# All Questions

0answers
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### Does there exist $P$ such that $PP^{\dagger}=\left(\begin{array}{cc} I & 0\\ 0 & -I \end{array}\right)$?

Does there exist $P$ such that $PP^{\dagger}=\left(\begin{array}{cc} I & 0\\ 0 & -I \end{array}\right)$? Here $P^{\dagger}$ is the hermitian of $P$, and $I$ means a $N\times N$ identity ...
1answer
7 views

### Infinite Sets Proof - Integer Sets

Let $Z^-$ be the set of negative numbers. Prove $Z^-$ ≈ $Z^+$ by finding a bijective function $f : Z^+-> Z^+$. Prove that the function is bijective. Could someone tell me how to get started on ...
0answers
5 views

### Find area of shaded area in curve with range of values for y

The parabola in the diagram has equation $y = 32 - 2x^2$ The shaded area lies between the lines $y=14$ and $y=24$ Looking at the graph, I only need to find half the area and multiply by ...
0answers
20 views

### How do you integrate $e^{-st}*t*cos(t)$?

I'm doing differential equations and specifically studying Laplace Transformations, where of course the Kernel is: $K(s,t) = e^{-st}$ And the Laplace Transformation $\mathcal{L}$ of a function ...
0answers
6 views

### Square root of a complex symmetric matrix?

Is it possible to express a complex symmetric matrix $A$ as square of a matrix $B$ (i.e. $A = B^2$)? If $A$ were Hermitian, we could use Spectral Theorem to get $A = UDU^{-1}$ where $D$ has diagonal ...
2answers
16 views

### Prove that $n^2 < n \cdot (n - 1) \cdot (n -2)$

How to prove or disprove that: $$n^2 < n \cdot (n - 1) \cdot (n -2)$$ for every $n > 0$
1answer
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### How can I express $i^{2i}$ in the form $x + iy$?

I'm not sure how to begin since this is not in the form $re^{i \theta}$.
0answers
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### What's the derivative of G(F(x))

Let $G(F(x))=F(x)-x\cdot e^{\frac{2F(x)-F(x)^2}{2-2F(x)}}$ I want to know $G'(F(x))=?$
1answer
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### What is the name of people who do algebra?

People who do topology is called topologists, people who do analysis is called analysts, people who do geometry is called geometers, then how about algebra?
0answers
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### Reciprocal of a quadratic form

I am working with an expression of the form $$\frac{x^TAx}{{x^TBx}}$$ and would like to simplify it. I understand that vectors do not have inverses, but viewing the bottom number as a 1 by 1 matrix, ...
0answers
14 views

### Area enclosed by the curve $\lfloor |x''| \rfloor +\lfloor |y''| \rfloor = 2$

The area enclosed by the curve $$\bigg\lfloor \frac{|x-1|}{|y-1|}\bigg\rfloor +\bigg\lfloor \frac{|y-1|}{|x-1|}\bigg\rfloor = 2\;,$$ Where $-2 \leq x,y\leq 0$ $\bf{My\; Try::}$ Let $x-1=x'$ and ...
0answers
3 views

### Condition for global cascade

Assume a unidirectional, unweighted network generated according to a degree distribution. Each node is given a value between 0 and 1 called threshold $\phi$. We topple some nodes, the neighbours will ...
4answers
36 views

### If $f'(x)\rightarrow \infty$ then $f(x)\rightarrow\infty$?

How to show: $f'(x)\rightarrow \infty$ as $x\rightarrow \infty$ then $f(x)\rightarrow \infty$, as $x\rightarrow \infty$ and $x>0$? Here $f'$ is derivative of $f$. Intuitively it is clear but got ...
1answer
14 views

### I'm trying to find a homogeneous equation for the following:

$$f(n) = 2f(n-1) + n$$ I don't know how to handle the $n$ by itself, any thoughts?
1answer
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### Modular Arithmetic and divisibility proof

I could use some help with this proof. Let $n, m ∈ Z^+$ and $a, b ∈ Z$. Suppose that $a ≡ b$(mod n) and $a ≡ b$(mod m) and $(m, n) = 1.$ Show that $a ≡ b$(mod mn). From what I understand, it is ...
1answer
8 views

### Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$.

Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$. $A$, $B$, and $C$ are square matrices of the same size. $\hat{c}_j$ is the $j$th column of $C$, $\hat{a}_k$ are the columns of $A$, ...
0answers
12 views

### What is the coefficient and constant term in the following sequence defined recursively?

Let $f_n(x)$ be a sequence of polynomials defined inductively as $f_1(x) = (x - 2)^2$ $f_{n+1}(x) = (f_n(x) - 2)^2$ $; n \ge 1$ Let $a_n$ and $b_n$ respectively denote the constant term and the ...
1answer
23 views

### How many integers from 43523 to 93107 contain at least one digit 7

How many integers from 43523 to 93107 contain the digit 7 at least once. I know that if we had 43000 to 93000, we would subtract integers that do not contain digit 7 from the total number. 50000 - (5 ...
3answers
29 views

### How to show that a continous function $f:\mathbb{R}^m \to \mathbb{R}$ has a maximum?

My task is this: Suppose $f:\mathbb{R}^m \to \mathbb{R}$ is a positive, continous function such that $\lim_{\mid \textbf{x}\mid \to \infty} f(\textbf{x}) = \textbf{0}$. Show that $f$ has a maximum. ...
1answer
16 views

### Sum of compact sets is compact without using continuity [duplicate]

I want to show that if $A$ and $B$ are compact sets, then $A+B$ (that is, the set $\{a+b : a \in A , b \in B\}$) is compact. I know that $A+B$ is bounded, but am having trouble showing that it is ...
0answers
15 views

### Exhaustion by compact sets in $\mathbb{C}^n$

Let $U\subseteq\mathbb{C}^n$ be open. For every $j\in\mathbb{N}$ define $$K_j:=\{z\in U:\left\|{z}\right\|_{\infty}\le j,d_{\infty}(z,\mathbb{C}^n\setminus U)\ge 1/j\}.$$ Then the following ...
0answers
30 views

### Prime number theory. [on hold]

If $a$ is coprime to $b$ and $y$ and $b$ are both coprime to $x$; then Prove that $ax+by$ is a coprime to $ab$.
1answer
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### AP Calculus BC - Antideriative of cos(1-x^2)/(x^2 + root(x))

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
1answer
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0answers
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### Complexity of modular multiplication

By considering the method long multiplication, how to informally prove that modular multiplication of two number of length $m$-bit each has a complexity of $O(m^3)$? Tried this Taking two number of ...
0answers
15 views

### Regarding a proof in Tu's 'Introduction to manifolds'

While reading Tu's differential geometry book, I came across a theorem which makes the following claim: Let $f$ be a $\mathcal{C}^\infty$ function on an open subset $U\in \mathbb{R}^n$, let $p\in U$, ...
0answers
17 views

### Number of ways to get from a point to another one in the plane

I was trying to solve the following problem related to "counting cases": Consider the point $(0,0)$ in the plane and another point $(m,n)$ with $m,n>0$ integers. Suppose you want to get from the ...
2answers
13 views

### normal approximation with continuity correction

a fair die is rolled 100 times. What is the probability that "6" appears more than 15 times? Use the normal approximation with continuity correction. I've found the mean to be $100/6$ or $50/3$ and ...
0answers
22 views

### 1-How my profesor reach this solution? 2-How can I use eigenvalues to compute betas?… if there is any way

this time I quite don't undertand how the profesor avoid using matrix algebra to solve this exercise. Statement: Below you can see a scatter plot of the data with the three regression lines ...
0answers
14 views

### Question regarding regular stochastic matrix

We say that a stochastic matrix is regular iff $\exists n\in \mathbb N$ such that $p_{ij}(n)>0$ for all states $i,j$ How many powers of a matrix do we need to compute at most in order to verify ...
0answers
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2answers
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### Find a matrix $M$ such that $M^TAM = I$

I have that my matrix $A = \begin{bmatrix}4&0&3\\0&1&0\\3&0&4\end{bmatrix}$, I've done diagonalization but now finding a matrix $M$ and its transpose acting as a conjugate for ...
0answers
5 views

$4 \times 4$ Transform Matrix with axis columns $XYZ$ left to right $X$-axis rotation: $$\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & \cos & -\sin & 0 \\ 0 & \sin & \cos ... 0answers 17 views ### Why is it that when n ≥ 1 the series is \le 1/4 So how is the series \sum_{n=1}^\infty \frac{1^2 * 3^2 * 5^2 ... (2n-1)^2}{2^2 * 4^2 * 6^2 ... (2n)^2} < 1/4 for n \ge 1 is it because the series is divergent outside of the interval of ... 2answers 15 views ### Adjoint of linear transformation T: \mathbb{M_n(C)} \rightarrow \mathbb{M_n(C)} Let V = {M_n(\mathbb C)} with inner product \langle A, B\rangle = \text{tr}\,(B^*A), A, B \in V. Let M \in {M_n(\mathbb C)}, Define T: V \rightarrow V by T(A) = MA. What is adjoint of ... 0answers 22 views ### For any function f in L^2(-π,π), is it true that ||f||_{L^2} \leq C||f||_{\infty}? For any function f in L^2(-\pi,\pi), is it true that ||f||_{L^2} \leq C||f||_{inf} ? I came up with this question because in An Introduction to Hilbert Space by N.Young, right before ... 5answers 40 views ### How to prove that \lim\limits_{n\to\infty}\sqrt[n]{p}=1? Could you tell me how to show if p>0 then\lim\limits_{n\to\infty}\sqrt[n]{p}=1? (+clues) 1.put \sqrt[n]{p}=1+h_{n} 2.Bernoulli's inequality If you don't mind, use the clues to prove it. 0answers 6 views ### How do I find the edge connectivity from (u,v)? Also how do you find the vertex connectivity from (u,v), the maximum size of a set of pairwise internally disjoint u,v paths, and the maximum size of a set of pariwise edge-disjoint u,v paths. I got ... 3answers 21 views ### Prove that if A, B, and C are sets then (A - B) \cup (A - C) = A - (B \cap C) Prove that if A, B, and C are sets then (A - B) \cup (A - C) = A - (B \cap C). I have the proof for the first direction: Let x \in (A - B) \cup (A - C) be given. Hence, x \in (A - B) or ... 0answers 8 views ### Taylor expand \ln(x) - \ln(1-y) with respect to \ln(x') and \ln(y') Can I taylor expand$$\ln(x) - \ln(1-y)$$around \ln(x') and \ln(y') such that I get$$ \ln(x') + \ln(y') + \frac{\partial (\ln(x) -\ln(1-y))}{\partial \ln(x')} (\ln(x) - \ln(x')) + ...

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