The No Retraction Theorem and a Generalization
... with the compactness of |K|, means that it is uniformly continuous. Let δ be such that if |x1 − x2 | < δ, then |r0 (x1 ) − r0 (x2 )| < 1/8. In particular, the images of triangles with diameter less than δ will have diameter less than 1/8. Let L be a subdivision of K all of whose simplices have diame ...
... with the compactness of |K|, means that it is uniformly continuous. Let δ be such that if |x1 − x2 | < δ, then |r0 (x1 ) − r0 (x2 )| < 1/8. In particular, the images of triangles with diameter less than δ will have diameter less than 1/8. Let L be a subdivision of K all of whose simplices have diame ...
Foundation - cndblessltd
... Solve more complex ratio and proportion problems, such as sharing out money between two groups in the ratio of their numbers ...
... Solve more complex ratio and proportion problems, such as sharing out money between two groups in the ratio of their numbers ...
Primary and Reciprocal Trig Ratios
... There are several special angles we can memorize the ratios for (Special triangles in grade 11). We refer to these as the related acute angles (R.A.A or R ) as we move forward in this unit. We memorize the ratios of these acute angles. ...
... There are several special angles we can memorize the ratios for (Special triangles in grade 11). We refer to these as the related acute angles (R.A.A or R ) as we move forward in this unit. We memorize the ratios of these acute angles. ...
Grade 11 or 12 Pre
... selecting and applying strategies (i.e., common factoring, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem) Determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), the connection between the real ro ...
... selecting and applying strategies (i.e., common factoring, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem) Determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), the connection between the real ro ...
Document
... B. $9.90 C. $16.50 D. $26.40 7. Emil bought a camera for $268.26, including tax. He made a down payment of $12.00 and paid the balance in 6 equal monthly payments. What was Emil’s monthly payment for this camera? A. $42.71 B. $44.71 C. $46.71 D. $56.71 8. A food company reduced the amount of salt in ...
... B. $9.90 C. $16.50 D. $26.40 7. Emil bought a camera for $268.26, including tax. He made a down payment of $12.00 and paid the balance in 6 equal monthly payments. What was Emil’s monthly payment for this camera? A. $42.71 B. $44.71 C. $46.71 D. $56.71 8. A food company reduced the amount of salt in ...
Math 396. The topologists` sine curve
... S = {(x, y) ∈ R2 | y = sin(1/x)} ∪ ({0} × [−1, 1]) ⊆ R2 , so S is the union of the graph of y = sin(1/x) over x > 0, along with the interval [−1, 1] in the y-axis. Geometrically, the graph of y = sin(1/x) is a wiggly path that oscillates more and more frequently (between the lines y = ±1) as we get ...
... S = {(x, y) ∈ R2 | y = sin(1/x)} ∪ ({0} × [−1, 1]) ⊆ R2 , so S is the union of the graph of y = sin(1/x) over x > 0, along with the interval [−1, 1] in the y-axis. Geometrically, the graph of y = sin(1/x) is a wiggly path that oscillates more and more frequently (between the lines y = ±1) as we get ...
Chapter 6 Halving segments
... First we describe the construction, then we verify its correctness, and finally we count the middle-level vertices. The construction We construct an infinite sequence L0 , L1 , L2 , . . . of sets of non-vertical lines in the plane in general position. Every line in every Lm , m ≥ 0, is of one of two ...
... First we describe the construction, then we verify its correctness, and finally we count the middle-level vertices. The construction We construct an infinite sequence L0 , L1 , L2 , . . . of sets of non-vertical lines in the plane in general position. Every line in every Lm , m ≥ 0, is of one of two ...
Course Outline
... describe the different types of intrinsic curvature as they relate to the angle-sums of triangles explain why the Euclidean property of similarity does not hold on spherical surfaces compare and contrast the Euclidean properties with spherical properties compare and contrast spherical geometry with ...
... describe the different types of intrinsic curvature as they relate to the angle-sums of triangles explain why the Euclidean property of similarity does not hold on spherical surfaces compare and contrast the Euclidean properties with spherical properties compare and contrast spherical geometry with ...
9.3 Hyperbolas
... Hyperbola set of all points (x, y) in a plane, the difference of whose distances from two distinct fixed points (foci) is constant. ...
... Hyperbola set of all points (x, y) in a plane, the difference of whose distances from two distinct fixed points (foci) is constant. ...
Connectivity, Devolution, and Lacunae in
... occurs when Sn is a disc of fixed radius rn . In this case the mosaic is formed by n identical copies of discs of radius rn distributed randomly in the unit disc. The graph that obtains is then an ordinary undirected graph with points connected by edges if, and only if, they are within Euclidean dis ...
... occurs when Sn is a disc of fixed radius rn . In this case the mosaic is formed by n identical copies of discs of radius rn distributed randomly in the unit disc. The graph that obtains is then an ordinary undirected graph with points connected by edges if, and only if, they are within Euclidean dis ...
homework 1
... 1. Prove that a topological manifold M is connected if and only if it is path-connected. Define an equivalence relation on Rn+1 \ {0} by x ∼ y ⇐⇒ y = tx, t ∈ R − {0}. The n-dimensional, real projective space RPn is the quotient of Rn+1 by the above equivalence relation. 2. Prove that RPn is second c ...
... 1. Prove that a topological manifold M is connected if and only if it is path-connected. Define an equivalence relation on Rn+1 \ {0} by x ∼ y ⇐⇒ y = tx, t ∈ R − {0}. The n-dimensional, real projective space RPn is the quotient of Rn+1 by the above equivalence relation. 2. Prove that RPn is second c ...
Relating Graph Thickness to Planar Layers and Bend Complexity
... we now draw the edges (vj , vj+1 ) one after another. Assume without loss of generality that x(pj ) < x(pj+1 ). We call a point p ∈ S between pj and pj+1 a visited point if the corresponding vertex v appears in v1 , . . . , vj , i.e., v has already been placed at p. We draw an x-monotone polygonal c ...
... we now draw the edges (vj , vj+1 ) one after another. Assume without loss of generality that x(pj ) < x(pj+1 ). We call a point p ∈ S between pj and pj+1 a visited point if the corresponding vertex v appears in v1 , . . . , vj , i.e., v has already been placed at p. We draw an x-monotone polygonal c ...
Here
... (a) X = Rn and the subset U ⊂ X is open if, for any x ∈ U , there is a real > 0 such that the open Euclidean ball B n (x, ) of radius and centred at x is contained in U . (b) X = Rn and the subset U ⊂ X is open if, for any x ∈ X \ U , there is a real > 0 such that the open Euclidean ball B n ...
... (a) X = Rn and the subset U ⊂ X is open if, for any x ∈ U , there is a real > 0 such that the open Euclidean ball B n (x, ) of radius and centred at x is contained in U . (b) X = Rn and the subset U ⊂ X is open if, for any x ∈ X \ U , there is a real > 0 such that the open Euclidean ball B n ...
on maps: continuous, closed, perfect, and with closed graph
... THEOREM 7. Let f:X Y be closed with closed (compact) fibers, where X is regular (Hausdorff) and one of the conditions (a), (b), (c) in theorem 4 is satisfied. Then f is continuous(perfect). Combining theorems and 4, we obtain the following relationship between continuous maps and maps with closed gr ...
... THEOREM 7. Let f:X Y be closed with closed (compact) fibers, where X is regular (Hausdorff) and one of the conditions (a), (b), (c) in theorem 4 is satisfied. Then f is continuous(perfect). Combining theorems and 4, we obtain the following relationship between continuous maps and maps with closed gr ...
Italo Jose Dejter
Italo Jose Dejter (Bahia Blanca, 1939) is an Argentine-born American mathematician of Bessarabian Jewish descent. He is a professor at the University of Puerto Rico (UPRRP) since 1984 and has conducted research in mathematics, particularly in areas that include algebraic topology, differential topology, graph theory, coding theory and design theory.He has an Erdos number of 2 since 1993.Dejter completed the Licentiate degree in Mathematics from University of Buenos Aires in 1967, and the Ph.D. degree in Mathematics from Rutgers University in 1975 under the supervision of Ted Petrie. He was a professor atFederal University of Santa Catarina, Brazil, from 1977 to 1984. Dejter has been a visiting scholar at a number of research institutions, including University of São Paulo,Instituto Nacional de Matemática Pura e Aplicada, Federal University of Rio Grande do Sul,University of Cambridge,National Autonomous University of Mexico, Simon Fraser University, University of Victoria, New York University, University of Illinois at Urbana–Champaign, McMaster University,DIMACS, Autonomous University of Barcelona, Technical University of Denmark, Auburn University, Polytechnic University of Catalonia, Technical University of Madrid, Charles University, Ottawa University, Simón Bolívar University, etc. The sections below describe the relevance of Dejter's work in the research areas mentioned in the first paragraph above, or in the box to the right.