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H: Variance of a piecewise pdf
There is a $95\%$ chance of event A happening and a $5\%$ chance of event B happening. Event A and B are exclusive so both cannot happen, it's one or the other. The pdf of event A is $f(x)=5e^{-5y}$ and the pdf of event B is $f(y)=7e^{-7y}$. What is the variance when an event is triggere... |
H: Proving DeMorgan's law for arbitrary unions/intersections
I am trying to prove DeMorgan's law for arbitrary unions and intersections using Munkres's notation. One of the laws takes the form
$$B - \bigcup\limits_{A \in \mathcal{A}} A = \bigcap\limits_{A \in \mathcal{A}} (B - A).$$
This is the not the notation I am a... |
H: nth term of a sequence which according to me is not an AP, GP or HP
The first term of a sequence is 2014 . Each succeeding term is the sum of the cubes of the digits of the previous term. Then the $2014^{\text {th }}$ term of the sequence is
I thought of doing it by writing recurrence relation but unable to do so
A... |
H: Calculus: derivative of logarithm with respect to logarithm
I have this expression:
$$ \dfrac{d\ln\left(\dfrac{x_2}{x_1}\right)}{d\ln(\theta)} $$
That I’m hoping to get some help solving,
Where $\ln\left(\dfrac{x_2}{x_1}\right)= \ln\left(\dfrac{1-a}{a}\right)+\ln(\theta)$
and
$\theta= \dfrac{a}{1-a} \cdot \dfrac{x_... |
H: Logic behind Pisano period
I just came to know about Pisano periods, and it's amazing application in computing modulus of large Fibonacci numbers. I know that a Pisano period periodically starts at $0,1$, but I haven't been able to figure out why this pattern occurs periodically.
Is there any underlying concept beh... |
H: Integral roots in quadratic equation
The smallest possible natural number $n$, for which the equation $x^{2}-n x+2014=0$ has integral roots, is
I know the discriminant will be a perfect square, but I am struck on equation of discriminant.
AI: The prime factorisation of $2014$ is $2 \times 19 \times 53$. $2$ is obv... |
H: Formula for first difference is not the derivative?
I was messing around in Desmos and wanted to create a chart to show the x values, function values, first differences, and second difference of some quadratics. The graph I was using is here. However, I had to create explicit formulas for the first differences and ... |
H: Determine the type of object (e.g. lines or plane) given the intersection
Given B =\begin{pmatrix} 1 & -1 & 1\\ 1 & 1 & 3\end{pmatrix}determine the general solution of the homogeneous system Bx=0. This describes the intersection of two objects (e.g. a line or a plane). Determine and find the Cartesian equation of t... |
H: Are finitely generated modules over a commutative local ring cancellative?
Let $M,N,P$ be finitely generated modules over a (Notherian) local ring $R$. If $M \oplus N \cong M \oplus P$, do we have $N \cong P$ ?
If not, what if we further assume that $M \cong R^n$ for some positive integer $n$, or even $M=R$ ?
AI: T... |
H: A symmetric point of the inverses
I have the graphs of $y = F(x) = e^x$, $y = G(x) = ln (x)$, and $L : y = x$ drawn.
Of course, $y = F(x)$ and $y = G(x)$ are inverses to each other and therefore they are symmetric about $L$.
Let $P(p, q)$ be a point on $y = G(x)$. Through $P$, I drop a perpendicular to $L$, cuttin... |
H: Let $T$ and $U$ be non-zero linear transformations from $V$ to $W$. If $R(T)\cap R(U) = \{0\}$, prove that $\{T,U\}$ is LI
Let $V$ and $W$ be vector spaces, and let $T$ and $U$ be non-zero linear transformations from $V$ to $W$. If $R(T)\cap R(U) = \{0\}$, prove that $\{T,U\}$ is a linearly independent subset of $\... |
H: Example of unequal iterated integral but that does not contradict Fubini's Theorem
Consider counting measure $\mu_1$ and $\mu_2$ on $X=Y=\mathbb{N}$
Define a function,
$$
f(x,y) = 2-2^{-x} \ \text{if} \ \ x=y \\
\text{and}\\ f(x,y) = -2 + 2^{-x} \ \text{if} \ \ x=y+1
$$
I showed that
$$
\int_X(\int_Y f(x,y)d\mu_2)... |
H: Verify Proof that N cannot be expressed as ${\frac{a}{b}} + {\frac{c}{d}}$ Given Certain Parameters
I'm trying to prove the statement for any integer, N:
N cannot be expressed as ${\frac{a}{b}} + {\frac{c}{d}}$
Given the two parameters:
a, b, c, and d are all different integers which are not 0
The fractions, ${\... |
H: Finding integer solutions to sum of reciprocals of x and y
$$\frac{1}{x}+\frac{1}{y}=\frac{1}{13}$$
Given the sum of reciprocals of $(x,y)$, what's a method to find integer solutions for an equation similar to the above? I've been wondering and I haven't really found something online.
If you could point me to res... |
H: Calculate the dot product when for one of the vectors we only know the sum
The problem is to calculate the following quantity:
$$ax+by$$
I know $a, b \in [0, 1]$, $x, y\in \mathbb Z^+$, and I know the sum of $x$ and $y$. Is it possible to calculate $ax+by$?
I have tried using substitution of one variable, trying no... |
H: Prove that there exists $n\in \mathbb{N}$ s.t. $x_n=\frac12$
Let $(x_n)_n$ a sequence given by $2x_{n+1}=2x_n^2-5x_n+3$ with $x_1\in \mathbb{Q}$. I know that the sequence is convergent. I know that the limit of the sequence should be $\dfrac{1}{2}$ or $3$.
I want to prove that there exists $n\in \mathbb{N}$ s.t. $x... |
H: Metrizability of RxR in the dictionary order topology
The question is one in Munkres where we are asked to prove the metrizability of RxR in the dictionary order topology.
My attempts of defining a metric seem to falter at the end. As for example, I have tried the standard bounded metric, usual metric etc...etc., b... |
H: Poisson arrival conditional probability
A meteorite shower is a poisson arrival with a rate of 16.6 per minute. Given that 7 meteorites were observed during the first minute, what is the expected value of the time passed until the 10'th meteorite is observed
Extracting info from the question gives us: $$N_t \sim ... |
H: Find $E[X\mid Y]$ for $y
I'm having a dillemma regarding the integral of a density:
Suppose: $$f_{X,Y}(x,y) = \frac{1}{2}xy\times\textbf{1}_{x\in[0,2]}\times\textbf{1}_{y\in[0,x]}$$
In order to find $E[X\mid Y]$ I know I want to solve the integral: $$\int_{\mathbb{R}}x\frac{f_{X,Y}(x,y)}{f_{Y}(y)}dx$$ And for that ... |
H: Multivariable Non-degenerate Critical Points Question
Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be a $C^2$ function, and the origin is a non-degenerate critical point and suppose $f(x,mx)$ is a local minimum at the origin for all $m$, then does $f$ have a local minimum at the origin?
I understand that if the func... |
H: Limit of sum function for infinite series $f(x)=\sum_{n=1}^{\infty}\frac{1}{n^6+x^4}$ as $x\rightarrow\infty$
As the title states I would like to determine the limit of $f(x)=\sum_{n=1}^{\infty}\frac{1}{n^6+x^4}$ as $x\rightarrow\infty$. My gut instinct here tells me that the limit should be 0 as each of the terms ... |
H: Prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational.
Prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational.
My steps so far: I found that the polynomial $y^3-6y-6=0$ has roots $\sqrt[3]{2} + \sqrt[3]{4}.$
Can I use this to prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational? If so, how? I was thinking of using Pro... |
H: Swapping variables in an equation
Consider this function,
$$ f\left(\frac{ax+by}{a+b} \right) = \frac{af(x) + bf(y)}{a+b}$$
would it be correct to write ,
$$ f\left( \frac{ay+bx}{a+b} \right) = \frac{ a f(y) + bf(x)}{a+b}$$
Reasoning: the equation should even if you switch variables
AI: Sure. Perhaps it might be ea... |
H: Extending the Dirac delta to $L^p$
In this question the Dirac delta is extended from $\mathcal C^0([-1,1])$ to $L^\infty([-1,1])$ by the Hahn-Banach theorem.
My question is: why can't it be extended to an arbitrary $L^p([-1,1])$ for $p \geq 1$?
AI: To extend $\delta$ to act on $L^p$ along the idea you mention, you ... |
H: Given two functions $f(x),g(x)$ so that $f(x)=-\frac{x^3}{3}+x^2+1,g(x)=5-2x$. Find the ranges of $x$ so that $f(g(x))
Given two functions $f(x), g(x)$ so that $f(x)= -\dfrac{x^{3}}{3}+ x^{2}+ 1, g(x)= 5- 2x$ . Find the ranges of $x$ so that
$$f\left ( g(x) \right )< g\left ( g(x) \right )$$
Is an easy Desmos pro... |
H: three variable inequality $x+y+z\le xyz+2$ with constraint $x^2+y^2+z^2=2$
Let $x$, $y$ and $z$ be three real numbers such that $x^2+y^2+z^2=2$.
it is asked to prove that
$$x+y+z \le xyz+2$$
I tried using Lagrange multipliers but I'm stuck with the following system
$$\begin{cases} x^2 + y^2 + z^2 = 2 \\ 1-yz = 2\l... |
H: How to calculate this conditional probability?
I was given a conditional probability task with the following wording:
The probability of having black hair is 60%. The probability of having blue eyes if your hair is not black is 40%. The probability of having blue eyes given that your hair is black is 10%. What is ... |
H: How to understand derivative of reciprocal function with taylor series
With taylor series, $f(x + h) = f(x) + \frac{df}{dx}h + O(h^2)$,
we know the derivative of $f(x)$ is the coefficient of the first order term $\frac{df}{dx}h$.
Using this definition of derivative, we can easily understand the linearity of derivat... |
H: Discrete Uniform Distribution with random variables
Let X be the random variable that records the number of “heads” when two coins are
tossed. Let Y be a random variable with the discrete uniform distribution on the probability space {1, 2, 3}.
Assume that X and Y are independent. Let U be the random
variable defin... |
H: Find the surface of a triangle between two slopes and x axis
Find the surface of the triangle between this two slopes and x axis.
$$f(x)=x+1$$
$$g(x)=(3-2\sqrt{2})x$$
So : $\text{k}_1= \arctan(1)=45$° and
$\text{k}_2=\arctan(3-2\sqrt{2})=?$
It's very hard to get this angle without a calculator, which I must not use... |
H: Maximize/Minimize $ \sum_{i=1}^n x_i^3$ subject to $\sum_{i=1}^n x_i = 0$ and $\sum_{i=1}^n x_i^2 =1 $
This question comes from a master thesis's course I followed in Optimization. The problem is the following
Maximize/Minimize $ \sum\limits_{i=1}^n x_i^3$ subject to $\sum\limits_{i=1}^n x_i = 0$ and $\sum\limi... |
H: Trying to understand the statement of Nakayama's lemma for coherent modules in Mumford' red book
Here is the statement of a version of Nakayama's lemma in Mumford's red book. Let $X$ be a noetherian scheme, $F$ a coherent $O_X$-module and $x \in X$. If $U$ is a neighbourhood of $x$ and $a_1, \ldots a_n \in \Gamma(U... |
H: $ a^2 <10^{\sqrt{a}}$ for $a\geq 2$.
How to show $ a^2 <10^{\sqrt{a}}$ for $a\geq 2 $ and $ a \in \mathbb{N}$?
Should I try considering a new function which is the difference and then differentiating it?
I could solve it using two case when $ a \in [10^{2m}, 10^{2m+1})$ or $[10^{2m-1}, 10^{2m})$. Is there any other... |
H: Condition for an operator to be compact
Suppose we have an Hilbert space $X$ a compact operator $T$ and an operator $S$ such that $TT^*-SS^*\geq 0$, then $S$ will be a compact operator. Using the condition I can see that $||Tx||\geq ||Sx|| ,\forall x\in X$, and we know that $T(B_X(0,1))$ is relative compact so if I... |
H: Proof of $\tan{x}>x$ when $x\in(0,\frac{\pi}{2})$
I have read Why $x<\tan{x}$ while $0<x<\frac{\pi}{2}$?
If I want to get $\tan{x}\gt x$ instead of weaker inequality $\tan{x} \ge x$. Do I need only to show that $\tan{x} \gt x$ when $x\to 0$? Because from @David Mitra 's picture, it is obvious to see $\tan{x}\gt x$ ... |
H: Finding $P(X>Y)$ where $X\sim U(0,2)$ and $Y\sim U(1,3)$ are independent
I have the following problem:
Two stochastic variables $X\sim U(0,2)$ and $Y\sim U(1,3)$ are independent. What is $P(X>Y)$?
The answers is $\frac18$, but I don't know how to solve.
I did the following:
I drew both distributions in one plot ... |
H: Residue fields at points on $\mathbb{A}^n$
Let $k=\bar k$ be a field. I'm trying to "write down" the residue fields at various points on $\mathbb{A}^n=\operatorname{Spec} k[x_1,\cdots, x_n]$, but am having some trouble with the non-closed points. The definition I'm using is that residue field at a point $x$ on an i... |
H: Convergence of $\int_0^1 x^p \ln^q \left(\frac{1}{x}\right)dx$ without using Gamma function
Determine all values of $p$ and $q$ that the integral convergens of $$\int_0^1 x^p \ln^q \left(\frac{1}{x}\right)dx$$ without using Gamma function.
First attempt: Since $$\int_0^1 x^p \ln^q \left(\frac{1}{x}\right)dx \le \... |
H: Does the heat equation have a unique solution with these mixed boundary conditions
Does the heat equation $u_t - u_{xx} = 0$ on the unit square with $\forall 0 \leq x \leq 1: u(x,0)=0$,
$\forall 0 \leq t \leq 1: u(0,t)=0$, $\forall 0 \leq t \leq 1: u_x(1,t)=0$ have a unique solution?
Here's my attempt:
Let $u, v$ b... |
H: Trying to understand example, determine fx and fy
I´m trying to learn 2nd order Taylor but i cant understand the example i have.
This is an example i have from a book,
$$y´=\frac{y}{2}+x$$
$$-1\le x \le 1 $$
$$y(-1)=1$$
How does fx = 1 and fy = 1/2?
$$f´(x,y)= fx+fyf = 1+\frac{1}{2}\left(\frac{y}{2}+x\right) = 1+... |
H: Inequality proof (perhaps inductive?)
Came up with this on my own and although it seems true (due to Desmos), I was interested to see a proof of it. I tried an inductive approach myself but unfortunately couldn't come up with anything concrete (just by assuming the statement, proving the base case and fiddling with... |
H: Dimension of intersection of distinct subspaces
Let $W_1, W_2, W_3$ be $3$ distinct subspaces of $\Bbb{R}$$^{10}$ such that each $W_i$ has dimension 9.
Let $W = W_1 \cap W_2 \cap W_3$. Then which of the following can we conclude?
$W$ may not be a subspace of $\Bbb{R}$$^{10}$
$\dim W \le 8$
$\dim W \ge 7$
$\dim W ... |
H: Change of variables in a sum
If we have the following function $$f(k) = \frac{1}{k-m} \sum_{j=k+1}^{m+n} \frac{1}{j-m}\ \text{where}\ k\in \{m,.....,m+n-1\}$$ How will the function look if we change the variables $i= j-m$ and $r=k-m$?
I have tried to resolve but still having trouble with the final result.
First, si... |
H: Why is the annual interest rate given if the principal isn’t compounded annualy?
Back in middle school, We learned that if the interest rate given, r, is annual, it must be divided By the number of times compounded per year, n, to get interest rare per period.
I was reviewing compound interest for a standardised te... |
H: Show equivalence with BPI: every Boolean algebra has a prime ideal
I want to prove that the following statements are equivalent:
(1) Every non-trivial Boolean algebra has a prime ideal.
(2) In every non-trivial Boolean algebra, every ideal is contained in a prime ideal.
(3) For every set $X$, is a filter on $X$ con... |
H: How to solve $T(n) = 4T(n-1) - 3T(n-2) +1$?
Which method should I use and how can I solve this recurrence to find the complexity (order) of the recurrence relation?
The equation is: $T(n) = 4T(n-1) - 3T(n-2) +1$
Find $O(T(n))$.
AI: We have
$$T(n) - 4T(n-1) + 3T(n-2) = 1. \qquad \cdots (1)$$
We solve it like we solv... |
H: Jack d'Aurizio's exercise on Chebyshev polynomials
I am working through Jack D'Aurizio's “Superior Mathematics from an Elementary point of view”,
and I found (Lemma 61) the following lemma:
$\sum_{k=1}^{n-1}\frac{1}{\sin^2(\pi k/n)}=(n^2-1)/3$. He does not provide a proof, but says that it follows by considering th... |
H: evaluate the limit using L'Hospitals rule
$$\lim_{x\to 0} (\csc x - \frac 1x)$$
i have tried using the L'Hopitals rule on it in 3 successive derivations and haven't been able to come to a solid conclusion. the denominator just keeps getting longer and harder to differentiate while the numerator keeps switching betw... |
H: Find all real values of $m$ such that all the roots of $f(x)=x^3-(m+2)x^2+(m^2+1)x-1$ are real
I have the following polynomial with real coefficients:
$$f(x)=x^3-(m+2)x^2+(m^2+1)x-1$$
I have to find all real $m$'s so that all of the roots of $f$ are real.
Trying to guess a root didn't get me anywhere.
I computed $x... |
H: Prove that there exists a positive integer $k$ such that $k2^n + 1$ is composite for every positive integer $n$.
Prove that there exists a positive integer $k$ such that $k2^n + 1$ is composite for every positive integer $n$. (Hint: Consider the congruence class of $n$ modulo 24 and apply the Chinese Remainder Theo... |
H: Prove that a polynomial has degree two
Let $f(x)$ be a polynomial that satisfies $x*f(x-2)=(x-4)*f(x)$. Proof that $f(x)$ has a degree of 2.
What I've tried was substituting $f(x)$ with $ax^2+bx+c$ and ended up with $4ax+2bx+4c=0$, which didn't help much.
AI: Substitute $x=0$, then $x=4$ to get that $f(0) = f(2) = ... |
H: Prove $\lim\limits_{n\to \infty}\frac{n!}{(n-k)!(n-a_n)^k}=1$
Let $a_n\to \lambda\in \mathbb{R}$ such that $\frac{a_n}{n}\to 0$ and let $k\in \mathbb{N_0}$.
I have to prove that $$\lim\limits_{n\to \infty}\frac{n!}{(n-k)!(n-a_n)^k}=1$$
I tried this:
$$\frac{n!}{(n-k)!(n-a_n)^k}=\frac{n!}{\sum_{i=0}^{k}(n-k)!\binom... |
H: Proof verification: $\mathbb{E}[X] = \int_0^\infty P(X > \alpha) d\alpha$
I want to show that given a probabilty density $P: \mathbb R^+ \rightarrow [0, 1]$, its expectation obeys the identity: $\mathbb{E}[X] = \int_0^\infty P(X > \alpha) d\alpha$.
We assume that the density $P$ is defined on $[0, u]$. We will get... |
H: Prove $\sum_{n=1}^{\infty}|x_n y_n|^p\leq\left(\sum_{n=1}^{\infty}|x_n |^p\right)\left(\sum_{n=1}^{\infty}| y_n|^p\right)$ for $1
Suppose that $(x_n)_{n=1}^{\infty}$ and $(y_n)_{n=1}^{\infty}$ be in $\ell_p$ for any $1<p<\infty$
prove that
$$\sum_{n=1}^{\infty}|x_n y_n|^p\leq\left(\sum_{n=1}^{\infty}|x_n |^p\ri... |
H: Using determinant to calculate surface of a triangle
$$\frac 12\left|\begin{matrix} -\frac{(\sqrt 2+1)}{2} &\frac{1-\sqrt 2}{2} & 1 \\ 0 & 0 & 1 \\ -1 & 0 & 1 \end{matrix}\right| $$
With this we calculate the area of a triangle that has vertices:
$$x=-\frac{(\sqrt 2+1)}{2} $$ and $$y=\frac{1-\sqrt 2}{2}$$ Is the po... |
H: Show the inequality $ |f(x)-p(x)| \leq h^{n+1}* \frac{||f^{n+1}||_{\infty}}{4(n+1)}$
Let $ f \in C^{n+1}([a,b])$ and a polynom p $\in P_{n} $.
The support points $ x_{i} = a +ih , i =0,...,n$ are equidistant with $ h \in \mathbb{R} $ so that $ x_{n}=b$.
Show
$ |f(x)-p(x)| \leq h^{n+1}* \frac{||f^{n+1}||_{\infty}}{4... |
H: Understanding definition of derivative of a scalar field
Let $f$ be a scalar field i.e., $f:S\subseteq\mathbb R^n \to \mathbb R$ is defined on a set $S\subseteq \mathbb R^n$. Let $B(a,r)=\{x\in S:\|x-a\|\lt r\}$ be an $n$-ball inside $S$. Let $v\in S$ be a vector such that $\|v\|\lt r$, so that $a+v\in B(a,r)$.
Th... |
H: Question regarding proof of a limit which equals e ( the compound interest one).
To prove the limit is e you do the following
$$
L = \lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^n
$$
\begin{align}
\ln L
&= \lim_{n \to \infty} \ln \left(1 + \frac{1}{n} \right)^n \\
&= \lim_{n \to \infty} n \ln \left(1 + \fra... |
H: Need help understanding step in the proof of multivariable chain rule
Theorem: Let $k\in\mathbb{N}$, $x_{0},x\in A\subseteq\mathbb{R}^{n}$, $x_{0}\neq x$ and $f:A\to\mathbb{R}$ satisfy that $\partial^{\alpha}f$ exists and is differentiable on $L=\{(1-t)x_{0}+tx\in\mathbb{R}^{n}\mid t\in[0,1]\}$ for $\lvert\alpha\rv... |
H: Can any undirected connected graph (UCG) with $N$ cycles be decomposed as 2 UCG with $N-1$ cycles?
Consider any (arbitrary) undirected connected graph $\mathcal{G}_{AB} = (V,E_{AB})$ which has $N$ cycles, $V$ is the set of vertices and $E_{AB}$ the set of edges.
I'm wondering if it is always possible to decompose $... |
H: Can a limit have 2 values?
Limit of a function is said to have only one value.
But say I have a function
$$\lim_{l\to \infty}\left(1+\frac 1{{(1+x^2)}^l}\right)$$
Is this limit not defined or is a function of $x$?
AI: There are several related but separate notions.
If you consider $x$ as standing for some fixed (bu... |
H: How do you prove that $\int_0^\infty \frac{\sin(2x)}{1-e^{2\pi x}} dx = \frac{1}{2-2e^2}$?
I know the following result thanks to the technique "Integral Milking":
$$\int_0^\infty \frac{\sin(2x)}{1-e^{2\pi x}} dx = \frac{1}{2-2e^2}$$
So I have a proof (I might list it here later, if it turns out this question seems ... |
H: Let $G$ and $X$ be groups with a surjective homomorphism $\phi : G \to X $. Show that if $H \trianglelefteq G$ then $\phi(H) \trianglelefteq X$
I proceeded as follows:
To show $\forall \bar{g} \in G, \bar{g^{-1}} H \bar{g} \subset H $
Let $$z \in \bar{g^{-1}} H \bar{g} \\
z=\bar{g^{-1}}h\bar{g} \ , \exists h\in H ... |
H: Convergence with respect to a metric in a locally convex space
Suppose $X$ is a locally convex space with topology generated by a countable family of seminorms $\mathcal{P}=\{||\cdot||_{k}\}_{k\in \mathbb{N}}$. Suppose $\{x_{n}\}_{n\in \mathbb{N}}$ is a sequence on $X$ which converges to $x \in X$ in the locally co... |
H: Compute norm of linear functional over $c_0$
I am given a functional $f$, that is defined as
\begin{align}
f:c_0 &\rightarrow \mathbb{R} \\
x &\mapsto \sum_{k=1}^\infty \frac{\xi_k}{3^k} ~~~x = (\xi_k)_{k\ge1}
\end{align}
and I am supposed to calculate ||f||.
I started by estimating an upper bound for $||fx||$:
... |
H: An old APMO problem involving combinatorial geometry
$\textbf{Question:}$
(APMO 1999.) Let
S
be a set of $2n+1$ points in the plane such that no three are
collinear and no four concyclic. A circle will be called
good
if it has 3 points of
S
on
its circumference,
n
−
1 points in its interior and
n
−
1 points in its ... |
H: The equation $x^2-x-1$ has no solution over finite fields of even order.
Is the equation $x^2-x-1$ has no solution over $GF(2^i)$ for all i. I can prove it trivially for some arbitrary chosen small even ordered fields, but can this be generalized?
AI: Note that in any characteristic $2$ field, $-1 = 1$ and the poly... |
H: Given $S =\{ x-y \mid (x,y) \in \Bbb R , x^2+y^2=1\}$, find $\max S$
My solution:
$x^2+y^2=1$ is the equation of a circle.
Let $x-y=k$. That becomes the equation of a line. Since $(x-y) \in S$,
the point $(x,y)$ satisfies both the circle and the line equation. So we know that the graph of the circle and the line ha... |
H: Is $e-1/e$ rational?
My intuition tells me that it's not rational, and even not algebraic (i.e., it's transcendental).
But I'm having a hard time showing it.
Taking it slightly further, $e-\frac1e=\frac{e^2-1}{e}=\frac{(e+1)(e-1)}{e}$, but I don't feel I'm in the right direction.
Any ideas how to tackle this one?
T... |
H: Example of dense sets
Provide an example of an infinity of dense subsets of a space $(X,d)$ such that the intersection of all of them is not a dense subset.
I have tried taking $(\mathbb{Q}, d_{usual})$ and as dense subsets the intervals $(0,1), (1,2), (2,3), ...$
But it seems to me that it does not work, since the... |
H: A property of a kind of product integral
Please, could you say to me which property was utilized here?
$$\int_0^\infty \int_0^x f(a)g(x-a)\,da\, dx=\int_0^\infty f(a)\,da\int_0^\infty g(x)\, dx$$
Many thanks!
AI: $\int_0^\infty \int_0^x f(a)g(x-a)\,da\, dx$=$\int_0^\infty \int_a^\infty f(a)g(x-a)\,dx\, da=\int_0^\i... |
H: Power of point hypothesis test
With $X_i$ i.i.d. $N(\theta,1)$ and $$H_0: \theta=0 vs. H_1: \theta=1$$
I got that the test statistic is $\bar{X_n} > \frac{q_{\alpha}}{\sqrt{n}}+\theta$, where $\theta$ is either 0 for $H_0$ or 1 for $H_1$.
The type 1 test statistic is given with $\alpha = 0.05$.
To compute the powe... |
H: Isomorphism of algebraic structures as an isomorphism of relational structures
In connection with the question Is a set together with an operation always a relational structure?,
I am trying to represent an isomorphism of algebraic structures as an isomorphism of relational structures.
Let's say a relational struct... |
H: If $\|x_1\| \leq \|y_1\|$ and $\|x_2\| \leq \|y_2\|$ then $\|\lambda x_1 + (1 - \lambda) x_2\| \leq \|\lambda y_1 + (1 - \lambda) y_2\|$
I have being running in circles in this problem. Let $x_1,x_2,y_1,y_2 \in \mathbb{R}^n_+$ such that $\|x_1\| \leq \|y_1\|$,$\|x_2\| \leq \|y_2\|$ and $y_1$ and $y_2$ are l.d.. Is ... |
H: Ways to write $n=p^k$ as a product of integers
Let's say that $F(n)$ is the number of ways to write $n$ as a product of integers greater than $1$. For example, $F(12)=4$ since $12=2\cdot 2 \cdot 3$, $12=2\cdot 6$, $12=3\cdot 4$ and $12=12$.
Given $n=p^k$ where $p$ is a prime number, what is the value of $F(n)$? I k... |
H: Is this a valid proof that the boundary of a set on a metric space is closed?
The definition that I have been given of the boundary of a subset $A$ of a metric space $X$ is:
$$\partial A=\{x\in X:\forall r\in \mathbb{R}, B_r(x)\cap A \neq\emptyset \text{ and } B_r(x)\cap A^c\neq\emptyset\}$$
So with this definition... |
H: Two vector sequences that are basis of a vector space
The sequence of vectors $(u_{1}, \cdots, u_{n})$ form a basis of the vector space $E$.
I need to show that there is a $k \in \{1, \cdots, n \}$ such that $ (u_{1}, \cdots, u_{k-1}, v, u_{k+1}, \cdots, u_{n}) $ is a base of $E$, with $v \in E \setminus \{ 0_{E} \... |
H: How is it possible that if $A \implies B $ is true then $ \lnot ( \lnot B \implies A )$ can be false?
While I was playing around with the material implication I made a proof by contradiction which I think it's wrong, but I don't find any mistake :
Say that $A \implies B $ is true , then suppose the truth of $ \lnot... |
H: If $\phi\in W^U$ and if $\psi\in W^V$ and if $W$ is a topological vector space then $f(u,v):=\phi(u)+_{_{W}}(-1)*_{_{W}}\psi(v)$ is continuous
Statement
Let be $W$ a topological vector space and $\phi:U\rightarrow W$ and $\psi:V\rightarrow W$ two continuous functions. So if we define $f:U\times V\rightarrow W$ thou... |
H: Small problem on minus sign $({-5}+{9}x)y'={4}x+{9}y$
I have the equation $$({-5}+{9}x)y'={4}x+{9}y$$
So I am trying to solve it in the way need to solve linear differential equation.
I recognise that I can write it as : $$({-5}+{9}x)y' -{9}y={4}x$$
now I dont know what I should do, I know that I could write $(({-5... |
H: analytical geometry problem with locus
A line of constant length 10 units moves with the end always on the -axis and
the end always on the line = 4. Find the equation of the locus of the midpoint of
.
How could I solve this problem?
AI: Let the point on x axis be $A(h,0)$ a general parametric point on the line ... |
H: Integrate $ \int_a^b \frac{1}{\sqrt{Ax-\frac{x^2}{2}}}dx$
From online integral calculators I am aware that:
$$ \int_a^b\frac{1}{\sqrt{Ax-\frac{x^2}{2}}}dx=\sqrt{2}\left[\arcsin\left(\frac{x}{A}-1\right)\right]\Bigg|_a^b$$
When I work backwards starting with: $$y=\sqrt{2}\left[\arcsin\left(\frac{x-A}{A}\right)\right... |
H: How to determine whether given nonlinear equation system cannot be solved analytically?
I am currently studying nonlinear equations that require numerical analysis methods to solve them. But I could not understand why can't I solve some equations analytically?
For example: x^2 + 4y^2 - 16 = 0 and x(y^2) - 3 = 0
How... |
H: Maximizing profits in a game with given number of shot and a given capacity
We are given $7$ shots to choose randomly (with uniform distribution) from the interval $[0,1]$. After choosing each number we can decide whether we want to keep this number or throw it away. Once we get to keeping $3$ numbers we stop playi... |
H: Show that two definitions for a subgroup are equivalent
Show that (1) $\Longleftrightarrow$(2):
(1) For $H \subseteq G$ with $H \ne \varnothing$ of a group $(G,\ast)$, $(H,\ast)$ is a subgroup of $(G,\ast)$ if:
(G1): $\forall a,b \in H: a \ast b \in H$
(G2): $\forall a \in H: a^{-1} \in H$
(2) For $H \subseteq G$ w... |
H: What's incorrect in this word problem solution?
I had a question and answered it, but I've been told that my solution is incorrect. What's the mistake here?
The Question
Runner A is running at the speed of x in a triangular path (each side of the triangle is of length a) and Runner B is running in the same track at... |
H: Sequence of vectors with specific conditions forming a basis
$E$ is a vector space with $e_{1}, e_{2}, e_{3}, e_{4}, e_{5} \in E$ with the following conditions:
$E = \langle e_{1}, e_{2}, e_{3}, e_{4}, e_{5} \rangle$.
$ e_{1}, e_{2}, e_{3}, e_{4}, e_{5} $ are linearly dependent.
$ e_{1} + e_{2}, e_{2} + e_{3}, e_{... |
H: Gilbert Strang 1.3 #4 Question about Linear Dependence
In Gilbert Strang's Linear Algebra Book 4th Edition, the question asks to find the zero vector with the combination of three vectors: $w_1$ = (1,2,3), $w_2$ = (4,5,6), $w_3$ = (7,8,9).
Working this out to reduced row echelon form, I get to a certain point where... |
H: The spectral radius of a $n\times n$ matrix
I would like to know which is the spectral radius of this $n\times n$ matrix:
$$
\begin{matrix}
0 & 1 & . & . & . &1 \\
1 & 0 & . & . & . &0 \\
. & . & . & & &. \\
. & . & & . & &. \\
. & . & & & . &. \\
1 & 0 & . & . & . &0 \\
... |
H: On the proof of $\;(1+x)^p\equiv1+x^p \pmod p$
I know the proof for $(1+x)^p\equiv 1+x^p\mod p$ using the binomial theorem. Moreover, I know that $x^p \equiv x \mod p$ due to Fermat's theorem.
Hence, is $(1+x)^p\equiv(1+x)\equiv1+x^p \mod p$ a correct proof of this relation?
After thinking about it for a bit, one o... |
H: Residues of $ \frac{z^4}{1+z^6} $
I am trying to compute all 6 the residues of $ \frac{z^4}{1+z^6} $. I tried the straightforward way first of finding all the points where the denominator is 0 etc but it became way too complicated. Any ideas?
AI: Let $a$ be a pole of $f(z)=z^4/(z^6+1)$. Then $a^6=-1$. Also $a$ is a... |
H: Prove the inconsistency of this definition of limit
I want to prove that the following fake definition of limit doesn't capture the property that $f(x)$ tends to infinity as $x$ does :
$$\lim_{x\rightarrow +\infty} f(x) = +\infty \iff \forall M > 0 \exists X > 0\mid( f(x) > M \implies x > X ) $$
If I take a strictl... |
H: a function that is perpendicular to another and goes through a given point
Find a linear function that goes through function $g(x)=(3-2\sqrt2){x}$ in point T($x_0$,1) and does that perpendicullary to the function..
$$\phi=\pi/2$$
$$\pi/2=\frac{(3-2\sqrt2-k)}{1+(3-2\sqrt2)k}$$
$$\frac{\pi}{2}(1+(3-2\sqrt2)k)-3+2\sqr... |
H: Can every square matrix be written as product of an invertible matrix and a projection matrix?
Let A be a square matrix.
Will there always be an invertible matrix B and a projection matrix P such that A = BP?
Thanks
AI: Sure: $A = (A+E)\Pi$ where $\Pi$ is the (orthogonal) projection onto $(\ker A)^\perp$ and E is a... |
H: Is my development about the continuity of the function correct?
I have the following statement:
Prove if $\sqrt{log(x^2+7)}$ is continuous at $x=-4$
My development was:
Let $g(x)=\log(x)$ and $ f(x)=x^2+7$
I will prove that $\log(x^2+7)$ is continuous at $x=-4$.
$\log(x^2 +7)$ is continuous at $x = -4 \iff \lim_{... |
H: Let $U$ be orthogonal. How can I prove that $||UA||_2=||A||_2$?
Let $U$ be orthogonal. How can I prove that $||UA||_2=||A||_2$?
I know that $||UA||_2\le||U||_2||A||_2$ and I also know that as $U$ is orthogonal, $U^{-1}=U^T$. But I don't know what else to do...
AI: Hint: since $U^TU=I$, $(UA)^TUA=A^TU^TUA=\cdots$ |
H: Probability of sum of IID variables
I have $X_1, X_2$ two IID random variables and I know $P[X_1<\epsilon]=P[X_2<\epsilon]\le c$.
Can I claim that $P[X_1+X_2<2\epsilon]=P[X_1<\epsilon]\le c$
I'm confused as it seems right but I don't know how to prove it.
AI: No, it's not true.
For example, consider tossing two fai... |
H: Can you claim $1+i>1$?
The title says it all: I know you can claim $i+1>i$. But can you also claim $1+i >1$? If not why can't I?
AI: As mentioned in the comments, there is no way to "order" $\mathbb{C}$ in a way that is compatible with the operations of addition and multiplication that $\mathbb{C}$ is equipped with... |
H: $\frac{dy}{dx} - {8} -{2}x^2+{4}y^2+y^2x^2 = 0.$ how should I procced from here
having the equation $$\frac{dy}{dx} - {8} -{2}x^2+{4}y^2+y^2x^2 = 0.$$
I am getting to the following $$\frac{1}{2^{\frac{3}{2}}}\ln \left(y+\sqrt{2}\right)-\frac{1}{2^{\frac{3}{2}}}\ln \left(y-\sqrt{2}\right)=\frac{x^3}{3}+4x+c$$ from h... |
H: How can we decide two elliptic curves over Q are isomorphic over Q?
How can we decide which of the following elliptic curves over $\mathbb{Q}$ are isomorphic over $\mathbb{Q}$?
$$E_1:y^2=x^3+1$$
$$E_2:y^2=x^3+2$$
$$E_3:y^2=x^3+x+1$$
AI: See Silverman's book and answer there namely given $$E_i/K:y^2=x^3+a_ix+b_i,\qq... |
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