Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...

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0
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2answers
15 views

If f1 and f2 are 2 continous functions then f = { f1(x), x is rational and f2(x) , x is irational is continous in x0 if only if f1(x0)=f2(x0)

I have to prove that. I know that i have to take a string with rational number and a string with irational numbers but i don;t know how to do next. Please help me !
0
votes
2answers
28 views

Numbers in a list which are perfect squares and perfect cubes of numbers

How many numbers in the list $$1,2,3,...,2001$$ are perfect squares and perfect cubes of whole numbers? My progress: Well I do know $$1,4,9,16,25,36,...$$ are perfect squares and $$1,8,27,64,...$$ ...
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votes
2answers
16 views

Perfect Square and Multiple question

The population of a village is a perfect square. Later, with an increase of 100, the population was 1 more than a perfect square. Now with an additional increase of 100, the population is a perfect ...
2
votes
2answers
37 views

Number of homomorphisms $\mathbb Z_3 \times \mathbb Z_3\to\mathbb Z_9$

I had this wonderful idea: $f$ is homomorphism: $G\to H$, $|G| = |\ker f| \cdot |\operatorname{im} f|$, $\ker f$ - subgroup of $G$, and $\operatorname{im} f$ - subgroup of $H$, so their orders must be ...
1
vote
1answer
20 views

Please assist me with proving this

Prove that if $a$,$b$ and $c$ are angles in a triangle,then $$ \tan\left(\frac{b-c}{2}\right) = \frac{b-c}{b+c}\cot\left(\frac{a}{2}\right) .$$
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votes
1answer
11 views

Integreal around a unit circle

I know that when $m \in \mathbb{Z} \backslash \{ 0 \}$, we have $$ \int_0^1 e^{2 \pi i m \beta} \ d \beta = 0. $$ I was wondering if there is a simple formula for the following similar integral, when ...
1
vote
0answers
10 views

Solving ODE numerically - getting local truncation error

Well I have NO idea how to do this or even where to start Compute the order of magnitude of the local truncation error of the following time integration scheme: $$y_{n+1} = y_{n-1} + 2h f(y_n)$$ H ...
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votes
1answer
11 views

Find the equation based on line segment

Suppose you draw a straight line segment between points $(a, 0)$ and $(0, 1 – a)$ for every real number $a\in (0, 1)$. All these segments taken together fill out a shape bounded by the positive ...
1
vote
4answers
30 views

Basic Probability Question (Expected Value)

We are given a bag of marbles containing 6 blue marbles, 7 red marbles, and 5 yellow marbles. We select 4 marbles without replacement. How can we calculate the expected value of the number of blue, ...
0
votes
3answers
63 views

How to prove $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ converges?

Could you please give me some hint how to prove this statement: If $0<a_n<1$ for each n, then $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ ...
0
votes
0answers
10 views

define the set of all affine real-valued functions

Define the set of all affine real-valued functions G := { $f$_a,b : a,b ∈ $R$ , a≠0}, where $f$_a,b : $R$ → $R$, x → ax + b. This is a group under composition ○. a) N := ...
0
votes
0answers
9 views

Bilinear Form with Constant condition show existence of bounded linear operator

The question states: Let $F:L^2(\mathbb{R}) \times L^2(\mathbb{R}) \to \mathbb{C}$ be a bilinear form (meaning that it is linear in each factor when the other is held fixed) such that there is a ...
1
vote
0answers
46 views

expand a rational function in a power series

$$\frac{4-x}{(2-x)(1-x)^2}$$ Expand in ascending powers of x, stating when the expansion is valid; also write down the coefficient of $x^n $
1
vote
1answer
14 views

Interest Accumulation - Geometric Sequence

Hello I have just worked a question in which I get an answer different to the answer in my book. The question states: If a person deposits 500 at the end of each month for 20 years at an AER of ...
0
votes
0answers
27 views

Derive three-point Gaussian quadrature formula [on hold]

Derive the three-point gaussian quadrature formula for
0
votes
1answer
21 views

Digit reversal arithmetic

How many three-digit odd numbers become bigger when their digits are reversed? $$abc<cba$$ and $c$ is either 1,3,5,7,9. This is the furthest I managed to reach.
2
votes
1answer
36 views

If every left coset of $H$ is a right coset the show that $H=aHa^{-1}$ for all a in G

$H$ is a subgroup of G. My attempt: $ha=ah' $ for every $h\in H$, where $h'\in H$ doesn't necessarily equal to $h$. So for each $h\in H$, $h=ah'a^{-1}\in aHa^{-1}$, so $H\subseteq aHa^{-1}$. Then how ...
3
votes
1answer
26 views

Determine $\text{Vect}_k(S^n)$

Let $\text{Vect}_k(X)$ denote the isomorphism classes of rank $k$ real vector bundles over smooth manifold $X$. Is there a rule for determining $\text{Vect}_k(X)$ over reasonably nice manifolds? ...
1
vote
2answers
52 views

Three-Digit numbers divisbile by 3

How many three digit numbers are divisible by 3 and have an additional property that the sum of of their digits is 4 times the middle digit? My approach: let the number be $abc$ so $$abc \equiv ...
0
votes
2answers
56 views

solutions of $x^2\equiv 1 \pmod p $ [duplicate]

If p is a prime, show that the only solutions of $x^2\equiv 1 \pmod p $ are $x=1$ and $x\equiv -1 \pmod p$. (from herstein's abstract algebra chapter2 section4 lagrange's theorem problem 15, this ...
2
votes
3answers
51 views

5 digit number $a6a41$ divisible by 9

In the 5-digit number $a6a41$ each of the a's represent the same number. If the number is divisible by 9, what is the digit represented by $a$? I first approached this by saying $$2a + 11$$ since ...
0
votes
0answers
12 views

supremums & functions

I'm am having some trouble with the below and was wondering if anyone would be kind enough to help me out. I have the following: $|L^n_{j}|\leq |\frac{\Delta t}{2} u_{tt}| + |\frac{ah}{2} u_{xx}|$ ...
1
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2answers
51 views

If $(1 + 2i)$ and $(3 - 2i)$ are two roots of $x^5 + ax^4 + bx^3 + cx^2 + dx + 4$, then $a$ =?

Consider the polynomial $x^5 + ax^4 + bx^3 + cx^2 + dx + 4$ where $a, b, c, d$ are real numbers. If $(1 + 2i)$ and $(3 - 2i)$ are two roots of this polynomial then what is the value of a? Well, I ...
6
votes
3answers
72 views

Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$.

Question : Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$. Hint: consider decimal expansions ...
0
votes
1answer
27 views

finite and infinite, vector space, linear transformation

I have to answer these questions for homework and I don't know if I'm answering these correctly. I think most of them are correct, but a double check would be much appreciated. a) If $S$ is a set ...
1
vote
0answers
7 views

Symetric powers of $sl_2$ representations

I'd like to understand some special things about representations of $sl_2$ (which is considered as a Lie algebra over $\mathbb{C}$). First, it can be shown that for each $n\in \mathbb{N}$ there is ...
1
vote
1answer
25 views

Energy Transfer in a Mechanical System - Standard Pulley Scenario

I understand that this is quite a basic question, I am new to dynamics and have trouble starting off questions, I found it quite difficult to find an example question alike to the one below thus I am ...
1
vote
0answers
39 views

How to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$?

Could you please give me some hint how to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$ when f(x) is continuous for $0<x\le1$ and $\lim_{x\to0^+}x^3f^2(x)=1$ ? I tried the usual way: ...
6
votes
1answer
25 views

Proof for Complex Analysis Inequality

This is a homework assignment that will be graded; so I'm not specifically asking for an answer. But I could use a hint, as it's been a few days and I'm still not sure if how I've proved it would ...
2
votes
2answers
47 views

Prove that if $y,z \in Q$ then $y^z \in A$

Question : Prove that if $y,z \in Q$ then $y^z \in A$ My attempt: Definition 2.7.8 states that a number s is an algebraic number when there exists some $p \in Z[x]$ such that $p(s) =0$. Let us ...
2
votes
0answers
21 views

Proof involving the Cauchy mean value theorem, Taylor series and induction.

I've been working on this problem for a long time and I'm stuck on it (nothing that I come up with is working). If someone could give me a hint or do the first part of the proof, that would be nice. ...
1
vote
0answers
11 views

Homomorphism from matrix to complex set.

Can anyone please help me simplify the homomorphism property. If $f(ab) = f(a)f(b)$ Let $f$ be a homomorphism. $f(A) = (az + b)/(cz + d)$, where $A$ is a determinant from the $GL(2,\mathbb C)$ and ...
-1
votes
0answers
23 views

How to show that there are as many left cosets as there are right cosets? [duplicate]

G is a finite group and H is a subgroup, How to show that there are as many distinct left cosets of H as there are right cosets? (If this is a duplicate, why not show me where is the original one, ...
2
votes
4answers
56 views

Proving for all n that $\sum_{i=0}^n \frac1{2^{i}} < 2$

Proving for all n $\in \mathbb N$, $$\sum_{i=0}^n \frac1{2^{i}} < 2$$ Hint. First prove that the left hand side can be expressed in closed form, i.e. without using the summation operator. This is ...
-1
votes
1answer
19 views

multivariable problems

Brand Z's annual sales are affected by the sales of related products X and Y as follows: Each $\$1 $ million increase in sales of brand X causes a $\$2.1$ million decline in sales of brand Z, whereas ...
2
votes
1answer
22 views

Prove that $\lim_{x\to a} f(x) = \infty \implies \lim_{x\to a} \frac{1}{f(x)} = 0$

So I have to prove that: $$\lim_{x\to a} f(x) = +\infty \implies \lim_{x\to a} \frac{1}{f(x)} = 0$$ By the pricese definition of a limit, if $\lim_{x\to a} f(x) = +\infty$, then: $$\forall M>0, ...
1
vote
1answer
15 views

Picture and length of an exponential complex curve

Problem 4: Draw a picture of the curve $\gamma(t) - e^{-t}e^{it}$ for $0\le t\le b$ and calculate the length of this curve. What happens if you let $b \to\infty$? Does this infinite curve have ...
0
votes
0answers
25 views

Math challenge(Infinite sets): Bottles of Beer

Imagine a counter with stools that stretch across the room(infinite). All the stools are occupied. Two drunks come in and want to squeeze in to sit down. Drunk #1 walks in and tells the person on the ...
1
vote
2answers
61 views

a question about limit, I am struggling with this!

Suppose that {$a_n$}is a sequence of positive numbers.For each n which is a natural number,let $b_n$=($a_1+a_2+......a_n$)/n,prove that $\sum b_n$ diverges to $+\infty$. This question is my homework, ...
0
votes
1answer
17 views

equation of horizontal/vertical line and changing to y=mx+c format

I've been given the equation 2x-3y=5 I was wondering whether this is a horizontal or vertical equation and how would I rearrange this to y=mx+c. I know that this is a fairly basic equation but the -3y ...
0
votes
1answer
24 views

General linear group/special linear group, isomorphic to R^(*)

Let $GL(n,\mathbb{R})$ be the group of invertible $n \times n$ real matrices, and let $SL(n,\mathbb{R})$ be the group of $n \times n$ real matrices of determinant $1$. And $\mathbb{R}^*$ be the group ...
6
votes
2answers
177 views

Definite integral problem

I was solving a definite integral problem which was reduced to : $$\int^{1}_{0} \frac{\ln(1+t)}{t} dt$$ I couldn't solve it and when I saw the solution, the answer was simply given as ...
0
votes
2answers
16 views

simplifying 3 different equations with 3 different variables

I am stuck trying to get the values for x, y, and z. I keep moving variables around but I end up getting answers like x = x or z = z and I do not think that is what I want. It's really just algebra ...
0
votes
1answer
24 views

Prove that f is 1 time continuously differentiable and express f' in terms of f

Suppose that $f$ is a continuous function on R that satisfies $$f(x) = 5 + 2\int_0^x f(t) \,dt$$ Prove that $f \in C{^1}(\mathbb{R})$ [ aka $f$ is one-time continuously differentiable on $\mathbb{R}$ ...
2
votes
0answers
12 views

d-dimensional integral of exponential and determinant

I'm working on a question from Stein and Shakarchi's Fourier Analysis. This is exercise 5 from chapter 6: Let $A$ be a $d \times d$ positive definite symmetric matrix with real coefficients. Show ...
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votes
1answer
18 views

Find tangent from other two trig ratios

I understand the six trig ratios, and know that tangent= opposite/adjacent. I feel like, on a website, I've seen people use either cosine/sine or sine/cosine (can't remember which) to find the ...
0
votes
1answer
15 views

Calcuate the test statistic

I am facing a little problem is this question. Can somebody please help e here A sample of 500 drivers was asked whether or not they speed while driving. The following table gives a two-way ...
0
votes
0answers
38 views

Thickness of G when G is a simple connected graph

The thickness of a simple graph G is the smallest number of planar subgraphs of G that have G as their union. Show that if G is a connected simple graph with v vertices and e edges, where v ≥ 3, then ...
0
votes
2answers
25 views

Having trouble showing the cardinality of two infinite sets is the same

We just learned about Aleph-naught today and I read about it on wikipedia but I do not know how to go about solving this problem in my homework: Prove that N(natural numbers) has the same ...
2
votes
1answer
33 views

Prove that if $v_1,v_2,…,v_r$ form a linearly independent set of vectors in $V$…

Let $S$ be a basis for an n-dimensional vector space $V$. Prove that if $v_1,v_2,...,v_r$ form a linearly independent set of vectors in $V$, then the coordinate vectors $(v_1)_S,(v_2)_S,...,(v_r)_S ...