All Questions

0
votes
0answers
5 views

Linear Combinations of Solutions to a Search Problem

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 $x$). ...
0
votes
0answers
3 views

Why $\dim(\ker T_z f)=\dim(T_z(f^{-1}(c)))$?

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 ...
0
votes
0answers
6 views

Cosets of the subgroup $(x,x) \in \mathbb{R}\times \mathbb{R}$

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\...
0
votes
0answers
11 views

How can I prove that the quotient of two periods of $\sin$ is b?

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 ...
0
votes
0answers
3 views

Automating the solution of pairs of polynomial inequalities as a bound

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- \...
0
votes
0answers
4 views

Moser iteration step

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^{\...
0
votes
0answers
3 views

Compartmental Model of a Time and Sampling Frequency Dependent System

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 ...
0
votes
0answers
6 views

A curious feature of envelope of one parametric family of lines

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^...
0
votes
0answers
12 views

Proof of combinatorial set

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
0
votes
0answers
5 views

On the mod 2 inner product between an unbalanced 0-1 vector and a random m choose d 0-1 vector

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 ...
0
votes
1answer
14 views

Simple existence and uniqueness proof

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} (...
1
vote
1answer
14 views

Permutations of the word $\text{TRIANGLE}$ with no vowels together.

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 ...
3
votes
2answers
22 views

Vectors added up to give null vector.

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 ...
4
votes
0answers
19 views

Can $A_n$ be generated by two elements $x,y$ with $x\ne 1$ arbitrarily chosen?

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:...
1
vote
0answers
14 views

Help showing Frenet-Serret Formulas with respect to $t$

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)}{||\...
0
votes
0answers
13 views

Nature of improper integral following values of $\alpha$

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 ...
0
votes
0answers
22 views

If f (n) and g(n) are injective then show if h(n) = f (n) + g(n) is injective?

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 ...
1
vote
1answer
15 views

Different topologies with the same continuous real maps

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 $X$ ...
0
votes
3answers
24 views

Proper dense open subset of 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 ...
-1
votes
0answers
12 views

What is the probability of the event that cannot represent by the X=x

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 $\...
0
votes
3answers
22 views

Regression computation $\sum_{i=1}^{n} (X_i - \bar{X})^2 = \sum_{i=1}^{n} X_i^2 - n \bar{X}^2$

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 &= \...
0
votes
1answer
15 views

A formal epsilon-delta proof for the Continuity Law for Composition

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 }...
0
votes
0answers
9 views

Probability of salary increment

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 ...
0
votes
0answers
16 views

Performances of Chebyshev's and Chernoff's inequalities on the sample mean

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 + ...
-1
votes
0answers
10 views

Total differential to calculate approximately the largest error

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 ...
0
votes
0answers
11 views

Boundary of a hollowed out sphere

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' ...
0
votes
0answers
12 views

Gaining intuition for Complex Integration by Parts

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 ...
-4
votes
1answer
48 views

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

Find all integers $a$ and $b$ such that $2a + 2b = ab$.
-1
votes
0answers
9 views

Recursive functions and recursive sets

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 ...
0
votes
0answers
7 views

Is this a valid argument? Independence of sign and magnitude in a random walk

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|\...
5
votes
2answers
46 views

In “An element of a set can never be a subset of itself”, what does ‘itself’ stand for?

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 ...
0
votes
0answers
13 views

Percentage higher than a value

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 ...
3
votes
2answers
75 views

If $\int_a ^b f df=0$ and f is continuous, then f is the function constant $0$

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 ...
0
votes
0answers
12 views

Sampling Methods Explaination: Pattern Recognition and Machine Learning Bishop

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 ...
0
votes
0answers
12 views

Are all convex optimization problems easy to solve?

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?
0
votes
0answers
9 views

Finding the density of $X$ selected from $[Y,1]$, where $Y$ is first selected in $[0,1]$

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 ...
0
votes
0answers
19 views

Rudin Ex. 4.22. Every closed set $A \subset X$ is $Z(f)$ for some continuous real $f$ on $X$.

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 ...
1
vote
1answer
11 views

What does it mean to integrate a distribution function with more than one variable?

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 $\...
0
votes
0answers
18 views

Why, intuitively, will multivariate (matrix-valued) Newton's method get us to a quick solution?

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+...
1
vote
1answer
22 views

Prove these two ODEs are equal given a base for their solution set

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 ...
-1
votes
2answers
35 views

determinant of a square matrix

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}...
0
votes
0answers
7 views

Poisson process example

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 $n^{th}$ ...
-1
votes
1answer
13 views

find the coordinates of the closest point on a cylinder to another point in 3D using lagrange multipliers

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?
0
votes
0answers
14 views

Mayer-Vietoris sequence for deruved homology

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)\...
0
votes
1answer
14 views

Joint distribution of X and XY

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 ...
-2
votes
0answers
28 views

What is this drawing where there is one or more points connecting to all the other points called?

I am sure that this circle is called something, but I do not know what it's called.
0
votes
3answers
54 views

Radius of Convergence of the series $\sum_{n=1}^{\infty}x^{n!}$

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 ...
0
votes
1answer
22 views

Arithmetic on infinite cardinal numbers

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 $\alpha$,...
0
votes
0answers
7 views

Is the spectrum of an operator relative to the spaces that contains its domain and range?

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. ...
1
vote
0answers
20 views

Let $I$ be an ideal of a ring $(R,+,\cdot)$. Prove that the quotient homomorphism $\varphi : R \rightarrow R / I$ is a ring homomorphism

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 $R$...

15 30 50 per page