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1
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1answer
65 views

Solving the differential equation $y' \tan y = \frac1x$

Express the differential equation $$\tan y\,\frac{dy}{dx}=\frac{1}{x}$$ in a form not involving $\frac{dy}{dx}$. I undersand the concept of a differential equation (though, as a student, I am ...
1
vote
2answers
1k views

Determine all primes p for which 5 is a quadratic residue modulo p

I need to determine all primes p for which 5 is a quadratic residue modulo p. I think I'll need to use quadratic recprocity laws to do this. Ie. I need to need to find numbers p where x^2 is congruent ...
4
votes
2answers
83 views

For every integer, some multiple of it is of the form $99 \ldots 900 \ldots 00$

The goal is to prove that for every positive integer $ z$ there exists a positive integer $a$ such that $az = 99 \ldots 9900 \ldots 00$. Let $a = \frac {99 \ldots 9900 \ldots 00}{z}$ That ...
-3
votes
1answer
41 views

Calculus use of integral [on hold]

Assume that the price of a product is at a constant value of $\$100$ per unit or the marginal function is $MR=f(x)=100,$ where $x$ equals the number of units sold $a)\ $ What is the total revenue ...
0
votes
3answers
48 views

How to convert an angle from degrees to radians? [closed]

How can I convert these angles to radians? $\begin{align}450^\circ \\ 630^\circ \\ {-}405^\circ \\ {-}675^\circ \\ {-}1035^\circ \\ {-}1125^\circ \\ 135^\circ \\ 335^\circ \end{align}$ Solved now ...
0
votes
1answer
49 views

Correct proof of supremum property?

Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$. Prove that $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$. ...
2
votes
1answer
295 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
2
votes
1answer
47 views

Let $a,b \in R^5$ be two independent vectors and let $W$ be the subspace spanned by $a,b$. Prove that $U\cap W$ contains a non-trivial vector

Let $a,b \in \Bbb R^5$ be two independent vectors and let $W$ be the subspace spanned by $a,b$. Let $U=\{(x_1, x_2, x_3, x_4, x_5) \in \Bbb R^5\colon x_1 + 2x_2 + x_3 - 3x_4 + x_5 = 0 \}$. Prove ...
1
vote
0answers
41 views

Using the method of characteristics to find a general solution to PDE

I want to find the general solution to $3U_x-4U_y = x^2$ using the method of characteristics. I'm given the answer which is $U(x,y)=\frac{x^3}{9}+F(3y+4x)$ but I'm having trouble getting to this ...
1
vote
4answers
82 views

How to compute 1/7 in base 8?

This is probably a very basic question but nonetheless I choked when I got in a math for programmers class. I was taught how to convert from base to base but I have no idea how to convert fractions to ...
0
votes
4answers
38 views

Prove that T = I with Linear Transformations.

Suppose that $T \in L(V)$ and $T^2 = I$ and -1 is not an eigenvalue of T. Prove that T = I. What I tried was: Suppose $\lambda$ is an eigenvalue of T such that $T(v) = \lambda v$ Then we know that ...
0
votes
0answers
19 views

Question about Boundary points of the sets in metric space

Let A be a metric spaces. Prove the following properties: The boundary of $A$ equals $A'-A$ The boundary of $A$ is the closed set. $A$ is closed if and only if it contains its boundary. Where ...
1
vote
1answer
126 views

Showing that planes intersect

let there be two planes $$2x-y-5z+11=0$$ and$$2x+2y+z-1=0 $$ show that they intersect attempt at a solution: If planes do not intersect they are parralel hence there is a $t\in R$ such that ...
-1
votes
2answers
55 views

Maclaurin series of the function $\frac{x^2}{2+3x^2}$

I got this question: Find the Maclaurin series of the function $\frac{x^2}{2+3x^2}$ and find its domain of convergence. I tried using the binomial series $(1+x)^m = 1 + \sum_{k=1}^{\infty}{m \choose ...
11
votes
3answers
456 views

Proving that $e$ is irrational

Show that $e$ is irrational. Recall $\mathrm{e} = \exp(1)$ so assume $\mathrm{e}$ is rational , then $$\sum\limits_{k=0}^\infty \frac{1}{k!} = \frac{a}{b}\quad \text{for some positive ...
1
vote
3answers
278 views

The Lebesgue Covering Lemma

Let $(X,\tau)$ a compact metric space and $\{ U_i : i \in I \}$ an open cover of $X$. Show that there is $r>0$ such that for all $a \in X$ there is an $i \in I$ such that $B_{r}(x) \subseteq ...
3
votes
1answer
44 views

Question about the Betti numbers

can someone explain me this definition from :http://en.wikipedia.org/wiki/Betti_number The $n^{th}$ Betti number represents the rank of the $n^{th}$ homology group, denoted $H_n$ "which tells us the ...
3
votes
2answers
438 views

The sum of two irrational square roots

This is very similar to this question, but I was wondering if there was a simpler proof. In particular, a proof that would prove that $\sqrt{x}+\sqrt{y}$ is an irrational number if both $\sqrt{x}$ ...
3
votes
1answer
213 views

How to Store a Banded Matrix by Diagonal

I'm taking a graduate level independent study course this semester in Matrix Computations. I'm not getting much support from the professor, so am turning to the excellent StackExchange community for ...
0
votes
3answers
71 views

The limit of $((1+x)^{1/x} - e+ ex/2)/x^2$ as $x\to 0$

$$\lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}=\,?$$ by directly substituting $x=0$ i got $\infty$ by using L-H's rule, i got $-1/8$ the given options are $a)\frac{24e}{11}$ ...
0
votes
2answers
58 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
0
votes
0answers
23 views

Find the Fixed points

Let $L=\mathcal P(\mathbb N)$ be a complete lattice of subsets of $\mathbb N$. Find the smallest and the greatest Fixed Point: 1) $F(X)=\left\{ x \mid x+1\in X \right\}$ 2) $F(X)=X \setminus \left\{ ...
0
votes
0answers
152 views

Convert expression to NAND only

Endless youtube videos and reading through notes later I am yet again stuck. I have to covert the following to NAND only $$\bar{A}\cdot\bar{B}\cdot\bar{C} + A\cdot\bar{B}\cdot C + A\cdot B\cdot ...
-2
votes
1answer
48 views

A sequence of functions converging to the Dirac delta

let $g_n(x)=\frac{1}{2}n $ for $|x|<\frac{1}{n}$ and for positive integer n. Prove that $$\lim_{n \to \infty} g_n(x)=\delta(x)$$ Pretty evident after a quick sketch, but I don't know how to show ...
-2
votes
1answer
58 views

Simplifying the sum of two consecutive binomial coefficients

I've been trying to figure out how they simplified the right side of the equation all the way down. $$\eqalign{ \binom nk + \binom n{k-1} & = \dfrac{n!}{(k)!(n-k)!}+\dfrac{n!}{(k-1)!(n-k+1)!} \\ ...
1
vote
1answer
210 views

Statistics Question of unbiased estimator

Could anyone give me a hint? or way to solve?
0
votes
0answers
33 views

What is the linear combination of B?

I have a problem where I am finding $A^n$B where B=$[3,1,1]^t$. I know the steps in solving, but I do not remember how to find linear combination. I do not see it. There has to be a way to calculate ...
2
votes
3answers
221 views

Summation of Infinite Geometric Series

Determine the sum of the following series: $$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} $$ My work: $$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} = \sum_{n=1}^{\infty } \frac{-1}{7} ...
2
votes
1answer
39 views

Finding conditional distribution

Let $X$ and $Y$ be independent $Exp(1)$-distributed random variables. Find the conditional distribution of $X$ given that $X + Y = c$ ($c$ is a positive constant). this is my idea: $$f_{X \mid ...
1
vote
1answer
27 views

Prove existence of Diagonalizable Matrix

Suppose R, T $\in L(F^3)$ each have 2, 6, 7 as eigenvalues. Prove that there exists an invertible operator S $\in L(F^3)$ such that $R=S^{-1}TS$. What I got so far is that since R and T have three ...
4
votes
1answer
76 views

If the same message is sent to Alice and Bob who are using different public keys, how can somoene following the channel find $m$

Alice and Bob are using different public keys, Alice is using ($N_{1,2}$) and Bob ($N_{2,2}$). A message, $m$ is sent to both of them using their RSA systems. It is also true that $N_1$ and $N_2$ are ...
2
votes
2answers
96 views

How to I write $\frac{7^{2n}}{4^{3n}}$ as a geometric series?

I am trying to write $$\frac{7^{2n}}{4^{3n}}$$ as a geometric series which has the form:$$\sum\limits_{i=0}^n{ar^n}$$. I'm not sure if I should get in the form $$\left(\frac{7}{4}\right)^{2n}$$ ...
0
votes
2answers
46 views

Geometric meaning of results obtained in (a) and (b)

The task: Plot the function $\sqrt{1-x^2}$. What does it look like? What is the geometric meaning of the results you obtained in (a) and (b)? Can anybody help me with geometric mean? I can't ...
1
vote
3answers
58 views

If one number is thrice the other and their sum is $16$, find the numbers

If one number is thrice the other and their sum is $16$, find the numbers. I tried, Let the first number be $x$ and the second number be $y$ Acc. to question $$ \begin{align} x&=3y &\iff ...
2
votes
2answers
129 views

Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero.

Let $f$ be an entire function. Suppose $f(z)$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero. How to prove? Totally I have no idea... Please give the solution in detail since ...
1
vote
3answers
75 views

How does $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ simplify to $1 - \sqrt 2 $?

I've the answer for a question in my textbook to be: $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ which i've then simplifed to: $-\sqrt {3 - 2\sqrt 2 } $ However my textbook states $-\sqrt ...
3
votes
2answers
2k views

Maximum likelihood estimation of $a,b$ for a uniform distribution on $[a,b]$

I'm supposed to calculate the MLE's for $a$ and $b$ from a random sample of $(X_1,...,X_n)$ drawn from a uniform distribution on $[a,b]$. But the likelihood function, ...
2
votes
4answers
34 views

Heaviside Unit Step Function

Convert to heaviside function: $$f(t) = \begin{cases}e^t ,& 0 \leq t \leq 1 \\0 ,& t > 1\end{cases}$$ My attempt: $f(t) = U(t) e^t - U(t-1) e^t $ I think my solution is not right because ...
3
votes
4answers
43 views

Prove a property about the centralisator

Let G be a group and $U \subseteq G$ a subgroup. Let $x \in G$ be arbitrary. How to show that $C_G(xUx^{-1})=xC_G(U)x^{-1}$ where $C_G(U):=\{g\in G : gu=ug$ $\forall u\in U\}$ For the first ...
2
votes
0answers
37 views

What is a good technique for evaluating this double integral?

The integral is: $ \int_0^1 \int_0^1 \frac{x^2 - y^2}{(x^2 + y^2)^2} dxdy $. I'm having difficulty finding an appropriate technique for evaluating it. I initially thought that polar coordinates ...
1
vote
1answer
42 views

Evaluating $\lim_{h\rightarrow 0}\frac{2^{8\cos(h)}}{8h}\left [ \sin^{8}(\pi/6+h))-\sin^{8}(\pi/6) \right ]$

$$\lim_{h\rightarrow 0}\frac{2^{8\cos(h)}}{8h}\left [ \sin^{8}(\pi/6+h))-\sin^{8}(\pi/6) \right ]$$ My Attempt: For $\lim_{h\rightarrow 0}\frac{\sin^{8}(\pi/6+h)-\sin^{8}(\pi/6)}{h}=f'(x)=8\cdot ...
3
votes
1answer
42 views

Finding derivative form the definition

I want to find the derivative of the function $f:\mathbb R^n\to \mathbb R^m$ at a point $x_0\in \mathbb R^n$, where $f(x)=c\in \mathbb R^m$, is a constant function. What I did is as follows: If $f$ ...
0
votes
1answer
234 views

Striking off a digit from each of the numbers written in seven rows, while preserving arithmetical operations

Problem Strike off any digit from each number in seven rows (need not be at same place) and combine the same operations with 3 digit numbers to get the same addition. After this strike off another ...
0
votes
3answers
63 views

Can you factor before finding derivative?

Say the function is $y=\frac{x^2-1}{x-1}$ Can you factor functions before finding the derivative or does that not work?
0
votes
1answer
37 views

Doubts about locus and its equation

Two points A and B with $(1,1)$ and $(-2,3)$ respectively are given.find the locus of point P.So that area of $\Delta$PAB is 9 square units. answer is :- $2x+3y+13=0$ or $2x+3y-23=0$. how i tried:- i ...
1
vote
2answers
39 views

Find the Fixed points (Knaster-Tarski Theorem)

Let $L=\mathcal P(\mathbb N)$ be a complete lattice of subsets of $\mathbb N$. a) Justify that the function $F(X)=\mathbb N \setminus X$ does not have a Fixed Point. I don't know how to solve this. ...
0
votes
2answers
33 views

Could anybody provide a more detailed explanation of a tangent equation in its general form?

In my textbook I'm currently at the topic of a tangent line to an ellipsis and hyperbola. And there I've encountered this statement: If a curve has an equation $$ y = f(x) $$ then an equation of a ...
1
vote
3answers
229 views

Find the limit of using L'Hopital's Rule

Could anyone explain to me how to calculate the limit $$\lim_{x \to 0} \frac{1}{\sqrt{x^3}} - \frac1{\sin x}$$
0
votes
1answer
70 views

How to simplify $3^{(2\log_335)}$

$3^{2\log_35}$ How do I simplify this? This is what I have done so far: $2\log_35=\log_35^2=\log_3(25)$ $3^{\log_3(25)}$ What do I do from here? And the answer is one of these mixed solutions: ...
0
votes
3answers
386 views

Proving by mathematical induction: $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+…+\frac{1}{\sqrt{n}}>2(\sqrt{n+1}-1)$ [duplicate]

Possible Duplicate: Proof of an inequality: $\sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}}$ Proving ...