NATURAL STRUCTURES in DIFFERENTIAL GEOMETRY Lucas
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HILBERT SPACE GEOMETRY Definition: a Vector Space Over Is a Set V (Whose Elements Are Called Vectors) Together with a Binary Operation
HILBERT SPACE GEOMETRY Definition: A vector space over is a set V (whose elements are called vectors) together with a binary operation +:V×V→V, which is called vector addition, and an external binary operation ⋅: ×V→V, which is called scalar multiplication, such that (i) (V,+) is a commutative group (whose neutral element is called zero vector) and (ii) for all λ,µ∈ , x,y∈V: λ(µx)=(λµ)x, 1 x=x, λ(x+y)=(λx)+(λy), (λ+µ)x=(λx)+(µx), where the image of (x,y)∈V×V under + is written as x+y and the image of (λ,x)∈ ×V under ⋅ is written as λx or as λ⋅x. Exercise: Show that the set 2 together with vector addition and scalar multiplication defined by x y x + y 1 + 1 = 1 1 + x 2 y2 x 2 y2 x λx and λ 1 = 1 , λ x 2 x 2 respectively, is a vector space. 1 Remark: Usually we do not distinguish strictly between a vector space (V,+,⋅) and the set of its vectors V. For example, in the next definition V will first denote the vector space and then the set of its vectors. Definition: If V is a vector space and M⊆V, then the set of all linear combinations of elements of M is called linear hull or linear span of M. It is denoted by span(M). By convention, span(∅)={0}. Proposition: If V is a vector space, then the linear hull of any subset M of V (together with the restriction of the vector addition to M×M and the restriction of the scalar multiplication to ×M) is also a vector space. -
LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
Arxiv:Gr-Qc/0611154 V1 30 Nov 2006 Otx Fcra Geometry
MacDowell–Mansouri Gravity and Cartan Geometry Derek K. Wise Department of Mathematics University of California Riverside, CA 92521, USA email: [email protected] November 29, 2006 Abstract The geometric content of the MacDowell–Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geomet- ric meaning to the MacDowell–Mansouri trick of combining the Levi–Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous ‘model spacetime’, including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti de Sitter, or other models. A ‘Cartan connection’ gives a prescription for parallel transport from one ‘tangent model spacetime’ to another, along any path, giving a natural interpretation of the MacDowell–Mansouri connection as ‘rolling’ the model spacetime along physical spacetime. I explain Cartan geometry, and ‘Cartan gauge theory’, in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell–Mansouri gravity, as well as its recent reformulation in terms of BF theory, in the arXiv:gr-qc/0611154 v1 30 Nov 2006 context of Cartan geometry. 1 Contents 1 Introduction 3 2 Homogeneous spacetimes and Klein geometry 8 2.1 Kleingeometry ................................... 8 2.2 MetricKleingeometry ............................. 10 2.3 Homogeneousmodelspacetimes. ..... 11 3 Cartan geometry 13 3.1 Ehresmannconnections . .. .. .. .. .. .. .. .. 13 3.2 Definition of Cartan geometry . ..... 14 3.3 Geometric interpretation: rolling Klein geometries . .............. 15 3.4 ReductiveCartangeometry . 17 4 Cartan-type gauge theory 20 4.1 Asequenceofbundles ............................. -
Does Geometric Algebra Provide a Loophole to Bell's Theorem?
Discussion Does Geometric Algebra provide a loophole to Bell’s Theorem? Richard David Gill 1 1 Leiden University, Faculty of Science, Mathematical Institute; [email protected] Version October 30, 2019 submitted to Entropy Abstract: Geometric Algebra, championed by David Hestenes as a universal language for physics, was used as a framework for the quantum mechanics of interacting qubits by Chris Doran, Anthony Lasenby and others. Independently of this, Joy Christian in 2007 claimed to have refuted Bell’s theorem with a local realistic model of the singlet correlations by taking account of the geometry of space as expressed through Geometric Algebra. A series of papers culminated in a book Christian (2014). The present paper first explores Geometric Algebra as a tool for quantum information and explains why it did not live up to its early promise. In summary, whereas the mapping between 3D geometry and the mathematics of one qubit is already familiar, Doran and Lasenby’s ingenious extension to a system of entangled qubits does not yield new insight but just reproduces standard QI computations in a clumsy way. The tensor product of two Clifford algebras is not a Clifford algebra. The dimension is too large, an ad hoc fix is needed, several are possible. I further analyse two of Christian’s earliest, shortest, least technical, and most accessible works (Christian 2007, 2011), exposing conceptual and algebraic errors. Since 2015, when the first version of this paper was posted to arXiv, Christian has published ambitious extensions of his theory in RSOS (Royal Society - Open Source), arXiv:1806.02392, and in IEEE Access, arXiv:1405.2355. -
APPLICATIONS of BUNDLE MAP Theoryt1)
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 171, September 1972 APPLICATIONS OF BUNDLE MAP THEORYt1) BY DANIEL HENRY GOTTLIEB ABSTRACT. This paper observes that the space of principal bundle maps into the universal bundle is contractible. This fact is added to I. M. James' Bundle map theory which is slightly generalized here. Then applications yield new results about actions on manifolds, the evaluation map, evaluation subgroups of classify- ing spaces of topological groups, vector bundle injections, the Wang exact sequence, and //-spaces. 1. Introduction. Let E —>p B be a principal G-bundle and let EG —>-G BG be the universal principal G-bundle. The author observes that the space of principal bundle maps from E to £G is "essentially" contractible. The purpose of this paper is to draw some consequences of that fact. These consequences give new results about characteristic classes and actions on manifolds, evaluation subgroups of classifying spaces, vector bundle injections, the evaluation map and homology, a "dual" exact sequence to the Wang exact sequence, and W-spaces. The most interesting consequence is Theorem (8.13). Let G be a group of homeomorphisms of a compact, closed, orientable topological manifold M and let co : G —> M be the evaluation at a base point x e M. Then, if x(M) denotes the Euler-Poincaré number of M, we have that \(M)co* is trivial on cohomology. This result follows from the simple relationship between group actions and characteristic classes (Theorem (8.8)). We also find out information about evaluation subgroups of classifying space of topological groups. In some cases, this leads to explicit computations, for example, of GX(B0 ). -
Spanning Sets the Only Algebraic Operations That Are Defined in a Vector Space V Are Those of Addition and Scalar Multiplication
i i “main” 2007/2/16 page 258 i i 258 CHAPTER 4 Vector Spaces 23. Show that the set of all solutions to the nonhomoge- and let neous differential equation S1 + S2 = {v ∈ V : y + a1y + a2y = F(x), v = x + y for some x ∈ S1 and y ∈ S2} . where F(x) is nonzero on an interval I, is not a sub- 2 space of C (I). (a) Show that, in general, S1 ∪ S2 is not a subspace of V . 24. Let S1 and S2 be subspaces of a vector space V . Let (b) Show that S1 ∩ S2 is a subspace of V . S ∪ S ={v ∈ V : v ∈ S or v ∈ S }, 1 2 1 2 (c) Show that S1 + S2 is a subspace of V . S1 ∩ S2 ={v ∈ V : v ∈ S1 and v ∈ S2}, 4.4 Spanning Sets The only algebraic operations that are defined in a vector space V are those of addition and scalar multiplication. Consequently, the most general way in which we can combine the vectors v1, v2,...,vk in V is c1v1 + c2v2 +···+ckvk, (4.4.1) where c1,c2,...,ck are scalars. An expression of the form (4.4.1) is called a linear combination of v1, v2,...,vk. Since V is closed under addition and scalar multiplica- tion, it follows that the foregoing linear combination is itself a vector in V . One of the questions we wish to answer is whether every vector in a vector space can be obtained by taking linear combinations of a finite set of vectors. The following terminology is used in the case when the answer to this question is affirmative: DEFINITION 4.4.1 If every vector in a vector space V can be written as a linear combination of v1, v2, ..., vk, we say that V is spanned or generated by v1, v2, ..., vk and call the set of vectors {v1, v2,...,vk} a spanning set for V . -
Math 704: Part 1: Principal Bundles and Connections
MATH 704: PART 1: PRINCIPAL BUNDLES AND CONNECTIONS WEIMIN CHEN Contents 1. Lie Groups 1 2. Principal Bundles 3 3. Connections and curvature 6 4. Covariant derivatives 12 References 13 1. Lie Groups A Lie group G is a smooth manifold such that the multiplication map G × G ! G, (g; h) 7! gh, and the inverse map G ! G, g 7! g−1, are smooth maps. A Lie subgroup H of G is a subgroup of G which is at the same time an embedded submanifold. A Lie group homomorphism is a group homomorphism which is a smooth map between the Lie groups. The Lie algebra, denoted by Lie(G), of a Lie group G consists of the set of left-invariant vector fields on G, i.e., Lie(G) = fX 2 X (G)j(Lg)∗X = Xg, where Lg : G ! G is the left translation Lg(h) = gh. As a vector space, Lie(G) is naturally identified with the tangent space TeG via X 7! X(e). A Lie group homomorphism naturally induces a Lie algebra homomorphism between the associated Lie algebras. Finally, the universal cover of a connected Lie group is naturally a Lie group, which is in one to one correspondence with the corresponding Lie algebras. Example 1.1. Here are some important Lie groups in geometry and topology. • GL(n; R), GL(n; C), where GL(n; C) can be naturally identified as a Lie sub- group of GL(2n; R). • SL(n; R), O(n), SO(n) = O(n) \ SL(n; R), Lie subgroups of GL(n; R). -
Notes on Principal Bundles and Classifying Spaces
Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG]. -
Worksheet 1, for the MATLAB Course by Hans G. Feichtinger, Edinburgh, Jan
Worksheet 1, for the MATLAB course by Hans G. Feichtinger, Edinburgh, Jan. 9th Start MATLAB via: Start > Programs > School applications > Sci.+Eng. > Eng.+Electronics > MATLAB > R2007 (preferably). Generally speaking I suggest that you start your session by opening up a diary with the command: diary mydiary1.m and concluding the session with the command diary off. If you want to save your workspace you my want to call save today1.m in order to save all the current variable (names + values). Moreover, using the HELP command from MATLAB you can get help on more or less every MATLAB command. Try simply help inv or help plot. 1. Define an arbitrary (random) 3 × 3 matrix A, and check whether it is invertible. Calculate the inverse matrix. Then define an arbitrary right hand side vector b and determine the (unique) solution to the equation A ∗ x = b. Find in two different ways the solution to this problem, by either using the inverse matrix, or alternatively by applying Gauss elimination (i.e. the RREF command) to the extended system matrix [A; b]. In addition look at the output of the command rref([A,eye(3)]). What does it tell you? 2. Produce an \arbitrary" 7 × 7 matrix of rank 5. There are at east two simple ways to do this. Either by factorization, i.e. by obtaining it as a product of some 7 × 5 matrix with another random 5 × 7 matrix, or by purposely making two of the rows or columns linear dependent from the remaining ones. Maybe the second method is interesting (if you have time also try the first one): 3. -
Notes on Euclidean Geometry Kiran Kedlaya Based
Notes on Euclidean Geometry Kiran Kedlaya based on notes for the Math Olympiad Program (MOP) Version 1.0, last revised August 3, 1999 c Kiran S. Kedlaya. This is an unfinished manuscript distributed for personal use only. In particular, any publication of all or part of this manuscript without prior consent of the author is strictly prohibited. Please send all comments and corrections to the author at [email protected]. Thank you! Contents 1 Tricks of the trade 1 1.1 Slicing and dicing . 1 1.2 Angle chasing . 2 1.3 Sign conventions . 3 1.4 Working backward . 6 2 Concurrence and Collinearity 8 2.1 Concurrent lines: Ceva’s theorem . 8 2.2 Collinear points: Menelaos’ theorem . 10 2.3 Concurrent perpendiculars . 12 2.4 Additional problems . 13 3 Transformations 14 3.1 Rigid motions . 14 3.2 Homothety . 16 3.3 Spiral similarity . 17 3.4 Affine transformations . 19 4 Circular reasoning 21 4.1 Power of a point . 21 4.2 Radical axis . 22 4.3 The Pascal-Brianchon theorems . 24 4.4 Simson line . 25 4.5 Circle of Apollonius . 26 4.6 Additional problems . 27 5 Triangle trivia 28 5.1 Centroid . 28 5.2 Incenter and excenters . 28 5.3 Circumcenter and orthocenter . 30 i 5.4 Gergonne and Nagel points . 32 5.5 Isogonal conjugates . 32 5.6 Brocard points . 33 5.7 Miscellaneous . 34 6 Quadrilaterals 36 6.1 General quadrilaterals . 36 6.2 Cyclic quadrilaterals . 36 6.3 Circumscribed quadrilaterals . 38 6.4 Complete quadrilaterals . 39 7 Inversive Geometry 40 7.1 Inversion . -
Mathematics of Information Hilbert Spaces and Linear Operators
Chair for Mathematical Information Science Verner Vlačić Sternwartstrasse 7 CH-8092 Zürich Mathematics of Information Hilbert spaces and linear operators These notes are based on [1, Chap. 6, Chap. 8], [2, Chap. 1], [3, Chap. 4], [4, App. A] and [5]. 1 Vector spaces Let us consider a field F, which can be, for example, the field of real numbers R or the field of complex numbers C.A vector space over F is a set whose elements are called vectors and in which two operations, addition and multiplication by any of the elements of the field F (referred to as scalars), are defined with some algebraic properties. More precisely, we have Definition 1 (Vector space). A set X together with two operations (+; ·) is a vector space over F if the following properties are satisfied: (i) the first operation, called vector addition: X × X ! X denoted by + satisfies • (x + y) + z = x + (y + z) for all x; y; z 2 X (associativity of addition) • x + y = y + x for all x; y 2 X (commutativity of addition) (ii) there exists an element 0 2 X , called the zero vector, such that x + 0 = x for all x 2 X (iii) for every x 2 X , there exists an element in X , denoted by −x, such that x + (−x) = 0. (iv) the second operation, called scalar multiplication: F × X ! X denoted by · satisfies • 1 · x = x for all x 2 X • α · (β · x) = (αβ) · x for all α; β 2 F and x 2 X (associativity for scalar multiplication) • (α+β)·x = α·x+β ·x for all α; β 2 F and x 2 X (distributivity of scalar multiplication with respect to field addition) • α·(x+y) = α·x+α·y for all α 2 F and x; y 2 X (distributivity of scalar multiplication with respect to vector addition). -
On Galilean Connections and the First Jet Bundle
ON GALILEAN CONNECTIONS AND THE FIRST JET BUNDLE JAMES D.E. GRANT AND BRADLEY C. LACKEY Abstract. We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations { sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces \laboratory" coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse { the \fundamental theorem" { that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion. 1. Introduction The geometry of a system of ordinary differential equations has had a distinguished history, dating back even to Lie [10]. Historically, there have been three main branches of this theory, depending on the class of allowable transformations considered. The most studied has been differ- ential equations under contact transformation; see x7.1 of Doubrov, Komrakov, and Morimoto [5], for the construction of Cartan connections under this class of transformation. Another classical study has been differential equations under point-transformations. (See, for instance, Tresse [13]). By B¨acklund's Theorem, this is novel only for a single second-order dif- ferential equation in one independent variable. The construction of the Cartan connection for this form of geometry was due to Cartan himself [2]. See, for instance, x2 of Kamran, Lamb and Shadwick [6], for a modern \equivalence method" treatment of this case.