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Geometry and Topology


Would like more details and explanations as to how the attached graph is solved. Create a graph with four odd vertices. See attached file for full problem description.

Geometry Proofs

Formalizing Proofs. See attached files for full problem description.

Geometry Word Problems, Angles, Lengths, Areas, Volumes and Graphs

Please see the attachment for the proper formatting and related diagrams. 1. DEF and GHI are complementary angles and GHI is eight times as large as DEF. Determine the measure of each angle. 2. If line p and q are parallel, find the measure of angle 2 3. (Using ABC, find the following: a) The length of s

Latin Squares

1) Show that for n less than or equal to 4, any Latin square of order n can be obtained from the multiplication table of a group by permuting rows, columns, and symbols. Show that this is not true for n=5 2) If n is an order for which mutually orthogonal Latin squares exist, does every Latin square of order n have an orthogo

Sets and Functions : The Symmetric Difference of Two Sets

The symmetric difference of two sets A and B, denoted by A Δ B, is defined by A Δ B = ( A - B ) U ( B - A ); it is thus the union of their differences in opposite orders. Show that A Δ ( B Δ C ) = ( A Δ B ) Δ C.

Affine Algebraic Sets and Topologies

1. Show that affine algebraic sets satisfy the axioms for the closed sets of a topology, i.e. show that the intersection of an arbitrary collection of algebraic sets is algebraic and the union of two algebraic sets is algebraic.

Mean Value Theorem and Maximizing Volume of an Open Box

The width of an open box is half its height and its surface area is 75 cm3. Find the dimensions to maximize its volume. Indicate which of the following functions satisfies the conditions of the mean value theorem on the given interval.

What is the volume of the solid revolution?

The region in the first quadrant bounded by the graphs of y = x and y = x^2/2 is rotated around the line y=x. Find (a) the centroid of the region and (b) the volume of the solid of revolution.

Cylindrial shells problem

Using the method of cylindrical shells to find the volume of the solid rotated about the line x=(-1) given the conditions: y=x^3 -x^2;y=0;x=0.

Tangent Plane to Parametric Surface

Find an equation of the tangent plane to the parametric surface x = -1rcos(theta), y = -5rsin(theta), z = r at the point (-1sqrt(2), -5 sqrt(2), 2), when r = 2 and theta = pi/4.

Circles and Cross Ratios

A) Let z1 and z2 be two points on a circle C. Let z3 and z4 be symmetric with respect to the circle. Show that the cross ratio (z1,z2,z3,z4) has absolute value 1. b)Let ad-bc=1, c not zero and consider T(z)=(az+b)/(cz+d). Show that it increases lengths and areas inside the circle|cz+d|=1 and decreases lengths and areas outsid

Fractional Transformations, Cross Ratios and Conformal Mapping

1. a) Let z1,z2,z3,z4 lie on a circle. Show that z1,z3,z4 and z2,z3,z4 determine the same orientation iff (z1,z2,z,3,z4)>0 b) Let z1,z2,z3,z4 lie on a circle and be consecutive vertices of a quadrilateral. Prove that |z1-z3|*|z2-z4|=|z1-z2|*|z3-z4|+|z2-z3|*|z1-z4|

Projective geometry problems

Projective Geometry Problem 1 i. Prove that a set of four points in a projective plane P (i.e. dim P = 2) form a projective frame if and only if no three of the points are collinear, i.e. no three lie on the same projective line. ii. Find a necessary and sufficient condition for five points to form a projective frame in a t