0
votes
0answers
2 views

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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 $n > 0$
1
vote
1answer
11 views

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 $re^{i \theta}$.
-1
votes
0answers
4 views

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 $G'(F(x))=?$
0
votes
1answer
20 views

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?
0
votes
0answers
4 views

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, ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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?
0
votes
1answer
12 views

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 ...
0
votes
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 $A$, ...
0
votes
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 ...
1
vote
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 ...
3
votes
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. ...
0
votes
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 ...
0
votes
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 ...
-2
votes
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 $ab$.
1
vote
1answer
24 views

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 ...
0
votes
1answer
18 views

Periodic solutions of $x'=x^2-1-\cos t$

Consider $x'=x^2-1-\cos t$. What can be said about the existence of periodic solutions for this equation? I'm not sure if periodic solutions exist, but if they do, they must have period equal to $ ...
1
vote
1answer
8 views

Differentiability of vector valued function?

The question is: Let $g$ and $h$ be real-valued functions on $R$ such that $g$ is differentiable at "a" and $h$ is differentiable at "b". Show that $f$: $R$2 → $R$ defined by $f$( x1, x2) = ...
0
votes
0answers
7 views

Are there any algorithm for schedule match

I am making a game. This game divide players into groups. Each group has N players in even number and will schedule player in the same group to fight each other in pair everyday I need to have every ...
0
votes
0answers
7 views

conditions on integrable function with counting measure

Let $P(N)$ be the power set of $N$ and $u$ be the counting measure on $N$. (a) Prove or disprove the measure space $(N, P(N), u)$ is complete? (b) Given function $g: N\rightarrow R$. Show that $g$ ...
0
votes
0answers
10 views

DFT confusion about complex conjugation in forward process

I learning about the DFT and there's one thing that's sort of confusing me, I hope it's not too dumb of a question! I understand that the DFT involves a process of taking an input signal vector and ...
0
votes
0answers
2 views

Reference that Explains Preconditioning

I would like to understand Preconditioning techniques and why they work. Could someone provide a good reference for this type of information?
0
votes
2answers
16 views

Power series solution (Why the constant of the recurrence relation can be chosen arbitrarily?)

Please help me understand this: Solve $y''-xy=0$ First, since there are no singular points, it can be guaranteed that we can always find two power series independent solution, centered at $0$, and ...
0
votes
2answers
42 views

Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function.Show that $\lim_{n \rightarrow \infty} n \int_{0}^{1}e^{-nx}f(x)dx=f(0)$

Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function.Show that $\lim_{n \rightarrow \infty} n \int_{0}^{1}e^{-nx}f(x)dx=f(0)$ I am making a claim is it really true? $\lim_{n \rightarrow ...
0
votes
0answers
18 views

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 ...
0
votes
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 $p\in U$, ...
1
vote
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
0answers
5 views

Decoupling coupled differential equations with time dependent coefficients

Consider the following system of coupled differential equation. $$\left[ \begin{array}{c} \frac{dc_1}{dt} \\ \frac{dc_2}{dt} \end{array} \right] = \begin{bmatrix} -B & -V(t) \\ -V(t) & B ...
1
vote
2answers
24 views

Find the sum using

The question is as follows: Find the sum: $1\cdot2 + 2\cdot3 + ... + (n-1)n$ What I have tried so far: We can write $(n-1)n$ as $\frac{(n+1)!}{(n-1)!}$ which we can also write as ...
1
vote
0answers
12 views

Adelic definition of “canonical divisor”

For a function field over a curve $F/K$, some book define the canonical divisor as the divisor of a map $\omega:\mathscr{A}_{F}\rightarrow K$ (where $\mathscr{A}_{F}$ is the pre-adele, ie. adele but ...
-1
votes
1answer
40 views

Conjecture: Partitioning $\Bbb N$ into parts that sum to $13^i$ [on hold]

Recently I was thinking and came up with a conjecture that goes as follows: Conjecture: There exists a $\Omega$ such that $$\Omega = \Bigg\{A_i \ \Bigg| \ \forall i,j:i\not=j, \ A_i\cap ...
1
vote
2answers
22 views

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 ...
0
votes
0answers
5 views

In a transformation matrix, why is $Y$-axis ($-\sin$) in left most column as opposed to right like X and Z

$4 \times 4$ 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 ...
-2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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.
0
votes
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 ...
1
vote
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 ...
0
votes
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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