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Linear Transformation

Assessing the forgetful functor

Let Phi : P -----> S be the forgetful functor. Suppose that Phi (f) is an isomorphism. Is f then an isomorphism? Alternatively, suppose that Phi (G) and Phi (G') are isomorphic in S. Are G and G' then isomorphic in P?

Homomorphisms are examined.

EDIT: G(p) = {x in G : |x| = p^k for some k greater than or equal to 0} ** Please see the attachment for the complete problem description **

Proofs for prime ideals

a. Let R^+ be the set of positive real numbers. Define operations on this set by the a + b = ab and a x b = e^(ln (a)ln (b)), where the right hand sides have the usual meaning in the real numbers. Prove that R^+ is a field with these operations by showing that exp : R -----> R^+ is an isomorphism, where exp is the usual expone

Characterization of abelian groups

a. Show that the function f : G ----> defined by f(x) = x^(-1) is a group homomorphism if and only if G is abelian b. Define a new group H to have the same elements as G, but the operation x # y = yx, where yx is defined by the operation in G. Show that the function f : G ----> H defined by f (x) = x ^ (-1) is an isomorphism

Finite dimensional extension field

Prove that a finite-dimensional extension field K of F is normal if and only if it has this property: Whenever L is an extension field of K and sigma : K ----> L an injective homomorphism such that sigma (c) = c for every c in F, then sigma (K) is contained in K.

Rank-Nullity Theorem Applied

Show that if c(A) is the vector space formed by the columns of an nxm matrix A where n>>m and N(c(A)) is the null space of c(A) - that is, N(c(A)) = {w|w'v=0 for all v∈c(A)}, then N(N(c(A)))=c(A). (Hint: Use the rank-nullity theorem).

Maximize Profit in Airline Tickets

Let q = demand for seats on a 500 seat airplane and p = price charged per ticket. Suppose that q = 600 - 3p and let's assume that the unit cost of flying a passenger is $50.00. To maximize profit from the flight, the airline should charge how much per ticket? a, $100 b, $125 c, $150 d, $175

Isomorphic Variable

Let H be a group and tau_1 : H ---->G_1, tau_2 : H ----> G_2, ... , tau_n : H -----> G_n homomorphism with this property: whenever G is a group and g_1 : G ---->G_1, g_2 : G ----G_2, ..., g_n : G ----> G_n are homomorphism, then there exists a unique homomorphism g* : G ----> H such that (tau)_i â?¢ g* = g_i for every i. Pro

Projection homomorphism: normal subgroup

Let M be a normal subgroup of a group G and let N be a normal subgroup of a group H. Use the first Isomorphism Theorem to prove that M X N is normal subgroup of G X H and that (G X H) / (M X N) is isomorphic to G/M X H/N

Characteristic group for automorphism

A) A subgroup N of a group is said to be characteristic if f(N) is contained in N for every automorphism f of G. Prove that every characteristic subgroup is normal. b) Prove that the converse is false.

Kernel of Phi and Homomorphism

Let phi is a homomorphism from Z30 onto a group order 3. Determine the kernel of phi. Find all generators of the kernel of phi.

Finite Abelian Group

Suppose that G is a finite Abelian group and G has no element of order 2. Show that the mapping g-->g^2 is an automorphism of G. Show, by example, that if G is infinite the mapping need not be an automorphism (hint: consider Z).

Minimization of a Function

What is the Value of W? What are the values of Y1 and Y2? What are the values of X1 and X2? What is the value of Z? X1 X2 S1 S2 Z K 0 1 2/7 -3/7 0 4 1 0 -1/7 5/7

Dirichlet Kernel

Let D_n (theta) = sum(k=-N to N) e^ik(theta)= sin ((N+1/2)theta)/sin(theta/2) and define L_n = 1/2Pi integral (from - Pi to Pi) |D_n (theta)| d(theta) prove that L_N is greater than or equal to c log (N) for some constant c>0 Hint: show that |D_n(theta)| is greater than or equal to c sin ((n+1/2)theta)/|theta| change variab

Model Formulation (Minimization)

Camel Trucking has a long term shipping contract with Hopeless Ventures Inc. Hopeless produces industrial quality generators at four manufacturing plants in Easton, Westville, Northbrook, and Southburg. Output from the four plants is shipped to warehouses in Singleton, Duce, Tripoli, Foura, Quincy, Six Gun City, Savannah, and Oc

linear fractional transformation

Consider the transformation w = (i - z) / (i + z). The upper half plane Im z > 0 maps to the disk |w| < 1 and the boundary of the half plane maps to the boundary of the circle |w| = 1. 1. Show that a point z = x is mapped to the point w = [(1 - x^2) / (1 + x^2)] + i[(2x) / (1 + x^2)], and use this to find the i

Determining Waiting Times

Question: McBurger's fast-food restaurant has a drive-through window with a single window with a single server who takes orders from an intercom and also is the cashier. The window operator is assisted by other employees who prepare the orders. Customers arrive at the ordering station prior to the drive-through window every 4.5

Solve in QM for Windows.

The Grandy Tire Company recaps tires. The weekly cost is $2,500., and the variable cost per tire is $9. Price is related to demand, according to the following linear equation: v = 200 - 4.75p Develop the nonlinear profit function for the tire company and determine the optim

Linear Programming for Maximum Profit

I need help with the following problem. Using the attached file please help with Constructing a model expressing the profit for X made in a day in terms of the decision variables. Solve the question graphically to determine the number of bowls and mugs to produce each day for maxinum profit. I know that Zmax is 40x+50y an

Linear Operations Solve Using QM for Windows

Solve Using QM for Windows Brooks City Transportation Problem: Brooks City has three consolidated high schools, each with a capacity of 1,200 students. The school board wants to determine the number of students to bus from each district to each school to minimize the total busing miles traveled: A. Formulate a linear programm

Countable linear ordering isomorphic to subset of rationals

Show that every countable linear ordering is isomorphic to some subset of the rationals under their usual order, but that omega_1 (the least uncountable ordinal) with its well order is not isomorphic to any set of reals under their usual ordering. The solution may use any algebraic facts about the reals. See attachments f

Demand, Revenue & Largest Reduction to Maximize Revenue: DVDs

200 DVD players are sold per week for $300 each. For each $10 rebate (reduction in sale price), 20 additional DVD players are sold. Find the demand, the revenue, and the largest reduction (multiple of 10) to maximize revenue. Cost: c(x) Marginal Cost: c'(x) Revenue: R(x)= x(p(x)) Demand: p(x) Marginal Revenue: R'(x) P

network flow problems

"Transportation Problems" are a subclass of network flow problems. You have a set of source cities and a set of demand cities, the amount of supply in each source city, the amount of demand in each demand city, and the shipping costs for each (source,demand) city pair. You must exactly meet demand in each demand city, and ea

Mathematics-Supply Greater Than Demand- Transportation Model

A company which owns hauling trucks moves materials between three sources (depots) and three destination projects. Project 1 needs 280 truckloads each week, Project 2 requires 400, and Project 3 needs 160. Depot 1 can supply 240 truckloads; Depot 2, 320, and Depot 3, 280 each week. Cost information is given below: To From

Determining Lowest Transportation Cost: Example Problem

Cammile, a fruit dealer, sells fruits to customers in Mani, Pampan, and Paciz. The monthly demand is 4000 kilos in Mani, 2500 kilos in Pampan, and 1000 kilos in Paciz. The fruits are shipped from three farms located in Laguna, Zamboa, and Caga. The monthly supply available in Laguna is 5600 kilos; in Zamboa, 900 kilos; and in

Minimize the risk of the investment

Referring to the information listed above, suppose the investor has changed his attitude about the investment and wishes to give greater emphasis to the risk of the investment. Now the investor wishes to minimize the risk of the investment as long as a return of at least 8% is generated. 1a. How much should be invested in

The Arden County, Maryland, superintendent of education is responsible for assigning students to the three high schools in his county. He recognizes the need to bus a certain number of students, for several sectors of the county are beyond walking distance to a school.

The Arden County, Maryland, superintendent of education is responsible for assigning students to the three high schools in his county. He recognizes the need to bus a certain number of students, for several sectors of the county are beyond walking distance to a school. The superintendent partitions the county into five geographi