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    Geometry and Topology

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    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

    Statement: If two lines are parallel then they do not intersect.

    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

    Al metal reacts with HCl produces AlCl3 and hydrogen gas.

    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

    The volume of the cross vault: Intersecting cylinders.

    Let two long circular cylinders, of diameter D, intersect in such a way that their symmetry axes meet perpendicularly. Let each of these axes be horizontal, and consider the "room" above the plane that contains these axes, common to both cylinders. (In architecture this room is called a "cross vault".) The floor of the cross vau

    Geometry : Solve for the angles of a parallelogram.

    Solve for the angles of a parallelogram. Please see the attached file for the fully formatted problems. PLEASE VIEW AT 150% it is easier to see. The 7y-2 should be in parenthesis (7y-2) and the same thing for 4x+1, i.e. (4x+1). I have tried solving this problem several times I need values for x and y. can you please e

    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. ---

    Submarine 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

    Homeomorphisms of a map

    (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. ---

    Surjective continuous 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 Descriptions

    (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. ---

    Calculating the amount of fencing required

    A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters. What dimensions would require the least amount of fencing if no fencing is needed along the river?

    Product space of the path connected topological space

    Let X=X1 x X2 x...x Xn be the product space of the path connected topological space X1, X2, ..., Xn. Prove that the product space X is also path connected. Please see the attached file for the fully formatted problems.

    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: Homeomorphism and Connectedness

    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? See the attached file.

    Connectedness, Continuity, Image, Antipodal Point & Borsuk-Ulam

    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 circ

    How Many Angles Are Formed?

    1. Consider a point P When one ray is drawn from the point P. there are no angles formed. When two rays are drawn, one angle (if we only count angles with measures less than 180°) is formed. How many angles (with measures less than 180°) are formed when 100 rays are drawn from point P? Show all work.

    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?

    Filling a swimming pool

    A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at 1/4 cubic meter per minute, and there is 1 meter of water at the deep end. (a) what percent of the pool is filled? (b) At what rate is the water level rising?

    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

    Finding the Dimensions (Side) of a Cube Given Volume

    A homemade loaf of bread turns out to be a perfect cube. Five slices of bread, each 0.6 in. thick, are cut from one end of the loaf. The remainder of the loaf now has a volume of 235 cu. in. What were the dimensions of the orginal loaf?

    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 wh

    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