Wk 6 Circle Properties Texts and Tasks
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Math.DG] 14 Dec 2004 Qain:Tertcladnmrclanalysis’
FILLING AREA CONJECTURE AND OVALLESS REAL HYPERELLIPTIC SURFACES1 VICTOR BANGERT!, CHRISTOPHER CROKE+, SERGEI V. IVANOV†, AND MIKHAIL G. KATZ∗ Abstract. We prove the filling area conjecture in the hyperellip- tic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu’s result in genus 0. We trans- late the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredi- ents. On the one hand, we exploit integral geometric comparison with orbifold metrics of constant positive curvature on real sur- faces of even positive genus. Here the singular points are Weier- strass points. On the other hand, we exploit an analysis of the combinatorics on unions of closed curves, arising as geodesics of such orbifold metrics. Contents 1. To fill a circle: an introduction 2 2. Relative Pu’s way 5 3. Outline of proof of Theorem 1.7 6 4. Nearoptimalsurfacesandthefootball 7 5. Finding a short figure eight geodesic 9 6. Proof of circle filling: Step 1 10 7. Proof of circle filling: Step 2 12 Appendix A. Ovalless reality and hyperellipticity 15 A.1. Hyperelliptic surfaces 15 arXiv:math/0405583v2 [math.DG] 14 Dec 2004 A.2. Ovalless surfaces 16 1Geometric and Functional Analysis (GAFA), to appear 1991 Mathematics Subject Classification. Primary 53C23; Secondary 52C07 . Key words and phrases. filling area, hyperelliptic surfaces, integral geometry, orbifold metrics, Pu’s inequality, real surfaces, systolic inequality. !Partially Supported by DFG-Forschergruppe ‘Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis’. +Supported by NSF grant DMS 02-02536 and the MSRI. -
Of Triangles, Gas, Price, and Men
OF TRIANGLES, GAS, PRICE, AND MEN Cédric Villani Univ. de Lyon & Institut Henri Poincaré « Mathematics in a complex world » Milano, March 1, 2013 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemannian (= non-Euclidean) geometry At each location, the units of length and angles may change Shortest path (= geodesics) are curved!! Geodesics can tend to get closer (positive curvature, fat triangles) or to get further apart (negative curvature, skinny triangles) Hyperbolic surfaces Bernhard Riemann 1826-1866 List of topics named after Bernhard Riemann From Wikipedia, the free encyclopedia Riemann singularity theorem Cauchy–Riemann equations Riemann solver Compact Riemann surface Riemann sphere Free Riemann gas Riemann–Stieltjes integral Generalized Riemann hypothesis Riemann sum Generalized Riemann integral Riemann surface Grand Riemann hypothesis Riemann theta function Riemann bilinear relations Riemann–von Mangoldt formula Riemann–Cartan geometry Riemann Xi function Riemann conditions Riemann zeta function Riemann curvature tensor Zariski–Riemann space Riemann form Riemannian bundle metric Riemann function Riemannian circle Riemann–Hilbert correspondence Riemannian cobordism Riemann–Hilbert problem Riemannian connection Riemann–Hurwitz formula Riemannian cubic polynomials Riemann hypothesis Riemannian foliation Riemann hypothesis for finite fields Riemannian geometry Riemann integral Riemannian graph Bernhard -
Analytic and Differential Geometry Mikhail G. Katz∗
Analytic and Differential geometry Mikhail G. Katz∗ Department of Mathematics, Bar Ilan University, Ra- mat Gan 52900 Israel Email address: [email protected] ∗Supported by the Israel Science Foundation (grants no. 620/00-10.0 and 84/03). Abstract. We start with analytic geometry and the theory of conic sections. Then we treat the classical topics in differential geometry such as the geodesic equation and Gaussian curvature. Then we prove Gauss’s theorema egregium and introduce the ab- stract viewpoint of modern differential geometry. Contents Chapter1. Analyticgeometry 9 1.1. Circle,sphere,greatcircledistance 9 1.2. Linearalgebra,indexnotation 11 1.3. Einsteinsummationconvention 12 1.4. Symmetric matrices, quadratic forms, polarisation 13 1.5.Matrixasalinearmap 13 1.6. Symmetrisationandantisymmetrisation 14 1.7. Matrixmultiplicationinindexnotation 15 1.8. Two types of indices: summation index and free index 16 1.9. Kroneckerdeltaandtheinversematrix 16 1.10. Vector product 17 1.11. Eigenvalues,symmetry 18 1.12. Euclideaninnerproduct 19 Chapter 2. Eigenvalues of symmetric matrices, conic sections 21 2.1. Findinganeigenvectorofasymmetricmatrix 21 2.2. Traceofproductofmatricesinindexnotation 23 2.3. Inner product spaces and self-adjoint endomorphisms 24 2.4. Orthogonal diagonalisation of symmetric matrices 24 2.5. Classification of conic sections: diagonalisation 26 2.6. Classification of conics: trichotomy, nondegeneracy 28 2.7. Characterisationofparabolas 30 Chapter 3. Quadric surfaces, Hessian, representation of curves 33 3.1. Summary: classificationofquadraticcurves 33 3.2. Quadric surfaces 33 3.3. Case of eigenvalues (+++) or ( ),ellipsoid 34 3.4. Determination of type of quadric−−− surface: explicit example 35 3.5. Case of eigenvalues (++ ) or (+ ),hyperboloid 36 3.6. Case rank(S) = 2; paraboloid,− hyperbolic−− paraboloid 37 3.7. -
Finsler Metric, Systolic Inequality, Isometr
DISCRETE SURFACES WITH LENGTH AND AREA AND MINIMAL FILLINGS OF THE CIRCLE MARCOS COSSARINI Abstract. We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of wall crossings, and the area of the surface is the number of crossings of the walls themselves. We show how to approximate a Riemannian (or self-reverse Finsler) metric by a wallsystem. This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere minimizes area among orientable Riemannian sur- faces that fill a circle isometrically. We introduce a discrete FAC: every square-celled surface that fills isometrically a 2n-cycle graph has at least n(n−1) 2 squares. We prove that our discrete FAC is equivalent to the FAC for surfaces with self-reverse metric. If the surface is a disk, the discrete FAC follows from Steinitz's algo- rithm for transforming curves into pseudolines. This gives a combinato- rial proof of the FAC for disks with self-reverse metric. We also imitate Ivanov's proof of the same fact, using discrete differential forms. And we prove a new theorem: the FAC holds for M¨obiusbands with self- reverse metric. To prove this we use a combinatorial curve shortening flow developed by de Graaf-Schrijver and Hass-Scott. The same method yields a proof of the systolic inequality for Klein bottles with self-reverse metric, conjectured by Sabourau{Yassine. Self-reverse metrics can be discretized using walls because every normed plane satisfies Crofton's formula: the length of every segment equals the symplectic measure of the set of lines that it crosses. -
Riemann, Schottky, and Schwarz on Conformal Representation
Appendix 1 Riemann, Schottky, and Schwarz on Conformal Representation In his Thesis [1851], Riemann sought to prove that if a function u satisfies cer- tain conditions on the boundary of a surface T, then it is the real part of a unique complex function which can be defined on the whole of T. He was then able to show that any two simply connected surfaces (other than the complex plane itself Riemann was considering bounded surfaces) can be mapped conformally onto one another, and that the map is unique once the images of one boundary point and one interior point are specified; he claimed analogous results for any two surfaces of the same connectivity [w To prove that any two simply connected surfaces are conformally equivalent, he observed that it is enough to take for one surface the unit disc K = {z : Izl _< l} and he gave [w an account of how this result could be proved by means of what he later [1857c, 103] called Dirichlet's principle. He considered [w (i) the class of functions, ~., defined on a surface T and vanishing on the bound- ary of T, which are continuous, except at some isolated points of T, for which the integral t= rx + ry dt is finite; and (ii) functions ct + ~. = o9, say, satisfying dt=f2 < oo Ox Oy + for fixed but arbitrary continuous functions a and ft. 224 Appendix 1. Riemann, Schottky, and Schwarz on Conformal Representation He claimed that f2 and L vary continuously with varying ,~. but cannot be zero, and so f2 takes a minimum value for some o9. -
104 Word Vocabulary List Geometry/Honors Geometry
104 Word Vocabulary List Geometry/Honors Geometry Summer Preparedness Project (“Math Summer Reading”) Summer 2021 – know and be prepared to define/discuss the following terms 1. Geometry - branch of mathematics that deals with points, lines, planes and solids and examines their properties. 2. Point – has no size; length, width, or height. It is represented by a dot and named by a capital letter. 3. Line – set of points which has infinite length but no width or height. A line is named by a lower case, cursive letter or by any two points on the line. 4. Plane – set of points that has infinite length and width but no height. We name a plane with a capital ‘funny font’ letter. 5. Collinear points – points that lie on the same line. 6. Noncollinear points – points that do not lie on the same line. 7. Coplanar points – points that lie on the same plane. 8. Noncoplanar points – points that do not lie on the same plane. 9. Segment – part of a line that consists of two points called endpoints and all points between them. 10. Ray- is the part of a line that contains an endpoint and all points extending in the other direction. 11. Congruent segments – segments that have the same length. 12. Bisector of a segment – line, ray segment, or plane that divides a segment into two congruent segments. 13. Midpoint of a segment – a point that divides the segment into two congruent segments. 14. Acute angle – angle whose measure is between 0 degrees and 90 degrees. 15. Right angle – angle whose measure is 90 degrees. -
Chapter 15 Measuring an Angle
Chapter 15 Measuring an Angle So far, the equations we have studied have an algebraic Cosmo S character, involving the variables x and y, arithmetic op- motors around erations and maybe extraction of roots. Restricting our Q the attention to such equations would limit our ability to de- circle R scribe certain natural phenomena. An important example P 20 feet involves understanding motion around a circle, and it can be motivated by analyzing a very simple scenario: Cosmo the dog, tied by a 20 foot long tether to a post, begins Figure 15.1: Cosmo the dog walking around a circle. A number of very natural ques- walking a circular path. tions arise: Natural Questions 15.0.1. How can we measure the angles ∠SPR, ∠QPR, and ∠QPS? How can we measure the arc lengths arc(RS), arc(SQ) and arc(RQ)? How can we measure the rate Cosmo is moving around the circle? If we know how to measure angles, can we compute the coordinates of R, S, and Q? Turning this around, if we know how to compute the coordinates of R, S, and Q, can we then measure the angles ∠SPR, ∠QPR, and ∠QPS ? Finally, how can we specify the direction Cosmo is traveling? We will answer all of these questions and see how the theory which evolves can be applied to a variety of problems. The definition and basic properties of the circular functions will emerge as a central theme in this Chapter. The full problem-solving power of these functions will become apparent in our discussion of sinusoidal functions in Chapter 19. -
Remote Sensing Fundamentals EM Radiation 2.1 W.D
Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.1 W.D. Philpot, Cornell University 2. ELECTROMAGNETIC RADIATION Electromagnetic (EM) radiation and its transfer from sources to objects or from objects to sensors is fundamental to remote sensing, and it is important that certain characteristics of this phenomenon be understood. For most of the passive systems that we consider, the sun is the primary source of EM radiation, but we will also consider emitted (thermal) radiation and active systems that provide their on radiation. In any case, EM radiation is energy and that energy might be in wave or particulate (i.e., photon or quantum) form. All electromagnetic radiation has wave properties; at all levels, radiation shows interference and diffraction. But studies also indicate that the energy carried by electromagnetic waves may, under certain conditions, be regarded as discontinuous rather than the continuously graded energy that would be expected from a wave. 2.1 Maxwell's Equations The fundamental description of Electromagnetic radiation begins with Maxwell's equations which describe propagating plane waves. Maxwell's equations are written: In the presence of charge In vacuum (free space) ρ ∇⋅E = ∇⋅E =0 (2.1) ε0 ∇⋅B =0 ∇⋅B =0 (2.2) ∇ ×EB = −∂/ ∂t ∇ ×EB = −∂/ ∂t (2.3) ∇×B =µ0 J+ ε 00 µ ∂ E/ ∂t ∇×BE =ε00 µ ∂/ ∂t (2.4) where: E = electric field B = magnetic-induction field ρe = electric charge density -12 2 2 ε0 = electrical permittivity of free space = 8.85x10 coulomb /Newton-meter -7 µ0 = magnetic permeability of free space = 12.57x10 weber/ampere-meter t = time ∇· = divergence (a spatial vector derivative operator) ∇ = curl (a spatial vector derivative operator) For our purposes, the key point to be gleaned from these equations is the symmetry between the electric and magnetic fields and the fact that they are always coupled. -
Basic Surveying – Theory and Practice 4 Hours
BASIC SURVEYING – THEORY AND PRACTICE 4 HOURS PDH ACADEMY PO BOX 449 PEWAUKEE, WI 53072 (888) 564-9098 www.pdhacademy.com Final Exam 1. This type of surveying in which the mean surface of the earth is considered as a plane, or in which its spheroidal shape is neglected, with regard to horizontal distances and directions. A. Hydrographic Surveying B. Plane Surveying C. Geodetic Surveying D. Construction Surveying 2. Who were the first known people to use some form of chaining in both land surveying and construction surveying? A. Egyptians B. Americans C. Chinese D. Canadians 3. When level chaining, what is the pointed weight on the end of a string called? A. Pea gun B. Break chaining C. Plumb bob D. Field measurement 4. A Common source of error in chaining is: A. Sag in the chain B. Changes in the temperature of the chain C. Variation in the tension on the chain D. All of the above 5. What decade was the first Electronic Distance Measuring (EDM) equipment developed? A. 1940’s B. 1950’s C. 1960’s D. 1970’s 6. These types of angles, right or left, are measured from an extension of the preceding course and the ahead line. A. Interior Angles B. Exterior Angles C. Deflection Angles D. Angles to the left 7. Which of the following is a type of Meridian? A. False B. Lock C. Saturn D. Magnetic 8. A ______________ is a succession of straight lines along or through the area to be surveyed. The direction and lengths of these lines are determined by measurements taken in the field. -
Filling Area Conjecture, Optimal Systolic Inequalities, and the Fiber
Contemporary Mathematics Filling Area Conjecture, Optimal Systolic Inequalities, and the Fiber Class in Abelian Covers Mikhail G. Katz∗ and Christine Lescop† Dedicated to the memory of Robert Brooks, a colleague and friend. Abstract. We investigate the filling area conjecture, optimal sys- tolic inequalities, and the related problem of the nonvanishing of certain linking numbers in 3-manifolds. Contents 1. Filling radius and systole 2 2. Inequalities of C. Loewner and P. Pu 4 3. Filling area conjecture 5 4. Lattices and successive minima 6 5. Precise calculation of filling invariants 8 6. Stable and conformal systoles 11 7. Gromov’s optimal inequality for n-tori 11 8. Inequalities combining dimension and codimension 1 13 9. Inequalities combining varieties of 1-systoles 15 arXiv:math/0412011v2 [math.DG] 23 Sep 2005 10. Fiber class, linking number, and the generalized Casson invariant λ 17 2000 Mathematics Subject Classification. Primary 53C23; Secondary 57N65, 57M27, 52C07. Key words and phrases. Abel-Jacobi map, Casson invariant, filling radius, fill- ing area conjecture, Hermite constant, linking number, Loewner inequality, Pu’s inequality, stable norm, systole. ∗Supported by the Israel Science Foundation (grants no. 620/00-10.0 and 84/03). †Supported by CNRS (UMR 5582). c 2005 M. Katz, C. Lescop 1 2 M. KATZ AND C. LESCOP 11. Fiber class in NIL geometry 19 12. A nonvanishing result (following A. Marin) 20 Acknowledgments 23 References 23 1. Filling radius and systole The filling radius of a simple loop C in the plane is defined as the largest radius, R> 0, of a circle that fits inside C, see Figure 1.1: FillRad(C R2)= R. -
A Theorem on an Analytic Mapping of Riemann Surfaces
A THEOREM ON AN ANALYTIC MAPPING OF RIEMANN SURFACES MINORU KURITA Recently S. S. Chern [1] intended an aproach to some problems about analytic mappings of Riemann surfaces from a view-point of differential geo- metry. In that line we treat here orders of circular points of analytic mapp- ings. The author expresses his thanks to Prof. K. Noshiro for his kind advices. 1. In the first we summarise formulas for conformal mappings of Rie- mannian manifolds (cf. [3], [4]). We take 2-dimensional Riemannian manifolds M and N of differentiate class C2 and assume that there exists a conformal mapping / of M into JV which is locally diffeomorphic. We take orthonormal frames on the tangent spaces of points of any neighborhood U of M. Then a line element ds of M can be represented as ds2 = ω\ + ω\ (1) with 1-forms ωi and ω2. When we choose frames on the tangent spaces of f(ϋ) corresponding to those on M by the conformal mapping /, we have for a line element dt of N df = π\ + it\ = <tds\ (2) where π\~aωu π%~ aω<L. (a>0) (3) A form O7i2 of Riemannian connection of M is defined in U by the relations d(ϊ)i — 472 Λ 0721, dϋ)2 = 07i Λ O7i2, ί«7j2 = ~ dhl- (4) When we put da/a ~ bιu)i + b2(ΰ2, (5) a connection form πi2 = - 7τ2i of Riemannian connection of N'mf(U) is given by Received April 28, 1961. 149 Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.42, on 01 Oct 2021 at 21:38:13, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. -
Analytic and Differential Geometry Mikhail G. Katz∗
Analytic and Differential geometry Mikhail G. Katz∗ Department of Mathematics, Bar Ilan University, Ra- mat Gan 52900 Israel E-mail address: [email protected] ∗Supported by the Israel Science Foundation (grants no. 620/00-10.0 and 84/03). Abstract. We start with analytic geometry and the theory of conic sections. Then we treat the classical topics in differential geometry such as the geodesic equation and Gaussian curvature. Then we prove Gauss's theorema egregium and introduce the ab- stract viewpoint of modern differential geometry. Contents Chapter 1. Analytic geometry 9 1.1. Sphere, projective geometry 9 1.2. Circle, isoperimetric inequality 9 1.3. Relativity theory 10 1.4. Linear algebra, index notation 11 1.5. Einstein summation convention 11 1.6. Matrix as a linear map 12 1.7. Symmetrisation and skew-symmetrisation 13 1.8. Matrix multiplication in index notation 14 1.9. The inverse matrix, Kronecker delta 14 1.10. Eigenvalues, symmetry 15 1.11. Finding an eigenvector of a symmetric matrix 17 Chapter 2. Self-adjoint operators, conic sections 19 2.1. Trace of product in index notation 19 2.2. Inner product spaces and self-adjoint operators 19 2.3. Orthogonal diagonalisation of symmetric matrices 20 2.4. Classification of conic sections: diagonalisation 21 2.5. Classification of conics: trichotomy, nondegeneracy 22 2.6. Quadratic surfaces 23 2.7. Jacobi's criterion 24 Chapter 3. Hessian, curves, and curvature 27 3.1. Exercise on index notation 27 3.2. Hessian, minima, maxima, saddle points 27 3.3. Parametric representation of a curve 28 3.4.