In mathematics, the Lichnerowicz conjecture is a generalization of a conjecture introduced by harvs txt last Lichnerowicz authorlink Andr Lichnerowicz year 1944 . Lichnerowicz s original conjecture was that locally harmonic 4 manifolds are locally symmetric, and was proved by harvtxt Walker 1949 . The Lichnerowicz conjecture usually refers to the generalization that locally harmonic manifolds are flat or rank 1 locally symmetric. References Citation last1 Lichnerowicz first1 Andr title Sur les espaces riemanniens compl tement harmoniques url http www.numdam.org item?id BSMF 1944 72 146 0 id MR 0012886 year 1944 journal Bulletin de la Soci t Math matique de France issn 0037 9484 volume 72 pages 146 168 Citation last1 Szab first1 Z. I. title The Lichnerowicz conjecture on harmonic manifolds url http projecteuclid.org getRecord?id euclid.jdg 1214444087 id MR 1030663 year 1990 journal Journal of Differential Geometry issn 0022 040X volume 31 issue 1 pages 1 28 Citation last1 Walker first1 A. G. title On Lichnerowicz s conjecture for harmonic 4 spaces doi 10.1112 jlms s1 24.1.21 id MR 0030280 year 1949 journal Journal of the London Mathematical Society. Second Series issn 0024 6107 volume 24 pages 21 28 Category Riemannian geometry ... more details
Beal s conjecture is a conjecture in number theory proposed by Andrew Beal in about 1993 a similar conjecture was suggested independently at about the same time by Andrew Granville . While investigating generalizations of Fermat s last theorem in 1993, Beal formulated the following conjecture If math A x B y C z, math where A , B , C , x , y , and z are positive integers with x , y , z 2 then A , B , and C must have a common prime divisor factor . Beal has offered a prize of USD US 100,000 for a proof of his conjecture or a counterexample . ref http www.math.unt.edu mauldin beal.html The Beal Conjecture Bot generated title ref Examples To illustrate, the solution 3 sup 3 sup 6 sup 3 sup 3 sup 5 sup has bases with a common factor of 3, and the solution 7 sup 6 sup 7 sup 7 sup 98 sup 3 sup has bases with a common factor of 7. Indeed the equation has infinitely many solutions, including for example math left a left a m b m right right m left b left a m b m right right m left a m b m right ... to the conjecture, since the bases all have the factor math a m b m math in common. The example 7 sup 3 sup 13 sup 2 sup 2 sup 9 sup shows that the conjecture is false if one of the exponents ... arithmetic , this conjecture has been verified for all values of all six Variable mathematics variables up to 1000. ref http www.norvig.com beal.html Beal s Conjecture A Search for Counterexamples ... 1000. A variation of the conjecture where x , y , z instead of A , B , C must have a common prime factor is not true. See, for example math 27 4 162 3 9 7 math . Beal s conjecture is a generalization ... a common factor, it can be divided out of each to yield an equation with smaller, coprime bases. The conjecture ... R. Daniel Mauldin title A Generalization of Fermat s Last Theorem The Beal Conjecture and Prize ... notices 199711 beal.pdf PlanetMath title Beal s Conjecture urlname BealsConjecture http mathoverflow.net questions 28764 status of beal tijdeman zagier conjecture DEFAULTSORT Beal s Conjecture Category ... more details
The Macdonald conjecture could be one of several conjectures due to harvtxt Macdonald 1982 Macdonald s conjectures about Macdonald polynomial s, Macdonald s generalization of the Dyson conjecture , Macdonald s generalization of the Mehta integral . References Citation authorlink Ian G. Macdonald last1 Macdonald first1 I. G. title Some conjectures for root systems doi 10.1137 0513070 id MathSciNet id 674768 year 1982 journal SIAM Journal on Mathematical Analysis issn 0036 1410 volume 13 issue 6 pages 988 1007 mathdab ... more details
orphan date November 2010 In combinatorics combinatorial mathematics, Toida s conjecture , due to Shunichi Toida in 1977, ref S. Toida A note on Adam s conjecture , J. of Combinatorial Theory B , pp. 239 246, October December 1977 ref is a refinement of the disproven d m s conjecture in 1967. Toida s conjecture states formally If S is a subset of math mathbb Z n math and math vec X vec X mathbb Z n S math then math vec X math is a CI digraph. Proofs The conjecture was proven in the special case where n is a prime power by Klin and Poschel in 1978, ref Klin, M.H. and R. Poschel The Konig problem, the isomorphism problem for cyclic graphs and the method of Schur rings, Algebraic methods in graph theory, Vol. I, II., Szeged, 1978, pp. 405 434. ref and by Golfand, Najmark, and Poschel in 1984. ref Golfand, J.J., N.L. Najmark and R. Poschel The structure of S rings over Z2m , preprint 1984 . ref The conjecture was then fully proven by Muzychuk, Klin, and Poschel in 2001 by using Schur algebra , ref Klin, M.H., M. Muzychuk and R. Poschel The isomorphism problem for circulant graphs via Schur ring theory, Codes and Association Schemes, American Math. Society, 2001. ref and simultaneously by Dobson and Morris in 2002 by using the Classification of finite simple groups . ref E. Dobson, J. Morris TOIDA S CONJECTURE IS TRUE, PhD Thesis, 2002. ref Notes reflist DEFAULTSORT Toida s Conjecture Category Combinatorics Category Conjectures numtheory stub ... more details
In mathematics, the Fatou conjecture , named after Pierre Fatou , states that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters. References Citation last1 wi tek first1 Grzegorz last2 Graczyk first2 Jacek title The real Fatou conjecture publisher Princeton University Press series Annals of Mathematics Studies isbn 978 0 691 00257 6 978 0 691 00258 3 id MR 1657075 year 1998 volume 144 Category Dynamical systems ... more details
Image Hexagons.jpg thumb right A regular hexagonal grid The honeycomb conjecture states that a regular hexagonal grid or honeycomb is the best way to divide a surface into regions of equal area with the least total perimeter. The conjecture was proposed by Pappus of Alexandria c. 290 c. 350 and proved by mathematician Thomas C. Hales . ref Cite web last Weisstein first Eric W. publisher MathWorld title Honeycomb Conjecture url http mathworld.wolfram.com HoneycombConjecture.html accessdate 27 Dec 2010 ref ref Cite journal last Hales first Thomas C. title The Honeycomb Conjecture date 8 Jun 1999 arxiv math 9906042 journal Discrete and Computational Geometry volume 25 pages 1 22 2001 ref References reflist Category Discrete geometry Category Euclidean plane geometry geometry stub es Conjetura del panal de abeja fr Th or me du nid d abeille ... more details
multiple image footer Graphical proof for Andrica s conjecture for the first a 100, b 200 and c 500 prime numbers. The function math A n math is always less than 1. align right direction vertical width 300 image1 Andrica s Conjecture.svg caption1 a The function math A n math for the first 100 primes. image2 Andrica s Conjecture2.svg caption2 b The function math A n math for the first 200 primes. image3 Andrica s Conjecture3.svg caption3 c The function math A n math for the first 500 primes. Andrica s conjecture named after Dorin Andrica is a conjecture regarding the prime gap gaps between prime number s. ref D. Andrica, Note on a conjecture in prime number theory. Studia Univ. Babes Bolyai Math. 31 1986 , no. 4, 44 48. ref The conjecture states that the inequality math sqrt p n 1 sqrt p n 1 math holds for all math n math , where math p n math is the n sup th sup prime number. If math g n p n 1 p n math denotes the n sup th sup prime gap , then Andrica s conjecture can also be rewritten as math g n 2 sqrt p n 1. math Empirical evidence Imran Ghory has used data on the largest prime gaps to confirm the conjecture for math n math up to 1.3002 x 10 sup 16 sup . ref Prime Numbers The Most Mysterious Figures in Math , John Wiley & Sons, Inc., 2005, p.13. ref The discrete function math A n ... likely the conjecture is true, although this has not yet been proven. Generalizations As a generalization of Andrica s conjecture, the following equation has been considered math p n 1 x p n x 1 ... x is conjectured to be x sub min sub 0.567148... OEIS id A038458 which occurs for n 30. This conjecture has also been stated as an inequality mathematics inequality , the generalized Andrica conjecture math p n 1 x p n x 1 math for math x x min . math See also Cram r s conjecture Legendre s conjecture ... Andrica s Conjecture at PlanetMath http planetmath.org ?op getobj&from objects&id 9636 Generalized Andrica conjecture at PlanetMath MathWorld urlname AndricasConjecture title Andrica s Conjecture ... more details
The Whitehead conjecture is a claim in algebraic topology . It was formulated by J. H. C. Whitehead in 1941. It states that every Connectedness connected subcomplex of a two dimensional Aspherical space aspherical CW complex is aspherical. In 1997, Mladen Bestvina and Noel Brady constructed a group G so that either G is a counterexample to the Eilenberg Ganea conjecture , or there must be a counterexample to the Whitehead conjecture. References http links.jstor.org sici?sici 0003 486X 28194104 292 3A42 3A2 3C409 3AOARTHG 3E2.0.CO 3B2 5 J. H. C. Whitehead, On adding relations to homotopy groups , Annals of Mathematics, 2nd Ser., 42 1941 , no. 2, 409 &ndash 428. http www.springerlink.com content nhj24dgb0vb7bx5p ?p 3b9c54d35a7c445587b1fc97576a6a83&pi 1 Mladen Bestvina, Noel Brady, Morse theory and finiteness properties of groups , Inventiones Mathematicae 129 1997 , no. 3, 445 &ndash 470. DEFAULTSORT Whitehead Conjecture Category Theorems in algebraic topology Category Conjectures ... more details
The mathematician Irving Kaplansky is notable for proposing numerous conjecture s in several branches of mathematics , including a list of ten conjectures on Hopf algebra s. They are usually known as Kaplansky s conjectures . NOTOC Kaplansky s conjecture on group rings Kaplansky s conjecture on group rings states that the complex group ring C G of a torsion free group G has no nontrivial idempotent s. It is related to the Richard Kadison Kadison idempotent conjecture, also known as the Kadison&ndash Kaplansky conjecture. Kaplansky s conjecture on Banach algebras This conjecture states that every algebra homomorphism from the Banach algebra C X where X is a compact space compact Hausdorff space Hausdorff topological space into any other Banach algebra, is necessarily continuous function continuous . The conjecture is equivalent to the statement that every algebra norm on C X is equivalent to the usual uniform norm . Kaplansky himself had earlier shown that every complete algebra norm on C X is equivalent to the uniform norm. In the mid 1970s, H. Garth Dales and J. Esterle independently proved that, if one furthermore assumes the validity of the continuum hypothesis , there exist compact Hausdorff spaces X and discontinuous homomorphisms from C X to some Banach algebra, giving counterexamples to the conjecture. In 1976, Robert M. Solovay R. M. Solovay proved building on work of H. Woodin that there is at least one model of ZFC Zermelo&ndash Fraenkel set theory axiom of choice in which Kaplansky s conjecture is true. Necessarily, in such a model the continuum hypothesis is false. Combined with the results of Dales and Esterle, this shows that the conjecture is independence mathematical logic independent of the axiom s of ZFC. See also List of statements undecidable in ZFC References H. G. Dales, Automatic continuity a survey . Bull. London Math. Soc. 10 1978 , no. 2, 129 ... conjecture for word hyperbolic groups . Invent. Math. 149 2002 , no. 1, 153 194. Category Ring ... more details
In matheamtics, Vojta s conjecture is a conjecture introduced by harvs txt last Vojta authorlink Paul Vojta year 1987 about heights of points on varieties over number field s. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory value distribution theory in complex analysis . It implies many of the other conjectures in diophantine approximation theory. References Citation last1 Vojta first1 Paul title Diophantine approximations and value distribution theory publisher Springer Verlag location Berlin, New York series Lecture Notes in Mathematics isbn 978 3 540 17551 3 doi 10.1007 BFb0072989 id MR 883451 year 1987 volume 1239 Category Number theory ... more details
The Smale Conjecture is the statement that the diffeomorphism diffeomorphism group of the 3 sphere has the homotopy type of its isometry group, the orthogonal group orthogonal group O 4 . Equivalent Statements There are several equivalent statements of the Smale Conjecture. One is that the component of the unknot in the space of smooth embeddings of the circle in 3 space has the homotopy type of the round circles, equivalently, orthogonal group O 3 . Another equivalent statement is that the group of diffeomorphisms of the Ball mathematics 3 ball which restrict to the identity on the boundary is contractible. References S.Smale, Diffeomorphisms of the 2 sphere, Proc. AMS 1959. Hatcher, A proof of the Smale conjecture, math scriptstyle mathrm Diff S 3 simeq mathrm O 4 math . Ann. of Math. 2 117 1983 , no. 3, 553 607. Category Smooth manifolds ... more details
In mathematics, the torsion conjecture or uniform boundedness conjecture for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field. References Citation last1 Merel first1 Lo c title Bornes pour la torsion des courbes elliptiques sur les corps de nombres url http dx.doi.org 10.1007 s002220050059 doi 10.1007 s002220050059 id MR 1369424 year 1996 journal Inventiones Mathematicae issn 0020 9910 volume 124 issue 1 pages 437 449 Category Abelian varieties ... more details
Catalan s conjecture or Mih ilescu s theorem is a theorem in number theory that was conjectured by the mathematician Eug ne Charles Catalan in 1844 and proven in 2002 by Preda Mih ilescu . 2 sup 3 sup and 3 sup 2 sup are two perfect power power s of natural number s, whose values 8 and 9 respectively are consecutive. The conjecture states that this is the only case of two consecutive powers. That is to say, that the only Diophantine equation solution in the natural numbers of x sup a sup y sup b sup 1 for x , a , y , b > 1 is x 3, a 2, y 2, b 3. History The history of the problem dates back at least to Gersonides , who proved a special case of the conjecture in 1343 where x and y were restricted to be 2 or 3. In 1976, Robert Tijdeman applied methods from the theory of transcendental number s to show that there is an effectively computable constant C so that the exponentiation exponents of all consecutive powers are less than C. As the results of a number of other mathematicians collectively had established a bound for the base dependent only on the exponents, this resolved Catalan s conjecture ... the proof of the theorem was nonetheless too time consuming to perform. Catalan s conjecture was proved ... Bilu in the S minaire Bourbaki . Pillai s conjecture Pillai s conjecture concerns a general difference ... St rmer s theorem Fermat Catalan conjecture Beal s conjecture References Catalan, Eugene. 1844 lang ... s Conjecture journal J. Reine angew. Math. volume 572 year 2004 pages 167 195 url http www.reference ... author Paulo Ribenboim authorlink Paulo Ribenboim title Catalan s Conjecture publisher Academic Press ... 03 00993 5 S0273 0979 03 00993 5.pdf format PDF title Catalan s conjecture another old Diophantine ... pages 43 57 doi 10.1090 S0273 0979 03 00993 5 cite journal author Yuri Bilu title Catalan s conjecture ... MathWorld urlname CatalansConjecture title Catalan s conjecture http www.maa.org mathland mathtrek ... A Cyclotomic Proof of Catalan s Conjecture Category Conjectures Category Diophantine equations ... more details
In mathematics , and in particular number theory , Grimm s conjecture states that to each element of a set of consecutive composite number s one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly , 76 1969 1126 1128. Formal statement Suppose n 1, n 2, &hellip , n k are all composite numbers , then there are k distinct primes p sub i sub such that p sub i sub Divisor divides n i for 1 &le i &le k . Weaker version A weaker, though still unproven, version of this conjecture goes If there is no prime in the interval math n 1, n k math , then math prod x le k n x math has at least k distinct prime divisor s. See also Prime gap References mathworld urlname GrimmsConjecture title Grimm s Conjecture Richard K. Guy Guy, R. K. Grimm s Conjecture. B32 in Unsolved Problems in Number Theory , 3rd ed., Springer Science Business Media , pp. 133 134, 2004. ISBN 0 387 20860 7 Category Conjectures about prime numbers es Conjetura de Grimm fr Conjecture de Grimm ... more details
In mathematics, Sendov s conjecture , sometimes also called Ilieff s conjecture , concerns the relationship between the locations of zero of a function roots and critical point mathematics critical point s of a polynomial function of a complex variable . It is named after Blagovest Sendov . The conjecture states that for a polynomial math f z z r 1 cdots z r n , qquad n ge 2 math with all roots r sub 1 sub , ..., r sub n sub inside the closed unit disk z 1, each of the n roots is at a distance no more than 1 from at least one critical point. The Gauss Lucas theorem says that all of the critical points lie within the convex hull of the roots. It follows that the critical points must be within the unit disk, since the roots are. The conjecture has not been proved for n 8. References G. Schmeisser, The Conjectures of Sendov and Stephen Smale Smale , Approximation Theory A Volume Dedicated to Blagovest Sendov B. Bojoanov, ed. , Sofia DARBA, 2002 pp. 353 369. External links http demonstrations.wolfram.com SendovsConjecture Sendov s Conjecture by Bruce Torrence with contributions from Paul Abbott at The Wolfram Demonstrations Project Category Complex analysis Category Conjectures ... more details
In number theory , Cram r s conjecture , formulated by the Sweden Swedish mathematician Harald Cram r in 1936, ref name Cram r1936 Citation last Cram r first Harald title On the order of magnitude of the difference between consecutive prime numbers url http matwbn.icm.edu.pl ksiazki aa aa2 aa212.pdf journal Acta Arithmetica volume 2 year 1936 pages 23 46 . ref states that math p n 1 p n O log p n 2 , math where p sub n sub denotes the n th prime number , O is big O notation , and log is the natural logarithm . Intuitively, this means the prime gap gaps between consecutive primes are always small, and it quantifies asymptotics asymptotically just how small they can be. This conjecture has not been proven or disproven. Heuristic justification Cram r s conjecture is based on a Probability theory probabilistic model essentially a heuristic of the primes, in which one assumes that the probability that a natural number x is prime is 1 log x . This is known as the Cram r model of the primes. Cram r proved that in this model, the above conjecture holds true with probability one. ref name Cram r1936 ... Cram r Granville conjecture Daniel Shanks conjectured asymptotic equality of record gaps, a somewhat stronger statement than Cram r s conjecture. ref Citation first Daniel last Shanks title On Maximal ... gaps gaps.html volume 68 year 1999 . ref He measures the quality of fit to Cram r s conjecture ... calculation, the Granville refinement of Cram r s conjecture seems to be a good fit to the data. See also Prime number theorem Legendre s conjecture and Andrica s conjecture , much weaker but still unproven upper bounds on prime gaps References Reflist External links mathworld title Cram r Conjecture urlname CramerConjecture mathworld title Cram r Granville Conjecture urlname Cramer GranvilleConjecture DEFAULTSORT Cramer s Conjecture Category Analytic number theory Category Conjectures about prime numbers es Conjetura de Cram r fr Conjecture de Cram r it Congettura di Cram r zh ... more details
of factors. Equivalently, it can be stated in terms of the summatory Liouville function , the conjecture ... Omega function counts the total number of prime factors of an integer. Disproof P lya s conjecture was disproven by C. Brian Haselgrove C. B. Haselgrove in 1958. He showed that the conjecture has a counterexample ... first C.B. last Haselgrove authorlink C. Brian Haselgrove year 1958 title A disproof of a conjecture ... pages 187 189 doi 10.3836 tjm 1270216093 mr 0584557 ref The P lya conjecture fails to hold for most ... title P lya Conjecture DEFAULTSORT Polya conjecture Category Disproved conjectures Category ... P lya fr Conjecture de P lya it Congettura di P lya pl Hipoteza P lyi ru sr ... more details
In mathematics , the Jacobian conjecture is a celebrated problem on polynomial s in several Variable mathematics variables . It was first posed in 1939 by Ott Heinrich Keller . It was later named and widely publicised by Shreeram Abhyankar , as an example of a question in the area of algebraic geometry that requires little beyond a knowledge of calculus to state. The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. As of 2011, there are no plausible ... conjecture is a strengthening of the Theorem Converse converse it states that if J is a non ... harvtxt Wang 1980 proved the Jacobian conjecture for polynomials of degree of a polynomial degree ... where the polynomials are of degree 3. harvtxt Moh 1983 checked the conjecture for polynomials of degree at most 100 in 2 variables. The Jacobian conjecture is equivalent to the Dixmier conjecture . ref ... conjecture Notes references References Citation last1 Bass first1 Hyman last2 Connell first2 Edwin H. last3 Wright first3 David title The Jacobian conjecture reduction of degree and formal expansion ... Belov Kanel first1 Alexei last2 Kontsevich first2 Maxim title The Jacobian conjecture is stably equivalent to the Dixmier conjecture arxiv math 0512171 mr 2337879 year 2007 journal Moscow Mathematical ... conjecture author A. van den Essen Citation last1 Keller first1 Ott Heinrich title Ganze Cremona ... Moh first1 T. T. title On the Jacobian conjecture and the configurations of roots url http resolver.sub.uni goettingen.de purl?GDZPPN002200376 id MR 691964 Preprint titled On the global Jacobian conjecture ... conjecture , ISBN 3 7643 6350 9 http emis.mi.ras.ru journals SC 1997 2 pdf smf sem cong 2 55 81.pdf . External links http www.math.purdue.edu ttm jacobian.html Web page of T. T. Moh on the conjecture DEFAULTSORT Jacobian Conjecture Category Polynomials Category Algebraic geometry Category Conjectures fr Conjecture jacobienne pms Congetura jacobian a ru zh ... more details
The Bunyakovsky conjecture or Bouniakowsky conjecture stated in 1857 by the Russian Empire Russian mathematician Viktor Bunyakovsky , claims that an irreducible polynomial of degree two or higher with integer coefficients generates for Natural number natural arguments either an infinite set of numbers with greatest common divisor gcd exceeding unity, or infinitely many prime number s. An example of the former case is the polynomial math f x x 2 x 4 math , which is irreducible but generates a set with gcd 2. A conjectured example of the latter case is the polynomial f x x sup 2 sup 1, for which some of the prime numbers generated are listed below border 1 cellpadding 2 cellspacing 0 style border collapse collapse bgcolor efefef x 1 2 4 6 10 14 16 20 24 26 36 x sup 2 sup 1 2 5 17 37 101 197 257 401 577 677 1297 The fifth Hardy Littlewood conjecture a special case of the Bunyakovsky conjecture states that math x 2 1 math generates infinitely many prime values for integer x 1. This special case goes back to Euler. To date, the Bunyakovsky conjecture has not been Mathematical proof proven correct, nor is a counterexample known. The Bunyakovsky conjecture can be seen as an extension of Dirichlet s theorem on arithmetic progressions Dirichlet s theorem , which states that irreducible degree one polynomials always generate an infinite number of primes. See also Integer valued polynomial Cohn s irreducibility criterion Schinzel s hypothesis H Bateman Horn conjecture Ulam spiral Hardy and Littlewood s Conjecture F Hardy and Littlewood s conjecture F References MathWorld urlname BouniakowskyConjecture title Bouniakowsky conjecture author Ed Pegg, Jr. cite arxiv last Rupert first Wolfgang M. title Reducibility of polynomials f x , y modulo p date 1998 08 05 eprint math 9808021 class math.NT cite journal last Bouniakowsky first V. title Nouveaux th or mes relatifs la distinction des nombres premiers et la d composition des entiers en facteurs journal M m. Acad. Sc ... more details
In mathematics , the Weinstein conjecture refers to a general existence problem for periodic orbit s of Hamiltonian flow Hamiltonian or Contact manifold Reeb vector field Reeb vector flow s. More specifically, the current understanding is that a regular compact contact type level set of a hamiltonian vector field Hamiltonian on a symplectic manifold should carry at least one periodic orbit of the Hamiltonian flow. The conjecture is stated for any Hamiltonian on any 2n dimensional symplectic manifold. By definition, a level set of contact type admits a contact form obtained by differential form Operations on forms contracting the Hamiltonian vector field into the symplectic form. In this case, the Hamiltonian flow is a Reeb vector field on that level set. It is a fact that any contact manifold M , can be embedded into a canonical symplectic manifold, called the symplectization of M , such that M is a contact type level set of a canonically defined Hamiltonian and the Reeb vector field is a Hamiltonian ... conjecture. Since, as is trivial to show, any orbit of a Hamiltonian flow is contained in a level set, the Weinstein conjecture is a statement about contact manifolds. It has been known that any ... to prove the Weinstein conjecture, though, because the Weinstein conjecture states that every ... which is only Isotopy isotopic to the given form. The conjecture was formulated in 1978 by Alan ... in the condition that the level set be of contact type. Weinstein s original conjecture included ... turned out to be unnecessary . The Weinstein conjecture has now been proven for all closed 3 dimensional ... related program of proving the Weinstein conjecture by showing that the Floer homology embedded ... eprint math 0310330 author1 Ginzburg title The Weinstein conjecture and the theorems of nearby and almost ... Witten equations and the Weinstein conjecture journal Geometry & Topology volume 11 pages 2117 ... M. year 2010 title Taubes s proof of the Weinstein conjecture in dimension three journal Bulletin ... more details
The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology .... More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums ... conjecture is one of the Clay Mathematics Institute s Millennium Prize Problems , with a prize of 1,000,000 for whoever can prove or disprove the Hodge conjecture using some argument . It was formulated ... H n k,n k X, mathbf C math . Loosely speaking, the Hodge conjecture asks Which cohomology classes in math H k,k X math come from complex subvarieties Z ? Statement of the Hodge conjecture Let math operatorname ... 2 k on X . The modern statement of the Hodge conjecture is Hodge conjecture. Let X be a projective ... way of phrasing the Hodge conjecture involves the idea of an algebraic cycle. An algebraic cycle ... algebraic . With this notation, the Hodge conjecture becomes Let X be a projective complex manifold. Then every Hodge class on X is algebraic. Known cases of the Hodge conjecture Low dimension and codimension The first result on the Hodge conjecture is due to Harvtxt Lefschetz 1924 . In fact, it predates the conjecture and provided some of Hodge s motivation. Theorem Lefschetz theorem on 1,1 classes ... geometry divisor on X . In particular, the Hodge conjecture is true for math H 2 math . A very quick ... transversality theorem shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties. By the Hard Lefschetz theorem , one can prove Theorem. If the Hodge conjecture holds for Hodge classes of degree p , p < n , then the Hodge conjecture holds for Hodge classes of degree 2 n &minus p . Combining the above two theorems implies that Hodge conjecture is true for Hodge classes of degree 2 n &minus 2. This proves the Hodge conjecture ... Hodge classes are generated by the Hodge classes of divisors, then the Hodge conjecture is true Corollary ... sup 1 sup X , then the Hodge conjecture holds for X . Abelian varieties For most abelian variety abelian ... more details
In mathematics , a smooth algebraic curve math C math in the complex projective plane , of degree math d math , has Genus mathematics Topology genus given by the formula math g d 1 d 2 2 math . The Thom conjecture , named after French mathematician Ren Thom , states that if math Sigma math is any smoothly embedded connected curve representing the same class in homology mathematics homology as math C math , then the genus math g math of math Sigma math satisfies math g geq d 1 d 2 2 math . In particular, C is known as a genus minimizing representative of its homology class. There are proofs for this conjecture in certain cases such as when math Sigma math has nonnegative self intersection number , and assuming this number is nonnegative, this generalizes to K hler manifold s an example being the complex projective plane . It was first proved by Peter B. Kronheimer Kronheimer Tomasz Mrowka Mrowka and John Morgan mathematician Morgan Zolt n Szab mathematician Szab Clifford Taubes Taubes in October 1994, using the then new Seiberg Witten invariant s. There is at least one generalization of this conjecture, known as the Symplectic geometry symplectic Thom conjecture which is now a theorem, as proved for example by Peter Ozsv th Ozsv th and Szab in 2000 ref Cite journal last Ozsv th first Peter last2 Szab first2 Zolt n year 2000 title The symplectic Thom conjecture journal Annals of Mathematics Ann. of Math. volume 151 issue 1 pages 93 124 arxiv math.DG 9811087 ref . It states that a symplectic surface of a symplectic 4 manifold is genus minimizing within its homology class. This would imply the previous result because algebraic curves complex dimension 1, real dimension 2 are symplectic surfaces within the complex projective plane, which is a symplectic 4 manifold. See also Adjunction formula algebraic geometry Adjunction formula References references DEFAULTSORT Thom Conjecture Category Four dimensional geometry Category 4 manifolds Category Algebraic surfaces Cate ... more details
The Ragsdale conjecture is a mathematics mathematical conjecture that concerns the possible arrangements of real algebraic curves embedded in the projective plane . It was proposed by Virginia Ragsdale several years after 1900 and was disproved in 1979. ref cite journal last Viro first Oleg Ya. authorlink Oleg Viro year 1980 title 7, 8 trans title Curves of degree 7, curves of degree 8 and the hypothesis of Ragsdale journal Doklady Akademii Nauk SSSR volume 254 issue 6 pages 1306 1309 Translated in cite journal journal Soviet Mathematics Doklady volume 22 pages 566 570 year 1980 ref Background Her dissertation dealt with Hilbert s sixteenth problem , which was proposed in the year 1900, along with Hilbert s problems 22 other unsolved problems of the 19th century . Ragsdale conjectured a particular upper bound on the number of topological circles of a certain type, along with the basis of evidence. The conjecture was held of high importance in the field of real algebraic geometry for nearly a century. Later Oleg Viro and Ilya Itenberg produced counterexamples to the Ragsdale conjecture, although the problem of finding a sharp upper bound remains unsolved. Conjecture Ragsdale s main conjecture is as follows. Assume that an algebraic curve of degree 2 k contains p even and n odd ovals. Ragsdale conjectured that math p le tfrac32 k k 1 1 quad text and quad n le tfrac32 k k 1 . math She also posed the inequality math 2 p n 1 le 3k 2 3k 1, math and showed that the inequality could not be further improved. This inequality was later proved by Ivan Petrovsky Petrovsky . Notes references References cite web url http www.agnesscott.edu LRiddle women ragsdale.htm title Virginia Ragsdale accessdate 2007 03 09 last De Loera first Jes s coauthors Frederick J. Wicklin year 2006 work Biographies of Women Mathematicians Category Conjectures Category Real algebraic geometry ... more details
In differential geometry in mathematics the Willmore conjecture is a conjecture about the Willmore energy of a torus , named after the England English mathematician Tom Willmore . Statement of the conjecture Let v M &rarr R sup 3 sup be a smooth function smooth immersion mathematics immersion of a compact space compact , orientability orientable surface of dimension two . Giving M the Riemannian metric induced by v , let H M &rarr R be the mean curvature the arithmetic mean of the principal curvature s &kappa sub 1 sub and &kappa sub 2 sub at each point . In this notation, the Willmore energy W M of M is given by math W M int M H 2 . math It is not hard to prove that the Willmore energy satisfies W M &ge 4 &pi , with equality if and only if M is an embedded round sphere . Calculation of W M for a few examples suggests that there should be a better bound for surfaces with genus topology genus g M > 0. In particular, calculation of W M for tori with various symmetries led Willmore to propose in 1965 the following conjecture, which now bears his name for any smooth immersed torus M in R sup 3 sup , W M &ge 2 &pi sup 2 sup . Announced proof On February 27, 2012, Fernando Cod Marques and Andr Neves announced a complete proof of the Willmore conjecture on a http arxiv.org abs 1202.6036 preprint posted to arxiv.org, using the min max theory of minimal surfaces. References cite journal last Fernando C. Marques first Andre Neves title Min Max theory and the Willmore conjecture Arxiv id 1202.6036 cite journal last Willmore first Thomas J. title Note on embedded surfaces journal An. ti. Univ. Al. I. Cuza Ia i Sec . I a Mat. N.S. volume 11B year 1965 pages 493&ndash 496 MathSciNet id 0202066 Category Conjectures Category Differential geometry Category Surfaces ... more details
In nonlinear control , Aizerman s conjecture states that a linear system in feedback with a sector nonlinearity would be stable if the linear system is stable for any linear gain of the sector. This conjecture was proven false but led to the valid circle criterion and Nonlinear control Absolute stability problem Popov criterion . ref name Atherton1977 cite journal author Atherton, D.P. coauthors Siouris, G.M. year 1977 title Nonlinear Control Engineering journal Systems, Man and Cybernetics, IEEE Transactions on volume 7 issue 7 pages 567 568 url http ieeexplore.ieee.org xpls abs all.jsp?arnumber 4309773 accessdate 2008 06 30 doi 10.1109 TSMC.1977.4309773 ref Mathematical statement of Aizerman s conjecture Aizerman problem Consider a system with one scalar nonlinearity math frac dx dt Px qf e , quad e r x quad x in R n, math where P is a constant n n matrix, q, r are constant n dimensional vectors, is an operation of transposition, f e is scalar function, and f 0 0. Suppose, that nonlinearity f e is from liner sector math k1 f e k2. math Then Aizerman s conjecture is that the system is stable in large i.e. unique stationary point is global attractor if all linear systems with f e ke, k k1,k2 are asymptotically stable. Strengthening of Aizerman s conjecture is Kalman s conjecture or Kalman problem where in place of condition on the nonlinearity it is required that the derivative of nonlinearity belongs to linear stability sector. References reflist Further reading cite journal author Bragin V.O., Vagaitsev V.I., Kuznetsov N.V., Leonov G.A. year 2011 title Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua s Circuits journal Journal of Computer and Systems Sciences International volume 50 number 4 pages 511&ndash 543 doi 10.1134 S106423071104006X External links cite web url http www.math.spbu.ru user nk PDF Harmonic balance Absolute stability.pdf title Counterexamples to Aizerman s and Kalman s conjectures and DF ... more details
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