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H: If cumulative probability distributions $(F_n)_n$ converge pointwise to a continuous cdf $F$, then $(F_n)_n$ converges uniformly to $F$.
I have a candidate proof for this result, but the solution given in the solution manual for this exercise seems much more complicated than mine so I wonder if I did something wron... |
H: Is it possible to construct a matrix norm that uses minimum instead of maximum over a compact convex set?
I'm reading a paper where the following matrix norm is used:
$$ ||A||_{C, 2} = \max_{x \in C} \|Ax\|_2, $$
where A is a $d \times q $ matrix, $C$ is a compact convex set in $\mathbb{R}^q$, and $\|.\|_2$is a sta... |
H: Finding $a$ such that $ \sqrt{\frac32x^2-xy+\frac32y^2}=x\cos a+y\sin a$ has at least one solution other than $(0,0)$
Find all values of the parameter $ a $ from the interval $ [0, 2 \pi) $, for which the equation
$$ \sqrt{\dfrac{3}{2}x^2 - xy + \dfrac{3}{2}y^2} = x \cos a + y \sin a $$
has at least one solution $ ... |
H: How to prove distance from foci on an ellipse is equal to twice the semi-major axis (for specific ellipse)
Prove that for any point (x,y) on the conic, the sum of the distances to the two foci is always twice the semi-major axis.
I know that this can be proven in general for all ellipses but the practice question s... |
H: Let $B$ be a collection of pairwise disjoint intervals $[a_i,b_i)$ where $a_i\in\Bbb R$ and $b_i\in\Bbb Q$. Can $B$ be uncountable?
In an exercise I am trying to solve the following question appeared:
Let $A_i$ denote the following interval: $[a_i,b_i) \subset \mathbb R$, with $a_i \in \mathbb R$ and $b_i \in \mat... |
H: Prove that a polynomial has no rational roots
Let $P(x)$ be an integer polynomial whose leading coefficient is odd. Suppose that $P(0)$
and $P(1)$ are also odd.
Prove that $P(x)$ has no rational roots.
I have been able to prove that there are no integer roots (using the binomial theorem), and I'm stuck.
AI: Let $P(... |
H: Question about cardinalities of sets
Let $A\cap C$ and $B\cap C$ be finite such that $\left |A\cap C \right|\ge \left |B\cap C \right|$. From this, can we conclude that $\left |\neg B\cap C \right|\ge \left |\neg A\cap C \right|$?
My gut feeling tells me that we can. Either $C$ is finite or not. If $C$ is finite, t... |
H: Condition in an inequality
I have an inequality that is reduced to :
$h(1 - 2v) \geq \frac{1-2v}{2}$
I need to find that :
if $v < 1/2$, then $h \geq 1/2 $
if $v > 1/2$, then $h \leq 1/2 $
But I am only able to find that :
$h(1 - 2v) \geq \frac{1-2v}{2}$
$h \geq \frac{1-2v}{(1 - 2v) 2}$
$h \geq \frac{1}{2}$
It m... |
H: No simple group of order 720
In his Notes on Group Theory, 2019 edition (http://pdvpmtasgaon.edu.in/uploads/dptmaths/AnotesofGroupTheoryByMarkReeder.pdf p. 83 and ff.)
Mark Reeder gives a proof of the non-existence of simple groups of order 720.
P. 83, before the proof, he says : "In the former case, where $n_3(G) ... |
H: How to evaluate the volume of tetrahedron bounded between coordinate planes and tangent plane?
Find the volume of the tetrahedron in $\mathbb{R}^3$ bounded by the coordinate planes $x =0, y=0, z=0$, and the
tangent plane at the point $(4,5,5)$ to the sphere $(x -3)^2 +(y -3)^2 +(z -3)^2 = 9$.
My attempt: I started ... |
H: Explicit solution to an ODE
Consider the nonlinear ODE
$$y'(t)=\frac{a(t)+b(t)}{a(t)b(t)}b(y(t)), \qquad y(0)=0, \qquad 0<t<1,$$
where $a,b \in C^0([0,1])$ are positive and Lipschitz.
Can I find $y$ explicitly in terms of $a,b$ ?
If $b(t)=b$ is constant, then obviously $y(t)=t+b\int_0^t \frac{1}{a(s)} \, ds$. What ... |
H: Use of Lim Sup in proof, rather than Lim
In my textbook on Advanced Probability it reads
"Definition: $X_n$ converges in probability to X if for all $\epsilon >0$, $\lim_{n\rightarrow \infty} P(|X_n-X| \geq \epsilon) = 0$"
Now in a lemma, we set out to prove that $X_n$ also converges in probability to $X$ if and on... |
H: Finding position vector of orthocentre
I wanted to know the position vector of orthocentre of a $\triangle ABC$. Given position vectors of vertices as $A(\mathbf a),\,B(\mathbf b),\,C(\mathbf c)$, can we find a general formula for orthocentre like for centroid it is $\displaystyle G\left(\frac{\mathbf{a+b+c}}{3}\ri... |
H: Continuous Images of Arc Connected spaces
Arc Connected: $X$ is arc connected if for any $x,y\in X$, $\exists$ homeomorphism $f:I\to X$ such that $f(0) = x, f(1) = y.$
If $g:X\to Y$ is a continuous surjective function and $X$ is arc connected, is $Y$ arc connected too?
I don't think so, but haven't been able to fin... |
H: If $Y\subseteq X:=\prod_{i\in I}X_i$ then there exist $Y_i\subseteq X_i$ for each $i\in I$ such that $Y=\prod_{i\in I}Y_i$
Statement
If $Y\subseteq X:=\prod_{i\in I}X_i$ then there exist $Y_i\subseteq X_i$ for each $i\in I$ such that $Y=\prod_{i\in I}Y_i$.
Defining $Y_i:=\pi_i[Y]$ for each $i\in I$ then clearly $Y\... |
H: Dirac Measure (weak limit)
I am wondering why Dirac measure is weak limit of the function?
AI: because for all $\varphi \in \mathcal C_c^\infty (\mathbb R^n)$, $$\lim_{r\to 0}\int_{\mathbb R^n}\varphi (x)f_r(x)\,\mathrm d x=\varphi (0)=\left<\varphi ,\delta _0\right>.$$ |
H: Making sense of linear transformations under change of basis
Let $T: V \rightarrow V$ be a linear transformation, where $V$ is some $n$-dimensional space. Let $A, B$ be two ordered bases for $V$. Let $T_A$ and $T_B$ represent the matrix representations of $T$ with respect to $A$ and $B$ respectively. Let $x_A$ and ... |
H: Find the remainder when $(x - 1)^{100} + (x - 2)^{200}$ is divided by $x^2 - 3x + 2$ .
Find the remainder when $(x - 1)^{100} + (x - 2)^{200}$ is divided by $x^2 - 3x + 2$ .
What I tried: In some step I messed up with this problem and so I think I am getting my answer wrong, so please correct me.
We have $x^2 - 3... |
H: Given a function $f(x)$ that is define by $f(x-1)$, by knowing $f(0)$ is it possible to rewrite $f(x)$ without using $f(x-1)$
Let a function $f(x)$ that is written using the function itself. Something like Fibonacci sequence $f(x)=f(x-2)+f(x-1)$. Now given enough result of $f(x)$ (in the example of Fibonacci sequen... |
H: Moving from log points to percentage points
I'm trying to understand the formula to move from log points to percentage points. I know the same question has already been asked here: How to interpret the difference in log points
and I can follow PaulB's answer easily until the taylor expansion, is the last step that... |
H: $M^{2\times 2}(\mathbb{Z})$ has identity $e_1$, but a certain subring $R$ has another identity $e_2$ and $e_1 \notin R$.
I'm a bit confused. Consider the following situation:
We know $M^{2\times 2}(\mathbb{Z})$ has an identity for $\cdot$ which is $$e_1 =\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}.$$
Now cons... |
H: Metric Space Question: is $H(x)$ in this neighborhood?
Let the metric $d$ be defined as
$$
d(f,g) =\sup_{x\in[0,1]}|f(x)-g(x)|,
$$
and let
$$
H(x) = \begin{cases} 0 \text{ if } x \leq \frac{1}{2}\\ 1 \text { if } x > \frac{1}{2} \end{cases}.
$$
Is $f(x) = x$ in $B_\frac{1}{2}(H)$ ?
My answer. No, because
$$
d(H(x),... |
H: Does multivariate polynomial over a finite field always have a solution (in the field)?
Let $K = F_{p^e}$ be a finite field. Say I have a single polynomial $f \in K[x_1,\ldots, x_n]$ of degree $d$.
Under what conditions on $n$ and $d$ can I claim that a root to $f$ always exists? In other words, do there exist poly... |
H: Algorithm to generate insecure random numbers
I would like an algorithm which can generate a list of random, uniformly distributed floating point numbers from a given seed, ideally also being able to specify the number of decimal places.
The use case if for randomly generating datasets for education, so does not ne... |
H: If $\frac{a}{b}$ is irreducible, then the quotient of the product of any $2$ factors of $a$ and any $2$ factors of $b$ are irreducible.
$a,b\in \mathbb{Z}$
Factors of $a$: $a_1,a_2,...,a_n$.
Factors of $b$: $b_1,b_2,...,b_m$
Prove that if $\frac{a}{b}$ is irreducible, then
$\frac{a_ia_j}{b_kb_l}$ is irreducible f... |
H: $6\times 6$ grid problem
[Edited to be consistent with the version I proposed in an answer below, which the OP agreed contained the essence of the problem.--John Hughes]
Some months ago, one friend proposed me a problem that I still do not find the solution. That is:
"You have a $6\times 6$ grid ($36$ squares) with... |
H: Combinatorics calculation
I am trying to solve a problem and stuck at an intermediate step. Let $s_M$ be average of elements of a set $M\subset N$, $|N|=n$. Find an average of all $s_M$.
I got result as :
(Sum of all set elements)*(1),where (1) is given below :
$$ \tag{1}
{n-1 \choose 0}/1+
{n-1 \choose 1}/2+
{n-1 ... |
H: a probabilistic series limit?
Could anyone tell me $\lim_{n\to \infty}\frac{1}{n+1} \sum\limits_{j=0}^{n}x(j)=?$ If I am given that $x(j)=1$ with probability $p$ and $x(j)=0$ with probability $q$, $p+q=1$
Thanks for helping.
AI: You cannot find the limit in general but if $x(j)$'s are indepedent then SLLN can be a... |
H: Solving $\frac{dy}{dx}=\sin(10x+6y)$. Why doesn't my approach work?
I do not want the solution to this question. I want to know why we cannot apply what I did.
A curve through origin satisfies $\frac{dy}{dx}=\sin(10x+6y)$. Find it.
My method:
Let $10x+6y=t$
This gives $\frac{dt}{dx}-6\sin t=10$
Integrating factor... |
H: Is this proof of $C[0,1]$ and $C[a,b]$ being isometric correct?
From the book Introductory Functional Analysis with Applications-Kreyszig:
Let $C[a,b]$ be the metric space of continuous, real valued functions defined on $[a,b]\subset \mathbb{R}$ with the metric $d(x,y)=\max_{t\in[a,b]}|x(t)-y(t)|$. Show that for a... |
H: Solve $\sqrt[4]{x}+\sqrt[4]{x+1}=\sqrt[4]{2x+1}$
Solve $\sqrt[4]{x}+\sqrt[4]{x+1}=\sqrt[4]{2x+1}$
My attempt:
Square both sides three times
$$\begin{align*}
36(x^2+x)&=4(\sqrt{x^2+x})(2x+1+\sqrt{x^2+x})\\
(\sqrt{x^2+x})(35\sqrt{x^2+x}-4(2x+1))&=0
\end{align*}$$
This means $0,-1$ are solutions but I can't make sure ... |
H: Probability of winning a ticket with a red dot
Question:
You have obtained some interesting information about the local lottery. There was a malfunction at the printer that accidentally marked a bunch of tickets with a red dot. This malfunction disproportionately affected winning lottery tickets. In total $40\%$ of... |
H: Infinite product limit and estimate
I came across this product series in research and need to understand and estimate it. It appears to be unbound, but what would be the law?
$$\lim_{n \to \infty} \prod_{k=1}^{n}\frac1{(1 - \frac1{2k+1})}$$
Many thanks if you know the answer.
AI: The first thing I would do is write... |
H: If $X_n$ converges to $X$ in probability, then for $f$ continuous, then $f(X_n)$ converges in probability to $f(X)$
The following is an exercise in the book Measure Theory and Probability, by Athreya and Lahire.
Let $X_n$ converge to $X$ in probability. If $f$ is continuous, then $f(X_n)$ converges in probability t... |
H: Why does definition of the inverse of a matrix involves having $AB=I=BA$?
So, I was reviewing the first course in Linear Algebra which I took and got curious about the reason behind defining the inverse of a matrix in the following way (from Wikipedia):
In linear algebra, an $n$-by-$n$ square matrix $A$ is called ... |
H: Evalution of a function where $t = x + \frac{1}{x}$
Consider a function $$y=(x^3+\frac{1}{x^3})-6(x^2+\frac{1}{x^2})+3(x+\frac{1}{x})$$ defined for real $x>0$. Letting $t=x+\frac{1}{x}$ gives: $$y=t^3-6t^2+12$$
Here it holds that $$t=x+\frac{1}{x}\geq2$$
My question is: how do I know that $t=x+\frac{1}{x}\geq2$ ?... |
H: How to Find Solutions to a Multivariate Polynomial System
I have a system of polynomials, where the first one is a multivariate linear polynomial, but the rest are univariate quadratic polynomials. How would I solve such a system (finding one or all solutions, or showing there are no solutions)? For example,
$$17x+... |
H: Some parts of the proof of downward Löwenheim–Skolem theorem I need to clarify
Downward Löwenheim–Skolem theorem states that, for every signature $\sigma$ of a first order language, every infinite $\sigma$-structure $\mathscr M$ with domain $M$ and every infinite cardinal number $\kappa \ge \vert\sigma\vert$, there... |
H: Integration of Sign(x)
I'm currently trying to create a function for a coding project. It takes a function of the form $ax^{b}$ and integrates it up until a value $c$ beyond which the value is 0.
From this I played around in desmos and came up with the following function:
$$
\frac{1}{2}ax^b(1-\operatorname{sgn}(x-c... |
H: derivative of multivariable recursive function
Given a recursive function
$$
f(x,y,z) = f(h(x),g(y,z),z)
$$
I want to get the derivative of the function to $z$
$$
{d\over dz } f(x,y,z) = ?
$$
My guess is
$$
{d \over dz}f(x,y,z) = f'(h(x),g(y,z),z) g'(y,z) {dy \over dz}
$$
But I'm not sure if I'm right, especially t... |
H: Prove $\sum_{k=1}^{\infty} \frac{{(-1)}^n}{k^2} \sum_{j=0}^{\infty} \frac{{(-1)}^j}{2k+j+1}=-\frac{\pi^2}{12}\ln{2}+\pi C-\frac{33}{16} \zeta(3)$
Prove $$\sum_{k=1}^{\infty} \frac{{(-1)}^k}{k^2} \sum_{j=0}^{\infty} \frac{{(-1)}^j}{2k+j+1}=-\frac{\pi^2}{12}\ln{2}+\pi C-\frac{33}{16} \zeta(3)$$
where C is catalan's c... |
H: What's meant by the number of "distinct $C^k$ differential structures" other than the amount of distinct maximal atlases?
When reading the Wiki page on differential structures, I'm struck by the exceptional case of $R = 4$.
However, the definition of differential structure leaves me nonplussed, as it seems to just ... |
H: Why does $I(\overline{S})=I(S)$?
Let $S\subset \operatorname{Spec}A$, where $A$ is a commutative ring with $1$. Define $I(S)$ to be the set of functions vanishing on $S$. In other words, $I(S)=\bigcap_{P\in S}P\subset A$. Why is it true that $I(\overline{S})=I(S)$? Here $\overline{S}$ denotes the Zariski closure of... |
H: Integrate over a set $B = \left\{ (x,y) \in \Bbb R^2: 2\leq x \leq y \leq 6 \right\}$.
How do I integrate over the following set?
$$B = \left\{ (x,y) \in \Bbb R^2: 2\leq x \leq y \leq 6 \right\}$$
This may seem trivial but I really am not sure how to find the bounds. I thought since $x \leq y$ it would indicate an ... |
H: Checking Presentations in GAP
If I have the following presentation for $A_5$ $$\langle x,y,z\mid x^3 = y^3= z^3 =(xy)^2=(xz)^2= (yz)^2= 1\rangle$$ with subgroup $$ H = \left\langle {x,y} \right\rangle$$ and let GAP apply coset enumeration to my generators and relations, as with the code below, is there a command I... |
H: Exercise with maximum and minimum between real numbers
Let $a, b, \alpha, \beta>1$ be four real numbers. Consider
$$\max\lbrace a^{-\alpha}, b^{-\beta}\rbrace.$$
I am looking for the right quantity C such that
$$\max\lbrace a^{-\alpha}, b^{-\beta}\rbrace\cdot C =1.$$
I guess that it is $\min\lbrace a^{\alpha}, b^{\... |
H: absolute convergence of the series, $\sum_{n=1}^\infty \frac{nz^{n-1}\{(1+n^{-1})^n-1\}}{(z^n-1)\{z^n-(1+n^{-1})^n\}}$
We need to prove the absolute convergence of the series, $\sum_{n=1}^\infty \frac{nz^{n-1}\{(1+n^{-1})^n-1\}}{(z^n-1)\{z^n-(1+n^{-1})^n\}}$.
Since
\begin{align*}
\sum_{n=1}^\infty \frac{nz^{n-1}\{(... |
H: If $K$ is compact then $K\cap Y$ is compact in $Y$ too for any closed $Y\subseteq X$
Definition
A subspace $K$ of a topological space $X$ is compact if every its open cover has a finite subcover.
Lemma
If $X$ is compact then any its closed subspace is compact too.
Proof. Omitted.
Theorem
If $K$ is compact and close... |
H: Finding the image of a matrix given two examples of transformations
I was given the following question:
Let $e_1=\left(\begin{matrix}1\\0\\\end{matrix}\right)$ and $e_2=\left(\begin{matrix}0\\1\\\end{matrix}\right)$, $y_1=\left(\begin{matrix}3\\5\\\end{matrix}\right)$ and $y_2=\left(\begin{matrix}-1\\8\\\end{matri... |
H: How to calculate $ \int_0^\infty \exp(-\frac{a^2}{x^2}-x^2)~\mathrm{d}x $
suppose $a>0$, how to integrate:
$$
\int_0^\infty e^{-a^2/x^2}e^{-x^2}~\mathrm{d}x
$$
AI: In general,
\begin{align}
\int_{-\infty}^{\infty}f\left(x-\frac{1}{x}\right) dx
&= \int_{-\infty}^{0}f\left(x-\frac{1}{x}\right) dx+ \int_{0}^{\infty}f... |
H: Finding the distribution of a sequence of random variables
I'm having some trouble with this problem:
Let $(X_n)_{n\ge1}$ a sequence of random variables that for all ${n\ge1}$, $X_n$ follows an exponential law with parameter $1/n$. Let $Y_n = X_n - \left\lfloor X_n\right\rfloor$.
$(Y_n)_{n\ge1}$ converges in distr... |
H: Example of an omitted type taken from Hodges' book
So when reading section 6.2 of Hodges' A shorter model theory I came across this example:
Now, what does it mean precisely "by symmetry"? What is the argument he is trying to make?
I tried to prove on my own that for each $s\subseteq\omega$ there exists a countabl... |
H: Let $a,b \in \Bbb C$, show that $\sum_{k=0}^{n-1} |a+w^kb|=\sum_{k=0}^{n-1} |b+w^ka|$ with $w=\exp\bigg(\dfrac{2i\pi}{n}\bigg)$
Let $a,b \in \Bbb C$, and let's denote $w=\exp\bigg(\dfrac{2i\pi}{n}\bigg)$
Show that for every $n\ge 2$ $$\sum_{k=0}^{n-1} |a+w^kb|=\sum_{k=0}^{n-1} |b+w^ka|$$
$\bullet~$ My attempts:
I t... |
H: Under what conditions does $ \ (a+b)^{n}=a^{n}+b^{n}$ for a natural number $ n \geq 2$?
Under what conditions does $ \ (a+b)^{n}=a^{n}+b^{n}$ holds for a natural number $ n \geq 2$?
My attempt at solving:
Using $(a+b)^2=a^2+2ab+b^2$; if $(a+b)^2=a^2+b^2$, $2ab=0$ therefore $a$ and/or $b$ must be $0$.
If $a$ and/or ... |
H: Difference between $\det(A) \neq 0$, the columns are linearly independent, and the rows are linearly independent?
The question is such that,
$A$ is an $m\times n$ matrix.
$x$ is an $n$ vector, $b$ is a $m$ vector.
We want the condition that ensures the existence of a solution for $$Ax=b$$
The options had,
$\det(A)... |
H: Finding sum of expressions involving coefficients of terms in the expansion $(1+x+x^2)^n$
We take:
$$(1+x+x^2)^n=a_0+a_1x+a_2x^2+a_3x^3+\cdots+a_{2n}x^{2n}$$
and we need to find the values of the expressions:
$$i)a_1+a_4+a_7+a_{10}+\cdots$$
$$ii)a_0-a_2+a_4-a_6+\cdots$$
I have solved similar expressions for eg.
$$1... |
H: Proof of the following equality with vectors
Let $\{v_1,v_2,\dots,v_n\}$ be an orthogonal set in $V$, and let $a_1,a_2,\dots,a_n$ be scalars. Prove that
$$\left\Vert \sum_{i=1}^na_iv_i \right\Vert^2=\sum_{i=1}^n|a_i|^2\Vert v_i\Vert^2$$
Here's what I've tried, but I don't know if it is correct: $$\left\Vert \sum_{i... |
H: Wrong proof: in a ring $R$ such that $r^n=r$ for every $r\in R$, there are no non-trivial ideals.
The claim is
Let $R$ be a commutative ring with an identity element $1\not=0$ with the property that for every $r\in R$ there is an $n\geq 2$ such that $r^n=r$. Then there are no non-trivial ideals.
My proof which I'... |
H: Incomplete information of player’s choice in Prisoner’s Dilemma
What happens if the players in a prisoner’s dilemma or stag hunt game don’t always have control over their choices?
Instead of deciding to cooperate or defect the players have to draw from a deck. There are an equal number of cooperate and defect cards... |
H: Conditions under which a series converges to another.
In a proof I'm reading they seem to be using a claim like this:
Let $(a_{s,n})_{s,n}$ and $(b_n)_n$ be sequences of real numbers. Suppose $\sum_n b_n$ converges and $\sum_n a_{s,n}$ converges for each fixed $s$. Suppose further that for fixed $n$ we have $a_{s,n... |
H: show that $v_n \leq 2u_n$
let $(u_n)_{n \geq 1}, \, (v_n)_{n \geq 1}$ such that :
$$\forall n\geq1, \,\,\,v_n \leq \frac{u_n}{(1-u_n)^2}$$
and $u_n \to 0$, prove that for $n$ large enough we have $v_n \leq 2u_n$.
now intuitively speaking I know why this would be the case, as $(1-u_n)^2$ would be close to $1$ and th... |
H: Is this proof of Bernoulli’s inequality correct?
I have to prove $(1+x)^{n} >1+nx$ for $n=2,3,4.... $and $x>-1$ and $x$ isnt 0. There are a lot of proofs of this but l want to know if this one works. If not, can u show where my reasoning is weak.
If $x>-1$ then $1+x>0$
Hence $ (1+x)^{n}>0$ for $n>=2$ and $x>-1$ . F... |
H: Proof of $\sum_{k=1}^n \left(\frac{1}{k^2}\right)\le \:\:2-\frac{1}{n}$
Proof of $\displaystyle\sum_{k=1}^n\left(\frac{1}{k^2}\right)\le \:\:2-\frac{1}{n}$
The following proof is from a book, however, there is something that I don't quite understand
for $k\geq 2$ we have:
(1): $\displaystyle\frac{1}{k^2}\le \frac{... |
H: Why does $V(I(S))=\overline{S}$?
Let $S\subset\operatorname{Spec}A$, where $A$ is a commutative ring with $1$. I am having trouble seeing why $V(I(S))=\overline{S}$, where $\overline{S}$ is the Zariski closure of $S$.
My attempt is as follows. It is not hard to see that $S\subset V(I(S))$: $V(I(S))$ is the set of a... |
H: Suppose that $f$ is surjective and relation preserving. Then $\mathcal{R}$ is reflexive iff $\mathcal{S}$ is reflexive.
This is a problem about relations from Proofs and Fundamentals by Ethan D. Bloch that I’m having some doubts and I would really appreciate if you could guide me.
The problem starts with the follow... |
H: Pathwise Connectification of Spaces
For any space $X$, let $Y=X\times I$, and topologize $Y$ by defining basic neighbourhoods of $(x,y)$ as -
$(x,y), y\neq0: U_{(x,y)} = \{x\}\times B_\epsilon(y)$
$(x,0): U_{(x,0)} = \{(x',z):z'\in U, 0\leq z < \epsilon_z \},$ where $U$ is a neighbourhood of $x$ in $X$, and $\eps... |
H: What is the opposite of "coprime integers"?
What do you call two integers that are not relatively prime? In my language, there is a clear term for that, but I can't seem to find one in English.
AI: There isn't a standard term for that notion.
"Integers with a nontrivial common factor" is probably the best you can d... |
H: When does the inequality hold?
I am trying to find a condition on $c$ such that the below inequality holds true
$$ \frac{1 - e^{-st}}{st} - \frac{1}{st+c} > 0 $$
where $s$, $c$ and $t$ are greater than $0$. I tried simpyfing it and got $c > (c + st) e^{-st}$, but I am not sure what do next.
AI: We have
$$\frac{1 - ... |
H: Integral of second-order derivative
How can I perform the integration of second-order T ($\int \partial^2T=0$), so that I can arrive at equation 5.85, where T is a variable of $\xi$ and $\eta$?
Here is what I get:
$$ \int \partial^2T=0 $$
$$ T\partial + C = 0 $$
and I'm not sure what to do with the $T\partial$. Do... |
H: $\lambda_{\max}(XDX^T)$ smaller than $\lambda_{\max}(XX^T)$?
$X\in\mathbb{R}^{n\times d}$ and $D$ is a $d$-dimensional diagonal matrix. All elements on the diagonal of $D$ are in $[0,1]$. I am wondering whether the largest eigenvalue $\lambda_{\max}(XDX^T)$ of $XDX^T$ is smaller or equal to $\lambda_{\max}(XX^T)$. ... |
H: Prove $A$ is dense in $C([0,1]\times[0,1])$
Given
$$
A=\left\{\sum^n_{k=0}f_k(x)g_k(y) : \ n \in \mathbb{Z}^+, \ f_k, g_k\in C[0,1]\right\}.
$$
I am trying to use the Stone-Weierstrass Theorem to prove that $A$ is dense in $C([0,1]\times[0,1])$.
It is easy to see that $A$ is an algebra. I know $A$ vanishes nowhere... |
H: $a_n$ is convergent.
Let $\{a_n\}$ be a bounded sequence of real numbers and $a_{n+1}\geq a_n - 2^{-n}$. Prove that $a_n $ is convergent.
My attempt: Suppose $a_n $ is not convergent then $\limsup a_n \neq \liminf a_n$. Let $\{x_n\}$ and $\{y_n\}$ converges to limsup and liminf. Then for some $x_p=a_{n_p}$ and $y... |
H: Finding the volume of a rectangular prism using only surface area
Three surfaces of a rectangular prism are 25 cm squared, 18 cm squared, and 8 cm squared. What is its volume?
Can someone please explain how to solve the problem without using guess and check? The book where I found this problem said the answer is 60... |
H: Is the summation $\sum_{i=1}^{n}\frac1{i} \binom{n}{i}$ possible?
I want to compute the following sum:
$$
\sum\limits_{i=1}^{n} \frac{{n\choose{i}}}{i}
$$
What I have done so far:
We know that $$(1+x)^n=\sum\limits_{r=0}^{n} {n\choose{r}}x^r$$
so, $$\frac{(1+x)^n-1}{x}=\sum\limits_{i=1}^{n} {{n\choose{i}}}x^{i-1}$... |
H: Laurent series of $\sin(-\frac{1}{z^2})$ radii of convergence
I am calculating radii of convergence of series for function:
$$
f(z)=\sin(-\frac{1}{z^2})
$$
I started with Taylor expansion for $\sin$ and then inserted $-\frac{1}{z^2}$. I got:
$$
-\frac{1}{z^2}+\frac{1}{3!}\frac{1}{z^6}-\frac{1}{5!}\frac{1}{z^{10}}+\... |
H: why is R matrix inversion and transposition the same, but matrix Q has different inversion and transposition results
$R ={\begin{bmatrix}0.9697253054707993 & 0.04804422035332832 & -0.2394255308445735\\-0.01069682073773017 & 0.9878712527451343 & 0.1549063137056192\\ 0.2439639521643922 & -0.1476554803940568&0.95847... |
H: k-partite Subgraph
I'm just working on a problem, but can only show, that the statement is true for $k=2$.
Let $G$ be a graph with $E(G)$ edges and $k \ge 2$. Show, that there is a $k$-partite subgraph $G*$ of $G$, so that $E(G*) \ge \frac{k-1}{k} E(G)$.
For $k = 2$, I solved the problem by induction over the numb... |
H: Find the smallest positive real $x$ such that $\lfloor{x^2}\rfloor - x\lfloor{x}\rfloor = 6$
Problem
Find the smallest positive real $x$ such that $\lfloor{x^2}\rfloor - x\lfloor{x}\rfloor = 6$.
What I've done
I set $m = \lfloor x \rfloor$ and $n = \{x\}$. Then I proceeded as below:
$\lfloor (m+n)^2 \rfloor -(m+n)m... |
H: Sheaf associated to a locally free presheaf of modules
Just checking here. It is true isn't it, that the sheaf associated to a locally free presheaf of $O_X$ modules (I suppose there is an example that is not a sheaf??) over a scheme $X$ is a locally free sheaf of $O_X$ modules? If the associated sheaf is coherent ... |
H: Uniform random variables question
Let U and V be independent random variables, both uniformly distributed on [0, 1]. Find
the probability that the quadratic equation $x^
2 + 2Ux + V = 0$ has two real solutions.
My solution:
The probability of two real solutions is the probability that $4U^2 - 4V > 0$.
$$
P(4U^2 - 4... |
H: The set $\{P \mid d(P,A) = k\cdot d(P,B)\}$ always represents a circle.
I'm trying to demonstrate that, given a real number $k$, with $k>0$ and $k\neq 1$, the set $\{P \mid d(P,A) = k\cdot d(P,B)\}$ always represents a circle.
I simply gave the coordinates $A=(m,n)$, $B=(p,q)$ and $P=(x,y)$ and put into $d(P,A) = k... |
H: How to divide with exponentiation?
Let's say I wanted to multiply but couldn't actually use the multiply operation. I could do this:
$$(a+b)^2 = a^2 + 2ab + b^2 \implies ab = \frac{(a+b)^2 - a^2 - b^2}{2} $$
Now, logically it should be possible to reverse this process, I should be able to divide two numbers without... |
H: Under what conditions will the covariance matrix be identical to the correlation matrix?
Under what conditions will the covariance matrix be identical to the correlation matrix?
I have been looking everywhere but no webpage or book seems to answer my question.
I just want to know when could this situation happen,... |
H: Show $\pi$ is the orthogonal projection of $W$ iff $\|\pi(u)\| \leq \|u\|$ for all $u \in V$
Let $V$ be an inner product space and $W \subseteq V$ a finite-dimensional subspace of $V$. Let $\pi \in \mathcal L(V,V)$ a projection with $W$ as image. Show that $\pi$ is the orthogonal projection $\operatorname{pr}_W$ o... |
H: Would my proof of induction be accepted in an intro Abstract Algebra Course. Self-studying and New to proofs.
Hello I'm self studying and I'm also new to proofs and would like to know whether my proof is rigorous enough for a first course in Abstract Algebra.
I'm asked to proof Induction of the second kind which st... |
H: Let $L/K$ be a finite Galois extension and $\alpha\in L\setminus K$. Then there exists $h\in G$ with prime power order not fixing $\alpha$.
Let $L/K$ be a finite Galois extension of fields and let $G=\text{Gal }L/K$. Let $\alpha\in L$ with $\alpha\notin K$. Show that there exists $h\in G$ with $h$ of prime power or... |
H: Law of large numbers question
Let $a, b, p \in (0, 1)$. What is the distribution of the sum of $n$ independent Bernoulli random variables with parameter $p$? By considering this sum and applying the weak law of large numbers, identify the limit
$$
\lim_{n \to \infty} \sum_{r \in \mathbb{N}:an<r<bn} \binom{n}{r} p^... |
H: Some Counterexamples on Connectedness
There are abundant counterexamples in literature of the $2$ statements -
$X$ is Path Connected $\implies$ $X$ is Locally Path Connected
$X$ is Arc Connected $\implies$ $X$ is Locally Arc Connected
In all of the counterexamples I've found, they hold as the space is Path/Arc Co... |
H: Little $o$ notation in the proof of central limit theorem.
In the proof im reading for the CLT, they seem to be using the following claim:
If $h(t)=o(t^2)$ then if $$g(n)\stackrel{\text{def}}{=}h\Big(\frac{t}{\sigma\sqrt{n}}\Big)$$ we have that $g(n)=o(\frac{1}{n})$.
More explicitly, I have a function $h$ that sa... |
H: Can any integer be expressed as sums of powers of three?
I heard a long time ago that any integer can be expressed as sums (or differences) of powers of three, using each power only once. Examples:
$5=9-3-1$
$6=9-3$
$22=27-9+3+1$
etc.
To my surprise, I couldn't find anything about this on the Internet. So..
Is thi... |
H: Spivak's Calculus: chapter 2, problem 18(c)
In Spivak's calculus book, I cannot understand the solution proposed for question (c) of problem 18 in chapter 2:
Prove that $\sqrt{2}+\sqrt[3]{2}$ is irrational. Hint: Start by
working out the first 6 powers of this number.
Working out the powers is quite easy:
$(2^\f... |
H: Appropriateness of Poisson distribution for low number of trials with the probability = 0.5 of success
I'm working on the following problem from Ross "A First Course in Probability" (9th edition):
People enter a gambling casino at a rate of 1 every 2 minutes. (a)
What is the probability that no one enters between ... |
H: When evaluating the limit of $f(x, y)$ as $(x, y)$ approaches $(x_0, y_0)$, should we consider only those $(x, y)$ in the domain of $f$?
When evaluating the limit of $f(x, y)$ as $(x, y)$ approaches $(x_0, y_0)$, we should or should not consider only those $(x, y)$ in the domain of $f(x, y)$ ? I am confused by diff... |
H: Why there is no suspension axiom for homology ? and why there is no excision axiom for cohomology theory?
Here are the axioms of reduced cohomology theory as given to me in the lecture:
1- $\tilde{H}^n(-;G): J_{*} \rightarrow Ab_{*}$ is a contravariant functor.
2- $\tilde{H}^n(X;G) \cong \tilde{H}^{n+1}(\sum X;G).$... |
H: Solving for matrix system using least squares quadratic
So I'm given the following coordinates below and I'm asked to set up a matrix system to solve for the least squares expressions.
I have the first question right, and I have matrix A of the second question correct. I'm a little stumped on how I would find the m... |
H: What does this mathematical expression mean?
I am reading a natural language processing paper and I came across this expression. I don't know what it means. Especially the unif part.
$$m_i \sim \operatorname{unif}\{1,n\}\text{ for } i = 1 \text{ to } k$$
AI: It means you have $k$ random variables, $m_1, m_2, \ldots... |
H: Derivative of a function of two variables
Let $f:\mathbb R^2\rightarrow \mathbb R $ be defined by
$f(x,y)=\exp(-\frac{1}{x^2+y^2})$ if $(x,y)\neq(0,0)$ and $f(0,0)=0$. Check whether $f$ is differentiate at (0,0) or not.
I have checked that $f$ is continuous at (0,0) and both partial derivatives of $f$ exists at (0,... |
H: Defining derivative of powers of $x$
We know that derivative of $x^n$ is $nx^{(n-1)}$ if $n$ is an integer.
My question is how do we define derivative of $x^r$ is $r$ is an irrational number. For example what is the derivative of $x^\sqrt2$ or $x^\pi$?
AI: We define $x^r$ as $x^r = e^{r \log x}$ so that
$$\begin{al... |
H: Why does AM>GM when applied on functions gives the absolute minima.
In some cases we use the relation AM>GM to find the minima for example take $f(x)=x+\frac1x$ $[x\gt 0]$ using the result AM>GM we can find the minima as $2$.It is the same minima which we get if we use the methods of derivative.
But why do we get ... |
H: Can a non-inner automorphism map every subgroup to its conjugate?
Let $G$ be a finite non-cyclic group. Can a non-inner automorphism map every subgroup to its conjugate? Namely, can there be a non-inner automorphism $\alpha$ that, for every $H\le G$, there exists some $g$ in $G$ such that $\alpha(H)=H^g$?
AI: Yes, ... |
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