E) Another pair are called missers. They are supposed to roll 3 & 7 more often than fair dice Below is the data from 16 rolls.

{ 10, 3, 5, 7, 3, 7, 8, 10, 6, 7, 7, 11, 3, 8, 2, 11 }

Can we say to a 10% that these dice do not roll 3 & 7 the way that fair dice are supposed to?

Remember there is a chance that the guy from gamblingcollectibles.com charged me $100 for a regular pair of dice.

For each of the subspaces ? and ? of ?4 (ℝ) defined below, find bases for ? + ? and ? ∩ ?, and verify that dim[?] + dim[? ] = dim[? + ? ] + dim[? ∩ ? ]

(a) ? = {(1, 3, 0, 1), (1, −2, 2, −2)}, ? = {(1, 0, −1, 0), (2, 1, 2, −1)}

(b) ? = {(1, 0, 1, 2), (1, 1, 0, 1)}, ? = {(0, 2, 1, 1), (2, 0, −1, 0)}

The resistance of blood flowing through an artery is

R = C

where L and r are the length and radius of the artery and C is a positive constant. Both L and r increase during growth. Suppose

r = 0.1 mm,

L = 1 mm,

and

C = 1.

(a) Suppose the length increases 10 mm for every mm increase in radius during growth. Use a directional derivative to determine the rate at which the resistance of blood flow changes with respect to a unit of growth in the r-L plane.

Cr4​

(b) Use a directional derivative to determine how much faster the length of the artery can change relative to that of its radius before the rate of change of resistance with respect to growth will be positive.


(c) Illustrate your answers to parts (a) and (b) with a sketch of the directional derivatives on a plot of the level curves of R. (Use u for the unit change described in part (a) and v for the unit change described in part (b).)

There are 1000 mailboxes at a post office, numbered 1, 2, 3, …, 1000. There are also 1000 mailbox owners, one for each mailbox. At the start of the Mailbox Challenge, all mailboxes are closed and the owners open and close the mailboxes according to the following rules:

Owner 1 opens every mailbox.

Owner 2 closes every second mailbox; that is, lockers 2, 4, 6, 8, …, 1000.

Owner 3 changes the state of every third locker, closing it if it is open and

opening it if it is closed.

Owner n changes the state of every nth mailbox, etc.

When all the owners have taken their turns, how many mailboxes are open?

Request to solve the second order differential equation by range Range kutta 4th order method 8d^2y/dx^2-x^2+2y^2=0 with initial conditions y(0)=1 and dy/dx(0)=0 compute y at 1 (Numerical Method)

2- Develop a simulink model for natural PWM inverter connected to a dc source of 100 V and an output frequency of 60 Hz. The load is a series RL load with R = 10 Ohm and L = 25 mH. Use the simulation to Determine (a) The frequency Ratio to eliminate the 11th harmonic . (b) The fundamental output Voltage V1 (first term of Fourier series) (c) The fundamental output current I1 (d) If the load requires a fundamental peak voltage V1=80V, find the necessary modulation index .

Your report must include screen capture of the Simulink model, scopes, displays in addition to solver and step time configuration.

Assignment 7: Congressional Vote Tracking Database

Description

Design an Extended E-R schema diagram for keeping track of information about votes taken in the U.S. House of Representatives and Senate during the current two-year congressional session.  The database needs to keep track of each U.S. STATE's Name (e.g. Texas, New York, Pennsylvania, etc.) and include the Region of the state (whose domain is {North-east, Midwest, Southeast, West}).  Each CONGRESSPERSON in the House of Representatives is described by his or her Name, plus the District represented, the StartDate and EndDate for each term that the congressperson was elected, and the political Party to which he or she belonged when elected (whose domain is {Republican, Democrat, Independent, Other}). Each CONGRESSPERSON in the Senate is elected statewide, 2 senators per state, for six-year terms. The database should capture each CONGRESSPERSON's participation on committees and track committee votes, House votes, and Senate votes on bills made by each CONGRESSPERSON. For each vote taken on a bill, the database should capture whether or not the vote passed, the numbers of Yeas, Nays, Abstains, and Absences. The database should also record the President's decision to either pass the bill into law or veto the bill.

The database keeps track of each BILL (i.e., proposed law), including the BillName, the DateOfVote on the bill, whether the bill PassedOrFailed (whose domain is {Yes, No}), and the Sponsor(s) (the congressperson(s) who sponsored - that is, proposed - the bill).  The database keeps track of how each congressperson voted on each bill (domain of vote attribute is {Yes, No, Abstain, Absent}).  Draw an Extended ER schema diagram for this database application.  Express all constraints such as cardinality ratios, disjoint vs. overlapping specializations, and full vs. partial participation constraints.  State clearly any assumptions you make.

Tasks & Deliverables

1. Draw the Extended E-R Diagram
2. Map EER to Relational Schema using the 8-Step Mapping Algorithm

A restaurant has the following table of values for some of its burrito sales during January from the previous 10 years

a. Find a cubic model for the price demand functions. What is the correlation coefficient?

b. Using your cubic model, find a model for the revenue

c. What price will the revenue be maximized

d. Does your answer from part c guarantee that the profit will be maximized? why or why not?

Let G be a group with the binary operation of juxtaposition and identity e. Let H be a subgroup of G.

(a) (4 points) Prove that a binary relation on G defined by a ∼ b if and only if a−1b ∈ H, is an equivalence.

(b) (3 points) For all a ∈ G, denote by [a] the equivalence class of a with respect to ∼ . Prove that [a] = {ah|h ∈ H}. We write [a] = aH and say that aH is a left coset of H in G. Denote by π : G → G/ ∼ the quotient map of ∼ . What is the value of π(a)?

(c) (3 points) Prove that the map λa : H → aH given by λa(h) = ah is one-to-one and onto. If H is finite, what can you say about the cardinalities |H| and |aH|?

(d) (4 points) (Lagrange’s Theorem) If G is a finite group then |H| divides |G|. The quotient [G : H] = |G| is called the index of H in G. What is the meaning of the index? Hint: the left

|H|
cosets of H in G form a partition of G.

(e) (1 point) Let K be a subgroup of G. Denote by ◃▹ the equivalence relation on G given by a ◃▹ b if and only if a−1b ∈ K, let σ : G → G/ ◃▹ be the quotient map of ◃▹ . What is the value of σ(a)?

(f) (1 point) Prove that if K ⊆ H then ◃▹ is finer than ∼ .

(g) (4 points) Suppose K ⊆ H and denote by g : G/ ◃▹−→ G/ ∼ the unique map satisfying π = gσ, see Corollary 8 of the file “Finer Equivalences and Lifting Maps.” For all a ∈ G, what is the value of g(aK)?

This question illustrates how bidding dishonestly can end up hurting the cheater. Four partners are dividing a million-dollar property using the lone-divider method. Using a map, Danny divides the property into four parcels s1, s2, s3, and s4. The following table shows the value of the four parcels in the eyes of each partner (in thousands of dollars): s1 s2 s3 s4 Danny $250 $250 $250 $250 Brianna $520 $170 $150 $160 Carlos $320 $350 $210 $120 Greedy $320 $300 $300 $80 Assuming all players bid honestly, which piece will Greedy receive? s1 s2 s3 s4 Assume Brianna and Carlos bid honestly, but Greedy decides to bid only for s1, figuring that doing so will get him s1. In this case there is a standoff between Brianna and Greedy. Since Danny and Carlos are not part of the standoff, they can receive their fair shares. Suppose Danny gets s3 and Carlos gets s2, and the remaining pieces are put back together and Brianna and Greedy will split them using the basic divider-chooser method. If Greedy gets selected to be the divider, what will be the value of the piece he receives?  

Ex 3. Consider the following definitions:

Definition: Let a and b be integers. A linear combination of a and b is an expression of the form ax + by, where x and y are also integers. Note that a linear combination of a and b is also an integer.

Definition: Given two integers a and b we say that a divides b, and we write a|b, if there exists an integer k such that b = ka. Moreover, we write a - b if a does not divide b.

For each proof state clearly which technique you used (direct proof, proof by contrapositive, proof by contradiction). Even if you are not able to prove some of the following claims, you can still use them in the proof of the following ones, if needed.

(a) Given the above definition, is it true that a|0 for all a in Z? Is it true that 0|a for all a in Z? Is it true that a|a for all a in Z? Explain your answers.

(b) Prove that if a and b are two integers such that b≠0 and a|b, then |a| ≤ |b|.

(c) Prove that if a, b and c are three integers such that c|a and c|b then c divides any linear combination of a and b.

(d) Let a be a natural number and b be an integer. If a|(b + 1) and a|(b − 1), then a = 1 or a = 2. (Hint: you may use a clever linear combination...)

(e) Prove that if a and b are two integers with a ≥ 2, then a - b or a - b + 1

How important is the use of an appropriate system of symbols to the development of a branch of mathematics?

June Watson is contributing ​$3,500 each year to a Roth IRA. The IRA earns 3.3% per year. How much will she have at the end of 35​years?

Solve the following optimization problem (Be sure to include the statement of the optimization problem and a graph of the feasible in your solution):

Jamie has joined a building contest. A dog shape requires 3 small blocks and one large block to build. A robot shape requires 5 small bricks and 5 large bricks to build. Jamie has a supply of 240 small bricks and 100 large bricks.

If a dog is worth 2 points and a robot is worth 7 points, how many shapes of each type should Jamie build to maximize the points?