Misplaced Pages

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55-615: 28 ( twenty-eight ) is the natural number following 27 and preceding 29 . Twenty-eight is a composite number and the second perfect number as it is the sum of its proper divisors: 1 + 2 + 4 + 7 + 14 = 28 {\displaystyle 1+2+4+7+14=28} . As a perfect number, it is related to the Mersenne prime 7, since 2 3 − 1 × ( 2 3 − 1 ) = 28 {\displaystyle 2^{3-1}\times (2^{3}-1)=28} . The next perfect number

110-469: A 6 × 6 {\displaystyle 6\times 6} square has a magic constant of 27. Including the null-motif, there are 27 distinct hypergraph motifs . There are exactly twenty-seven straight lines on a smooth cubic surface , which give a basis of the fundamental representation of Lie algebra E 6 {\displaystyle \mathrm {E_{6}} } . The unique simple formally real Jordan algebra ,

165-487: A bijection. If D f x {\displaystyle Df_{x}} is a bijection at x {\displaystyle x} then f {\displaystyle f} is said to be a local diffeomorphism (since, by continuity, D f y {\displaystyle Df_{y}} will also be bijective for all y {\displaystyle y} sufficiently close to x {\displaystyle x} ). Given

220-432: A compact subset of M {\displaystyle M} , this follows by fixing a Riemannian metric on M {\displaystyle M} and using the exponential map for that metric. If r {\displaystyle r} is finite and the manifold is compact, the space of vector fields is a Banach space . Moreover, the transition maps from one chart of this atlas to another are smooth, making

275-464: A differentiable manifold that is second-countable and Hausdorff . The diffeomorphism group of M {\displaystyle M} is the group of all C r {\displaystyle C^{r}} diffeomorphisms of M {\displaystyle M} to itself, denoted by Diff r ( M ) {\displaystyle {\text{Diff}}^{r}(M)} or, when r {\displaystyle r}

330-482: A homeomorphism, f {\displaystyle f} and its inverse need only be continuous . Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism. f : M → N {\displaystyle f:M\to N} is a diffeomorphism if, in coordinate charts , it satisfies the definition above. More precisely: Pick any cover of M {\displaystyle M} by compatible coordinate charts and do

385-646: A multiple of 27. For example, 378, 783, and 837 are all divisible by 27. In senary (base six), one can readily test for divisibility by 43 (decimal 27) by seeing if the last three digits of the number match 000, 043, 130, 213, 300, 343, 430, or 513. In decimal representation, 27 is located at the twenty-eighth (and twenty-ninth) digit after the decimal point in π : 3.141 592 653 589 793 238 462 643 383 27 9 … {\displaystyle 3.141\;592\;653\;589\;793\;238\;462\;643\;383\;{\color {red}27}9\ldots } If one starts counting with zero, 27

440-542: A smooth map from dimension n {\displaystyle n} to dimension k {\displaystyle k} , if D f {\displaystyle Df} (or, locally, D f x {\displaystyle Df_{x}} ) is surjective, f {\displaystyle f} is said to be a submersion (or, locally, a "local submersion"); and if D f {\displaystyle Df} (or, locally, D f x {\displaystyle Df_{x}} )

495-434: A stress-induced transformation is called a deformation and may be described by a diffeomorphism. A diffeomorphism f : U → V {\displaystyle f:U\to V} between two surfaces U {\displaystyle U} and V {\displaystyle V} has a Jacobian matrix D f {\displaystyle Df} that is an invertible matrix . In fact, it

550-402: A sum of the first primes ( 2 + 3 + 5 + 7 + 11 {\displaystyle 2+3+5+7+11} ), and a sum of the first nonprimes ( 1 + 4 + 6 + 8 + 9 {\displaystyle 1+4+6+8+9} ), and it is unlikely that any other number has this property. There are twenty-eight oriented diffeomorphism classes of manifolds homeomorphic to

605-477: A well-defined inverse if and only if D f x {\displaystyle Df_{x}} is a bijection. The matrix representation of D f x {\displaystyle Df_{x}} is the n × n {\displaystyle n\times n} matrix of first-order partial derivatives whose entry in the i {\displaystyle i} -th row and j {\displaystyle j} -th column

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660-442: Is σ {\displaystyle \sigma } -compact, there is a sequence of compact subsets K n {\displaystyle K_{n}} whose union is M {\displaystyle M} . Then: The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of C r {\displaystyle C^{r}} vector fields ( Leslie 1967 ). Over

715-921: Is ∂ f i / ∂ x j {\displaystyle \partial f_{i}/\partial x_{j}} . This so-called Jacobian matrix is often used for explicit computations. Diffeomorphisms are necessarily between manifolds of the same dimension . Imagine f {\displaystyle f} going from dimension n {\displaystyle n} to dimension k {\displaystyle k} . If n < k {\displaystyle n<k} then D f x {\displaystyle Df_{x}} could never be surjective, and if n > k {\displaystyle n>k} then D f x {\displaystyle Df_{x}} could never be injective. In both cases, therefore, D f x {\displaystyle Df_{x}} fails to be

770-401: Is Then the image ( d u , d v ) = ( d x , d y ) D f {\displaystyle (du,dv)=(dx,dy)Df} is a linear transformation , fixing the origin, and expressible as the action of a complex number of a particular type. When ( dx ,  dy ) is also interpreted as that type of complex number, the action is of complex multiplication in

825-414: Is 496 , the previous being 6 . Though perfect, 28 is not the aliquot sum of any other number other than itself; thus, it is not part of a multi-number aliquot sequence . The next perfect number is 496. Twenty-eight is the sum of the totient function for the first nine integers. Since the greatest prime factor of 28 2 + 1 = 785 {\displaystyle 28^{2}+1=785}

880-464: Is multiply transitive ( Banyaga 1997 , p. 29). In 1926, Tibor Radó asked whether the harmonic extension of any homeomorphism or diffeomorphism of the unit circle to the unit disc yields a diffeomorphism on the open disc. An elegant proof was provided shortly afterwards by Hellmuth Kneser . In 1945, Gustave Choquet , apparently unaware of this result, produced a completely different proof. The (orientation-preserving) diffeomorphism group of

935-535: Is simply connected , a differentiable map f : U → V {\displaystyle f:U\to V} is a diffeomorphism if it is proper and if the differential D f x : R n → R n {\displaystyle Df_{x}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} is bijective (and hence a linear isomorphism ) at each point x {\displaystyle x} in U {\displaystyle U} . Some remarks: It

990-455: Is 157, which is more than 28 twice, 28 is a Størmer number . Twenty-eight is a harmonic divisor number , a happy number , the 7th triangular number , a hexagonal number , a Leyland number of the second kind ( 2 6 − 6 2 {\displaystyle 2^{6}-6^{2}} ), and a centered nonagonal number . It appears in the Padovan sequence , preceded by

1045-476: Is a diffeomorphism f {\displaystyle f} from M {\displaystyle M} to N {\displaystyle N} . Two C r {\displaystyle C^{r}} -differentiable manifolds are C r {\displaystyle C^{r}} -diffeomorphic if there is an r {\displaystyle r} times continuously differentiable bijective map between them whose inverse

1100-533: Is a homeomorphism, given a pair of manifolds which are diffeomorphic to each other they are in particular homeomorphic to each other. The converse is not true in general. While it is easy to find homeomorphisms that are not diffeomorphisms, it is more difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2 and 3, any pair of homeomorphic smooth manifolds are diffeomorphic. In dimension 4 or greater, examples of homeomorphic but not diffeomorphic pairs exist. The first such example

1155-417: Is also r {\displaystyle r} times continuously differentiable. Given a subset X {\displaystyle X} of a manifold M {\displaystyle M} and a subset Y {\displaystyle Y} of a manifold N {\displaystyle N} , a function f : X → Y {\displaystyle f:X\to Y}

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1210-512: Is always metrizable . When the manifold is not compact, the strong topology captures the behavior of functions "at infinity" and is not metrizable. It is, however, still Baire . Fixing a Riemannian metric on M {\displaystyle M} , the weak topology is the topology induced by the family of metrics as K {\displaystyle K} varies over compact subsets of M {\displaystyle M} . Indeed, since M {\displaystyle M}

1265-683: Is an extension of f {\displaystyle f} ). The function f {\displaystyle f} is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth. Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the Hadamard-Caccioppoli theorem: If U {\displaystyle U} , V {\displaystyle V} are connected open subsets of R n {\displaystyle \mathbb {R} ^{n}} such that V {\displaystyle V}

1320-399: Is bijective at each point, f {\displaystyle f} is not invertible because it fails to be injective (e.g. f ( 1 , 0 ) = ( 1 , 0 ) = f ( − 1 , 0 ) {\displaystyle f(1,0)=(1,0)=f(-1,0)} ). Since the differential at a point (for a differentiable function) is a linear map , it has

1375-500: Is differentiable as well. If these functions are r {\displaystyle r} times continuously differentiable, f {\displaystyle f} is called a C r {\displaystyle C^{r}} -diffeomorphism. Two manifolds M {\displaystyle M} and N {\displaystyle N} are diffeomorphic (usually denoted M ≃ N {\displaystyle M\simeq N} ) if there

1430-496: Is essential for V {\displaystyle V} to be simply connected for the function f {\displaystyle f} to be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the "realification" of the complex square function Then f {\displaystyle f} is surjective and it satisfies Thus, though D f x {\displaystyle Df_{x}}

1485-459: Is injective, f {\displaystyle f} is said to be an immersion (or, locally, a "local immersion"). A differentiable bijection is not necessarily a diffeomorphism. f ( x ) = x 3 {\displaystyle f(x)=x^{3}} , for example, is not a diffeomorphism from R {\displaystyle \mathbb {R} } to itself because its derivative vanishes at 0 (and hence its inverse

1540-464: Is not differentiable at 0). This is an example of a homeomorphism that is not a diffeomorphism. When f {\displaystyle f} is a map between differentiable manifolds, a diffeomorphic f {\displaystyle f} is a stronger condition than a homeomorphic f {\displaystyle f} . For a diffeomorphism, f {\displaystyle f} and its inverse need to be differentiable ; for

1595-529: Is required that for p {\displaystyle p} in U {\displaystyle U} , there is a neighborhood of p {\displaystyle p} in which the Jacobian D f {\displaystyle Df} stays non-singular . Suppose that in a chart of the surface, f ( x , y ) = ( u , v ) . {\displaystyle f(x,y)=(u,v).} The total differential of u

1650-622: Is said to be smooth if for all p {\displaystyle p} in X {\displaystyle X} there is a neighborhood U ⊂ M {\displaystyle U\subset M} of p {\displaystyle p} and a smooth function g : U → N {\displaystyle g:U\to N} such that the restrictions agree: g | U ∩ X = f | U ∩ X {\displaystyle g_{|U\cap X}=f_{|U\cap X}} (note that g {\displaystyle g}

1705-457: Is the Euler–Mascheroni constant ; this hypothesis is true if and only if this inequality holds for every larger n . {\displaystyle n.} In decimal , 27 is the first composite number not divisible by any of its digits, as well as: Also in base ten, if one cyclically rotates the digits of a three-digit number that is a multiple of 27, the new number is also

28 (number) - Misplaced Pages Continue

1760-554: Is the cube of 3 , or the 2nd tetration of 3: 2 3 = 3 3 = 3 × 3 × 3 {\displaystyle ^{2}3=3^{3}=3\times 3\times 3} . It is divisible by the number of prime numbers below it ( nine ). The first non-trivial decagonal number is 27. 27 has an aliquot sum of 13 (the sixth prime number) in the aliquot sequence ( 27 , 13 , 1 , 0 ) {\displaystyle (27,13,1,0)} of only one composite number, rooted in

1815-522: Is the second self-locating string after 6 , of only a few known. Twenty-seven is also: Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987), p. 106. Diffeomorphism In mathematics , a diffeomorphism is an isomorphism of differentiable manifolds . It is an invertible function that maps one differentiable manifold to another such that both

1870-605: Is twice that of F 4 {\displaystyle \mathrm {F_{4}} } in 104 dimensions) is included. In Robin's theorem for the Riemann hypothesis , twenty-seven integers fail to hold σ ( n ) < e γ n log ⁡ log ⁡ n {\displaystyle \sigma (n)<e^{\gamma }n\log \log n} for values n ≤ 5040 , {\displaystyle n\leq 5040,} where γ {\displaystyle \gamma }

1925-422: Is understood, Diff ( M ) {\displaystyle {\text{Diff}}(M)} . This is a "large" group, in the sense that—provided M {\displaystyle M} is not zero-dimensional—it is not locally compact . The diffeomorphism group has two natural topologies : weak and strong ( Hirsch 1997 ). When the manifold is compact , these two topologies agree. The weak topology

1980-571: The 13 -aliquot tree. In the Collatz conjecture (i.e. the 3 n + 1 {\displaystyle 3n+1} problem), a starting value of 27 requires 3 × 37 = 111 steps to reach 1, more than any smaller number. 27 is also the fourth perfect totient number — as are all powers of 3 — with its adjacent members 15 and 39 adding to twice 27. A prime reciprocal magic square based on multiples of 1 7 {\displaystyle {\tfrac {1}{7}}} in

2035-469: The Lie bracket of vector fields . Somewhat formally, this is seen by making a small change to the coordinate x {\displaystyle x} at each point in space: so the infinitesimal generators are the vector fields For a connected manifold M {\displaystyle M} , the diffeomorphism group acts transitively on M {\displaystyle M} . More generally,

2090-597: The outer automorphism group of the fundamental group of the surface. William Thurston refined this analysis by classifying elements of the mapping class group into three types: those equivalent to a periodic diffeomorphism; those equivalent to a diffeomorphism leaving a simple closed curve invariant; and those equivalent to pseudo-Anosov diffeomorphisms . In the case of the torus S 1 × S 1 = R 2 / Z 2 {\displaystyle S^{1}\times S^{1}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}} ,

2145-749: The 7-sphere. There are 28 non-equivalent ways of expressing 1000 as the sum of two prime numbers. Twenty-eight is the smallest number that can be expressed as the sum of four nonzero squares in (at least) three ways: 5 2 + 1 2 + 1 2 + 1 2 {\displaystyle 5^{2}+1^{2}+1^{2}+1^{2}} , 4 2 + 2 2 + 2 2 + 2 2 {\displaystyle 4^{2}+2^{2}+2^{2}+2^{2}} or 3 2 + 3 2 + 3 2 + 1 2 {\displaystyle 3^{2}+3^{2}+3^{2}+1^{2}} (see image). Twenty-eight is: 27 (number) Twenty-seven

2200-439: The appropriate complex number plane. As such, there is a type of angle ( Euclidean , hyperbolic , or slope ) that is preserved in such a multiplication. Due to Df being invertible, the type of complex number is uniform over the surface. Consequently, a surface deformation or diffeomorphism of surfaces has the conformal property of preserving (the appropriate type of) angles. Let M {\displaystyle M} be

2255-463: The ball B n {\displaystyle B^{n}} . For manifolds, the diffeomorphism group is usually not connected. Its component group is called the mapping class group . In dimension 2 (i.e. surfaces ), the mapping class group is a finitely presented group generated by Dehn twists ; this has been proved by Max Dehn , W. B. R. Lickorish , and Allen Hatcher ). Max Dehn and Jakob Nielsen showed that it can be identified with

28 (number) - Misplaced Pages Continue

2310-409: The circle is pathwise connected. This can be seen by noting that any such diffeomorphism can be lifted to a diffeomorphism f {\displaystyle f} of the reals satisfying [ f ( x + 1 ) = f ( x ) + 1 ] {\displaystyle [f(x+1)=f(x)+1]} ; this space is convex and hence path-connected. A smooth, eventually constant path to

2365-415: The diffeomorphism group acts transitively on the configuration space C k M {\displaystyle C_{k}M} . If M {\displaystyle M} is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space F k M {\displaystyle F_{k}M} and the action on M {\displaystyle M}

2420-408: The diffeomorphism group into a Banach manifold with smooth right translations; left translations and inversion are only continuous. If r = ∞ {\displaystyle r=\infty } , the space of vector fields is a Fréchet space . Moreover, the transition maps are smooth, making the diffeomorphism group into a Fréchet manifold and even into a regular Fréchet Lie group . If

2475-448: The exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions , is 27-dimensional; its automorphism group is the 52-dimensional exceptional Lie algebra F 4 . {\displaystyle \mathrm {F_{4}} .} There are twenty-seven sporadic groups , if the non-strict group of Lie type T {\displaystyle \mathrm {T} } (with an irreducible representation that

2530-510: The function and its inverse are continuously differentiable . Given two differentiable manifolds M {\displaystyle M} and N {\displaystyle N} , a differentiable map f : M → N {\displaystyle f\colon M\rightarrow N} is a diffeomorphism if it is a bijection and its inverse f − 1 : N → M {\displaystyle f^{-1}\colon N\rightarrow M}

2585-534: The identity gives a second more elementary way of extending a diffeomorphism from the circle to the open unit disc (a special case of the Alexander trick ). Moreover, the diffeomorphism group of the circle has the homotopy-type of the orthogonal group O ( 2 ) {\displaystyle O(2)} . The corresponding extension problem for diffeomorphisms of higher-dimensional spheres S n − 1 {\displaystyle S^{n-1}}

2640-838: The images of ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } . The map ψ f ϕ − 1 : U → V {\displaystyle \psi f\phi ^{-1}:U\to V} is then a diffeomorphism as in the definition above, whenever f ( ϕ − 1 ( U ) ) ⊆ ψ − 1 ( V ) {\displaystyle f(\phi ^{-1}(U))\subseteq \psi ^{-1}(V)} . Since any manifold can be locally parametrised, we can consider some explicit maps from R 2 {\displaystyle \mathbb {R} ^{2}} into R 2 {\displaystyle \mathbb {R} ^{2}} . In mechanics ,

2695-560: The manifold is σ {\displaystyle \sigma } -compact and not compact the full diffeomorphism group is not locally contractible for any of the two topologies. One has to restrict the group by controlling the deviation from the identity near infinity to obtain a diffeomorphism group which is a manifold; see ( Michor & Mumford 2013 ). The Lie algebra of the diffeomorphism group of M {\displaystyle M} consists of all vector fields on M {\displaystyle M} equipped with

2750-433: The mapping class group is simply the modular group SL ( 2 , Z ) {\displaystyle {\text{SL}}(2,\mathbb {Z} )} and the classification becomes classical in terms of elliptic , parabolic and hyperbolic matrices. Thurston accomplished his classification by observing that the mapping class group acted naturally on a compactification of Teichmüller space ; as this enlarged space

2805-427: The same for N {\displaystyle N} . Let ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } be charts on, respectively, M {\displaystyle M} and N {\displaystyle N} , with U {\displaystyle U} and V {\displaystyle V} as, respectively,

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2860-547: The terms 12, 16, 21 (it is the sum of the first two of these). It is also a Keith number , because it recurs in a Fibonacci -like sequence started from its decimal digits: 2, 8, 10, 18, 28... There are 28 convex uniform honeycombs . Twenty-eight is the only positive integer that has a unique Kayles nim-value . Twenty-eight is the only known number that can be expressed as a sum of the first positive integers ( 1 + 2 + 3 + 4 + 5 + 6 + 7 {\displaystyle 1+2+3+4+5+6+7} ),

2915-457: Was constructed by John Milnor in dimension 7. He constructed a smooth 7-dimensional manifold (called now Milnor's sphere ) that is homeomorphic to the standard 7-sphere but not diffeomorphic to it. There are, in fact, 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is the total space of a fiber bundle over the 4-sphere with the 3-sphere as the fiber). More unusual phenomena occur for 4-manifolds . In

2970-491: Was homeomorphic to a closed ball, the Brouwer fixed-point theorem became applicable. Smale conjectured that if M {\displaystyle M} is an oriented smooth closed manifold, the identity component of the group of orientation-preserving diffeomorphisms is simple . This had first been proved for a product of circles by Michel Herman ; it was proved in full generality by Thurston. Since every diffeomorphism

3025-451: Was much studied in the 1950s and 1960s, with notable contributions from René Thom , John Milnor and Stephen Smale . An obstruction to such extensions is given by the finite abelian group Γ n {\displaystyle \Gamma _{n}} , the " group of twisted spheres ", defined as the quotient of the abelian component group of the diffeomorphism group by the subgroup of classes extending to diffeomorphisms of

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