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Algebraic Geometry

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

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).

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)

Geometric and arithmetic series, pulleys in parallell

Resistances in series can be reduced to a unique resistance R such that eq(1) R= r1 + r2 +...+ rn in Parallel , we have eq (2) 1/ (1/r1 +1/r2 +...+ 1/rn) For the pulleys , to reduce the effort to keep a block and tackle (which has a mass M at he end) in equilibrium , the necessary force F to

Enlargement

Phil and Fran are photographers who develop their own pictures and also restore old photographs. They have an enlargement and reducing machine that can change the size of photographs. A customer asks Fran to enlarge a 3 inch by 5 inch photograph to 8 inch by 10 inch. Can this be done without cutting or distorting the picture?