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Geometry and Topology

Frieze patterns

Q 1. A frieze pattern has one and only one direction of translation. The translation isometry is denoted by T. As we have noticed in lectures the symmetry group of the fundamental pattern consists of all powers of T and is thus isomorphic to which group? Q 2. Now use Lemma F from your lecture notes to prove the following r

Diagonals, Squares Circles and Endpoints

6. If a square's diagonal has endpoints (3,4) and (7,8) , find the endpoints of the other diagonal. (please illlustrate) What is the length of the diagonal? (Please give formula) What is the perimeter of the square? Write the equation of the circle that has the endpoint (3,4) as its center and goes throught the two c

Can you lease check my work

I have attached a file regarding contrapositives and inverse and converse statements --- Statement: If two lines are parallel then they do not intersect. True Converse: If 2 lines do not intersect then they are parallel False Inverse: If 2 lines are not parallel then they intersect

Space of functions is sequentially compact

Let C_0 be the space of functions f:R --> R such that lim f(x) = 0 as x goes to infinity and negative infinity C_0 becomes a metric space with sup-norm ||f|| = sup { |f(x)| : x in R } Prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions

Trivial Topology, Continuity and Connectedness

Let X and Y be topological spaces, where the only open sets of Y are the empty set and Y itself, i.e., Y has the trivial topology. ? Show that any map X --> Y is continuous ? Show that Y is path connected and simply connected. ---

Time And Distances

The movement of two submarines are being followed by a tracking system, and the positions of the submarines are modelled by points. The position of Sub A at time t is (2t+2, 2t+1) and the position of Sub B at time t is (4-t, t+5) (distance in Kilometers) 1.How would I go about eliminating t from each pair of coordinates, and

Homeomorphism

(See attached file for full problem description with proper symbols) --- Let and a map, given by . Let ~ be the equivalence relation on Xx[0,1] defined by and all other points are equivalent only to themselves. Show that Xx[0,1]/~ is homeomorphic to the Moebius strip. ---

Continuous and identification maps

(See attached file for full problem description with proper symbols and equation) --- Let be a surjective continuous map between topological spaces. Show that: a) If f is an identification mp, then for any pace Z and any map the composition is continuous if and only if g is continuous. b) If, for any space Z and any

Identification map

(See attached file for full problem description with proper symbols and equations) --- ? Let be the subspace of of all positive real numbers. Show that the map defined by is an identification map. ---

Break Even Point and Capital Budgeting

2. (Payback period, net present value, profitability index, and internal rate of return calculations) You are considering a project with an initial cash outlay of $160,000 and expected free cash flows of $40,000 at the end of each year for 6 years. The required rate of return for this project is 10 percent. a.) What is the pr

Topology

Compact spaces and path connectedness (see attachment). --- ? Show that if is a homeomorphism between topological spaces, then X is path connected if and only if Y is path connected. Using open cover definition: 1) is a compact subset? 2) Is a compact subset? ---

Connectedness, Continuity, Image, Antipodal Point and Borsuk-Ulam Theorem

Show that, if X is a connected topological space and is continuous, then the image f(X) is an n interval. Show that, if is a continuous map, then if given a,b,c in with a < b and c between f(a) and f(b), there exists at least one with a and f(x)=c Let be a continuous map. Show that there exists a point in the ci

Compact Set, Convergent Sequences and Subsequences and Accumulation Points

Prove that a set A, a subset of the real numbers, is compact if and only if every sequence {an} where an is in A for all n, has a convergent subsequence converging to a point in A. For the forward direction, I know that a compact set is closed and bounded, thus every sequence in A is bounded, and so has a convergent subsequen

Geometry

Exactly how many minutes is it before eight o'clock, if 40 minutes ago, it was three times as many minutes past four o'clock?

Use double inegrals to find the volume of a region.

Find the volume of the following region in space: The first octant region bounded by the coordinate planes and the surfaces y=1-x^2, z=1-x^2. This question is #12 (section 9.3) in Advanced engineering mathmatics (8th ed.) by Kreyszig. This section deals with the evaluation of double integrals.

Use the cylindrical shell method to find volume of solid of revolution.

Show step by step work using cylindrical shell method to find the volume of the solid formed by revolving the given region about the y-axis. 22) the region bounded by the curve y=SQRT(x), the y-axis, and the line y=1. 24) the region bounded by the parabolas y=x^2, y=1-x^2, and y axis for x&#8805;0. 26) the region ins

Volumes of Solids of Revolution and Sketches of Bounded Regions

Sketch the given region and then find the volume of the solid whose base is the given region and which has the property that each cross section perpendicular to the x-axis is a square. 2) the region bounded by the x-axis and the semi circle y = SQRT (16-x^2). Sketch the given region and then find the volume of the solid

Volume of a Tetrahedron

Find the volume of a tetrahedron with height h and base area B. Hint: B=(ab/2)sin(theta) Also, please see the attached document for the provided diagram of the tetrahedron. I already know that the answer is V=(Bh/3). I am simply looking for how my teacher came to this answer. Please show as many steps as possible so that

Topologies

Determine, for each of these topologies, which of the others in contains. --- (See attached file for full problem description)

Continuity proofs

Show that if {Aa} is a finite collection of sets... --- (See attached file for full problem description)

Topology proofs

Let A be a set. Let {Xa} be a family of spaces.... --- (See attached file for full problem description)

Clock problem involving angles

I am studying for a geometry test and am having trouble with a review problem at the end of the chapter. This is not homework. One problem asks the following: At 3:00, the hands of a clock form an angle of 90 degrees. To the nearest second, at what time will the hands of the clock next form a 90 degree angle? I figure t

General and Differential Topology

Find conditions under which the box topology is strictly finer than the product topology. See attached file for full problem description.