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# Topic: Partition function (number theory)

###### In the News (Tue 22 Aug 17)

 11: Number theory Number theory is one of the oldest branches of pure mathematics, and one of the largest. For example, "additive number theory" asks about ways of expressing an integer N as a sum of integers a_i in a set A. If we set f(z) = Sum exp(2 pi i a_i z), then f(z)^k has exp(2 pi i N z) as a summand iff N is a sum of k of the a_i. Questions in algebraic number theory often require tools of Galois theory; that material is mostly a part of 12: Field theory (particularly the subject of field extensions). www.math.niu.edu /~rusin/known-math/index/11-XX.html   (2587 words)

 in theory We shall construct a partition $P= (B_1,\ldots,B_m)$ of $\{0,1\}^n$ such that: (i) given $x$, we can efficiently compute what is the block $B_i$ that $x$ belongs to; (ii) the number $m$ of blocks does not depend on $n$; (iii) $g$ restricted to most blocks $B_i$ behaves like a random function of the same density. A number of proofs of the hard core theorem are known, and connections have been found with the process of boosting in learning theory and with the construction and the decoding of certain error-correcting codes. Marge, I agree with you - in theory. in-theory.blogspot.com   (7009 words)

 Arithmetic, Numeration, Number Theory - Numericana This allows the GCD of two commensurable numbers to be defined as well: Two real numbers are commensurable iff they are proportional to two integers; Their GCD is simply the GCD of those integers times the common scaling factor. The next two numbers in the list, the 13th and 14th Mersenne primes, are much larger (corresponding to n=521 and n=607) and were both discovered the same day (January 30, 1952, around 22:00 PST and shortly before midnight) by Raphael Mitchel Robinson (1911-1995), at the dawn of the computer age. This coefficient is indeed obtained by counting the number of ways there is to choose an exponent multiple of 1 from the first factor, a multiple of 2 from the second factor, a multiple of 3 from the third, etc. so these exponents add up to n. home.att.net /~numericana/answer/numbers.htm   (7639 words)

 Info on Numerical Partitions The number of partitions of n is denoted p(n) and the number of partitions of n into k parts is denoted p(n,k). The number of numerical partition for n = 1,2,...,15, is p(n) = 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176. A Ferrers diagram is a pictorial representation of a partition. www.theory.csc.uvic.ca /~cos/inf/nump/NumPartition.html   (594 words)

 PlanetMath: partition function is defined to be the number of partitions of the integer This is version 6 of partition function, born on 2006-06-09, modified 2006-09-30. (Combinatorics :: Enumerative combinatorics :: Partitions of integers) planetmath.org /encyclopedia/PartitionFunction2.html   (138 words)

 Ivars Peterson's MathLand One can prove that for a given whole number, the number of partitions in which all the parts are odd always equals the number of partitions in which all the parts are distinct. In general, the partition function p(n) is the number of partitions of n. Here are the values of the partition function for the integers from 1 to 21: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, and 792. www.maa.org /mathland/mathland_3_24.html   (816 words)

 Rational Numbers: Toward a Semantic Analysis - Emphasis On The Operator Construct Both present a notion of rational number as an exchange function-that is, they exchange the operand quantity of a rational number operator to a conceptually new quantity that has a ratio to the original quantity equal to the numerator-to-denominator ratio. These include: (a) ability to partition quantities, (b) flexibility in formation and re-formation of units, (c) understanding of and ability to perform partitive division, (d) an understanding of the concept of function as a mapping, and (e) skill with and understanding of multiplication as repeated addition. For the stretcher/shrinker operator interpretation of rational number, we consider the numerator to be a stretcher and the denominator a shrinker. www.education.umn.edu /rationalnumberproject/93_1.html   (7206 words)

 BIBLIOGRAPHY On the theorems of Watson and Dragonette for Ramanujan's mock theta functions. On the geometry of numbers in elementary number theory. On a conjecture of Guinand for the plane partition function. www.math.psu.edu /andrews/biblio.html   (2652 words)

 Abstracts   (Site not responding. Last check: ) This function is well defined for those integral vectors b that lie in the nonnegative span of the columns of A (of which we assume that it does not contain a line). Using a bijection between partitions and vacillating tableaus, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. As a corollary, the number of $k$-noncrossing partitions is equal to the number of $k$-nonnesting partitions. www.theoryofnumbers.com /CANT/2005/abstracts.htm   (4450 words)

 Partition function - Wikipedia, the free encyclopedia In number theory, see partition function (number theory). In statistical mechanics, see partition function (statistical mechanics). In quantum field theory, see partition function (quantum field theory). en.wikipedia.org /wiki/Partition_function   (89 words)

 Number Theory - Euler Number Theory is the area of mathematics concerned primarily with integer (sometimes rational) solutions to expressions. Fermat "prime" is not at all prime, and this one counterexample shatters Fermat's far-reaching conjecture. If we begin with a distinct partition of n, we may write each of the addends as a power of 2 multiplied by an odd number, and then write n as the sum of these odd numbers repeated an appropriate number of times. members.aol.com /tylern7/math/euler-6.html   (1564 words)

 TGD abstracts One general idea which results as an outcome of the generalized notion of number is the idea of a universal function continuable from a function mapping rationals to rationals or to a finite extension of rationals to a function in any number field. Number theoretic entropies can be negative and provide genuine information measures, and it turns that bound states should correspond in TGD framework to entanglement coefficients which belong to a finite-dimensional extension of rationals and have negative number theoretic entanglement entropy. Since Riemann ζ function allows interpretation as a thermodynamical partition function for a quantum field theoretical system consisting of bosons labelled by primes, it is interesting to look Riemann hypothesis from the perspective of physics. www.helsinki.fi /~matpitka/tgdnumber/atgdnumber.html   (10285 words)

 The Operator Construct of Rational Number: A Refinement of the Concept The denominator as a partition-reducer suggests that its effect on an operand is to partition that whole quantity into a number of parts equal to the denominator and then to adjust the quantity to the size of one of its parts. It suggests that 3/1 and 1/4 are 3-for-1 and 1-for-4 exchange functions; and consequently, that 3/4 is a 3-for-4 exchange function. The partition to go from step d to e (Figure 1) is more complex with 3/1 applied first because of perceptual distractors, reunitizing the 3(8-unit)s to (24-unit)s and then to 4(6-unit)s might be less complex. education.umn.edu /rationalnumberproject/91_1.html   (1541 words)

 Periodbot output starting #22762360 The formulation of Euler's generating function is a special case of a [[q-series]] and is similar to the product formulation of many [[modular form]]s, and specifically the [[Dedekind eta function]]. Some works of science fiction advance a derivative of the theory as a rationalization for the improbable tendency of fictional extra-terrestrials to be strongly humanoid in form as well as living on earth-compatible worlds (see [[Class M planet]]). A distinction is sometimes made between anti-microbial ''preservatives'' which function by inhibiting the growth of [[insect]]s, [[bacteriumbacteria]] and [[fungusfungi]], and ''[[antioxidant]]s'', which inhibit the [[oxidation]] of food constituents. www.cs.cmu.edu /~tom7/periodbot/377.html   (2469 words)

 SCISEARCH - A CITED REFERENCE SCIENCE DATABASE [34,434] Second example includes the "weight" (002) assigned to the RF Code (indicates number of citations in common with the original Research Front cluster). Title, Genuine Article Number, Number of References, and Publication Date If the accession number of a specific record is known, it can be used to display the record directly. library.dialog.com /bluesheets/html/bl0034.html   (869 words)

 Statistical mechanics: the Riemann zeta function interpreted as a partition function One of the earliest, and perhaps most significant, examples of number theory influencing the development of physics was the application of Pólya's work on the Riemann zeta function to the theory of phase transitions by Lee and Yang in the early 1950's. In the theory of the distribution of primes, the fundamental object is the Riemann zeta function. A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the single particle density. www.secamlocal.ex.ac.uk /people/staff/mrwatkin/zeta/physics2.htm   (8406 words)

 [No title]   (Site not responding. Last check: ) Dennis Eichhorn - California State University "Does the number of t-cores rise as t increases in size?" Abstract: In 1999, Stanton conjectured that for t > 3 and n > t + 1, the number of (t+1)-core partitions of n is greater than or equal to the number of t-core partitions of n. The upper bound for the number of non—zero coefficients in the polynomial numerators of Hilbert series $H({\bf d}^m;z)$ of graded subrings for non--symmetric semigroups ${\sf S} ({\bf d}^m)$ of dimension, $m\geq 4$, is established. As a corollary, the number of $k$-non-crossing partitions is equal to the number of $k$-non-nesting partitions. www.theoryofnumbers.com /CANT/2005/cant2005_abstract.doc   (2488 words)

 No Title For Fourier the notion of function was rooted in the 18 For Fourier, a general function was one whose graph is smooth except for a finite number of exceptional points. Theorem I. is a primitive function; that is F'=f www.math.tamu.edu /~don.allen/history/riemann/riemann.html   (1521 words)

 [No title] The free Riemann gas is a surprisingly natural concept, and its partition function is identical to the zeta function. In statistical mechanics, the partition function is the fundamental mathematical object of study; in the analytic theory of the distribution of primes, the zeta function is the fundamental object. Hence this unorthodox interpretation of the zeta function as a partition function points to a possible link of fundamental significance between the distribution of primes and this branch of physics. www.lycos.com /info/prime-numbers.html   (517 words)

 The On-Line Encyclopedia of Integer Sequences a(n) = number of partitions of n (the partition numbers). Number of distinct Abelian groups of order p^n, where p is prime (the number is independent of p). The n-th partition number arises as the number of terms in the numerator of the expression for c_n: The coefficient c_n of the inverted power series is a fraction with b_0^(n+1) in the denominator and in its numerator having a(n) products of n coefficients b_i each. www.research.att.com /projects/OEIS?Anum=A000041   (1495 words)

 OUP: UK General Catalogue In combinatorial number theory, exact formulas for number-theoretic quantities are derived from relations between analytic functions. Farkas and Kra, well-known masters of the theory of Riemann surfaces and the analysis of theta functions, uncover here interesting combinatorial identities by means of the function theory on Riemann surfaces related to the principal congruence subgroups $\Gamma(k)$. The authors also obtain a variant on Jacobi's famous result on the number of ways that an integer can be represented as a sum of four squares, replacing the squares by triangular numbers and, in the process, obtaining a cleaner result. www.oup.com /uk/catalogue/?ci=9780821813928   (589 words)

 math.html She found them using the theory of modular forms in a completely new and clever way. A partition function essentially counts the number of ways you can write a positive integer as a positive integer, according to Bruce Berndt, professor of mathematics at the University of Illinois, who is known as a leading expert on the work of Ramanujan. The partitions are the number four by itself, three plus one, two plus two, two plus one plus one, and one plus one plus one plus one. www.psu.edu /ur/archives/intercom_2000/May18/math.html   (770 words)

 Science News Online - Ivars Peterson's MathLand - 3/22/97 Indeed, "anytime the number of ways of writing a number as the sum of other numbers arises, the theory of partitions can't be far off," says number theorist George E. Andrews of Pennsylvania State University. Recently, Ken Ono of the Institute for Advanced Study in Princeton, N.J., proved some results involving the parity of the partition function and sets of integers known as arithmetic progressions. This effort formed the basis of his submission to the 56th Westinghouse Science Talent Search, from which he emerged as the third-place winner when the awards were announced earlier this month. www.sciencenews.org /pages/sn_arc97/3_22_97/mathland.htm   (828 words)

 numbertheory.html   (Site not responding. Last check: ) Andrew Odlyzko: Papers on Number Theory (that are not primarily computational nor deal with zeros of zeta functions) Erdos and A. Odlyzko, J. Number Theory, 11 (1979), pp. MacWilliams and A. Odlyzko, J. Combinatorial Theory A, 22 (1977), pp. www.dtc.umn.edu /~odlyzko/doc/numbertheory.html   (486 words)

 Amazon.ca: An Introduction to the Theory of Numbers, 5th Edition: Books: Ivan Niven,Herbert S. Zuckerman,Hugh L. ...   (Site not responding. Last check: ) by Ivan Niven (Author), Herbert S. Zuckerman (Author), Hugh L. Montgomery (Author) "The theory of numbers is concerned with properties of the natural numbers 1,2,3,4,... The Fifth Edition of one of the standard works on number theory, written by internationally-recognized mathematicians. The theory of numbers is concerned with properties of the natural numbers 1,2,3,4,... www.amazon.ca /Introduction-Theory-Numbers-5th/dp/0471625469   (717 words)

 NTU Info Centre: List of mathematical topics (P-R)   (Site not responding. Last check: ) ]] -- [[Partition of an interval]] [[Talk:Partition of an interval ]] -- [[Partition of unity]] [[Talk:Partition of unity ]] -- [[Persistence of a number]] [[Talk:Persistence of a number www.nowtryus.com /article:List_of_mathematical_topics_(P-R)?source=true   (1327 words)

 Transition State Theory The molecular partition function gives an indication of the average number of states that are thermally accessible to a molecule at the temperature of the system. The prime again denotes the fact that the transitional partition function, and hence the overall molecular partition function, is on per unit volume. is the partition function per unit volume with the partition function for the vibration frequency for crossing removed. www.engin.umich.edu /~CRE/03chap/html/transition   (2545 words)

 The Top Twenty: Partitions The number of (unrestrict) partiton of n, denoted p(n), s the number of ways of writing the integer n as a sum of positive integers. Wright, An introduction to the theory of numbers, Oxford University Press, ISBN 0198531702. Ono, "Distribution of the partition function modulo m," Ann. primes.utm.edu /top20/page.php?id=54   (255 words)

 Math 571 Analytic Number Theory I Another is to show that every large odd number is the sum of three prime numbers (the ternary Goldbach (1690-1764) problem). A related open question is whether every even number is prime or the sum of two primes (the binary Goldbach problem), and an associated question is whether there are infinitely many primes p such that p + 2 is also prime (the twin prime problem). Fundamental to many of the analytic methods in number theory are questions as to how closely a given real number can be approximated by a rational number with denominator not exceeding a given quantity, and generalisations of this are related to Minkowski's theorem in the geometry of numbers. www.math.psu.edu /rvaughan/Math571F04.html   (447 words)

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