Bromhead#0
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Sir Edward Thomas ffrench Bromhead, 2nd Baronet FRS FRSE (26 March 1789 – 14 March 1855) was a British landowner and mathematician, best remembered as patron of the mathematician and physicist George Green and mentor of George Boole. == Life == Born the son of Gonville Bromhead, 1st Baronet Bromhead (grandfather of the British second in command of the same name at Rorke's Drift) and Lady Jane ffrench, Baroness ffrench, in Dublin. Bromhead was educated at the University of Glasgow and later at Caius College, Cambridge ( B.A. 1812, M.A. 1815) before taking up the study of law at the Inner Temple in London. He was elected a Fellow of the Royal Society in 1817. Returning to Lincolnshire, he became High Steward of Lincoln. He became the 2nd Bromhead baronet, of Thurlby Hall in 1822. While at Cambridge, Bromhead was a founder of the Analytical Society, a precursor of the Cambridge Philosophical Society, together with John Herschel, George Peacock and Charles Babbage, with whom he maintained a close and lifelong friendship. While he was, by all accounts, a gifted mathematician in his own right (although ill-health prevented him from pursuing his studies further), his greatest contribution to the subject is at second hand: having subscribed to the first publication of self-taught mathematician and physicist George Green, he encouraged Green to continue his research and to write further papers (which Bromhead sent on to be published in the Transactions of the Cambridge Philosophical Society and those of the Royal Society of Edinburgh). Bromhead repeated his success by encouraging the young George Boole from Lincoln. Bromhead was President of the Lincoln Mechanics Institute in the Lincoln Greyfriars, where George Boole's father was the curator. Boole first came to public notice when he gave a lecture on the work of Sir Isaac Newton on 5 February 1835. The young Boole's development was fed by books that Bromhead supplied. Bromhead lost his sight when he was old and he died unmarried at his home of Thurlby Hall in Thurlby, North Kesteven on 14 March 1855. == Arms == == Selected publications == X. Remarks on the present state of botanical classification Philosophical Magazine Series 3 Volume 11, Issue 64-65, 1837 XXVIII. Memoranda on the origin of the botanical alliances Philosophical Magazine Series 3 Volume 11, Issue 67, 1837 Bromhead, Edward Ffrench (1838). "An Attempt to ascertain Characters of the Botanical Alliances". Edinburgh New Philosophical Journal. 25: 123–134. Retrieved 21 April 2015. == References == == Bibliography == Edwards, A. W. F. "Bromhead, Sir Edward Thomas Ffrench". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/37224. (Subscription or UK public library membership required.) Cannel, D. M. and Lord, N. J. (March 1993). "George Green, mathematician and physicist 1793–1841". The Mathematical Gazette. 77 (478): 26–51. doi:10.2307/3619259. JSTOR 3619259. S2CID 238490315.{{cite journal}}: CS1 maint: multiple names: authors list (link) Mentions Bromhead's role in the career of George Green.
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Wikipedia:Edward Charles Titchmarsh#0
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Edward Charles "Ted" Titchmarsh (June 1, 1899 – January 18, 1963) was a leading British mathematician. == Education == Titchmarsh was educated at King Edward VII School (Sheffield) and Balliol College, Oxford, where he began his studies in October 1917. == Career == Titchmarsh was known for work in analytic number theory, Fourier analysis and other parts of mathematical analysis. He wrote several classic books in these areas; his book on the Riemann zeta-function was reissued in a 1986 edition edited by Roger Heath-Brown. Titchmarsh was Savilian Professor of Geometry at the University of Oxford from 1932 to 1963. He was a Plenary Speaker at the ICM in 1954 in Amsterdam. He was on the governing body of Abingdon School from 1935-1947. == Awards == Fellow of the Royal Society, 1931 De Morgan Medal, 1953 Sylvester Medal, 1955 Berwick Prize winner, 1956 == Publications == The Zeta-Function of Riemann (1930); Introduction to the Theory of Fourier Integrals (1937) 2nd. edition(1939) 2nd. edition (1948); The Theory of Functions (1932); Mathematics for the General Reader (1948); The Theory of the Riemann Zeta-Function (1951); 2nd edition, revised by D. R. Heath-Brown (1986) Eigenfunction Expansions Associated with Second-order Differential Equations. Part I (1946) 2nd. edition (1962); Eigenfunction Expansions Associated with Second-order Differential Equations. Part II (1958); == References ==
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Wikipedia:Edward Jan Habich#0
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Edward Jan Habich (Spanish: Eduardo de Habich) (31 January 1835, Warsaw – 31 October 1909, Lima, Peru) was a Polish engineer and mathematician. In 1876, he founded the National University of Engineering (Spanish: Universidad Nacional de Ingeniería), a renowned engineering school in Lima, Peru. He was a member of the Peruvian Geographic Society and an Honorary Citizen of Peru. In his native Poland he took part in the January Uprising against the Russian Empire in 1863. == Burial == Edward Jan Habich is buried at the Cementerio Presbítero Matías Maestro, Lima, Peru. == Gallery == == References == == External links == Media related to Edward Jan Habich at Wikimedia Commons
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Wikipedia:Edward McWilliam Patterson#0
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Edward McWilliam Patterson, FRSE (30 July 1926 – 5 April 2013) was an English mathematician. He was born in Whitby, North Yorkshire, the son of parents from Northern Ireland, and educated at the local Lady Lumley's school and Leeds University, where he graduated B.Sc in mathematics and was awarded a Ph.D. on the subject of differential geometry. From 1959 to 1951 he was a demonstrator at Sheffield University before moving to St Andrews in to take up a post as lecturer for five years. After three further years as a lecturer in Leeds, he returned to Scotland in 1959 as a senior lecturer at the University of Aberdeen. The same year he was elected a fellow of the Royal Society of Edinburgh. In 1965 he was made professor of mathematics at Aberdeen and in 1974 became head of department, a position he held alternately with Professor John Hubbuck until his retirement in 1989. From 1981 to 1984 he also served as dean of science. His mathematical work was originally geometry-based, and he published a textbook entitled Topology in 1956. He later switched to algebra, especially ring theory and Lie algebra, and published two textbooks, Elementary Abstract Algebra in 1965, in collaboration with Professor Dan Rutherford, and Vector Algebra in 1968. He was awarded the Makdougall Brisbane Prize by the Royal Society of Edinburgh for 1960–1962. He was president of the Edinburgh Mathematical Society from 1964 to 1965 and served as a councillor for The Royal Society of Edinburgh from 1966 to 1968. He also served on the council of The London Mathematical Society. He died in Aberdeen in 2013. He had married twice: firstly Joan Maddick, with whom he had a daughter, Christine and secondly, after her death, Elizabeth Hunter. == References ==
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Wikipedia:Edward Norman Dancer#0
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Edward Norman Dancer FAA (born 29 December 1946, Bundaberg, Queensland, Australia) is an Australian mathematician, specializing in nonlinear analysis. Dancer received in 1969 a Bachelor of Science with Honours (BSc (Hons)) from the Australian National University and in 1972 a PhD from the University of Cambridge with thesis advisor Frank Smithies and thesis Bifurcation in Banach Spaces. As a postdoc Dancer was from 1971 to 1972 at the University of Newcastle-upon-Tyne, UK and from 1972 to 1973 at the Institute of Advanced Studies at the Australian National University. At the University of New England in New South Wales, he was from 1973 to 1975 a lecturer, from 1976 to 1981 a senior lecturer, from 1981 to 1987 an associate professor, and from 1987 to 1993 a full professor. From 1993 to the present, he has been a full professor and chair of the school of mathematics and statistics at the University of Sydney. His present research interests include nonlinear analysis, especially degree theory, Morse theory and Conley index; applications to nonlinear ordinary and partial differential equations, including singular perturbations; bifurcation theory. Dancer is also a part time professor at Swansea University. == Honours and awards == 1996 — elected a Fellow of the Australian Academy of Science (FAA) 2002 — Humboldt Research Award received from the Humboldt Foundation and Helmholtz Association 2009 — Hannan Medal awarded by the Australian Academy of Science 2010 — Invited Speaker at the International Congress of Mathematicians in Hyderabad 2017 — Schauder Medal == Selected publications == Dancer, E. N. (1973). "Global solution branches for positive mappings". Archive for Rational Mechanics and Analysis. 52 (2): 181–192. Bibcode:1973ArRMA..52..181D. doi:10.1007/BF00282326. S2CID 122278304. with Ralph S. Phillips: Dancer, E. N.; Phillips, Ralph (1974). "On the structure of solutions of non-linear eigenvalue problems". Indiana University Mathematics Journal. 23 (11): 1069–1076. doi:10.1512/iumj.1974.23.23087. JSTOR 24890776. Dancer, E. N. (1977). "On the Dirichlet problem for weakly non-linear elliptic partial differential equations". Proceedings of the Royal Society of Edinburgh, Section A. 76 (4): 283–300. doi:10.1017/S0308210500019648. S2CID 123218340. Dancer, E.N. (1983). "On the indices of fixed points of mappings in cones and applications". Journal of Mathematical Analysis and Applications. 91 (1): 131–151. doi:10.1016/0022-247X(83)90098-7. Dancer, E.N (1985). "On positive solutions of some pairs of differential equations, II" (PDF). Journal of Differential Equations. 60 (2): 236–258. Bibcode:1985JDE....60..236D. doi:10.1016/0022-0396(85)90115-9. Dancer, E.N (1990). "The effect of domain shape on the number of positive solutions of certain nonlinear equations, II". Journal of Differential Equations. 87 (2): 316–339. Bibcode:1990JDE....87..316D. doi:10.1016/0022-0396(90)90005-A. Dancer, E. N. (1991). "On the existence and uniqueness of positive solutions for competing species models with diffusion". Trans. Amer. Math. Soc. 326 (2): 829–859. doi:10.1090/S0002-9947-1991-1028757-9. with Peter Hess: "Stability of fixed points for order-preserving discrete-time dynamical systems". J. Reine Angew. Math. 1991 (419): 125–139. 1991. doi:10.1515/crll.1991.419.125. S2CID 120712971. Dancer, E.N. (1992). "Some notes on the method of moving planes". Bulletin of the Australian Mathematical Society. 46 (3): 425–434. doi:10.1017/S0004972700012089. Dancer, E. N. (1998). "Some remarks on a boundedness assumption for monotone dynamical systems" (PDF). Proc. Amer. Math. Soc. 126 (3): 801–807. doi:10.1090/S0002-9939-98-04276-2. with Shusen Yan: "Multipeak solutions for a singularly perturbed Neumann problem". Pacific Journal of Mathematics. 182 (2): 241–262. 1999. with Yihong Du: Dancer, E. N.; Du, Yihong (2003). "Some remarks on Liouville type results for quasilinear elliptic equations" (PDF). Proc. Amer. Math. Soc. 131 (6): 1891–1899. doi:10.1090/S0002-9939-02-06733-3. with Juncheng Wei and Tobias Weth: Dancer, E.N.; Wei, Juncheng; Weth, Tobias (2010). "A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system". Annales de l'Institut Henri Poincaré C. 27 (3): 953–969. Bibcode:2010AIHPC..27..953D. doi:10.1016/j.anihpc.2010.01.009. with Kelei Wang and Zhitao Zhang: Dancer, E. N.; Wang, Kelei; Zhang, Zhitao (2012). "Dynamics of strongly competing systems with many species". Trans. Amer. Math. Soc. 364 (2): 961–1005. doi:10.1090/S0002-9947-2011-05488-7. == References ==
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Wikipedia:Edwina Rissland#0
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Edwina Luane Rissland (also published as Edwina Luane Rissland Michener) is a retired American mathematician and computer scientist. Initially focusing on knowledge representation and the philosophy of mathematics, her later research in artificial intelligence included work on case-based reasoning and the applications of AI in legal work. She is a professor emerita in the Manning College of Information & Computer Sciences at the University of Massachusetts Amherst. In her retirement she has worked as a photographer and art curator. == Education and career == Rissland majored in applied mathematics as an undergraduate at Brown University, where her undergraduate studies included a computer science course from Andries van Dam. She graduated in 1969. After a 1970 master's degree at Brandeis University, she completed a Ph.D. in mathematics at the Massachusetts Institute of Technology in 1977. Her dissertation, Epistemology, Representation, Understanding and Interactive Exploration of Mathematical Theories, was supervised by Seymour Papert. She joined the UMass Amherst faculty in 1979. Her interest in law began around this time after reading a book on US Supreme Court decisions. In 1982 and 1983 she was affiliated with the Harvard Law School as a Fellow of Law and Computer Science, and from 1985 to 1986 she returned to the Harvard Law School as a lecturer. She also served two terms as a program director for artificial intelligence and cognitive science at the National Science Foundation, from 2003 to 2007 and again from 2010 to 2012, helped found the International Association for Artificial Intelligence and Law, and later served as its president. She retired in 2013. == Recognition == Rissland was named a Fellow of the Association for the Advancement of Artificial Intelligence in 1991. In 2023 she received the CodeX prize of the Stanford Center for Legal Informatics for her contributions to computational law, together with her doctoral student, Kevin Ashley, and in particular for their HYPO CBR system for legal case-based reasoning. == Selected publications == Rissland is a coauthor of the textbook Cognitive Science: An Introduction (MIT Press, 1988; 2nd ed., 1995). Her research publications include: Michener, Edwina Rissland (October 1978), "Understanding understanding mathematics", Cognitive Science, 2 (4): 361–383, doi:10.1207/s15516709cog0204_3, hdl:1721.1/5735 Rissland, Edwina L. (1983), "Examples in legal reasoning: Legal hypotheticals" (PDF), in Bundy, Alan (ed.), Proceedings of the 8th International Joint Conference on Artificial Intelligence. Karlsruhe, FRG, August 1983, William Kaufmann, pp. 90–93 Rissland, Edwina L.; Ashley, Kevin D. (1987), "A case-based system for trade secrets law", Proceedings of the First International Conference on Artificial Intelligence and Law, ICAIL '87, Boston, MA, USA, May 27–29, 1987, ACM, pp. 60–66, doi:10.1145/41735.41743, ISBN 0-89791-230-6 Rissland, Edwina L. (1990), "Artificial intelligence and law: Stepping stones to a model of legal reasoning", The Yale Law Journal, 99 (8): 1957–1981, doi:10.2307/796679, hdl:20.500.13051/16697, JSTOR 796679 Rissland, Edwina L.; Skalak, David B. (1991), "CABARET: Rule interpretation in a hybrid architecture", International Journal of Man-Machine Studies, 34 (6): 839–887, doi:10.1016/0020-7373(91)90013-W Skalak, David B.; Rissland, Edwina L. (1992), "Arguments and cases: An inevitable intertwining", Artificial Intelligence and Law, 1 (1): 3–44, doi:10.1007/BF00118477 Rissland, Edwina L.; Ashley, Kevin D.; Branting, Karl (2005), "Case-based reasoning and law", The Knowledge Engineering Review, 20 (3): 293–298, doi:10.1017/S0269888906000701 == References == == External links == Rissland Arts, Rissland's photography web site The case-based reasoning group, Rissland's research group at UMass Amherst
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Wikipedia:Effective dimension#0
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In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notions of effective dimension) of which the most common is effective Hausdorff dimension. Dimension, in mathematics, is a particular way of describing the size of an object (contrasting with measure and other, different, notions of size). Hausdorff dimension generalizes the well-known integer dimensions assigned to points, lines, planes, etc. by allowing one to distinguish between objects of intermediate size between these integer-dimensional objects. For example, fractal subsets of the plane may have intermediate dimension between 1 and 2, as they are "larger" than lines or curves, and yet "smaller" than filled circles or rectangles. Effective dimension modifies Hausdorff dimension by requiring that objects with small effective dimension be not only small but also locatable (or partially locatable) in a computable sense. As such, objects with large Hausdorff dimension also have large effective dimension, and objects with small effective dimension have small Hausdorff dimension, but an object can have small Hausdorff but large effective dimension. An example is an algorithmically random point on a line, which has Hausdorff dimension 0 (since it is a point) but effective dimension 1 (because, roughly speaking, it can't be effectively localized any better than a small interval, which has Hausdorff dimension 1). == Rigorous definitions == This article will define effective dimension for subsets of Cantor space 2ω; closely related definitions exist for subsets of Euclidean space Rn. We will move freely between considering a set X of natural numbers, the infinite sequence χ X {\displaystyle \chi _{X}} given by the characteristic function of X, and the real number with binary expansion 0.X. === Martingales and other gales === A martingale on Cantor space 2ω is a function d: 2ω → R≥ 0 from Cantor space to nonnegative reals which satisfies the fairness condition: d ( σ ) = 1 2 ( d ( σ 0 ) + d ( σ 1 ) ) {\displaystyle d(\sigma )={\frac {1}{2}}(d(\sigma 0)+d(\sigma 1))} A martingale is thought of as a betting strategy, and the function d ( σ ) {\displaystyle d(\sigma )} gives the capital of the better after seeing a sequence σ of 0s and 1s. The fairness condition then says that the capital after a sequence σ is the average of the capital after seeing σ0 and σ1; in other words the martingale gives a betting scheme for a bookie with 2:1 odds offered on either of two "equally likely" options, hence the name fair. (Note that this is subtly different from the probability theory notion of martingale. That definition of martingale has a similar fairness condition, which also states that the expected value after some observation is the same as the value before the observation, given the prior history of observations. The difference is that in probability theory, the prior history of observations just refers to the capital history, whereas here the history refers to the exact sequence of 0s and 1s in the string.) A supermartingale on Cantor space is a function d as above which satisfies a modified fairness condition: d ( σ ) ≥ 1 2 ( d ( σ 0 ) + d ( σ 1 ) ) {\displaystyle d(\sigma )\geq {\frac {1}{2}}(d(\sigma 0)+d(\sigma 1))} A supermartingale is a betting strategy where the expected capital after a bet is no more than the capital before a bet, in contrast to a martingale where the two are always equal. This allows more flexibility, and is very similar in the non-effective case, since whenever a supermartingale d is given, there is a modified function d' which wins at least as much money as d and which is actually a martingale. However it is useful to allow the additional flexibility once one starts talking about actually giving algorithms to determine the betting strategy, as some algorithms lend themselves more naturally to producing supermartingales than martingales. An s-gale is a function d as above of the form d ( σ ) = e ( σ ) 2 ( 1 − s ) | σ | {\displaystyle d(\sigma )={\frac {e(\sigma )}{2^{(1-s)|\sigma |}}}} for e some martingale. An s-supergale is a function d as above of the form d ( σ ) = e ( σ ) 2 ( 1 − s ) | σ | {\displaystyle d(\sigma )={\frac {e(\sigma )}{2^{(1-s)|\sigma |}}}} for e some supermartingale. An s-(super)gale is a betting strategy where some amount of capital is lost to inflation at each step. Note that s-gales and s-supergales are examples of supermartingales, and the 1-gales and 1-supergales are precisely the martingales and supermartingales. Collectively, these objects are known as "gales". A gale d succeeds on a subset X of the natural numbers if lim sup n d ( X | n ) = ∞ {\displaystyle \limsup _{n}d(X|n)=\infty } where X | n {\displaystyle X|n} denotes the n-digit string consisting of the first n digits of X. A gale d succeeds strongly on X if lim inf n d ( X | n ) = ∞ {\displaystyle \liminf _{n}d(X|n)=\infty } . All of these notions of various gales have no effective content, but one must necessarily restrict oneself to a small class of gales, since some gale can be found which succeeds on any given set. After all, if one knows a sequence of coin flips in advance, it is easy to make money by simply betting on the known outcomes of each flip. A standard way of doing this is to require the gales to be either computable or close to computable: A gale d is called constructive, c.e., or lower semi-computable if the numbers d ( σ ) {\displaystyle d(\sigma )} are uniformly left-c.e. reals (i.e. can uniformly be written as the limit of an increasing computable sequence of rationals). The effective Hausdorff dimension of a set of natural numbers X is inf { s : s o m e c . e . s − g a l e s u c c e e d s o n X } {\displaystyle \inf\{s:\mathrm {some\ c.e.} \ s\mathrm {-gale\ succeeds\ on\ } X\}} . The effective packing dimension of X is inf { s : s o m e c . e . s − g a l e s u c c e e d s s t r o n g l y o n X } {\displaystyle \inf\{s:\mathrm {some\ c.e.} \ s\mathrm {-gale\ succeeds\ strongly\ on\ } X\}} . === Kolmogorov complexity definition === Kolmogorov complexity can be thought of as a lower bound on the algorithmic compressibility of a finite sequence (of characters or binary digits). It assigns to each such sequence w a natural number K(w) that, intuitively, measures the minimum length of a computer program (written in some fixed programming language) that takes no input and will output w when run. The effective Hausdorff dimension of a set of natural numbers X is lim inf n K ( X | n ) n {\displaystyle \liminf _{n}{\frac {K(X|n)}{n}}} . The effective packing dimension of a set X is lim sup n K ( X | n ) n {\displaystyle \limsup _{n}{\frac {K(X|n)}{n}}} . From this one can see that both the effective Hausdorff dimension and the effective packing dimension of a set are between 0 and 1, with the effective packing dimension always at least as large as the effective Hausdorff dimension. Every random sequence will have effective Hausdorff and packing dimensions equal to 1, although there are also nonrandom sequences with effective Hausdorff and packing dimensions of 1. == Comparison to classical dimension == If Z is a subset of 2ω, its Hausdorff dimension is inf { s : s o m e s − g a l e s u c c e e d s o n a l l e l e m e n t s o f Z } {\displaystyle \inf\{s:\mathrm {some} \ s\mathrm {-gale\ succeeds\ on\ all\ elements\ of\ } Z\}} . The packing dimension of Z is inf { s : s o m e s − g a l e s u c c e e d s s t r o n g l y o n a l l e l e m e n t s o f Z } {\displaystyle \inf\{s:\mathrm {some} \ s\mathrm {-gale\ succeeds\ strongly\ on\ all\ elements\ of\ } Z\}} . Thus the effective Hausdorff and packing dimensions of a set X {\displaystyle X} are simply the classical Hausdorff and packing dimensions of { X } {\displaystyle \{X\}} (respectively) when we restrict our attention to c.e. gales. Define the following: H β := { X ∈ 2 ω : X h a s e f f e c t i v e H a u s d o r f f d i m e n s i o n β } {\displaystyle H_{\beta }:=\{X\in 2^{\omega }:X\ \mathrm {has\ effective\ Hausdorff\ dimension\ } \beta \}} H ≤ β := { X ∈ 2 ω : X h a s e f f e c t i v e H a u s d o r f f d i m e n s i o n ≤ β } {\displaystyle H_{\leq \beta }:=\{X\in 2^{\omega }:X\ \mathrm {has\ effective\ Hausdorff\ dimension\ } \leq \beta \}} H < β := { X ∈ 2 ω : X h a s e f f e c t i v e H a u s d o r f f d i m e n s i o n < β } {\displaystyle H_{<\beta }:=\{X\in 2^{\omega }:X\ \mathrm {has\ effective\ Hausdorff\ dimension\ } <\beta \}} P β := { X ∈ 2 ω : X h a s e f f e c t i v e p a c k i n g d i m e n s i o n β } {\displaystyle P_{\beta }:=\{X\in 2^{\omega }:X\ \mathrm {has\ effective\ packing\ dimension\ } \beta \}} P ≤ β := { X ∈ 2 ω : X h a s e f f e c t i v e p a c k i n g d i m e n s i o n ≤ β } {\displaystyle P_{\leq \beta }:=\{X\in 2^{\omega }:X\ \mathrm {has\ effective\ packing\ dimension\ } \leq \beta \}} P < β := { X ∈ 2 ω : X h a s e f f e c t i v e p a c k i n g d i m e n s i o n < β } {\displaystyle P_{<\beta }:=\{X\in 2^{\omega }:X\ \mathrm {has\ effective\ packing\ dimension\ } <\beta \}} A consequence of the above is that these all have Hausdorff dimension β {\displaystyle \beta } . H β , H ≤ β {\displaystyle H_{\beta },H_{\leq \beta }} and H < β {\displaystyle H_{<\beta }} all have packing dimension 1. P β , P ≤ β {\displaystyle P_{\beta },P_{\leq \beta }} and P < β {\displaystyle P_{<\beta }} all have packing dimension β {\displaystyle \beta } . == References ==
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Wikipedia:Effective domain#0
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In convex analysis, a branch of mathematics, the effective domain extends of the domain of a function defined for functions that take values in the extended real number line [ − ∞ , ∞ ] = R ∪ { ± ∞ } . {\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}.} In convex analysis and variational analysis, a point at which some given extended real-valued function is minimized is typically sought, where such a point is called a global minimum point. The effective domain of this function is defined to be the set of all points in this function's domain at which its value is not equal to + ∞ . {\displaystyle +\infty .} It is defined this way because it is only these points that have even a remote chance of being a global minimum point. Indeed, it is common practice in these fields to set a function equal to + ∞ {\displaystyle +\infty } at a point specifically to exclude that point from even being considered as a potential solution (to the minimization problem). Points at which the function takes the value − ∞ {\displaystyle -\infty } (if any) belong to the effective domain because such points are considered acceptable solutions to the minimization problem, with the reasoning being that if such a point was not acceptable as a solution then the function would have already been set to + ∞ {\displaystyle +\infty } at that point instead. When a minimum point (in X {\displaystyle X} ) of a function f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} is to be found but f {\displaystyle f} 's domain X {\displaystyle X} is a proper subset of some vector space V , {\displaystyle V,} then it often technically useful to extend f {\displaystyle f} to all of V {\displaystyle V} by setting f ( x ) := + ∞ {\displaystyle f(x):=+\infty } at every x ∈ V ∖ X . {\displaystyle x\in V\setminus X.} By definition, no point of V ∖ X {\displaystyle V\setminus X} belongs to the effective domain of f , {\displaystyle f,} which is consistent with the desire to find a minimum point of the original function f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} rather than of the newly defined extension to all of V . {\displaystyle V.} If the problem is instead a maximization problem (which would be clearly indicated) then the effective domain instead consists of all points in the function's domain at which it is not equal to − ∞ . {\displaystyle -\infty .} == Definition == Suppose f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} is a map valued in the extended real number line [ − ∞ , ∞ ] = R ∪ { ± ∞ } {\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}} whose domain, which is denoted by domain f , {\displaystyle \operatorname {domain} f,} is X {\displaystyle X} (where X {\displaystyle X} will be assumed to be a subset of some vector space whenever this assumption is necessary). Then the effective domain of f {\displaystyle f} is denoted by dom f {\displaystyle \operatorname {dom} f} and typically defined to be the set dom f = { x ∈ X : f ( x ) < + ∞ } {\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)<+\infty \}} unless f {\displaystyle f} is a concave function or the maximum (rather than the minimum) of f {\displaystyle f} is being sought, in which case the effective domain of f {\displaystyle f} is instead the set dom f = { x ∈ X : f ( x ) > − ∞ } . {\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)>-\infty \}.} In convex analysis and variational analysis, dom f {\displaystyle \operatorname {dom} f} is usually assumed to be dom f = { x ∈ X : f ( x ) < + ∞ } {\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)<+\infty \}} unless clearly indicated otherwise. == Characterizations == Let π X : X × R → X {\displaystyle \pi _{X}:X\times \mathbb {R} \to X} denote the canonical projection onto X , {\displaystyle X,} which is defined by ( x , r ) ↦ x . {\displaystyle (x,r)\mapsto x.} The effective domain of f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} is equal to the image of f {\displaystyle f} 's epigraph epi f {\displaystyle \operatorname {epi} f} under the canonical projection π X . {\displaystyle \pi _{X}.} That is dom f = π X ( epi f ) = { x ∈ X : there exists y ∈ R such that ( x , y ) ∈ epi f } . {\displaystyle \operatorname {dom} f=\pi _{X}\left(\operatorname {epi} f\right)=\left\{x\in X~:~{\text{ there exists }}y\in \mathbb {R} {\text{ such that }}(x,y)\in \operatorname {epi} f\right\}.} For a maximization problem (such as if the f {\displaystyle f} is concave rather than convex), the effective domain is instead equal to the image under π X {\displaystyle \pi _{X}} of f {\displaystyle f} 's hypograph. == Properties == If a function never takes the value + ∞ , {\displaystyle +\infty ,} such as if the function is real-valued, then its domain and effective domain are equal. A function f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} is a proper convex function if and only if f {\displaystyle f} is convex, the effective domain of f {\displaystyle f} is nonempty, and f ( x ) > − ∞ {\displaystyle f(x)>-\infty } for every x ∈ X . {\displaystyle x\in X.} == See also == Proper convex function Epigraph (mathematics) – Region above a graph Hypograph (mathematics) – Region underneath a graph == References == Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. Vol. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.
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Wikipedia:Efim Zelmanov#0
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Efim Isaakovich Zelmanov (Russian: Ефи́м Исаа́кович Зе́льманов; born 7 September 1955) is a Russian-American mathematician, known for his work on combinatorial problems in nonassociative algebra and group theory, including his solution of the restricted Burnside problem. He was awarded a Fields Medal at the International Congress of Mathematicians in Zürich in 1994. == Biography == Zelmanov was born on 7 September 1955 into a Jewish family in Khabarovsk. He entered Novosibirsk State University in 1972, when he was 17 years old. He obtained a doctoral degree at Novosibirsk State University in 1980, and a higher degree at Leningrad State University in 1985. He had a position in Novosibirsk until 1987, when he left the Soviet Union.In 1990, he moved to the United States, becoming a professor at the University of Wisconsin–Madison. He was at the University of Chicago in 1994/5, then at Yale University. In 1996, he became a Distinguished Professor at the Korea Institute for Advanced Study and in 2002, he became a professor at the University of California, San Diego. In 2011 he got hon DSc from QUB (Belfast) In 2022, he moved to the People's Republic of China and joined the Southern University of Science and Technology in Shenzhen, China. He served as a chair professor and the scientific director of the SUSTech International Center for Mathematics. Zelmanov was elected a member of the U.S. National Academy of Sciences in 2001, becoming, at the age of 47, the youngest member of the mathematics section of the academy. He is also an elected member of the American Academy of Arts and Sciences (1996) and a foreign member of the Korean Academy of Science and Technology and of the Spanish Royal Academy of Sciences. In 2012, he became a fellow of the American Mathematical Society. Zelmanov gave invited talks at the International Congress of Mathematicians in Warsaw (1983), Kyoto (1990) and Zurich (1994). He delivered the 2004 Turán Memorial Lectures. He was awarded Honorary Doctor degrees from the University of Hagen, Germany (1997), the University of Alberta, Canada (2011), Taras Shevchenko National University of Kyiv, Ukraine (2012), the Universidad Internacional Menéndez Pelayo in Santander, Spain (2015), the University of Lincoln, UK (2016), and the Vrije Universiteit Brussel, Belgium (2023). Zelmanov's early work was on Jordan algebras in the case of infinite dimensions. He was able to show that Glennie's identity in a certain sense generates all identities that hold. He then showed that the Engel identity for Lie algebras implies nilpotence, in the case of infinite dimensions. == Notable publications == Zelʹmanov, E.I. Solution of the restricted Burnside problem for groups of odd exponent. Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 42–59, 221. English translation in Math. USSR-Izv. 36 (1991), no. 1, 41–60. doi:10.1070/IM1991v036n01ABEH001946 Zelʹmanov, E.I. Solution of the restricted Burnside problem for 2-groups. Mat. Sb. 182 (1991), no. 4, 568–592. English translation in Math. USSR-Sb. 72 (1992), no. 2, 543–565. doi:10.1070/SM1992v072n02ABEH001272 == References == == External links == Efim Zelmanov at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., "Efim Zelmanov", MacTutor History of Mathematics Archive, University of St Andrews The Work of Efim Zelmanov (Fields Medal 1994) by Kapil Hari Paranjape.
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Wikipedia:Egyptian Mathematical Leather Roll#0
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The Egyptian Mathematical Leather Roll (EMLR) is a 10 × 17 in (25 × 43 cm) leather roll purchased by Alexander Henry Rhind in 1858. It was sent to the British Museum in 1864, along with the Rhind Mathematical Papyrus, but it was not chemically softened and unrolled until 1927 (Scott, Hall 1927). The writing consists of Middle Kingdom hieratic characters written right to left. Scholars date the EMLR to the 17th century BCE. == Mathematical content == This leather roll is an aid for computing Egyptian fractions. It contains 26 sums of unit fractions which equal another unit fraction. The sums appear in two columns, and are followed by two more columns which contain exactly the same sums. Of the 26 sums listed, ten are Eye of Horus numbers: 1/2, 1/4 (twice), 1/8 (thrice), 1/16 (twice), 1/32, 1/64 converted from Egyptian fractions. There are seven other sums having even denominators converted from Egyptian fractions: 1/6 (listed twice–but wrong once), 1/10, 1/12, 1/14, 1/20 and 1/30. By way of example, the three 1/8 conversions followed one or two scaling factors as alternatives: 1. 1/8 x 3/3 = 3/24 = (2 + 1)/24 = 1/12 + 1/24 2. 1/8 x 5/5 = 5/40 = (4 + 1)/40 = 1/10 + 1/40 3. 1/8 x 25/25 = 25/200 = (8 + 17)/200 = 1/25 + (17/200 x 6/6) = 1/25 + 102/1200 = 1/25 + (80 + 16 + 6)/1200 = 1/25 + 1/15 + 1/75 + 1/200 Finally, there were nine sums, having odd denominators, converted from Egyptian fractions: 2/3, 1/3 (twice), 1/5, 1/7, 1/9, 1/11, 1/13 and 1/15. The British Museum examiners found no introduction or description to how or why the equivalent unit fraction series were computed. Equivalent unit fraction series are associated with fractions 1/3, 1/4, 1/8 and 1/16. There was a trivial error associated with the final 1/15 unit fraction series. The 1/15 series was listed as equal to 1/6. Another serious error was associated with 1/13, an issue that the examiners of 1927 did not attempt to resolve. == Modern analysis == The original mathematical texts never explain where the procedures and formulas came from. This holds true for the EMLR as well. Scholars have attempted to deduce what techniques the ancient Egyptians may have used to construct both the unit fraction tables of the EMLR and the 2/n tables known from the Rhind Mathematical Papyrus and the Lahun Mathematical Papyri. Both types of tables were used to aid in computations dealing with fractions, and for the conversion of measuring units. It has been noted that there are groups of unit fraction decompositions in the EMLR which are very similar. For instance lines 5 and 6 easily combine into the equation 1/3 + 1/6 = 1/2. It is easy to derive lines 11, 13, 24, 20, 21, 19, 23, 22, 25 and 26 by dividing this equation by 3, 4, 5, 6, 7, 8, 10, 15, 16 and 32 respectively. Some of the problems would lend themselves to a solution via an algorithm which involves multiplying both the numerator and the denominator by the same term and then further reducing the resulting equation: 1 p q = 1 N × N p q {\displaystyle {\frac {1}{pq}}={\frac {1}{N}}\times {\frac {N}{pq}}} This method leads to a solution for the fraction 1/8 as appears in the EMLR when using N=25 (using modern mathematical notation): 1 / 8 = 1 / 25 × 25 / 8 = 1 / 5 × 25 / 40 = 1 / 5 × ( 3 / 5 + 1 / 40 ) {\displaystyle 1/8=1/25\times 25/8=1/5\times 25/40=1/5\times (3/5+1/40)} = 1 / 5 × ( 1 / 5 + 2 / 5 + 1 / 40 ) = 1 / 5 × ( 1 / 5 + 1 / 3 + 1 / 15 + 1 / 40 ) = 1 / 25 + 1 / 15 + 1 / 75 + 1 / 200 {\displaystyle =1/5\times (1/5+2/5+1/40)=1/5\times (1/5+1/3+1/15+1/40)=1/25+1/15+1/75+1/200} == Modern conclusions == The EMLR has been considered a student scribe test document since 1927, the year that the text was unrolled at the British Museum. The scribe practiced conversions of rational numbers 1/p and 1/pq to alternative unit fraction series. Reading available Middle Kingdom math records, RMP 2/n table being one, modern students of Egyptian arithmetic may see that trained scribes improved conversions of 2/n and n/p to concise unit fraction series by applying algorithmic and non-algorithmic methods. == Chronology == The following chronology shows several milestones that marked the recent progress toward reporting a clearer understanding of the EMLR's contents, related to the RMP 2/n table. 1895 – Hultsch suggested that all RMP 2/p series were coded by aliquot parts. 1927 – Glanville concluded that EMLR arithmetic was purely additive. 1929 – Vogel reported the EMLR to be more important (than the RMP), though it contains only 25 unit fraction series. 1950 – Bruins independently confirms Hultsch's RMP 2/p analysis (Bruins 1950) 1972 – Gillings found solutions to an easier RMP problem, the 2/pq series (Gillings 1972: 95–96). 1982 – Knorr identifies RMP unit fractions 2/35, 2/91 and 2/95 as exceptions to the 2/pq problem. 2002 – Gardner identifies five abstract EMLR patterns. 2018 – Dorce explains the pattern of RMP 2/p. == See also == Egyptian mathematical texts: Akhmim Wooden Tablet Berlin Papyrus 6619 Lahun Mathematical Papyri Moscow Mathematical Papyrus Reisner Papyrus Other: Liber Abaci Sylvia Couchoud (in French) == References == == Further reading == Brown, Kevin S. The Akhmin Papyrus 1995 – Egyptian Unit Fractions 1995 Bruckheimer, Maxim and Y. Salomon. "Some Comments on R. J. Gillings' Analysis of the 2/n Table in the Rhind Papyrus." Historia Mathematica 4 Berlin (1977): 445–452. Bruins, Evert M. "Platon et la table égyptienne 2/n". Janus 46, Amsterdam, (1957): 253–263. Bruins, Evert M. "Egyptian Arithmetic." Janus 68, Amsterdam, (1981): 33–52. Bruins, Evert M. "Reducible and Trivial Decompositions Concerning Egyptian Arithmetics". Janus 68, Amsterdam, (1981): 281–297. Daressy, Georges. "Akhmim Wood Tablets", Le Caire Imprimerie de l'Institut Francais d'Archeologie Orientale, 1901, 95–96. Dorce, Carlos. "The Exact Computation of the Decompositions of the Recto Table of the Rhind Mathematical Papyrus", History Research, Volume 6, Issue 2, December 2018, 33–49. Gardner, Milo. "Mathematical Roll of Egypt", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, Nov. 2005. Gillings, Richard J. "The Egyptian Mathematical Leather Roll". Australian Journal of Science 24 (1962): 339–344, Mathematics in the Time of the Pharaohs. Cambridge, Mass.: MIT Press, 1972. New York: Dover, reprint 1982. Gillings, Richard J. "The Recto of the Rhind Mathematical Papyrus: How Did the Ancient Egyptian Scribe Prepare It ?" Archive for History of Exact Sciences 12 (1974), 291–298. Gillings, Richard J. "The Recto of the RMP and the EMLR", Historia Mathematica, Toronto 6 (1979), 442–447. Gillings, Richard J. "The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?" (Historia Mathematica 1981), 456–457. Gunn, Battiscombe George. Review of "The Rhind Mathematical Papyrus" by T. E. Peet. The Journal of Egyptian Archaeology 12 London, (1926): 123–137. Annette Imhausen. "Egyptian Mathematical Texts and their Contexts", Science in Context, vol 16, Cambridge (UK), (2003): 367–389. Legon, John A.R. "A Kahun Mathematical Fragment". Discussions in Egyptology, 24 Oxford, (1992). Lüneburg, H. "Zerlgung von Bruchen in Stammbruche" Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, Wissenschaftsverlag, Mannheim, 1993. 81–85. Rees, C. S. "Egyptian Fractions", Mathematical Chronicle 10, Auckland, (1981): 13–33. Roero, C. S. "Egyptian mathematics" Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences" I. Grattan-Guinness (ed), London, (1994): 30–45. Scott, A. and Hall, H.R., "Laboratory Notes: Egyptian Mathematical Leather Roll of the Seventeenth Century BC", British Museum Quarterly, Vol 2, London, (1927): 56. Sylvester, J. J. "On a Point in the Theory of Vulgar Fractions": American Journal of Mathematics, 3 Baltimore (1880): 332–335, 388–389. == External links == "Egyptian Mathematical Leather Roll." Gardner, Milo. MathWorld. "Egyptian Mathematical Leather Roll." Gardner, Milo. PlanetMath.org.
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Wikipedia:Egyptian algebra#0
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In the history of mathematics, Egyptian algebra, as that term is used in this article, refers to algebra as it was developed and used in ancient Egypt. Ancient Egyptian mathematics as discussed here spans a time period ranging from c. 3000 BCE to c. 300 BCE. There are limited surviving examples of ancient Egyptian algebraic problems. They appear in the Moscow Mathematical Papyrus (MMP) and in the Rhind Mathematical Papyrus (RMP), among others. == Fractions == Known mathematical texts show that scribes used (least) common multiples to turn problems with fractions into problems using integers. The multiplicative factors were often recorded in red ink and are referred to as Red auxiliary numbers. == Aha problems, linear equations and false position == Aha problems involve finding unknown quantities (referred to as Aha) if the sum of the quantity and part(s) of it are given. The Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. Problem 19 asks one to calculate a quantity taken 1 and one-half times and added to 4 to make 10. In modern mathematical notation, this linear equation is represented: 3 2 x + 4 = 10. {\displaystyle {\frac {3}{2}}x+4=10.} Solving these Aha problems involves a technique called method of false position. The technique is also called the method of false assumption. The scribe would substitute an initial guess of the answer into the problem. The solution using the false assumption would be proportional to the actual answer, and the scribe would find the answer by using this ratio. == Pefsu problems == 10 of the 25 problems of the practical problems contained in the Moscow Mathematical Papyrus are pefsu problems. A pefsu measures the strength of the beer made from a heqat of grain. pefsu = number loaves of bread (or jugs of beer) number of heqats of grain . {\displaystyle {\mbox{pefsu}}={\frac {\mbox{number loaves of bread (or jugs of beer)}}{\mbox{number of heqats of grain}}}.} A higher pefsu number means weaker bread or beer. The pefsu number is mentioned in many offering lists. For example, problem 8 translates as: (1) Example of calculating 100 loaves of bread of pefsu 20 (2) If someone says to you: "You have 100 loaves of bread of pefsu 20 (3) to be exchanged for beer of pefsu 4 (4) like 1/2 1/4 malt-date beer (5) First calculate the grain required for the 100 loaves of the bread of pefsu 20 (6) The result is 5 heqat. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer (7) The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain. (8) Calculate 1/2 of 5 heqat, the result will be 21⁄2 (9) Take this 21⁄2 four times (10) The result is 10. Then you say to him: (11) Behold! The beer quantity is found to be correct. == Geometrical progressions == The use of the Horus eye fractions shows some (rudimentary) knowledge of geometrical progression. One unit was written as 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/64. But the last copy of 1/64 was written as 5 ro, thereby writing 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + (5 ro). These fractions were further used to write fractions in terms of 1 / 2 k {\displaystyle 1/2^{k}} terms plus a remainder specified in terms of ro as shown in for instance the Akhmim wooden tablets. == Arithmetical progressions == Knowledge of arithmetic progressions is also evident from the mathematical sources. == References ==
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Wikipedia:Egyptian fraction#0
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An Egyptian fraction is a finite sum of distinct unit fractions, such as 1 2 + 1 3 + 1 16 . {\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{16}}.} That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The value of an expression of this type is a positive rational number a b {\displaystyle {\tfrac {a}{b}}} ; for instance the Egyptian fraction above sums to 43 48 {\displaystyle {\tfrac {43}{48}}} . Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including 2 3 {\displaystyle {\tfrac {2}{3}}} and 3 4 {\displaystyle {\tfrac {3}{4}}} as summands, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics. == Applications == Beyond their historical use, Egyptian fractions have some practical advantages over other representations of fractional numbers. For instance, Egyptian fractions can help in dividing food or other objects into equal shares. For example, if one wants to divide 5 pizzas equally among 8 diners, the Egyptian fraction 5 8 = 1 2 + 1 8 {\displaystyle {\frac {5}{8}}={\frac {1}{2}}+{\frac {1}{8}}} means that each diner gets half a pizza plus another eighth of a pizza, for example by splitting 4 pizzas into 8 halves, and the remaining pizza into 8 eighths. Exercises in performing this sort of fair division of food are a standard classroom example in teaching students to work with unit fractions. Egyptian fractions can provide a solution to rope-burning puzzles, in which a given duration is to be measured by igniting non-uniform ropes which burn out after a unit time. Any rational fraction of a unit of time can be measured by expanding the fraction into a sum of unit fractions and then, for each unit fraction 1 / x {\displaystyle 1/x} , burning a rope so that it always has x {\displaystyle x} simultaneously lit points where it is burning. For this application, it is not necessary for the unit fractions to be distinct from each other. However, this solution may need an infinite number of re-lighting steps. == Early history == Egyptian fraction notation was developed in the Middle Kingdom of Egypt. Five early texts in which Egyptian fractions appear were the Egyptian Mathematical Leather Roll, the Moscow Mathematical Papyrus, the Reisner Papyrus, the Kahun Papyrus and the Akhmim Wooden Tablet. A later text, the Rhind Mathematical Papyrus, introduced improved ways of writing Egyptian fractions. The Rhind papyrus was written by Ahmes and dates from the Second Intermediate Period; it includes a table of Egyptian fraction expansions for rational numbers 2 n {\displaystyle {\tfrac {2}{n}}} , as well as 84 word problems. Solutions to each problem were written out in scribal shorthand, with the final answers of all 84 problems being expressed in Egyptian fraction notation. Tables of expansions for 2 n {\displaystyle {\tfrac {2}{n}}} similar to the one on the Rhind papyrus also appear on some of the other texts. However, as the Kahun Papyrus shows, vulgar fractions were also used by scribes within their calculations. === Notation === To write the unit fractions used in their Egyptian fraction notation, in hieroglyph script, the Egyptians placed the hieroglyph: (er, "[one] among" or possibly re, mouth) above a number to represent the reciprocal of that number. Similarly in hieratic script they drew a line over the letter representing the number. For example: The Egyptians had special symbols for 1 2 {\displaystyle {\tfrac {1}{2}}} , 2 3 {\displaystyle {\tfrac {2}{3}}} , and 3 4 {\displaystyle {\tfrac {3}{4}}} that were used to reduce the size of numbers greater than 1 2 {\displaystyle {\tfrac {1}{2}}} when such numbers were converted to an Egyptian fraction series. The remaining number after subtracting one of these special fractions was written as a sum of distinct unit fractions according to the usual Egyptian fraction notation. The Egyptians also used an alternative notation modified from the Old Kingdom to denote a special set of fractions of the form 1 / 2 k {\displaystyle 1/2^{k}} (for k = 1 , 2 , … , 6 {\displaystyle k=1,2,\dots ,6} ) and sums of these numbers, which are necessarily dyadic rational numbers. These have been called "Horus-Eye fractions" after a theory (now discredited) that they were based on the parts of the Eye of Horus symbol. They were used in the Middle Kingdom in conjunction with the later notation for Egyptian fractions to subdivide a hekat, the primary ancient Egyptian volume measure for grain, bread, and other small quantities of volume, as described in the Akhmim Wooden Tablet. If any remainder was left after expressing a quantity in Eye of Horus fractions of a hekat, the remainder was written using the usual Egyptian fraction notation as multiples of a ro, a unit equal to 1 320 {\displaystyle {\tfrac {1}{320}}} of a hekat. === Calculation methods === Modern historians of mathematics have studied the Rhind papyrus and other ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form 2 n {\displaystyle {\tfrac {2}{n}}} in the Rhind papyrus. Although these expansions can generally be described as algebraic identities, the methods used by the Egyptians may not correspond directly to these identities. Additionally, the expansions in the table do not match any single identity; rather, different identities match the expansions for prime and for composite denominators, and more than one identity fits the numbers of each type: For small odd prime denominators p {\displaystyle p} , the expansion 2 p = 1 ( p + 1 ) / 2 + 1 p ( p + 1 ) / 2 {\displaystyle {\frac {2}{p}}={\frac {1}{(p+1)/2}}+{\frac {1}{p(p+1)/2}}} was used. For larger prime denominators, an expansion of the form 2 p = 1 A + 2 A − p A p {\displaystyle {\frac {2}{p}}={\frac {1}{A}}+{\frac {2A-p}{Ap}}} was used, where A {\displaystyle A} is a number with many divisors (such as a practical number) between p 2 {\displaystyle {\tfrac {p}{2}}} and p {\displaystyle p} . The remaining term ( 2 A − p ) / A p {\displaystyle (2A-p)/Ap} was expanded by representing the number 2 A − p {\displaystyle 2A-p} as a sum of divisors of A {\displaystyle A} and forming a fraction d A p {\displaystyle {\tfrac {d}{Ap}}} for each such divisor d {\displaystyle d} in this sum. As an example, Ahmes' expansion 2 37 = 1 24 + 1 111 + 1 296 {\displaystyle {\tfrac {2}{37}}={\tfrac {1}{24}}+{\tfrac {1}{111}}+{\tfrac {1}{296}}} fits this pattern with A = 24 {\displaystyle A=24} and 2 A − p = 11 = 8 + 3 {\displaystyle 2A-p=11=8+3} , as 1 111 = 8 24 ⋅ 37 {\displaystyle {\tfrac {1}{111}}={\tfrac {8}{24\cdot 37}}} and 1 296 = 3 24 ⋅ 37 {\displaystyle {\tfrac {1}{296}}={\tfrac {3}{24\cdot 37}}} . There may be many different expansions of this type for a given p {\displaystyle p} ; however, as K. S. Brown observed, the expansion chosen by the Egyptians was often the one that caused the largest denominator to be as small as possible, among all expansions fitting this pattern. For some composite denominators, factored as p ⋅ q {\displaystyle p\cdot q} , the expansion for 2 p q {\displaystyle {\tfrac {2}{pq}}} has the form of an expansion for 2 p {\displaystyle {\tfrac {2}{p}}} with each denominator multiplied by q {\displaystyle q} . This method appears to have been used for many of the composite numbers in the Rhind papyrus, but there are exceptions, notably 2 35 {\displaystyle {\tfrac {2}{35}}} , 2 91 {\displaystyle {\tfrac {2}{91}}} , and 2 95 {\displaystyle {\tfrac {2}{95}}} . One can also expand 2 p q = 1 p ( p + q ) / 2 + 1 q ( p + q ) / 2 . {\displaystyle {\frac {2}{pq}}={\frac {1}{p(p+q)/2}}+{\frac {1}{q(p+q)/2}}.} For instance, Ahmes expands 2 35 = 2 5 ⋅ 7 = 1 30 + 1 42 {\displaystyle {\tfrac {2}{35}}={\tfrac {2}{5\cdot 7}}={\tfrac {1}{30}}+{\tfrac {1}{42}}} . Later scribes used a more general form of this expansion, n p q = 1 p ( p + q ) / n + 1 q ( p + q ) / n , {\displaystyle {\frac {n}{pq}}={\frac {1}{p(p+q)/n}}+{\frac {1}{q(p+q)/n}},} which works when p + q {\displaystyle p+q} is a multiple of n {\displaystyle n} . The final (prime) expansion in the Rhind papyrus, 2 101 {\displaystyle {\tfrac {2}{101}}} , does not fit any of these forms, but instead uses an expansion 2 p = 1 p + 1 2 p + 1 3 p + 1 6 p {\displaystyle {\frac {2}{p}}={\frac {1}{p}}+{\frac {1}{2p}}+{\frac {1}{3p}}+{\frac {1}{6p}}} that may be applied regardless of the value of p {\displaystyle p} . That is, 2 101 = 1 101 + 1 202 + 1 303 + 1 606 {\displaystyle {\tfrac {2}{101}}={\tfrac {1}{101}}+{\tfrac {1}{202}}+{\tfrac {1}{303}}+{\tfrac {1}{606}}} . A related expansion was also used in the Egyptian Mathematical Leather Roll for several cases. == Later usage == Egyptian fraction notation continued to be used in Greek times and into the Middle Ages, despite complaints as early as Ptolemy's Almagest about the clumsiness of the notation compared to alternatives such as the Babylonian base-60 notation. Related problems of decomposition into unit fractions were also studied in 9th-century India by Jain mathematician Mahāvīra. An important text of medieval European mathematics, the Liber Abaci (1202) of Leonardo of Pisa (more commonly known as Fibonacci), provides some insight into the uses of Egyptian fractions in the Middle Ages, and introduces topics that continue to be important in modern mathematical study of these series. The primary subject of the Liber Abaci is calculations involving decimal and vulgar fraction notation, which eventually replaced Egyptian fractions. Fibonacci himself used a complex notation for fractions involving a combination of a mixed radix notation with sums of fractions. Many of the calculations throughout Fibonacci's book involve numbers represented as Egyptian fractions, and one section of this book provides a list of methods for conversion of vulgar fractions to Egyptian fractions. If the number is not already a unit fraction, the first method in this list is to attempt to split the numerator into a sum of divisors of the denominator; this is possible whenever the denominator is a practical number, and Liber Abaci includes tables of expansions of this type for the practical numbers 6, 8, 12, 20, 24, 60, and 100. The next several methods involve algebraic identities such as a a b − 1 = 1 b + 1 b ( a b − 1 ) . {\displaystyle {\frac {a}{ab-1}}={\frac {1}{b}}+{\frac {1}{b(ab-1)}}.} For instance, Fibonacci represents the fraction 8/11 by splitting the numerator into a sum of two numbers, each of which divides one plus the denominator: 8/11 = 6/11 + 2/11. Fibonacci applies the algebraic identity above to each these two parts, producing the expansion 8/11 = 1/2 + 1/22 + 1/6 + 1/66. Fibonacci describes similar methods for denominators that are two or three less than a number with many factors. In the rare case that these other methods all fail, Fibonacci suggests a "greedy" algorithm for computing Egyptian fractions, in which one repeatedly chooses the unit fraction with the smallest denominator that is no larger than the remaining fraction to be expanded: that is, in more modern notation, we replace a fraction x/y by the expansion x y = 1 ⌈ y x ⌉ + ( − y ) mod x y ⌈ y x ⌉ , {\displaystyle {\frac {x}{y}}={\frac {1}{\,\left\lceil {\frac {y}{x}}\right\rceil \,}}+{\frac {(-y)\,{\bmod {\,}}x}{y\left\lceil {\frac {y}{x}}\right\rceil }},} where ⌈ ⌉ represents the ceiling function; since (−y) mod x < x, this method yields a finite expansion. Fibonacci suggests switching to another method after the first such expansion, but he also gives examples in which this greedy expansion was iterated until a complete Egyptian fraction expansion was constructed: 4/13 = 1/4 + 1/18 + 1/468 and 17/29 = 1/2 + 1/12 + 1/348. Compared to ancient Egyptian expansions or to more modern methods, this method may produce expansions that are quite long, with large denominators, and Fibonacci himself noted the awkwardness of the expansions produced by this method. For instance, the greedy method expands 5 121 = 1 25 + 1 757 + 1 763 309 + 1 873 960 180 913 + 1 1 527 612 795 642 093 418 846 225 , {\displaystyle {\frac {5}{121}}={\frac {1}{25}}+{\frac {1}{757}}+{\frac {1}{763\,309}}+{\frac {1}{873\,960\,180\,913}}+{\frac {1}{1\,527\,612\,795\,642\,093\,418\,846\,225}},} while other methods lead to the shorter expansion 5 121 = 1 33 + 1 121 + 1 363 . {\displaystyle {\frac {5}{121}}={\frac {1}{33}}+{\frac {1}{121}}+{\frac {1}{363}}.} Sylvester's sequence 2, 3, 7, 43, 1807, ... can be viewed as generated by an infinite greedy expansion of this type for the number 1, where at each step we choose the denominator ⌊ y/x ⌋ + 1 instead of ⌈ y/x ⌉, and sometimes Fibonacci's greedy algorithm is attributed to James Joseph Sylvester. After his description of the greedy algorithm, Fibonacci suggests yet another method, expanding a fraction a/b by searching for a number c having many divisors, with b/2 < c < b, replacing a/b by ac/bc, and expanding ac as a sum of divisors of bc, similar to the method proposed by Hultsch and Bruins to explain some of the expansions in the Rhind papyrus. == Modern number theory == Although Egyptian fractions are no longer used in most practical applications of mathematics, modern number theorists have continued to study many different problems related to them. These include problems of bounding the length or maximum denominator in Egyptian fraction representations, finding expansions of certain special forms or in which the denominators are all of some special type, the termination of various methods for Egyptian fraction expansion, and showing that expansions exist for any sufficiently dense set of sufficiently smooth numbers. One of the earliest publications of Paul Erdős proved that it is not possible for a harmonic progression to form an Egyptian fraction representation of an integer. The reason is that, necessarily, at least one denominator of the progression will be divisible by a prime number that does not divide any other denominator. The latest publication of Erdős, nearly 20 years after his death, proves that every integer has a representation in which all denominators are products of three primes. The Erdős–Graham conjecture in combinatorial number theory states that, if the integers greater than 1 are partitioned into finitely many subsets, then one of the subsets has a finite subset of itself whose reciprocals sum to one. That is, for every r > 0, and every r-coloring of the integers greater than one, there is a finite monochromatic subset S of these integers such that ∑ n ∈ S 1 n = 1. {\displaystyle \sum _{n\in S}{\frac {1}{n}}=1.} The conjecture was proven in 2003 by Ernest S. Croot III. Znám's problem and primary pseudoperfect numbers are closely related to the existence of Egyptian fractions of the form ∑ 1 x i + ∏ 1 x i = 1. {\displaystyle \sum {\frac {1}{x_{i}}}+\prod {\frac {1}{x_{i}}}=1.} For instance, the primary pseudoperfect number 1806 is the product of the prime numbers 2, 3, 7, and 43, and gives rise to the Egyptian fraction 1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1806. Egyptian fractions are normally defined as requiring all denominators to be distinct, but this requirement can be relaxed to allow repeated denominators. However, this relaxed form of Egyptian fractions does not allow for any number to be represented using fewer fractions, as any expansion with repeated fractions can be converted to an Egyptian fraction of equal or smaller length by repeated application of the replacement 1 k + 1 k = 2 k + 1 + 2 k ( k + 1 ) {\displaystyle {\frac {1}{k}}+{\frac {1}{k}}={\frac {2}{k+1}}+{\frac {2}{k(k+1)}}} if k is odd, or simply by replacing 1/k + 1/k by 2/k if k is even. This result was first proven by Takenouchi (1921). Graham and Jewett proved that it is similarly possible to convert expansions with repeated denominators to (longer) Egyptian fractions, via the replacement 1 k + 1 k = 1 k + 1 k + 1 + 1 k ( k + 1 ) . {\displaystyle {\frac {1}{k}}+{\frac {1}{k}}={\frac {1}{k}}+{\frac {1}{k+1}}+{\frac {1}{k(k+1)}}.} This method can lead to long expansions with large denominators, such as 4 5 = 1 5 + 1 6 + 1 7 + 1 8 + 1 30 + 1 31 + 1 32 + 1 42 + 1 43 + 1 56 + 1 930 + 1 931 + 1 992 + 1 1806 + 1 865 830 . {\displaystyle {\frac {4}{5}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{30}}+{\frac {1}{31}}+{\frac {1}{32}}+{\frac {1}{42}}+{\frac {1}{43}}+{\frac {1}{56}}+{\frac {1}{930}}+{\frac {1}{931}}+{\frac {1}{992}}+{\frac {1}{1806}}+{\frac {1}{865\,830}}.} Botts (1967) had originally used this replacement technique to show that any rational number has Egyptian fraction representations with arbitrarily large minimum denominators. Any fraction x/y has an Egyptian fraction representation in which the maximum denominator is bounded by O ( y log y ( log log y ) 4 ( log log log y ) 2 ) , {\displaystyle O\left(y\log y\left(\log \log y\right)^{4}\left(\log \log \log y\right)^{2}\right),} and a representation with at most O ( log y ) {\displaystyle O\left({\sqrt {\log y}}\right)} terms. The number of terms must sometimes be at least proportional to log log y; for instance this is true for the fractions in the sequence 1/2, 2/3, 6/7, 42/43, 1806/1807, ... whose denominators form Sylvester's sequence. It has been conjectured that O(log log y) terms are always enough. It is also possible to find representations in which both the maximum denominator and the number of terms are small. Graham (1964) characterized the numbers that can be represented by Egyptian fractions in which all denominators are nth powers. In particular, a rational number q can be represented as an Egyptian fraction with square denominators if and only if q lies in one of the two half-open intervals [ 0 , π 2 6 − 1 ) ∪ [ 1 , π 2 6 ) . {\displaystyle \left[0,{\frac {\pi ^{2}}{6}}-1\right)\cup \left[1,{\frac {\pi ^{2}}{6}}\right).} Martin (1999) showed that any rational number has very dense expansions, using a constant fraction of the denominators up to N for any sufficiently large N. Engel expansion, sometimes called an Egyptian product, is a form of Egyptian fraction expansion in which each denominator is a multiple of the previous one: x = 1 a 1 + 1 a 1 a 2 + 1 a 1 a 2 a 3 + ⋯ . {\displaystyle x={\frac {1}{a_{1}}}+{\frac {1}{a_{1}a_{2}}}+{\frac {1}{a_{1}a_{2}a_{3}}}+\cdots .} In addition, the sequence of multipliers ai is required to be nondecreasing. Every rational number has a finite Engel expansion, while irrational numbers have an infinite Engel expansion. Anshel & Goldfeld (1991) study numbers that have multiple distinct Egyptian fraction representations with the same number of terms and the same product of denominators; for instance, one of the examples they supply is 5 12 = 1 4 + 1 10 + 1 15 = 1 5 + 1 6 + 1 20 . {\displaystyle {\frac {5}{12}}={\frac {1}{4}}+{\frac {1}{10}}+{\frac {1}{15}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{20}}.} Unlike the ancient Egyptians, they allow denominators to be repeated in these expansions. They apply their results for this problem to the characterization of free products of Abelian groups by a small number of numerical parameters: the rank of the commutator subgroup, the number of terms in the free product, and the product of the orders of the factors. The number of different n-term Egyptian fraction representations of the number one is bounded above and below by double exponential functions of n. == Open problems == Some notable problems remain unsolved with regard to Egyptian fractions, despite considerable effort by mathematicians. The Erdős–Straus conjecture concerns the length of the shortest expansion for a fraction of the form 4/n. Does an expansion 4 n = 1 x + 1 y + 1 z {\displaystyle {\frac {4}{n}}={\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}} exist for every n? It is known to be true for all n < 1017, and for all but a vanishingly small fraction of possible values of n, but the general truth of the conjecture remains unknown. It is unknown whether an odd greedy expansion exists for every fraction with an odd denominator. If Fibonacci's greedy method is modified so that it always chooses the smallest possible odd denominator, under what conditions does this modified algorithm produce a finite expansion? An obvious necessary condition is that the starting fraction x/y have an odd denominator y, and it is conjectured but not known that this is also a sufficient condition. It is known that every x/y with odd y has an expansion into distinct odd unit fractions, constructed using a different method than the greedy algorithm. It is possible to use brute-force search algorithms to find the Egyptian fraction representation of a given number with the fewest possible terms or minimizing the largest denominator; however, such algorithms can be quite inefficient. The existence of polynomial time algorithms for these problems, or more generally the computational complexity of such problems, remains unknown. Guy (2004) describes these problems in more detail and lists numerous additional open problems. == See also == List of sums of reciprocals 17-animal inheritance puzzle == Notes == == References == == External links == Brown, Kevin, Egyptian Unit Fractions. Eppstein, David, Egyptian Fractions. Knott, Ron, Egyptian fractions. Weisstein, Eric W., "Egyptian Fraction", MathWorld Giroux, André, Egyptian Fractions and Zeleny, Enrique, Algorithms for Egyptian Fractions, The Wolfram Demonstrations Project, based on programs by David Eppstein.
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Wikipedia:Egyptian geometry#0
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Egyptian geometry refers to geometry as it was developed and used in Ancient Egypt. Their geometry was a necessary outgrowth of surveying to preserve the layout and ownership of farmland, which was flooded annually by the Nile river. We only have a limited number of problems from ancient Egypt that concern geometry. Geometric problems appear in both the Moscow Mathematical Papyrus (MMP) and in the Rhind Mathematical Papyrus (RMP). The examples demonstrate that the ancient Egyptians knew how to compute areas of several geometric shapes and the volumes of cylinders and pyramids. == Area == The ancient Egyptians wrote out their problems in multiple parts. They gave the title and the data for the given problem, in some of the texts they would show how to solve the problem, and as the last step they verified that the problem was correct. The scribes did not use any variables and the problems were written in prose form. The solutions were written out in steps, outlining the process. Egyptian units of length are attested from the Early Dynastic Period. Although it dates to the 5th dynasty, the Palermo stone recorded the level of the Nile River during the reign of the Early Dynastic pharaoh Djer, when the height of the Nile was recorded as 6 cubits and 1 palm (about 3.217 m or 10 ft 6.7 in). A Third Dynasty diagram shows how to construct a circular vault using body measures along an arc. If the area of the Square is 434 units. The area of the circle is 433.7. The ostracon depicting this diagram was found near the Step Pyramid of Saqqara. A curve is divided into five sections and the height of the curve is given in cubits, palms, and digits in each of the sections. At some point, lengths were standardized by cubit rods. Examples have been found in the tombs of officials, noting lengths up to remen. Royal cubits were used for land measures such as roads and fields. Fourteen rods, including one double-cubit rod, were described and compared by Lepsius. Two examples are known from the Saqqara tomb of Maya, the treasurer of Tutankhamun. Another was found in the tomb of Kha (TT8) in Thebes. These cubits are 52.5 cm (20.7 in) long and are divided into palms and hands: each palm is divided into four fingers from left to right and the fingers are further subdivided into ro from right to left. The rulers are also divided into hands so that for example one foot is given as three hands and fifteen fingers and also as four palms and sixteen fingers. Surveying and itinerant measurement were undertaken using rods, poles, and knotted cords of rope. A scene in the tomb of Menna in Thebes shows surveyors measuring a plot of land using rope with knots tied at regular intervals. Similar scenes can be found in the tombs of Amenhotep-Sesi, Khaemhat and Djeserkareseneb. The balls of rope are also shown in New Kingdom statues of officials such as Senenmut, Amenemhet-Surer, and Penanhor. Triangles: The ancient Egyptians knew that the area of a triangle is A = 1 2 b h {\displaystyle A={\frac {1}{2}}bh} where b = base and h = height. Calculations of the area of a triangle appear in both the RMP and the MMP. Rectangles: Problem 49 from the RMP finds the area of a rectangular plot of land Problem 6 of MMP finds the lengths of the sides of a rectangular area given the ratio of the lengths of the sides. This problem seems to be identical to one of the Lahun Mathematical Papyri in London. The problem also demonstrates that the Egyptians were familiar with square roots. They even had a special hieroglyph for finding a square root. It looks like a corner and appears in the fifth line of the problem. Scholars suspect that they had tables giving the square roots of some often used numbers. No such tables have been found however. Problem 18 of the MMP computes the area of a length of garment-cloth. The Lahun Papyrus Problem 1 in LV.4 is given as: An area of 40 "mH" by 3 "mH" shall be divided in 10 areas, each of which shall have a width that is 1/2 1/4 of their length. A translation of the problem and its solution as it appears on the fragment is given on the website maintained by University College London. Circles: Problem 48 of the RMP compares the area of a circle (approximated by an octagon) and its circumscribing square. This problem's result is used in problem 50. That this octagonal figure, whose area is easily calculated, so accurately approximates the area of the circle is just plain good luck. Obtaining a better approximation to the area using finer divisions of a square and a similar argument is not simple. Problem 50 of the RMP finds the area of a round field of diameter 9 khet. This is solved by using the approximation that circular field of diameter 9 has the same area as a square of side 8. Problem 52 finds the area of a trapezium with (apparently) equally slanting sides. The lengths of the parallel sides and the distance between them being the given numbers. Hemisphere: Problem 10 of the MMP computes the area of a hemisphere. == Volumes == Several problems compute the volume of cylindrical granaries (41, 42, and 43 of the RMP), while problem 60 RMP seems to concern a pillar or a cone instead of a pyramid. It is rather small and steep, with a seked (slope) of four palms (per cubit). A problem appearing in section IV.3 of the Lahun Mathematical Papyri computes the volume of a granary with a circular base. A similar problem and procedure can be found in the Rhind papyrus (problem 43). Several problems in the Moscow Mathematical Papyrus (problem 14) and in the Rhind Mathematical Papyrus (numbers 44, 45, 46) compute the volume of a rectangular granary. Problem 14 of the Moscow Mathematical Papyrus computes the volume of a truncated pyramid, also known as a frustum. == Seked == Problem 56 of the RMP indicates an understanding of the idea of geometric similarity. This problem discusses the ratio run/rise, also known as the seked. Such a formula would be needed for building pyramids. In the next problem (Problem 57), the height of a pyramid is calculated from the base length and the seked (Egyptian for slope), while problem 58 gives the length of the base and the height and uses these measurements to compute the seked. In Problem 59 part 1 computes the seked, while the second part may be a computation to check the answer: If you construct a pyramid with base side 12 [cubits] and with a seked of 5 palms 1 finger; what is its altitude? == References == == Bibliography == Clagett, Marshall (1999). Ancient Egyptian Science: A Source Book, Vol. III: Ancient Egyptian Mathematics. Memoirs of the APS, Vol. 232. Philadelphia: American Philosophical Society. ISBN 978-0-87169-232-0. Lepsius, Karl Richard (1865). Die Alt-Aegyptische Elle und Ihre Eintheilung (in German). Berlin: Dümmler.
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Wikipedia:Egyptian numerals#0
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The system of ancient Egyptian numerals was used in Ancient Egypt from around 3000 BC until the early first millennium AD. It was a system of numeration based on multiples of ten, often rounded off to the higher power, written in hieroglyphs. The Egyptians had no concept of a positional notation such as the decimal system. The hieratic form of numerals stressed an exact finite series notation, ciphered one-to-one onto the Egyptian alphabet. == Digits and numbers == The following hieroglyphs were used to denote powers of ten: Multiples of these values were expressed by repeating the symbol as many times as needed. For instance, a stone carving from Karnak shows the number 4,622 as: Egyptian hieroglyphs could be written in both directions (and even vertically). In this example the symbols decrease in value from top to bottom and from left to right. On the original stone carving, it is right-to-left, and the signs are thus reversed. == Zero == There was no symbol or concept of zero as a placeholder in Egyptian numeration and zero was not used in calculations. However, the symbol nefer (nfr𓄤, "good", "complete", "beautiful") was apparently also used for two numeric purposes: in a papyrus listing the court expenses, c. 1740 BC, it indicated a zero balance; in a drawing for Meidum Pyramid (and at other sites), nefer is used to indicate a ground level: height and depths are measured "above nefer" or "below nefer" respectively. According to Carl Boyer, a deed from Edfu contained a "zero concept" replacing the magnitude in geometry. == Fractions == Rational numbers could also be expressed, but only as sums of unit fractions, i.e., sums of reciprocals of positive integers, except for 2⁄3 and 3⁄4. The hieroglyph indicating a fraction looked like a mouth, which meant "part": Fractions were written with this fractional solidus, i.e., the numerator 1, and the positive denominator below. Thus, 1⁄3 was written as: Special symbols were used for 1⁄2 and for the non-unit fractions 2⁄3 and, less frequently, 3⁄4: If the denominator became too large, the "mouth" was just placed over the beginning of the "denominator": == Written numbers == As with most modern day languages, the ancient Egyptian language could also write out numerals as words phonetically, just like one can write thirty instead of "30" in English. The word (thirty), for instance, was written as while the numeral (30) was This was, however, uncommon for most numbers other than one and the signs were used most of the time. == Hieratic numerals == As administrative and accounting texts were written on papyrus or ostraca, rather than being carved into hard stone (as were hieroglyphic texts), the vast majority of texts employing the Egyptian numeral system utilize the hieratic script. Instances of numerals written in hieratic can be found as far back as the Early Dynastic Period. The Old Kingdom Abusir Papyri are a particularly important corpus of texts that utilize hieratic numerals. Boyer proved 50 years ago that hieratic script used a different numeral system, using individual signs for the numbers 1 to 9, multiples of 10 from 10 to 90, the hundreds from 100 to 900, and the thousands from 1000 to 9000. A large number like 9999 could thus be written with only four signs—combining the signs for 9000, 900, 90, and 9—as opposed to 36 hieroglyphs. Boyer saw the new hieratic numerals as ciphered, mapping one number onto one Egyptian letter for the first time in human history. Greeks adopted the new system, mapping their counting numbers onto two of their alphabets, the Doric and Ionian. In the oldest hieratic texts the individual numerals were clearly written in a ciphered relationship to the Egyptian alphabet. But during the Old Kingdom a series of standardized writings had developed for sign-groups containing more than one numeral, repeated as Roman numerals practiced. However, repetition of the same numeral for each place-value was not allowed in the hieratic script. As the hieratic writing system developed over time, these sign-groups were further simplified for quick writing; this process continued into Demotic, as well. Two famous mathematical papyri using hieratic script are the Moscow Mathematical Papyrus and the Rhind Mathematical Papyrus. == Egyptian words for numbers == The following table shows the reconstructed Middle Egyptian forms of the numerals (which are indicated by a preceding asterisk), the transliteration of the hieroglyphs used to write them, and finally the Coptic numerals which descended from them and which give Egyptologists clues as to the vocalism of the original Egyptian numbers. A breve (˘) in some reconstructed forms indicates a short vowel whose quality remains uncertain; the letter 'e' represents a vowel that was originally u or i (exact quality uncertain) but became e by Late Egyptian. == See also == Egyptian language Egyptian mathematics == References == == Bibliography == Allen, James Paul (2000). Middle Egyptian: An Introduction to the Language and Culture of Hieroglyphs. Cambridge: Cambridge University Press. Numerals discussed in §§9.1–9.6. Allen, James P. (2014). Middle Egyptian: An Introduction to the Language and Culture of Hieroglyphs (Third ed.). Cambridge University Press. ISBN 978-1-107-66328-2. Gardiner, Alan Henderson (1957). Egyptian Grammar; Being an Introduction to the Study of Hieroglyphs. 3rd ed. Oxford: Griffith Institute. For numerals, see §§259–266. Goedicke, Hans (1988). Old Hieratic Paleography. Baltimore: Halgo, Inc. Hoffmann, Friedhelm (2024-03-11). "Aspects of Zero in Ancient Egypt". In Gobets, Peter; Lawrence Kuhn, Robert (eds.). The Origin and Significance of Zero: An Interdisciplinary Perspective. Brill. pp. 64–81. doi:10.1163/9789004691568_007. ISBN 978-90-04-69156-8. Joseph, G.G. (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton University Press. ISBN 978-0-691-13526-7. Retrieved 2024-05-03. Möller, Georg (1927). Hieratische Paläographie: Die Ägyptische Buchschrift in ihrer Entwicklung von der Fünften Dynastie bis zur römischen Kaiserzeit. 3 vols. 2nd ed. Leipzig: J. C. Hinrichs Schen Buchhandlungen. (Reprinted Osnabrück: Otto Zeller Verlag, 1965) == External links == Introduction to Hieroglyphs Numbers and Fractions at the Wayback Machine (archived September 29, 2007) Numbers and dates at the Wayback Machine (archived March 4, 2001) Egyptian Numbers at the Wayback Machine (archived January 12, 2004) Egyptian Math History
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Wikipedia:Ehrenpreis's fundamental principle#0
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In mathematical analysis, Ehrenpreis's fundamental principle, introduced by Leon Ehrenpreis, states: Every solution of a system (in general, overdetermined) of homogeneous partial differential equations with constant coefficients can be represented as the integral with respect to an appropriate Radon measure over the complex “characteristic variety” of the system. == References ==
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Wikipedia:Ehud Hrushovski#0
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Ehud Hrushovski (Hebrew: אהוד הרושובסקי; born 30 September 1959) is a mathematical logician. He is a Merton Professor of Mathematical Logic at the University of Oxford and a Fellow of Merton College, Oxford. He was also Professor of Mathematics at the Hebrew University of Jerusalem. == Early life and education == Hrushovski's father, Benjamin Harshav (Hebrew: בנימין הרשב, né Hruszowski; 1928–2015), was a literary theorist, a Yiddish and Hebrew poet and a translator, professor at Yale University and Tel Aviv University in comparative literature. Ehud Hrushovski earned his PhD from the University of California, Berkeley in 1986 under Leo Harrington; his dissertation was titled Contributions to Stable Model Theory. He was a professor of mathematics at the Massachusetts Institute of Technology until 1994, when he became a professor at the Hebrew University of Jerusalem. Hrushovski moved in 2017 to the University of Oxford, where he is the Merton Professor of Mathematical Logic. == Career == Hrushovski is well known for several fundamental contributions to model theory, in particular in the branch that has become known as geometric model theory, and its applications. His PhD thesis revolutionized stable model theory (a part of model theory arising from the stability theory introduced by Saharon Shelah). Shortly afterwards he found counterexamples to the Trichotomy Conjecture of Boris Zilber and his method of proof has become well known as Hrushovski constructions and found many other applications since. One of his most famous results is his proof of the geometric Mordell–Lang conjecture in all characteristics using model theory in 1996. This deep proof was a landmark in logic and geometry. He has had many other famous and notable results in model theory and its applications to geometry, algebra, and combinatorics. === Honours and awards === He was an invited speaker at the 1990 International Congress of Mathematicians and a plenary speaker at the 1998 ICM. He is a recipient of the Erdős Prize of the Israel Mathematical Union in 1994, the Rothschild Prize in 1998, the Karp Prize of the Association for Symbolic Logic in 1993 (jointly with Alex Wilkie) and again in 1998, In 2007, he was honored with holding the Gödel Lecture. In his absence, a lecture on his work titled Algebraic Model Theory was given by Thomas Scanlon. In 2019 he was awarded the Heinz Hopf Prize and in 2022 the Shaw Prize in Mathematical Sciences. Hrushovski is a fellow of the American Academy of Arts and Sciences (2007), and Israel Academy of Sciences and Humanities (2008). He was elected a Fellow of the Royal Society in 2020. == References == == External links == Homepage Prof. Ehud Hrushovski Ehud Hrushovski at the Hebrew University of Jerusalem
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Wikipedia:Ehud de Shalit#0
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Ehud de Shalit (Hebrew: אהוד דה שליט; born 16 March 1955) is an Israeli number theorist and professor at the Hebrew University of Jerusalem. == Biography == Ehud de Shalit was born in Rehovot. His father was Amos de-Shalit. He completed his B.Sc. at the Hebrew University in 1975, and his Ph.D. at Princeton University in 1984 under the supervision of Andrew Wiles. == Academic career == De Shalit joined the faculty of Hebrew University in 1987 and was promoted to full professor in 2001. He is an editor for the Israel Journal of Mathematics. == Published works == De Shalit, Ehud (2001). "Residues on buildings and de Rham cohomology of p {\displaystyle p} -adic symmetric domains". Duke Mathematical Journal. 106 (1): 123–191. doi:10.1215/s0012-7094-01-10615-7. De Shalit, Ehud (1989). "Eichler cohomology and periods of modular forms on p {\displaystyle p} -adic Schottky groups". Journal für die reine und angewandte Mathematik. 1989 (400): 3–31. doi:10.1515/crll.1989.400.3. S2CID 118777849. Coleman, Robert; de Shalit, Ehud (1988). " p {\displaystyle p} -adic regulators on curves and special values of p {\displaystyle p} -adic L {\displaystyle L} -functions". Inventiones Mathematicae. 93 (2): 239–266. Bibcode:1988InMat..93..239C. doi:10.1007/bf01394332. S2CID 122242212. De Shalit, Ehud (1987). Iwasawa theory of elliptic curves with complex multiplication. Perspectives in Mathematics. Boston: Academic Press. ISBN 978-0-12-210255-4. OCLC 256787655. De Shalit, Ehud (1985). "Relative Lubin-Tate Groups" (PDF). Proceedings of the American Mathematical Society. 95 (1): 1–4. doi:10.2307/2045561. JSTOR 2045561. == References == == External links == Ehud de Shalit at the Mathematics Genealogy Project
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Wikipedia:Eigengap#0
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In linear algebra, the eigengap of a linear operator is the difference between two successive eigenvalues, where eigenvalues are sorted in ascending order. The Davis–Kahan theorem, named after Chandler Davis and William Kahan, uses the eigengap to show how eigenspaces of an operator change under perturbation. In spectral clustering, the eigengap is often referred to as the spectral gap; although the spectral gap may often be defined in a broader sense than that of the eigengap. == See also == Eigenvalue perturbation == References ==
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