All Questions

Let $P_1=(X_1,U_1)$ be a search problem with the domain of search $$X_1=\{x \in Z_2^n | wt(x)\leq k\}$$ and the set of admissable tests be $U=Z_2^n$ (where $wt(x)$ is the hamming weight of $x$). ...

I am studying submanifolds and I have some problems with the proof of a claim about Rank Theorem. Rank Theorem: Let $f: U \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a smooth map and each ...

This question originates from Chapter 13, B5 of the 2nd edition of A Book of Abstract Algebra by Charles C. Pinter. Describe the cosets of the subgroup $H = \{(x,y): x=y\}$ of $\mathbb{R}\times\...

Let $y(x)=a\sin(bx)+d$ be the function we want to use to describe periodic events. In this context $a > 0$ is supposed to be the amplitude and d the crossing point on the y-axis. Now I read that b ...

Let's consider the following pair of quadratic inequalities: $x^2+x \geq a$ and $x^2-x < a$ The solution of the first inequality is $x \geq \frac{-1+ \sqrt{1+4a}}{2}$ or $x \geq \frac{-1- \...

Can you kindly help me with the first step Moser Iteration on page 14, namely $$ (\int_{Q_{k+1}}u^q\,d\mu\,dt)^\frac{1}{q}\leq (\frac{C\gamma ^k}{(\sigma-\sigma')})^\frac{p}{\rho_{k-1}}(\int_{Q_k}u^{\...

Suppose I have a three compartments (See here) system. While concentrations of two of the compartments change with time (t), the third compartment's concentration changes with number of system ...

Envelope of one parametric family of lines $y=mx+f(m)$ is claimed to touch the given family of lines. I find/list five cases of lines and their respective envelopes below: \begin{matrix} [1]: y=mx+e^...

How to give a combinatorial proof from $$\sum_{i=1}^n {n\choose i}^2={2n\choose n}$$ I have tried to give an argument with 2n set elements colored red but i got stuck on this Thanks in advance

There are $m$ balls with $(0.5-p)m$ black balls and others white, and we randomly select $d$ balls without replacement. What is the probability that an odd number of black balls are selected? While a ...

I came across the following exercise: "Prove that there is a unique real number $x$ such that for every real number $y$, $xy+x-4 = 4y$". In other words: $\exists!x\in \mathbb{R} \forall \mathbb{R} (...

First of all, $\text{TRIANGLE}$ has $8$ distinct letters, $3$ of which are vowels($\text{I, A, E}$) and rest are consonants($\text{T, R, N, G, L}$). While attempting this, I came up with the idea of ...

I should begin by saying that I'm only beginning to study $11$th grade Physics. Recently I figured out that two vectors which gets added up to a null vector, must at most lie in a line. Well that ...

We have this theorem Theorem. Let $x$ be any nontrivial element of the symmetric group $S_n$. If $n\ne 4$, then there exists an element $y\in S_n$ such that $S_n = \langle x,y\rangle$. My question:...

The Frenet-Serret Formulas involve a unit tangent vector, $$\hat T = \frac{\vec r’(t)}{||\vec r’(t)||}$$ and a unit normal vector (differentiating with respect to $t$) $$\hat N = \frac{\vec T’(t)}{||\...

It is asked to study the nature of the improper integral $$ \displaystyle\int_0^{+\infty} \dfrac{\ln(\arctan(x))}{x^\alpha} dx $$ At $+\infty$, it seems it converges for $\alpha > 1$ since the ...

Let $f,g:\mathbb Z\to \mathbb Z$. If $f$(n) and $g$(n) are injective then show if $h$(n) = $f$(n) + $g$(n) is injective? Prove or disprove. I think it's true, but I'm having a hard time ...

For any topological space $(X,\tau)$, denote $C(X,\tau)$ the algebra of all continuous real-valued functions. Is the following true or false : For any set $X$, and topologies $\tau$, $\tau'$ on $X$ ...

$X$ be a topological space and $U$ be a proper dense open subset of $X$. Then pick the correct statement from the following: If $X$ is connected then $U$ is connected. If $X$ is compact then $U$ is ...

Directly state my question by an Example: A probability space $(\Omega,\mathcal{F}, P)$ , $\Omega$ is $\{a,b,c,d\}$, take $\mathcal{F}$ be the power set of $\Omega$, the probability function $P$ is $\...

Prove that $\sum_{i=1}^{n}\left(X_i - \bar{X}\right)^2 = \sum_{i=1}^{n}\left(X_i^2\right) - n \bar{X}^2$ Here's what I have so far: $$\begin{align} \sum_{i=1}^{n}\left(X_i - \bar{X}\right)^2 &= \...

The continuity law for composition states, informally, that: $$\text{IF } f \text{ is continuous at } g \ \text{ AND }\ g \text{ is continuous at } f(a) \text{ THEN } g(f(a)) \text{ is continuous at }...

I got this problem, The average salary increment amount is 300,000 VND per month for a Development Facilitator. Standard Deviation is 200,000 VND per month. What is the probability that a certain ...

Given a random variable $X$ taking values in $[0,M]$ and with finite variance, I want to know how many iid samples $x_1, \dots, x_n \sim X$ have to be drawn such that the sample mean $\bar{x} = (x_1 + ...

I have the following problem: Use the total differential to calculate approximately the largest error at determine the area of a triangle rectangle (right triangle) from the lengths of the cathetus if ...

If I let M be a manifold such that M is a hollowed out sphere in $\mathbb{R}^3 - \{0\}$, then what is the boundary of $M$? Specifically if I'm integrating a form over M, and I want to use Stokes' ...

I'm trying to gain some intuition for integration by parts in the complex case. Suppose we integrate on $[0,1]$ for $f,g : [0,1] \rightarrow \mathbb{C}$ that are differentiable at all points on the ...

Find all integers $a$ and $b$ such that $2a + 2b = ab$.

Given the definition of recursive function as below, I need to prove that: i)Prove that a function N → N is recursive if and only if its graph is a recursive subset of $N^2$. (ii) Now we ask about ...

If $X_i$ are independent, symmetric random variables with mean $0$, and form the basis of a random walk $S_n =\sum_{i=1}^n X_i $, I am wondering if it's correct to say that the events: $$A=\{|X_i|\...

I have just begun learning about sets. My first language isn't English. I'm in high school. Here's an example problem I found in my textbook: Example 11: Let A, B and C be three sets. If A∈B and ...

I am doing an experiment where I get two values in two cases. Let's say the values are 10 in case 1 and 15 in case 2. For case 2, I want to report how high it is compared to case 1 in terms of ...

I have been studied some properties of Riemann Stieltjes integral, and i found this: If $\int_a ^b f df=0$ and f is continuous for every $a<b$ in $R$, then f is the function constant $0$ without ...

I am reading chapter 11, sampling methods, from the book 'Pattern Recognition and Machine Learning' by Bishop. In the introduction, in short, he evaluates the expectation of some function $f(z)$ with ...

Or can I say that, if I can prove it is a convex optimization problem, there must be an efficient method to solve it using the convex optimization technique?

Based on the problem statement: $f_{X|Y}(x|y) = \frac{1}{1-y}$ for $0<y<x<1$. $f_X(x) = \int_{-\infty}^{\infty} f_{X|Y}(x|y)f_Y(y)dy = \int_{-\infty}^{\infty}= \frac{1}{1-y}dy$ My ...

I am confused about the English of this sentence. (Since I am self-studying, I may be unfamiliar with some conventions). But what does this mean by ``Every closed set $A\subset X$ is $Z(f)$''? What in ...

It's easy to understand that when calculating integrals of distribution functions of one variable, we can get the following equation, which is corresponding to the calculation of differentials like $\...

Suppose I want to find the solution to the matrix-square root equation $f(X) = A - X^2 = 0$. Taking $Df(X)(H) = XH + HX$, we'll eventually obtain the recurrence $X_k H_k + H_k X_k = A - X_k^2$, $X_{k+...

I've been going at this question in my homework for a while now and I can't think of a way to prove it. Any hints or nudges in the right direction would be appreciated! Let $B:=\{y_{1},y_{2}\}$ be a ...

When a square matrix is squared. Then why doesn't its determinant be negative? For e.g in the 2*2 matrix $$ \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pmatrix} ^2=\begin{pmatrix}...

My question is about an example involving Poisson process inter arrival times. First some background: A Poisson process refers to the number of arrivals at time $t$ denoted by $N_t$. The $n^{th}$ ...

The cylinder has the equation $y^2 + z^2 = 25$ The point is $(2,9,12)$ How do I find the coordinates of the closest point on the cylinder to the point $(2,9,12)$ using lagrange multipliers?

I am trying to convince myself that the Mayer Vietoris sequence of reduced homology for a topological space $X$ covered by $int U$ and $int V$ is the following: $\ldots H_1(U) \oplus H_1(V)\...

Suppose X and Y, independent, are distributed as Uniform[0,1]. (Disclaimer: this relates to a homework problem, but is not itself a homework problem. The problem itself asks to find the conditional ...

I am sure that this circle is called something, but I do not know what it's called.

I was trying to find the Radius of Convergence of the following series: $\sum_{n=1}^{\infty}x^{n!}$. I wrote the sum as $\sum_{n=1}^{\infty}x^{((n-1)!)^n}$ and replaced $x^{(n-1)!}$ with $y$. Then the ...

I am stuck on the following problem that says: Assuming the Generalized Continuum Hypothesis (GCH), that is, the statement $2^{\aleph_{\alpha}}$ = $\aleph_{\alpha+1}$ for every ordinal $\alpha$,...

I am trying to learn spectral theory by myself from Kreyzig's book and got confused about the definition of the spectrum. Consider an operator $T: D(T) \longrightarrow X$ where $X$ is a normed space. ...

This is an exercise from textbook Analysis I by Amann/Escher. I would like to verify if my attempt is correct. Thank you for your help! Let $(R,+,\cdot)$ be a ring with unity and $I$ an ideal of $R$...