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H: Prove a set is numerable
I have 4 different sets:
a) $\{ [x]: x \in \mathbb{R}\}$
b) $\mathbb Q \cup (2,+\infty)$
c) $\{n^2: n \in \mathbb N\}$
d) $\mathbb Q \cap (2,+\infty)$
A set is numerable if exists a bijective function from $\mathbb N$ to $A$
the c) is numerable because exists $f:\mathbb N \rightarrow A$ so ... |
H: we have 1000 copies of Hunter's, Rosen's, Liu's and Epp's discrete mathematics books and 1000 students .
We have 1000 copies of Hunter's,1000 copies of Rosen's, 1000 of Liu's and 1000 copies of Epp's discrete mathematics books and 1000 distinct students . Every student should take exactly one copy of some discret... |
H: Why are we always assuming normality?
For a great number of statistical testing (ANOVA or discriminant analysis for example), we suppose that the variables follow a normal law. How many times did I see this supposition ?
But I recently learnt that we could calculate the Kurtosis (the measure of the "tailedness") an... |
H: Closedness of a subset of complex numbers under addition
If we have two angles $$\phi_1,\phi_2\in[0,2\pi]$$ such that $$\phi_1\le\phi_2$$ and we perform standard addition on complex numbers from the subset of $$S=\{ z\in \mathbb{C} : arg(z) \in[\phi_1 ,\phi_2]\}$$
do we get a closed algebraic structure $(S,+)$? In ... |
H: Markov Property definition via conditional expectation
In several textbooks I have seen the following equivalent statement for the Markov property:
Let $\{X\}_{t \geq 0}$ be a stochastic process, $\mathcal{F}_u^v = \sigma\{X_s, s \in [u,v]\}$. Then $\{X_s\}_s$ has the Markov property iff for all $0 \leq t < s$ and ... |
H: Two natural numbers are selected at random, what is the probability that the sum is divisible by 11?
This same question to check divisibility by 10 is quite straightforward but when we have to check whether it is divisible by 11 there is one case that has to be excluded, I would like it if someone explains it intui... |
H: $\frac{dy}{dx}=\frac{y^3}{e^x +y^2}$
The curve $f(x,y)=0$ passes through $(0,2)$ and satisfies
$$\frac{dy}{dx}=\frac{y^3}{e^x +y^2}$$
The line $x=\ln 5$ intersects the curve at $y=a$ and $y=b$. Find the value of $\frac{4(a^2+b^2)}{53}$
It is the differential equation that I'm unable to crack. I'm almost sure that... |
H: $\sum_{n=1}^\infty\frac{(-1)^n\sin ^2n}{n}$ Is the following solution wrong ?; Does $\sum\frac{(-1)^n\cos 2n}{2n}$ converge?
$$\sum_{n=1}^\infty\frac{(-1)^n\sin ^2n}{n}$$
Solution from the lecture notes :
$$\frac{(-1)^n\sin ^2n}{n}=\frac{(-1)^n(1-\cos
> 2n)}{2n}=\frac{(-1)^n}{2n}-\frac{(-1)^n\cos 2n}{2n}$$
$\frac{... |
H: Show that $ x\cdot\cos(x)+\sin(x)/2=\sum_{n=2}^\infty (-1)^n\cdot\frac{2n}{n^2-1}\cdot\sin(nx)$ when $x\in [-\pi,\pi]$
To show this, I have used definitions for $\cos(x)$ and $\sin(x)$:
$$x\cdot \cos(x)+1/2\sin(x)=x\cdot \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}\cdot x^{2n}+1/2\cdot \sum_{n=0}^\infty \frac{(-1)^n}{(2n... |
H: How to show $x_{k}=\frac{(I-\alpha Q)^{k} x_{0}}{\left\|(I-\alpha Q)^{k} x_{0}\right\|}$ where Q is positive definite, converges to optimal solution
I'm trying to show $\overrightarrow x_{k}=\frac{(I-\alpha Q)^{k} \overrightarrow x_{0}}{\left\|(I-\alpha Q)^{k} \overrightarrow x_{0}\right\|}$ where $\overrightarrow ... |
H: Why when $t\equiv n\pmod s$, $n$ is also $t - s\lfloor \frac{t}{s}\rfloor$?
I'd like to know why when $t\equiv n\pmod s$, $n$ is also $t - s\lfloor \frac{t}{s}\rfloor$?
Here, $n\in[0;s[$
I can't find a way to prove $n$.
Thank you for your help!
AI: $n$ is the remainder when $t$ is divided by $s$, which you can writ... |
H: Is $( \mathbb{ Z}^*,\cdot) \rightarrow (\mathbb{Z}_5^{*},\cdot), n \mapsto n \pmod 5 $ well-defined?
Is $( \mathbb{ Z}^*,\cdot) \rightarrow (\mathbb{Z}_5^{*},\cdot), n \mapsto n \pmod 5 $ well-defined?
So what I think is that it is not well -defined because the non-zero multiples of 5 in $\mathbb{ Z}^*$ map to $[0... |
H: Defining the logarithm of a function
I have a contractible open subset $U$ of a smooth manifold $M$, and a smooth $f:U\to\mathbb C^*$ (I don't know if I need all of these properties, but it is what I am working with).
Can I define $\log(f):U\to\mathbb C$, such that $\exp\circ\log(f)=f$?
AI: Yes you can define $\log... |
H: Why is a solution colinear to two rational solutions of a Weierstrass-form elliptic curve also rational?
I learned this fact and it blew my mind: given an equation
$y^{2}=x^{3}+ax+b$
and two rational solutions:
$(x_1, y_1), (x_2, y_2)$
with $x_1, y_1, x_2, y_2 \in \mathbb{Q}$, then any other solution colinear with ... |
H: Expected value of a sum of die rolls
You roll a standard six-sided die, then roll n more six-sided die, where n was the first roll.
What is the expected value of the sum of all the die you rolled?
I did some calculations, but I'm not sure.
Let expected value of the sum of all the die rolls be E[X], then
$$E[X] = \f... |
H: Prove $2\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z} \overset{\sim}{=} \mathbb{Z}/2\mathbb{Z}$
I need to prove that $2\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z} \overset{\sim}{=} \mathbb{Z}/2\mathbb{Z}$.
I know that $\mathbb{Z} \otimes_{\mathbb{Z}}\mathbb{Z}/2\mathbb{Z} \overset{\sim}{=} \math... |
H: What is $\sin{ω}$?
I am reading about hyperreal numbers defined as (to my understanding) certain equivalence classes on all sequences of real numbers. $ω$ is defined as $(1, 2, 3, ...)$, and all functions are applied element-wise. This makes sense for sequences that have an infinite limit, like $e^ω$, which is simp... |
H: Which of the following is divergent? $\sum\frac1n\sin^2\frac1n$, $\sum\frac1{n^2}\sin\frac1n$, $\sum\frac1n\log n$, $\sum\frac1n\tan\frac1n$
Which of the following is divergent?
(a) $\displaystyle \quad\sum_{n=1}^{\infty}\frac{1}{n}\sin^2(\frac{1}{n})$
(b) $\displaystyle \quad\sum_{n=1}^{\infty}\frac{1}{n^2}\sin(\... |
H: Prove that $\int_{\Omega}fd\mu \int_{\Omega}gd\mu\geq 1.$
Suppose that $\mu(\Omega)=1$ and suppose $f$ and $g$ are positive measurable functions on $\Omega$ such that $fg\geq 1$. Prove that
$$\int_{\Omega}fd\mu \int_{\Omega}gd\mu\geq 1.$$
AI: $\phi (x)=\frac 1 x$ is a convex function on $(0,\infty)$. By Jensen's i... |
H: Is $(\mathbb{Z_4},+) \rightarrow (\mathbb{Z_5^{*}},\cdot), n\bmod 4 \mapsto n \bmod 5 $ a homomorphism?
For the following relation
$(\mathbb{Z_4},+) \rightarrow (\mathbb{Z_5^{*}},\cdot), n\bmod 4 \mapsto n \bmod 5 $
Determine if it is well-defined and an homomorphism
So I think it is not well-defined because $6\eq... |
H: Limit of Sequence...
Suppose $a>0$, define sequence $a_n$ by
$$a_n=\sqrt{n}\left(\sqrt[n]{ea}-\sqrt[n]{a}\right).$$ Find $\lim_{n\to \infty} a_n$ if it exists.
By observing the speed of $\left(\sqrt[n]{ea}-\sqrt[n]{a}\right)$ I thought that limit is $0$.
AI: $a_n=a^{1/n} \sqrt n (e^{1/n}-1)$. Now $a^{1/n} \to 1$ an... |
H: Regarding a result about the degree of an element in a family of finite sets.
We have a family $F = \{S_1, S_2... S_m\} $ of $m$ subsets of $\{1,2...n\} $, all with the same cardinality. We're given that $ \forall a,b \in \cup S_i $, number of subsets $S_i$ containing both $a$ and $b$ is the same, or, $|\{S_i : S_i... |
H: Why is the vector $(u_x, u_y, - 1)$ normal to the surface $u=u(x, y) $?
Studying the method of characteristics, the argument goes as follows:
We are interested in the equation: $a(x, y)u_x+b(x, y) u_y=f(x, y, u)$;
$(a(x, y), b(x, y), f(x, y, u))(u_x, u_y, - 1) =(a,b,f)\nabla{F}=0$, where $F =u(x, y) - u=0$.
Hence, ... |
H: Every set $A\subseteq\mathbb{R}\;$ is isomorphic to some subset of $[a,b]\;$ for all $b>a$
I want to prove a more general theorem.
I know that for some ordinal $\alpha$ there exists an isomorphism between $\alpha$ and some $A\subseteq\mathbb{R}\;$
I want to show that for all $b>a\;$ I can find a subset of $[a,b]\;... |
H: Relation between $(km)!$ and few others
Show that $(m!)^k, (k!)^m, k!m!$ are all less equal $(km)!$ where $k$ and $m$ are integers greater than $1$.
I tried it by inducting on $k$ keeping $m$ fixed but got stuck at inductive step.
AI: For any $0\leq j\leq k-1$,
$$
\prod_{t=1}^m (jm+t)\geq \prod_{t=1}^m t=m!
$$
So
$... |
H: Prove that every random variable has a cumulative distribution function
The answer by this post started by stating that every random variable has a cumulative distribution function (CDF).
Question: How to prove that the existence of CDF of every random variable?
I understand that some random variable might not h... |
H: First order logic: structural induction
I am having trouble trying to prove the following.
$\mathcal{S}\subset \mathcal{P}(\mathbb{N})$ is the minimal set that satisfies the following:
$\mathbb{N} ∈ \mathcal{S} $
for every $a \in \mathbb{N}$, $\mathbb{N} \setminus \{a\} ∈ \mathcal{S}$
for every $A,B \in \mathcal... |
H: Asymptotic behavior of $\sum_{k=1}^{n^2}\frac{q^{n^2-k}(1-q^k)}{k(1-q^{n^2})}$ when $q=1-\frac{\log(n)}{n}$
Let $q=1-\dfrac{\log(n)}{n}$.
Numerical simulations indicate that that
\begin{align*}
\lim_{n \to \infty}\displaystyle\sum_{k=1}^{n^2}\dfrac{q^{n^2-k}(1-q^k)}{k(1-q^{n^2})} = 0
\end{align*}
in a monotone decr... |
H: Solving an algebraic equation with fractions
How do I rework the following equation to solve for P when I know the other variables? (Apologies, I tried to Google but just couldn't get the right search terms.)
T = ( 1/AP - 1/P ) * Q
I got this far, but don't know the next step(s):
T/Q = 1/AP - 1/P
Cheers.
AI: $$T =... |
H: If $0
Sorry! This may appear an easy question for you but this is definitely not easy for me. Please suggest how to approach this problem. Thanks in advance.
AI: Let $f(x,y,z)= \frac{\sin x+\sin y+\sin z}{\cos x+\cos y+\cos z}$ and its range is
$$0=\lim_{z\to y\to x\to 0}f(x,y,z)< \frac{\sin x+\sin y+\sin z}{\cos x... |
H: A clarification on the definitions of Topological space
I have a question regarding the following statement from the definitions of Topological Space.
“ The intersection of a finite number of sets in T is also in T.”
Here, what is the role of “finite” here? I do understand the proof of the statement but have no ide... |
H: Use the notion of Integers to find a solution for $a+x=b$, $a,b,x \in \mathbb{N}$.
In my syllabus for "Scientific Computing" it states the following idea.
Consider the case $a \leq b$, $a,b \in \mathbb{N}$. This is essentially the same as $a + x = b$, $x \in \mathbb{N}$. Therefore, the only solutions to this equat... |
H: Is this function $L^1(]0,1])$?
Background:
I was asking myself if in order to prove that given two random variables $X,Y \in L^1$ (i.e. with finite mean value) then $\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]$. I was asking myself if we can avoid to require $\mathbb{E}[|XY|]<+\infty$.
I tried in this way:
Define $\m... |
H: show that $|x|^{-(n-2)}$ is harmonic function
let $f:\mathbb{R}^n \to \mathbb{R}$, and $f: \mathbf{x} \mapsto \Vert\mathbf{x}\Vert^{-(n-2)}$
show that $f$ is harmonic.
I tried to take derivative by $x_1$ twice, and I got this result:
$1/4 (-1 + 4 n^2) (x + x_{2...n})^{(-3/2 - n)}$ this of course doesn't sum up to z... |
H: Real positive solution to specific system of equations
In the course of an optimization problem, I have encountered the following system of equations for $\alpha, \beta > 0$:
$$
\begin{align}
a_1 + a_2 + b_1 + b_2 &= \alpha + \beta\\
a_1b_1 + \frac{1}{4} a_2 b_2 &= \alpha \beta.
\end{align}
$$
Is there a general ap... |
H: Calculate the following integral $\iint_{T}\frac{x^2\sin(xy)}{y}\,dx\,dy$
Calculate the following integral $$\iint_{T}\frac{x^2\sin(xy)}{y}\,dx\,dy\,,$$ where $$T=\{(x,y)\in\mathbb{R}^2:x^2<y<2x^2,y^2<x<2y^2\}$$
I found the $1/2\leq x\leq 1,1/2\leq y\leq 1$ but i got stuck on how set the limits of the integral
AI: ... |
H: What is the Meaning of $Q[u_1] \cong Q[x]/\langle f(x)\rangle$?
Suppose that $f(x) = ax^2 + bx + c \in Q[x]$ is irreducible.
Then $\sqrt{(b^2 − 4ac)} \neq 0$, so that $f$ has two distinct roots -
$$\begin{array}{*{20}c} {u_1 = \frac{{ - b + \sqrt {b^2 - 4ac} }}{{2a}}}{, u_2 = \frac{{ - b - \sqrt {b^2 - 4ac} }}{{2a}... |
H: Why is $(1 - \frac{1}{n^{1-\epsilon}})^{n} < e^{-n^{\epsilon}}$ for $0 < \epsilon < 1$?
This argument appears in one proof in my lecture and I don't know why this holds. Maybe someone knows
a theorem that implies this inequality? Thanks for help.
AI: If we take $x=e^{-n^\varepsilon}$ then the inequality to be prove... |
H: Find a homomorphism $f : H \to S_n$ such that $N = \ker(f)$.
I am studying for an exam, and I got stuck on question on finding a certain homomorphism.
For a group $G$, and a finite subset $A \subset G$, with $n=\#A$, consider subgroup $H=\{
g \in G \mid \text{for all }a \in A, gag^{−1}\in A\}$, and $N=\{g\in G\mid ... |
H: Formula about Closure of Sets
I am reading the proof in link here in which I found a formula confusing.
It says that ${\rm Cl}_X(W_0\cap A)={\rm Cl}_X(W_0\cap{\rm Cl}_X(A))$ due to $W_0$ is open.
One direction ${\rm Cl}_X(W_0\cap A)\subseteq{\rm Cl}_X(W_0\cap{\rm Cl}_X(A))$ is trivial, but another direction seems n... |
H: Given a finite set $X$, which of $\mathcal{P}(X\times X)\times\mathcal{P}(X\times X)$ and $\mathcal{P}({\mathcal{P}(X)})$ has more elements?
This is a problem that I found in the book "Proofs and Fundamentals", by E. Bloch.
Problem: Let $X$ be a finite set. Which of the two sets $\mathcal{P}(X \times X) \times \ma... |
H: Why not to calculate the limit of individual terms when the number of terms approach infinity due to limiting variable?
My instructor was using this example to illustrate his point-
$$ \lim_{n\to \infty} \frac{1+2+3+...+n}{n^2}$$
The numerator is an AP and the limit can be easily calculated this way to be a finite ... |
H: Prove that the integral $\int_a^b \frac{\sin(x)}{x}dx$ is bounded uniformly.
How do I show that exists a constant $M>0$ such that, for all $0\leq a \leq b < \infty$,
$$\left|\int_a^b \frac{\sin(x)}{x}dx\right| \leq M.$$
I just read on Richard Bass book's that is enough to prove the uniformly boundeness of the integ... |
H: Integrality and normality of ideals
Consider a ring $R$, an idela $I$. An element$z\in R$ is integral over $I$ if $z$ satisfies the equation
$$z^n+a_1z^{n-1}+\ldots+a_{n-1}z+a_n=0,$$
where $a_i\in I^i$.
We define the integral closure $\overline{I}$ of an ideal as the set of elements in $R$ which are integral over $... |
H: Standard etale maps are etale
We define a finitely presented $R$-algebra $A$ to be etale if for every $R$-algebra $S$, every ideal $I\subset S$ such that $I^2=0$ and every homomorphism
$$\phi: A\to S/I$$
there exists a unique homomorphism
$$\psi: A\to S$$ such that $\pi\circ\psi=\phi$, where
$$\pi: S\to S/I$$
is th... |
H: Existence of Euler path in $K_5$, the complete graph with five vertices
Construct an Eulerian path in $K_5$
I tried with the aid of the theorem:
A graph contains an Eulerian path if and only if there are at most two vertices of odd degree.
But I became stuck while ending the walk at initial.
AI: If you put the fi... |
H: How Lebesgue integration solved the problem of a function being integrable but its limit is not integrable?
My professor gave us the following form of Dirichlet function as an example of the problems we faced in Riemann integration:
$\{r_{n}\}$ enumeration $\mathbb{Q} \cap [0,1]$
$$ f_{n}(x) = \begin{cases}
... |
H: Regarding closed half spaces in $\mathbb{C}^n$
I know that a closed convex set $\Omega\subset \mathbb{R}^n$ is the intersection of the half spaces containing it. Here a half space is of the form $H=\{x\in: \mathbb{R}^n: f(x)\geq0 \}$, where $f: \mathbb{R}^n \longrightarrow \mathbb{R}$ is an affine map. And $f$... |
H: A doubt regarding the delta complex structure for a circle
I'm currently self-learning algebraic topology from Hatcher. In page 106, Hatcher computes $H_{n}^{\Delta}(S^1)$. To do this, he says "let $X = S^1$ with one vertex and one edge". The diagram given in the book is clear but I'm not able to see how this is a ... |
H: Harmonic series "fulfills" Cauchy criterion
Let $S_n=\sum_{k=1}^n \frac{1}{k}$ the $n$-th partial sum in the harmonic series. Once can prove that
$$
\lim_{n\rightarrow\infty}|S_{n+p}-S_n|=0,
$$
for all $p\in\mathbb{N}$. On the other hand, the harmonic series diverges. So the question is why the above limit "fulfill... |
H: Why Folland's Advanced Calculus is so strict about uniform convergence?
Folland's Advanced Calculus uses uniform convergence to justify the interchange of limits (i.e. to change order of integration and summation). But actually uniform convergence is far powerful than only justifying such an action. For example Fub... |
H: Why does $\sum_{i=n}^{2n-1}\binom{i-1}{n-1}2^{1-i}$ computes the probability of $n$ head or tails
$\sum_{i=n}^{2n-1}\binom{i-1}{n-1}2^{1-i}$
For $i = n,n+1,\ldots, 2n - 1$, the sum above computes $P(E_i)$, the probability that i
tosses of a fair coin are required before obtaining $n$ heads or $n$ tails.
I am not as... |
H: Proving the open unit disc in $\Bbb R^2$ is open
I'm going through Rudin chapter two on Topology, and have been working on open sets, closed sets and material related to that. I understand the concepts, but am struggling to apply it to some examples. In the book it has the following:
The set of all complex numbers ... |
H: Compact and $T_2$ - space question
Give $X$ is compact $T_2$-space and $(F_n)$ is the sequence of closed sets that $F_n \supset F_{n+1}$.
I have proved that $\displaystyle F=\bigcap_{n=1}^{+\infty}F_n \neq \varnothing$. So, i feeling that if $G$ is open contain $F$ then exists $n_0 \in \mathbb{N}$ that $F_{n_0} \su... |
H: $(\lvert a\rvert +\lvert b\rvert)^{p}\leq 2^{p}(\lvert a\rvert^{p} +\lvert b\rvert^{p})$ for $p > 1$ and are absolute values necessary?
Is it true that, for any $a,b\in \mathbb R$ and $p \geq 1$, we have
$(\lvert a\rvert +\lvert b\rvert)^{p}\leq 2^{p}(\lvert a\rvert^{p} +\lvert b\rvert^{p})$
If $p=2$ we have an ex... |
H: How to convert coeffecient multiplying $t$ in a trig function (e.g. $\sin(bt)$) into Hertz?
How to convert coeffecient multiplying $t$ in a trig function (e.g. $\sin(bt)$) into Hertz?
For example, if I want a machine to produce waves in my bathtub such that the waves follow the function $\sin(4.224 t)$, but the mac... |
H: In $\triangle ABC$, $D$ bisects $AB$, $E$ trisects $BC$ and $\angle ADC=\angle BAE$. Find $\angle BAC$
Giving a $\triangle ABC$ having point $D$ as the mid point of $AB$ and $E$ as the trisection point on $BC$ such that $BE>CE$. If $\angle ADC=\angle BAE$. Find $\angle BAC$
I can't solve this problem but I was ab... |
H: Orbit Space of Action of $SO(n)$ on $\mathbb{E}^n$
I'm trying to solve problem 28, chapter 4 of M. A. Armstrong's Basic Topology :
Describe the orbits of the natural action of $SO(n)$ on $\mathbb{E}^n$ as a group of linear
transformations, and identify the orbit space.
I take the norm function as an identificatio... |
H: Simplifying $\ln{U}=-2\sin x+\ln{\cos x}+\ln{C}$ to find $U$.
I need to simplify this expression and find my $U$. What should I do?
$$\ln{U}=-2\sin x+\ln{\cos x}+\ln{C}$$
AI: It is $$U=e^{-\sin(x)+\ln(C\cos(x))}$$ |
H: Inequality proof including $n!$
$n^{n}e^{-n+1} \le n! \le n^{n}e^{-n+1} n$,
$n \in \mathbb{N}$
I'm struggling solving the inequality above, I have tried AM-GM, Bernoulli but I guess now that the proof is based maybe on induction.The squeeze theorem can make maybe also sense. I appreciate any help.
AI: $1. \log$ and... |
H: Suppose $a$, $b$, $c$, and $d$ are real numbers, $00$. Prove that if $ac\geq bd$ then $c>d$.
Not a duplicate of
Prove that if $a$, $b$, $c$, and $d$ are real numbers and $0 < a < b$ and $d > 0$ and $ac ≥ bd$ then $c > d$
This is exercise $3.1.11$ from the book How to Prove it by Velleman $(2^{nd}$ edition$)$:
Suppo... |
H: Quotient lattice: definition, references
I apologize in advance for this dumb question, but I don't know where to look and I'd like to understand.
Consider the lattice $\mathbb{Z}^n$ (for me a lattice is a free abelian group of finite rank), and consider an element $v=(v_1,\ldots,v_n)\in\mathbb{Z}^n$.
Consider the ... |
H: Does $\iint_{\mathbb{R}^2}\frac{\sin(x^2+y^2)}{x^2+y^2}\,dx\,dy$ converge?
Consider $$\iint_{\mathbb{R}^2}\frac{\sin(x^2+y^2)}{x^2+y^2}\,dx\,dy$$
My try: changing to polar coordinates and then calculate the integral $$\int_{0}^{2\pi}\int_{0}^{\infty}\frac{\sin(r^2)}{r^2}\,dr\,d\theta\underset{r^2=u}{=}\pi\int_{0}^{... |
H: Convergence of simple non-negative functions to a measurable function
The question is from "Measure theory and probability theory (pag. 51)" by Krishna and Soumendra that lacks of a demonstration.
Let $(\Omega, \mathcal{F}, \mu)$ be a measure space and $f:\Omega \rightarrow [0, \infty]$ a non-negative measurable fu... |
H: Can I deduce from $ \frac{a}{b} = \frac{x}{y} $ that a = x and b = y where $ a, x \in Z $ and $ b, y \in Z^{+} $?
Can I deduce from $ \frac{a}{b} = \frac{x}{y} $ that a = x and b = y where $ a, x \in Z $ and $ b, y \in Z^{+} $?
I'm wanting to prove that the function $ f(x, y): \frac{x}{y} $ where $(x , y) \in \math... |
H: Prove a function is continuous (using partial derivative)?
Given:
$|f(x,y)_x'|<M, |f(x,y)_y'|<N near (0,0)$-partial derivative-
How may I prove that $f(x,y)$ is continuous in (0,0)?
I tried to prove that using $\epsilon$ but got to the end of the road since partial Derivative are not related. Any help?
AI: Note tha... |
H: A dense subset, $E$, of $[0,1]$ with measure $\frac{1}{2}$, and no proper subset of $E$ is an interval?
I'm trying to find a subset of the unit interval that's analogous to the irrationals in some sense; it's dense in $[0,1]$, no subset of it is an interval, but it has a strictly smaller measure than the irrationa... |
H: Associative Law for infinitely many sets
I know the associative law for (union or intersection) of two sets and I know why it works.
Applying this rule many times, will show us intuitively that this rule also holds for infinitely many sets.
Although, what would be an adequate proof for this generalization? Is it su... |
H: Confusion between power series and taylor series
Given some power series $f(x) = \sum_{n=1}^{\infty} a_n x^n$, is it generally true that $a_n = \frac{f^{(n)}}{n!}$? If so, why? We get this form when we develop a taylor series, but why is this case for every power series? Can't the $a_n$ terms be arbitrary? I'm a bi... |
H: Solution to an infinite series
Given an infinite series in the form of:
$$a \cdot a^{2\log(x)} \cdot a^{4\log^2(x)} \cdot a^{8\log^3(x)} \dotsb = \frac{1}{a^7} $$
find the solution for all positive and real $a$ other than $1$.
The textbook, from which this problem was taken, says that there are no solutions to thi... |
H: A simple group of order 168 doesn't have subgroups of order 14
Let $G$ be a simple group of order 168. Prove that $G$ doesn't have subgroups of order 14.
I know there are eight 7-Sylows in $G$ and that the normalizer of a 7-Sylow has order 21, it was a previous exercise.
My idea: Assume there is a subgroup $H<G$ of... |
H: Does this series over $SL_2(\mathbb{Z})$ converge?
Consider the following series of positive numbers
$$
\sum_{\substack{a,b,c,d \in \mathbb{Z} \\ ad-bc=1}}\frac{1}{((a-b)^2 + (c-d)^2 + 1)^2}
$$
Does it converge? How to prove it?
AI: The answer to your literal question is "no", but for a reason which may clarify wha... |
H: Probability of 2 consecutive heads before a tails
What's the probability of 2 consecutive heads before a tails, given that we keep flipping until we get either of these items? Is this approach correct?
Notation: P2h = probability of 2 consecutive heads; P1h = probability of 1 head
P2h = (1/2)*P1h
P1h = (1/2)
Thus, ... |
H: Probability distribution for a function of random variables
I'm very new in the Statistic Math field, so this question maybe be a bit trivial for you guys. Anyway, I'd appreciate any guidance in this matter.
I was thinking about whether is possible to find the distribution for a function $f(x_i)$, with random varia... |
H: Proving the open disc in $\mathbb R^2$ is not closed
I'm trying to get use to calculating limit points in metric spaces, but am getting stuck.
Consider the following subset of $\mathbb{R}^{2}$
The set of all complex $z$ such that $|z| < 1$
I'm trying to show that this set is not closed.
To do this, I need to show t... |
H: Group homomorphism from finite abelian group to $\mathbb{C}\setminus\{0\}$
Let $G$ be a finite abelian group, $g\in G$ be an element of order $n$ and $\xi = \exp (\frac{2\pi i}{n})$.
I want to show that for each $0\leq i\leq n-1$, there are exactly $\frac{|G|}{n}$ group homomorphisms $G\to \mathbb{C}\setminus\{0\}... |
H: How to substitute for the variables in $f^{-1}$ of $f(x,y)$
By the horizontal line test I think that I can tell $f(x,y)=x^{2}+y^{2}-1$ is not supposed to have an inverse function, however the screenshot that I have added shows two inverse functions $ f^{-1}= \pm \sqrt{x-y^{2}+1} $. When this function just maps to $... |
H: Simplifying Axler's proof that every operator on an odd-dimensional real vector space has an eigenvalue (linear algebra done right, 2nd ed)
This is how the proof starts:
From here, the author introduces another operator to get around the issue that $W$ might not be invariant under $T$ and follows up with some mor... |
H: Finding derivative of integral $I(a)=\int_{0}^{1} \frac{dx}{1+ae^x} $
Given $$I(a)=\int_{0}^{1} \frac{dx}{1+ae^x} $$
How may I calculate $$\frac{\partial}{\partial a}I(a)=\int_{0}^{1} \frac{dx}{1+ae^x} \ ?$$
Is there any rule for this?
AI: Notice, it's Leibniz integral rule which gives
$$\frac{\partial}{\partial a}... |
H: Differentiation in the geodesic problem
Consider the geodesic problem on a flat plane. We have to find the extremal function $y$ such that:
$$L[y] = \int_{a}^{b} \sqrt{1+(y')^2} \mathrm{d} x$$
is minimized, where $y(a) = y_1, y(b) = y_2$.
Applying the Euler-Lagrange equation to $L(x,y,y')$.
$$\frac{\partial L}{\par... |
H: Show that $\sum_{n=1}^{\infty}\frac{x^{2}}{x^{2}+n^{2}}$ does not converge uniformly on $(-\infty,\infty)$.
I am trying to prove that this infinite series $\sum_{n=1}^{\infty}\frac{x^{2}}{x^{2}+n^{2}}$ does not converge uniformly on $(-\infty,\infty)$.
I can definitely show that this series converges uniformly on $... |
H: Is it acceptable to think about dual vector space in terms of operations on different vector spaces?
I'm looking into abstract algebra.
It seems like with actual examples of covectors, the things we do the vector space operations on (addition and scalar multiplication) are objects like: $\begin{pmatrix}a&b&c\end{pm... |
H: Fourier transform of $\frac{x}{x^2 + x + 1}$
Apparently the expression of the result changes depending on the sign of $\omega$, I'm not sure why.
$$\mathcal{F}[f](\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{xe^{-i\omega x}}{x^2+x+1}dx$$
So I consider this complex function:
$$\frac{ze^{-i\omega z}}{z... |
H: Partial derivative of $f(x,y)= \int_{x}^{y}e^{t^-2}dt$
I want to compute the partial derivative of the following function:
$f(x,y)= \int_{x}^{y}e^{-{t^2}}dt$
Since $f$ is a continous function it has an antiderivative. Let $F$ be the antiderivative of f.
Then:
$\int_{x}^{y}e^{-t^2}dt = F(y) - F(x)$.
Now computing th... |
H: Tangent plane equation in the point of intersection with $\space y$-axis
I've encountered one tiny problem - I have to find tangent plane equation of the function $\space f(x,y)=y^3 - \sqrt{1-x^2y^2} \space$ in the point where it intersects with $\space y$-axis. So $\space f(0,y) = y^3... \space$ and that's not ex... |
H: Is the following restatement of convexity wrong?
They claim equation 5 is restatement of convexity. What am I missing? $\lambda = 0, \lambda'=1$ seems wrong no?
https://www.scihive.org/paper/1702.04877
AI: Take $F(x) = x$ on the reals, $p=0,q=1, \lambda = 1, \lambda' = 1$ then the above says $F(q)=1 \le F(p) = 0$. |
H: Conditional Expectation Formula on discrete random Variable Using Indicator Function
Conditioned on a discrete random variable, the conditional expectation is given by the formula :
$$E(X|Y=y)=\sum xp(x|Y=y)$$
However I've found another formula in Wikipedia that given an event H:
$$E(X|H)=\frac{E(X 1_H)}{p(H)}$$
Ca... |
H: An Explicit isomorphism between the orthogonal Lie algebras $\mathfrak{so}_n$ and the Lie algebras of type $B_n$ or $D_n$.
I know from many sources, including Samelson's Notes on Lie Algebras, that the Lie algebras of even size skew-symmetric matrices, $\mathfrak{so}_{2n}$, are isomorphic to the Lie algebras of typ... |
H: Evaluate $\oint_{|z|=1} \frac{dz}{\sqrt{6z^2-5z+1}} $
I want to evaluate $\oint_{|z|=1} \frac{dz}{\sqrt{6z^2-5z+1}} $ (Joseph Bak's chapter 12 example 2)
According to the textbook, I need to change the contour to $R$ instead of $1$. But I couldn't find reason why $\oint_{|z|=1} \frac{dz}{\sqrt{6z^2-5z+1}} = \oint_{... |
H: If $xM$
I am wondering how I would
Prove for every pair of positive integers $x$ $y$ where such $x<y$ there exist a natural number $M$ such that if $n$ is a natural number and $n>M$ then $\frac{1}{n}<(y-x)$
So I did $y>x$ so then $y-x>0$
let $M=\frac{1}{y-x}+1$ and I guess you could use floor function on the $\fr... |
H: Is this subset a subspace of $\mathbb{R}^3$?
$\{(x, y, z)\} \space$ with $\space x + y + z = 0$
Working through some problems in a textbook and I'm not very confident about checking if subsets are subspaces. I know that for a subset to be a subspace of $\space \mathbb{R}^3 \space$ it must be closed under addition a... |
H: Symmetry proof in expectation
Let $X_1, \dots, X_n$ be iid random variables with positive mean. Let $W_i=X_i/(X_1+\cdots+X_n)$, find $E(W_i)$.
The proof goes like this: since $E(W_1+\cdots+W_n)=E(1)$, by symmetry and linearity of expectation, the answer is $1/n$.
My question is, how does symmetry work here? I don't... |
H: Find the orders of all elements in the group $D_3×Q$.
So to start, I know that the order of an element in a group is the power that it needs to be raised to in order to equal the identity element - i.e., for $\mathbb{Z_{11}}$, the element 6 will have an order of 10 because 6^10 is the first time that it mods to 1. ... |
H: Prove that the 2 sequences are nested intervals and give the element of the nested interval
$$ a_{n+1} = \frac {2a_{n}b_{n}}{a_{n}+b_{n}}, b_{n+1}= \frac{a_{n}+b_{n}}{2}$$
where $0<a_{1}<b_{1}$ and $n \in \mathbb{N}$ and proven is that the sequence $[a_{1} b_{1}], [a_{2}, b_{2}]... $makes a nested interval and givi... |
H: A sequence on $\{x_n\} \subset (0,1)$ such that $\lim_{n \to \infty} x_n=1$ with nice minimum distance properties.
I am looking for a nice example of sequence $\{x_n\} \subset (0,1)$ such that
$\lim_{n \to \infty} x_n=1$.
$|x_n-x_k|= f(|n-k|)$ for some explicit function $f$.
AI: Your conditions are impossible to f... |
H: Is $8x^2-8y^2$ factorable or prime?
Edit: From Blitzer Intermediate Algebra 6th edition, page 371
In exercises 23 - 48, factor completely, or state that the polynomial is prime
Edit 2: These set of problems focus on factoring using sum and difference of cubes, although that's not explicitly stated in the instruct... |
H: Linear regression ML denotations
Could somebody explain what does it mean this denotation:
$$\min_w ||Xw - y||^2_2$$
P.S. Linear regression is described this way in Machine Learning.
AI: For a matrix $X$ choose $w$ such that $Xw$ is close to $y$ in the usual Euclidean distance sense. |
H: There exist a compact not rectificable set
What shown below is a reference from "Analysis on manifolds" by James R. Munkres
So Munkres gives a counterexample of open and bounded set that is not rectificable and then as exercise he asks to find a closed and bounded set that is not rectificable. Unfortunately I can'... |
H: A sequence of $x_n \in (0,1)$ such that $x_n \to 1$ with explicit $|x_n-x_k|$.
I am looking for a sequence of $x_n \in (0,1)$ such that
1.$x_n \to 1$
we have an explicit form for $|x_n-x_k|$ for every $n$ and $k
AI: $$x_n := 1 - \frac{1}{n},$$
$$|x_n - x_k| = |1 - \frac{1}{n} - 1 + \frac{1}{k}| = \frac{|n - k|... |
H: How to show that a series of random variables, $\sum\limits_{n\ge1}X_n$, converges almost surely.
Let $X_n$ be random variables such that for some $a_n\in \mathbb{R}$:
\begin{align}
\sum\limits_{n\ge1}\mathbb{P}(X_n\ne a_n)<\infty \quad \text{and} \quad \sum\limits_{n\ge1}a_n \ \ \text{converges}
\end{align}
Show... |
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