Pengguna:Dedhert.Jr/Uji halaman 17: Perbedaan antara revisi

 

== Table of problems ==

{| class="wikitable sortable" style="text-align:left"

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!style="text-align:center;" width=44% | Status

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|style="text-align:center;"| 1940, 1963

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| {{partial|{{sort|2|}}There is no consensus on whether results of [[Kurt Gödel|Gödel]] and [[Gerhard Gentzen|Gentzen]] give a solution to the problem as stated by Hilbert. Gödel's [[Gödel's incompleteness theorems|second incompleteness theorem]], proved in 1931, shows that no proof of its consistency can be carried out within arithmetic itself. Gentzen proved in 1936 that the consistency of arithmetic follows from the [[well-founded relation|well-foundedness]] of the [[epsilon numbers (mathematics)|ordinal ''ε''₀]].}}

|style="text-align:center;"| 1931, 1936

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|style="text-align:center;"| 1900

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| {{dunno|{{sort|4|}}Too vague to be stated resolved or not.{{refn|According to Gray, most of the problems have been solved. Some were not defined completely, but enough progress has been made to consider them "solved"; Gray lists the fourth problem as too vague to say whether it has been solved.|group=lower-alpha}} }}

|style="text-align:center;"| —

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| Are continuous [[group (mathematics)|groups]] automatically [[Lie group|differential groups]]?

| {{partial|{{sort|2|}}Resolved by [[Andrew Gleason]], assuming one interpretation of the original statement. If, however, it is understood as an equivalent of the [[Hilbert–Smith conjecture]], it is still unsolved.}}

|style="text-align:center;"| 1953?

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| Mathematical treatment of the [[axiom]]s of [[physics]]<br /><br />(a) axiomatic treatment of probability with limit theorems for foundation of [[statistical physics]]<br /><br />(b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"

| {{partial|{{sort|2|}}Partially resolved depending on how the original statement is interpreted.<ref>{{cite journal |last1=Corry |first1=L. |year=1997 |title=David Hilbert and the axiomatization of physics (1894–1905) |journal=Arch. Hist. Exact Sci. |volume=51 |issue=2 |pages=83–198 |doi=10.1007/BF00375141|s2cid=122709777 }}</ref> Items&nbsp;(a) and (b) were two specific problems given by Hilbert in a later explanation.<ref name=Hilbert_1902/> [[probability axioms|Kolmogorov's axiomatics]] (1933) is now accepted as standard. There is some success on the way from the "atomistic view to the laws of motion of continua."<ref>{{cite journal |last1=Gorban |first1=A.N. |author-link=Alexander Nikolaevich Gorban |last2=Karlin |first2=I. |year=2014 |title=Hilbert's 6th Problem: Exact and approximate hydrodynamic manifolds for kinetic equations |journal=Bulletin of the American Mathematical Society |volume=51 |issue=2 |pages=186–246 |arxiv=1310.0406 |doi=10.1090/S0273-0979-2013-01439-3| doi-access= free}}</ref>}}

|style="text-align:center;"| 1933–2002?

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| Is ''a^b'' [[transcendental number|transcendental]], for [[algebraic number|algebraic]] ''a'' ≠ 0,1 and [[irrational number|irrational]] algebraic ''b'' ?

| {{yes|{{sort|1|}}Resolved. Result: Yes, illustrated by [[Gelfond's theorem]] or the [[Gelfond–Schneider theorem]].}}

|style="text-align:center;"| 1934

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| The [[Riemann hypothesis]]<br />("the real part of any non-[[Trivial (mathematics)|trivial]] [[root of a function|zero]] of the [[Riemann zeta function]] is {{frac|1|2}}")<br />and other prime number problems, among them [[Goldbach's conjecture]] and the [[twin prime conjecture]]

| {{no|{{sort|3|}}Unresolved.}}

|style="text-align:center;"| —

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| Find the most general law of the [[Quadratic reciprocity|reciprocity theorem]] in any [[algebra]]ic [[number field]].

| {{partial|{{sort|2|}}Partially resolved.{{refn|Problem 9 has been solved by [[Emil Artin]] in 1927 for [[Abelian extension]]s of the [[rational numbers]] during the development of [[class field theory]]; the non-abelian case remains unsolved, if one interprets that as meaning [[non-abelian class field theory]].|group=lower-alpha}} }}

|style="text-align:center;"| —

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|style="text-align:left;"| Find an algorithm to determine whether a given polynomial [[Diophantine equation]] with integer coefficients has an integer solution.

| {{yes|{{sort|1|}}Resolved. Result: Impossible; [[Matiyasevich's theorem]] implies that there is no such algorithm.}}

|style="text-align:center;"| 1970

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| Solving [[quadratic form]]s with algebraic numerical [[coefficient]]s.

| {{partial|{{sort|2|}}Partially resolved.<ref>{{cite book|first=Michiel|last=Hazewinkel|date= 2009|title= Handbook of Algebra|publisher=Elsevier|page=69|isbn=978-0080932811|volume=6}}</ref>}}

|style="text-align:center;"| —

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| Extend the [[Kronecker–Weber theorem]] on Abelian extensions of the [[rational number]]s to any base number field.

| {{partial|{{sort|2|}}Partially resolved.<ref>{{cite web|url=https://www.quantamagazine.org/mathematicians-find-polynomial-building-blocks-hilbert-sought-20210525/|first=Kelsey|last=Houston-Edwards|title=Mathematicians Find Long-Sought Building Blocks for Special Polynomials|date=25 May 2021 }}</ref>}}

|style="text-align:center;"| —

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| Solve [[septic equation|7th degree equation]] using algebraic (variant: continuous) [[mathematical function|functions]] of two [[parameter]]s.

| {{no|{{sort|3|}}Unresolved. The continuous variant of this problem was solved by [[Vladimir Arnold]] in 1957 based on work by [[Andrei Kolmogorov]], but the algebraic variant is unresolved.{{refn|1=It is not difficult to show that the problem has a partial solution within the space of single-valued analytic functions (Raudenbush). Some authors argue that Hilbert intended for a solution within the space of (multi-valued) algebraic functions, thus continuing his own work on algebraic functions and being a question about a possible extension of the [[Galois theory]] (see, for example, Abhyankar<ref>{{cite book |url=http://www.emis.de/journals/SC/1997/2/pdf/smf_sem-cong_2_1-11.pdf |first=Shreeram S. |last=Abhyankar |title=Hilbert's Thirteenth Problem |series=Séminaires et Congrès |volume=2 |publisher=Société Mathématique de France |date=1997}}</ref> Vitushkin,<ref>{{cite journal |last1=Vitushkin |first1=Anatoliy G. |title=On Hilbert's thirteenth problem and related questions |journal=Russian Mathematical Surveys |date=2004 |volume=59 |issue=1 |pages=11–25 |doi=10.1070/RM2004v059n01ABEH000698 |publisher=Russian Academy of Sciences}}</ref> Chebotarev,<ref>{{cite journal |last1=Morozov |first1=Vladimir V. |title=О некоторых вопросах проблемы резольвент |journal=Proceedings of Kazan University |date=1954 |volume=114 |issue=2 |pages=173-187 |url=http://www.mathnet.ru/php/getFT.phtml?jrnid=uzku&paperid=406&what=fullt&option_lang=eng |publisher=Kazan University |language=ru |trans-title=On certain questions of the problem of resolvents}}</ref> and others). It appears from one of Hilbert's papers<ref>{{cite journal |first=David |last=Hilbert |title=Über die Gleichung neunten Grades |journal=Math. Ann. |volume=97 |year=1927 |pages=243–250|doi=10.1007/BF01447867 |s2cid=179178089 }}</ref> that this was his original intention for the problem.

|style="text-align:center;"| —

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| Is the [[invariant theory|ring of invariants]] of an [[algebraic group]] acting on a [[polynomial ring]] always [[Finitely generated algebra|finitely generated]]?

| {{yes|{{sort|1|}}Resolved. Result: No, a counterexample was constructed by [[Masayoshi Nagata]].}}

|style="text-align:center;"| 1959

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| Rigorous foundation of [[Schubert's enumerative calculus]].

| {{partial|{{sort|2|}}Partially resolved.{{Citation needed|date=November 2019}}}}

|style="text-align:center;"| —

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| Describe relative positions of ovals originating from a [[real numbers|real]] [[algebraic curve]] and as [[limit cycle]]s of a polynomial [[vector field]] on the plane.

| {{no|{{sort|3|}}Unresolved, even for algebraic curves of degree 8.}}

|style="text-align:center;"| —

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| Express a nonnegative [[rational function]] as [[quotient]] of sums of [[Square (algebra)|squares]].

| {{yes|{{sort|1|}}Resolved. Result: Yes, due to [[Emil Artin]]. Moreover, an upper limit was established for the number of square terms necessary.}}

|style="text-align:center;"| 1927

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| (a) Are there only finitely many essentially different space groups in n-dimensional Euclidean space? <br/> <br/> (b) Is there a polyhedron that admits only an [[anisohedral tiling]] in three dimensions?<br /><br />(c) What is the densest [[sphere packing]]?

| {{yes|{{sort|1|}}(a) Resolved. Result Yes (by [[Ludwig Bieberbach]]) <br/> <br/> (b) Resolved. Result: Yes (by [[Karl Reinhardt (mathematician)|Karl Reinhardt]]).<br /><br />(c) Widely believed to be resolved, by [[computer-assisted proof]] (by [[Thomas Callister Hales]]). Result: Highest density achieved by [[Close-packing of equal spheres|close packings]], each with density approximately 74%, such as face-centered cubic close packing and hexagonal close packing.{{refn|Gray also lists the 18th problem as "open" in his 2000 book, because the sphere-packing problem (also known as the [[Kepler conjecture]]) was unsolved, but a solution to it has now been claimed.|group=lower-alpha}}}}

|style="text-align:center;"| {{sort|1928|(a) 1910 <br/> <br/> (b) 1928<br /><br />(c) 1998}}

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| Are the solutions of regular problems in the [[calculus of variations]] always necessarily [[Analytic function|analytic]]?

| {{yes|{{sort|1|}}Resolved. Result: Yes, proven by [[Ennio de Giorgi]] and, independently and using different methods, by [[John Forbes Nash]].}}

|style="text-align:center;"| 1957

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| Do all [[calculus of variations|variational problems]] with certain [[boundary condition]]s have solutions?

| {{yes|{{sort|1|}}Resolved. A significant topic of research throughout the 20th century, culminating in solutions for the non-linear case.}}

|style="text-align:center;"| ?

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| Proof of the existence of [[linear differential equation]]s having a prescribed [[monodromic group]]

| {{partial|{{sort|2|}} Partially resolved. Result: Yes/No/Open depending on more exact formulations of the problem.}}

|style="text-align:center;"| ?

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| Uniformization of analytic relations by means of [[automorphic function]]s

| {{partial|{{sort|2|}}Partially resolved. [[Uniformization theorem]]}}

|style="text-align:center;"| ?

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| Further development of the [[calculus of variations]]

| {{dunno|{{sort|4|}}Too vague to be stated resolved or not.}}