Elliptic Hypergeometric Functions 25 2.1 Threelevelsofhypergeometry
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Methods of Measuring Population Change
Methods of Measuring Population Change CDC 103 – Lecture 10 – April 22, 2012 Measuring Population Change • If past trends in population growth can be expressed in a mathematical model, there would be a solid justification for making projections on the basis of that model. • Demographers developed an array of models to measure population growth; four of these models are usually utilized. 1 Measuring Population Change Arithmetic (Linear), Geometric, Exponential, and Logistic. Arithmetic Change • A population growing arithmetically would increase by a constant number of people in each period. • If a population of 5000 grows by 100 annually, its size over successive years will be: 5100, 5200, 5300, . • Hence, the growth rate can be calculated by the following formula: • (100/5000 = 0.02 or 2 per cent). 2 Arithmetic Change • Arithmetic growth is the same as the ‘simple interest’, whereby interest is paid only on the initial sum deposited, the principal, rather than on accumulating savings. • Five percent simple interest on $100 merely returns a constant $5 interest every year. • Hence, arithmetic change produces a linear trend in population growth – following a straight line rather than a curve. Arithmetic Change 3 Arithmetic Change • The arithmetic growth rate is expressed by the following equation: Geometric Change • Geometric population growth is the same as the growth of a bank balance receiving compound interest. • According to this, the interest is calculated each year with reference to the principal plus previous interest payments, thereby yielding a far greater return over time than simple interest. • The geometric growth rate in demography is calculated using the ‘compound interest formula’. -
The Chiral De Rham Complex of Tori and Orbifolds
The Chiral de Rham Complex of Tori and Orbifolds Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urMathematik und Physik der Albert-Ludwigs-Universit¨at Freiburg im Breisgau vorgelegt von Felix Fritz Grimm Juni 2016 Betreuerin: Prof. Dr. Katrin Wendland ii Dekan: Prof. Dr. Gregor Herten Erstgutachterin: Prof. Dr. Katrin Wendland Zweitgutachter: Prof. Dr. Werner Nahm Datum der mundlichen¨ Prufung¨ : 19. Oktober 2016 Contents Introduction 1 1 Conformal Field Theory 4 1.1 Definition . .4 1.2 Toroidal CFT . .8 1.2.1 The free boson compatified on the circle . .8 1.2.2 Toroidal CFT in arbitrary dimension . 12 1.3 Vertex operator algebra . 13 1.3.1 Complex multiplication . 15 2 Superconformal field theory 17 2.1 Definition . 17 2.2 Ising model . 21 2.3 Dirac fermion and bosonization . 23 2.4 Toroidal SCFT . 25 2.5 Elliptic genus . 26 3 Orbifold construction 29 3.1 CFT orbifold construction . 29 3.1.1 Z2-orbifold of toroidal CFT . 32 3.2 SCFT orbifold . 34 3.2.1 Z2-orbifold of toroidal SCFT . 36 3.3 Intersection point of Z2-orbifold and torus models . 38 3.3.1 c = 1...................................... 38 3.3.2 c = 3...................................... 40 4 Chiral de Rham complex 41 4.1 Local chiral de Rham complex on CD ...................... 41 4.2 Chiral de Rham complex sheaf . 44 4.3 Cechˇ cohomology vertex algebra . 49 4.4 Identification with SCFT . 49 4.5 Toric geometry . 50 5 Chiral de Rham complex of tori and orbifold 53 5.1 Dolbeault type resolution . 53 5.2 Torus . -
Math Club: Recurrent Sequences Nov 15, 2020
MATH CLUB: RECURRENT SEQUENCES NOV 15, 2020 In many problems, a sequence is defined using a recurrence relation, i.e. the next term is defined using the previous terms. By far the most famous of these is the Fibonacci sequence: (1) F0 = 0;F1 = F2 = 1;Fn+1 = Fn + Fn−1 The first several terms of this sequence are below: 0; 1; 1; 2; 3; 5; 8; 13; 21; 34;::: For such sequences there is a method of finding a general formula for nth term, outlined in problem 6 below. 1. Into how many regions do n lines divide the plane? It is assumed that no two lines are parallel, and no three lines intersect at a point. Hint: denote this number by Rn and try to get a recurrent formula for Rn: what is the relation betwen Rn and Rn−1? 2. The following problem is due to Leonardo of Pisa, later known as Fibonacci; it was introduced in his 1202 book Liber Abaci. Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year? 3. Daniel is coming up the staircase of 20 steps. He can either go one step at a time, or skip a step, moving two steps at a time. -
On Hodge Theory and Derham Cohomology of Variétiés
On Hodge Theory and DeRham Cohomology of Vari´eti´es Pete L. Clark October 21, 2003 Chapter 1 Some geometry of sheaves 1.1 The exponential sequence on a C-manifold Let X be a complex manifold. An amazing amount of geometry of X is encoded in the long exact cohomology sequence of the exponential sequence of sheaves on X: exp £ 0 ! Z !OX !OX ! 0; where exp takes a holomorphic function f on an open subset U to the invertible holomorphic function exp(f) := e(2¼i)f on U; notice that the kernel is the constant sheaf on Z, and that the exponential map is surjective as a morphism of sheaves because every holomorphic function on a polydisk has a logarithm. Taking sheaf cohomology we get exp £ 1 1 1 £ 2 0 ! Z ! H(X) ! H(X) ! H (X; Z) ! H (X; OX ) ! H (X; OX ) ! H (X; Z); where we have written H(X) for the ring of global holomorphic functions on X. Now let us reap the benefits: I. Because of the exactness at H(X)£, we see that any nowhere vanishing holo- morphic function on any simply connected C-manifold has a logarithm – even in the complex plane, this is a nontrivial result. From now on, assume that X is compact – in particular it homeomorphic to a i finite CW complex, so its Betti numbers bi(X) = dimQ H (X; Q) are finite. This i i also implies [Cartan-Serre] that h (X; F ) = dimC H (X; F ) is finite for all co- herent analytic sheaves on X, i.e. -
AN EXPLORATION of COMPLEX JACOBIAN VARIETIES Contents 1
AN EXPLORATION OF COMPLEX JACOBIAN VARIETIES MATTHEW WOOLF Abstract. In this paper, I will describe my thought process as I read in [1] about Abelian varieties in general, and the Jacobian variety associated to any compact Riemann surface in particular. I will also describe the way I currently think about the material, and any additional questions I have. I will not include material I personally knew before the beginning of the summer, which included the basics of algebraic and differential topology, real analysis, one complex variable, and some elementary material about complex algebraic curves. Contents 1. Abelian Sums (I) 1 2. Complex Manifolds 2 3. Hodge Theory and the Hodge Decomposition 3 4. Abelian Sums (II) 6 5. Jacobian Varieties 6 6. Line Bundles 8 7. Abelian Varieties 11 8. Intermediate Jacobians 15 9. Curves and their Jacobians 16 References 17 1. Abelian Sums (I) The first thing I read about were Abelian sums, which are sums of the form 3 X Z pi (1.1) (L) = !; i=1 p0 where ! is the meromorphic one-form dx=y, L is a line in P2 (the complex projective 2 3 2 plane), p0 is a fixed point of the cubic curve C = (y = x + ax + bx + c), and p1, p2, and p3 are the three intersections of the line L with C. The fact that C and L intersect in three points counting multiplicity is just Bezout's theorem. Since C is not simply connected, the integrals in the Abelian sum depend on the path, but there's no natural choice of path from p0 to pi, so this function depends on an arbitrary choice of path, and will not necessarily be holomorphic, or even Date: August 22, 2008. -
Abelian Varieties
Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties . -
Arithmetic and Geometric Progressions
Arithmetic and geometric progressions mcTY-apgp-2009-1 This unit introduces sequences and series, and gives some simple examples of each. It also explores particular types of sequence known as arithmetic progressions (APs) and geometric progressions (GPs), and the corresponding series. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: • recognise the difference between a sequence and a series; • recognise an arithmetic progression; • find the n-th term of an arithmetic progression; • find the sum of an arithmetic series; • recognise a geometric progression; • find the n-th term of a geometric progression; • find the sum of a geometric series; • find the sum to infinity of a geometric series with common ratio r < 1. | | Contents 1. Sequences 2 2. Series 3 3. Arithmetic progressions 4 4. The sum of an arithmetic series 5 5. Geometric progressions 8 6. The sum of a geometric series 9 7. Convergence of geometric series 12 www.mathcentre.ac.uk 1 c mathcentre 2009 1. Sequences What is a sequence? It is a set of numbers which are written in some particular order. For example, take the numbers 1, 3, 5, 7, 9, .... Here, we seem to have a rule. We have a sequence of odd numbers. To put this another way, we start with the number 1, which is an odd number, and then each successive number is obtained by adding 2 to give the next odd number. -
Sequences, Series and Taylor Approximation (Ma2712b, MA2730)
Sequences, Series and Taylor Approximation (MA2712b, MA2730) Level 2 Teaching Team Current curator: Simon Shaw November 20, 2015 Contents 0 Introduction, Overview 6 1 Taylor Polynomials 10 1.1 Lecture 1: Taylor Polynomials, Definition . .. 10 1.1.1 Reminder from Level 1 about Differentiable Functions . .. 11 1.1.2 Definition of Taylor Polynomials . 11 1.2 Lectures 2 and 3: Taylor Polynomials, Examples . ... 13 x 1.2.1 Example: Compute and plot Tnf for f(x) = e ............ 13 1.2.2 Example: Find the Maclaurin polynomials of f(x) = sin x ...... 14 2 1.2.3 Find the Maclaurin polynomial T11f for f(x) = sin(x ) ....... 15 1.2.4 QuestionsforChapter6: ErrorEstimates . 15 1.3 Lecture 4 and 5: Calculus of Taylor Polynomials . .. 17 1.3.1 GeneralResults............................... 17 1.4 Lecture 6: Various Applications of Taylor Polynomials . ... 22 1.4.1 RelativeExtrema .............................. 22 1.4.2 Limits .................................... 24 1.4.3 How to Calculate Complicated Taylor Polynomials? . 26 1.5 ExerciseSheet1................................... 29 1.5.1 ExerciseSheet1a .............................. 29 1.5.2 FeedbackforSheet1a ........................... 33 2 Real Sequences 40 2.1 Lecture 7: Definitions, Limit of a Sequence . ... 40 2.1.1 DefinitionofaSequence .......................... 40 2.1.2 LimitofaSequence............................. 41 2.1.3 Graphic Representations of Sequences . .. 43 2.2 Lecture 8: Algebra of Limits, Special Sequences . ..... 44 2.2.1 InfiniteLimits................................ 44 1 2.2.2 AlgebraofLimits.............................. 44 2.2.3 Some Standard Convergent Sequences . .. 46 2.3 Lecture 9: Bounded and Monotone Sequences . ..... 48 2.3.1 BoundedSequences............................. 48 2.3.2 Convergent Sequences and Closed Bounded Intervals . .... 48 2.4 Lecture10:MonotoneSequences . -
1 Complex Theory of Abelian Varieties
1 Complex Theory of Abelian Varieties Definition 1.1. Let k be a field. A k-variety is a geometrically integral k-scheme of finite type. An abelian variety X is a proper k-variety endowed with a structure of a k-group scheme. For any point x X(k) we can defined a translation map Tx : X X by 2 ! Tx(y) = x + y. Likewise, if K=k is an extension of fields and x X(K), then 2 we have a translation map Tx : XK XK defined over K. ! Fact 1.2. Any k-group variety G is smooth. Proof. We may assume that k = k. There must be a smooth point g0 G(k), and so there must be a smooth non-empty open subset U G. Since2 G is covered by the G(k)-translations of U, G is smooth. ⊆ We now turn our attention to the analytic theory of abelian varieties by taking k = C. We can view X(C) as a compact, connected Lie group. Therefore, we start off with some basic properties of complex Lie groups. Let X be a Lie group over C, i.e., a complex manifold with a group structure whose multiplication and inversion maps are holomorphic. Let T0(X) denote the tangent space of X at the identity. By adapting the arguments from the C1-case, or combining the C1-case with the Cauchy-Riemann equations we get: Fact 1.3. For each v T0(X), there exists a unique holomorphic map of Lie 2 δ groups φv : C X such that (dφv)0 : C T0(X) satisfies (dφv)0( 0) = v. -
Section 6-3 Arithmetic and Geometric Sequences
466 6 SEQUENCES, SERIES, AND PROBABILITY Section 6-3 Arithmetic and Geometric Sequences Arithmetic and Geometric Sequences nth-Term Formulas Sum Formulas for Finite Arithmetic Series Sum Formulas for Finite Geometric Series Sum Formula for Infinite Geometric Series For most sequences it is difficult to sum an arbitrary number of terms of the sequence without adding term by term. But particular types of sequences, arith- metic sequences and geometric sequences, have certain properties that lead to con- venient and useful formulas for the sums of the corresponding arithmetic series and geometric series. Arithmetic and Geometric Sequences The sequence 5, 7, 9, 11, 13,..., 5 ϩ 2(n Ϫ 1),..., where each term after the first is obtained by adding 2 to the preceding term, is an example of an arithmetic sequence. The sequence 5, 10, 20, 40, 80,..., 5 (2)nϪ1,..., where each term after the first is obtained by multiplying the preceding term by 2, is an example of a geometric sequence. ARITHMETIC SEQUENCE DEFINITION A sequence a , a , a ,..., a ,... 1 1 2 3 n is called an arithmetic sequence, or arithmetic progression, if there exists a constant d, called the common difference, such that Ϫ ϭ an anϪ1 d That is, ϭ ϩ Ͼ an anϪ1 d for every n 1 GEOMETRIC SEQUENCE DEFINITION A sequence a , a , a ,..., a ,... 2 1 2 3 n is called a geometric sequence, or geometric progression, if there exists a nonzero constant r, called the common ratio, such that 6-3 Arithmetic and Geometric Sequences 467 a n ϭ r DEFINITION anϪ1 2 That is, continued ϭ Ͼ an ranϪ1 for every n 1 Explore/Discuss (A) Graph the arithmetic sequence 5, 7, 9, ... -
Intermediate Jacobians and Abel-Jacobi Maps
Intermediate Jacobians and Abel-Jacobi Maps Patrick Walls April 28, 2012 Intermediate Jacobians are complex tori defined in terms of the Hodge structure of X . Abel-Jacobi Maps are maps from the groups of cycles of X to its Intermediate Jacobians. The result is that questions about cycles can be translated into questions about complex tori . Introduction Let X be a smooth projective complex variety. The result is that questions about cycles can be translated into questions about complex tori . Introduction Let X be a smooth projective complex variety. Intermediate Jacobians are complex tori defined in terms of the Hodge structure of X . Abel-Jacobi Maps are maps from the groups of cycles of X to its Intermediate Jacobians. Introduction Let X be a smooth projective complex variety. Intermediate Jacobians are complex tori defined in terms of the Hodge structure of X . Abel-Jacobi Maps are maps from the groups of cycles of X to its Intermediate Jacobians. The result is that questions about cycles can be translated into questions about complex tori . The subspaces Hp;q(X ) consist of classes [α] of differential forms that are representable by a closed form α of type (p; q) meaning that locally X α = fI ;J dzI ^ dzJ I ; J ⊆ f1;:::; ng jI j = p ; jJj = q for some choice of local holomorphic coordinates z1;:::; zn. Hodge Decomposition Let X be a smooth projective complex variety of dimension n. The Hodge decomposition is a direct sum decomposition of the complex cohomology groups k M p;q H (X ; C) = H (X ) ; 0 ≤ k ≤ 2n : p+q=k Hodge Decomposition Let X be a smooth projective complex variety of dimension n. -
Abelian Varieties and Theta Functions
Abelian Varieties and Theta Functions YU, Hok Pun A Thesis Submitted to In Partial Fulfillment of the Requirements for the Degree of • Master of Philosophy in Mathematics ©The Chinese University of Hong Kong October 2008 The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate School. Thesis Committee Prof. WANG, Xiaowei (Supervisor) Prof. LEUNG, Nai Chung (Chairman) Prof. WANG, Jiaping (Committee Member) Prof. WU,Siye (External Examiner) Abelian Varieties and Theta Functions 4 Abstract In this paper we want to find an explicit balanced embedding of principally polarized abelian varieties into projective spaces, and we will give an elementary proof, that for principally polarized abelian varieties, this balanced embedding induces a metric which converges to the flat metric, and that this result was originally proved in [1). The main idea, is to use the theory of finite dimensional Heisenberg groups to find a good basis of holomorphic sections, which the holo- morphic sections induce an embedding into the projective space as a subvariety. This thesis will be organized in the following way: In order to find a basis of holomorphic sections of a line bundle, we will start with line bundles on complex tori, and study the characterization of line bundles with Appell-Humbert theorem. We will make use of the fact that a positive line bundle will have enough sections for an embedding. If a complex torus can be embedded into the projective space this way, it is called an abelian variety, and the holomorphic sections of an abelian variety are called theta functions.