1
vote
0answers
12 views

What is the probability of getting exactly one two and one three in a 5 card draw?

In a 52 cards deck, what is the probability of getting exactly one 2 and one 3 if 5 cards are drawn. I'm wondering what is the difference between doing it the following two ways. Intuitively I would ...
0
votes
0answers
4 views

Constructing a family of sets

I am completely stuck at the following question. Suppose $X$ is an infinite set. Show that there is a family $\mathcal{F}$ of subsets of $X$ satisfying the following: (a) If $A \subseteq X$ is ...
0
votes
0answers
3 views

Text introducing $T^{i,j}$-tensor algebra

I'm reading a lecture note here : http://www.cis.upenn.edu/~cis610/diffgeom7.pdf It introduces $T^{•,•}(M)$ the tensor algebra and says that this is a necessary tool in differential geometry. Well, ...
0
votes
0answers
2 views

Approximating Lipschitz Functions by $C^1$ functions

According to Evans-Gariepy as a corollary of the Whitney's Extension Theorem we have the following Theorem (Approximating Lipschitz Functions) Suppose $f: \mathbb R^n \to \mathbb R$ is Lipschitz ...
0
votes
0answers
3 views

Proving that something equals the commutator subgroup and conjugacy classes/normal subgroups

I've learned that the commutator subgroup is generated by the commutators. Now this says little about its elements (to me) because I don't see how they need to be commutators themselves. I'm ...
0
votes
0answers
7 views

Problem solving a PDE using separation of variables and Fourier expansion

I am trying to solve the heat equation: $$\frac{\partial \theta}{\partial t}=\frac{\partial^2 \theta}{\partial x^2}$$ The boundary conditions are: $$\frac{\partial \theta}{\partial x}(x=0)=0$$ ...
0
votes
0answers
21 views

What are the practical applications of this trigonometric identity?

On various occasions people have asked here how to prove that $$ \text{if } \alpha+\beta+\gamma = \pi \text{ then } \frac{\sin(2\alpha) + \sin(2\beta) + \sin(2\gamma)} 2 = ...
3
votes
3answers
16 views

For some arbitrarily fast growing function $f$ and a strictly sublinear function $g$, can $g \circ \cdots \circ g \circ f$ always grow polynomially?

Given two functions $f, g: \mathbb{R}_{≥ 0} \to \mathbb{R}_{≥ 0}$ that are monotonically growing, with $g(x) \in o(x)$ (i.e. $g$ grows strictly sublinear), does there always exist an $m \in ...
3
votes
0answers
42 views

Alternate proof of the integral: $\int_0^1 x^x(1-x)^{2x}\,dx\neq3/8$

I am looking into the integral: $$I=\int_0^1 x^x(1-x)^{2x}\,dx\neq\frac{3}{8}$$ How might you prove this to be true? What's tough is that the integral $$3/8\lt I<0.37503$$ numerically. I managed ...
0
votes
0answers
12 views

Converting a sum over squared norms to Quadratic Programming problem

I'm trying to minimize the following function which is a function of a set of vectors $\vec{x}_i \in \mathbb{R}^2$, and is parametrized by the fixed parameters $\alpha, \beta,\gamma_i,\vec{u}_i$ where ...
0
votes
0answers
10 views

Find $T_1(\langle (1,2,3,4,5,6,7,8,9) \rangle )$

$T_k(G)=\{g \in G : o(g) \, \,|\, \, p^k \}$ I am unsure what to do. Let that long permutation be $b$. Do we just find calculations of $b$ like $b^2$ or $b^{-1}$ or $b^b$ etc, that would give an ...
0
votes
0answers
23 views

Let $f_n$ uniformly convergent to$ f$. Proof that $f$ is integrable and $\int_A f_n \rightarrow \int_A f$

(When I write integrable I mean Riemann-integrable) Let $A \subseteq \mathbb{R}^m$ be a block, $f_n:A\rightarrow R$ be integrable functions and $f_n \rightarrow_{un} f$. Proof that $f$ is integrable ...
0
votes
0answers
14 views

Calculate modulo expression

How does one evaluate the expression; $$5\times52^{366} \mod367^1$$ I can infer from the exercise solutions that $52^{366} ≡ 1 \mod 367$, but why?
0
votes
1answer
13 views

Evaluate limit of a sum that includes summed term

I am trying to determine whether the limit $\lim_{n \to \infty} \sum_{k=2}^n (\frac{n-k}{n-2})^{2k} (\frac{l-1}{2})^k$ exists and is finite. No idea how to approach this. Please let me know if you ...
1
vote
2answers
34 views

Dude with vectorial spaces (Basis and dimension)

Good night, i'm working in a problem, i need an basis and the dimension of the space. $a_{1}=(1,0,0,-1),\:a_{2}=(2,1,1,0),\:a_{3}=(1,1,1,1),\:a_{4}=(1,2,3,4),\:a_{5}=(0,1,2,3)$ I make this: $\left[ ...
0
votes
0answers
27 views

Combinatorics: numbers on a blackboard with integer sums

$N\geq9$ distinct real numbers are written on a blackboard. All these numbers are nonnegative, and all are less than $1$. It happens that for very $8$ distinct numbers on the board, the board contains ...
0
votes
0answers
14 views

Combinatorics: Find the maximal number of positive integers in this circle

$100$ integers are arranged in a circle. Each number is greater than the sum of the two subsequent numbers (in a clockwise order). Determine the maximal possible number of positive numbers in such ...
0
votes
0answers
17 views

Polar coordinate - Write an equivalent rectangular equation…

Write an equivalent rectangular equation $y=-x$ It should be simple, but in my attempt, I always come up short with coordinates $\sqrt{2}$ for both $x$ and $y$. My $r$-variable is $2\sqrt{2}$. Is my ...
0
votes
0answers
9 views

Explicit Morse functions on closed surfaces

Let $M_g$ a surface of genus $g$, why can we find a explicit Morse function on $M_g$? For the torus we have $f(x,y)=\cos(2\pi x) + \cos(2\pi y)$, for sphere $f(x,y)=x^2+2y^2$ and this induces an ...
0
votes
0answers
13 views

Change of variables in Power series

Hello StackExchange community, this is probably a dumb question, but suppose you have a function that have the following power series $f= \sum_{i = 0}^{\infty} f_i r^i$. where $f:\mathbb{R} ...
0
votes
0answers
6 views

Can you recover a distribution from mollification?

Let $f\in \mathcal S'(\mathbb R)$ be a Schwartz distribution. Given $\rho \in C^\infty_c(\mathbb R)$ define the convolution as the function $$x\mapsto (f\ast\rho)(x):=\langle f, \rho (\cdot ...
0
votes
0answers
18 views

Is there a nice proof of this zeta function integral

Could some one link me to a good proof of this integral written on the first line here. All the sites I've seen so far just introduce with the definition without explaining it. If not, could anyone at ...
1
vote
2answers
24 views

Row replacement operation not changing the determinant

Can someone prove why a row replacement operation does not change the determinant of a matrix? **row replacement operation being adding one row to another or something of that sort
-2
votes
2answers
23 views

Statistics basics

Given that $X$ has mean $a$ and variance $b$. Then $E(X^2) = a^2 + b^2$. Why is this true? Please provide a proof alongside any other relevant information. Thanks in advance.
0
votes
1answer
22 views

When a set of functions becomes complete?

I know that a set of functions are said to form a complete basis on an inteval if any function on that interval can be expressed as a linear combination of the functions in the set. I also know that ...
0
votes
1answer
7 views

the sum operation in a normed space is continuous

If $ (E, \parallel \parallel) $ is a normed space , then the function $ + : E \times E \rightarrow E $ , $ ( x , y) \mapsto x + y $ is continuous. Question: to solve this exercise ... Which it is ...
0
votes
0answers
6 views

Methods for detecting kinks along a set of points (for interpolating)

Can anyone point to some literature on methods for detecting kinks/sharp turns with given a set of points. To give some background, I'm trying to get estimates of some function based on randomly ...
0
votes
1answer
26 views

What is the order of

What is the order of the following: $$\frac{(33x^{7}+6)(x^{2}+3)}{\sqrt{x^3+7x^2-x+5}}$$ Would it be $$\Theta (x^{\frac{17}{2}})$$
2
votes
2answers
9 views

Checking injectivity of a certain function from a union of a family indexed by $K$ to $K$

Let $A = \{ A_k | k \in K \}$ be a family of sets indexed by $K$. By Zermelo's theorem, $K$ can be well-ordered. Now, let $\leq$ be a well-order on $K$. Define $j: \bigcup\limits_{k \in K} A_k \to ...
0
votes
0answers
17 views

Three games of two-players each being played by three players simultaneously

Has the game theory literature considered situations wherein there are three two-player games being played by three players concurrently with each other; and the outcomes of those games may impact the ...
0
votes
0answers
11 views

Linear stability of an ODE $\frac{dN}{dt}=-\mu N(t)+\mu N(t-T)\left(1+q\left(1-\left[\frac{N(t-T)}{K}[\right]^z\right)\right)$

This is a part of exercise: Consider the following equation: $$ \frac{dN}{dt}=-\mu N(t)+\mu N(t-T)\left(1+q\left(1-\left[\frac{N(t-T)}{K}[\right]^z\right)\right) $$ where all involved constants are ...
1
vote
3answers
55 views

Use $\epsilon$, $\delta$ to prove that $\lim\limits_{x\to\ b} \frac{1}{a+x}$ = $\frac{1}{a+b}$.

I've been working on this epsilon delta proof for the longest time now, and I can't quite get it. Let $a>0$ and $b>0$. Use $\epsilon$, $\delta$ to prove that $\lim\limits_{x\to\ b} ...
1
vote
1answer
20 views

Combinatorics [sandwich toppings]

I'm trying to figure out how to solve this excercise; You can order a sandwich from 5 different types of bread, you can have butter, lettuce or neither. You get to choose from 3 types of meat, and ...
0
votes
1answer
16 views

Sum of two normal operators.

We have euclidean space, $A$ and $B$ are normal operators (definition: $A$ is normal when $AA^*=A^*A$) and we know that $Im(A)$ and $Im(B)$ are orthogonal. How can I prove that $A+B$ is normal too? I ...
0
votes
1answer
13 views

Isometry and Extreme points

If $X$ is a Hilbert space and either $T$ or $T^*$ is an isometry, show that T is an extreme point of the closed unit ball of $B(X)$ where $B(X)$ is bounded linear functionals on $X$. Can I get some ...
0
votes
1answer
17 views

Volume and Average Height

How do you calculate the average number of floors of buildings across a city block when all the buildings are varying heights. Followings in a listing of what the area looks like and the number of ...
0
votes
0answers
27 views

Congruent number 23

The set of Congruent numbers are all the integer areas of rational sided right triangles. This means that if g is a Congruent number there exists some integer n such that g*n^2 is the integer area of ...
4
votes
0answers
35 views

Fourier transformation formula

Let $f$ be absolutely integrable over all $\mathbb{R}$ and continuous and $\hat{f}(s)=0$, where $s\in \mathbb{R}$ and $|s|>a$, where $a>0$. Prove that following formula holds ...
0
votes
3answers
22 views

Inequality for quadratic function and exponent

I might be a but rusty but while doing probability tasks i got stuck on some inequalities from analysis. The task is to prove that there exists $K>0$, so that: $$ \left|\frac{1}{1+e^{3x}} ...
0
votes
0answers
15 views

What is computational complexity of a coding technique

In my previous Question Help in understanding a coding technique based on inverse mapping of a dynamical system I learnt how to apply chaotic map in coding theory in communications. Based on the ...
3
votes
2answers
32 views

Using the Weierstrass M-test, show that the series converges uniformly on the given domain

$\sum_{k \geq 0} \frac{z^k}{z^k+1}$ on the domain $\overline{D}[0, r]$, where $0 \leq r < 1$ I'm honestly not sure how to do this. My text mentions the Weierstrass M-test but the example they ...
0
votes
2answers
25 views

How to find slope at a point where the derivative is indeterminate

How should I find the slope of a curve at origin whose derivative at the origin is indeterminate. My original problem is to calculate the equation of tangent to a curve at origin. But for the equation ...
0
votes
1answer
26 views

Show $T_k(G)=G$

Suppose $G = C_n$ is an abelian $p$-group (so that $n = p^a$ for some $a$). Show that $T_k(G) = G$ if $a ≤ k$ and $T_k(G) = C_{p^k}$, otherwise. $T_k(G)=\{g \in G : o(g) \, \,|\, \, p^k \}$ If $a ...
1
vote
2answers
19 views

Let $f\in L^p(0,1)$ and define $f_h$

Let $f\in L^p(0,1)$ ($1\leq p<\infty$) and define $f_h$ as $$f_h(x)=\begin{cases}f(x+h)&\text{ for } x+h\in [0,1]\\ 0 &\text{ for } x+h\not\in[0,1]\end{cases}$$ Prove that for all ...
0
votes
0answers
11 views

Permuting a cycle when assignments exist.

There are $n$ agents total and $n$ objects total. Each agent ranks objects uniformly at random independent of other agents. Suppose I fix a subset of $k$ agents $a_1,..,a_k$, and $k$ objects ...
0
votes
0answers
12 views

Method of Characteristics for $(1+x^2)u_x+2u_y=y\cdot(1+u^2)$

Suppose I had a problem like $$(1+x^2)u_x+2u_y=y\cdot(1+u^2),\;\; u(1,y)=1$$ This is of the form $$a(x,y)u_x+b(x,y)u_y+c(x,y,u)$$ So I would use the Method of Characteristics: ...
0
votes
1answer
12 views

Upper bound of a bilinear form

Suppose I have a form $|X^TBY|$ where $X \in R^n, Y \in R^m$ and $B \in R^{n \times m}$ is a matrix whose elements are bounded. Is there an upper bound for the whole expression of the form $|X^TBY|\le ...
0
votes
1answer
10 views

Why are most Lagrange multipliers zero in the SVM solution?

I read everywhere that a non-zero Lagrange multiplier $\lambda_i$ signifies that the corresponding point $x_i$ is a support vector, but I can't see how a support vector and a non-support vector have a ...
0
votes
0answers
14 views

why am I getting this result on a sample size calculation tool

I found an online calculator for sample size needed to perform an A/B test to a specific metric. Given a requested z-score of 1.96, a minimum detectable effect (MDE) of 5% and an estimated ...
0
votes
4answers
46 views

Chain rule to differentiate $\sin ^2\frac{x}{2}$

I have this equation $$\sin ^2(\frac{x}{2})$$ Using the chain rule $ M'(N(x)).N'(x)$: $$\begin{align*} &M= (\sin \frac{x}{2})^2 \\ &N= \frac{x}{2}\end{align*}$$ That makes $$2\sin ...

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