Mathematical Physics, Lecture 9

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Mathematical Physics, Lecture 9 Hoshang Heydari Fysikum April 25, 2012 Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

2 Table of contents 1 Differentiable manifolds 2 Differential maps and curve 3 Tangent, cotangent and tensor spaces 4 Tangent map and submanifolds 5 Differential forms 6 Integration of forms Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

3 Introduction In this lecture we give a short introduction to differentiable manifolds, differential forms and integration on manifolds. This topics are discussed in chapters of the main text book: Szekeres, Peter - A Course In Modern Mathematical Physics - Groups, Hilbert Spaces And Differential Geometry (2004) and in Dariusz Chruscinski, Andrzej Jamiolkowski, Geometric Phases in Classical and Quantum Mechanics, Progress in Mathematical Physics, Birkhäuser, Berlin Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

4 Differentiable manifolds Differentiable manifolds Differentiable manifolds A topological manifold M of dimension n = dim M is a Hausdorff space in which every point x has a neighboring homeomorphic to an open subset of R n. A coordinate chart at a point of p of M is the pair (U, φ), where domain of the chart U is open subset of M and φ : U φ(u) R n is a homeomorphism between U and its image φ(u) which is also an open subset of R n. Let pr i : R n R be projection maps. Then the map or functions x = pr i φ : U R for i = 1, 2,..., n are called coordinate functions. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

5 Differentiable manifolds Differentiable manifolds Differentiable manifolds A family of charts A = {(U α, φ α ) : α A} with the coordinate neighborhoods U α that cover M and any pair of charts from A are C -compatible is called an atlas on M. If A and B are two atlas on M, then so is their union A B. Any atlas could be extended to maximal atlas by adding all charts that are C -compatible with charts of A. The maximal atlas is called a differentiable structure on M. A pair (M, A), M is a topological manifold of dimension n and A is a differential structure on M is called differential manifold. Example R n is a trivial manifold, since the charts (U = R n, φ = id) covers it and generates a unique atlas. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

6 Differentiable manifolds Differentiable manifolds Differentiable manifolds Example Any open subspace of R n is a differentiable manifold formed by giving it the relative topology and the differentiable structure is generated by the chart (U, id U : U R n ). Let (V, ψ) be a charts on R n. Then every charts on U is the restriction of a coordinate neighborhood and coordinate map on R n to the open region U, that is (U V, ψ U V ). This manifold is called open submanifold of R n. Example The circle S 1 R 2 defined by x 2 + y 2 = 1 is a one dimensional manifold (exercise). S 1 is not homeomorphic to R. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

7 and φ ± i = U ± i R n defined by Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42 Differentiable manifolds Differentiable manifolds Differentiable manifolds Example The sphere S 2 R 3 defined by x 2 + y 2 + z 2 = 1 is a two dimensional differentiable manifold (exercise). Example The n-sphere S n R n+1 defined by S n = {x R n+1 : (x 1 ) 2 + (x 2 ) (x n+1 ) 2 = 1} is a n dimensional differentiable manifold. Let U + i = {x S n : x i > 0}, U i = {x S n : x i < 0}

8 which provides φ : M(n, R) with Hausdorff topology inherited from R n2. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42 Differentiable manifolds Differentiable manifolds Differentiable manifolds Example φ ± i (x) = (x 1, x 2,..., x i 1, x i+1,..., x n ). Then a set of charts providing an atlas is the set of rectangular on hemispheres (U + i, φ ± i ) and (U i, φ ± i ). Example There is a one-to-one correspondence between the set of n n real matrices M(n, R) and the points of R n2 through the map φ : M(n, R) R n2 defined by φ(a = (a ij )) = (a 11, a 12,..., a 1n, a 21, a 22,..., a nn )

9 Differentiable manifolds Differentiable manifolds Differentiable manifolds Example Moreover, the differential structure generated by (M(n, R), φ) converts M(n, R) into a differentiable manifolds of dimension n 2. Let M and N be differential manifolds of dimensions m and n respectively. Then their product M N is also differential manifold (exercise). Example The topological 2-torus T 2 = S 1 S 1 has a differential structure as a n times { }} { product manifold. And in general n-torus T n = S 1 S 1 S 1 is a product of n circles. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

10 Differential maps and curve Differential maps and curve Differential maps and curve Let M be a differential manifold of dimension n. Then a map f : M R is called differentiable at a point p M if for some coordinate chart (U, φ; x i ) at p the function f = f φ 1 : φ(u) R is differentiable at φ(p) = (x 1 (p), x 2 (p),..., x n (p)) = x(p). The definition is independent of choice of the chart at p. The set of all real- valued functions on M that are differentiable at p M are denoted by F p (M). Let V be an open subset of M. Then a real-valued function f : M R is called differentiable or smooth if it is differentiable at every point p V and it is denoted by F(V ). One can show that F(V ) is both a ring and a real vector space (exercise). Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

11 Differential maps and curve Differential maps and curve Differential maps and curve If M and N are differential manifolds, then a map α : M N is differentiable at p M if for any pair of coordinate charts (U, φ; x i ) and (V, ψ; y j ) covering p and α(p) resp., its coordinate representation ψ α φ 1 : φ(u) ψ(v ) is differentiable at φ(p). A diffeomorphism is map α : M N that is injective and α and its inverse are differentiable. Two manifolds M and N are said to bee diffeomorphic, M = N if there exists a diffeomorphism α and m = dim M = dim N = n. A smooth parametrized curve on an manifold M is a differentiable map Λ : (a, b) M, where (a, b) R is open interval. The curve is said to pass through p at t = t 0 if Λ(t 0 ) = p for a < t 0 < b. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

12 Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Tangent vectors A directional derivative of a differentiable function f : R n R along the curve at x 0 is defined by Xf = df (x(t)) dt t=t0 = dx i (t) f (x) t=t0 dt x i x=x0, where X is a linear differential operator X = dx i (t) t=t0 dt x i x=x 0 X is a real-valued map on the algebra of differentiable function at x 0. The map X : F x0 (R n ) R has the following properties: Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

13 Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Tangent vectors 1 It is linear on the space F x0 (R n ): for any pair of functions f and g we have X (af + bg) = axf + bxg, where a, b R. 2 It satisfies the Leibnitz rule X (fg) = f (x 0 )Xg + g(x 0 )Xf. A tangent vector X p at any point p of a differential manifold M is a linear map X p : F x0 (R n ) R that satisfies 1 Linearity: X p (af + bg) = ax p f + bx p g, where a, b R. 2 Leibnitz rule: X p (fg) = f (p)x p g + g(p)x p f. The set of tangent vector at p form a vector space T p (M) called tangent space at p. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

14 Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Tangent vectors If (U, φ) is any chart at p with coordinate functions x i, then the operators defined by ( x i ) p = x i p : F p (M) R ( x i ) p f = x i pf = f (x 1, x 2,..., x n ) x i x=φ(p), where f = f φ 1 : R n R. Thus any linear combination X p = X i x i p = n i=1 X i x i p, Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

15 Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Tangent vectors where X i R is a tangent vector. The coefficient X j is computed by the action of X on coordinate functions x j X p x j = X i x j x i x=φ(p) = X i δ j i = X j. Theorem If (U, φ; x i ) is a chart at p M, then the operators ( x i ) p defined by ( x i ) p f = x i p f = f (x 1,x 2,...,x n ) x i x=φ(p), form a basis of tangent space T p M and its dimension is n = dim M. For the proof see page Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

16 Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Cotangent and tensor spaces The cotangent space at p is the dual space T p (M) associated to the tangent space at p M and it consists of all linear functionals on T p (M), also called covectors or 1-forms at p. The action of covector ω p at p on tangent vector X p is denoted by ω p (X p ), ω p, X p or X p, ω p. Note also that dim T p (M) = dim T p (M) = dim M. If f is differentiable function at p, then its differential at p is defined by (df ) p which acts on tangent vector X p as follows (df ) p, X p = X p f. For a chart (U, φ; x i ) at p, the differential of the coordinate functions have the following property (dx i ) p, X p = X p x i = X i, Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

17 Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Cotangent and tensor spaces where X i are components of X p = X i ( x i ) p. Applying (dx i ) p to ( x j ) p we get (dx i ) p, ( x j ) p = x j px i = x i x j φ(p) = δj i. Thus the linear functional (dx 1 ) p, (dx 2 ) p,..., (dx n ) p are the dual basis that span cotangent space. Every covector ω p has a unique expansion where w i = ω p, ( x i ) p. ω p = w i (dx i ) p, Vector and tensor field A vector field X is an assignment of tangent vector X p at each point Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

18 Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Vector and tensor field p M. Or X is a map from M to the set p M T p(m) with the property that the image of every point, X (p) belong to the tangent space T p (M) at p. The vector field is called differentiable or smooth if for every differentiable function f F(M) the function Xf defined by defined by (Xf )(p) = X p f is differentiable, that is f F(M) = Xf F(M). The set of all differentiable vector fields on M will be denoted by T (M). Every smooth vector field defines a map X : F(M) F(M) which satisfies the following properties Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

19 Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Vector and tensor field 1 Linearity: X (af + bg) = axf + bxg, where a, b R and f, g F(M). 2 Leibnitz rule: X (fg) = fxg + gxf. Conversely any map X with above properties defines a smooth vector field. The constructions of tangent and cotangent spaces enable we to introduce an arbitrary tensor field on M. Thus we call a smooth map x T (M) T (k,l) x M = T x M T x M T x M T x M, a tensor field of type (k, l). A vector field is a tensor field of type (1, 0). Any tensor field is uniquely defined by its components T = i 1i 2 i k j 1 j 2 j l x i 1 x i 2 x i k dx j 1 dx j l. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

20 Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Tensor bundles The tangent bundle on a manifold is defined by TM = T p (M). p M And there is a natural projection map π : TM M defined by π(u) = p for u T p (M). Moreover, for chart (U, φ; x i ) on M we define a chart (π 1 (U), φ) on TM where the coordinate map φ : π 1 (U) R 2n is defined by φ(v) = (x 1 (p),..., x n (p), v 1,..., v n ), where p = π(v) and v = n i=1 v i x i p. The cotangent bundle Tp (M) is defined in similar manner. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

21 Tangent map and submanifolds Tangent map and submanifolds The tangent map and pullback of a map Tangent map Let α : M N be a differentiable map. Then α induces a map α : T p (M) T α(p) (N) called the tangent map of α, where the tangent vector Y α(p) = α X p is defined by Y α(p) f = (α X p )f = X p (f α) for any function f F α(p) (N). Pullback of a map The map α : M N also induces a map Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

22 Submanifolds Let α : M N be a differentiable map such that m = dim M n = dim N. Then the map α is called immersion if the tangent map Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42 Tangent map and submanifolds Tangent map and submanifolds The tangent map and pullback of a map Pullback of a map α : T α(p) (N) T p (M) between cotangent spaces which is called the pullback induced by α. The pullback of a 1-form ω α(p) is defined by for arbitrary tangent vectors X p. α ω α(p), X p = ω α(p), α X p

23 Tangent map and submanifolds Tangent map and submanifolds The tangent map and pullback of a map Submanifolds α : T p (M) T α(p) (N) is injective at every point p M, e.g., α is non-degenerate linear map everywhere. If the map α and the tangent map α are injective, then the map is called an embedding and the pair (M, α) are called an embedded submanifold of N. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

24 Differential forms Differential forms Differential forms Next we will review a class of tensor fields on differential manifold that play an important role in physical applications. A skew-symmetric tensor of type (0, k) is called a differential form of order k or a k-form. Let Λ k (M) be the space of k-form on M with Λ k (M) = { } for k > n. Then, the space of differential forms on M is defined by Λ k (M) = n Λ k (M) with Λ 0 (M) = C (M). The space Λ(M) equipped with the following operations: k=0 Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

25 Differential forms Differential forms Differential forms 1 A wedge product: : Λ k (M) Λ l (M) Λ k+l (M). 2 The exterior derivative: d : Λ k (M) Λ k+1 (M) defined by dα = 1 k! α i1 i k x j dx j dx i 1 dx i k. Example A function on a differential manifold M is a zero-form which has the following one form as its exterior derivative df = f dx i. note also that x i d(df ) = 0. If M = R n and (x 1, x 2,..., x n ) are cartesian coordinates, then df gives the components of gradf. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

26 Differential forms Differential forms Differential forms Proposition The exterior derivative satisfies the following properties d 2 α = d(dα) = 0 for any α Λ(M). d(α β) = dα β + ( 1) k α dβ, for α, β Λ(M). Proposition The pullback operation commutes with exterior derivative and wedge product, that is φ (dα) = d(φ α), φ (α β) = φ α φ β for any differential form α and β on N. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

27 Integration of forms Integration of forms Integration of forms Let f : R n R be a function and (x 1,..., x n ) be cartesian coordinates of R n. Then, an n-dimensional integral id defined by f fdv = f (x 1,..., x n )dx 1 dx n. After a coordinate transformation we have =J { ( }} ) { x d x 1 d x n i = det x j dx 1 dx n, where J is the Jacobian of transformation. An n-dimensional manifold M is orientable if and only if there exists a nowhere vanishing n-form on it. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

28 Integration of forms Integration of forms Example A Riemannian manifold (M, g) is differentiable manifold M equipped with a smooth metric tensor g of type (0, 2) such that g is symmetric and for each x M, the bilinear form g x : T x M T x M R is nondegenerate. We call a Riemannian manifold proper if g x (v, v) > 0 v T x M, v 0. Otherwise a manifold is called pseodu-riemannian. The n dimensional Euclidean space R n is a proper Riemannian manifold and the Minkowski space R 1,3 is pseudo-riemannian. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

29 Integration of forms Integration of forms Hodge star Consider the space of differential forms on M with n = dim M. The so called Hodge operation of Hodge star : Λ k (M) Λ n k (M) defined by ( α) i1 i n k = 1 k! g ɛi1 i n k j 1 j k α j 1 j k, where g = det(g ij ) and α j 1 j k = g j 1m1 g j km k α m1 m k. The form α is called the Hodge dual of α and g il g lj = δj i. Let τ = g dx 1 dx n. Then if M is compact we define volume of M by Vol(M) = τ. M Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

30 Integration of forms Integration of forms Hodge star Let K be a k dimensional orientable submanifold of M and j : K M be a canonical embedding. If (y 1,..., y k ) are local coordinates on K then j is defined by x 1 = x 1 (y 1,..., y k ) x 2 = x 2 (y 1,..., y k )... x n = x n (y 1,..., y k ) If α is a k-form on M, then j α is a k-form on K. Thus we can define an integral of K over j α (K, α) = j α K = 1 k! x i 1 ik x α i1 i k y 1 y k dy 1 dy k, Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

31 Integration of forms Integration of forms Hodge star where α = 1 k! α i 1 i k dx i 1 dx i k. Let R n + = {(x 1,..., x n ) R n : x 1 0}. The we define the boundary of R n + by R n + = {(x 1,..., x n ) R n : x 1 = 0}. We call M a manifold with a boundary when there exists an open covering (U i, φ i ) such that φ i (U i ) defines an open subset of R n + and the boundary M is defined by M = i φ 1 i ( R n +). Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

32 Integration of forms Integration of forms Hodge star Example As an example we consider the unit ball B n in R n B n = {(x 1,..., x n ) R n : n (x i ) 2 1}. i=1 The boundary of B n is a (n 1)-dimensional sphere B n = S n 1. Note also that for any manifold M = ( M) = { } and in particular S n = ( B n+1 ) = {}. A manifold without boundary is called a closed manifold. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

33 Integration of forms Integration of forms Hodge star Stokes theorem Let M be a n dimensional manifold with boundary M and let ω Λ n 1 (M). The we have dω = ω. M M Example Let Σ be two-dimensional surface in R 3 and A be a vector field. Then we have curla ds = A dl, where ds denotes a surface elements on σ. Σ C= Σ Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

34 Integration of forms Integration of forms De Rham cohomology We call a k-form α a closed form if dα = 0, and an exact form if α = dβ for any (k 1)-form β. For a manifold M, the set of closed form is defined by Z k = {α Λ k (M) : dα = 0} and the set of exact form is defined by B k = {α Λ k (M) : β Λ k 1 (M), dβ = α}. Next we define the following equivalence relation in Λ k (M) α 1 α 2 β Λ k 1 (M), dβ = α 1 α 2 } Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

35 Integration of forms Integration of forms De Rham cohomology Thus α 1 α 2 = dα 1 = dα 2 and we can define the space of equivalence classes by H k (M) = Z k (M) B k (M), is called the de Rham cohomology group of M and means that H k (M) is the set of closed forms that differ only by an exact k-form. de Rham cohomology group is an abelian group where the operation is the addition of k-forms. [ω] is the equivalence class containing ω and is called a cohomology class of ω. If [ω 1 ] H k (M) and [ω 2 ] H l (M), then we have Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

36 defined by φ # ([ω]))[φ ω]. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42 Integration of forms Integration of forms De Rham cohomology In R n all closed forms are exact. [ω 1 ω 2 ] = [ω 1 ] [ω 2 ] H k+l (M) Poincare lemma Let M be a differentiable manifold. Then any closed form on M is locally exact, that is if dα = 0, then for any x M, there is a neighborhood U containing x such that α = dβ on U. Thus only global properties ofm decide if de Rham cohomology grops are trivial or not. The smooth map φ : M N induces a linear transformation φ # : H K (M) H k (N)

37 Integration of forms Integration of forms De Rham cohomology If φ is a homeomorphism, then the induced map φ # is an isomorphism. Thus topologically equivalent manifolds have isomorphic cohomology groups and in particular we have b k (M) = dim H k (M) = dim H k (N) = b k (N), k = 0, 1, 2,..., n are called Betti numbers. From Betti numbers we can construct Euler characteristic n χ(m) = ( 1) k b k (M). k=0 The topological equivalent manifolds have same Euler characteristic. Example Let M = R 2. Then H 1 (R 2 ) = H 2 (R 2 ) = 0, H 0 (R 2 ) = R, and χ(r 2 ) = 1. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

38 Integration of forms Integration of forms De Rham cohomology Example Let M = S 2. Then H 0 (S 2 ) = H 2 (S 2 ) = R and all other cohomology groups are trivial. We have also χ(s 2 ) = 2. Example Let M = T 2 = S 1 S 1. Then H 0 (T 2 ) = H 2 (T 2 ) = R and all other cohomology groups are trivial (exercise). We have also χ(t 2 ) = 0. Proposition A contractible manifold M by which we mean a manifold that may be continuously contracted to single point has trivial de Rham cohomology groups H k (M) for all k 1. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

39 Integration of forms Integration of forms De Rham cohomology Proposition Let M be a manifold which is compact, connected, and orientable with n = dim M. Then we have H k (M) = H n k (M), k = 0, 1, 2,..., n. This important result is called the Poincare duality. Lie derivative Let X be a vector field on a manifold M. Then the flow of the X is the collection of maps F t : M M which satisfy d dt F t(x) = X (F t (x)), x X, t R. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

40 Integration of forms Integration of forms Lie derivative By fixing a point x X we obtain a map t R F t (x) M which defines a curve in M called the integral curve of X passing a point x. The flow F t also satisfies the following property F t F s = F t+s. Example Let A : R n R n be a vector field on R n defined by x Ax T x R n = R n, where x R n and the flow satisfies the following equation d dt F t(x) = AF t (x) = F t (x) = e At x. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

41 where the Lie bracket is defined by [X, Y ] i = X k k Y i Y k k X i. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42 Integration of forms Integration of forms Lie derivative Let T be a tensor field on a manifold M and X be a vector field on M. Then the Lie derivative of T with respect to X is defined by (L X T )(x) = d dt (F t T )(x) t=0, where F t is the flow of vector field X. If f is a function on M then L X f is given by L X f = d dt (F t f )(x) t=0 = d dt (f F tt )(x) t=0 = X (f ), is the directional derivative of f along X. If X and Y are vector fields on M, then L X Y = [X, Y ],

42 Problems Integration of forms Problem 15.4 This is the first problem of the home assignment three. Problem 16.4 This is the second problem of the home assignment three. Problem 17.7 This is the third problem of the home assignment three. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, / 42

8.1 Examples, definitions, and basic properties

8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

Math 225A, Differential Topology: Homework 3

Math 225A, Differential Topology: Homework 3 Ian Coley October 17, 2013 Problem 1.4.7. Suppose that y is a regular value of f : X Y, where X is compact and dim X = dim Y. Show that f 1 (y) is a finite

MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI

14.11. Geodesic Lines, Local Gauss-Bonnet Theorem

14.11. Geodesic Lines, Local Gauss-Bonnet Theorem Geodesics play a very important role in surface theory and in dynamics. One of the main reasons why geodesics are so important is that they generalize

Chapter 2. Parameterized Curves in R 3

Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,

DIVISORS AND LINE BUNDLES

DIVISORS AND LINE BUNDLES TONY PERKINS 1. Cartier divisors An analytic hypersurface of M is a subset V M such that for each point x V there exists an open set U x M containing x and a holomorphic function

Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

Introduction to Topology

Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

Fiber Bundles and Connections. Norbert Poncin

Fiber Bundles and Connections Norbert Poncin 2012 1 N. Poncin, Fiber bundles and connections 2 Contents 1 Introduction 4 2 Fiber bundles 5 2.1 Definition and first remarks........................ 5 2.2

Definition 12 An alternating bilinear form on a vector space V is a map B : V V F such that

4 Exterior algebra 4.1 Lines and 2-vectors The time has come now to develop some new linear algebra in order to handle the space of lines in a projective space P (V ). In the projective plane we have seen

Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

A Maxwellian Formulation by Cartan s Formalism

Applied Mathematical Sciences, Vol. 8, 2014, no. 11, 499-505 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.312694 A Maxwellian Formulation by Cartan s Formalism Elmo Benedetto Mathematics

Proof. The map. G n i. where d is the degree of D.

7. Divisors Definition 7.1. We say that a scheme X is regular in codimension one if every local ring of dimension one is regular, that is, the quotient m/m 2 is one dimensional, where m is the unique maximal

Homotopy groups of spheres and low-dimensional topology

Homotopy groups of spheres and low-dimensional topology Andrew Putman Abstract We give a modern account of Pontryagin s approach to calculating π n+1 (S n ) and π n+2 (S n ) using techniques from low-dimensional

LECTURE 1: DIFFERENTIAL FORMS. 1. 1-forms on R n

LECTURE 1: DIFFERENTIAL FORMS 1. 1-forms on R n In calculus, you may have seen the differential or exterior derivative df of a function f(x, y, z) defined to be df = f f f dx + dy + x y z dz. The expression

INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS

INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Abstract. We classify the left-invariant metrics with nonnegative

The Tangent Bundle. Jimmie Lawson Department of Mathematics Louisiana State University. Spring, 2006

The Tangent Bundle Jimmie Lawson Department of Mathematics Louisiana State University Spring, 2006 1 The Tangent Bundle on R n The tangent bundle gives a manifold structure to the set of tangent vectors

Lecture 18 - Clifford Algebras and Spin groups

Lecture 18 - Clifford Algebras and Spin groups April 5, 2013 Reference: Lawson and Michelsohn, Spin Geometry. 1 Universal Property If V is a vector space over R or C, let q be any quadratic form, meaning

Remarks on Lagrangian singularities, caustics, minimum distance lines

Remarks on Lagrangian singularities, caustics, minimum distance lines Department of Mathematics and Statistics Queen s University CRM, Barcelona, Spain June 2014 CRM CRM, Barcelona, SpainJune 2014 CRM

An Introduction to Riemannian Geometry

Lecture Notes in Mathematics An Introduction to Riemannian Geometry Sigmundur Gudmundsson (Lund University) (version 1.0304-30 March 2016) The latest version of this document can be found at http://www.matematik.lu.se/matematiklu/personal/sigma/

COBORDISM IN ALGEBRA AND TOPOLOGY

COBORDISM IN ALGEBRA AND TOPOLOGY ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ aar Dedicated to Robert Switzer and Desmond Sheiham Göttingen, 13th May, 2005 1 Cobordism There is a cobordism equivalence

Tensors on a vector space

APPENDIX B Tensors on a vector space In this Appendix, we gather mathematical definitions and results pertaining to tensors. The purpose is mostly to introduce the modern, geometrical view on tensors,

BILINEAR FORMS KEITH CONRAD The geometry of R n is controlled algebraically by the dot product. We will abstract the dot product on R n to a bilinear form on a vector space and study algebraic and geometric

4. Expanding dynamical systems

4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

The cover SU(2) SO(3) and related topics

The cover SU(2) SO(3) and related topics Iordan Ganev December 2011 Abstract The subgroup U of unit quaternions is isomorphic to SU(2) and is a double cover of SO(3). This allows a simple computation of

PROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin

PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles

College Meetkunde/Geometry, 3de en 4de jaar, najaar/fall 2002, 12 weken/weeks.

College Meetkunde/Geometry, 3de en 4de jaar, najaar/fall 2002, 12 weken/weeks. Bas Edixhoven, Universiteit Leiden. December 15, 2002 The writing of this syllabus is in progress, as one can see from the

A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

Extrinsic geometric flows

On joint work with Vladimir Rovenski from Haifa Paweł Walczak Uniwersytet Łódzki CRM, Bellaterra, July 16, 2010 Setting Throughout this talk: (M, F, g 0 ) is a (compact, complete, any) foliated, Riemannian

NOTES ON MINIMAL SURFACES

NOTES ON MINIMAL SURFACES DANNY CALEGARI Abstract. These are notes on minimal surfaces, with an emphasis on the classical theory and its connection to complex analysis, and the topological applications

Invariant Metrics with Nonnegative Curvature on Compact Lie Groups

Canad. Math. Bull. Vol. 50 (1), 2007 pp. 24 34 Invariant Metrics with Nonnegative Curvature on Compact Lie Groups Nathan Brown, Rachel Finck, Matthew Spencer, Kristopher Tapp and Zhongtao Wu Abstract.

The Euler class groups of polynomial rings and unimodular elements in projective modules

The Euler class groups of polynomial rings and unimodular elements in projective modules Mrinal Kanti Das 1 and Raja Sridharan 2 1 Harish-Chandra Research Institute, Allahabad. Chhatnag Road, Jhusi, Allahabad

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS

TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS OSAMU SAEKI Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday Abstract. We classify singular fibers of C stable maps of orientable

The Topology of Fiber Bundles Lecture Notes. Ralph L. Cohen Dept. of Mathematics Stanford University

The Topology of Fiber Bundles Lecture Notes Ralph L. Cohen Dept. of Mathematics Stanford University Contents Introduction v Chapter 1. Locally Trival Fibrations 1 1. Definitions and examples 1 1.1. Vector

DEFORMATION OF DIRAC STRUCTURES ALONG ISOTROPIC SUBBUNDLES. and MARCO ZAMBON

Vol. 65 (2010) REPORTS ON MATHEMATICAL PHYSICS No. 2 DEFORMATION OF DIRAC STRUCTURES ALONG ISOTROPIC SUBBUNDLES IVÁN CALVO Laboratorio Nacional de Fusión, Asociación EURATOM-CIEMAT, E-28040 Madrid, Spain

RANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis

RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, applied

Almost Quaternionic Structures on Quaternionic Kaehler Manifolds. F. Özdemir

Almost Quaternionic Structures on Quaternionic Kaehler Manifolds F. Özdemir Department of Mathematics, Faculty of Arts and Sciences Istanbul Technical University, 34469 Maslak-Istanbul, TURKEY fozdemir@itu.edu.tr

How to minimize without knowing how to differentiate (in Riemannian geometry)

How to minimize without knowing how to differentiate (in Riemannian geometry) Spiro Karigiannis karigiannis@maths.ox.ac.uk Mathematical Institute, University of Oxford Calibrated Geometries and Special

Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

Metrics on SO(3) and Inverse Kinematics

Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction

A new viewpoint on geometry of a lightlike hypersurface in a semi-euclidean space

A new viewpoint on geometry of a lightlike hypersurface in a semi-euclidean space Aurel Bejancu, Angel Ferrández Pascual Lucas Saitama Math J 16 (1998), 31 38 (Partially supported by DGICYT grant PB97-0784

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

A numerable cover of a topological space X is one which possesses a partition of unity.

Chapter 1 I. Fibre Bundles 1.1 Definitions Definition 1.1.1 Let X be a topological space and let {U j } j J be an open cover of X. A partition of unity relative to the cover {U j } j J consists of a set

What is complete integrability in quantum mechanics

INTERNATIONAL SOLVAY INSTITUTES FOR PHYSICS AND CHEMISTRY Proceedings of the Symposium Henri Poincaré (Brussels, 8-9 October 2004) What is complete integrability in quantum mechanics L. D. Faddeev St.

3. Prime and maximal ideals. 3.1. Definitions and Examples.

COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,

LECTURE III. Bi-Hamiltonian chains and it projections. Maciej B laszak. Poznań University, Poland

LECTURE III Bi-Hamiltonian chains and it projections Maciej B laszak Poznań University, Poland Maciej B laszak (Poznań University, Poland) LECTURE III 1 / 18 Bi-Hamiltonian chains Let (M, Π) be a Poisson

Chapter 10. Abstract algebra

Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and

BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

CURVATURE. 1. Problems (1) Let S be a surface with a chart (φ, U) so that. + g11 g 22 x 1 and g 22 = φ

CURVTURE NDRÉ NEVES 1. Problems (1) Let S be a surface with a chart (φ, U) so that. = 0 x 1 x 2 for all (x 1, x 2 ) U. Show that [ 1 K φ = 2 g 11 g 22 x 2 where g 11 = x 1. x 1 ( ) x2 g 11 + g11 g 22 x

Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver

Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point

Finite dimensional C -algebras

Finite dimensional C -algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for self-adjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n

Fiber Bundles. 4.1 Product Manifolds: A Visual Picture

4 Fiber Bundles In the discussion of topological manifolds, one often comes across the useful concept of starting with two manifolds M ₁ and M ₂, and building from them a new manifold, using the product

INTRODUCTION TO TOPOLOGY

INTRODUCTION TO TOPOLOGY ALEX KÜRONYA In preparation January 24, 2010 Contents 1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology

The Euler class group of a polynomial algebra

The Euler class group of a polynomial algebra Mrinal Kanti Das Harish-Chandra Research Institute, Allahabad. Chhatnag Road, Jhusi, Allahabad - 211 019, India. e-mail : mrinal@mri.ernet.in 1 Introduction

Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces

Communications in Mathematical Physics Manuscript-Nr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan Bechtluft-Sachs, Marco Hien Naturwissenschaftliche

Mitschrieb zur Vorlesung: Riemannsche Geometrie

Mitschrieb zur Vorlesung: Riemannsche Geometrie Priv.Doz. Dr. Baues Vorlesung Wintersemester 2005/2006 Letzte Aktualisierung und Verbesserung: 18. Februar 2006 Mitschrieb der Vorlesung Riemannsche Geometrie

PRESIDENCY UNIVERSITY, KOLKATA

PRESIDENCY UNIVERSITY, KOLKATA Syllabus for Three Year B.Sc. MATHEMATICS (GenEd) Course (With effect from the Academic Session 2013-14) Module Structure Semester Module No. Name of the Module Marks I M11

1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points

Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a

Milnor s Fibration Theorem for Real Singularities

Milnor s Fibration Theorem for Real Singularities av NIKOLAI B. HANSEN MASTEROPPGAVE for graden Master i Matematikk (Master of Science) Det matematisk- naturvitenskapelige fakultet Universitetet i Oslo

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

An Advanced Course in Linear Algebra. Jim L. Brown

An Advanced Course in Linear Algebra Jim L. Brown July 20, 2015 Contents 1 Introduction 3 2 Vector spaces 4 2.1 Getting started............................ 4 2.2 Bases and dimension.........................

POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS

POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS N. ROBIDOUX Abstract. We show that, given a histogram with n bins possibly non-contiguous or consisting

1 Introduction to Matrices

1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

tr g φ hdvol M. 2 The Euler-Lagrange equation for the energy functional is called the harmonic map equation:

Notes prepared by Andy Huang (Rice University) In this note, we will discuss some motivating examples to guide us to seek holomorphic objects when dealing with harmonic maps. This will lead us to a brief

A GENERAL FRAMEWORK FOR NONHOLONOMIC MECHANICS: NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS

A GENERAL FRAMEWORK FOR NONHOLONOMIC MECHANICS: NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS DAVID IGLESIAS, JUAN C. MARRERO, D. MARTÍN DE DIEGO, AND DIANA SOSA Abstract. This paper presents a geometric description

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the r-th

Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres textbook. John Rognes

Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres textbook John Rognes November 29th 2010 Contents Introduction v 1 Set Theory and Logic 1 1.1 ( 1) Fundamental Concepts..............................

Singular fibers of stable maps and signatures of 4 manifolds

359 399 359 arxiv version: fonts, pagination and layout may vary from GT published version Singular fibers of stable maps and signatures of 4 manifolds OSAMU SAEKI TAKAHIRO YAMAMOTO We show that for a

LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

The Full Pythagorean Theorem

The Full Pythagorean Theorem Charles Frohman January 1, 2010 Abstract This note motivates a version of the generalized pythagorean that says: if A is an n k matrix, then det(a t A) = I det(a I ) 2 where

Chapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces

Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z

The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

Introduction to Characteristic Classes

UNIVERSITY OF COPENHAGEN Faculty of Science Department of Mathematical Sciences Mauricio Esteban Gómez López Introduction to Characteristic Classes Supervisors: Jesper Michael Møller, Ryszard Nest 1 Abstract

A NEW CONSTRUCTION OF 6-MANIFOLDS

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 8, August 2008, Pages 4409 4424 S 0002-9947(08)04462-0 Article electronically published on March 12, 2008 A NEW CONSTRUCTION OF 6-MANIFOLDS

INTRODUCTION TO MANIFOLDS IV. Appendix: algebraic language in Geometry

INTRODUCTION TO MANIFOLDS IV Appendix: algebraic language in Geometry 1. Algebras. Definition. A (commutative associatetive) algebra (over reals) is a linear space A over R, endowed with two operations,

and s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space

RAL ANALYSIS A survey of MA 641-643, UAB 1999-2000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σ-algebras. A σ-algebra in X is a non-empty collection of subsets

CLASSIFICATIONS OF STAR PRODUCTS AND DEFORMATIONS OF POISSON BRACKETS

POISSON GEOMETRY BANACH CENTER PUBLICATIONS, VOLUME 51 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 000 CLASSIFICATIONS OF STAR PRODUCTS AND DEFORMATIONS OF POISSON BRACKETS PHILIPP E BO

The Math Circle, Spring 2004

The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the

Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

NOTES on LINEAR ALGEBRA 1

School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter

University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

CLOSED AND EXACT DIFFERENTIAL FORMS IN R n

CLOSED AND EXACT DIFFERENTIAL FORMS IN R n PATRICIA R. CIRILO, JOSÉ REGIS A.V. FILHO, SHARON M. LUTZ paty@ufmg.br, j019225@dac.unicamp.br, sharon.lutz@colorado.edu Abstract. We show in this paper that

MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

Analytic cohomology groups in top degrees of Zariski open sets in P n

Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction

EE 580 Linear Control Systems VI. State Transition Matrix

EE 580 Linear Control Systems VI. State Transition Matrix Department of Electrical Engineering Pennsylvania State University Fall 2010 6.1 Introduction Typical signal spaces are (infinite-dimensional vector

Allen Back. Oct. 29, 2009

Allen Back Oct. 29, 2009 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )). Notation:(anachronistic) Let the coefficient

Lecture 4 Cohomological operations on theories of rational type.

Lecture 4 Cohomological operations on theories of rational type. 4.1 Main Theorem The Main Result which permits to describe operations from a theory of rational type elsewhere is the following: Theorem

Samuel Omoloye Ajala

49 Kragujevac J. Math. 26 (2004) 49 60. SMOOTH STRUCTURES ON π MANIFOLDS Samuel Omoloye Ajala Department of Mathematics, University of Lagos, Akoka - Yaba, Lagos NIGERIA (Received January 19, 2004) Abstract.