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Geometry and Topology

Finite Axiomatic Geometry

Consider the following axiom system. The undefined terms are point, line, and on. Axioms: I. Given any two distinct points, there exactly one line on both of them. II. Given any line, there is at least one point not on it. III. Given any line, there are at least five points on it. IV. There is at least one line. Questio

Hits on a rectangular board

On a 12 by 20 rectangular board, three plane figures are drawn: a square 6 on a side , a circle with radius 4 and an equilateral triangle that is 8 on a side. If a dart is thrown that does hit the large rectangle what is theprobability that it hits inside one of the three plane figures? (Landing on the side of a figure counts

Topologies : Open Sets

? Let X:={a,b,c} be a set of three elements. A certain topology of X contains (among others) the sets {a}, {b}, and {c}. List all open sets in the topology T. ? Let X':={a,b,c,d,e} be a set of five elements. A certain topology T' on X' contains (among others) the sets {a,b,c}, {c,d} and {e}. List any other open set in T' which

Topology : Open Unit Balls

Please see the attached file for the fully formatted problems. Let , and denote the three metrics defined on . What are the open unit balls , and with respect to these three metrics? Make a sketch and describe them algebraically.

Related rates

The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the rate of 0.2 cm/min. What is the rate of change of the volume of the cylinder, in cm^3/min, when the radius is 2 cm and the height is 3 cm? (Note: The volume of a right circular cylinder is V = p r^2h.)

Euler Path Problem : Traveler's Dilemma - Cross Bridges No More Than Once

Please see the attached file for the fully formatted problem. Traveler's Dilemma One day, travelers in a faraway land came upon a river with an island in the middle. On the other side of the island, the river continued but it formed two branches. The travalers also saw seven bridges that crossed the river in seven different

Topology : Subspace

Suppose (X,T) is a topological space. Let Y be non-empty subset of X. The the set J={intersection(Y,U) : U is in T} is called the subspace toplogy on Y. Prove that J indeed a toplogy on Y i.e., (Y,J) is a topological space.

Topological Space : Subspace

19. Let X be a topological space and let Y be a subset of X. Check that the so-called subspace topology is indeed a topology of Y. (question is also included in attachment)

Topology

14. Make a Mà¶bius strip out of a rectangle of paper and cut it along its central circle. What is the result? 15. Cut a Mà¶bius strip along the circle which lies halfway between the boundary of the strip and the central circle. Do the same for the circle which lies one-third of the way in from the boundary. What are the r

Undergrad 400 level Topology.

Find a tree in the polyhedron of figure 1.3 which contains all the vertices. Construct the dual graph Г and show that Г contains loops. (You don't have to construct the graph, but please describe it to me how it looks like.) (SEE ATTACHMENT)

Undergrad Topology - 400 Level

1. Prove that v(Г) - e(Г) = 1 for any tree T. (v :vertices and e : edges) 2. Even better, show that v(Г) - e(Г) ≤ 1 for any graph Г, with equality precisely when Г is a tree.

Topology : Connected Spaces and Explosion Point

Please see the attached file for the fully formatted problems. B6. (a) Define what it means for a topological space to be connected. (b) Suppose that A and B are subspaces of a topological space X, and that U C A fl B is open in both A and B in the relative topologies. Show that U is open in A U B in the relative topology.

Fubini Type I : Volume

Find the volume of the region in R3 bounded by z = 1 - x2, z = x2-1, y + z =1 and y= 0.

Minimize Cost for a Cylindrical Can

The metal used to make the top and bottom of a cylindrical can costs 4 cents/in^2, while the metal used for the sides costs 2 cents/in^2. The volume of the can is to be exactly 100 in^3. What should the dimensions of the can be to minimize the cost of making it? Could you please show all work so I can better grasp the conce

Maltitudes, Circumcircles and Circumcenters

For any quadrilateral one can define the so-called maltitudes. A maltitude on a side of a quadrilateral is defined as the line through the midpoint of the side and perpendicular to the opposite side. Generally the four maltitudes of a quadrilateral are not concurrent, but if the quadrilateral is cyclic they are. Prove that i

Compact and perfect sets

If P is a perfect set and K is compact is the intersection P intersection K always compact?always perfect?.

Sets

Let S = {(x,y,z):2x-y+3z=6},p=(1,0,-1) belonging to R3, and let f:S->R be defined by f(q)=d(q,p), ie, f(x,y,z)=sqrt((x-1)^2+y^2+(z+1)^2) Question: Is S compact? Please verify if q belongs to S and q does NOT belong to [-5,5]^3, then f(q) >= 4. ALSO prove that f attains its absolute minimum value at some point q0 belongi

Dv/dt

If V is the volume of a cube with edge length x and the cube expands as time passes, find dV/dt in terms of dx/dt.

Find all possible lengths, widths & heights of a given volume of a prism

I need to know how to find all possible lengths, widths & heights of a given volume of a rectangular prism. I'm writing a program in Java that takes the user inputted volume of a rectangular prism and then tells the user all of the possible lengths, widths & heights are for that given volume. I just don't know the calculations t

Volume

If 2400 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Volume = cubic centimeters.

Dimensions of a Rectangle Given the Area

James wanted a photo frame 3 in. longer than it was wide. The frame he chose extended 1.5 in. beyond the picture on each side. Find the outside dimensions of the frame if the area of the unframed picture is 70 in.^2 (square inch.)

Minimization of area

A piece of wire 35cm long is to be cut into two pieces,one piece will be bent into a equilateral triangle and the other a circle.How should the wire be cut so that the enclosed area is minimized?

Cylindrical area and volume

Solve the following word problem, you will need to know that a cylinder of radius r and height h has volume V = Pi r^2 h. Also, a circle of radius r has area A = Pi r^2 and circumference C = 2 Pi r. A closed cylindrical can has a radius t and height h. a) If the surface area S of the can is a constant, express the volume

Complex Variable (Circle; Contour)

See attachment for better formula representation and theorem 5. For each of the following use Theorem 5. to establish the indicated estimate. a. If C is the circle |z| = 3 traversed once then: | Integral C ( dz/ z^2 - i) | < 3pi / 4

Angles and Degrees

Determine the angle when (6 angle (120deg))*(-4+jb+2*exp(j*15))=a+jb Note: These angles are in degrees. Possible choices: a. -21.3 b. -12.1 c. -3 d. -2.07