text stringlengths 83 79.5k |
|---|
H: Prove $\lim_{n\to \infty}\int\limits_0^1 \sin(x^n)dx=0$
Prove $$\lim_{n\to \infty}\int\limits_0^1 \sin(x^n)dx=0$$
I found that $\sin(x^n)$ converges uniformly to $f(x)=0$ on $[0, a]$ for every $0<a<1$.
Therefore I know that for every $0<a<1$, $\lim_{n\to \infty}\int\limits_0^a \sin(x^n)dx = \int\limits_0^a 0dx=0$... |
H: Metric space where every bounded sequence is contained in a compact
Let $(X,d)$ be a metric space such that, for every bounded sequence $(x_n)$ taking values in $X$, there exists a compact set $K\subseteq X$ such that $x_n \in K$ for all $n$. Do such spaces have a particular name?
This includes the case of metric s... |
H: Chi-squared distribution question
Here is a problem that I just cannot figure out how to answer:
$z_1, z_2, \cdots, z_7$ are all independent, and are all normal random variables with $N(0,1)$. What is the probability of this occurring?
$$
P(Z_1^2 + Z_2^2 + Z_3^2) \leq 4.943(Z_4^2 + Z_5^2 + Z_6^2 + Z_7^2)
$$
AI: Ex... |
H: Prove that field $\mathbb{Z}[x]/(x,3)$ is isomorphic to field $\mathbb{Z}/3\mathbb{Z}$.
I don't realize what to do. Could you give me some hints or point to solution?
I tried to construct surjective homomorphism with $Ker(\phi)=(x, 3)$ by intuition to use First Isomorphism Theorem for Rings, but couldn't find such.... |
H: Variance of Poisson distribution multiplied by a constant
Assume I have a Poisson distribution $x=Poisson(X)$. It is clear that the mean and the variance will be equal to X.
In the case we have of $B *Poisson(X)$ being B a constant, I would like to see a small proof showing why the variance is $B^2*X$ and the mean... |
H: Area of a trapezium inscribed in a circle?
A circle, having center at $(2, 3)$ and radius $6$, crosses $y$-axis at the points $P$ and $Q$. The straight line with equation $x= 1$ intersects the radii $CP$ and $CQ$ at points $R$ and $S$ respectively. Find the area of the trapezium $PQSR$.
I am getting stu... |
H: Maximum number of possible intersections between tangent line and function $x^3$
I'm struggling with a problem but can't find a way how to solve it:
Calculate maximum number of possible intersections between tangent line of function $f(x) = x^3$ and function $f(x) = x^3$ where $x \in \mathbb{R}$.
I know I should us... |
H: Determine whether series converge.
Determine if the series $\Sigma^{\infty}_{n=1} \ln\left(1+\frac{(-1)^n}{n} \right)$ converges.
$$\Sigma^{\infty}_{n=1} \ln\left(1+\frac{(-1)^n}{n} \right) = \Sigma^{\infty}_{n=1} (\ln(n +(-1)^n) - \ln n ) \\ S_n = \ln(0) - \ln(1) + \ln(3) - \ln(2) + \ln(2) - \ln(3) + \ln(5) - \ln(... |
H: If $X$ is a 2x3 matrix such that $|XX’|\not =0$ and $A=I-X(X’X)^{-1}X’$, then find $A^2$
$$A=I-X(X^{-1}X^{-1’})X’$$
$$A=I-I=O$$
So $A^2=O$
But the correct answer is $A$. What am I doing wrong?
AI: $A^2=(I-X(X'X)^{-1}X')(I-X(X'X)^{-1}X') =$
$ I^2-IX(X'X)^{-1}X'-X(X'X)^{-1}X'I+X(X'X)^{-1}X'X(X'X)^{-1}X'=$
$I-2X(X'X)^... |
H: Why can't we have these alternative cross product relationships?
I am currently studying Introduction to Electrodynamics, fourth edition, by David J. Griffiths. Chapter 1.1.2 Vector Algebra: Component Form says the following:
$$\mathbf{\hat{x}} \times \mathbf{\hat{x}} = \mathbf{\hat{y}} \times \mathbf{\hat{y}} = \... |
H: $\mathbb Z [x]$ module structure on $\mathbb Z$
For each integer $n \in \mathbb Z$, define the ring homomorphism
$$φ_n :\mathbb Z [x]\to \mathbb Z, \ \ φ_n(f)=f(n).$$
This provides a $\mathbb Z[x]$-module structure on $\mathbb Z$ given by $$f ◦ a = f(n) · a$$ for all $f \in \mathbb Z[x]$ and $a ∈ Z$. Given two inte... |
H: What is the probability of equal number of Heads when coin is tossed large number of times (say 10000)
As per basic probability theorem getting equal number of heads and tails when coin is tossed $10000$ times is
= $\frac{\binom{n}{r}}{2^n} = \frac{\binom{10000}{5000}}{2^{10000}}$
This value is not equal to or clo... |
H: Cossine of a standard Normal distribution
Let $X \sim \mathcal{N}(0,1) $ have a standard normal distribution. How do I compute
$$\mathbb{E}[cos(X)] \text{ ?}$$
I know that $$\mathbb{E}[sin(X)] = 0$$
because $$f(x)=sin(x) \frac{e^{-x^2}}{\sqrt{2 \pi}}$$
is an odd function. Can I use this fact to compute $\mathbb{E}... |
H: Prove: $R$ is local $\iff$ $R$ has exactly one maximal ideal.
We have that a commutative ring $R$ with $1$ is called local if $R − R^×$ is an ideal of $R$. I have to proof the following:
$R$ is local $\iff$ $R$ has exactly one maximal ideal.
We have that every ideal $I$ $\subseteq$ $R$ is contained in a unique maxi... |
H: When can we use Weiterstrass M-test and when not?
For $x \in \mathbb{R}$ consider the series
$$
S = \sum_{n=1}^\infty \frac{2x}{x^2+n^2}
$$
Then I have to show that $S$ converges uniformly. I have been told before in here that we are not able to use Weiterstrass M-test when the convergent series we find depend on $... |
H: Vanishing of cohomology of affine scheme
In EGA I 5.1, more specifically the proof of 5.1.9, which states that $X$ is affine iff the closed subscheme defined by a quasi-coherent sheaf of ideals $\mathscr{I}$ such that $\mathscr{I}^n = 0$ for some $n$ is also affine, it is proved in a nice way that the first cohomol... |
H: $R$ be local and suppose $x ∈ R$ satisfies $x^2 = x$
If we let $R$ be local and suppose $x ∈ R$ satisfies $x^2 = x$, then I have to show that $x = 0$ or $x = 1$.
A commutative ring $R$ with $1$ is called local if $R − R^×$ is an ideal of $R$.
If we let $x, y ∈ R$, then I was thinking of the expansion of the equatio... |
H: Converting base 2 to Base 10?
How is $0.11 \cdot 2^{1}$ converted to $1.5$ in base 10? I am unable to understand the theory behind this, since I believe $0.11 * 2^{1}$ corresponds to $1.1$ but I dont see the conversion process?
AI: $0.11\cdot 2^1$ does indeed correspond to $1.1$. In binary. Which is to say, it mean... |
H: Finding discrete logarithm of composite numbers
I started to learn discete logarithm the definition says that:suppose that "p" is a prime number , "r" is a primitive root (modulo p) and "a" is an integer between "1 and p-1" inclusive.If r^e (modp)=a and "e" is in [0,p-1] then the "e" is discrete logarithm of "a ... |
H: Pairing, Hahn-Banach theorem
I'd like to solve the following problem for part (ii). $X^*$ denotes the space of bounded linear functionals on the normed vector space $X$, and $\left<\ ,\right>$ stands for the pairing between a space and its dual. I desire to proceed with part (i); thus, I consider two cases separat... |
H: About non-normal nilpotent subgroups
Suppose that all non-normal abelian subgroups of a finite group $G$ are cyclic. What can I say about non-normal nilpotent subgroups of $G$?
Is it true that such supgroups are cyclic?
I appreciate your help.
AI: The finite p-groups all of whose non-normal abelian subgroups are cy... |
H: Show that $\lnot\exists x\in A P(x) \equiv\forall x\in A\lnot P(x)$
I need to show that: $$\lnot\exists x\in A P(x) \equiv\forall x\in A\lnot P(x)$$
when I start from LHS, should I write(1): $$\lnot\exists x (x\in A \to P(x))$$
or(2): $$\lnot\exists x (x\in A \land P(x))$$
because after simplification in first case... |
H: Finding the $\lim \limits_{x \to 0} {1 - \cos(x)\over \sin(x) \ln(1+x)}$ using Taylor's series.
I am a bit stuck. This is what I have so far and I am not sure how to simplify it further:
$${{x^2\over 2} - o(x^4)\over (x - {x^3 \over 6} + o(x^5))(x - {x^2 \over 2} +o(x^3))} $$
How do I proceed further?
AI: What you... |
H: Showing $a$ is a primitive root modulo $p \Leftrightarrow -a$ is a primitive root modulo $p$ if $p \equiv 1 \ (\mathrm{mod} \ 4)$
Let $p$ be a prime number with $p \equiv 1 \ (\mathrm{mod} \ 4)$ and $a \in \mathbb{N}$ with $p \nmid a$.
How to show that
$a$ is a primitive root modulo $p \Leftrightarrow -a$ is a prim... |
H: Function with local extreme value at a point but not monotone in any of its neighborhood
In a question a function is given as-
$f(x)=\begin{cases}x^4(2+sin(\frac{1}{x})) & \text{if x$\neq$ 0} \\[2ex] 0 & \text{if x=0}\end{cases}$
In this function, $f'(0)$ comes out to be 0 (and f has minima at 0) and in every neigh... |
H: A filter that has no countable basis
Conside a straight line $L$ in the plane $\mathbb{R}^{2}$. The filter of neighborhods of $L$ in $\mathbb{R}^{2}$ is the filter formed by the sets which contain an open set containing $L$. Prove that there is no basis of this filter which in countable.
First of all, I noticed tha... |
H: the proper ideal of a quotient ring of a Dedekind domain $ / ^{n+1}$
I don't understand what it says in the hint of EXERCIZE 5 in the 23p of this pdf.
For example, $\mathcal O = \mathbb Z$ and $\mathcal a = (4)$.
$\mathbb Z/(4) = \mathbb Z/4\mathbb Z = \{[0],[1],[2],[3]\}$
and $a = (4) = \{ 4n | n \in \mathbb Z\} =... |
H: $\sqrt {144} = 12$ How can I find it without calculator?
$$12\times 12 = 144$$ If there's a question that asks me to find square roots manually, how will I find that?
I know a way that's really long and boring. I actually know about the "Tree" method. Is there any other interesting and effective way to find square... |
H: Probability of a negative random variable in normal distribution.
It is given that $X \sim \mathcal{N}(1.5,(3.2)^2)$.
Find the probability that the random chosen value of $X$ is less than $-2.4$.
Solution:
$Pr(X<-2.4)=1-Pr(X<2.4)=1-Pr\left[Z < \cfrac{2.4-1.5}{3.2}\right]=1-Pr[Z < 0.28]=1-0.61026=0.389$
where $Z \si... |
H: Expectation of Stopping Time for a Brownian Motion with a drift
Let $a,b>0$ and define the stopping time $T_{a,b}$ for Brownian Motion as
$$T_{a,b}:=inf\{t>0:B(t)=at-b\}$$
Compute $E[T_{a,b}]$.
My idea:
I think $E[T_{a,b}]=\infty$.
If that was not the case, then by Wald´s Lemma
$E[T_{a,b}]=Var(B(T_{a,b}))=Var(aT_{... |
H: Difference between a number and a set with one number
I am studying Analysis in Tao's textbook and he mentions the following:
The set $\{3, \{3, 4\}, 4\}$ is a set of three distinct elements, one of which happens to itself be a set of two elements. However, not all objects are sets; for instance, we typically do ... |
H: Sum $\sum_{k=1}^n{k(k-1)}$
I have a series as
$$n(n-1)+(n-1)(n-2)+(n-2)(n-3)+ \ldots +n\ \text{terms}$$
How to provide a summation of this series? I think it would be similar to:
$$\sum_{i=0}^{N}(n-i)(n-(i+1)).$$
AI: This is
$$\sum_{k=1}^n{k(k-1)}=\sum_{k=1}^n{(k^2-k)}=\sum_{k=1}^n k^2-\sum_{k=1}^nk = \frac{n(n+1... |
H: Expected value of different dice roll results given number of rolls
A die is rolled $N$ times where $N\sim \operatorname{Pois}(\lambda)$. $X =$ number of different results in the rolls. For example: $(1,1,6,1)$ has 2 different results and $(1,2,3,4)$ has 4. Calculate $E(X)$
Define an indicator: $$I_i = \begin{cas... |
H: Show that $f(x)=\lfloor x\rfloor$ is discontinuous exactly at all $\xi \in \mathbb{Z}$
Hello I again had a little task to show whats described in my title.
It would be helpful, if someone could again check my reasoning and if my task was successful or not :) and if not, give me some constructive advice :)
Also I do... |
H: Which derivative to use for an integral?
Does anyone know of a rule to tell which derivative to use when faced with an integral like this? (C is any constant, eg. 2400 or 4, etc).
$$\int \:\frac{C}{x}\mathrm{d}x$$
I know that $\ln(x) = \int\frac{1}{x}\mathrm{d}x$ so then $\int \:\frac{C}{x}\mathrm{d}x$ could equal ... |
H: commenting on whether $f'(x)$ is even or odd.
The question is as follows:
Let $f(x)$ be a differentiable function $\forall x,y \in \Bbb R$ and $$f(x-y),f(x),f(y),f(x+y)$$ are in AP then comment whether $f'(x)$ is even or odd (given $f(0) \neq 0)$.
My approach :
Since the given sequence is an AP, hence we can say t... |
H: Integral of a function with complex numbers
I'm trying to understand how the Fourier Transform can give the coefficients of epicycloids for any arbitrary closed shape.
Given a "periodic function on the complex plane $f(t)$" (as I understand it, it's a function that returns a complex number for any $t$), one might... |
H: A vowel is followed by a consonant 80% of the time and a consonant is followed by a vowel 50% of the time. Proportion between cs and vs?
A text is such that the probability that a vowel is followed by a
consonant is 80% and a consonant is followed by a vowel 50% of the
time. What is the proportion of vowels in... |
H: Difficulty interpreting high order derivatives in $\mathbb{R}^n$
If $f:U\subseteq\mathbb{R}^m\to \mathbb{R}^n$ is differentiable function then its derivative
$$
f':U\to M_{n\times m}(\mathbb{R})\simeq\mathcal{L}(\mathbb{R}^m;\mathbb{R}^n)
$$
can be seen, for each $x\in U$, as a linear transformation $f'(x):\mathbb{... |
H: Arithmetic of quotients
I was wondering whether for a ring $R$ and two elements $a,b$ that $R/(a,b)$ is isomorphic to $(R/(a))/(b)$ (with the parenthesis representing the smallest ideal containing the respective element/s).
I have seen this trick applied to polynomial rings on multiple occations.
Thank you
AI: Th... |
H: Show $2$ is not prime in $\{f\in\mathbb{Q}[x]:f(m)\in\mathbb{Z}\text{ for all }m\in\mathbb{N}\}$
Consider the subring $\text{Int}(\mathbb{Z})=\{f\in\mathbb{Q}[x]:f(m)\in\mathbb{Z}\text{ for all }m\in\mathbb{N}\}$. Show $2$ is irreducible but not prime in $\text{Int}(\mathbb{Z})$.
I was able to show $2$ is irreduc... |
H: Show that $h(t) =\int_0^t \int_0^s f(x) dxds$ is strictly convex
I am working on the following exercise:
Let $f: (0,1) \rightarrow \mathbb{R}_{\ge0}$, be a continuous function such that the set $\{x \in (0,1) \mid f(x) = 0\}$ is a nowhere dense set. Prove that the iterated integral $$h(t) =\int_0^t \int_0^s f(x) d... |
H: Cauchy-Euler equation - wronskian and $e^u$ substitution
Wikipedia says te following about Cauchy-Euler equation:
$$a_{n} x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + \cdots + a_0 y(x) = 0$$
is called Cauchy-Euler equation. Substitution $x=e^u$ (where $u = ln(x)$; for $x < 0$ we replace $x$ with $|x|$ to exte... |
H: How many points are needed to uniquely specify a box?
In $\mathbb{R}$, to specify a line segment, we need two points. In $\mathbb{R}^2$, to specify a given rectangle, all we need are three points. Then the fourth point is determined, and there is only one rectangle with those three points as vertices. In $\mathbb{R... |
H: Proving a linear form to be continuous
In the Hilbert space $L^2(R)$ I have seen that the following form is linear, however, I need to check if it is continuous and find the associated vector using the Riesz-Fréche Theorem. I have tried to prove that it is bounded but I was not able to do it.
$$F(f) = \int_0^1 f(x^... |
H: Question about $x \in \text{Img}(X)$
When I studied the expectation of a discrete random variable $X$, I saw it is defined as $$\mathbb{E}(X) = \sum_{x \in \text{Img}(X)} x \cdot \mathbb{P}(X=x)$$
What exactly does $x \in \text{Img}(X)$ mean? Does it mean $x$ can be all possible values of $X$?
AI: The set of all po... |
H: Set Theoretic naming for a union property
I am introducing a new operation for countable sets, say $\mathcal{X}(\cdot)$ is the operation. I have proved that for two sets $S_1, S_2$ we have:
$$\mathcal
X(S_1 \cup S_1) = \mathcal{X}(S_1) \cup \mathcal{X}(S_2).$$
How can I name this property in set-theory language?
A... |
H: Compactness of the set $S=\{\sin(2^nx)\mid n\in\mathbb{N}\}$.
Let $$S=\{\sin(2^nx)\mid n\in\mathbb{N}\}$$ in the metric space $L^2([-\pi,\pi])$ of Lebesgue square integrable functions. On $[-\pi,\pi]$ with metric defined as $$ d(f,g)=[\int_{-\pi}^{\pi}(f-g)^{2}dx]^{\frac{1}{2}}$$ Then which of the following are tr... |
H: Solving a three variable exponential equation
I have the following function:
$$
y=a.x^b+c
$$
And I have a finite set of already calculated values for the x and the y. Using this data set I need to calculate the values of a, b, and c. Is there a way to do so?
I didn't manage to get an answer using what I remembered... |
H: what is the limit of $\sum_{n>0}^\infty p^ncos(n\zeta)$
Let $p\in \mathbb{R}$ with $|p|<1$ and $\zeta \in \mathbb{R}$.
$\sum_{n>0}^\infty p^ncos(n\zeta)$ and $\sum_{n>0}^\infty p^nsen(n\zeta)$
I need to find the limit of this series, I need it for a theorem that I am proving, but I could not find it
AI: Call these... |
H: Prove that set is closed and ideal of another set
Let $(X; d)$ be a compact metric space. For a given $x_0 \in X$, define $C_{x_0}(X,\mathbb{R})$ by
${C}_{x_0} = \{ f \in \mathcal{C}(X,d): f(x_0) = 0 \}$.
I need to prove that $C_{x_0}(X,\mathbb{R})$ is closed in $\mathcal{C}(X,\mathbb{R})$ and that $C_{x_0}(X,\ma... |
H: Find $x(t)$ given $\frac{dx}{dt}$ and $\frac{dy}{dt}$
For $0 \leq t \leq 1$ , a particle is moving along a curve so that its position at time $t$ is $(x(t),y(t))$. At time $t=0$ , the particle is at position $(0,0)$ .
We are given that
$$\frac{dx}{dt} = \frac{t}{\sqrt {1+t^2}}$$
$$\frac{dy}{dt} = \sqrt {\frac... |
H: Finding the remainder of $123^{456}+78^9$ divided by $7$
Find the remainder when $123^{456}+78^9$ is divided by $7$.
This would be $123^{456}+78^9\pmod7,$ but is there any way to find some slick congruence? I don't really understand how to find congruences.
AI: $123^{456}\equiv1\bmod 7$ by Fermat's little theore... |
H: growth of population- solving initial value problem
I want to solve the initital value problem
$$ P'= \alpha P^{ \beta} , \space \space P(0)=P_0 > 0 $$
with
$ \alpha \in \mathbb{R}^{ +} ,\beta \in [1, \infty[ $
and show, that there exists a time T for $ \beta >1$ so that
$$ lim_{ t \rightarrow T} P(t)= \infty $$ ... |
H: Astrodin parametric equation derivation
I know, that the astroid with equation
$$
x^\frac{2}{3}+y^\frac{2}{3}=a^\frac{2}{3}
$$
can be parametricly expressed as:
$$
x=a*cos^3(t)
$$
$$
y=a*sin^3(t).
$$
I was wandering if there is a way, to derive parametric equations from formula at the top.
AI: Usually when we see t... |
H: Monotonicity of the Lebesgue Integral
I am working through Baby Rudin, and encountered the following remark:
If $f$ and $g\in\mathcal{L}(\mu)$ on $E$, and if $f(x)\leq g(x)$ for $x\in E$, then
$$\int_{E}fd\mu\leq \int_E gd\mu.$$
Given a measurable function $f$, $\int_{E}fd\mu$ is always defined, but $f$ only b... |
H: Minimizing Type I and Type II errors Simultaneously
I am trying to understand why I can't simultaneously minimize Type I error and Type II error simultaneously. A non-randomized Hypothesis test is essentially a deterministic partition of the Range space of our Random Variable.
$$R_0=\{x\in R: Reject Null\}\hspace{... |
H: Prove by contradiction that $A\cup B=B\cup A.$
For arbitrary sets $A$ and $B$, assume the opposite of our conclusion: $A\cup B \neq B \cup A$.
Thus, either there exists $x\in (A \cup B)$ such that $x \notin (B\cup A)$ or there exists $x'\in (B\cup A)$ such that $x' \notin (A\cup B)$.
Without loss of generality, ass... |
H: Divisibility of $a_{24}$ by 7. ($a_n=\underbrace{999\cdots9 }_{n \text{ times}})$
Question: By which number is $a_{24}$ divisible by?
Where $a_n=\underbrace{999\cdots9 }_{n \text{ times}}$
The solution says the answer is $7$. Here's what is given:
$$a_{24}=\underbrace{999\cdots9 }_{24 \text{ times}}$$
$$=9(\u... |
H: If $f$ is differentiable at $[a,+\infty$) and $\lim_{x\to\infty}(f(x)+f'(x))=c\in\mathbb{R}$. Prove that $lim_{x\to\infty}f(x)=c$.
I figured if I proved that $f'(x)=0$ then that would be it because, if first derivative is zero, than the function has constant value for every point of its domain. But I don't know how... |
H: How to find a vector given the scalar triple product
Scalar triple product formula: $a (b \times c)$
If I know the value of the product and I know what vectors $b$ and $c$ are, how do I find what vector $a$ is equal to?
AI: Unfortunately, you can't. Your real question is even simpler: if you know $\mathbf{x}$ and $... |
H: Cramer-Rao lower bound for exponential distribution
Given a sample $X_1,\dots , X_n$ from a population $X\sim \operatorname {Exp} (\lambda )$, I have to calculate Cramer-Rao bounds for the estimation of $\lambda$ and $\frac 1 \lambda$; I also must determine if there are estimators that limit.
Now, we have that $\f... |
H: Question about unclear definition of Ackermann-Péter function in Stanford Encyclopedia of Philosophy
I'm reading Recursive Functions at Stanford Encyclopedia of Philosophy (section 1.4). The following paragraph defines function β which is then used to define variant of Ackermann-Péter function:
What exactly is the... |
H: Doesn't $0.\overline9=1$ lead to consequences like $a-0.\overline01=a$ and $2=1.\overline931415926$?
I'm just starting to learn calculus, but this was the first idea presented:
$$0.\overline9=1$$
This would mean that this is true:
$$a-0.\overline01=a$$
When I thought about it, then I realised that if the above was ... |
H: $\lVert x_k \rVert \rightarrow \infty \quad \Longrightarrow \quad f(x_k) \rightarrow+\infty$ implies existence of global minimiser
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a continuous function sucht that for every sequence $(x_k)_k \subset \mathbb{R}^n$ holds
$$\lVert x_k \rVert \rightarrow \infty \quad \L... |
H: Prove that $\mathbb{Q}[X]/I\cong Q\times Q$
Let $f(X)=(X^2-2)(X^4-X)$ and $g(X)=(X^2-1)X\in \mathbb{Q}[X]$.
Let $I=(f,g)$ the ideal generated by $f$ and $g$.
Prove that $\mathbb{Q}[X]/I\cong Q\times Q$
Using the reasoning of this answer I have computed that $I=X(X-1)$.
The next step is to define an epimorph... |
H: Poisson kernel equivalence
I'm currently reading "Real and Complex Analysis" by Rudin.
It was shown that
$$
\sum_{n = -\infty}^{\infty} r^{|n|} e^{in \theta} = \frac{1 - r^2}{1 - 2r \cos{\theta} + r^2 } \quad 0 \leq r < 1,\; \theta \in \mathbb{R}
$$
I'm having trouble verifying this equality. The textbook just ... |
H: Find $g(x)$ if $f(x)= \begin{cases} -1, & -2 \leq x \leq 0 \\ x-1, & 0 < x \leq 2 \end{cases}$ and $g(x) = |f(x)| + f(|x|)$
$$f:[-2,2] \rightarrow \Bbb R$$
$$\text {and }f(x)= \begin{cases} -1, & -2 \leq x \leq 0 \\ x-1, & 0 < x \leq 2 \end{cases}$$
And, let $g(x)$ be equal to $|f(x)|+f(|x|)$
We need to find the v... |
H: Show the feasible set of convex functions is convex.
Problem Statement
Let $g_{1},\dots, g_{M}$ be convex functions and let $b_{1},\dots, b_{M}$ be real numbers. Show that $F:= \{ \boldsymbol{x} \in \mathbb{R}^{d} : g_{i}(\boldsymbol{x}) \leq b_{i}, $ for all $i = 1,\dots , M \}$ is a convex set.
Attempted Proof
... |
H: Limit of sequence of series $a_n=\frac{1}{n}\sum_{j=1}^n\sqrt{\frac{j}{n}}$
Given the series (sequence of series, really) $$a_n = \frac{1}{n}\sum_{j=1}^n \sqrt{\frac{j}{n}}$$
what is its limit as $n\rightarrow \infty$? (I happen to know its 2/3, but how does one show it?)
(Note I've seen a few solutions using some... |
H: Prove the following two statements about differentiability are equivalent
Let $E$ be a subset of $\textbf{R}$, $f:E\to\textbf{R}$ be a function, $x_{0}\in E$, and $L\in\textbf{R}$. Then the following two statements are equivalent
(a) $f$ is differentiable at $x_{0}$, and $f'(x_{0}) = L$
(b) We have $\displaystyle\l... |
H: Operator norm inequality $\|XY\|\geq\frac{\|X\|}{\|Y^{-1}\|}$
Let $X, Y$ and $Y^{-1}$ be linear operators on a normed space. How to prove the inequality $$\|XY\|\geq\frac{\|X\|}{\|Y^{-1}\|}?$$
I already know that $\|XY\|\leq\|X\|\|Y\|$ but I don't see how I can use this fact right now.
AI: Simply note that $$\|XY\... |
H: Why are the $p$-oldforms $f(z)$ and $f(pz)$ linearly independent at level $\Gamma_0(pN)$?
Let $f$ be a newform (normalized eigenform) of weight $k$ and level $\Gamma_0(N)$. Fix $p$ not dividing $N$ and set $f_p(z)=f(pz)$.
Viewing $f$ and $f_p$ at level $\Gamma_0(pN)$, why are they linearly independent?
Here is ... |
H: How do I prove that sum of reciprocals of first $2^n$ natural nos is always greater than $\frac{n+1}{2}$
How can prove this inequality
\begin{equation}
\sum_{r=1}^{2^n} \frac{1}{r}\geq \frac{n+1}{2}
\end{equation}
Without using induction... Want to have an insight on summation inequalities
AI: For non-negative inte... |
H: Calculating $E[Z(Z-1)(Z-2)(Z-3)]$ where $Z$ is Poisson
If $Z$ is a variable that distributes by Poisson, with Expected value, $E(Z) = 2.5$.
I need to solve:
$E[Z(Z-1)(Z-2)(Z-3)]$
So What I thought to do is first:
$E[(Z^2-Z)(Z^2-2Z)(Z^2-3Z)]$
$[E(Z^2)-E(Z)][E(Z^2)-2E(Z)][E(Z^2)-3E(Z)]$
From here I'm not sure how ca... |
H: On $\sum_{x=0}^\infty x^n r^x$
I am looking for either a closed form or recursive expression for $$\sum_\limits{x=0}^\infty x^n r^x\ $$ that does not include differential operators, where $n \in \mathbb N_0$. It is clear to me that $$\sum_\limits{x=0}^\infty x^n r^x = r \cfrac{\text d}{\text dr}\sum_\limits{x=0}^\i... |
H: $(x,y,1)$ is perpendicular to $(1,2,-7)$. What is the relationship between $x$ and $y$?
$(x,y,1)$ is perpendicular to $(1,2,-7)$. What is the relationship between $x$ and $y$?
I think that there is no relationship between $x$ and $y$ because they could be any number. I know that the dot product of both $x$ and $y... |
H: Is this representation isomorphic to the group?
Let's consider s representation $\rho: G \rightarrow GL(V)$ such that there is a bijection between every linear operator of the image of $\rho$ and the elements of $G$. Is the image of $\rho$ isomorphic to $G$?
AI: No. Consider the representation $\rho:GL(n,\Bbb C)\t... |
H: Show that $\lvert \lvert (x,y) \rvert \rvert \rightarrow \infty$ implies $f(x,y) = x^2-4xy+4y^2+y^4-2y^3+y^2 \rightarrow \infty$
I am working on the following exercise:
Consider the function
$$f(x,y) = x^2-4xy+5y^2+y^4-2y^3.$$
Show that $$\lvert \lvert (x,y) \rvert \rvert \rightarrow \infty \quad \Longrightarrow... |
H: Simple function that returns a Gaussian curve?
I need a simple function that it's output is a Gaussian for $\mathbb{R} \to [0,1]$.
Any tips? Thanks.
AI: $$f_\sigma(x)=e^{-\frac{x^2}{2\sigma^2}}$$
Here $\sigma$ is controlling the width of the Gaussian |
H: If $\sum_{n>0}^\infty c_{n}z^{n}$ has a radius of convergence R then the series $\sum_{n>0}^\infty \frac {c_n}{n!} $ has a radius of convergence +∞.
Suppose the series $\sum_{n>0}^\infty c_{n}z^{n}$ has a radius of convergence $R \in \mathbb{R}^{+}$ Justify that the series $\sum_{n>0}^\infty \frac {c_n}{n!} $ has ... |
H: throwing 2 dice probability
I am throwing $2$ dices and I want the sum of the two dices to be $> 8$
Now the dice is from $1$ to $6$ so to get $>8$ we have $9 - 12$
$ 6 + 6$ $ 5 + 6$ $ 4 + 6$
$ 6 + 5$ $ 5 + 5$ $ 4 + 5$
$ 6 + 4$ $ 5 + 4$ $ 3 + 6$
$ 6 + 3$
so it should be that you only have $4$ options out of $6$ fo... |
H: Deriving polar coordinate form of ellipse. Issue with length of a distance to a foci.
I am reading through in Spivak on how to obtain the polar coordinate form of the ellipse. I'm given the following diagram:
All I'm trying to do is establish that the distance between $(x,y)$ and $(-2\epsilon a, 0)$ is as is stat... |
H: Proving intersection of POsets is a POset - Reflexive
I need to prove that the intersection of 2 POsets R and S is a POset.
So we basically want to prove that if $R$ and $S$ are POsets then $R \cap S$ is reflexive, transitive and anti-symmetric.
The problem is in the reflexive, I proved it like this:
We need ... |
H: Sum of orthogonal projection operators
Let $X$ be an Hilbert space, I am trying to see that if $P$ and $Q$ are orthogonal projection operators then the following are equivalent:
$(1) Im Q\subseteq KerP$
$(2)P+Q$ is an orthogonal projection operator.
I was able to do $(1) \implies (2)$ but I cant seem to make the ... |
H: Rademacher complexity of linear function class with compact parameter space and support
Consider the class of functions $\mathcal{F}$ of the form $f_\theta(x) = \langle x,\theta \rangle$ where $\theta \in \{\theta \in R^d : ||\theta||_1 \leq r\}$ and x has support $\{x \in R^d : ||x||_\infty \leq M\}$.
Give an uppe... |
H: Let $X,Y\sim U(0,1)$ and suppose $X$ and $Y$ Are independent. How to find explicitly the distribution of $Z=XY$ Using conditional expectation?
I found this problem interesting but I don't see clearly how to solve it. Any hint or ideas on how to get this one started?
AI: For example like that:
Take any $t \in (0,1)$... |
H: Area between curves with respect to $x$ or $y$
Sketch the region enclosed by the curves given below. Decide whether to integrate with respect to $x$ or $y$. Then find the area of the region. $y=3\cos(x)$ , $y=3−\frac{6x}{\pi}$
Need help with solving this question. I did $\int_0^{\pi/2} [3\cos(x)- (3−\frac{6x}{\pi}... |
H: General rule of thumb to convert between number systems.
I would like to know what is a general rule of thumb to convert between the following number systems:
Decimal
Octal
Hexadecimal
Binary
I know if I want to convert from octal to hexadecimal, I first convert the octal number to binary with groups of 3 bits an... |
H: Calculate maximum of Gini impurity
Gini impurity is defined as :
$$G = 1 - \sum_{i=1}^{k}p_i^2$$
In my book of statistics it's written that it has a maximum when $p_1 = p_2 = … = p_k = \frac{1}{k} $ but there is no derivation . How can one prove that it has a maximum in this case ? I tried something but without su... |
H: Recovering the covariance matrix from directional variance
How to find the covariance matrix $K$ (in some basis) of dimension $n$, given $\{$projected variance of $K$ along $V \enspace:\enspace \forall \enspace n$-dimensional vector $V\}$?
In other words, if we know the variance along any direction, how can a covar... |
H: Can an average of an overall set be equal to the average of a subset?
I read in a publication that the average salary for lawyers in America is $\bar x=\$163,595$. Of these salaries, the average for men is $\bar x_m=\$183,687$, and for women, it is $\bar x_w=\$163,595$. I'm thinking, how is it possible for the aver... |
H: If $X$ is a random variable, and $Y= 2X$, then why isn't it enough to multiply the density function of $X$ by $2$ to find the density function of $Y$?
This may be a dumb question, and I've tried searching online for answers, but I can't seem to wrap my head around it.
So say I have a random variable $X$ and $Y = 2... |
H: Locally compact subsests of R
I'm looking for an example of two locally compact subsets of the real line R, but their union isn't locally compact.
I know that generally it is not true that such union is locally compact, as we can take $\left \{ (x,y):x>0 \right \}\cup \left \{ (0,0) \right \}\
$, where the origin ... |
H: Given a sequence of i.i.d. random variables, prove a result involving conditional expectation by means of symmetry argument
Let $(X_n)_{n\geq1}$ be an i.i.d. sequence with $\mathbb{E}\{|X_1|\}<\infty$. Let $S_n=X_1+\cdots+X_n$ and $\mathcal{F}_{-n}=\sigma(S_n,S_{n+1},\ldots)$. Then, one can state that $$M_{-n}=\mat... |
H: If $(\alpha_i,\alpha_j)=(\beta_i,\beta_j)$ for $0
Let $V$ be an inner product space over $\mathbb{R}$ with dimension $n$. Let $S$ be a subset of $V$ and $L(S)$ denote the span of $S$. If $(\alpha_i,\alpha_j)=(\beta_i,\beta_j)$ for every $i, j\in \{1, ...,n\}$, Then
$$L(\alpha_1, ..., \alpha_n)\cong L(\beta_1, ...,... |
H: Application of Burnside's Lemma on the vertices of a cube
Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are considered distinguishable if neither can be rota... |
H: Prove that $\left(\dfrac{b}{a}+\dfrac{d}{c}\right)\cdot\left(\dfrac{a}{b}+\dfrac{c}{d}\right)\geq4$ with $a>0, b>0 , c> 0$ and $d>0.$
Prove that
$\left(\dfrac{b}{a}+\dfrac{d}{c}\right)\cdot\left(\dfrac{a}{b}+\dfrac{c}{d}\right)\geq4$
with $a>0, b>0 , c> 0$ and $d>0.$
My attempt:
$$\begin{align*}\left(\dfrac{b... |
H: for all $p\in [0,1]$ there is a constant $M(p)$ such that $|f(z)|
Let $f(z)=\sum_{n>0} \frac{c_n}{n!}z^n$ proof that for all $p\in [0,1]$ there is a constant $M(p)$ such that $|f(z)|<M(p)\exp\left(\frac{|z|}{pR}\right)$ where $R$ is the convergence radius of $\sum_{n>0} c_n z^n$ and $z\in \mathbb{C}$.
I need to use... |
H: Find all functions $f(x) = c[T(f)](x)$ for a linear transformation $T$
I'm working through Apostol's calculus vol 2, and I'm not sure how to finish the following question (2.4, #29, part f):
Let $V$ be the set of all real functions continuous on the interval of $[-\pi, \pi]$ and let $T: V \to V$ be the linear tran... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.