A Locus in Human Extrastriate Cortex for Visual Shape Analysis
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Descartes and Coordinate, Or “Analytic” Geometry April 19 and 21, 2017
MONT 107Q { Thinking about Mathematics Discussion { Descartes and Coordinate, or \Analytic" Geometry April 19 and 21, 2017 Background In 1637, the French philosopher and mathematician Ren´eDescartes published a pamphlet called La G´eometrie as one of a series of discussions about particular sciences intended apparently as illustrations of his general ideas expressed in his work Discours de la m´ethode pour bien conduire sa raison, et chercher la v´erit´edans les sciences (in English: Discourse on the method of rightly directing one's reason and searching for truth in the sciences). This work introduced other mathematicians to a new way of dealing with geometry. Via the introduction (at first only in a rather indirect way) of numerical coordinates to describe points in the plane, Descartes showed that it was possible to define geometric objects by means of algebraic equations and to apply techniques from algebra to deduce geometric properties of those objects. He illustrated his new methods first by considering a problem first studied by the ancient Greeks after the time of Euclid: Given three (or four) lines in the plane, find the locus of points P that satisfy the relation that the square of the distance from P to the first line (or the product of the distances from P to the first two of the lines) is equal to the product of the distances from P to the other two lines. The resulting curves are called three-line or four-line loci depending on which case we are considering. For instance, the locus of points satisfying the condition above for the four lines x + 2 = 0; x − 1 = 0; y + 1 = 0; y − 1 = 0 is shown at the top on the back of this sheet. -
Chapter 14 Locus and Construction
14365C14.pgs 7/10/07 9:59 AM Page 604 CHAPTER 14 LOCUS AND CONSTRUCTION Classical Greek construction problems limit the solution of the problem to the use of two instruments: CHAPTER TABLE OF CONTENTS the straightedge and the compass.There are three con- struction problems that have challenged mathematicians 14-1 Constructing Parallel Lines through the centuries and have been proved impossible: 14-2 The Meaning of Locus the duplication of the cube 14-3 Five Fundamental Loci the trisection of an angle 14-4 Points at a Fixed Distance in the squaring of the circle Coordinate Geometry The duplication of the cube requires that a cube be 14-5 Equidistant Lines in constructed that is equal in volume to twice that of a Coordinate Geometry given cube. The origin of this problem has many ver- 14-6 Points Equidistant from a sions. For example, it is said to stem from an attempt Point and a Line at Delos to appease the god Apollo by doubling the Chapter Summary size of the altar dedicated to Apollo. Vocabulary The trisection of an angle, separating the angle into Review Exercises three congruent parts using only a straightedge and Cumulative Review compass, has intrigued mathematicians through the ages. The squaring of the circle means constructing a square equal in area to the area of a circle. This is equivalent to constructing a line segment whose length is equal to p times the radius of the circle. ! Although solutions to these problems have been presented using other instruments, solutions using only straightedge and compass have been proven to be impossible. -
Analytic Geometry
INTRODUCTION TO ANALYTIC GEOMETRY BY PEECEY R SMITH, PH.D. N PROFESSOR OF MATHEMATICS IN THE SHEFFIELD SCIENTIFIC SCHOOL YALE UNIVERSITY AND AKTHUB SULLIVAN GALE, PH.D. ASSISTANT PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF ROCHESTER GINN & COMPANY BOSTON NEW YORK CHICAGO LONDON COPYRIGHT, 1904, 1905, BY ARTHUR SULLIVAN GALE ALL BIGHTS RESERVED 65.8 GINN & COMPANY PRO- PRIETORS BOSTON U.S.A. PEE FACE In preparing this volume the authors have endeavored to write a drill book for beginners which presents, in a manner conform- ing with modern ideas, the fundamental concepts of the subject. The subject-matter is slightly more than the minimum required for the calculus, but only as much more as is necessary to permit of some choice on the part of the teacher. It is believed that the text is complete for students finishing their study of mathematics with a course in Analytic Geometry. The authors have intentionally avoided giving the book the form of a treatise on conic sections. Conic sections naturally appear, but chiefly as illustrative of general analytic methods. Attention is called to the method of treatment. The subject is developed after the Euclidean method of definition and theorem, without, however, adhering to formal presentation. The advan- tage is obvious, for the student is made sure of the exact nature of each acquisition. Again, each method is summarized in a rule stated in consecutive steps. This is a gain in clearness. Many illustrative examples are worked out in the text. Emphasis has everywhere been put upon the analytic side, that is, the student is taught to start from the equation. -
Analysis of the Cut Locus Via the Heat Kernel
Unspecified Book Proceedings Series Analysis of the Cut Locus via the Heat Kernel Robert Neel and Daniel Stroock Abstract. We study the Hessian of the logarithm of the heat kernel to see what it says about the cut locus of a point. In particular, we show that the cut locus is the set of points at which this Hessian diverges faster than t−1 as t & 0. In addition, we relate the rate of divergence to the conjugacy and other structural properties. 1. Introduction Our purpose here is to present some recent research connecting behavior of heat kernel to properties of the cut locus. The unpublished results mentioned below constitute part of the first author’s thesis, where they will proved in detail. To explain what sort of results we have in mind, let M be a compact, connected Riemannian manifold of dimension n, and use pt(x, y) to denote the heat kernel for 1 the heat equation ∂tu = 2 ∆u. As a special case of a well-known result due to Varadhan, Et(x, y) ≡ −t log pt(x, y) → E(x, y) uniformly on M × M as t & 0, where E is the energy function, given in terms of the Riemannian distance by 1 2 E(x, y) = 2 dist(x, y) . Thus, (x, y) Et(x, y) can be considered to be a geomet- rically natural mollification of (x, y) E(x, y). In particular, given a fixed x ∈ M, we can hope to learn something about the cut locus Cut(x) of x by examining derivatives of y Et(x, y). -
Traveling Cortical Netwaves Compose a Mindstream
bioRxiv preprint doi: https://doi.org/10.1101/705947; this version posted April 29, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC 4.0 International license. Traveling cortical netwaves compose a mindstream Ernst Rudolf M. Hülsmann 26 Rue de Bonn. CH-3186 Guin. Switzerland. [email protected] ABSTRACT The brain creates a physical response out of signals in a cascade of streaming transformations. These transfor- mations occur over networks, which have been described in anatomical, cyto-, myeloarchitectonic and functional research. The totality of these networks has been modelled and synthesised in phases across a continuous time- space-function axis, through ascending and descending hierarchical levels of association1-3 via changing coalitions of traveling netwaves4-6, where localised disorders might spread locally throughout the neighbouring tissues. This study quantified the model empirically with time-resolving functional magnetic resonance imaging of an imperative, visually-triggered, self-delayed, therefor double-event related response task. The resulting time series unfold in the range of slow cortical potentials the spatio-temporal integrity of a cortical pathway from the source of perception to the mouth of reaction in and out of known functional, anatomical and cytoarchitectonic networks. These pathways are consolidated in phase images described by a small vector matrix, which leads to massive simplification of cortical field theory and even to simple technical applications. INTRODUCTION On a first sight it seems to be self-evident that a local perturbation within the cortical sheet should spread locally. -
Mapping Striate and Extrastriate Visual Areas in Human Cerebral Cortex EDGAR A
Proc. Natl. Acad. Sci. USA Vol. 93, pp. 2382-2386, March 1996 Neurobiology Mapping striate and extrastriate visual areas in human cerebral cortex EDGAR A. DEYOE*, GEORGE J. CARMANt, PETER BANDETTINIt, SETH GLICKMAN*, JON WIESER*, ROBERT COX§, DAVID MILLER1, AND JAY NEITZ* *Department of Cellular Biology and Anatomy, and Biophysics Research Institute, The Medical College of Wisconsin, 8701 Watertown Plank Road, Milwaukee, WI 53226; tThe Salk Institute for Biological Studies, La Jolla, CA 92037; tMassachusetts General Hospital-NMR Center, Charlestown, MA 02129; §Biophysics Research Institute, The Medical College of Wisconsin, Milwaukee, WI 53226; and lBrown University, Providence, RI 02912 Communicated by Francis Crick The Salk Institute for Biological Sciences, San Diego, CA, November 14, 1995 (received for review August 2, 1995) ABSTRACT Functional magnetic resonance imaging represented by each active site in the brain (12). A similar (fMRI) was used to identify and map the representation of the technique has been described by Engel et al. (15). visual field in seven areas of human cerebral cortex and to To enhance activation of extrastriate cortex (12) and to help identify at least two additional visually responsive regions. maintain attention and arousal, the subject was required to The cortical locations of neurons responding to stimulation detect a small target appearing at a random position super- along the vertical or horizontal visual field meridia were imposed on the checkered hemifield or annulus. A new target charted on three-dimensional models of the cortex and on was presented every 2 sec and the subject activated one of two unfolded maps of the cortical surface. -
The Cut Locus of Riemannian Manifolds: a Surface of Revolution
KMITL Sci. Tech. J. Vol.16 No.1 Jan-Jun. 2016 The cut locus of Riemannian manifolds: a surface of revolution Minoru Tanaka Department of Mathematics, Tokai University, Hiratsuka City, Kanagawa Pref. 259- 1292, Japan Abstract This article reviews the structure theorems of the cut locus for very familiar surfaces of revolution. Some properties of the cut locus of a point of a Riemannian manifold are also discussed. Keywords: Riemannian manifolds, surface of revolution, homeomorphic 1. Definition of a cut point and the cut locus Let γ :[0,aM ] → denote a minimal geodesic segment emanating from a point p on a complete connected Riemannian manifold M. The end point γ (a) is called a cut point of p along the minimal geodesic segment γ if any geodesic extension γ :[0,bM ] → , where b > a, of γ is not minimal anymore. Definition 1.1 The cut locus Cp of a point p is the set of all cut points of p along minimal geodesic segments emanating from p. It is very difficult to determine the structure of the cut locus of a point in a Riemannian manifold. The cut locus for a smooth surface is not a graph anymore, although it was proved by Myers in [1] and [2] that the cut locus of a point in a compact real analytic surface is a finite graph. In fact, Gluck and Singer [3] proved that there exists a 2-sphere of revolution admitting a cut locus with infinitely many branches. Their result implies that one cannot improve the following Theorem 1.3 without any additional assumption. -
ANALYTIC GEOMETRY - GOOD PROBLEMS from Alex Pintilie
ANALYTIC GEOMETRY - GOOD PROBLEMS from Alex Pintilie Together: 1)a) Consider the fixed points A(-a,0) and B(a,0). Find the equation of the locus of the points M with the property MA2 - MB2 = k, where k is a constant. Identify the locus. b) Consider the points A (-2, 0) B(2,0). Find the equation of the locus of the points M with the property MA2 - MB2 = 12. Draw a diagram. 2)a) Consider the fixed points A(-a,0), B(a,0). Find the equation of the locus of the points M with the property MA = k MB, where k is a constant. Identify the locus. b) Consider the fixed points A(-4,0), B(4,0). Find the equation of the locus of the points M with the property MA = 2 MB. Draw a diagram. 2 2 2 2 3) Consider the concentric circles with equations x y 4 and x y 9. A radius from the centre O intersects the inner circle at P and the outer circle at Q. The line parallel to the x- axis through P meets the line parallel to the y-axis through Q at the point R. Prove that R lies on x2 y2 1 the ellipse 9 4 . Class work: 1) Let P be a variable point on the line x=1. We construct the triangle OPQ with O the origin, O=90o, OP=OQ. Find the locus of the point Q. A 2) For the equilateral triangle ABC, side BC is the diameter of a semicircle. -
Cytoarchitectonic Mapping of the Human Dorsal Extrastriate Cortex
Brain Struct Funct (2013) 218:157–172 DOI 10.1007/s00429-012-0390-9 ORIGINAL ARTICLE Cytoarchitectonic mapping of the human dorsal extrastriate cortex Milenko Kujovic • Karl Zilles • Aleksandar Malikovic • Axel Schleicher • Hartmut Mohlberg • Claudia Rottschy • Simon B. Eickhoff • Katrin Amunts Received: 23 September 2011 / Accepted: 31 January 2012 / Published online: 22 February 2012 Ó The Author(s) 2012. This article is published with open access at Springerlink.com Abstract The dorsal visual stream consists of several superimposed in the anatomical MNI space, and probabi- functionally specialized areas, but most of their cytoar- listic maps were calculated. They show a relatively high chitectonic correlates have not yet been identified in the intersubject variability in volume and position. Based on human brain. The cortex adjacent to Brodmann area 18/V2 their location and neighborhood relationship, areas hOc3d was therefore analyzed in serial sections of ten human post- and hOc4d are putative anatomical substrates of function- mortem brains using morphometrical and multivariate ally defined areas V3d and V3a, a hypothesis that can now statistical analyses for the definition of areal borders. be tested by comparing probabilistic cytoarchitectonic Two previously unknown cytoarchitectonic areas (hOc3d, maps and activation studies of the living human brain. hOc4d) were detected. They occupy the medial and, to a smaller extent, lateral surface of the occipital lobe. The Keywords Cytoarchitecture Á Visual cortex Á hOc3d Á larger area, hOc3d, is located dorso-lateral to area V2 in the hOc4d Á V3 Á V3a Á Probabilistic maps Á Human brain atlas region of superior and transverse occipital, as well as parieto-occipital sulci. -
On the Compatibility of Derived Structures on Critical Loci
University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2019 On The Compatibility Of Derived Structures On Critical Loci Antonijo Mrcela University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Mathematics Commons Recommended Citation Mrcela, Antonijo, "On The Compatibility Of Derived Structures On Critical Loci" (2019). Publicly Accessible Penn Dissertations. 3441. https://repository.upenn.edu/edissertations/3441 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/3441 For more information, please contact [email protected]. On The Compatibility Of Derived Structures On Critical Loci Abstract We study the problem of compatibility of derived structures on a scheme which can be presented as a critical locus in more than one way. We consider the situation when a scheme can be presented as the critical locus of a function w∈?(S) and as the critical locus of the restriction w|ₓ∈?(X) for some smooth subscheme X⊂S. In the case when S is the total space of a vector bundle over X, we prove that, under natural assumptions, the two derived structures coincide. We generalize the result to the case when X is a quantized cycle in S and also give indications how to proceed when X⊂S is a general closed embedding. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Mathematics First Advisor Tony Pantev Keywords Batalin–Vilkovisky formalism, compatibility of derived structures, -
Treatise on Conic Sections
OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA fY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA 9se ? uNliEHSITY OF CUIFORHU LIBRARY OF THE UNIVERSITY OF CUIfORHlA UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA APOLLONIUS OF PERGA TREATISE ON CONIC SECTIONS. UonDon: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE RIARIA LANE. ©Inaooh): 263, ARGYLE STREET. lLftp>is: P. A. BROCKHAUS. 0fto goTit: MACMILLAN AND CO. ^CNIVERSITT• adjViaMai/u Jniuppus 9lnlMus%^craticus,naufmato cum ejcctus aa UtJ ammaScrudh Geainetncn fJimmta defer,pta, cxcUnwimJIe cnim veiiigia video. coMXiXcs ita Jiatur^hcnc fperemus,Hominnm APOLLONIUS OF PEEGA TEEATISE ON CONIC SECTIONS EDITED IN MODERN NOTATION WITH INTRODUCTIONS INCLUDING AN ESSAY ON THE EARLIER HISTORY OF THE SUBJECT RV T. L. HEATH, M.A. SOMETIME FELLOW OF TRINITY COLLEGE, CAMBRIDGE. ^\€$ .•(', oh > , ' . Proclub. /\b1 txfartl PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. MANIBUS EDMUNDI HALLEY D. D. D. '" UNIVERSITT^ PREFACE. TT is not too much to say that, to the great majority of -*- mathematicians at the present time, Apollonius is nothing more than a name and his Conies, for all practical purposes, a book unknown. Yet this book, written some twenty-one centuries ago, contains, in the words of Chasles, " the most interesting properties of the conies," to say nothing of such brilliant investigations as those in which, by purely geometrical means, the author arrives at what amounts to the complete determination of the evolute of any conic. The general neglect of the " great geometer," as he was called by his contemporaries on account of this very work, is all the more remarkable from the contrast which it affords to the fate of his predecessor Euclid ; for, whereas in this country at least the Elements of Euclid are still, both as regards their contents and their order, the accepted basis of elementary geometry, the influence of Apollonius upon modern text-books on conic sections is, so far as form and method are concerned, practically nil. -
Straight Lines
Chapter 10 STRAIGHT LINES 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope of the line and is denoted by m. The slope of a line passing through points P(x1, y1) and Q (x2, y2) is given by y− y m =tan θ = 2 1 x2− x 1 10.1.2 Angle between two lines The angle θ between the two lines having slopes m1 and m2 is given by ()m− m tanθ= ± 1 2 1+ m1 m 2 m1− m 2 If we take the acute angle between two lines, then tan θ = 1+ m1 m 2 If the lines are parallel, then m1 = m2. If the lines are perpendicular, then m1m2 = – 1. 10.1.3 Collinearity of three points If three points P (h, k), Q (x1, y1) and R (x2,y2) y− k y − y are such that slope of PQ = slope of QR, i.e., 1= 2 1 x1− h x 2 − x 1 or (h – x1) (y2 – y1) = (k – y1) (x2 – x1) then they are said to be collinear. 10.1.4 Various forms of the equation of a line (i) If a line is at a distance a and parallel to x-axis, then the equation of the line is y = ± a. (ii) If a line is parallel to y-axis at a distance b from y-axis then its equation is x = ± b 18/04/18 166 EXEMPLAR PROBLEMS – MATHEMATICS (iii) Point-slope form : The equation of a line having slope m and passing through the point (x0, y0) is given by y – y0 = m (x – x0) (iv) Two-point-form : The equation of a line passing through two points (x1, y1) and (x2, y2) is given by y2− y 1 y – y1 = (x – x1) x2− x 1 (v) Slope intercept form : The equation of the line making an intercept c on y-axis and having slope m is given by y = mx + c Note that the value of c will be positive or negative as the intercept is made on the positive or negative side of the y-axis, respectively.