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0
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2answers
56 views

How does $\sqrt {\frac{{4 + \sqrt {15} }}{8}} = \frac{{\sqrt {8 + 2\sqrt {15} } }}{4}$

I have the follow answering to a question from my textbook: $\sqrt {\frac{{4 + \sqrt {15} }}{8}}$ However my textbook simplifies it to: $\frac{{\sqrt {8 + 2\sqrt {15} } }}{4}$ I've checked and my ...
1
vote
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 ...
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 $u-\epsilon < a$. ...
0
votes
0answers
30 views

The set of rational numbers, each point is point accumulation

Please let us help someone by telling you a precise formulation is below, and then someone please tell me solution that has since become like that with a few days my friend we debates, here my ...
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 ...
0
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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 ...
-3
votes
1answer
40 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 ...
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0answers
39 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
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 ...
1
vote
3answers
110 views

Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$

How to evaluate the following integral $$\int_0^\infty\frac{dl}{(r^2+l^2)^{\large\frac32}}$$ The solution is supposed to look like this, unfortunately I can't derive it. $$ ...
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 ...
0
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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
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 ...
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 $\sqrt{x}$ ...
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 ...
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} ...
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 ...
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 $$\left(\frac{7}{4}\right)^{2n}$$ ...
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 ...
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
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 ...
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 ...
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 ...
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 $ ...
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
60 views

Every Cauchy sequence is bounded

Please help me to understand step-by-step how this example is proven. The statement is follows: Every Cauchy sequence is bounded. since I do not understand how verified, please help me, thank ...
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 ...
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
38 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
3answers
70 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
55 views

Grade 4 level arithmetic maths problem that I can't work out

My grade 4 son was given this problem as homework, except I can't even seem to work it out! Any help is appreciated. Each student in year 4 was asked to bring a paint tin, a paint brush and glue for ...
2
votes
1answer
39 views

Question on continuity with [x]

given the function $f(x)=\frac{2[x]}{3x-[x]}$ the question is to find continuity of the function at $x=1$ and $x=\frac{-1}{2}$ note: [x] denotes the largest integer which is less than or equal to x. ...
2
votes
3answers
36 views

inverse of quadratic log functions

Can a Log function with a quadratic have an inverse function? The specific question is to find the inverse of $$f(x) = \log_2(x^2-3x-4)$$ The function already fails the horizontal line test, but ...
1
vote
1answer
22 views

If $K_1,…,K_n$ are compact convex sets then ${\bar conv}(K_1,…,K_n)= conv(K_1,…,K_n)$

If $X$ is a locally convex space and $K_1,...,K_n$ are compact convex subsets of $X$, then ${\bar conv}(K_1,...,K_n)= conv(K_1,...,K_n)$ and this convex hull is compact. Unfortunately I do not have ...
0
votes
1answer
28 views

Diagonalization of Skew symmetric matrix

I have a skew symmetric matrix $$C=\left( \begin{array}{ccc} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \\ \end{array} \right).$$. and we have the relation $C=UDU^{-1} ...
0
votes
10answers
76 views

Complex numbers and their imaginary parts

Question: If $$z = \left(\frac{\sqrt3}{2} + \frac{i}{2}\right)^{107} + \left(\frac{\sqrt3}{2} - \frac{i}{2}\right)^{107} $$ Show that Im(z) = 0 I have no idea how to even start the question. Please ...
0
votes
4answers
60 views

Give a Counterexample if V is infinite dimensional

$V = nullT \oplus rangeT$ if and only if $V = null T + rangeT$. Where $T \in L(V)$ I'm having alot of trouble coming up with an example for this. Shouldn't both cases always fail if V is infinite ...
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 ...
2
votes
2answers
38 views

If G is a group and N is normal in G with index d, then $x^d \in N$

I want to show the statement in the title. If $G$ is a group and $N$ is normal in $G$ with $[G:N]=d$, then $x^d \in N$ for all $x \in G$ I want to consider the image $xN$ of $x$ in $G/N$ $G/N$ has ...
1
vote
2answers
29 views

Applying a polynomial to an operator?

Suppose $T \in L(V)$ and $\exists$ a positive integer n such that $T^n = 0$. Prove that $(I-T)$ is invertible and that $(I-T)^{-1} = I + T + \dots + T^{n-1}$. I wish I could say that I attempted ...
1
vote
2answers
28 views

Prove that these have the same eigenvalues

Suppose $T \in L(V)$. Suppose $S \in L(V)$ is invertible. Prove that $T$ and $S^{-1}TS$ have the same eigenvalues. What is the relationship between the eigenvectors of T and the eigenvectos of ...
0
votes
1answer
30 views

Why do these vectors not span the given space?

I need some help understanding this solution to a problem. I am working on the problem above. I know that in order for a set of vectors to be a basis it must be linearly independent and span the ...
2
votes
2answers
54 views

Convergence/Divergence of a the series $\sum_{k=1}^{\infty} a_k$, where $a_1=1$ and $\forall 1\leq k\in\mathbb{N},a_{k+1}=\cos(a_k)$

I got this question: Determine wether the series $\sum_{k=1}^{\infty} a_k$ absolutely converges, conditionally converges or diverges, where $a_1=1$ and for each $1\leq k \in\mathbb{N}$, ...
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
30 views

Bounded from below or not?

if i have that a functional $J$ defined on a Hilbert space is weakly lower semi continuous and coercive is it bounded from below ??? Please help me Thank you
0
votes
0answers
20 views

How does inserting N objects one at a time into an ordered AVL tree yield an efficient sorting algorithim

If we assume reblalancing an AVL tree of height n after an insertion or deletion takes O(n) operations. How does inserting N objects one at a time into an ordered AVL tree yield an efficient sorting ...
1
vote
3answers
228 views

Find the limit of using L'Hopital's Rule

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

Show a set is dense in $C(X)$

Let $X$ be a totally discontinuous compact space. Show that the algebra generated by $$\{f_F; ~f_F=\chi_F-\chi_{X/F},F \text{ is a clopen subset of }X\}$$ is dense in $C(X)$. My attempt: Suppose ...