1
vote
3answers
24 views

Prove that $\{(a,b):a,b\in\mathbb N, a\geq b\}$ is denumerable.

If $S=\{(a,b):a,b\in\mathbb N, a\geq b\}$, how do I prove that $S$ is denumerable? Work: Since $S \subseteq\mathbb{N\times N}$ I know that $S$ is denumerable. But I don't know how to structure the ...
-3
votes
1answer
22 views

Integrate $\cot^2x-\frac{\cos^2x}{\tan^2x}$

Integrate $\int{\cot^2x-\frac{\cos^2x}{\tan^2x}}dx$
1
vote
1answer
15 views

If $l(a, b, c) = l(a', b', c')$, then $(a, b, c) = (a ', b', c')k$ for some $k \in F$?

Let $F$ be a division ring. Define $l(a, b, c) = \{(x, y, z) \in F^3 : xa + yb + cz = 0\}$. Question: If $l(a, b, c) = l(a', b', c')$ is it true that $(a, b, c) = (a', b', c')k$ for some $ k \in F$? ...
1
vote
1answer
19 views

Using trial division in $\mathbb{Z}/2\mathbb{Z}[x]$, factor $x^6+x^4+x$ into a product of irreducible polynomials.

I know how to normally factor this, but I am hazy on the idea of irreducible polynomials. I know that $x^6+x^4+x=x(x^5+x^3+1)$ but I am not sure how to tell if the second factor is irreducible, or if ...
0
votes
1answer
15 views

Sequence of complex numbers, having throuble with this problem.

The question: Supose $a,b \in \mathbb{C}$ with $\lvert a \rvert = \lvert b\rvert > 1$. If the sequence $\{a^n - b^{n}\}_{n \in \mathbb{N}}$ is limited, prove that a = b. I was thinking in use the ...
-3
votes
1answer
34 views

Integrate $\int^{1}_{0}{\sin^2x}$ [on hold]

What is the value of this integration ? $\int^{1}_{0}{\sin^2x}dx $
0
votes
1answer
24 views

Prove that this sequence converges

I need to show that $ |r^n|$ converges for $0<|r|<1$. I know this converges to $0$. The problem that I have is that I need to use the observation that $\lim_{x\to inf}|r^{n+1}|=\lim_{n\to ...
0
votes
1answer
10 views

${\bf E}[Y]$ of a joint distribution

So, I have that a joint pdf is given by the formula: $$ 5e^{-5x} / x, \quad 0 < y < x < \infty $$ and I have to find the $Cov(X,Y)$. I know that $Cov(X,Y) = {\bf E}[XY] - {\bf E}[X]{\bf ...
0
votes
2answers
15 views

How many bit strings of length $7$ either begin with two $1's$ or end with three $1's$?

So for the first case (beginning with 2 $1's$) there are: $2*2*2*2*2=32$ ways Second case (end with three $1's$): $2*2*2*2=16$ And then we can just sum it $32+16=48$ different bit string of length 7 ...
1
vote
1answer
24 views

Russian Roulette and conditional probability

Let's say you play Russian roulette with a 6-chamber gun and there is only one bullet in it. Your friend spins and pulls the trigger, he's still alive, and then he gives the gun to you and you need to ...
0
votes
2answers
21 views

How to expand 1/(1+z^2) in powers of (z-a)?Here z is a complex number.

How to expand 1/(1+z^2) in powers of (z-a)?Here z is a complex number. I know for people who knows how to do this this is a stupid problem.But I am just a beginner.Differentiating 1/(1+x^2) seems not ...
1
vote
0answers
8 views

How is the upper bound of a minimisation IP determined during branch-and-bound?

When using the branch-and-bound algorithm to solve an Integer Programming (IP) problem, the entire enumeration tree doesn't need to be evaluated and that's where the speed-up is achieved. Suppose the ...
0
votes
0answers
9 views

What is meant by a Rank 1 Update?

What is meant by a rank-1 update in linear algebra? I've performed the calculations but I don't understand why it is called a rank-1 update.
0
votes
2answers
10 views

Combinations questions

a. How many different 4 letter codes can there be? b. What if letters cannot be repeated? c. What if, in addition, 2 of the letters are x and y? For a, it would simply be $26*26*26*26=456976$ For ...
0
votes
1answer
18 views

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact.

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact. Some helpful definitions: bounded - A subset $S$ of a ...
0
votes
3answers
73 views

What is the proof of $\cosh{x}=\frac{e^x+e^{-x}}{2}$? [on hold]

What is the proof of this equation ? $$\cosh{x}=\dfrac{e^x+e^{-x}}{2}$$
1
vote
1answer
14 views

Question over combinations

A t-shirt is being sold in 8 colors, 4 sizes, collared or tee, and long sleeve or short sleeve. a. How many different shirts are being sold? b. What if collared shirts only come in 5 colors and 2 ...
0
votes
1answer
13 views

Probability: arithmetic on Random Variables

I have a question about the arithmetic on random variable in probability. Question: Are the events $\{X=Y\}$, $\{Y=Z\}$,$\{Z=X\}$ independent? My solution: $$ P(X=Y,Y=Z,Z=X) = {(0.5^2 ...
1
vote
1answer
23 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
0
votes
0answers
14 views

What about uniqueness of general solution?

I found some info about uniqueness for inital value problem. But what about uniqueness of general solution? Is it right that ODE $y'=y$ has two general solutions? 1) $y=Ce^x$ 2) $y=e^{(x+C)}$ Or ...
0
votes
1answer
13 views

A continuity question

Find a non-zero value for the constant k that makes $f(x)=\begin{Bmatrix} \dfrac{\tan(kx)}{x} ,& x<0 \\[6pt] 3x+2k^{2}, & x\geqslant 0 \end{Bmatrix}$ continous at $x=0$. I've been trying ...
0
votes
0answers
14 views

How can I write this in Divergence form

Consider the PDE $u_{xx}-(yu_y)_x-y(u_x)_y+yu_y+(y^2+\frac{1}{H^2(x)})u_{yy}$ I need to write this in divergence form. That is, I need to write it in the form $\sum_{i,j}\frac{\partial}{\partial ...
0
votes
1answer
14 views

Prove that the limit exists of an increasing and bounded function

This was an exam question I had last year but wasn't able to answer it (and still can't). Suppose $a<b$, and $f : (a,b) \to \mathbb{R}$ is a function that is both increasing and bounded. Prove ...
1
vote
1answer
29 views

feedback on my answer regarding set intersections.

Prove or find a counter-example to the claim that for all sets $A,B,C$ if $A\cap B = B \cap C = A \cap C = \emptyset$, then $A \cap B \cap C=\emptyset $. the above statement is not true so i need a ...
1
vote
2answers
25 views

Verify :$\cos^2x=\cot^2x-\frac{\cos^2x}{\tan^2x}$

$$\cos^2x=\cot^2x-\frac{\cos^2x}{\tan^2x}$$ How can I solve it?
0
votes
0answers
8 views

Dynamic Statistics Equation

There are two data sets, Alpha and Bravo. Every data set in this problem has the same structure. A data set consists of six places (a, b, c, d, e, f) and each place in each data set contains an ...
0
votes
2answers
17 views

how to find standard deviation when given a percentage?

I'm stuck on this question, can anyone help? An electronics company's biggest seller is a talking toy. Of the toys produced, 4% have a defective voicebox. A quality control tech randomly collects 250 ...
0
votes
1answer
27 views

Let $f(x)=x^2+bx+4$ in $\mathbb{R}[x]$. For each $b \in \mathbb{R}$, factor $f(x)$ into a product of irreducible polynomials in $\mathbb{R}[x]$.

I know that for a polynomial to be irreducible, this means that if it is factored then one of the factors has to be a unit. I am confused by what this question is asking because there are an infinite ...
1
vote
0answers
29 views

Logical expression

Please can someone give me feedback on my answer to the question below. Question. Surf the internet and find a theorem of number theory. State the claim of the theorem, and then express it in logical ...
0
votes
0answers
14 views

Argument Principle solving the quesion

1/2pi i ∫ c(0,12) f'(z)/f(z) dz where f(z) = (z-5+12i)^4(z-7)^6 / 12(z-8)^6(z-i+6)^7 dz. How do I go about doing this kind of question. I only know its argument principle. I googled it but still cant ...
0
votes
0answers
12 views

Finite element spaces question

When using a square mesh, is it true that Q1 is, in general, a more useful finite element space than P1? Here, Q1=span{1,x,y,xy} and P1=span{1,x,y}.
3
votes
3answers
43 views

$\int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x$

My try, using $x = sec(u)$ substitution: $$ \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x = \int \frac{\sqrt{sec^2(u) - 1}}{sec(u)}tan(u)sec(u) \mathrm{d}u = \int tan^2(u) \mathrm{d}u = tan(u) - u + C = ...
1
vote
1answer
35 views

Product of Differences of nth Roots of Unity

I'm trying to show that $$\prod_{j=1}^{n-1}\left(1-e^{2\pi ij/n}\right)=n$$ but am finding it surprisingly difficult. I know by symmetry that $$\prod_{j=1}^{n-1}\left(1-e^{2\pi ...
3
votes
1answer
21 views

Is C(E)a dual of any linear norm space?

E is a closed bounded set of R. Is C(E)a dual of any linear norm space?
0
votes
0answers
9 views

How to calculate the confidence interval of variance [on hold]

n=80 mean=825 SD= 48.5 whats the formula for calculating 95% confidence interval of variance?
1
vote
0answers
11 views

Subshifts of finite type; No fixed or period 2 points

I'm working out of Devaney's Introduction to Chaotic Systems, and one of the problems I'm working on is to construct a subshift of finite type in $\Sigma_3$ with no fixed or period two points, but ...
0
votes
1answer
24 views

Separability of functions with compact support

Let $X$ be a locally compact metric space which is also $\sigma$-compact. Let $C_{c}(X)$ be the continuous functions on $f$ from $X$ to $\mathbb{R}$ with compact support. Is $C_{c}(X)$ separable? My ...
0
votes
0answers
18 views

Cauchy-Riemann Equations

Let $f(z)=\frac{x^{3}(1+i)-y^{3}(1+i)}{x^{2}+y^{2}}$ at $z$ not equal to zero and $f(z)=0$ at $z=0$. I want to show that the Cauchy-Riemann equations are satisfied at the origin but not analytic. ...
2
votes
2answers
51 views

Proving that (0,1) and [0,1] are numerically equivalent.

as the title suggests, I need help proving that the cardinality of $(0,1)$ and $[0,1]$ are the same. Here is my work: $f:[0,1] \rightarrow (0,1)$ Let $n\in N$ Let $A=\{\frac{1}{2}, \frac{1}{3}, ...
0
votes
1answer
13 views

Show that in Z/2Z[x] two polynomials are associates if and only if they are equal.

I believe that I should show the forward direction by first showing the factorization of two polynomials, f and g, such that f=p1 . . . ps and g=q1 . . . qs, where each pi and qj are irreducible ...
1
vote
1answer
12 views

Dimensions in field extensions

How would I be able to determine $[\Bbb Q(\sqrt{42}, \sqrt{-42}):\Bbb Q]$? So far, I think the dimension might be four as the root equation could be $(x^2 + 42)(x^2 - 42)$.
1
vote
1answer
25 views

How to find I(t)?

I'm working with a SIS model for diseases. Where S stands for susceptibles, and I stands for infected. I have a situation that is modeled by the system: $$S'(t)=\frac{dS}{dt}=-\beta SI-\lambda S$$ ...
0
votes
1answer
28 views

Prove that $a^4 \equiv 1 \bmod 5$ if $\space a \neq 5$

Prove that $a^4 \equiv 1 \bmod 5$ if$ \space a \neq 5$ I've tried showing this by induction. Clearly if $ a = 5$ then $ a \equiv 0 \bmod 5$ now if $a = 1$ then $a^4 - 1 = 0$ which is divisible by ...
3
votes
2answers
27 views

What's the relation between prime spectrum and affine space?

Let $A$ be a ring ,$X$ be the set of all prime ideal of $A$.For each subset $E$ of $A$,let $V(E)$ denoted the set of all prime ideals of $A$ which contain $E$. we have: ...
4
votes
1answer
33 views

Need help solving exponential equation $2\mathrm{e}^x=5-\mathrm{e}^{-x}$

I need help solving $2\mathrm{e}^x=5-\mathrm{e}^{-x}$. I've tried many ways of solving it but I keep getting the wrong answer. By the way, my book says the solutions are $x=-1.518$ and $x=0.825$ ...
1
vote
1answer
9 views

finding conditional expectation under binomial distribution.

Suppose X and Y independent and are both binomial random variables with parameter N, p Compute E(X|X+Y).
0
votes
1answer
8 views

Solving a recurrence relation using forward substitution.

How can I solve this? $$T(n)=3T\left(\frac{n}{4}\right),$$ for $n>1$, $n$ a power of $4$, and $T(1)=3$.
1
vote
2answers
28 views

How do i prove this? (Cauchy's theorem)

Dummit-Foote p.96 Exercise 9 Let $G$ be a finite group divisible by a prime $p$. Define $S=\{(x_1,\cdots,x_p)\in G^n: x_1\cdots x_n = 1\}$ THen, $S$ has $|G|^{p-1}$ elements. HOw do i ...
0
votes
2answers
26 views

proving discrete mathematics or giving counter example

Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational. can anyone explain to me or give a hint on how to start ...
0
votes
0answers
9 views

Subfields of irreducible polynomial fields with known dimensions

Let's say we have an irreducible polynomial, $h(x) = x^4 + x + 1 \in \Bbb F_2[x]$, and that L is a field equal to $\Bbb F_2 [x]/(h(x))$. How would I go about finding a subfield K such that $[L : K] = ...

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