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# Normal Section In Differential Geometry

 The notion of vector is a bit more delicate. Differential geometry of surfaces: Surface, tangent plane and normal, equation of tangent plane, equaiton of normal, one parameter family of surfaces, characteristic of surface, envelopes, edge of regression, equation of edge of regression, developable surfaces, osculating developable, polar developable, rectifying developable. The Schwarzschild metric remains valid inside the Schwarzschild radius. Differential equations have a remarkable ability to predict the world around us. Math 431 Section 6 -- Ordinary Differential equations Math 621 -- Complex Analysis. Differential Geometry and Its Applications, 2nd Edition. Three coordinates su ce. Then I filled time by discussing the “normal form” for a curve. 1b) (3 pts) Deﬁne Gaussian Curvature of a surface in R3. Email: tfei[at]math[. My lectures at the Tsukuba workshop were supplemented by talks by T. Under certain conditions, every continuous section of a holomorphic fibre bundle can be deformed to a holomorphic section. Questions are taken from the pre 2010 exam papers. Differential Geometry: Helgason The above were the textbooks my professors used for my first 3 years when I was in college! What is wrong to have a textbook that students can read and learn? Ah, I forgot that a mostly used teaching trick: adopting a "hard" textbook, and teaching from the contents of another textbook. Exterior differential forms, Stokes theorem. Lambert (1728-77). AN INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH USE OF THE TENSOR CALCULUS By LUTHER PFAHLER EISENHART. combined with geometry in 513; class analysis combined with geometry in 515; class affine differential geometry, projective differential geometry in 516. The length of x¨ will be the curvature κ. If you're seeing this message, it means we're having trouble loading external resources on our website. Differential Geometry of Curves and Surfaces - CRC Press Book Differential Geometry of Curves and Surfaces, Second Edition takes both an analytical/theoretical approach and a visual/intuitive approach to the local and global properties of curves and surfaces. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The present excerpt covers the area of Geometry (minus differential geometry). Clelland, "Geometry of conservation laws for a class of parabolic partial differential equations," Selecta Mathematica, New Series 3 (1997), 1-77. See also there at differential cohesion – G-Structure. Differential Geometry of Curves The differential geometry of curves and surfaces is fundamental in Computer Aided Geometric Design (CAGD). Mathematics Section Chennai Mathematical Institute Siruseri, Kelambakkam, India. 5 Notes on Simple Surfaces, Unit Normal, Tangent Vectors, Tangent Planes Notes on the first fundamental form (metric) Text: Richard S. arise are as level sets of a smooth function, say f(x,y,z) = c, at a non-critical1. MATH 443 Differential Geometry (3) NW Further examines curves in the plane and 3-spaces, surfaces in 3-space, tangent planes, first and second fundamental forms, curvature, the Gauss-Bonnet Theorem, and possible other selected topics. Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. ABOUT THE CLASS: This course will be roughly broken into three parts: (1) differential geometry (with an emphasis on curvature), (2) special relativity, and (3) general relativity. We will see the differential geometry material come to the aid of gravitation theory. I will make a list of the best books for differential geometry textbooks in my subsequent lines and you will be amazed to find them all on this free mathematics and differential geometry books site. Towards Riemannian Geometry 7. We present a novel point rendering primitive, called Differential Point (DP), that captures the local differential geometry in the vicinity of a sam-pled point. Differential Geometry and Its Applications- The journal publishes original research papers and survey papers in differential geometry and in all interdisciplinary areas in mathematics that use differential geometric methods and investigate geometrical structures. Based on Kreyszig's earlier book Differential Geometry , it is presented in a simple and understandable manner with many examples illustrating the ideas, methods, and results. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. Now we can formulate the rst generalization of the fundamental theorem. Its projections in the xy-,. Time permitting, Penrose's incompleteness theorems of general relativity will also be discussed. BASIC DIFFERENTIAL GEOMETRY: RIEMANNIAN IMMERSIONS AND SUBMERSIONS WERNER BALLMANN Introduction Immersions and submersions between SR-manifolds which respect the SR-structures are called Riemannian immersions respectively Riemannian submer-sions. (Of course, for a decreasing function, or a function whose graph is below the x-axis, the picture will look a bit different, but the definitions are the same. Proofs of the inverse function theorem and the rank theorem. He does just the right thing: assuming the language and background developed in the first volume, he goes through the material on curves and surfaces that one typically meets in a first elementary course. There is also a section that derives the exterior calculus version of Maxwell's equations. The courses 120A and 120B deal with differential geometry in a special context, curves and surfaces in 3-space, which has a firm intuitive basis, and for which some remarkable and striking theorems are available. The task of studying the local structure of a surface can be reduced to the same task for the family of curves formed by the normal sections of the surface at a given point in various directions (see Curvature; Normal curvature). Then the torsion-free Levi-Civita connection is introduced. It builds on the course unit MATH31061/MATH41061 Differentiable Manifolds. DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces Preliminary Version Summer, 2016 Theodore Shifrin University of Georgia Dedicated to the memory of Shiing-Shen Chern, my adviser and friend c 2016 Theodore Shifrin No portion of this work may be reproduced in any form without written permission of the author, other than. Differential geometry is the field of mathematics that studies geometrical structures on differentiable manifolds by using techniques of differential calculus, integral calculus, and linear algebra. This book is intended for advanced students and young researchers interested in the analysis of partial differential equations and differential geometry. We define the differential monomial a(x,y,ψ) as the product of powers of coordinates x, y, ψ and derivatives ∂ k+l ψ/∂x k ∂y l. MATH 286 Intro to Differential Eq Plus credit: 4 Hours. Find out more by clicking on the image above. The other two are characterized by the property of having Euler normal number +4 or -4. Given a function $$y = f\left( x \right)$$ we call $$dy$$ and $$dx$$ differentials and the relationship between them is given by,. We first consider the problem which suggests the notion of absolute differentiation and later (in Section 76) the problem which leads to the displacement of Levi-Civita. Here we introduce the normal curvature and explain its relation to normal sections of the surface. Rodrigue's formula with proof in differential geometry Welcome to the Golden Section - Duration: 8:13. DIFFERENTIAL GEOMETRY IMAGES. Everyday low prices and free delivery on eligible orders. Geometry of conservation laws for a class of parabolic partial differential equations, Selecta Math. Based on Kreyszig's earlier book Differential Geometry , it is presented in a simple and understandable manner with many examples illustrating the ideas, methods, and results. It was written by Silvio Levy and is reproduced here with permission. txt) or view presentation slides online. Preface This volume documents the full day course Discrete Differential Geometry: An Applied Introduction presented at S. Clelland, "Geometry of conservation laws for a class of parabolic PDEs II: normal forms for equations with conservation laws," Selecta Mathematica, New Series 3 (1997), 497-515. General existence theorem 4 2. Differential Geometry of Surfaces Jordan Smith and Carlo Sequin´ CS Division, UC Berkeley 1 Introduction These are notes on differential geometry of surfaces based on read-ing [Greiner et al. Differential Geometry of Contents Index 2. We will see the differential geometry material come to the aid of gravitation theory. Students who have not taken at least an undergraduate class in Topology, may skip problems 1. If you need to meet with a faculty member, here are their office hours. Introduction to Riemannian geometry in higher dimensions. Three coordinates su ce. All contributors to this book are close friends, colleagues and students of Gu Chaohao. to discrete & computational differential geometry and optimization • Develop new tools to explore the variety of feasible / optimized designs through links to the geometry of shape spaces • Advance the theory (discrete differential geometry, shape spaces, …) through novel concepts motivated by applications. 19 The Shape of Di erential Geometry in Geometric Calculus 5 Thus GC uni es the familiar concepts of \divergence" and \curl" into a single vector derivative, which could well be dubbed the \gradient", as it reduces to the usual gradient when the eld is scalar-valued. The Second Fundamental Form of a Surface The main idea of this chapter is to try to measure to which extent a surface S is diﬀerent from a plane, in other words, how "curved" is a surface. Parker Elements of Differential Geometry, Prentice-Hall, 1977. This normal vector is always perpendicular to the tangent vector T, so the vector T must be in the plane perpendicular to N. If we take the unit normal at each point of the curve, and put its tail at the origin, the head. of n pairwise transversal and orthogonal foliations of connected submanifolds of dimension 1. Bolton and L. Achetez neuf ou d'occasion. Courses tought in Fall 99: Math 563 -- Introduction to Differential Geometry. Math 431 Section 6 -- Ordinary Differential equations Math 621 -- Complex Analysis. One of the basic principles in differential geometry is try to (1) compute things locally via differential calculus and (2) ﬁnd a way to patch local information together to get global results. Discrete differential geometry aims to preserve selected structure when going from a continuous abstraction to a finite representation for computational purposes. But it was in an 1827 paper that C. ISBN: 9781461477310 ID: 9781461477310. Differential Geometry: Review Ramesh Sridharan and Matthew Johnson Quick Reference f : U → Rn+1, where U is an open set in Rn, parametrizes an n-dimensional submanifold in n+1 dimensions. with 1% BSA and 5% normal donkey serum at 37 °C for 45. Wolf) Relativity (T. Math 240ABC, Introduction to Differential Geometry and Riemannian Geometry; Geometry Examination. As pointed out by Serre in [161] (Chapter III, Section 8), the relationship between the first definition and the third definition of the tangent space at p is best described by a nondegenerate pairing which shows that Tp(M) is the dual of the space of point derivations at p that vanish on stationary germs. Differential geometry prefers to consider Euclidean geometry as a very special kind of geometry of zero curvature. (Of course, for a decreasing function, or a function whose graph is below the x-axis, the picture will look a bit different, but the definitions are the same. I am a Junior Professor in mathematics at the University of Freiburg since April 2016. He was (among many other things) a cartographer and many terms in modern di erential geometry (chart, atlas, map, coordinate system, geodesic, etc. Visualization of Differential Geometry 131 Let S be a surface without parabolic points, that is points of vanishing Gaussian curvature, and without umbilical points, that is points with κ1 = κ2 for the principal curvatures κ1 and κ2. A quantity that characterizes the deviation of the surface at a point in the direction from its tangent plane and is the same in absolute value as the curvature of the corresponding normal section. Differential geometry is basically the complete physics: spacetime isn't Euclidean, everything is written in Lagrangians and differential equations, resp. Honors Abstract Algebra, Section C Math 428. The Geometry of Surfaces 28 3. DIFFERENTIAL GEOMETRY IMAGES. Unit01_Lecture01 - Free download as Powerpoint Presentation (. The classical approach of Gauss to the differential geometry of surfaces was the standard elementary approach which predated the emergence of the concepts of Riemannian manifold initiated by Bernhard Riemann in the mid-nineteenth century and of connection developed by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early twentieth century. Geometry of conservation laws for a class of parabolic PDEs II: Normal forms for equations with conservation laws. 6: Definition of normal curvature In order to quantify the curvatures of a surface , we consider a curve on which passes through point as shown in Fig. This book is intended for advanced students and young researchers interested in the analysis of partial differential equations and differential geometry. The normal curvature in the direction is where is the curvature of the normal section in the direction. They studied the Kobayashi metric of the domain bounded by an ellipsoid in C2, and their calculations showed that the. OMICS International organises 3000+ Global Conferenceseries Events every year across USA, Europe & Asia with support from 1000 more scientific Societies and Publishes 700+ Open Access Journals which contains over 50000 eminent personalities, reputed scientists as editorial board members. Curves in space are the natural generalization of the curves in the plane which were discussed in Chapter 1 of the notes. Applications of Differential Geometry to Econometrics. Estimating Parameters; Linear Normal Model Examples; Testing Hypotheses About Linear Normal Models; Uniform Normal Distribution; One-way Analysis of Variance; Two-way Analysis of Variance; Linear Regression; Comparing Regression Slopes; Two Dimensional Normal Distribution. An excellent reference for the classical treatment of diﬀerential geometry is the book by Struik [2]. Whitehead, Volume 1: Differential Geometry contains all of Whitehead's published work on differential geometry, along with some papers on algebras. NASA Astrophysics Data System (ADS) Grandi, Nicolás; Sturla, Mauricio. 2, include a proof of Sard’s Theorem in Section 1. General existence theorem 4 2. Level or difficulty as indicated by:. Free linear first order differential equations calculator - solve ordinary linear first order differential equations step-by-step. Klein , 1 Charles L. Copies are available from the Maths office, the electronic version can be found on duo; M. zero cross-section of E* V is strongly pseudo-convec (cf. A Differential Approach to Geometry 1. Figure 3 Normal curvatures when α is a normal section in point p. BASIC DIFFERENTIAL GEOMETRY: CONNECTIONS AND GEODESICS WERNER BALLMANN Introduction I discuss basic features of connections on manifolds: torsion and curvature tensor, geodesics and exponential maps, and some elementary examples. 1) rather than the function-theoretic definition (cf. Normal curvature Given a regular surface and a curve within that surface, the normal curvature at a point is the amount of the curve's curvature in the direction of the surface normal. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature. 3 Second fundamental form II (curvature) Figure 3. Warner) Homogeneous spaces (J. 2 C k manifolds. We look at applications of this idea to asymptotic analysis, the relationship between geometry and information theory and the problem of the choice of parameterisation. Starting from some classical examples (open sets in Euclidean spaces, spheres, tori, projective. Geometry of conservation laws for a class of parabolic partial differential equations, Selecta Math. Complex differential geometry (S. Differential Geometry. 5 is a short section on systems of ordinary differential equations, and Section 7. Read this book using Google Play Books app on your PC, android, iOS devices. The notion of point is intuitive and clear to everyone. The part of differential geometry that studies properties of geometrical forms, in particular curves and surfaces, "in the small". We define the differential monomial a(x,y,ψ) as the product of powers of coordinates x, y, ψ and derivatives ∂ k+l ψ/∂x k ∂y l. Snapshot 2: hyperbolic paraboloid. Namely, a curve can be thought of as a length of string that is twisted this way and that, in a smooth manner. It is required that such assignment of vectors is done in a smooth way so that there are no major "changes" of the vector eld between nearby points. We look at applications of this idea to asymptotic analysis, the relationship between geometry and information theory and the problem of the choice of parameterisation. Then there is a chapter on tensor calculus in the context of Riemannian geometry. Actually, there are a couple of applications, but they all come back to needing the first one. We will see the differential geometry material come to the aid of gravitation theory. Geometry? 1. Here we introduce the normal curvature and explain its relation to normal sections of the surface. Section 5 considers the most important tool that a differential geometric approach offers: the affie connection. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature. This normal vector is always perpendicular to the tangent vector T, so the vector T must be in the plane perpendicular to N. Differential Geometry: Review Ramesh Sridharan and Matthew Johnson Quick Reference f : U → Rn+1, where U is an open set in Rn, parametrizes an n-dimensional submanifold in n+1 dimensions. , SoCG '03 • "On the convergence of metric and geometric properties of polyhedral surfaces", Hildebrandt et al. Characterization of tangent space as derivations of the germs of functions. Description. This differs from the usual approach in that the results. We define the differential monomial a(x,y,ψ) as the product of powers of coordinates x, y, ψ and derivatives ∂ k+l ψ/∂x k ∂y l. example of the concept of a natural bundle. Differential geometry, analysis, and physics. The discipline owes its name to its use of ideas and techniques from differential calculus , though the modern subject often uses algebraic and purely geometric techniques instead. Corbyn did 1. differential coordinates, even classical physics as fluid mechanics. Differential Geometry in Toposes. A point of a surface S which is either a circular point or a planar point of S. Section-Il (3/7) Theory Of Spaec Curves Introduction, Index notation and summation convention Space curves, Arc length, Tangent, Normal and binormal Osculating, Normal and rectifying planes Curvature and torsion The Frenet-Serret theorem Natural equation Of a curve Involutes and evolutes. and methods from differential geometry, in particular for 2- and 3-manifolds, in a discrete rather than discretized setup. He was led to his Theorema Egregium (see 5. General existence theorem 4 2. It has a rich history. NOTES ON DIFFERENTIAL GEOMETRY 3 the ﬁrst derivative of x: (6) t = dx/ds = x˙ Note that this is a unit vector precisely because we have assumed that the parameterization of the curve is unit-speed. Starting from some classical examples (open sets in Euclidean spaces, spheres, tori, projective. But it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that dif-. Written primarily for students who have completed the standard first courses in calculus and linear algebra, Elementary Differential Geometry, Revised 2nd Edition, provides an introduction to the geometry of curves and surfaces. (1)Phil Tynan is the TF, who isn’t here (2)email: [email protected] This book is a collection of papers in memory of Gu Chaohao on the subjects of Differential Geometry, Partial Differential Equations and Mathematical Physics that Gu Chaohao made great contributions to with all his intelligence during his lifetime. By endowing the surface with a normal vector ﬁeld, which is a transversal vector ﬁeld given by the ambient Birkhoﬀ orthogonality, we get an ana-logue of the Gauss map. Take a look at all Open University courses. (Guillemin 65, section 4) In higher differential cohesive geometry. While the depth of knowledge involved is not beyond the contents of the textbooks for graduate students, discovering the solution of the problems requires a. Explicit formulas, projections of a space curve onto the coordinate planes of the Frenet basis, the shape of curve around one of its points, hypersurfaces, regular hypersurface, tangent space and unit normal of a hypersurface, curves on hypersurfaces, normal sections, normal curvatures, Meusnier's theorem. Stoker makes this fertile branch of mathematics accessible to the nonspecialist by the use of three different notations: vector algebra and calculus, tensor calculus, and the notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called. Chapter 4: calculus on surfaces in R3 ( unfinished, beware some typos on last couple pages). If you are studying differential equations, I highly recommend Differential Equations for Engineers If your interests are matrices and elementary linear algebra, have a look at Matrix Algebra for Engineers And if you simply want to enjoy mathematics, try Fibonacci Numbers and the Golden Ratio Jeffrey R. The final section focuses on physical applications, covering gravitational equations and general relativity. Plotting Surfaces in MAPLE. , depending on my mood when I was writing those particular lines. Concluding this section is a general framework, used in the remaining sections, for deriving ﬁrst and second order operators at the vertices of a mesh. Differential geometry of submanifolds with planar normal sections Article (PDF Available) in Annali di Matematica Pura ed Applicata 130(1):59-66 · January 1982 with 34 Reads How we measure 'reads'. This note explains the following topics: From Kock-Lawvere axiom to microlinear spaces, Vector bundles,Connections, Affine space, Differential forms, Axiomatic structure of the real line, Coordinates and formal manifolds, Riemannian structure, Well-adapted topos models. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. 1923] DIFFERENTIAL GEOMETRY OF AN »»-DIMENSIONAL MANIFOLD 1Q5 Also* Yihh = 0 is the condition that the family of curves fa be geodesies. Questions are taken from the pre 2010 exam papers. Bolton and L. made relative to the local tangent plane or normal. Email: tfei[at]math[. We will see the differential geometry material come to the aid of gravitation theory. The normal vector, ν(x) is orthogonal to the tangent space to Y at x. The normal curvature of S is the same in all directions on S at an umbilical point of S. Chapter 3 Concepts Solutions DOC Chapter 3 Concepts Solutions PDF. A normal section of a surface S at a given point M on the surface is the curve of intersection of S with a plane drawn through the normal at the point M. Learn how surfaces in 3-space are curved. Previously, I was a PostDoc in Josef Teichmann's working group in financial mathematics at ETH Zürich and in Harvard EdLabs. Differential Geometry of Curves 1 Mirela Ben The torsion indicates how much the normal changes, in the direction orthogonal to. engineering drawing. This book is intended for advanced students and young researchers interested in the analysis of partial differential equations and differential geometry. , normal vector and principal curvature tensor) of a smooth surface and the ﬁrst- and second-order derivatives (i. A normal section of a surface S at a given point M on the surface is the curve of intersection of S with a plane drawn through the normal at the point M. Be aware that differential geometry as a means for analyzing a function (i. As pointed out by Serre in [161] (Chapter III, Section 8), the relationship between the first definition and the third definition of the tangent space at p is best described by a nondegenerate pairing which shows that Tp(M) is the dual of the space of point derivations at p that vanish on stationary germs. Numerical treatment of geodesic differential equations 21 The system of differential equations 3. We will also look at an application of this new notation. March 13 - March 17, The 2nd OCAMI-KOBE-WASEDA Joint International Workshop on Differential Geometry and Integrable Systems, Osaka City University, Japan. Department of Mathematics University of Washington. jp Room 413, Bldg. 2 Principal Curvatures Planes that contain the surface normal at P are called normal planes. Full curriculum of exercises and videos. The notion of point is intuitive and clear to everyone. By the variational principle, we obtain generalized coupled PB equation (10) and potential driven geometric flow equation (10). Burdujan Clifford-Kähler manifolds 15' 4. 2 C k manifolds. A Quick and Dirty Introduction to Exterior Calculus 45 4. Its length can be approximated by a chord length , and by means of a Taylor expansion we have. These three points determine a plane. com: Coulomb Frames in the Normal Bundle of Surfaces in Euclidean Spaces: Topics from Differential Geometry and Geometric Analysis of Surfaces (Lecture Notes in Mathematics, Vol. In the next section, we will encounter several unit normal vector fields naturally associated with a given space curve. Catalog Description. Differential geometry contrasts with Euclid's geometry. Welcome! This is one of over 2,200 courses on OCW. This page contains sites relating to Differential Geometry. The Geometry of Curves 34 3. This is the osculating plane at f(u). Math study guide Discrete a point is a dot T geometry a line is a set of dots in a row T a line has no thickness F 2 crossing lines intersect in a point F Synthetic a point is a physical dot F (exact location) Geometry a line is a set of points going in both directions going the shortest paths T A line has no thickness T A line extends 2 directions T Plane coordinate a point is an ordered pair. Then is a parametric curve lying on the surface. From Wikibooks, open books for an open world. For surfaces in 3-dimensional space the exercises cover the unit normal vector, the coefficients of the first and second fundamental forms, and the normal curvature. 1)(a)We know the following to holdN u = a 11 X u + a 21 X vN v = a 12 X u + a 22 X vwhere (a ij ) is the matrix. In fact, the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. Applications of Differential Geometry to Econometrics. The discipline owes its name to its use of ideas and techniques from differential calculus , though the modern subject often uses algebraic and purely geometric techniques instead. Which brings me in a roundabout way to the blue paperback before me titled Lectures On Differential Geometry by Iskander A. This is the osculating plane at f(u). According to (4. Pnk associated with the surface normal nk such that: Bin picking system based on EGI Photometric Stereo Set-up Bin-Picking System Summary Differential Geometry Computer Vision #8 Differential Geometry 1. Differential geometry, branch of mathematics that studies the geometry of curves, surfaces, and manifolds (the higher-dimensional analogs of surfaces). Bolton and L. Some problems in differential geometry and topology S. The Genesis of Differential Methods 2. You meet its language all of the time, so the better you understand it the easier will be physics. First variation of area functional 5 2. Differential Geometry in Physics (Lecture Notes). differential geometry. Prerequisite: CDS 201 or AM 125a Basic differential geometry, oriented toward applications in control and dynamical systems. Be aware that differential geometry as a means for analyzing a function (i. We will spend about half of our time on differential geometry. differential coordinates, even classical physics as fluid mechanics. From the 19th century it has grown, considering more. Mabuchi (on Donaldson's work) and by M. Find materials for this course in the pages linked along the left. He was (among many other things) a cartographer and many terms in modern di erential geometry (chart, atlas, map, coordinate system, geodesic, etc. Email: tfei[at]math[. I will make a list of the best books for differential geometry textbooks in my subsequent lines and you will be amazed to find them all on this free mathematics and differential geometry books site. combined with geometry in 513; class analysis combined with geometry in 515; class affine differential geometry, projective differential geometry in 516. Much of this theory generalizes to manifolds of arbitrary dimension, but this is too abstract for an intermediate level course. The final section focuses on physical applications, covering gravitational equations and general relativity. 6: The tangent, normal, and binormal vectors define an orthogonal coordinate system along a space curve. Gauss made the big breakthrough that allows DG to answer the question of whether the annular strip can be made into the strake without distortion. Preface: Since 1909, when my Differential Geometry of Curves and Surfaces was published, the tensor calculus, which had previously been invented by Ricci, was adopted by Einstein in his General Theory of Relativity, and has been developed further in the study of Riemannian Geometry and various. If you are studying differential equations, I highly recommend Differential Equations for Engineers If your interests are matrices and elementary linear algebra, have a look at Matrix Algebra for Engineers And if you simply want to enjoy mathematics, try Fibonacci Numbers and the Golden Ratio Jeffrey R. Curvature of curve 2. BURKE University of California, Santa Cruz The right of the University of Cambridge ta print and sell all manner of books was granted by H. Jump to navigation Jump to search. Differential geometry unit 1 lec 4( normal plane, rectifying plane. Title: Differential Geometry 1 Differential Geometry 2 normal section non-normal section normal curvature Principal directions and principal curvatures 9. Plane Curves 3. These are manifolds (or. Helices Fundamental existence theorem of space curves. Banchoff over the course of four summers, 2000-2003. One may formalize the concept of integrable G G-structure in the generality of higher differential geometry, formalized in differential cohesion. Differential geometry arguably offers the smoothest transition from the standard university mathematics sequence of the first four semesters in calculus, linear algebra, and differential equations to the higher levels of abstraction and proof encountered at the upper division by mathematics majors. But it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that dif-. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Math 405/538 Differential Geometry Final Exam January 7, 2013 1a) (3 pts) Deﬁne torsion of a regular curve in R3. Differential Geometry Of Three Dimensions Volume 1 " Curvature of normal section MeUnier'e theorem Examples IV. 6 is on the Third Lie Theorem, which is fundamental in the study of Lie groups. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hypersurfaces in Euclidean space. Curves in space are the natural generalization of the curves in the plane which were discussed in Chapter 1 of the notes. Do Carmo, Differential Geometry of Curves and Surfaces. Notes on Diﬁerential Geometry with special emphasis on surfaces in R3 Markus Deserno May 3, 2004 Department of Chemistry and Biochemistry, UCLA, Los Angeles, CA 90095-1569, USA Max-Planck-Institut fur˜ Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany These notes are an attempt to summarize some of the key mathe-. DIFFERENTIAL GEOMETRY OF THREE DIMENSIONS BY E WEATHERBURN PDF - Full text of "Weatherburn C. DIFFERENTIAL GEOMETRY Young Wook Kim⁄ and Hyoung Yong Lee Abstract. Explicit formulas, projections of a space curve onto the coordinate planes of the Frenet basis, the shape of curve around one of its points, hypersurfaces, regular hypersurface, tangent space and unit normal of a hypersurface, curves on hypersurfaces, normal sections, normal curvatures, Meusnier's theorem. An introduction to the differential geometry of surfaces in the large provides students with ideas and techniques involved in global research. The theory of surfaces and principle normal curvatures was extensively developed by the French Geometry school led by Gaspard Monge (1746-1818). Differential Geometry (2) Find The Frenet Apparatus Of The Spherical Indicatrix Of The Principle Question: Differential Geometry (2) Find The Frenet Apparatus Of The Spherical Indicatrix Of The Principle Normal Of A Regular Curve. Snapshot 1: elliptical paraboloid. Mark Meyer, Mathieu Desbrun, Peter Schroder, and Alan H. Examples of Wedge and Star in Rn 52 4. Math 136: Differential Geometry (Tu, Th 10-11:30) The exterior differential calculus and its applications to curves and surfaces in 3-space and to various notions of curvature. Chapter 8 is on applications to differential geometry. 410-516-6089; [email protected] 23 do Carmo pg 212 problem 11Let X be a parametrization of a surface with normal N. AN INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH USE OF THE TENSOR CALCULUS By LUTHER PFAHLER EISENHART. Among mathematical disciplines it is probably the least understood (1). We will discuss gravitational redshift, precessions of orbits, the "bending of light," black holes, and the global topology of the universe. There are no exercises for this chapter. (Then the question would be whether the specific quadric out of the family of second order osculation quadrics which is equal to the surface built by the set of the osculating conics at a point p is wellknown/has been studied before. In one of the examples, I assume some familiarity with some elementary di erential geometry as in SE. Property 2: Principal Directions The normal of the two. A special case in point is the inter-esting paper [11]. K-FAC, mirror descent and the natural gradient also derive from or are closely connected to work in information geometry. From the Hitchin section to opers through nonabelian Hodge: Olivia Dumitrescu. The fact that the exterior derivative d transforms sections of Λ k T ∗ M into sections of Λ k+1 T ∗ M for every manifold M can be expressed by saying that d is an operator from Λ k T ∗ M into Λ k+1 T ∗ M. If you are new to university level study, find out more about the types of qualifications we offer, including our entry level Access courses and Certificates. Example of integral manifold: pdf. normal section • normal curvatures ortho-normal frame fields in classical differential geometry: Canonical name: ClassicalDifferentialGeometry: Date of. Hence the fiurvature of a curve on the variety consists of two parts, a vector part In h normal to the variety, which is the same for all curves having the. Characterization of tangent space as derivations of the germs of functions. 4 1 Geometry of the Ellipsoid 1. ∀x ∈ E there exists (U, ϕ) with U open and x ∈ U , such that ϕ : U → ϕ(U ) is a homeomorphism. How is a differential equation different from a regular one? Well, the solution is a function (or a class of functions), not a number. Chapter 8 is on applications to differential geometry. It is fine to perform mathematical calculations using the Schwarzschild metric. Covers all the MATH 285 plus linear systems. The classical approach of Gauss to the differential geometry of surfaces was the standard elementary approach which predated the emergence of the concepts of Riemannian manifold initiated by Bernhard Riemann in the mid-nineteenth century and of connection developed by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early twentieth century. We will then take a "break" and address special relativity. Here is the definition: The book says that the normal vector. Normal Section. Kazdan and F. txt) or view presentation slides online. Differential Geometry: Review Ramesh Sridharan and Matthew Johnson Quick Reference f : U → Rn+1, where U is an open set in Rn, parametrizes an n-dimensional submanifold in n+1 dimensions. Hence the mesh can be stored with a single ﬂoat per ver-tex. Suppose that in three-dimensional Euclidean space a. It is better described as Riemannian geometry without the quadratic re-striction (2). 2 Differential Geometry of Surfaces Differential geometry of a 2D manifold or surface embedded in 3D. This is the Past Exam of Math Tripos which includes General Relativity, Galaxies, Formation, Experimental Design and Applied Multivariate Analysis, Environmental Fluid Dynamics, Elliptic Functions and Elliptic Integrals etc. DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 2. The sequence of papers on projective geometry, linear algebra and Lie groups make important improvements and extensions of the concepts and methods in the book Clifford Algebra to Geometric Calculus (CA to GC).
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## Study23 Group Title PLEASE, HELP been waiting nearly and hour... – Drawing a vector displacement problem, need help! After being kidnapped and placed in the trunk of a car, you discover you are able to pry open the trunk. The GPS tells you that you're moving 35 mph in a direction 66 north of east at a location 1.33 mi west of the police station. 5 min later, you take another reading that tells you are now 2.1 mi from the police station in a direction 20 degrees east of north and are moving E at a speed of 10 mph. Find the avg. velocity and acceleration during the trip. one year ago one year ago

1. AllTehMaffs

|dw:1381583505376:dw| The second set of points was meaner to find because it gave them to you in polar form. Remember that polar coordinates in cartesian can be found by x = r cos (theta) y = r sin (theta) But only when theta is measured from quadrant 1. If you used the 20 degrees they gave you, the cos and the sin would be switched around. Remember that the average velocity is the change in position - that's the vector from point A to point B - over time 5 min. The average acceleration is the change in velocities. Whereas you can calculate displacement vector just by making a right triangle using your two points, the change in velocity is most easily done component-wise. Hope this helps :)

2. AllTehMaffs

* from the origin in quadrant 1 anticlockwise

3. Study23

Hey @AllTehMaffs I'm having trouble finding average velocity. By the way, I switches cos and sin around because I'm using 20 degrees. Could you help me out?

4. Study23

How would I use components to find the avg. velocity @AllTehMaffs

5. loremipsum

Kidnapped? What a horrifying question.

6. AllTehMaffs

Hi. Sure. For the average velocity you need to find the magnitude of the vector that goes from from the first GPS point to the second. So |dw:1381698737663:dw|

7. Study23

Okay... I used 20° though

8. Study23

Does that matter?

9. AllTehMaffs

That won't change the x,y position, just make sure that where I have a sin 70, you have a cos 20 and vice versa

10. Study23

So how would I use components?

11. AllTehMaffs

whoops, I definitely forgot my trig identities. scratch what I just said about the swapping. But we'll get back to that in a sec. It's not immediately important.

12. AllTehMaffs

Can you see how to get the lengths of the sides of that triangle?

13. Study23

Using sin and cos?

14. AllTehMaffs

|dw:1381699218098:dw| You shouln't need to use sin and cos for the displacement vector. You technically don't know the angles right off the bat :P

15. Study23

So to find the deltas I would...?

16. AllTehMaffs

:) Not yet. you already have the points. What's your change in x?

17. Study23

2.1sin20-(-1.33)

18. AllTehMaffs

right, and you find your y the same way, then use Pythagoras to find C

19. AllTehMaffs

I think I contradicted myself by saying we don't need angles, but you're good so far

20. Study23

This gives the displacement so I would have to divide by time?

21. AllTehMaffs

yeah, and then that's your average velocity between those points

22. Study23

So then as far as acceleration goes?

23. AllTehMaffs

That's where you need those other angles decided to overheat

24. AllTehMaffs

wow, don't know how those messages got mashed up :P Sorry, my computer decided to overheat

25. AllTehMaffs

So! Acceleration. You have to divide your velocities up into components, too. The first one, anyway; the second is only in 1 direction.

26. AllTehMaffs

|dw:1381700640583:dw|

27. Study23

Sorry 35 mph?

28. AllTehMaffs

that's the angle. V1 was 35 mph

29. AllTehMaffs

yeah

30. AllTehMaffs

your acceleration in the x direction is $a_{x}=\frac{ v _{2x}-v_{1x} }{ \Delta t }$ and in the y it's $a_{y}=\frac{ v _{2y}-v_{1y} }{ \Delta t }\ 31. Study23 For v2 I would use 10sin20? 32. AllTehMaffs :/ \[a_{y}=\frac{ v _{2y}-v_{1y} }{ \Delta t }]$

33. AllTehMaffs

Not quite. Remember that when the person made the measurement, the car was going straight east.

34. AllTehMaffs

So all of V2 is in the x direction

35. Study23

So what would it be?

36. AllTehMaffs

my computer is being crashy. So calculate the change of velocity in the x direction, and then calculate it in the y

37. AllTehMaffs

What's v2x?

38. Study23

I'm not sure

39. AllTehMaffs

How much of V2 is moving in the x direction?

40. Study23

2.1sin20?

41. AllTehMaffs

|dw:1381701681185:dw| |dw:1381701807535:dw|

42. AllTehMaffs

Does that help?

43. Study23

I can't see the v2 stuff on the dawning

44. AllTehMaffs

nooooo!! bummer. k |dw:1381702127697:dw|

45. AllTehMaffs

Does it make sense why we're not using the 20 degrees for this part of the calculation?

46. Study23

Yes

47. Study23

So 2.1cos0

48. Study23

?

49. AllTehMaffs

Not that either. In the problem, what does it say about his velocity when he measures it a second time?

50. AllTehMaffs

2.1 miles is the distance away from the police station

51. Study23

??

52. AllTehMaffs

"tells you are now 2.1 mi from the police station in a direction 20 degrees east of north and are moving E at a speed of 10 mph"

53. Study23

2.1?

54. AllTehMaffs

moving E at a speed of 10 mph"

55. AllTehMaffs

V2 = ?

56. Study23

10

57. AllTehMaffs

V2x - V1x = ?

58. AllTehMaffs

Where did you get 1.33?

59. Study23

10-35cos66°

60. AllTehMaffs

yes!! Now V2y - V1y = ?

61. Study23

0-35sin66

62. AllTehMaffs

yes!!

63. AllTehMaffs

now (V2x - V1x) / t = ax (V2y - V1y)/t = ay a^2 = (ax)^2 + (ay)^2

64. AllTehMaffs

and a is your average acceleration

65. Study23

Should I use | |?

66. AllTehMaffs

??

67. Study23

Abs value

68. AllTehMaffs

for acceleration? It has to be positive anyway because you're squaring the term

69. Study23

Why is a squared?

70. AllTehMaffs

You're finding the magnitude of vector a, which is kinda like a right triangle. It's defined as a = sqrt ((ax)^2 + (ay)^2)

71. Study23

When I divide by t it's 5/60 right?

72. AllTehMaffs

ummm , yeah, I think so

73. AllTehMaffs

5 min (1 hour/60 min)

74. AllTehMaffs

yeah

75. AllTehMaffs

I gotta run to work. This problem was mean!! :/

76. Study23

Ok so I get 387. What units should it be?

77. AllTehMaffs

that'd be miles per hour, so something is really wrong there. Because that's insane. I'll look at it again when I get back

78. Study23

Ok!

79. AllTehMaffs

Or, maybe I way over complicated it. It's going 35 miles per hour to start, and it ends up going ten miles per hour

80. AllTehMaffs

the magnitude to the deceleration is (10-35)/5min

81. AllTehMaffs

it might just be that, I don't know. Good mathin'!

82. Study23

When you have a chance just double check for me! :)

83. AllTehMaffs

First, units on acceleration here should be miles/(hour^2) (acceleration always has units distance/time^2), so sorry I mistyped that before. Second, I think that the answer is right, and the just 10-35 / t is definitely wrong. Not used to working in mph, so the numbers are bigger than I had thought they would be... I got 387miles/h^2 too. It would be negative, since both x and y components are getting smaller, and has a direction. But I don't think you need that for your answer. Hope this was the right method they were looking for! :)

84. AllTehMaffs

|dw:1381728262451:dw|


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## anonymous 4 years ago Which of the following is the vertex of the quadratic equation y = 6x2 + 24x + 16?

1. anonymous

Use the formula: -b/2a to find your axis of symmetry, sub that in, to find the y coordinate.

2. anonymous

was this originally a multiple choice question? Your question implies it is, but there are no follow up options.

3. anonymous

i got (2, 8)

4. apoorvk

vertex of a quadratic equation' parabola = (-b/2a , -D/4a) where, D = Discriminant. $D= \sqrt{b^2 - 4ac}$

5. anonymous

(–2, –8) (–2, 8) (2, –8) (2, 8)

6. anonymous

these are the options

7. anonymous

Try again

8. apoorvk

Err.. a wee bit of a mistake. should be (-2, -8)

9. anonymous

can you work it out please? I don't understand

10. anonymous

|dw:1334421718973:dw|

11. anonymous

That gives you your axis of symmetry

12. anonymous

|dw:1334421758630:dw|

13. anonymous

(–2, –8) is the ans y=6(x^2+4x+4)-8 (y+8)=6(x+3)^2 therefore –2, –8) is the ans


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[OS X TeX] bold italic vector expressions in Plain TeX

Daniel Drucker aa1856 at wayne.edu
Tue Jun 30 14:54:36 EDT 2020

Dear All,

I am using TeXShop 4.44 on a mid-2011 iMac running Mac OS 10.13.6 (High Sierra).
That’s the latest version of the Mac OS that’s compatible with my iMac. I also
have a fairly new MacBook Pro, but I like using TeX on the iMac because it has
a 27” screen.

I run into problems when I try to use bold extended text italic in Plain TeX’s
math mode. I’d appreciate any suggestions you might have in how to use bold
extended text italic painlessly in math mode.

I’m attaching a short document that illustrates the behavior I’m seeing and the
problems I’m having.

Thanks,
Dan Drucker

*************************************************************************************

\magnification\magstep2
\nopagenumbers
\font\bx=cmbxti10
\def\ts{\thinspace}
\def\bi{{\bx i}} \def\bj{{\bx j}} \def\bk{{\bx k}}
\def\bv{{\bx v}} \def\bw{{\bx w}}
\def\bvv{\hbox{\bx v}} \def\bww{\hbox{\bx w}}
\def\bvvv{\hbox{\bx v\ts}} \def\bwww{\hbox{\bx w\ts}}

{\bx This looks fine, but it's plain text, not math.}

$3\bi + 5\bj$. Math mode has overridden bold italic here. Why?

{\bx 3\bi\ $+$ 4\bj}. The numbers are also in bold italic.

3\bi\ + 4\bj\ + 5\bk. This looks fine, but it's not in math mode.

4\bv\ - 5\bw. The minus sign needs to be in math mode.

4\bv\ $-$ 5\bw. Now the minus sign looks right.

$4v-5w$. Here's the spacing in ordinary math mode without bold italic.

$c$\bv\ $-d$\bw. Math mode doesn't fix the spacing after the minus sign.

$c$\bv\ $-$ $d$\bw. This looks better, but the coding is awkward.

$c\bvv+d\bww$. hboxes help, but the spacing isn't quite right.

$c\bvvv+d\bwww$. Adding thinspaces still doesn't yield correct spacing.

$c\bvv-d\bww$. Here's the same experiment, but with a minus sign.

$c\bvvv-d\bwww$. Spacing is still a problem.

\bye

*************************************************************************************


text:
# Is a particular element of a particular ring a nonzerodivisor?

Let $A$ be the ring $\Bbbk[\alpha_0, \alpha_1, \alpha_2, x_0, x_1, x_2]$ (where $\Bbbk$ is an infinite field, algebraically closed if it matters). Let $g \in \Bbbk[\alpha_0, \alpha_1, \alpha_2]$ be a homogeneous polynomial of degree at least one, such that $\alpha_0$ does not divide $g$. Let $$I = (g, \sum_i \alpha_i x_i).$$

Is $\alpha_0$ necessarily a nonzerodivisor in $A/I$?

A little bit of motivation: I've shown that $I$ has a certain nice property that I would like to carry over to its localization $I[\alpha_0^{-1}] \subset A[\alpha_0^{-1}]$. This will hold automatically if $I[\alpha_0^{-1}] \cap A = I$, which is true iff $\alpha_0$ is a nonzerodivisor in $A/I$. (Each of these is clearly equivalent to the statement: if $\alpha_0 f \in I$, then $f \in I$.)

I can show that at least $\alpha_0$ is not nilpotent, by showing that $(A/I)[\alpha_0^{-1}]$ is not the zero ring.

-

Let's try to think of the question geometrically. The polynomial $g$ defines a curve $C$ in $\mathbb{P}^2$ with coordinates $\alpha_i$ (I'd prefer these to be $x$'s and your $x$'s to be $\alpha$'s, but we won't change your notation). The bihomogeneous polynomial $\sum \alpha_ix_i=0$ defines the codimension $1$ locus in $\mathbb{P}^2\times (\mathbb{P}^2)^\ast$ of pairs $(p,\ell)$ where $p$ is a point in $\mathbb{P}^2$ and $\ell$ is a line containing $p$. Then $I$ defines the codimension $2$ locus consisting of pairs $(p,\ell)$ with $p\in C$ and $p\in \ell$. The irreducible components of this locus are in bijective correspondence with the irreducible components of $C$, and a polynomial $f\in k[\alpha_i]$ will be a zero divisor in $A/I$ if and only if $gcd(f,g)\neq 1$, so that $f$ vanishes on a component of the locus defined by $I$.
An additional note on embedded points: $C$ has no embedded points since it is a projective hypersurface, and the locus defined by $I$ is flat over $C$, hence maps associated points to associated points, hence has no embedded points by a dimension count. (All the fibers over $C$ are isomorphic to $\mathbb{P}^1$.) –  Charles Staats May 28 '11 at 20:19


text:
# 85 hours in days

## Conversion

85 hours is equivalent to 3.54166666666667 days.[1]

## Conversion formula How to convert 85 hours to days?

We know (by definition) that: $1\mathrm{hr}\approx 0.041666667\mathrm{d}$

We can set up a proportion to solve for the number of days.

$1  hr 85  hr ≈ 0.041666667  d x  d$

Now, we cross multiply to solve for our unknown $x$:

$x\mathrm{d}\approx \frac{85\mathrm{hr}}{1\mathrm{hr}}*0.041666667\mathrm{d}\to x\mathrm{d}\approx 3.541666695\mathrm{d}$

Conclusion: $85  hr ≈ 3.541666695  d$

## Conversion in the opposite direction

The inverse of the conversion factor is that 1 day is equal to 0.282352941176471 times 85 hours.

It can also be expressed as: 85 hours is equal to $\frac{1}{\mathrm{0.282352941176471}}$ days.

## Approximation

An approximate numerical result would be: eighty-five hours is about three point five four days, or alternatively, a day is about zero point two eight times eighty-five hours.

## Footnotes

[1] The precision is 15 significant digits (fourteen digits to the right of the decimal point).

Results may contain small errors due to the use of floating point arithmetic.


text:
# zbMATH — the first resource for mathematics

Comparing Heegaard and JSJ structures of orientable 3-manifolds. (English) Zbl 0959.57010
The authors prove that the Heegaard genus of an irreducible closed orientable 3-manifold limits the number and complexity of the pieces that arise in the Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. Some earlier work in this direction was done by T. Kobayashi [Osaka J. Math. 24, 173-215 (1987; Zbl 0665.57010)].
Reviewer: J.Hebda (St.Louis)

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
Heegaard genus; canonical tori; three-manifolds
Full Text:
##### References:
 [1] A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), no. 3, 275 – 283. · Zbl 0632.57010 · doi:10.1016/0166-8641(87)90092-7 · doi.org [2] Wolfgang Haken, Some results on surfaces in 3-manifolds, Studies in Modern Topology, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1968, pp. 39 – 98. [3] P. Heegaard, Forstudier til en Topologiskteori for de Algebraiske Aladers Sammenhaeng, Ph. D. thesis, Copenhagen, 1898. [4] William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. · Zbl 0433.57001 [5] Tsuyoshi Kobayashi, Structures of full Haken manifolds, Osaka J. Math. 24 (1987), no. 1, 173 – 215. · Zbl 0665.57010 [6] Yoav Moriah and Hyam Rubinstein, Heegaard structures of negatively curved 3-manifolds, Comm. Anal. Geom. 5 (1997), no. 3, 375 – 412. · Zbl 0890.57025 · doi:10.4310/CAG.1997.v5.n3.a1 · doi.org [7] A. Przybyszewska, in Knot theory from Vandermonde to Jones by J. Przytycki, Mathematics Institute, Odense University, Preprint 43, 1993. [8] Hyam Rubinstein and Martin Scharlemann, Comparing Heegaard splittings — the bounded case, Trans. Amer. Math. Soc. 350 (1998), no. 2, 689 – 715. · Zbl 0892.57009 [9] M. Scharlemann, Heegaard splittings of compact $$3$$-manifolds, to appear in Handbook of Geometric Topology, ed by R. Daverman and R. Sherr, Elsevier Press. · Zbl 0985.57005 [10] M. Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology and its applications 90 (1998), 135-147. · Zbl 0926.57018 [11] M. Scharlemann and J. Schultens, The tunnel number of the sum of $$n$$ knots is at least $$n$$, Topology 38 (1999), 265-270. CMP 98:05 · Zbl 0929.57003 [12] J. Schultens, Additivity of tunnel number, to appear. · Zbl 0972.57007 [13] Peter Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401 – 487. · Zbl 0561.57001 · doi:10.1112/blms/15.5.401 · doi.org [14] Martin Scharlemann and Abigail Thompson, Thin position for 3-manifolds, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 231 – 238. · Zbl 0818.57013 · doi:10.1090/conm/164/01596 · doi.org [15] Friedhelm Waldhausen, Heegaard-Zerlegungen der 3-Sphäre, Topology 7 (1968), 195 – 203 (German). · Zbl 0157.54501 · doi:10.1016/0040-9383(68)90027-X · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
Free body diagram

1. Homework Statement
When drawing a freebody diagram for just an elevator, not the people in it, what are all the forces that we have to consider? I just used Fg down and Ft up, but because it is accelerating up, I made Ft greater.

Related Introductory Physics Homework Help News on Phys.org
Just because the elevator is moving up, doesn't mean it is always accelerating up. It may only accelerate when it first starts moving and right before it stops at your floor. Otherwise, it may be moving with uniform velocity upwards.

Well, in the question it stated that it was accelerating... So was I right to have only the two forces Ft and Fg with Ft bigger?

Well, in the question it stated that it was accelerating... So was I right to have only the two forces Ft and Fg with Ft bigger?
In that case, your diagram is correct to have Ft greater in magnitude than Fg, since Ft + Fg = Fnet = ma and a must be non-zero, and I'm assuming the elevator is accelerating in the direction of Ft. ;)

Last edited:


text:
# Graphing amplitudes

1. Nov 19, 2013

### cutesteph

Say x(n) = n for 0 <= n < 4 and 0 o.w.

So X(W) = Sum n=-∞ to ∞ x(n) exp(-inw) = sum from n=0 to 3 nexp(-inw)
= 0 + exp(-iw) + 2exp(-j2w) + 3 exp(-j3w)

get do I get the amplitude and phase?

Also if the signal was continuous x(t) = t for the same interval

X(W)= integral 0 to 4 texp(-iwt) dy = (4exp(-i4w)/ -iw) - (exp(-iw4) -1)/w^2

How would I graph the amplitude and phase?

Last edited: Nov 19, 2013
2. Nov 19, 2013


text:
# Math Help - differentiation

1. ## differentiation

if the graph of y=f(x) contains the point (0,2), dy/dx=-x/ye^(x^2) and f(x)>0 for all x then f(x) = ?

2. Originally Posted by lbj2322
if the graph of y=f(x) contains the point (0,2), dy/dx=-x/ye^(x^2) and f(x)>0 for all x then f(x) = ?
$y'=-\frac{x}{ye^{x^2}} \implies yy' = -\frac{x}{e^{x^2}}$

Integrate both sides. (Use u-substitution with $u=x^2$ and $du=2x\,dx$ on the righthand integral.)

$\int yy'\,dx = -\int\frac{x}{e^{x^2}}\,dx \implies \frac{1}{2}y^2 = \frac{1}{2}e^{-x^2}+c \implies y = \sqrt{e^{-x^2}+C}$

Plug in $(0,2)$: $2=\sqrt{1+C} \implies C=3$

So $y=f(x)=\sqrt{e^{-x^2}+3}$

3. thanks alot

i was confused on what to do with the coordinates but now i understand. Thanks again


text:
10. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Given: Two circles are drawn with sides AC and AB of $$\triangle{ABC}$$ as diameters. Both circles intersect each other at D.

To prove: D lies on BC.

$$\angle{ADB}$$ = $$90^\circ$$ ...(i)(Angles in a semi-circle)
And $$\angle{ADC}$$ = $$90^\circ$$ ...(ii)(Angles in a semi-circle)
$$\angle{ADB}$$ + $$\angle{ADC}$$ = $$90^\circ$$ + $$90^\circ$$ = $$180^\circ$$


text:
Conference paper Open Access

# Single Frequency Networks for 5G Broadcast: a Software Defined Radio Experiment

Alvaro Ibanez; Borja Inesta; Manuel Fuentes; David Gomez-Barquero; Diarmuid Collins; Joao F. Santos

### JSON-LD (schema.org) Export

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"inLanguage": {
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"description": "
Broadcast is a mechanism created for the delivery of the same content to many users simultaneously. This technology has been considered in 3GPP as part of 4G and recently 5G. One of the key solutions to enhance coverage when using 5G broadcast is the use of Single Frequency Networks (SFN), where all transmitters deliver the same content at the same frequency in a synchronized manner. This paper is derived from the 5GROSE project, which works towards the transmission of 5G broadcast services over SFN virtualized networks, using opensource software. The work is divided into three topics. The first part is the development and testing of the first virtualized Multimedia Broadcast Multicast Service over Single Frequency Network (MBSFN) transmission. The second part consists of the introduction of 5G physical layer Release (Rel)-16 specific components. The third part is about the combination of both unicast and broadcast transmissions by means of network slicing. This paper describes in detail the implemented technologies, the testbed facility where these are integrated, as well as the implementation and setup to carry out the experiment.
",
"creator": [
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"affiliation": "Universitat Politecnica de Valencia",
"@type": "Person",
"name": "Alvaro Ibanez"
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"affiliation": "Universitat Politecnica de Valencia",
"@type": "Person",
"name": "Borja Inesta"
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{
"affiliation": "Universitat Politecnica de Valencia",
"@type": "Person",
"name": "Manuel Fuentes"
},
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"affiliation": "Universitat Politecnica de Valencia",
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"name": "Single Frequency Networks for 5G Broadcast: a Software Defined Radio Experiment"
}
64
71
views


text:
# FX hedging: forward rate and implied forward rate

It is argued that the forward rate that a corporation receives from entering a forward contract (let's call it $F$) is the same as the implied forward rate from issuing foreign debt (let's call it $\hat{F}$).

Under the assumption that CIP holds, I disagree.

In particular, assume a european corporation that wants to hedge a future exposure (for example coming from future dollar receipts) by buying euros forwards. Then the direct way to do this would be to enter a forward contract and receive euros at the forward rate, $F$; where according to CIP, $F$ is given by:

$F = S \frac{(1+i_{euro})}{(1+i_{dollar})}$

where $S$ is the spot exchange rate (in terms of euros per dollar) and $i_{euro}$ is the interest rate in euro and $i_{dollar}$ is the interest rate in dollar). Because the corporation is presumably entering a transaction with a bank and it is posting a margin collateral, the interest rates could be assumed to be equal to the LIBOR rates.

The paper argues that $F$ can be derived synthetically: the corporation could issue a zero coupon dollar bond, swap the notional in euro at spot $S$ and invest it at the euro rate. The implied forward rate $\hat{F}$ would then be equal to $F$.

My objection is that in this latter case the interest rate at which the corporation is borrowing in dollars must be different from the USD LIBOR rate; in particular I expect that it would reflect a spread due to a risk premium specific to the corporation issuing the dollar bond.

My conclusion is therefore that $\hat{F} < F$ and hence that it would be cheaper for the corporation to hedge the FX exposure by entering into a forward contract.

• Thank You. Are you referring to this strategy in the context of FX hedging? In the sense, that the firm by buying back her own debt can in fact save an amount of interest which will then go in the determination of an implied forward rate: $\hat{F} = S \frac{1+i_e^c}{1+i_d^c}$, where $i_e^c$ is the corporate bond rate in euros and $i_e^c$ is the corporate bond rate in dollars? – night_owl89 Aug 24 '18 at 13:24


text:
Calculations of Pic^0, Pic, NS of surfaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T09:08:02Z http://mathoverflow.net/feeds/question/40555 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40555/calculations-of-pic0-pic-ns-of-surfaces Calculations of Pic^0, Pic, NS of surfaces Justin Shih 2010-09-29T23:37:46Z 2010-09-30T15:09:02Z
I'm looking for an example in the literature where $\mbox{Pic}^0(X)$, $\mbox{Pic}(X)$, and $NS(X)$ of a projective surface $X$ over a field are calculated. I want them for an example I'm trying to work out, so ideally $X$ would be relatively simple, perhaps a cubic hypersurface in $\mathbb{P}^3$, or something along those lines. I know it's out there, but googling and browsing arXiv and MathSciNet haven't quite panned out.
 http://mathoverflow.net/questions/40555/calculations-of-pic0-pic-ns-of-surfaces/40563#40563 Answer by Yuri Zarhin for Calculations of Pic^0, Pic, NS of surfaces Yuri Zarhin 2010-09-30T01:12:37Z 2010-09-30T03:31:04Z
Manin's book "Cubic forms" contains the calculations of these groups when $X$ is a smooth projective cubic surface. In particular, $\operatorname{Pic}^0(X)={0}$ and $\operatorname{Pic}(X)=\operatorname{NS}(X)$ is a free commutative group of rank 7.

Another class of examples is provided by products $X=E \times E'$ of two elliptic curves. In particular, if $E=E'$ has no complex multiplication then $NS(E\times E)$ is a free commutative group of rank 3 generated by the classes of $E \times {0}$, ${0}\times E$ and the diagonal while $\operatorname{Pic}^0(E\times E)=E \times E$. See Mumford's Abelian Varieties.
 http://mathoverflow.net/questions/40555/calculations-of-pic0-pic-ns-of-surfaces/40573#40573 Answer by Chandan Singh Dalawat for Calculations of Pic^0, Pic, NS of surfaces Chandan Singh Dalawat 2010-09-30T04:42:51Z 2010-09-30T04:42:51Z
There are also some computations for conic bundles by Sansuc in

MR0695346 (85d:14014) Sansuc, Jean-Jacques À propos d'une conjecture arithmétique sur le groupe de Chow d'une surface rationnelle.
 http://mathoverflow.net/questions/40555/calculations-of-pic0-pic-ns-of-surfaces/40580#40580 Answer by Remke Kloosterman for Calculations of Pic^0, Pic, NS of surfaces Remke Kloosterman 2010-09-30T07:37:33Z 2010-09-30T07:37:33Z
For surfaces in $\mathbb{P}^3$ of degree at most 3 the calculation of $Pic(X)$ is relatively easy: In this case $X$ is rational, hence $NS(X)$ modulo torsion equals $H^2(X,\mathbb{Z})$. (If you work over the complex numbers you might also apply Lefschetz (1,1)-Theorem.)

Starting from degree 4 the calculation of $NS(X)$ is much more involved. The difficulty depends on how you present $X$. In case you give $X$ just by an equation it is not so easy to calculate $NS(X)$, at least if you work in characteristic 0. See e.g., http://pjm.math.berkeley.edu/ant/2007/1-1/p01.xhtml where an example is given of a quartic surfaces with $\mathrm{rank} Pic(X)=\mathrm{rank} NS(X)=1$.

Over a finite fields (or over $\overline{\mathbb{F}_p}$) you can find an upper bound for the rank of $NS(X)$ in terms of the zeta function of $X$ (see loc. cit.). If you believe the Tate conjecture then this upper is the actual rank of $NS(X)$. In concrete examples you might try to use this upper bound, try to find sufficiently many curves on $X$ and then use the intersection pairing to prove that the classes of these curves are independent in $NS(X)$.
 http://mathoverflow.net/questions/40555/calculations-of-pic0-pic-ns-of-surfaces/40598#40598 Answer by Daniel Loughran for Calculations of Pic^0, Pic, NS of surfaces Daniel Loughran 2010-09-30T10:48:57Z 2010-09-30T10:48:57Z
In general, if $X$ is a smooth complex projective variety which is simply connected, then we have $\rm{Pic}^0(X)=0$. Indeed we have $H^1(X,\mathbb{Z})=0$, and then Hodge theory implies that $H^1(X,\mathcal{O}_X)=0$. The exponential sheaf sequence http://en.wikipedia.org/wiki/Exponential_sheaf_sequence then implies that the natural map $\rm{Pic}(X) \to H^2(X, \mathbb{Z})$ is injective.

In particular, any hypersuface of dimension greater than $1$ is simply connected (by the Lefschetz hyperplane section theorem), and so $\rm{Pic}^0(X)$ is always trivial in this case.
 http://mathoverflow.net/questions/40555/calculations-of-pic0-pic-ns-of-surfaces/40627#40627 Answer by Laie for Calculations of Pic^0, Pic, NS of surfaces Laie 2010-09-30T15:09:02Z 2010-09-30T15:09:02Z
For Reid's list of 95 K3 surfaces Picard lattices have been computed by Belcastro. Her paper can be downloaded from the arXiv at http://arxiv.org/PS_cache/math/pdf/9809/9809008v2.pdf . She has also made her thesis available, which can be downloaded at http://www.toroidalsnark.net/sm_thesis.pdf .


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How the simulator work?

Recently I focused on how to simulate in classical computer, and I found Qiskit offers qasmsimulator and statevector simulator. And others such as project Q also can simulate on classical computer. So is there a general way to simulate on classical computer? And what are differences among simulators? I mean actually it seems like we just need to multiply gate operation as matrixes.

The second level of simulation is State Vector simulation. In this case, we are basically doing what you mentioned in your question. We take an input state (of size $$2^n$$, where n is the number of qubits) and then apply gates to it through matrix multiplication. Due to the exponential size of the state, this requires resources exponential in the system size to be simulated, and as a result is not considered classically efficient. The only restriction on these simulations is that the gates must all be unitary and all states must be pure.
The last level is Density Matrix simulation. Here, we store the full $$2^n \times 2^n$$ density matrix of the state. As a result we can simulate any quantum channel, and mixed states are permissible. However we now have an even bigger object to work with. These simulations are often necessary for doing work on simulating physical noise or other non-unitary processes, but are extremely limited in size due to their exponential resource requirements.


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# Overview

A shortcode is code similar to HTML, but is Pressbooks-specific and allows you to easily insert formatting or content elements. Shortcodes are typically easier to use than HTML and can be used both in the Visual Editor and in the Text editor.

For most users, inserting content via the Visual Editor will be the easiest option. However, for users whose workflow involves importing Word documents into Pressbooks or copy/pasting from a manuscript written in a different editor, shortcodes are a convenient way to insert or format content elements that are Pressbooks-specific.

Most of these shortcodes support custom classes. For more information on applying custom classes, see the “Applying Custom Classes” section at the end of this chapter.

# Available Shortcodes

Caution: please do not directly copy/paste the code displayed on this page as it contains special characters that force the shortcodes to be displayed as text instead of being properly rendered. Instead, retype the shortcode in your chapter.

[heading]Top-level heading inside chapter (H1)[/heading] [subheading]Second-level heading inside chapter (H2)[/subheading]

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 Example Code Example Output The code can also be a block. 4 spaces before the line here             and here more code [/code]” width=”370″ height=”108″>

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[equation]latex code here[/equation] OR [latex]latex code here[/latex]
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 Example Code Example Output [equation]e^{\i \pi} + 1 = 0[/equation]

## Blockquote

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## Footnotes

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In the visual editor, the footnote text, wrapped in the shortcode, should be placed where the footnote reference number should be located, and not at the end of the chapter or page.

 Example Code Example Output This leads to a footnote[footnote]This is the footnote content. [/footnote].

## Anchor

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The preset “shaded” class or custom classes may be applied to the textbox shortcode in order to change the appearance upon display.

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[columns]Your text here[/columns] OR [columns count=2]Your text will be displayed in 2 columns[/columns] OR [columns count=3]Your text will be displayed in 3 columns[/columns]

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## Media

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 Example Code Example Output [gallery ids="68,34,264"]

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If you are familiar with CSS, you may apply custom classes and use custom CSS to modify the appearance of some of the elements. Shortcodes that support custom classes are indicated in the descriptions above. Custom classes in the shortcodes work the same way as custom classes in HTML, by adding class="".

For example, adding a custom class to a textbox:

[textbox class=custom_class]Your text here[/textbox]

# Nested Shortcodes

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Almost every shortcode has an alternate styling method available from the visual editor toolbar. The only shortcode elements that do not have a non-shortcode alternative are:

• Footnotes
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We cannot guarantee that all combinations of nested shortcodes will function correctly in webbooks and various exports, but the following have been tested and verified:

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• [footnote] shortcodes inside of [textbox] shortcodes


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# Time series price prediction and linear regression: using high/low rather than last quotes price

Discrete time series regression models, like ARIMA, are usually built around the assumption that we only have 1 available price for each period t, which I will call the Close.

In reality asset time series (bid, ask) are continuos processes and we might have more data than just the Close for each interval. Let's say we have, for each period, an Open, High, Low and Close price. t1(Low) will therefore be the lowest asset price during t1.

Now let's also assume we want to build a mean reversion model. We have reasons to believe that some of our predictors have significant negative correlation to the next period price, although we can't be confident that mean reversion will materialize precisely at t1(Close). If we want to make R-squared more meaningful, does it make sense to use the next period Close as the DV? Shouldn't we use the Low/High? That is, if the asset price at interval t0 as travelled from an Open of 10 to a Close of 20, we will want to regress t0 IVs against t1(Low), not t1(Close).

-
Please edit your question for clarity. You say you are predicting prices but then define Y as a return in T1. Also you say you are predicting price at "any time" -- not sure that what means. Maybe you are trying to build a model that achieves some target return over the course of a horizon -- not clear from your question. – Quant Guy Dec 18 '11 at 15:32
I am, if anything, even more confused as to what you are asking after your most recent edits. At least, I am confused as to what you are proposing. I feel your initial goal of "maximizing prediction value" may be misguided. A model that overly depends on the specific regression you run is not very robust. Also, if I understand you right, rounding your dependent variable, particularly if you are trying to predict prices or returns, is rarely a good idea. Better to use robust regression, if outliers are a problem. – Tal Fishman Dec 18 '11 at 18:59
Alright, let's make it simple and I will walk you through: in time series analysis, be it ARIMA or any other forecasting model, what is the generally used observed value Y in the regression? If the answer is the last available differenced price at time t1 for the continuous process, why is that? Why not to regress against the differenced high or low price for the period t0-t1? And why to close the question anyway, other people might provide more specific critique than general "confused" comments. – Robert Kubrick Dec 18 '11 at 21:29
@RobertKubrick Closing need not be permanent. Closing only says that, as it stands right now, at least 5 active users believe your question is not clear. If you edit your question to be clearer, then it will be re-opened. If you wish, you could make your last comment, "Why not regress against the differenced high or low price?" your entire question, and perhaps that is good enough. – Tal Fishman Dec 18 '11 at 22:42
@RobertKubrick following your latest edits, is your question now "shouldn't we use high/low?" Please make it clear that that is your ultimate question. As it stands, the question still seems a bit overly chatty, more like giving your opinion than asking a question. – Tal Fishman Dec 19 '11 at 13:51
show 1 more comment

$$\frac{O+C+\frac{H+L}2}3$$


text: Welcome to Chem Zipper.com......: When 3.2 g of sulphur is vapourised at 450°C and 723 mm pressure, the vapours occupy a volume of 780 ml. What is the molecular formula of sulphur vapours under these conditions?


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# Inverse of a function

If a function is bijective, then its inverse exists. Is there any example that inverse of a function exists but the original function is not bijective?

Homework Helper
A function has to be bijective in order to have an inverse.

A function can be locally bijective, so it's inverse exists only in some finite interval.

For example $$x^{2}$$ is not a bijective in any interval containing x=0 (since f'(0)=0) but if you restrict yourself to x>0, then you off course have the inverse
$$f(x)=\sqrt{x}$$ or in x<0 the inverse is $$f(x)=-\sqrt{x}$$.

Last edited:
HallsofIvy
Homework Helper
Another example. f(x)= ex, as a function from R to R, is not surjective, so not and does not have an inverse. In particular, there is no f-1(-1). But if I consider it as a function from R to R+, the positive real numbers, then it is bijective and f-1(x)= ln(x).

A function, from A to B, has an inverse if and only if it is bijective.

Thanks a lot! It may be a typo in my textbook.

A function can be locally bijective, so it's inverse exists only in some finite interval.

For example $$x^{2}$$ is not a bijective in any interval containing x=0 (since f'(0)=0) but if you restrict yourself to x>0, then you off course have the inverse
$$f(x)=\sqrt{x}$$ or in x<0 the inverse is $$f(x)=-\sqrt{-x}$$.
However, why is x^2 not bijective if we define the domain containing zero(like x=>0)? It is a continous function so isn't it bijective at this interval though being not strictly increasing? Thanks

Mark44
Mentor
f(x) = x2 on [0, infinity) is strictly increasing. If a and b are any two numbers in this interval such that a < b, then f(a) < f(b).

Thanks a lot! It may be a typo in my textbook.

However, why is x^2 not bijective if we define the domain containing zero(like x=>0)? It is a continous function so isn't it bijective at this interval though being not strictly increasing? Thanks

I had a little confusion in defining the inverse sorry.

For $$x\geq 0$$ the inverse of $$y=x^{2}$$ is $$x=f^{-1}(y)=\sqrt{y}$$

For $$x\leq 0$$ the inverse of $$y=x^{2}$$ is $$x=f^{-1}(y)=-\sqrt{y}$$

Notice of course that the inverse is defined only over $$y\geq 0$$, since the range of x^2 is only the non-negative real numbers.

Also, the interval where the function can be define as a bijection may inclue x=0, but only as a boundary point.

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## Half life

$\frac{d[R]}{dt}=-k; [R]=-kt + [R]_{0}; t_{\frac{1}{2}}=\frac{[R]_{0}}{2k}$

Niyanta Joshi 1F
Posts: 30
Joined: Thu Jul 13, 2017 3:00 am

### Half life

Can we use the half life formulas for zero order and second order in practical life examples?

Michelle Chernyak 1J
Posts: 33
Joined: Fri Sep 29, 2017 7:06 am

### Re: Half life

yes

Lily Guo 1D
Posts: 64
Joined: Fri Sep 29, 2017 7:03 am

### Re: Half life

Yes, just plug your given values into the equations and solve!

Sabah Islam 1G
Posts: 50
Joined: Sat Jul 22, 2017 3:01 am

### Re: Half life

Yes you can. You can use the half life formulas and also plug in your values for time and concentration into the zero order and second order equations as well.

Vincent Kim 2I
Posts: 50
Joined: Fri Sep 29, 2017 7:04 am
Been upvoted: 2 times

### Re: Half life

Yes, first order half lives apply to radioactive decay. Zero order and Second order half lives exist in real life with other processes

Priyanka Bhakta 1L
Posts: 50
Joined: Fri Sep 29, 2017 7:04 am

### Re: Half life

Also, be sure to use the appropriate half life equation based on the order!

Juanalv326
Posts: 49
Joined: Fri Sep 29, 2017 7:04 am

### Re: Half life

Yeah, unless otherwise given in another form

Christopher Anisi 2K
Posts: 30
Joined: Fri Sep 28, 2018 12:21 am

### Re: Half life

Yes, just be sure to include use the proper equation for the corresponding order.


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back to index | new

$\textbf{Multiplication}$

In the multiplication problem below, $A$, $B$, $C$, and $D$ are different digits. What is $A+B$? $$\begin{array}{cccc}& A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}$$

$\textbf{Lying Politicians}$

Suppose $125$ politicians sit around a conference table. Each politician either always tells the truth or always lies. (Statements of a liar are never completely true, but can be partially true.) Each politician now claims that the two people beside him or her are both liars. What are the maximum possible number and minimum possible number of liars?

$\textbf{Lily Pads}$

There are $24$ lily pads shown below. A toad can jump from one pad to an adjacent one either horizontally or vertically, but not diagonally. Can this toad visit all the pads without stopping at a pad for more than once? It can choose any pad to start its journey.

There are $2015$ people standing in a circle, counting $1$ and $2$ in turn continuously. Those who count $2$ will be out. For example, people who stand at initial positions of $2, 4, \dots, 2014, 1, 3, \dots$ etc will be out. The game goes on until there is only one person remaining in the circle. What is his initial position?

$\textbf{Cutting Pizza}$

Assume you have a magical pizza in the shape of an infinite plane. You have a magical pizza cutter that can cut an infinite line, but it can only be used $14$ times. To share with as many of your friends as possible, you cut the pizza in a way that maximizes the number of pieces (the pizza is too heavy to be lifted up). How many finite pieces of pizza do you have?

To build roads between $16$ cities so that one can travel from any city to any other city by passing through at most one other city. What is the minimum number of roads that the city with the most road has?

As shown in the figure, in a regular triangular house, all the rooms are in the shape of regular triangles. There are doors between adjacent rooms. Starting from one of the rooms, go through the doors to visit other rooms, without repeating rooms or leaving the house. Including the starting room, how many rooms can be visited?

There are $7$ pieces of paper on a table. Cut some of them into $7$ each and put them back to the table. Repeat this process as many times as you wish. Is it possible to have $1990$ pieces of paper on the table at some moment?

$\textbf{Cover the Board}$

Joe cuts off the top left corner and the bottom right corner of an $8\times 8$ board, and then tries to cover the remaining board using thirty-one $1\times 2$ smaller pieces. Is it possible? Note: a smaller piece can be rotated, but cannot be further broken up.

$\textbf{Cover the Board (II)}$

Joe cuts off a $2\times 2$ corner from an $8\times 8$ board, and then tries to cover the remaining part using $15$ L-shaped grids made of $4$ grids as shown. Is it possible?

$\textbf{Cards Game (Sum Fifteen)}$

There are nine cards laid out on a table, numbered $1$ through $9$. Two players, Joe and John, take turns to pick up one card a time (and once a card is picked up, it is out of play). As soon as one of these two players has the sum of three of cards equal $15$ , that player wins. Who will win if both players adopt the best strategy?

$\textbf{Heaps of Beans}$

A game starts with four heaps of beans, containing $3$, $4$, $5$ and $6$ beans, respectively. The two players move alternately. A move consists of taking either one bean from a heap, provided at least two beans are left behind in that heap, or a complete heap of two or three beans. The player who takes the last bean wins. Does the first or second player have a winning strategy?

$\textbf{Passing the Bridge}$

It is a dark and stormy night. Four people must evacuate from an island to the mainland. The only link is a narrow bridge which allows passage of two people at a time. Moreover, the bridge must be illuminated, and the four people have only one lantern among them. After each passage to the mainland, if there are still people on the island, someone must bring the lantern back. When they cross the bridge individually, the four people take $2$, $4$, $8$ and $16$ minutes, respectively. Crossing the bridge in pairs, the slower speed is used. What is the minimum time for the entire evacuation?

Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?

Initially, a chip is placed in the upper-left corner square of a $n\times m$ grid of squares as shown. The chip can move in an $L$-shaped pattern, moving two squares in one direction (up, right, down or left) and then moving one square in a corresponding perpendicular direction. What is the minimum number of $L$-shaped moves needed to move the chip from its initial location to the square marked “$X$”?

$\textbf{Prisoners' Problem}$

One hundred prisoners will be lined up. Each one will be assigned either a red hat or a blue hat. No one can see the color of his or her own hat. However, each person is able to see the color of the hat worn by every person in front of him or her. That is, for example, the last person in line can see the colors of the hats on $99$ people in front of him or her; and the first person, who is at the front of the line, cannot see the color of any hat. Beginning with the last person in line, and then moving to the $99^{th}$ person, the $98^{th}$, etc., each will be asked to name the color of his or her own hat. He or she can only answer "red" or "blue". If the color is correctly named, the person lives; if not, the person is shot dead on the spot. Everyone in line is able to hear all the responses but not the gunshots. Before being lined up, the $100$ prisoners are allowed to discuss a strategy aiming to save as many of them as possible. How many people can be saved if they can agree on a good strategy?

As a more challenging question, what if the hats can have $100$ known different colors instead of $2$?

$\textbf{Flip the Grid}$

Given two grids shown below, is it possible to transform ($a$) to ($b$) after a series of operations? In each operation, one can change all the signs in either one entire row or one entire column.

There are $100$ lights lined up in a long room. Each light has its own switch and is currently off. The room has an entry door and an exit door. There are $100$ people lined up outside the entry door. Each light is numbered consecutively from $1$ to $100$. So is each person.

Person No. $1$ enters the room, switches on every light, and exits. Person No. $2$ enters and flips the switch on every second light (i.e. turn off lights $2$, $4$, $6$...). Person No. $3$ enters and flips the switch on every third light (i.e. toggle lights $3$, $6$, $9$...). This continues until all $100$ people have passed through the room. How many of the lights are on at the end?

$\textbf{Eel}$

An eel is a polyomino formed by a path of unit squares that makes two turns in opposite directions. (Note that this means the smallest eel has four cells.) For example, the polyomino shown below is an eel. What is the maximum area of a $1000\times 1000$ grid of unit squares that can be covered by eels without overlap?

$\textbf{Key Set}$

A sensitive location is protected by a door with multiple locks. This place has $11$ workers. The regulation requires that any combination of six workers can open all the locks, but any combination of five cannot. What is the minimal number of locks and how to distribute the keys?

$\textbf{Two Doormen}$

Two doormen are guarding two rooms. One room contains tons of gold and the other is empty. Among these two doormen, one is honest who always tells the truth and the other is a liar who always gives false answers. While they know each other well, you do not know who is honest and who is not. If you are given just one chance to ask one question to one of them, what can you do in order to find out which room contains the gold?

$\textbf{Toggler's Problem}$

Among a group of $100$ people, only one is a truth teller and the rest $99$ are togglers. A truth teller always tells the truth. A toggler will tell the truth and a lie in an alternating fashion. That is, after he or she tells the truth the first time, this person will tell a lie next time. However, if his or her first answer is false, then the next answer will be true. It is unknown whether a toggler's first answer is the truth or a lie.

If all these people know who is the truth teller, how many questions do you need to ask in order to identify the truth teller?

$\textbf{Connect the Lights}$

You are in a control room which has three switches. Each switch controls one of three lights in another room. Once you leave the control room, you can not touch the switches again. How can you figure out which switch controls which light?

$\textbf{What Bear}$

Joe leaves his campsite and hikes south for $3$ miles. He then turns east and hikes for $3$ miles. Finally he turns north and hikes for $3$ miles. At this moment he sees a bear inside his tent eating his food! What color is the bear?

$\textbf{Unique Number}$

What makes the number $8549176320$ unique?


text:
# Plotting a line over 'histogram-like' bins

I have to convert the following JPEG image into a TeX-equivilant diagram using tikz and pgfplots.

This is what I have sofar:

\documentclass[border=3mm]{standalone}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
width=\linewidth,
height=8cm,
ymin=0,
ytick={0,10,20,30,40,50,60,70,80,90,100},
ymax=100,
ylabel={Percentage},
symbolic x coords={0-15,15-59,60-64,65-69,70-74,75-79,80-84,85-89,90+},
xtick=data,
nodes near coords align={vertical}
]
\addplot[draw={rgb:red,36;green,105;blue,179},ultra thick, smooth, nodes near coords]
coordinates {(0-15,0.7) (15-59,0.6) (60-64,1.9) (65-69,3.0) (70-74,5.0) (75-79,9.8) (80-84,21.0) (85-89,38.2) (90+,64.4)};
\end{axis}
\end{tikzpicture}
\end{document}

and this is what it looks like:

The graph almost resembles the aforementioned JPEG image. However I am having problems with the following:

A. How do I draw vertical lines between the bins as seen in the first JPEG Image?

B. How do I remove the x and y axes ticks as seen in my TeX Graph?

C. How do I visualize the big blue dots at the numbered value as seen in the first JPEG Image?

D. How do I shift the visualized numbers at the coords slightly upwards so that they do not interfere with the drawn line?

Any help would be greatly appreciated...

B. Use option tick style={draw=none}.

A. For the gray lines use a grid at minor ticks:

minor x tick num=1,
minor ytick={10,30,90},
grid=minor,

C. Use option mark* for the plot and change draw to color.

\addplot[color={rgb:red,36;green,105;blue,179},mark=*,...] ...

D. Use option nodes near coords style to shift the numbers and change the color back to black:

nodes near coords style={yshift=1em,black},

Additionally I would use a smaller font size.

Code:

\documentclass[border=3mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.14}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
width=\linewidth,
height=8cm,
ymin=0,
ytick={0,10,20,30,40,50,60,70,80,90,100},
ymax=100,
ylabel={Percentage},
symbolic x coords={0-15,15-59,60-64,65-69,70-74,75-79,80-84,85-89,90+},
xtick=data,
font=\footnotesize,
tick style={draw=none},
minor x tick num=1,
minor ytick={10,30,90},
grid=minor,
nodes near coords style={yshift=1em,black},
]
\addplot[color={rgb:red,36;green,105;blue,179},mark=*,ultra thick, smooth, nodes near coords]
coordinates {(0-15,0.7) (15-59,0.6) (60-64,1.9) (65-69,3.0) (70-74,5.0) (75-79,9.8) (80-84,21.0) (85-89,38.2) (90+,64.4)};
\end{axis}
\end{tikzpicture}
\end{document}

• for B draw=none is better – percusse Nov 8 '16 at 18:10
• @percusse Ok, I have changed this. – esdd Nov 8 '16 at 18:27


text:
# How to extend Dirichlet distribution to Dirichlet process

For a Dirichlet process, there are two parameter $\alpha$ and $H$, and the Dirichlet process $X$ is defined as $$(X(B_1),\cdots,X(B_n))\sim Dir(\alpha H(B_1),\cdots,\alpha H(B_n))$$ where$\{B_i\}_{i=1}^n$is a partition of the measurable space.

While the Dirichlet distribution is like that $$f(x_1,x_2, \cdots ,x_{K - 1};\alpha _1,\alpha _2, \cdots ,\alpha _{K - 1},\alpha _K) = \frac{1}{B(\vec \alpha )}\prod\limits_{i = 1}^K x_i^{\alpha_i - 1}$$

My question is:

If I want to view the Dirichlet distribution as a special case of the Dirichlet process, then how should I set the parameters $\alpha$ and $H$ in the definition of Dirichlet process?

Here $\alpha$ is a number, and $H$ is a measure.
I believe that you should take a finite space, say, $\{1,...,K\}$. Then for any partition of $\{1,...,K\}$, your first formula will hold. $H$ measures the asymmetry of your Dirichlet process, as the parameters $\alpha_1,...,\alpha_K$ do in the second formula.
• if I partition {1,...,K} into {1},{2}...,{K}, should $\alpha$ equals to $\sum _{i=1}^K \alpha_i$ and $H(i)$ equals to $\alpha_i/\alpha$ ? – henrysupercool Sep 18 '14 at 0:59


text:
# Congruence uncertainty

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• Apr 24th 2010, 09:53 AM
empressA88
Congruence uncertainty
Suppose f(x)and g(x)∈Z_7 [x]. Suppose also that f(x)≡g(x)(mod x). What can we say about f(0)and g(0)?

With this problem i am jus not sure if i am over thinking it because we have done congruences with ideals and in F[x].. would fall under just dealing with F[x]
• Apr 24th 2010, 12:50 PM
HallsofIvy
Quote:

Originally Posted by empressA88
Suppose f(x)and g(x)∈Z_7 [x]. Suppose also that f(x)≡g(x)(mod x). What can we say about f(0)and g(0)?

With this problem i am jus not sure if i am over thinking it because we have done congruences with ideals and in F[x].. would fall under just dealing with F[x]

f and g are 7 th degree polynomials in in x. Saying that they are congruent "mod x" f- g is a multiple x which, in turn, means f- g has no constant term.
• Apr 24th 2010, 02:15 PM
empressA88
is that f - g, or are you say f and g?
• Apr 24th 2010, 05:52 PM
chiph588@
Quote:

Originally Posted by HallsofIvy
f and g are 7 th degree polynomials in in x. Saying that they are congruent "mod x" f- g is a multiple x which, in turn, means f- g has no constant term.

Doesn't $f,g\in\mathbb{Z}_7[x]$ mean $f$ and $g$ are polynomials with coefficients in $\mathbb{Z}_7$?

If so then $f\equiv g\bmod{x} \implies x\mid f-g\implies$ the constant term of $f-g\equiv0\bmod{7}$.

But this constant term is equal to $f(0)-g(0)$. Therefore $f(0)\equiv g(0) \bmod{7}$.