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# Geometry and Topology

### Binormal Twisted Curve

Suppose a twisted curve is defined in terms of the arclength s by r(s)=sech(?)cos(s)i+sech(?)sin(s)j+tanh(?)k, where ? is a constant parameter. Determine (i) the tangent t(s), (ii) the normal n(s), (iii) the binormal b(s), (iv) the curvature ?, (v) the torsion Ï?.

### the amplitude, period and frequency of the oscillations

Assume that the particle under the influence of an attractive force F(r), where r is the distance from a fixed point O, then it moves in a plane through O and the equation of motion is ((d^2)u/d?^2)+u=F(1/u)/(h^2)(u^2), where h is a constant, u=1/r and ? is the angle the line from O to the particle makes with some fixed direc

### Finding an Open Subset

Let g: R -> R by: g(x) = { x^2 + 2 ... if x <= 1, 5 - x ... if x > 1 } Find an open subset of R (w/ respect to the usual topology) whose preimage under g is not an open subset of R.

### Extending Continuity of Restrictions

Let X refer to a general topological space. Suppose X = A1 ï?? A2 ï?? â?¦, where An ï? Ã...n+1 for each n. If f : X --> Y is a function such that, for each n, f |An : An --> Y is continuous with respect to the induced topology on An, show that f itself is continuous. Please note that Ã...n+1 (A with a small

### Topology - open or closed susbset

Please help with the following problem. For the following, I'm trying to decide (with proof) if A is a closed subset of Y with respect to the topology, T (i) Y = N, T is the finite complement topology, A = {n e N | n^2 - 2011n+1 < 0}. (ii) Y = R, T is the usual topology, A is the set of irrational numbers between 0 and

### properties of discrete spaces

I am looking at the properties of discrete spaces, particularly this one: Every discrete space is first-countable; it is moreover second-countable if and only if it is countable. How would this be proved?

### Solve: Topological Space

Let X be a normed space with the topology induced by the norm. Show that || || : X ---> R is a continuous function on X. Please show all of your work. Thank you.

### Determination of Orthogonal Contrasts

An experiment on sugar beets compared times and methods of applying artificial (N-P-K) fertilizer, using a completely randomized design. The four treatments were: Treatment 1 - no artificials(control) Treatment 2 - artificials applied in January and ploughed into soil Treatment 3 - artificials applied in January and bro

### Is set X a Hausdorff space with this topology?

8. Let X be any set of infinite cardinality. Consider the set A = { A SUBSET X : X - A is finite } UNION { PHI , X }. a) Prove that A is a topology on X. b) Describe the neighborhoods of a point P BELONGING_TO X. c) Is X a Hausdorff space with this topology? Please see the attached image for question with appropriate sym

### Finding Characteristics of a Set with Topology Induced

Find the interior, the closure, the accumulation points, the isolated points and the boundary points of the set A = { x = (x_1, x_2 ...) e l^2 : -1 <x_1 < 1} with the topology induced by || ||_2.

### Topology induced

Question: Find the interior, the closure, the accumulation points, the isolated point and the boundary points of the following sets. a) X = [(0,1) in R with the topology induced by d(x,y) = |x-y|] b) X = Q in R with the same topology as above. c) X = {(x,y) : |y| < x^2} U {(0,y), y E R} in R^2, with the topology induced by

### Compact Subsets

Define a new metric d on X = (0, 1/2)^2 by d((a,b), (r,s)) = 1 if a is not equal to r Or |b - s| if a = r. a) Show that d is a metric on X. b) What are the compact subsets of X? Prove your statement.

### discrete topology covering map

A covering map is a map p: E -> B with the property that each point b, an element of B, has a neighborhood U such that p^-1(U) is a disjoint union of open sets V_alpha such that, for each alpha, the restriction of p to V_alpha is a homeomorphism of V_alpha onto U. Show that, if Y has the discrete topology and if p: X x Y --

### Problems exemplifying congruency

1) Suppose n belongs to Z. (a) Prove that if n is congruent to 2 (mod 4), then n is not a difference of two squares. (b) Prove that if n is not congruent to 2 (mod 4), then n is a difference of two squares. 3) Let n = 3^(t-1). Show that 2^n is congruent to -1 (mod 3^t). 5) Let p be an odd prime, and n = 2p. Show that a^(

### Compact and closed

Let S be the set [0,1] and define a subset F of S to be closed if either it is finite or is equal to S. Prove that this definition of closed set yields a topology for S. Show that S with this topology is compact, but S is not a Hausdorff space. Show that each subset of S is compact and that therefore there are compact subs

### De Morgan's Laws - FIP, Hausdorff and Compact

1.Let X be a set and T and T' are two topologies on X. Prove that if T subset of T' and (X,T') is compact, then (X,T) is compact. Prove that if (X,T) is Hausdorff and (X,T') is compact with T subset of T', then T=T'. 2.Let X be a topological space. A family {F_a} with a in I of subsets of X is said to have the finite

### Geometry

The importance of geometry's role in the math curriculum is debated in many high schools and colleges. Some schools offer the course while others have done away with it. Based on what you have learned within this unit, do you think geometry is a valuable tool for students to learn? Choose one side of this debate, state your view

### intersection of the collection of open intervals

1. Prove if F is a subset of R^n and if d(x,F)=inf(||x-z||:z in F}=0 then x belongs to F. 2.The intersection of two open sets is compact iff it is empty. Can the intersection of an infinite collection of open sets be a non-empty compact set?

### Using Differentials to Estimate

1. In the manufacturing process, ball bearings must be made with radius of 0.5mm, with a maximum error in the radius of +/- 0.016 mm. Estimate the maximum error in the volume of the ball bearing. Solution: The formula for the volumen of the sphere is _____. If an error deltaR is made in measuring the radius of the sphere

### Intervals with rational endpoints as a basis for topology

Prove that the family of open intervals with rational end points is a basis for the topology of the real line. A rigorous proof with detailed explanations is given.

### Interior of a set in a topological subspace

Let Y be a subspace of X and let A be a subset of Y. Denote by Int_X(A) the interior of A in the topological space X and by Int_Y(A) the interior of A in the topological space Y. Prove that Int_X(A) is a subset of Int_Y(A). Illustrate by an example the fact that in general Int_X(A) not Int_Y(A).

### Boundary of the n-dimensional ball

In R^n with the usual topology, let A be the set of points x=(x_1,x_2,...,x_n) such that x_1^2+x_2^2+...+x_n^2 le 1. Prove that Bdry(A) is the (n-1)-dimensional sphere S^n-1.ie. x in Bdry(A) iff x_1^2+x_2^2+...+x_n^2=1. Intuitively it is easy.. but I am not sure where to start. Probably looking at bdry(A)=Cl(A) intersect w

### Volume of the Parallelepiped

Calculate the given quantities if a = <1,1,-2>, b= <3,-2,1>, and c=<1,1,-5>. a) 2a + 3b b) |b| c) Unit vector in the direction of b. d) a (dot) b e) a x b f) comp a/b g) proj a/b h) Volume of the parallelepiped determined by a , b , and c. i) Determine whether a and b are parallel, perpendicular or

### Geometric concepts and children

How might you involve children in learning geometric concepts? Which geometric concept do you think will be most difficult for children to learn and why?

### Altitude of parallelepiped

Find the altitude of a parallelepiped determined by vectors A, B, and C if the base is taken to be the parallelogram determined by A and B and A=i+j+k B=2i+4j-k C=i+j+3k.

### Calculating the amount of material for finishing a basement

Question 11A Guest needs material to finish a room in a basement. The room is square and one wall measures 15'. The height of the room is 8'. There is one door that measures 3' wide and 7' high, and two windows that measure 3' wide and 4' high. The Guest is covering the walls with 1x6 carsiding (we will say that it covers 5"

### Parabola Vertex Equation

Find the equation of the parabola: 1. Vertex at (-2, 3), Focus at (-4, 3) 2. Vertex at (2, 4), directrix at x = -3 3. Vertex at (-1, -2), axis vertical and passing (3, 6) 4. Directrix = 4y, focus at (3, 0) 5. Axis horizontal, parabola passing through: (1, 1), (1, -3), and (-2, 0) 6. Latus rectum joining the poi

### equation in standard form.

Transform the equation to standard form and find the center and radius of the circle: 1. (x - 3)^2 = 9 - y^2 2. x^2 + y^2 = 6x - 8y 3. 2x^2 + 2y^2 = 2x + 2y - 1 4. (x + 1)^2 + (y - 2)^2 = 25

### Inductive Reasoning to Predict Numbers

Glenda Everson Math 1002 Practice Problems 1. Use inductive reasoning to predict the next three numbers in the pattern (or sequence). 4, −20, 100, −500, . . . Step 1: Form a Hypothesis: Step 2: Make observations related to the hypothesis: Step 3: Come to a conclusion: 2. Flying West New York City is on easter

### Coordinates and Altitude

Let A = (0,3) and B = (4,0) and AB = 5 = c Let D be the altitude to the hypotenuse. a) Find the coordinates of D b) Find the lengths of AD, BD, and CD