Write a function or script that will solve linear systems of any size by Gaussian elimination with partial pivoting in Python.

6. Find all monic irreducible polynomials of degree ≤ 3 over Z3. Using your list, write each of the following polynomials as a product of irreducible polynomials over Z3:

(a) x^4 + 2x^2 + 2x + 2.

(b) 2x^3 − 2x + 1.

(c) x^4 + 1.

GRAPH THEORY

Prove/Show that a connected Graph G is not separable if and only if it is nonseparable.

Definitions for Reference: A connected Graph G is called nonseparable if it has no cut vertices (A vertex v in a connected graph G is caled a cut vertex if G-v is disconnected)

A connected graph G is called separable if there exist subgraphs H1, H2 ⊂ G. with E(H1) ∪ E(H2) = E(G) and E(H1) ∩ E(H2) = ∅. V (H1) ∪ V (H2) =V (G) and V (H1) ∩ V (H2) containing a single vertex.

A plane in R3 can be given by an equation Ax + By + Cz = D where A, B, C, and D areconstants and A, B, and C are not all zero. Suppose two planes are given by equations A1x+B1y+C1z = D1 and A2x+B2y+C2z = D2. The intersection of the two planes can be empty, or it can be a line, or the planes could be identical. How can the correct possibility be determined from the constants in the equations? Explain! (Hint: Think in terms of solving linear systems.)

Math of Finance

Alex is paying 1184.87 in monthly mortgage payments. She wants to refinance her existing mortgage loan of $100,000 at 14% interest for 30 years that she obtained 4 years ago. Her mortgage officer informed her that he could get her a rate of 10% but the finance cost would include paying a prepayment penalty on the existing loan, which is equal to 6 months interest on the balance of the loan, plus a closing cost of $2000. Would it be worth if for Alex to refinance?

MATLAB please make it simple to understand

P4.2.5 Write a script that draws an equilateral triangle that is partitioned into four smaller equilateral

triangles by connecting the midpoints of its sides. The four little triangles should be colored

differently.

make a first order differential equation from a problem in real life and give an explanation and derivative (the proof) of the formula

Explain in what sense composition of linear fractional transformations corresponds withmatrix multiplication

Consider the following family of sets: For any n∈N, we define Q_n={ q \in Q | q= \frac{m} {n^k} for some k, m \in Z}

(That is, Q_n is the set of rational numbers expressed with the denominator that is a power of n.)

a) Is this set closed under addition, subtraction, multiplication, and division?

b) Decide the truth value of the following statement: Q 12 ⊆ Q 18 . Justify.

7. Let C be the oriented curve consisting of four line segments from (0, 0, 0) to (0, 2, 0), from (0, 2, 0) to (0, 0, 1), from (0, 0, 1) to (1, 0, 0), from (1, 0, 0) to (0, 0, 0).

Consider the vector field F (x, y, z) = <2z(1 + y) + e^x^10 , 3xz, 4(x + 1)y>.

(a) Compute the curl of F .

(b) Compute the line integral C F dr . (Hint: start by sketching the curve C.)

A topological space X is zero − dimensional if it has a basis B consisting of open sets which are simultaneously closed. (a) Prove that the set C = {0, 1}^N with the product topology is zero-dimensional. (b) Prove that if (X, d) is a metric space for which |X| < |R|, that is the cardinality of X is less than that of R, then X is zero-dimensional.

Use the simplex algorithm to solve the following LP:

??? ? = ?1 + ?2

?.?. ?1 − ?2 ≤ 1

?1 + ?2 ≤ 2

?1 , ?2 ≥ 0

Use the following information to complete the concrete mixture designs for questions 4.

The following materials are available for a concrete mixture design:

ASTM C 150 Type I/II Cement

            Relative density of 3.15

Coarse Aggregate: well-graded ¾ in. maximum-size aggregate (MSA), crushed limestone, angular

            Oven-dry specific gravity: 2.45

            Absorption: 0.80%

            Oven-dry rodded bulk density: 97 lb/ft³

            Moisture content of coarse stockpile: 0.63%

Fine Aggregate: well-graded natural sand

            Oven-dry specific gravity: 2.64

            Absorption: 1.4%

            Moisture content of fine stockpile: 2.8%

            Fineness Modulus: 3.00

An air entraining admixture will provide enough air for the design mixture at a dosage rate of 3 oz/ft³, if the system needs to be air entrained.

Question 4.

Concrete is required for an 8 in thick exterior concrete slab in central Texas. A specified compressive strength, f’_c , of 5700 psi is required at 28 days using an ASTM C 150 Type II portland cement. The design calls for a minimum of 2 in. of concrete cover over the reinforcing steel. The minimum distance between reinforcing bars is 4 in. A slump of 6 inches should be the target. The sidewalk is located in an environment that does not require air entraining. The concrete is required to have low permeability when exposed to water and moderate sulfates. The concrete will be exposed water soluble sulfates in soil at 1.14%. No statistical data on previous mixes are available. Determine:

1. The theoretical mixture proportions of all concrete constituents (fine aggregate, coarse aggregate, water, cement, chemical admixtures) based on 1 yd³ of concrete on an oven dry basis for aggregates.  

2. The as-batched mixture proportions based on the moisture contents provided in the materials available section.

3. How much of each material would be required to make a 20 ft long by 15 ft slab from the concrete mixture described above?

The function given possesses real roots:

x^3 – 1.7x^2 + 0.84x – 0·108

Using a precision of 1% of a root, find the estimates of all roots using the secant method. After finding a root, reduce the polynomial using polynomial reduction before finding the next root with the secant method.

PLEASE USE MATLAB