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Canonical representation genetic programming

John R. Woodward, Ruibin Bai
2009 Proceedings of the first ACM/SIGEVO Summit on Genetic and Evolutionary Computation - GEC '09  
When the search space consists of canonical representations it is called a canonical search space, and when Genetic Programming is applied to this search space, it is called Canonical Representation Genetic  ...  It is argued that, if a search space can be constructed where only unique representations of a function are permitted, then this will be more successful than employing multiple representations.  ...  Primitive Set {+, −, * , /} ∪ {1} Given {+, −, * , /} ∪ {1}, we can generate rational numbers. Rational numbers, as we have seen, have a number of canonical representations.  ... 
doi:10.1145/1543834.1543914 dblp:conf/gecco/WoodwardB09 fatcat:ljidrgczpnhyziaxq47dvsaa44

Free actions of finite groups on rational homology 3-spheres

D. Cooper, D.D. Long
2000 Topology and its Applications  
We show that any finite group can act freely on a rational homology 3-sphere.  ...  The number of times a simple module appears (up to isomorphism) in a decomposition of M into simple submodules is independent of the decomposition [2, p. 440 ].  ...  If we consider the decompositions of both sides into simple submodules and compare the number of times each simple module appears, we deduce that ker(i * ) ∼ = Im(i * ) ∼ = (L G ) k .  ... 
doi:10.1016/s0166-8641(98)00116-3 fatcat:nxmw33g25rag7mjh4tps4cyx6m

Page 4001 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews  
This paper presents the character tables for the irreducible rational representations of SL(2, p”) for all values of the prime p and exponent n.  ...  of V®”", so the problem of calculating the Kazhdan-Lusztig polynomials is a special case of the problem of calculating the canonical basis in a tensor power of the funda- mental representation of U,sl  ... 

LCF: A Lexicographic Binary Representation of the Rationals [chapter]

Peter Kornerup, David W. Matula
1996 J.UCS The Journal of Universal Computer Science  
in the representation of any reasonably sized rational number.  ...  Worst case representation-induced precision loss for any real number by a xed length representable number of the system is shown to be at most 19 of bit word length, with no precision loss whatsoever induced  ...  Acknowledgements This paper reports some previously unpublished results presented as part of an invited talk entitled "Rational Number Systems and Arithmetic", given by the rst author at the symposium  ... 
doi:10.1007/978-3-642-80350-5_41 fatcat:3bok4cu4mrb7tnfxwn23g72moa

A McKay Correspondence in Positive Characteristic [article]

Christian Liedtke
2022 arXiv   pre-print
In dimension 2, we establish simultaneous resolutions of singularities of these canonical lifts via G-Hilbert schemes.  ...  We discuss linearly reductive quotient singularities and canonical lifts over the ring of Witt vectors.  ...  In this case, the number of isomorphism classes of simple representations is equal to the number of conjugacy classes.  ... 
arXiv:2207.06286v1 fatcat:k56czlrhtbd73j2zllygl6nc5u

Euler and algebraic geometry

Burt Totaro
2007 Bulletin of the American Mathematical Society  
In section 1, we reach a major open problem of algebraic geometry: which representations of the fundamental group are summands of the cohomology of some family of algebraic varieties?  ...  from rational curves, and not only in a negative way.  ...  in all the rational numbers t.  ... 
doi:10.1090/s0273-0979-07-01178-0 fatcat:faenzkrkv5fnthovcirdagixga

Dibaryons as canonically quantized biskyrmions [article]

T. Krupovnickas, E. Norvaišas, D.O. Riska
2000 arXiv   pre-print
The ground state of the skyrmion with baryon number 2 is axially symmetric, and is well approximated by a simple rational map, which represents a direct generalization of Skyrme's hedgehog ansatz for the  ...  If the Lagrangian density is canonically quantized this configuration may support excitations that lie close and possible below the threshold for pion decay, and therefore describe dibaryons.  ...  Research supported in part by the Academy of Finland by grants No. 43982 and 44903.  ... 
arXiv:nucl-th/0011063v1 fatcat:n6y7k5c2cjeszhcplvxj53i5iq

Page 2054 of Mathematical Reviews Vol. , Issue 90D [page]

1990 Mathematical Reviews  
(S @ k)) be the canonical /-adic representation (/ a prime number). Then the canonical map NS(S @, k) @z Q; — H?(S @, k)(1)4°'™) is an isomorphism (NS = Néron-Severi group).  ...  Surfaces of type K 3 over number fields, and /-adic representations. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 1252-1271, 1328; translation in Math.  ... 

Page 2802 of Mathematical Reviews Vol. , Issue 97E [page]

1997 Mathematical Reviews  
The canonical local height is given in terms of a local integral of a simple polynomial function over the complete curve.  ...  The paper under review provides and discusses bounds for the number of Q-rational elliptic curves with good reduction outside of a finite set S of rational primes.  ... 

Representation, simplification and display of fractional powers of rational numbers in computer algebra [article]

Albert D. Rich, David R. Stoutemyer
2013 arXiv   pre-print
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra.  ...  Since they are such a restricted subset of algebraic numbers, it might seem that good simplification of them must already be implemented in all widely used computer algebra systems.  ...  One way to make it canonical is: Whenever an input or a tentative result is a rational number times a fractional power of a rational number: 1.  ... 
arXiv:1302.2169v1 fatcat:gzvoeosmyvd4pftng32dq4oj6i

On Laguerre Series

R. E. Greenwood
1940 Proceedings of the National Academy of Sciences of the United States of America  
of like elements in the canonical (basis) representation of integers of the ring. 4.  ...  The passage from simple to compound systems in the ring of rational integers depends only on the existence of a G. C. D. for rational integers.  ... 
doi:10.1073/pnas.26.7.466 pmid:16588385 pmcid:PMC1078211 fatcat:67yqrw72dnce5esjsoozu6znne

Compound Multiplicative Diophantine Systems

E. T. Bell
1940 Proceedings of the National Academy of Sciences of the United States of America  
of like elements in the canonical (basis) representation of integers of the ring. 4.  ...  The passage from simple to compound systems in the ring of rational integers depends only on the existence of a G. C. D. for rational integers.  ... 
doi:10.1073/pnas.26.7.462 pmid:16588384 pmcid:PMC1078210 fatcat:mhfdyp7gfzgqhgwyhfbjthtd3m

The rational invariants of the tame quivers

Claus Michael Ringel
1980 Inventiones Mathematicae  
Let n, be the number of simple regular representations in ~,. Let V = V~|174 ~ be a direct sum decomposition with V i indecomposable.  ...  We denote by n t the number of simple regular representations in ~t-We will also have to consider the special quiver A=o~ (which has an oriented cycle).  ...  Using again the multiplicativity of F on extensions, one sees immediately, that for any regular representation ((p~.,,) we find a representation (~.,) in  ... 
doi:10.1007/bf01390253 fatcat:rr4io2k46nakvke3a7richlhri

Page 665 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 50, Issue 9 [page]

1944 American Mathematical Society. Bulletin of the American Mathematical Society  
Let S[n, g(x) ]=>02",e(3)/p*, where the summation is over numbers prime to p, and p is a rational prime.  ...  The question of the invariance of the representation is discussed in the case (1).  ... 

REPRESENTATIONS OF THE COMMUTATION RELATIONS

L. Garding, A. Wightman
1954 Proceedings of the National Academy of Sciences of the United States of America  
The canonical representation is the only one for which the number of particles operator 2 ak*ak exists. Letting M be concentrated on a Ar, we obtain a translated canonical representation.  ...  Let A denote the set of "rational" sequences, i.e., those with only a finite number of components different from zero.  ... 
doi:10.1073/pnas.40.7.622 pmid:16589534 pmcid:PMC528006 fatcat:7fo2vl75zfcs7k3plbk4twlwcm
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