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# Algebraic Geometry

### Collineations and Translations

What are the fixed points of a translation T? A fixed point is a point that is not moved by a given collineation, i.e. P = TP

### Students Enrolled in Classes

500 students enrolled in at least 2 of 3 classes: English, Math, History. 150 students enrolled in both history and English, 300 enrolled in math & history, 170 enrolled in both math and English. How many of the 500 students are enrolled in all three classes?

### Path Connected Subsets

Give proofs or counter-example for the following statements: i)If X and Y are path connected subsets of Z and X/Y ( X intersection Y) is non empty then X/Y is path connected. ii) If X and Y are path connected subsets of Z the X/Y(union) is path-connected. iii)If X and Y are path connected subsets of Z and X/Y ( X in

### Finding the sum of the first n terms of a geometric sequence.

Please see the attached file for the fully formatted problems.

### Geometric sequence

Please show how you arrived at the following solution Use the geometric sequence of numbers 1, 2, 4, 8,..... to find the following : a. What is r, the ratio between 2 consecutive terms b. Using the formula for the nth term of a geometric sequence, what is the 24th term.

### Geometric Series : Pennies and Checkerboard Problem

A salesman challenged a stanger to a checker game. He told the stranger to put one penny on the first square. Then place two pennies on the next square. Then place four pennies on the third square. Continue this until all 64 squares are covered with pennies." As he'd been saving pennies for over 25 years, the stranger did n

### Infinite Series : Convergent or Divergent?

Show whether the following series converges or diverges by performing a test for convergence. &#8734; &#8721; e^(-n) n=1

### Sequences and Series

I need assistance in discerning sequences and series including the classical grains and checkboard problem

### Equivalent Paths

Let f,g ; I-->X be two paths with initial point x0 and terminal point x1. Prove that f g iff f g-bar is equivalent to the constant path at x0 . Note: the path g-bar is obtained by traversing the path g in the opposite direction. See the attached file.

### Series problem

I got a answer but it didn't make sense here is my question again. A farmer is going to grant a sales man a reward for good work. So, the salesman said, "If you insist, I do not want much. Get your checkerboard and place one penny on the first square. Then place two pennies on the next square. Then place four pennies on the th

### Geometric Series and Sums : Checkerboard Story Problem - Get your checkerboard and place one penny on the first square. Then place two pennies on the next square. Then place four pennies on the ....

CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulled a young lady out of the flaming truck. Farmer Brown came out and gratefully thanked the traveling salesman for saving his daughter's life. Mr. Brown

### Geometric series: Good learning

Use the geometric series of numbers 1, 2, 4, 8,...to find the following: What is r, the ratio between 2 consecutive terms? Can you please show how you get the answers in detail to help me Thanks. Using the formula for the nth term of a geometric series, what is the 24th term? Using the formula for the sum of a geometr

### Path components

(See attached file for full problem description with proper symbols and equations) --- Let X be a topological space. Mapping a point to the path component which contains x establishes a map . Show that for any continuous map between topological spaces, there exists a map such that the following holds: ? ? for two co

### Dual Space and Isometrically Isomorphic Spaces

Let c be the set of all sequences *see attachment* , such that limit alpha_n exists. Let be the dual space of c , and c consist of all functions f -> F, F or such that for every e>0 {n E : |f(n)| >E} is finite. Show that c* is isometrically isomorphic to l'. Are c* and c isometrically isomorphic? Please see the att

### Homomorphisms, Bijection Map and Continuous Map

1) Prove that the map GIVEN BY is a homomorphism between the real line and open interval (-1,1). 2) Let be the map given by a) show that f is a bijection map b) show that f is a continuous map c) If f a homomorphism? Justify your answer. Please see the attached file for the fully formatted problems.

### Geometric Series of Numbers : a1=-2,r=2

Determine the sixth term of the geometric series, a1=-2,r=2

### Multipliers and Geometric Series: Application to U.S. Economy

Suppose that, throughout the U.S. economy, individuals spend 90% of every additional dollar that they earn. Economists would say that an individual's marginal propensity to consume is 0.90. For example, if Jane earns an additional dollar, she will spend 0.9(1)=\$0.90 of it. The individual that earns \$0.90(from Jane) will spend

### Infinite Series: Geometric Progression

Summation of series from n=2 to infinity of (pi/4)^(n/2) I don't know if I should set the whole thing up as a power of e and then calculate it, or calculate it as geometric series.

### Dimensions of the building

Three buildings abut as shown in the diagram below. What are the dimensions of the courtyard and what is the perimeter of the building? See attachment for diagram

### Sum of infinite geometric series

A regular hexagon is inscribed in a circle of radius 100. The area of that hexagon outside the square is shaded. Another circle is inscribed in the hexagon, and hexagon is inscribed in the second circle. The area of the second circle which lies outside the hexagon is shaded. This process continues to infinity. What is the

### Function Terminology of 'Onto' and 'One to One'

Assume f:A->B and g:B->C. a) Show that if f and g are both onto, then g o f is onto b) Show that if g o f is one-to-one, then f is one-to-one c) Show that if g o f is onto and g is one-to-one, then f is onto

### Infinite Geometric Series: Square Inscribed in Circle Radius

A square is inscribed in a circle of radius 100. The area of that circle which lies outside of the square is shaded. Another circle is inscribed in the square, and then a second square is inscribed in that second circle. The area of the second circle which lies outside of the second square is shaded. This process is continued t

### Volume and water displaced.

When a chunk of iron is completely submerged into a cylindrical tank of water, the water rises 6 inches. The diameter of the tank is 18 inches. What is the volume of the chunk of iron?

### How long is the edge of a cube whose total area is numerically equal to it's volume?

How long is the edge of a cube whose total area is numerically equal to it's volume?

### Find the location of a pole in a rectangle.

Johnny green has a rectangular piece of ground. He places a pole in the gound five feet from the upper right corner. The same pole is nine feet from the lower right corner. It is also thirteen feet from the lower left corner. How far is the pole from the upper left corner?

### Make a model for a Klein bottle

27. Make a model for a Klein bottle as shown below.... see attachment

### Annulus: X and Y cannot be Homomorphic

24. Let X,Y be the subspace of the plane shown as below. Under the assumption that any homomorphism from the annulus to itself must send the points of the two boundary circles among themselves, argue that X and Y cannot be homomorphic. (Question is also included in attachment).

### Determine the Metric Space

See attached file.

### Metric space

Let X be a metric space and x0 in X. Define a function f: X --> R (all real numbers) by f(x) = d(x,x0). Show that f is continuous. HINT: Prove the variant of the triangle inequality which says |d(x,z)-d(y,z)|< d(x,y) for any x,y,z in X

### Sets

Note: C = containment int = interior ext = exterior cl = closure Could you please prove if S C T, then a)int(S) C int(T) b)ext(T) C ext(S) c)cl(S) C cl(T)