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0
votes
1answer
6 views

How do I plot the likelyhood of an event occurring after X chances?

Let's say there's a 1/10 chance of something happening. Given 1 chance, there's a 0.1 chance of the thing happening. Given 2 chances, there's a 0.1 + (0.9 * 0.1) chance of the thing happening. ...
0
votes
1answer
34 views

Can three sticks make a triangle? - statistical method of solving

I was challenged to provide as many solutions as I could to the triangle problem: Given a stick of arbitrary length broken into three pieces of independently random lengths, what is the probability ...
2
votes
2answers
16 views

Find the subgroup of $S_4$ generated by $(1,2,4)$. What is the order of $(1,2,4)$?

I am literally just being introduced to group theory this is all new. I know that the permutations that arise are $(4,1),(1,2),(2,4),(1,2,4)$ This is the same as saying that $4$ goes to $1$ then $1$ ...
1
vote
1answer
16 views

limit of the sequence $s_n=\sum_{k=1}^n (-1)^{k+1}a_k

Consider the sequence $s_n=\sum_{k=1}^n (-1)^{k+1}a_n$, with $a_k$ for every $k\geq1$, with $\lim_{k\rightarrow\infty}a_k=0$. Suppose that the sequence $s_{2n+1}$, $n\geq0$, is decreasing and the ...
0
votes
1answer
14 views

If $X$ and $Y$ are finite sets and $ X \cap Y \neq \emptyset \land X \not\subseteq Y \land Y \not\subseteq X$ , what is the cardinality of $X \cup Y$?

This is an exercise inspired by Terence Tao's Analysis I book. Note: If $X$ is a finite set, Tao uses $\text{#}(X)$ to denote the cardinality of $X$. If $X$ and $Y$ are finite sets, what is the ...
2
votes
3answers
37 views

Finding the norm of linear operator $A:X\to X$ as $Ax=f(x)y$ where $y\in X$ and $f\in X^*$

Let $X$ be a linear normed space, $y\in X$ and $f\in X^*$. Define $A:X\to X$ as $Ax=f(x)y$. Prove that $A\in \mathcal{L}(X)$ and find its norm. Hint: when finding $||A||$, you may use the corollary of ...
0
votes
0answers
5 views

If $e$ is powered to n-th multiplier of $2\pi$ should the real part of the multiplied by this multiplier or it must be left as it is?

Hopefully the title is not too confusing, but imagine there is an expression $e^{2+i(\ln(3)-10\pi k)}$ I am not sure does the $-10\pi k$ automatically reduces as a n-th multiply of $2\pi$ or is there ...
88
votes
7answers
45k views
+200

What is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6?

Problem 6 of the 1988 International Mathematical Olympiad notoriously asked: Let $a$ and $b$ be positive integers and $k=\frac{a^2+b^2}{1+ab}$. Show that if $k$ is an integer then $k$ is a perfect ...
0
votes
1answer
5 views

How to prove the graph of a continuous on a closed interval is measurable?

Let $f:[a,b]\rightarrow\mathbb{R}$ be a continuous function on the closed interval $[a,b]$. Let $\mathcal{C}=\{(x,f(x))\in\mathbb{R}^2|x\in[a,b]\}$ be its graph. Prove that $\mathcal{C}$ is measurable ...
-4
votes
0answers
23 views

Good Inequality [closed]

Please prove this inequality $$(a^2 + 2)(b^2 + 2)(c^2 + 2) \geq 9(ab+bc+ca)$$
0
votes
0answers
9 views

Question in proof of a theorem in Field Theory

I am self studying Field Theory from Thomas Hungerford and I have a question in the proof of this theorem. In line 6th of proof, why there shouldn't be F = π(K(x)) isomorphic to π(K[x]) isomorphic ...
-4
votes
0answers
11 views

please help …solve

find a recurrence relation such that f(2)=1 , f(3)=5 , f(4)=31, f(5)=307 , f(6)=3161
0
votes
2answers
1k views

Finding Partial Derivatives at Point in Equation

Find the partial derivative of $z$ with respect to partial derivative of $x$ at point $(1,1,1)$ in equation $xy-z^3x-2yz = 0.$ If I am not mistaken, after simplification of the partial derivative, ...
2
votes
4answers
90 views

Computing $\lim_{n \to \infty} \left[\left(\prod_{i=1}^{n}i!\right)^{1\over n^{2}} (n^{x})\right] $ if exists for certain $x\in\mathbb R$

I came across this problem on my exam and even after three hours of trying I am not able to get through the problem. I think it can be expressed as the limit of a sum but I am not sure and all my ...
-2
votes
1answer
8 views

gradient descent - getting the basics right

So, I found this question: Given that $w_{j+1}=w{j}-v∇Q(w{j})$ Let $Q(w_{1},w_{2})=\frac{1}{2}(w^2_{1}+w^2_{2}).$ Suppose $w_{0}=(1,0)$ and $v=2$, what is $w_{2}$? The answer given ...
0
votes
1answer
13 views

Which variables to choose in a change of variables for a PDE

I want to solve the following PDE: $$\frac{\partial f}{\partial x}-3 \frac{\partial f}{\partial y}=x+y$$ with boundary condition: $$f(x,-2 x)=\sin \left(x^{2}\right)$$ Using a change of variables. How ...
0
votes
1answer
36 views

Law of Cosines and Heron's Formula in inequalities

I had the following question: Suppose $a$, $b$, and $c$ are non-zero real numbers, and $x$, $y$, and $z$ satisfy the equations $$ bx + ay = c, cx + az = b, cy + bz = a. $$ Prove that $-1 < x,...
0
votes
0answers
8 views

Is this possible(pythagorean triples)

My maths teacher gave me some problems and one of them was that I have a right triangle with side 6 and area 30.. So my questions are: 1.Is this even possible and 2.should I assume that most of the ...
0
votes
0answers
6 views

Which of the following statements about f is true?

For the function $f (x)$ on the real line $\mathbb{R}$ defined below, which of the following statements about $f$ is true?Choose all the correct options: $$f (x) :=\sum_{n\ge 1}\frac{\text{sin}(x/n)}...
0
votes
0answers
7 views

Copy and pasting issue PDF to Excel

I have a PDF file. It has included person wise details. I want to copy those details from PDF and paste it into Excel. Few details can copy and paste the same as PDF. The serial number can paste the ...
0
votes
1answer
25 views

Book recommendation for Olympiad inequalities?

I have searched a lot about books on Olympiad inequalities.. And I have finalized 3 books 1) Secrets in inequalities by pham Kim hung 2) inequalities a problem solving approach by bj Venkatachala ...
-1
votes
0answers
8 views

Justification for the convolutional formula for the density of two random variables $X$ and $Y$

I was wondering if someone could tell me if my justification using the towering property of the conditional expectation below is rigorous enough: By the definition of the density, $h(z)$ is a density ...
0
votes
1answer
7 views

Characterization of weak convergence for a Hilbert space

Suppose $H$ is a Hilbert space, $L\subset H$ is a total subset, i.e. $\overline{span(L)}=H$. Let $\{f_n, n\geq 1\}\subset H^∗$ be a sequence of linear continuous functionals such that $\forall y\in L$ ...
1
vote
4answers
35 views

Knowing $a>1$ and $b>1$, how can we prove the limit of $\frac{x^a}{b^x}$ when $x$ goes to infinity?

The limit is $0$, but I want to show it by the definition. I know that, in order to do that, i must show that, given $\epsilon>0$, then there is a $M$ such that, for every $x>M$, we have $|\...
0
votes
1answer
30 views

Equivalent definition of brownian motion

I am having trouble proving two definition of brownian motions are equivalent. Let $(\Omega, F, (F_t), P)$ be a filtered probability space satisying the usual conditions. Let $(X_t)$ be a continuos ...
0
votes
0answers
29 views

How to resolve this equation $\exp(\frac{-tI}{RC}) = cos(wtI)$

Is this possible to resolve this equation. I m interested in expressing tI in function of R, C and w. Here is the equation : $\exp(\frac{-tI}{RC}) = cos(wtI)$ where tI is between 3T/4 and T. Thank ...
1
vote
2answers
43 views

$\Sigma_{\tau \in S} 2^{-|\tau|} = 2^{-|\sigma|}$, for any prefix-free $S \subseteq 2^{<\omega}$ s.t. $[S] = [\sigma]$.

The Kolmogorov Inequality gives me $\Sigma_{\tau \in S} 2^{-|\tau|} \leq 2^{-|\sigma|}$, for any prefix-free $S$ extending $\sigma$. But equality seems to hold when $[S] = [\sigma]$. Notation: $[\...
0
votes
0answers
5 views

Trace of positive operator

Let $A$ be a bounded $\ge 0$ operator on separable complex Hilbert space $\mathscr{H}$. By the spectral theorem, we know that there exists a unitary transform $U:\mathscr{H} \rightarrow L^2 (\mu)$ and ...
0
votes
2answers
13 views

Let $(X, T_X)$ and $(Y, T_Y)$ be topological spaces with bijective function $h:X \to Y$ continuous and open. Then $h$ is a topological equivalence.

Problem: Let $(X, T_X)$ and $(Y, T_Y)$ be topological spaces with bijective function $h:X \to Y$ continuous and open. Then $h$ is a topological equivalence between $X$ and $Y$. It says that, $h:X \to ...
1
vote
0answers
7 views

How many the number of the vectors for given this linear transformation?

Here is the my lecturer's question given to our class $Q)$ Say Linear transforamtion, $T : P_3(\mathbb{R}) \to \mathbb{R}^4$ by $T(f(x)) = (f(a_1),f(a_2),f(a_3),f(a_4))$ Here $f(x) \in P_3(\mathbb R)...
0
votes
1answer
25 views

If (x,y,z) is a Pythagorean triple such that each of x,y,z can be written as sum of two squares then prove that 180|xyz

This is a problem based on Pythagorean Triples. If (x,y,z) is a Pythagorean triple such that each of x,y,z can be written as sum of two squares then prove that 180|xyz Any ideas of how to start ...
-1
votes
0answers
12 views

Finding a relation and the number of 132 avoiding permutations

Can someone guide me to find the arelation، for finding the number of 132 avoiding permutations,using generating functions and the kernel method.
1
vote
2answers
21 views

How to find the common factor

I am currently stomped at this problem $(2 x^2-(y+z) (y+z-x))/(2 y^2-(z+x) (z+x-y))$ I am supposed to factor it and I can't find a way how to get to the answer. It says that $(2 x^2+x y-y^2+x z-2 y ...
0
votes
0answers
5 views

ab product of t disjoint cycles -> ba also product of t disjoint cycles

Let a,b belong to $S_{n}$. Suppose that ab is the product of t disjoint cycles of lengths k1,k2,..,kt. Prove that ba is also the product of t disjoint cycles of lengths k1,k2,..,kt. \ How would you ...
0
votes
1answer
52 views

Describe all orthogonal matrices in $G$

If $G = \{ A \in GL(3,\mathbb R): Ax = x \}$, where $x = \begin{bmatrix} 1 \\ 0\\ 0 \\ \end{bmatrix}$, describe all orthogonal matrices in $G$.
2
votes
1answer
26 views

The Newton's method

Using the Newton's method the function $f(x)=\frac{1}{x^2}-a$ calculate approximation of $\frac{1}{\sqrt a}$. a) Is this method locally convergent squarely or cubically? b) Find $x_0$ such that ...
-1
votes
0answers
19 views

Let I be an ideal in an algebra R. Verify in detail that R/I is an algebra. [closed]

An associative k-algebra is a k-vectorspace R together with a bilinear map mult : R×R → R satisfying the associativity axiom r(st) = (rs)t, r, s, t ∈ R. We assume that there is a 1 = 1R ∈ R which is a ...
0
votes
0answers
3 views

Find a GFG for L = {aˆnbˆm: n != m-1, m >= 0, n >= 0}

I can't figure it out. So I came to the conditions: n > m-1 and n < m-1. Find a context free grammar for $L = \{ a^n b^m : n \neq m-1, m \ge 0, n \ge 0 \}$
-1
votes
0answers
17 views

Number of solutions for the given condition [closed]

Given $X,p,a,b$, we need to find out how many $n\in\mathbb N,\;(1\leqslant n\leqslant X)\;$ satisfy the following condition: $$na^n\equiv b\pmod{p}$$ Constraints: $$\begin{aligned}\begin{...
0
votes
2answers
434 views

Probability question about cards being dealt and probability of getting a particular suit [closed]

In this card game, the entire standard deck of $52$ cards is dealt out to $4$ players. What is the probability that (a) one of the players receives all $13$ spades? (b) each player receives $...
0
votes
1answer
7 views

Entropy and KL divergence $H(\hat{\theta}|D_n) - H(\hat{\theta}|D_n, Z_{n+1})$ where $D_n = \{Z_i\}_{i= 1}^n$

Consider a stream of data $D_n = \{Z_i\}_{i= 1}^n$ and the goal is to estimate a parameter $\theta$ based on the observed data. In a lecture note I am reading it says that, if $H(X)$ denotes the ...
2
votes
1answer
21 views

Automorphism group of finite field extension has a trivial stabilizer

We have this theorem. Let $L|K$ a field extension with $[L:K]<\infty$ and $G=\text{Aut}(L|K)$. We let $G$ act on $L$. Then there is a trivial stabilizer. The proof is the following, I would ...
0
votes
1answer
19 views

Find the orders of $M_3(\mathbb Z_2)$ and $GL_3(\mathbb Z_2)$ [duplicate]

$|M_3(\mathbb Z_2)| = 2^{3^2} = 64$ $|GL_3(\mathbb Z_2)| = 64$ minus all singular matrices in $M_3(\mathbb Z_2)$ or all matrices with linearly independent columns. My question is how do I find all ...
8
votes
1answer
70 views

Proving $\sin(\tanh x) \ge \tanh(\sin x)$, for $x \in [0,\pi/2]$

Earlier, a very interesting proof of an inequality has been proposed at MSE: How prove this inequality $\tan{(\sin{x})}>\sin{(\tan{x})}$ Here the question is: How to prove that $$\sin(\tanh x) \ge ...
0
votes
0answers
5 views

Uniqueness of subspaces dual-basis

Consider the vector-space $\mathbb{R}^3$. Now think of an arbitrary 1D-subspace of this vectorspace, lets say e.g. the subspace spanned by the basis vector $(1,1,0)^T$. This is a finite vector-space ...
0
votes
4answers
26 views

Show that the series $\sum^\infty_{n=1}(-1)^n\frac{n}{n^2+1}$ is conditionally convergent

I have to show that the series $\sum^\infty_{n=1}(-1)^n\frac{n}{n^2+1}$ is conditionally convergent. I am first going to show the series is convergent by the alternating series which states that a ...
0
votes
2answers
20 views

Check my argument for this proof for Quotient Rule (limits)

I have seen that calculus books have other ways to prove this theorem (involving triangle inequality). Just wanted to know if this reasoning is okay or I have done some circular reasoning in this ...
1
vote
0answers
12 views

What are some metrics to compare which line is more “curved”?

Let's say we have two lines. I want to compare which line is more "curved" over the same interval. For example, let's say: $f1(x) = x^2$ $f2(x) = sin(3x)$ Over the interval x = [0, 1], it is ...
0
votes
0answers
15 views

Does Stewart's calculus consider endpoints of the domain of a (nice) function to be critical numbers?

This is a question of convention - specifically the convention used in Stewart's Calculus. In Stewart's calculus (the latest version), in chapter 4.1, definition 6 defines a critical number of a ...
0
votes
1answer
14 views

Given $n\in \mathbb{N}$ find $x\in (0,1)$ such that $p(n,x)<10^{-12}$

Given $n\in \mathbb{N}$ find $x\in (0,1)$ such that $p(n,x)<10^{-12}$, if: $$p(n,x)=x(1-x)\frac{x^{n-2}}{1-x^2}+x(1-x)\frac{(1-x)^{n-2}}{1-(1-x)^2}.$$ Attempt. We may also write $$p(n,x)=\frac{...

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