A Formula for the Intersection Angle of Backbone Arcs with the
... side of the line, all of whose points are that distance from the given line. A Petrie polygon is a polygonal path of edges in a regular tessellation traversed by alternately taking the left-most and right-most edge at each vertex. ...
... side of the line, all of whose points are that distance from the given line. A Petrie polygon is a polygonal path of edges in a regular tessellation traversed by alternately taking the left-most and right-most edge at each vertex. ...
274 Curves on Surfaces, Lecture 5
... as being composed of two ideal triangles, we can slide the ideal triangles against each other by moving the opposite vertex, and this does not change the triangles themselves up to isometry. Ordinary Euclidean triangles do not have this property. We would like to make ideal geometry look more like o ...
... as being composed of two ideal triangles, we can slide the ideal triangles against each other by moving the opposite vertex, and this does not change the triangles themselves up to isometry. Ordinary Euclidean triangles do not have this property. We would like to make ideal geometry look more like o ...
Non-Euclidean Geometry
... A circle can be constructed when a point for its centre and a distance for its radius are given. ...
... A circle can be constructed when a point for its centre and a distance for its radius are given. ...
The Euler characteristic of an even
... answer this yet, we will comment on it anyway. In the graph case, where traditional tensor calculus is absent, it is natural to look at the Einstein tensor T (v, e) = R(e) − S(v) which involves the Ricci curvature R, the average of wheel graph curvatures through an edge e and scalar curvature S(v) t ...
... answer this yet, we will comment on it anyway. In the graph case, where traditional tensor calculus is absent, it is natural to look at the Einstein tensor T (v, e) = R(e) − S(v) which involves the Ricci curvature R, the average of wheel graph curvatures through an edge e and scalar curvature S(v) t ...
Hyperbolic
... Lines AC and BD are then parallel by Proposition 27 and have a common perpendicular by Theorem H37. Let the common perpendicular meet lines AC and BD at points F and G, respectively, and let E be the midpoint of F G. Then by Theorem H38, E is on line AB and AE = EB by one of the homework problems. S ...
... Lines AC and BD are then parallel by Proposition 27 and have a common perpendicular by Theorem H37. Let the common perpendicular meet lines AC and BD at points F and G, respectively, and let E be the midpoint of F G. Then by Theorem H38, E is on line AB and AE = EB by one of the homework problems. S ...
Partial Metric Spaces - Department of Computer Science
... Then (X, p) is a partial metric space, and p(x, x) = |x|. Conversely, if (X, p) is a partial metric space, then (X, d p , | · |), where (as before) d p (x, y) = 2 p(x, y) − p(x, x) − p(y, y) and |x| = p(x, x), is a weighted metric space. It can be seen that from either space we can move to the other ...
... Then (X, p) is a partial metric space, and p(x, x) = |x|. Conversely, if (X, p) is a partial metric space, then (X, d p , | · |), where (as before) d p (x, y) = 2 p(x, y) − p(x, x) − p(y, y) and |x| = p(x, x), is a weighted metric space. It can be seen that from either space we can move to the other ...
Exercises - Durham University
... Starred problems due on Tuesday, 10 February 13.1. Draw in each of the two conformal models (Poincaré disc and upper half-plane): (a) (b) (c) (d) (e) ...
... Starred problems due on Tuesday, 10 February 13.1. Draw in each of the two conformal models (Poincaré disc and upper half-plane): (a) (b) (c) (d) (e) ...
Uniform Continuity in Fuzzy Metric Spaces
... (1) For each fuzzy metric space (Y, N, ⋆) any continuous mapping from (X,τM ) to (Y, τN ) is uniformly continuous as a mapping from (X, M, ∗) to (Y, N, ⋆). (2) Every real valued continuous function on (X, τM ) is R-uniformly continuous on (X, M, ∗). (3) Every real valued continuous function on (X, τ ...
... (1) For each fuzzy metric space (Y, N, ⋆) any continuous mapping from (X,τM ) to (Y, τN ) is uniformly continuous as a mapping from (X, M, ∗) to (Y, N, ⋆). (2) Every real valued continuous function on (X, τM ) is R-uniformly continuous on (X, M, ∗). (3) Every real valued continuous function on (X, τ ...
Unit 9 − Non-Euclidean Geometries When Is the Sum of the
... Euclid’s fifth postulate (the parallel postulate) is usually stated as follows: “Through a point not on a line, there exists exactly one line parallel to the line.” This version of the parallel postulate is known as Playfair’s postulate (1795). It is logically equivalent to Euclid’s original fifth p ...
... Euclid’s fifth postulate (the parallel postulate) is usually stated as follows: “Through a point not on a line, there exists exactly one line parallel to the line.” This version of the parallel postulate is known as Playfair’s postulate (1795). It is logically equivalent to Euclid’s original fifth p ...
METRIC SPACES AND UNIFORM STRUCTURES
... function xy on the factor space establishes a homeomorphism between this space and the real line. On the other hand, there is no natural way to define a metric in the factor space based on the Euclidean metric in the plane. Any two elements of the factor contain points arbitrary close to each other ...
... function xy on the factor space establishes a homeomorphism between this space and the real line. On the other hand, there is no natural way to define a metric in the factor space based on the Euclidean metric in the plane. Any two elements of the factor contain points arbitrary close to each other ...
1 An introduction to homotopy theory
... A cell complex, otherwise known as a CW complex, is a topological space constructed from disks (called cells), step by step increasing in dimension. The basic procedure in the construction is called “attaching an n-cell”. An n-cell is the interior en of a closed disk Dn of dimension n. How to attach ...
... A cell complex, otherwise known as a CW complex, is a topological space constructed from disks (called cells), step by step increasing in dimension. The basic procedure in the construction is called “attaching an n-cell”. An n-cell is the interior en of a closed disk Dn of dimension n. How to attach ...
WHAT IS HYPERBOLIC GEOMETRY? Euclid`s five postulates of
... It is natural to ask what the geodesics of this metric are. That is, given distinct points in Hn , is there a smooth curve realizing the distance between them? In fact, the geodesics in Hn are precisely the lines defined in Section 1. One can show that this would not be the case if we measured the l ...
... It is natural to ask what the geodesics of this metric are. That is, given distinct points in Hn , is there a smooth curve realizing the distance between them? In fact, the geodesics in Hn are precisely the lines defined in Section 1. One can show that this would not be the case if we measured the l ...
Topology Proceedings - Topology Research Group
... pseudo-sequence-covering compact mappings equivalent to sequentially-quotient compact mappings? This question is still open, which arouses our interest in the relations between pseudo-sequencecovering compact images and sequentially-quotient compact images for these metric domains. In [16], P. Yan p ...
... pseudo-sequence-covering compact mappings equivalent to sequentially-quotient compact mappings? This question is still open, which arouses our interest in the relations between pseudo-sequencecovering compact images and sequentially-quotient compact images for these metric domains. In [16], P. Yan p ...
ON THE EXISTENCE OF UNIVERSAL COVERING SPACES FOR
... That (1) and (3) are equivalent is not surprising since a connected, locally path connected space has a universal cover if and only if it is LC1 (or, in another notation, semilocally simply connected) and there is a well-developed theory of ANR-like extensions of maps from spaces of prescribed dimen ...
... That (1) and (3) are equivalent is not surprising since a connected, locally path connected space has a universal cover if and only if it is LC1 (or, in another notation, semilocally simply connected) and there is a well-developed theory of ANR-like extensions of maps from spaces of prescribed dimen ...
Geometric and Solid Modeling Problems - Visgraf
... understanding of algorithmic modeling and its relation with geometric and solid modeling” ...
... understanding of algorithmic modeling and its relation with geometric and solid modeling” ...
Probability of an Acute Triangle in the Two
... The statistical theory of shape can be traced back from 1893 when Charles Dodgson (Lewis Carroll) proposed the following question. Question: Find the probability that a triangle formed by choosing three points at random on an infinite plane would have an obtuse triangle. In [5], by introducing the C ...
... The statistical theory of shape can be traced back from 1893 when Charles Dodgson (Lewis Carroll) proposed the following question. Question: Find the probability that a triangle formed by choosing three points at random on an infinite plane would have an obtuse triangle. In [5], by introducing the C ...
15 the geometry of whales and ants non
... happen, but they would not be the least bit surprising to whales. The sum of the angles of a triangle is less than 180 degrees. Rectangles (four-sided figures with all right angles) do not exist; however, right-angled pentagons ...
... happen, but they would not be the least bit surprising to whales. The sum of the angles of a triangle is less than 180 degrees. Rectangles (four-sided figures with all right angles) do not exist; however, right-angled pentagons ...
Wu_Y_H
... • Quasi-local idea is to define gravitational energymomentum associated to a closed 2-surface • We examine (1) Komar integral, (2) Brown-York expression and (3) Dogan-Mason mass expression in ...
... • Quasi-local idea is to define gravitational energymomentum associated to a closed 2-surface • We examine (1) Komar integral, (2) Brown-York expression and (3) Dogan-Mason mass expression in ...
Remedial topology
... Prove that the quotient space is Hausdorff. Exercise 1.28. Let f : X −→ Y be a continuous map of topological spaces, with X compact. Prove that f (X) is also compact. Exercise 1.29. Let Z ⊂ Y be a compact subset of a Hausdorff topological space. Prove that it is closed. Exercise 1.30. Let f : X −→ Y ...
... Prove that the quotient space is Hausdorff. Exercise 1.28. Let f : X −→ Y be a continuous map of topological spaces, with X compact. Prove that f (X) is also compact. Exercise 1.29. Let Z ⊂ Y be a compact subset of a Hausdorff topological space. Prove that it is closed. Exercise 1.30. Let f : X −→ Y ...
properties of fuzzy metric space and its applications
... every F-open sphere S (A,r) contains element of G. which implies A is an F-adherent point of G. conversely if A is an F-adherent point of G. then wither A is an F-isolated point of G case d(A.G) =O hence ...
... every F-open sphere S (A,r) contains element of G. which implies A is an F-adherent point of G. conversely if A is an F-adherent point of G. then wither A is an F-isolated point of G case d(A.G) =O hence ...
Topology vs. Geometry
... As a mathematical branch, topology has its origin in the work of Henri Poincaré towards the end of the nineteenth century, but can also be linked to the great Gauss. In what is now called the GaussBonnet theorem, Gauss, and later Bonnet, showed that the sum of the curvature for every point on a ball ...
... As a mathematical branch, topology has its origin in the work of Henri Poincaré towards the end of the nineteenth century, but can also be linked to the great Gauss. In what is now called the GaussBonnet theorem, Gauss, and later Bonnet, showed that the sum of the curvature for every point on a ball ...
THE UNIFORMIZATION THEOREM AND UNIVERSAL COVERS
... of developing an intuitive view of what curvature is. Once we have built up this notion of curvature, we will introduce the Gauss-Bonnet theorem, the genus, and the Euler characteristic, and use these notions to show that the genus of a compact, orientable Riemann surface, thanks to its curvature, e ...
... of developing an intuitive view of what curvature is. Once we have built up this notion of curvature, we will introduce the Gauss-Bonnet theorem, the genus, and the Euler characteristic, and use these notions to show that the genus of a compact, orientable Riemann surface, thanks to its curvature, e ...
Topology Proceedings 7 (1982) pp. 279
... respect to {C n ; n E N} with Cn c Cn +1' by the proof of F. Siwiec [12; Proposition 2(a)], we show that f is a closed map without any topological property of F • n each Fn is now Lasnev, so is M. Then X is Lasnev. ...
... respect to {C n ; n E N} with Cn c Cn +1' by the proof of F. Siwiec [12; Proposition 2(a)], we show that f is a closed map without any topological property of F • n each Fn is now Lasnev, so is M. Then X is Lasnev. ...
WHAT IS HYPERBOLIC GEOMETRY? - School of Mathematics, TIFR
... H is called the Poincare upper half plane in honour of the great French mathematician who discovered it. Motivation, an aside: Without any motivation, the model described above seems to have come out of thin air. This is the only place where, during the course of this article, some more mathematical ...
... H is called the Poincare upper half plane in honour of the great French mathematician who discovered it. Motivation, an aside: Without any motivation, the model described above seems to have come out of thin air. This is the only place where, during the course of this article, some more mathematical ...
Baire Spaces and the Wijsman Topology
... Theorem (Zsilinszky, 20??): Let X be an almost locally separable metrizable space. Then X is Baire if and only if (F(X),Twd) is Baire for each compatible metric d on X. A space is almost locally separable, provided that the set of points of local separability is dense. In a metrizable space, this ...
... Theorem (Zsilinszky, 20??): Let X be an almost locally separable metrizable space. Then X is Baire if and only if (F(X),Twd) is Baire for each compatible metric d on X. A space is almost locally separable, provided that the set of points of local separability is dense. In a metrizable space, this ...