text stringlengths 83 79.5k |
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H: If $A^3=I_n$ (the identity matrix), then what are the possible minimal polynomials for $A$?
I just got done with an exam and one question was to determine the possible minimal polynomials of $A$, if $A^3$ is the identity matrix. Note that $A$ is just some square matrix over $\mathbb{Q}$. My thoughts were that since... |
H: prove if $f(x)$ has a finite limit m then limit of $\frac{1}{f(x)}$ is $\frac{1}{m}$
I want to proove that if $\lim_{x\to c}$f(x) =$m$ then $\lim_{x\to c} \frac{1}{f(x)}$ = $1/m$ where $m\not= 0$.
I tried solving it using the epsilon delta definition but was stuck and not able to move forward using the definition f... |
H: If a member $\{a,b\}$ of $S$ is picked at random, what is the probability that $a+b$ is even?
Let $S$ be the set of all unordered pairs of distinct two digit integers. If a member $\{a,b\}$ of $S$ is picked at random, what is the probability that $a+b$ is even?
Solution:
The pairs (unordered are): $$\{10,11\},\{1... |
H: Strategy to calculate $ \frac{d}{dx} \left(\frac{x^2-6x-9}{2x^2(x+3)^2}\right) $.
I am asked to calculate the following: $$ \frac{d}{dx} \left(\frac{x^2-6x-9}{2x^2(x+3)^2}\right). $$
I simplify this a little bit, by moving the constant multiplicator out of the derivative:
$$ \left(\frac{1}{2}\right) \frac{d}{dx} \l... |
H: Non-exact differential equation where does not exist integrating factor dependent on x and y only
Our calculus professor gave us a few supplementary problems on differential equations and I'm trying to solve the non-exact differential equation
$$(6y+x^2y^2)+(8x+x^3y)y'=0$$
I tried finding both $\frac{M_y-N_x}{N}$ ... |
H: showing $I_{c,d}$ is not a prime ideal
Given the ring $R:=\{f:\mathbb{R}\rightarrow \mathbb{R}, f $ is continous$\}$ we define for $c,d \in \mathbb{R}$ where $c\neq d$ , the ideal $I_{c,d}:=\{f\in R|f(c)=f(d)=0\} \subset R$. I have to show that $I_{c,d}$ is not a prime ideal.
My attempt is to find a counterexample:... |
H: Sum of the product of binomial coefficients
I am trying to prove $\sum_{j=1}^{n-k}{n \choose j}{n-k-1 \choose j-1}={2n-k-1 \choose n-k}$. I tried to apply Vandermonde's Identity, however I have not been able to.
AI: If you rewrite the left-hand side as $$\sum_{j=1}^{n-k}\binom{n}{j} \binom{n-k-1}{n-k-j},$$ you can ... |
H: If $f(n)$ $\in$ $O(g(n))$ and $O(h(n))$ $\subset$ $O(g(n))$, Can we say that in general $f(n)$ $\not\in$ $O(h(n))$
I have this problem because I want to prove this.
For some $k\in \mathbb{Z}^+$, let $f(n) = \sum_{i=1}^{n} i^k$ and let $g(n) = n^k$, approve or disapprove that $f(n) \in O(g(n))$.
I did this. (I do no... |
H: Rudin proves $b^{x+y} = b^x b^y$ for $x$, $y \in \mathbb{R}$ and $b>1$. But what about $b\leq 1$?
The title pretty much sums up my question. Rudin only has the exercise for $b>1$, but as I recall it is true that $b^{x+y} = b^x b^y$ for $x$, $y \in \mathbb{R}$ and for each $b\in \mathbb{R}$. So what would the rest o... |
H: Calculate the expected cost
A headphone manufacturer guarantees that the company will repair any headphone that is defective within one year warranty period for free. Past records showed that 5% of their headphone needed repair during the warranty period. The company sold 30 headphones last January. If the average ... |
H: Understanding transitivity of set elements
The following is a very simple exericse in Rudin.
Let $E$ be a nonempty subset of an ordered set; suppose $\alpha$ is a lower bound of $E$ and $\beta$ is an upper bound of $E$. Prove that $\alpha \leq \beta$.
The proof goes something like this.
Let $x \in E$ (since $E... |
H: $f(x+1) = f(x)$ for all $x ∈ \mathbb{R}$. Let $g(t)=\int_{0}^tf(x)dx$ $h(t)=\lim_{n→\infty}\frac{g(t+n)}n, t∈\mathbb{R}$
QUESTION: Consider a real valued continuous function $f$ satisfying $f(x+1) = f(x)$ for all $x ∈ \mathbb{R}$. Let
$$g(t)=\int_{0}^tf(x)dx$$
$$h(t)=\lim_{n→\infty}\frac{g(t+n)}n, t∈\mathbb{R}$$
... |
H: Norm of a linear map: Alternative definition?
Let $T: V \to W$ be a bounded linear map between normed spaces. I'm wondering if it is true that
$$\Vert T \Vert = \sup\{\Vert Tv\Vert : v \in V, \Vert v \Vert < 1\}$$
Note the strict inequality. In the usual definition this is $\leq $.
I could not prove it nor provid... |
H: Condition for collinearity of points?
Under which of the following conditions will the points $A, B, C$ with position vectors $\vec a$, $\vec b$ and $\vec c$ respectively be collinear?
(a) $\vec c-\vec a = 2(\vec b-\vec a)$
(b) $|\vec c-\vec a|=2|\vec b-\vec a|$
(c) $\vec a=2(\vec b+\vec c)$
(d) $2\vec a+\vec b=\v... |
H: Definition of Symplectic manifold
Let $(M,\omega)$ be a symplectic manifold. I'm trying to see why the non-degeneracy of $\omega$ is equivalent to say that the contraction map $X \longmapsto i_{X}\omega$ defines an isomorphism between vector fields on $M$ and differential $1$-forms on $M$.
So clearly the condition... |
H: Relation between annihilators and exact sequence
This is repeat post from relation of annihilators on exact sequence and the hint seems unclear.
$0 \rightarrow M^{\prime} \rightarrow M \rightarrow M^{\prime \prime} \rightarrow 0$ is an exact sequence of $k[x_0, \dotsc, x_n]$-modules. To prove: ${\mathrm{Ann}}~M = {... |
H: Consider the sequence $a_1 = 24^{1/3}$ $a_{n+1} = (a_n + 24)^{1/3},n ≥ 1.$ Then what is the integer part of $a_{100}$?
QUESTION: Consider the sequence
$a_1 = 24^{1/3}$
$a_{n+1} = (a_n + 24)^{1/3},n ≥ 1.$
Then what is the integer part of $a_{100}$ ?
MY APPROACH: I tried this one really hard but couldn't get the tri... |
H: why absolute square of matrix equal matrix by transpose
I want to understand why is $ |e^2| = e^T * e $ I find this formula a lot while studying deep learning like in here
AI: It isn't $\lvert e^2\rvert$, but $\lvert e\rvert^2$, and it is more about column vectors ($(n\times 1)$ matrices) than it is about matrices.... |
H: Proofs about infinite numbers of primes of different forms
I am not an expert of primes numbers. But I noticed the following. It is known that there are infinitely many primes that have a linear form-this is implied from Dirichlet theorem on primes in arithmetic progression or the Green-Tao theorem. But when primes... |
H: {$A°, Fr(A),(X-A)°$} is partition of metric space
could you help me with the following problem please:
Prove that if $A$ is a subset of a metric space $(X, d)$, then the
{$A°, Fr(A),(X-A)°$}
family forms a partition of space X. Where $A°$={$x\in A| x$ it's inside $A$}, and $Fr(A)$ denotes the border of $A$
I am a ... |
H: Evaluating the following integral $\int_{0}^{\infty} \frac{\ln(x^{2}+1)} {(x(x^{2}+1))} dx$
I have tried many different methods on this particular integral, none of them yielding any fruitful results. Here was an attempt
I(t) = $\int_{0}^{\infty} \frac{ln(tx^{2}+1)} {(x(x^{2}+1))}dx$, I(1) = $\int_{0}^{\infty} \fr... |
H: Functional equation involving composition and exponents
Do there exist functions $f,g : R → R$ such that $f (g(x)) = x^2$ and $g( f (x)) = x^3 \text{ , }\forall x ∈ R$.
Simply applying $g$ on both sides of equation $1$ and $f$ on equation $2$ respectively, we get
$g(x)^3=g(x^2)$ and $f(x)^2=f(x^3)$.
It does seem l... |
H: Which Bernstein is behind "Bernstein's Connected Sets"?
Steen and Seebach in "Counterexamples in Topology" give "Bernstein's Connected Sets" as item 124.
"Let $\{C_\alpha|\alpha\in [0,\Gamma)\}$ be the collection of all nodegenerate closed connected subsets of the Euclidean plane $\mathbb R^2$ well-ordered by $\Gam... |
H: Proof of "For transitive closure of a relation $R$ on a finite set with $n$ elements it is sufficient to find $R^*=\bigcup_{k=1}^n R^k$"
I was reading "Discrete Mathematics and its Applications" by Kenneth Rosen where I came across the following lemma and I felt that is incomplete to some extent.
Lemma: Let $A$ be... |
H: Probability: proving $E(X)=np$
This was the proof that user T_M kindly submitted to me about why the expected number of times a result comes up when you are, say, rolling a die is equal to $P(\text{result}) \times\text{Number of trials}=np$.
\begin{align}
\mathbb{E}(X) &= \sum_{k=0}^n \binom{n}{k}p^k(1-p)^{n-k} k \... |
H: How to prove below?
Given
\begin{split}
A =~& \theta(X) \cdot D + g(X) + \epsilon ~~~&~~~ \mathbf E[\epsilon | X] = 0 \\
D =~& f(X) + \eta & \mathbf E[\eta \mid X] = 0 \\
~& \mathbf E[\eta \cdot \epsilon | X] = 0
\end{split}
How to prove below?
\begin{split}
A - \mathbf E[A | X] = \theta(X) \cdot (D - \mathb... |
H: Positive eigenvalues and Schur complements
For a symmetric matrix,
$$M = \left(\begin{array}{cc}
A & C\\
C^{\top} & D
\end{array}\right)$$
it is well known that $M$ is positive definite if and only if $A$ and the Shur complement $M\backslash A = D-CA^{-1}C^T$ are positive definite.
Is there a generalization of ... |
H: Show that $n^2
Show that $n^2<n!$ for all $n\geq 4$
What I did was
Base case:
$4^2<4!\iff16<24$
Inductive hypothesis:
Being that for a $n\leq k$, the equation remains true, we prove for $k+1$
$$\begin{align*}k^2+(k+1)^2&<(k+1)!\\
\iff k^2+(k+1)^2&<(k+1)*k!\end{align*}$$
as $k!>k^2$, summing $(k+1)^2$ will be also l... |
H: Proof of A[i, j] matrix's inverse is A[i, j] matrix with 1/A[i,j] entries
A matrix A is diagonal if A[i, j] = 0 for i≠j Entries on the diagonal are not required to
be nonzero, however, for this problem, assume that A[i, j] ≠ 0 for 1 ≤ i ≤ n. Show that the
inverse matrix to A is a diagonal matrix with entries 1/
A... |
H: How do you approach doing research on an unsolved mathematics problem?
I have difficulty gathering all the information I need on open questions. How do I obtain all of the historical and technical details of a problem (e.g. a millennium problem)?
AI: Take a three-pronged approach.
Find important articles in the li... |
H: Using three points to create a quadratic equation produces an equation, which doesn't seem to pass through the original points
I've got three (x,y) points from which I am trying to create a quadratic equation, which are:
(2325, 5500)
(1880, 3700)
(1400, 2360)
Using those three points, I create three simultaneous e... |
H: Who was Hans Bauer who worked on the Perron integral?
I'm referring to the Hans Bauer who is the author of this article from 1915 (H. Bauer, Der Perronsche Integralbegriff und seine Beziehung auf Lebesguesschen, Monatsh. Math. Phys., 26 (1915), 153–198). When was this man born and when did he die? I'm asking becaus... |
H: Distribution of square numbers in intervale
The square numbers are of the form $n^{2}$ $(1,4,9,16,...)$
My question is there some formula to know how many square numbers up to $x$? or a least approximation formula ?
AI: If $n^2\le x<(n+1)^2$, then $n\le \sqrt x\lt n+1$, which means $n=\lfloor\sqrt x \rfloor$,
whe... |
H: Determine a matrix from terms of Leibniz formula
Let $A$ is an $n\times n$ matrix with nonzero real entries $a_{ij}$. Say one is given
$$
X_{\sigma}:=\text{sgn}(\sigma)\cdot\prod_{i=1}^na_{i\,\sigma(i)}
$$
for each $\sigma\in S_n$. Is this information enough to determine $A$? I believe the answer is no by the fo... |
H: Understanding the proof of: $p$ is prime and $a \in \mathbb{Z}_p \implies x^p - a \in \mathbb{Z}_p$ is reducible.
I am trying to understand the proof of: $p$ is prime and $a \in \mathbb{Z}_p \implies x^p - a \in \mathbb{Z}_p$ is reducible. In the proof, it is shown that $a$ is a zero of $x^p - a$ which implies that... |
H: A question about dimension on exponential terms
In an article I've studying, I have $f$, a function with unit $[\text{length}]^{-1}$, and the exponential $e^{\int_0^x f(s)d(s)}$, where $x$ is a length.
This exponential is a real number.
I'd like to understand better how could I drop there the units, because they ar... |
H: How does $\frac{x^2 + 6x + 5}{x^2 - x - 2}$ simplify to $\frac{(x + 5)(x + 1)}{(x - 2)(x + 1)}$?
I stumbled upon following rational, where the right hand of equation is a simplification:
$$\frac{x^2 + 6x + 5}{x^2 - x - 2} = \frac{(x + 5)(x + 1)}{(x - 2)(x + 1)}$$
I can't understand how do you derive this simplific... |
H: For any square matrix A of rank $n$ and size $2n$: $KerA \subset ImA$
I'm trying to prove the following:
Let $A$ be a $2n\times 2n$ matrix, $\operatorname{rank}A = n$, and there exists a vector $X$ such that $AX = 0$. Then there exists a vector $Y\left(Y \neq X\right)$ such that $X = AY$.
The only thing I've come u... |
H: Two doubts on expectation inequalities
Let's define a probability space $\left(\Omega\text{, }\mathcal{F}\text{, } \mathbb{P}\right)$. Suppose to have two random variables $X$ and $Y$ defined on it.
Two questions:
1. Could I state that $\vert E\{X-Y\}\vert\le E\{\vert X-Y\vert\}$ by Jensen's inequality? If not, whi... |
H: how to simplify the given integrals?
If $\frac{\int_{0}^{1}(1-x^3)^{49}dx}{\int_{0}^{1}(1-x^3)^{50}dx}$ =$\frac{m}{n}$ where m and n are relatively prime then find 2m-n.
I thought that i could use by parts to solve denominator in form of numerator but i was stuck since a term x^3 is causing trouble.Are there any ot... |
H: Proof $\lim \limits_{n \to \infty} \frac{1}{\sqrt[n]{n!}} = 0$ using the exponential series.
I am trying to proof that
$$\lim \limits_{n \to \infty} \frac{1}{\sqrt[n]{n!}} = 0$$
using the exponential series
$$E(x)=\sum_{n=0}^\infty \frac{x^{n}}{n!}$$
I am aware that I can accomplish the proof using the Taylor serie... |
H: convergence of a succession of functions.
Can you help me with this problem?
Let $f: \mathbb{R} \to \mathbb{R}$ a continuous function, and for all $n \in \mathbb{N}$
define $f_{n} : \mathbb{R} \to \mathbb{R}$ as $f_{n}(x)=f(\frac{x}{n})$, $x \in \mathbb{R}$.
a)Find $g= \lim_{n\to \infty} f_{n}$
b) Prove the conve... |
H: An identity relating partial derivatives to a function
I saw somewhere that given a function f(x,y) you can approximate with the following
Δf≈ ∂f/∂x * Δx + ∂f/∂y * Δy
Can someone explain why this works and where this came from?
Thanks
AI: Using Taylor on the first variable,
$$g(x,y):=f(x+\Delta x, y)\approx f(x, ... |
H: In the described game, can the rubies be divided into 105 piles of one?
Problem
In a pirate ship, there is a chest with 3 sacks containing 5, 49, and 51 rubies respectively. The treasurer of the pirate ship is bored and decides to play a game with the following rules:
He can merge any two piles together into one p... |
H: Percentage of $M$ picks (with replacement) that are unique out of $N$ unique choices
I have $N$ distinct objects to choose from. I can make $M$ choices with replacement.
I would like to figure out the percentage of the $M$ choices that are distinct. We know that the probability of one of the distinct $N$ objects no... |
H: mapping on the complex plane
find the image inside the circle $|z-2|=2$ under the mapping $f(z)=\frac{z}{2z-8}$, I don't know what it turns into, I would appreciate any help.
AI: This is a Mobius transformation. They are determined by what they do to $3$ points. And they map generalized circles to generalized cir... |
H: Convergence of $\sum^{\infty}_{k=1}k^2\tan\frac{k+2}{k^2+5}$
Convergence of series $$\sum^{\infty}_{k=1}k^2\tan\frac{k+2}{k^2+5}$$
What I have tried: using $\tan x>x$ for $x>0$
$$\tan\frac{k+2}{k^2+5}>\frac{k+2}{k^2+5}$$
$$\sum^{\infty}_{k=1}k^2\tan\frac{k+2}{k^2+5}>\sum^{\infty}_{k=1}\frac{k^2(k+2)}{k^2+5}$$
AI:... |
H: Is a set of integers unbounded?
I needed to confirm this for a proof. Since a set of positive integers is unbounded above and bounded below and I think same goes for a set of negative integers but just the opposite, so does this mean that a set of integers is unbounded both above and below or is there an exception ... |
H: Let $Y$ be a proper subspace of $(X, \| \cdot \|)$. Is $\text{dist}(x,Y) > 0$ for $x \in X \setminus Y$?
Let $(X, \| \cdot \|)$ be an (infinite-dimensional) normed space, $Y \subset X$ a proper subspace and $x \in X \setminus Y$. Is $\text{dist}(x, Y) := \inf_{y \in Y} \| x - y \| > 0$?
My attempt.
Assume that $\... |
H: Are two distinct prime factorizations relatively prime to each other?
Consider two integers $x$ and $y$ and their respective prime factorizations
\begin{equation}
x = p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}} \dots p_{n}^{\alpha_{n}}
\end{equation}
and
\begin{equation}
y = q_{1}^{\beta_{1}}q_{2}^{\beta_{2}} \dots q_{m}^... |
H: Probability of winning at least one prize from playing two independent lotteries
Consider Lottery A which has a probability $0.8$ of winning a prize and Lottery B where the probability of winning a prize is $0.9$. If I buy one ticket for Lottery A and one ticket for Lottery B, what is the probability that I will w... |
H: If $\omega$ is a differential $4$-form on a $10$-manifold $M$ then $\omega \wedge d\omega$ is exact
Let $\omega$ be a differential $4$-form on a $10$-manifold $M$. I am trying to show that $\omega \wedge d\omega $, which is a $9$-form, is exact.
Clearly $\omega \wedge d\omega$ is closed, because $d\omega \wedge d\... |
H: Difference between Boolean operator symbols for AND and OR
I feel like an idiot for asking this because I SHOULD know the answer, but what is the difference between the Boolean logic operators for AND and OR? For instance, I know AND can have the symbols $\&$, $\land$, and ·. I also know OR can have the symbols $|$... |
H: Why integration of $\dot{x}\ddot{x}$ is equal to $4\dot{x}^2$?
I know how to calculate the basic integral such as $x$ or $x^2$ but I don't understand why the integral of $\dot{x}\ddot{x}$ is equal to $4\dot{x}^2$.
I know this is a very elementary question but can someone teach me the integration process?
AI: Integ... |
H: prove that $x=ax+b$ has solution for all values of $a$ and $b$
Let $\Bbb R_+$ be the set of positive real numbers (including $\infty$). $(\Bbb R_+, +, \cdot, \leq)$ is an ordered semiring endowed with usual addition and multiplication in $\Bbb R_+$. Then it is easy to verify that the equation $$a+x=b$$ has a soluti... |
H: Are there any explicit solutions of $yy' = 5x$ that pass through the origin?
First part: Use the fact that $5x^2 − y^2 = c$ is a one-parameter family of solutions of the differential equation: $y y'= 5x$ to find an implicit solution of the initial-value problem:
$$y \dfrac {dy}{dx} = 5x \\ y(2) = −6$$
I obtained th... |
H: Differentiate between basic and binomial probability
So I saw a question under the topic for Binomial Distributions which asks that what is the probability of making 4 out of 7 free throws where the $P(making a free throw) = 0.7$. Why can't the answer be a simple $ (0.7)^4 $? Why would it be $7C4 * (0.7)^4 * (0.3)^... |
H: Isometries between metric spaces
Could you help me with the next exercise please:
Let $(X,d_{x})$ and $(Y,d_{y})$ be metric spaces. If there is a function $f:X \rightarrow Y$ such that $d_{x}(x,x')=d_{y}(f(x),f(x'))$ for each $x, x' \in X$, prove that then $X$ is isometric to a subspace of $Y$.
I have tried applyin... |
H: Irreducible elements in the ring of formal power series
I have to show that
If $f(x)$ is an irreducible element in the ring of formal power series over $\mathbb{C}$ then $f(x)$ and $x$ are associates. Also the constant term has to be $0.$
I tried by writing$f(x) = g(x)h(x)$, where one of $g(x)$ or $h(x)$ is a un... |
H: Does the perimeter of a semicircle include the diameter?
Is the perimeter of a semicircle $(\pi)(\text{radius})$ or $(\pi)(\text{radius})+\text{diameter}$?
AI: (Transferring from a comment.)
You'll find in mathematics that terminology can vary from author to author. (We can't even agree on whether the natural nu... |
H: proability mass function binomial distribution excersice
i have to solve this excercise, but i'm confussed how to find the probability mass function and if it is an binomial distribution with a bernoulli trials.
A mouse is placed in a model with five doors. Each door opens when the mouse is detected by a sensor, th... |
H: Using the letter $P$ to represent an event
It is known that in statistics $P(X)$ represents the probability of $X$. My question, is it WRONG to use the letter $P$ to represent an event, where $P(P)$ represents the probability of $P$.
For example, let $P$ represent the event that having Pizza for dinner. Then, $P(P)... |
H: Operating on sets to split into individual elements
There are four piles of coins, of size $165$, $243$, $273$ and $455$, respectively. You have two legal operations:
You can merge any two piles to form a new pile
If a pile has an even number of coins you can split it into two piles of equal size.
Show a sequenc... |
H: Sets for defining functions
Let $$f:X\rightarrow Y$$
Then
$$f\subseteq X \times Y$$
Also, the set of all functions from $X$ to $Y$ is written as $Y^X$ so $$f \in Y^X$$
Then $X\times Y \in Y^X$ ?
That last statement seems surprising to me. I get that since $f \subset X\times Y$ that the last statment is probably not... |
H: How to check if a number is of the form $2^n - 1$
Numbers which are of the form $2^n-1$ are $1, 3, 7, 15, 31...$
Can we find directly using a formula that a number is of the form $2^n-1$?
Please help.
AI: Let $k$ be the number in question. $$k=2^n -1$$ $$\therefore k+1=2^n$$ $$\therefore n=\log_2(k+1).$$ If $n$ is ... |
H: What topic can be studied in knot theory?
I'm a high school student from China. And I'm going to start a project about knot theory with my friend, but haven't had a topic yet. Could you guys give me some advice. Thanks a lot!
(Sorry, I'm not good at writing in English)
AI: Good for you starting a project like this ... |
H: Show that $p(x)=2x^6+12x^5+30x^4+60x^3+8x^2+30x+45$ has no real roots
The following question appears in a book for high school students who do not know any calculus but know the basic theory of polynomials:
Show that the equation $$p(x)=2x^6+12x^5+30x^4+60x^3+8x^2+30x+45=0$$ has no real roots.
My thoughts:
Clear... |
H: how many queries does it take to determine which 2 cards from a set of M are marked, if you can only query a subset at a time?
Suppose you have M face-down cards on a table, and 2 of them have an X on the hidden side. You can pick any subset of the cards and query an oracle which will tell you "Yes" or "No" as to ... |
H: Degree of dual variety of complete intersection
Let $X$ be a complete intersection in $\mathbb P^n$, does anyone know how to calculate the degree of a dual variety of $X$?
Some preliminaries can be found in 3264 and all that, chapter 10. For instance, such dual varieties are always codimension one (except when $X$... |
H: convergence $\sum^{\infty}_{k=1}\frac{k^2+3k+1}{k^3-2k-1}$
Finding whether the series $$\sum^{\infty}_{k=1}\frac{k^2+3k+1}{k^3-2k-1}$$ is converges or diverges.
What i try
$$\frac{k^2+3k+1}{k^3-2k-1}\approx\frac{k^2}{k^3}=\frac{1}{k}$$
So our series seems to ne diverges.
But i did not understand How do i use ine... |
H: Prove that any subspace of $V$ that contains both $W_{1}$ and $W_{2}$ must also contain $W_{1}+W_{2}$.
Let $W_{1}$ and $W_{2}$ be subspaces of a vector space $V$.
(a) Prove that $W_{1}+W_{2}$ is a subspace of $V$ that contains both $W_{1}$ and $W_{2}$.
(b) Prove that any subspace of $V$ that contains both $W_{1}$ a... |
H: To prove that there are infinitely many prime numbers using topology
Proof
Let $\tau$ denote that collection of $S(a,b)$. We show $\tau$ is topology. $\varnothing \in \tau$ is automatic. Next, since $\mathbb{Z} = \bigcup \{ n \} $ and $\{ n \} = S(1,0)$, then it is in $\tau$. Now, take a collection of $\{ S(a,b) \... |
H: Notation for a vector with superscript and subscript
In this equation where x is a vector (Euclidean norm):
https://i.stack.imgur.com/alSnj.png
How do I read this part:
https://i.stack.imgur.com/iEcQt.png
I think its Einstein notation but I haven't been able to find anything about a vector with a superscript and ... |
H: Meaning of Exact Transformation and $K$-Automorphism in the context of Ergodic Theory/Mixing
I am reading MAGIC$010$ Ergodic Theory course. In this course's lecture $4$ notes, it is mentioned that
$1)$ Let $T$ be an exact transformation of the probability space $(X,B,μ)$ .Then $T$ is strong-mixing.
$2)$ Let $T$ be ... |
H: A field is a commutative division ring
Herstein's Topics in Algebra defines a ring $(R,+,\cdot)$ as having the following properties:
$(R,+)$ is an abelian group, and its identity is denoted by $0$.
$(R,\cdot)$ is a semigroup, which means multiplication is associative and $R$ is closed under it.
Distributivity: $(a... |
H: a set of finite measure is almost bounded
How can I show:
Let $E\subseteq \Bbb R$ be of finite Lebesgue measure.
For any $\epsilon>0,$ there exists $M>0$ such that $m(E\setminus[-M,M])<\epsilon$
Any answer would be appreciated.
AI: Hint:
$I_0=(-1,1), I_n = (-(n+1),-n] \cup [n,n+1)$ and note that $I_0,I_1,...$ form ... |
H: A permutation problem involving round a table
The question:
There are $4$ man, $2$ woman and $1$ child sitting in a round table. Find the number of ways of arranging the $7$ people if the child is seated:
i) Between two women.
ii) Between two men.
My attempt:
I've drawn a diagram to help me visualize it first:
I... |
H: Solve $(x+1)^2y + (y+1)^2x=0, x, y \in \mathbb{Z} $
Solve $$(x+1)^2y + (y+1)^2x=0, x, y \in \mathbb{Z} $$
I kind of know the only integer solutions are $\{0, 0\}, \{-1, -1\}$ but I don't know how to prove that
AI: Working in modulo $x$, we have $y \equiv 0 \pmod x$, i.e. $x$ divides $y$. By symmetry, $y$ divides $x... |
H: Proof that two hermitian commutating operators have the same eigenbasis
$A$ has the eigenbasis $\{| a \rangle \}$.
Let $\lambda$ be the eigenvalue of $B$ and $v \in V_{\lambda}$ an eigenvector ($Bv = \lambda v$).
Since $A$ and $B$ commute and are hermitian, following must be true:
$A^\dagger = A$, $B^\dagger = ... |
H: Definition of Equivalence Relation
I was going through the text "Discrete Mathematics and its Application" by Kenneth Rosen (5th Edition) where I am across the definition of equivalence relation and felt that it is one sided.
Definition: A relation on a set A is called an equivalence relation if it is reflexive, s... |
H: Positiveness of a continuous function on an interval
Consider a continuous function $f:D\subset \mathbb{R}\rightarrow \mathbb{R}$. Let $I \subset D$ be an interval. $f$ has a property that for each pair $x_{1},~x_{2}\in I$ with $x_{2} \ge x_{1}$, if $f(x_{1}) > 0$, then $f(x_{2}) > 0$.
Suppose $\inf I \in D$ and $f... |
H: Convex formulation of the smallest distance to a point outside of a polyhedron
Consider a polyhedron $S$ whose set of extreme points (vertices) is $\{v_1, v_2,\dots,v_k\}$. Given a point $y \notin S$, we would like to find the point with the smallest distance to $y$. Provide a convex optimization formulation and j... |
H: Show that $X + Y$ and $|X − Y |$ are uncorrelated
Be $X$ and $Y$ Independent Bernoulli-distributed random variables with parameter $p= \frac{1}{2}$.
Show that $X + Y$ and $|X − Y |$ are uncorrelated.
So I have to show $cov(X + Y,|X − Y |)= 0$
$cov(X + Y,|X − Y |)= \mathbb{E}[(X+Y)|X-Y|]-\mathbb{E}[X+Y] \mathbb{E}[... |
H: What differential equations do these solutions belong to?
Find the differential equation whose solution is given by $y=Ae^{2x}+Be^{-2x}$ and $y=c(x-c)^2$ where $c$ is a constant.
The first one is easy. It is $y’’-4y=0$ but I’m having trouble with the second one. Thanks for the help.
AI: Basically, we have to elimin... |
H: Permutations Password Problem
Password of length $10$ to be made from letters ($a$-$z$) and numbers ($0$-$9$). No of passwords of length $10$ that can be made with $3$ letters and $7$ digits, and at most one $9$.
Is $26^3 \times 9^6 \times 1$ (for one $9$) $+ 26^3 \times 9^7$(for no $9$'s) correct?
AI: You can eith... |
H: An alternate motivation 1988 IMO question #6 (the infamous one)
This is an especially famous problem that you can check out an alternative asking and full solutions to here and, more formally, here. This post is not asking for solutions or full fledged proofs; rather, I am wondering whether an alternative motivatio... |
H: gcd and divisibility in ℤ
I'm studying the divisibility properties in ℤ, there's a passage in my manual I find difficult to grasp. I don't understand why it is consequent that $d_1$ divides $d_2$ in the following statement:
Let $d_1 = \gcd(b, c)$ and $d_2 = \gcd(ab, c)$. Now $d_1 = \gcd(b, c)$
⇒[($d_1$ divides $... |
H: Expected value of number of different cards
I have this question:
From a deck of $10$ cards, Tom draws two cards and places them back in the deck. Now Jerry draws two cards from the deck.
Let $X$ be the number of cards that was chosen by only one of Tom or Jerry. What is $\Bbb E[X]$ ?
I've figured that the range ... |
H: Lagrange's theorem to prove $b^{p-1}=1$
My problem:
(a) Let $p$ be a prime and let $b$ be a nonzero element of the field $Z_p$. Show that $b^{p-1}=1$.
Hint: Lagrange.
(b) Use (a) to prove that if $p$ is a prime and $a$ is an integer then $p$ divides $a^p-a$. This result is known as Fermat’s Little Theorem.
Well I ... |
H: Prove a set is totally disconnected
Let $I=[0,1]$ be a set with the topology induced from the usual topology of $\mathbb{R}$ and $D \subset I$ an open and dense subset. Prove that $I-D$ is totally disconnected.
AI: You want to prove that all the connected subsets of $I \setminus D$ are singletons.
By hypothesis $I\... |
H: one function for 3 scenarios
Just wondering, is there a function that depends and a small number of parameters and can model these 3 scenarios (depending on the chosen parameters):
The first is a constant, the second a log-normal (?) and the 3rd a decay function. Thanks!
AI: Let's say you have two functions $f_0, ... |
H: Finding $\iint _De^{\frac{x-y}{x+y}} dy\,dx$ where $D=\{(x,y):x\geq 0,y\geq 0,1\leq x+y\leq 2\}$
Integrate $e^{\frac{x-y}{x+y}} dy\,dx$ over the set $D=\{(x,y):x\geq 0,y\geq 0,1\leq x+y\leq 2\}$.
I tried to do substitution $u=x-y$ and $v=x+y$ so i know that $ 1\leq v\leq 2$ but i couldn't figure how to get the boun... |
H: Banach Algebra: $r_\sigma(x)=\|x\| \iff \|x^2\|=\|x\|^2$
I am trying to see that if we have a complex Banach algebra with unity, we will have that $$r_\sigma(x)=\|x\| \iff \|x^2\|=\|x\|^2.$$
I was able to do the first implication: Since we know that $\|x^2||\leq \|x||^2$ always and we have that $r_\sigma(x^2)=r_\... |
H: How do I proceed from this algebra problem?
The question:
Formatted question
Find the values of p and q for which the expression $12x^4 + 16x^3 + px^2 + qx - 1$ is divisible by $4x^2 - 1$. Hence, find the other factors of the expression.
My problems with this question:
1) I don't understand what the question means ... |
H: True meaning of negation of a proposition
$$\text{If p = "This device is of excellent quality" then is it safe to say that }\\ \neg p = \text{"This device is of terrible quality" ?} $$
When taking the negation, it means that the original proposition (p) is not true. So, It's not true that this device is of excellen... |
H: Tensor products and linear maps
Let $R$ be a commutative ring (with unity) and let $M,M',N,N'$ be $R$-modules. I know that there is a standard linear map $$\varphi:\,Hom_R(M,M')\oplus Hom_R(N,N')\longrightarrow Hom_R(M\otimes_R N,\, M'\otimes_R N')$$ sending $(\alpha,\beta)$ to $\alpha\otimes\beta$ and this last ma... |
H: Lower bound on sum of two matrices
Let $A\in\mathbb{R}^{m\times n}$ and let $B\in\mathbb{R}^{n\times n}$ be such that $\alpha I\preceq B\preceq\beta I$ for some $\beta\geq\alpha>0$ (here, $I$ refers to the identity). I wonder whether there always exists a constant $\gamma>0$ such that the matrix inequality
\begin{e... |
H: prove a field is a divisible group
I saw a statement: every field of characteristic 0, with its underlying additive group structure, is divisible.
I met with troubles in checking the above statement.
Suppose $\operatorname{char}(F)=0$, for each $g\in F$ and for each $n$, how to find $h\in F$ such that $nh=g$
AI: T... |
H: Generating function given the recurrence relation
Let the recurrence relation be given by $$a_{n}=2a_{n-1}+3$$ and let $a_0=1$ find the generating function in closed form. $$\text{Attempt}$$ the general equation of the form $a_n=ca_{n-1}+d$ has the solution as $a_n=c^n(a_0+\frac{d}{c-1})-\frac{d}{c-1}$ thus the rel... |
H: Find value of $\cot^620^\circ-9\cot^420^\circ+11\cot^220^\circ$
What I've tried is factoring out $\cot^220^\circ$
$$\cot^220^\circ(\cot^420^\circ-9\cot^220^\circ+11)$$
but this quadratic can't be factor with integers so I'm stuck here.
AI: As $$\cot3x=\dfrac{3c-c^3}{1-3c^2}$$
where $c=\cot x$
Set $3x=60^\circ$
$$\i... |
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