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H: Product of $L^p$-convergent sequences are Cauchy
I'm working on showing:
If $\|f_n - f\|_p \to 0$ with $1≤p<\infty$ and $h_n \to h$ pointwise with $|h_n|<M$ for all $n$, then $\{f_nh_n\}$ is Cauchy in $L^p$
I've shown that $\|h_n-h\|_p \to 0$, but I'm having a hard time showing the product is Cauchy. I tried to man... |
H: What does $f:\mathbb R \rightarrow \mathbb R$ mean?
This is simply a basic notation question: what is the meaning of
$$f:\mathbb R \rightarrow \mathbb R$$
I imagine it's some sort of function to do with the set of real numbers, perhaps some sort of mapping. Until now I've only encountered functions of the form
$$f(... |
H: Does the independence of the axiom of choice imply Gödel's incompleteness theorem?
I recently wrote this answer describing Gödel's completeness and incompleteness theorems, in which I came to the conclusion that a theory is (syntactically) complete if and only if all its models are elementarily equivalent, that is ... |
H: How to compute the smallest face of a polytope containing a given point
Let $S\subseteq\mathbb{R}^d$ be a finite set and $P=conv(S)$ its convex hull so $P$ is a convex polytope. Let $p\in P$ be given. Note that if $F_1$ and $F_2$ are two faces of $P$ containing $p$, then $F_1\cap F_2$ is also a face of $P$ containi... |
H: All solutions of $f(w+z)=f(w)f(z)$, $f(1)=e$
Let $w=u+iv$ and $z=x+iy$ with $u,v,x,y\in\mathbb{R}$. Is the function
$$f:\, \mathbb{C}\to\mathbb{C},\, f(z)=e^x\cos y+ie^x\sin y$$
the only solution of the functional equation
$$f:\, \mathbb{C}\to\mathbb{C},\, f(w+z)=f(w)f(z),\, f(1)=e?$$
I was only able to prove that ... |
H: $K(H)^{**}= B(H)$
Let $H$ be a Hilbert space. Is it true that we have an isometric linear isomorphism
$$B(H) \cong K(H)^{**}$$
where $K(H)^{**}$ is the bidual of the compact operators on $H$.
I think I proved this but I could not find a reference on the internet. Is the result true?
AI: Yes, this is completely sta... |
H: Strong Law of Large Numbers with randomly many summands
I'm wondering about a SLLN where the number of summands is also allowed to be a random variable.
More specifically assume that:
$\{X_i\}$ are iid with $\mathbb{E}[|X_1|] < \infty $
$N=\{N_n\}_{n\in\mathbb{N}}$ is a sequence of random positive integers
indepen... |
H: calculate $\oint_{|z|=1} \left(\frac{z}{z-a}\right)^n \, dz$
calculate $\oint_{|z|=1}\left(\frac{z}{z-a}\right)^{n}$
whereas a is different from 1, and n is integer.
My try:
\begin{align}
& \oint_{|z|=1}\left(\frac{z}{z-a}\right)^n \, dz\\[6pt]
& \oint_{|z|=1}\frac{z^n \, dz}{\sum_{k=0}^n z^k(-a)^{n-k}}\\[6pt]
& \o... |
H: Can reflexive partial orders be defined from strict partial orders without using equality?
Consider a reflexive partial order $\leq$. We can define its strict counterpart without using equality, like so: $x \leq y$ $\land$ $\neg y \leq x$. But from a strict partial order $<$, can we define its reflexive counterpart... |
H: Calculating expected value of $X$ with the density function $f(x)=16xe^{-4x}$
Suppose, $X$ be a random variable with probability density funciton,
$$
f(x) = \begin{cases}
16xe^{-4x}, & x \geq 0; \\
0, & \text{otherwise}
\end{cases}
$$
(source)
I tried to find the expected value of $X$, so I integrated $16x^2 e^{-... |
H: Find partial derivatives for $\sqrt[3]{(x^3 + y^3)}$ at the point $(0,0)$.
For the function: $\sqrt[3]{(x^3 + y^3)}$ I'm requested to find both partial derivatives at the point $(0,0)$. I believe this is a problem that uses the limit definition of a partial derivative, however I always get 0 as an answer, which is ... |
H: Question about convexity: how do we prove that $\displaystyle \sum_{i=1}^{k}p_{i}b_{i}\geq\prod_{i=1}^{k}b^{p_{i}}_{i}$?
Let $b_{1},b_{2},\ldots,b_{k}$ be nonnegative numbers and $p_{1} + p_{2} + \ldots + p_{k} = 1$ where each $p_{i}$ is positive. Then
\begin{align*}
\sum_{i=1}^{k}p_{i}b_{i}\geq\prod_{i=1}^{k}b^{p_... |
H: Properties of a continuous submartingale which is a function of Brownian motion
I'm attempting to solve the following question but I'm really unsure of what my approach should be. I've some progress but I need a lot of help, as I'm not sure if I'm anywhere near the right track:
For part (a) intuitively, $$\textbf... |
H: Given $AB\cdot F=JG$, $AD^E=AHJ$, ..., find the value of $G^E$
In the problem above, I know that the digits are all between 1 and 9, inclusive, at least most of them are, but I can't figure out a way other than trial and error to get the answer. Can someone please help? Thanks!
AI: To solve these types of probl... |
H: Show that a Hibert-Schmidt operator is compact.
Let $H$ be a Hilbert space and $u: H \to H$ a Hilbert-Schmidt operator, i.e. for an orthonormal basis $E$
$$\Vert u\Vert_2^2:=\sum_{x \in E} \Vert u(x) \Vert^2 < \infty$$
(this does not depend on the choice of basis).
I want to show that $u$ is compact. I looked this ... |
H: Half-open topology on $\mathbb R$ is separable, and $A \setminus \hat A$ is countable
This is part of Exercise 9 in Section 2.2 of Topology and Groupoids, by Brown.
For each $x \in \mathbb R, N \subseteq \mathbb R$ is a neighborhood of $x$ if and only if there are real numbers $x^{\prime}, x^{\prime \prime}$ such t... |
H: How to prove $\sqrt{2}$ is irrational in Type Theory?
Using Intuitionistic Type Theory, how would one go about proving $\sqrt{2}$ is irrational?
I read that we can not use law of excluded middle. (So does this mean we cannot use proof by contradiction).
So I assume we want to prove $\forall a,b\in \mathbb{N^2}: \ln... |
H: A graph theoretic definition of functions?
The graph(1) $G$ of the function $f:X\to Y$ is the set $\{(x,f(x)):x \in X\}$.
Unless I'm mistaken, this can be interpreted as a directed, bipartite graph(2) where every vertex in one of the partitions (domain) has out-degree $1$ and in-degree $0$. The other partition is t... |
H: Differentiation while transforming the expression by introducing polar coordinates
The expression that is to be transformed by polar coordinates is $\Big(\frac{\partial z}{\partial x}\Big)^2+\left(\frac{\partial z}{\partial y}\right)^2$.
So I defined $x=x(r,\phi)=r\cos\phi$ and $y=y(r,\phi)=r\sin\phi$, where $r>0$... |
H: The second largest eigenvalue
Let $A$ be an $n\times n$ positive semidefinite real matrix with the trace of $A$ is $2$. Suppose there exist unit vectors $y,z$ satisfying $|Ay|=|Az|=1$ and $y\cdot z=0$.
How is it showed that the second largest eigenvalue of $A$ is greater than or equal to $1$?
AI: It's false. Let $A... |
H: Can you explain the behavior of $\frac{1}{x} \sin(\frac{1}{x})$ and $|\frac{1}{x} \sin(\frac{1}{x})|$ as $x$ approaches $0$
When I graph it out, I can see what the behavior of the graph is, but my intuition is failing to grasp why it acts this way.
As $x\to 0$, $f(x) = \frac{1}{x} \sin(\frac{1}{x})$ only oscillates... |
H: What is the name of sets of nodes of a tree that cover/span/cut the entire width of the tree?
What are the following sets called?
They may be constructed by repeating the following recipe.
Choose a parent node
Remove the parent node's descendents
Repeat as desired while parent nodes remain.
Here is a tree in grap... |
H: Solving Exponential Equation $x^2 \cdot 2^{x+1} + 2^{|x-3|+2} = x^2 \cdot 2^{|x-3|+4} + 2^{x-1} $
The solution for the Exponential Equation considering $x\in\mathbb{R}$ is $\boxed{x\in[3, \infty)\cup \left\{ -\frac12;\frac12 \right\} }$
My try for solving $x^2 \cdot 2^{x+1} + 2^{|x-3|+2} = x^2 \cdot 2^{|x-3|+4} + ... |
H: How does this statement follow algebraically from the last one? (Algebraic eqations)
I was reviewing some combinatorics notes and i found the following.
Can someone please help me understand how in the world does that statement follow from the monstrosity above it?
In case it matters, the task is attempting to sol... |
H: Area under the graph of a convex function
Consider the following problem:
Suppose that $f$ is a twice differentiable real function such that
$f''(x)>0$ for all $x\in[a,b]$. Find all numbers $c\in[a,b]$ at which
the area between the graph $y=f(x)$, the tangent to the graph at
$(c,f(c))$, and the lines $x=a$, $x=b$,... |
H: Revenue and quadratic formula - for every x increase in price there are y fewer sales
"When a shoe costs $\$80.00$, there are $300$ sales. Every $\$5.00$ increase in price will result in 10 fewer sales. Find the price that will maximize income."
I am able to solve the question just fine, but I am confused about the... |
H: Range of Convergence of $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n-1}}{n \ 3^n (x-5)^n}$
$$\sum\limits_{n=1}^{\infty} \frac{(-1)^{n-1}}{n \ 3^n (x-5)^n}$$
I am trying to use the alternating series test to find a range of $x$ for which $(1) b_n > b_{n+1}$ and $ (2) \lim_{n \to \infty} \frac{1}{n \ 3^n (x-5)^n} = 0$. ... |
H: Doubt on showing that when $p$ is prime, the only unipotent class is $p-1$?
I am trying to understand the solution of the following exercise:
A number $a$ is unipotent if $a\neq1$ and $a^2\equiv 1 \pmod{p}$.
Show that when $p$ is prime, the only unipotent class is $p-1$.
The answer is:
$(p-1)x\equiv1 \pmod{p}... |
H: Calculating limits of a system
For the following system
$$x'=x(-x^2-y+4)=f(x,y)$$
$$y'=y(y^2+8x-1)=g(x,y)$$
I need to find the location of the critial points and determine each points type and stability.
I then need help finding calculating the limits
$$\lim\limits_{t \to \infty} x(t), \lim\limits_{t \to \infty} y(... |
H: How to compute $\sum_{n=1}^\infty{\frac{n}{(2n+1)!}}$?
In a calculus book I am reading I have encountered the following problem:
$$\sum_{n=1}^\infty{\frac{n}{(2n+1)!}}$$
The hint is to use Taylor series expansion's for $e^x$. I tried to express the sum as the form $$e^x=\sum_{n=0}^{\infty}{\frac{x^n}{n!}}$$
But I c... |
H: Completeness of $(\mathcal M (2,\mathbb R),\lVert \cdot\rVert)$
Let $\mathcal M (2,\mathbb R)$ be the space of $2\times2$ matrices with inner product $ \langle A,B\rangle=\operatorname{tr}(AB^T) $
We define the normed space : $ \bbox[1px,border:1px solid green] { X:=(\mathcal M (2,\mathbb R),\lVert \cdot\rVert)}... |
H: What is this topological space called?
Given any topological space $T=(X,\tau)$ and any map $f:X'\to X$ note we can define the space: $T_f=(X',\{f^{-1}[S]:S\in \tau\})$ what is this topological space called with respect to $T$ and $f$?
AI: The topology on $X'$ you've described is called the initial topology on $X'$... |
H: How to show that two reccurence relations are equal?
I encountered the following problem:
I've never seen this on tutorials or similar. How could i go about proving this? I tried using induction but its looking pretty grim.
AI: Induction works, with just one tricky bit. I’ll use $G$ to denote the function with the... |
H: Proving the Marginal Distribution of a $\text{Gamma}(2,\lambda)$ Joint Distribution follows an Exponential
I derived the bivariate joint distribution with a transformation from two exponential distributions with $\lambda = 1.5$ for $X,Y$ and $U=X+Y$ and $V=X-Y$. With $X=(U+V)/2$ and $Y=(U-V)/2$ I derived the joint ... |
H: $\sigma(n)$ is injective?
I was reading about the sum function of divisors $\sigma(n)$, and the question arose as to whether this is injective. Does anyone know the answer? in case of being injective some idea for the demonstration? Thanks in advance. And I apologize if it turns out to be an obvious question, I hav... |
H: Possible mistake in the solution of Baby Rudin Ch. 6 Ex. 11
This is exercise 11, Chapter 6 in Baby Rudin:
Let $\alpha$ be a fixed increasing function on $[a, b]$. For $u \in \mathscr{R}(\alpha)$, define
$$ \lVert u \rVert_2 = \left\{ \int_a^b \lvert u \rvert^2 \ \mathrm{d} \alpha \right\}^{1/2}. $$
Suppose $f, g, ... |
H: Calculate: $\int_0^\infty [x]e^{-x} \, dx$ where $[x]:=\max \{k\in\mathbb{Z}:k\leq x\}$
Calculate: $\int_0^\infty [x]e^{-x} \, dx$ where $[x]:=\max \{k\in\mathbb{Z}:k\leq x\}$
Solution: $$\int_0^\infty[x]e^{-x} \, dx = \sum_{k=0}^\infty \int_k^{k+1}[x]e^{-x} \, dx = \sum_{k=0}^\infty k\int_k^{k+1}e^{-x} \, dx = \su... |
H: Let $F$ be a field of characteristic $p$. Do the morphism $x^p-a$ always has a root in $F$ for every $a \in F$?
Do you have an example in which $x^p - a$ does not have a solution in $F$?
AI: Take the field $\Bbb F_p(t)$ of rational functions in $t$ with coefficients in $\Bbb F_p$, then $x^p-t$ does not have a root.... |
H: Determining whether the function is exponential?
Here's a screenshot of the problem:
Because the $x$-value is an exponent, then this must be an exponential function. By definition, an exponential function is where the independent variable (the $x$-value) is the exponent.
To write the function in the form $K(x) = a... |
H: Linear program for flow into a node should exit all on one edge
Suppose I'm given a problem where I want to route some flow from a set of sources to a set of sinks in a directed graph; however, as opposed to the standard flow constraints, I also want to constrain some nodes in the following way: all flow into the n... |
H: Proof-verification: Any prime $p>3$ can be expressed in the form $(6n+1)$ or $(6n+5)$
I have already seen answers to this question, but I would like to get my own proof verified.
Proposition: Any prime $p>3$ can be expressed in the form $(6n+1)$ or $(6n+5)$.
Definition: A prime $p$ is a natural number greater th... |
H: Seeking Starting Point for $f(a) = \int_0^1 \frac{\ln\left(ax^2 + 1\right)}{x + 1}\:dx$
I would like to solve the following parameterised definite integral.
$$
f(a) = \int_0^1 \frac{\ln\left(ax^2 + 1\right)}{x + 1}\:dx
$$
Where $a \in \mathbb{R}^+$
I have tried a few different methods that haven't resulted in anyth... |
H: Spivak's Calculus Assumptions 2
Spivak's Calculus is known for being the best Calculus book when it comes to rigour, but I think I've found an assumption he makes during his very first proof!
The axioms are:
P1 (Associativity of Addition): (a + b) + c = a + (b + c)
P2 (Additive Identity): a + 0 = a
P3 (Additive Inv... |
H: Quetion about this irrationality proof.
I found this proof on mathoverflow. It's about the irrationality of $(\arcsin 1/4)/\pi$.
My question is: Assuming that we don't know the value of $(\arcsin 1/2)$, and only knew that $\sin(\arcsin 1/2) = 1/2$, then this proof wouldn't work just the same to establish the irrat... |
H: Does $\lim_{n \to \infty}\sum_{k=1}^n \left[\zeta\left(2k-1-\frac{1}{2n}\right) + \zeta(2k)\right]$ equal the Euler-Mascheroni constant?
Let $\zeta(s)$ be the Riemann zeta function and $\gamma$ be the Euler-Mascheroni constant. Is the following formula for the Euler-Mascheroni constant true?
$$
\lim_{n \to \infty}\... |
H: Homeomorphism between circle and ellipse
I want to show that the circle represented by $x^2+y^2=1$ is homeomorphic to the ellipse $x^2/4+y^2=1$
They are both subsets of $R^2$ of course.
I just don’t know what the functions would look like
What is the function taking the circle to the ellipse and vice versa and how... |
H: Kernel of matrix $\begin{pmatrix}x&y&z\end{pmatrix}$ over $k[x,y,z]$
Let $R=k[x,y,z]$ and let $M=\langle f_1,f_2,f_3\rangle$ where
$$
f_1=\begin{pmatrix}y\\-x\\0\end{pmatrix},\;f_2=\begin{pmatrix}z\\0\\-x\end{pmatrix},\;f_3=\begin{pmatrix}0\\z\\-y\end{pmatrix}.
$$
Show $M=\ker A$ where $A=\begin{pmatrix}x&y&z\end{... |
H: If $z \in \mathbb C$ such that $|z|+|z-2019|=2019$ then $z \in \mathbb R$
Let $z$ be a complex number such that $|z|+|z-2019|=2019$. Note that
$$|z+(2019-z)|=2019=|z|+|z-2019|=|z|+|2019-z|$$
This equality occurs when $0,z,2019-z$ are collinear.
But, how to show that z is a real number from that?
Note. By using the ... |
H: Are there elements of sets that are no sets in Zermelo Fraenkel Set theory?
I've seen the ZFC formalised in a lecture where the lecturer introduced, part by part, propositional logic, 1. order logic, and then zermelo-fraenkel-set theory. The lecturer didn't introduce any notion of identity ("=") in the part about 1... |
H: Two players until one player wins three games in a row. Each player will win with probability $\frac{1}2$. How many games will they play?
QUESTION: Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player... |
H: Convergence of reciprocal of a linear function
For what values of $m$ does the integral $\int_m^\infty \frac{1}{4x-16}\;dx$ converge?
I began by taking the following limit $$\lim\limits_{k\rightarrow\infty}\frac{1}{4}\int_m^k\frac{1}{x-4}\;dx=\lim\limits_{k\rightarrow\infty}\left(\frac{1}{4}\ln{|x-4|}\right)\bigg|_... |
H: Given joint pdf, how can I find $\operatorname{Var}(X)$ without using the marginal distribution of $X$?
$(X,Y)$ has joint pdf $\frac{1}{y}$ for $0<x<y<1$.
I would usually use the marginal pdf to get the expected value. But the question doesn't let me use the marginal distribution of $X$. I think this means that I a... |
H: Number of subsets with $m$ elements of a set with $n$ elements is $\frac{n!}{m!}$
I wonder if my proof of the statement in the title is correct:
Let $n,m\in\mathbb{N}: m<n$.
Let $X_n$ be a set with $n$ elements.
Let $A_m\subset X_n$ with $m$ elements.
Let $B_m$ be a set of all possible sets $A_m$.
$A_0=\emptyset\im... |
H: Prove that $10^n + 1 \equiv 0 \ \mod \ 1 \ldots 1, n \geqslant 2$ has no solutions.
Prove that $$10^n + 1 \equiv 0 \ \mod \ 1 \ldots 1 = \dfrac{10^k-1}{9}, k \geqslant 3, n \geqslant 2$$ has no solutions. Where the number of units is greater than or equal to $3$.
AI: Let $m=\frac{10^k-1}{9}$ so that $10^k=9m+1\equ... |
H: Linear independence in $\mathbb{Z}_2$ and $\mathbb{R}$
Let $v_1,...,v_k$ be vectors whose entries are all in $\{ 0, 1 \}$. These vectors can be considered both as elements of $\Bbb{Z}_2^n = \{0,1\}^n$ (with modulo 2 addition), and as elements of $\Bbb{R}^n$ (with regular addition).
Question: Is it true that $v_1,..... |
H: Comparison of Product-Like Topologies
Let $X$ be a Banach space, let $\{w_n\}_{n=1}^{\infty}$ be a sequence of (strictly) positive real numbers, and consider the two associated topological spaces $\prod_{n=1}^{\infty} X$ and $$
X_1:=\left\{
x \in \prod_{n=1}^{\infty} X:\, \sum_{n=1}^{\infty} w_n \|x\|_X < \infty
\r... |
H: What is the probability that player A rolls a larger number if player B is allowed to re-roll (20-sided die)?
The problem statement is:
2 players roll a 20-sided die. What is the probability that player A rolls a larger number if player B is allowed to re-roll a single time?
The question is a bit ambiguous, but I a... |
H: Subset of a normal subgroup is a subgroup?
I am studying Dummit & Foote abstract algebra. This question is what I have a problem.
Let $G$ be a finite group, $H$ be a subgroup of $G$, $N$ be a normal subgroup of $G$. If $|H|$ and $|G:N|$ are relatively prime, prove that $H$ is a subgroup of $N$.
I founded that $H$... |
H: Distinct subrings of cardinality $p$ of the field $\mathbb F_{p^2} $
This particular question was asked in masters entrance of a university for which I am preparing.
Question Let $p$ be a prime number. How many distinct subrings ( with unity) of cardinality $p$ does the field $\mathbb F_{p^2}$ have?
$0$
$1$
$p... |
H: Is $R(T)$ a closed subspace of $\ell^2$?
Let $T$ be the diagonal operator on the Hilbert space $\ell^2$ such that
$$T=\text{diag}(0,\frac{1}{2},\frac{1}{3},\ldots).$$
Clearly the range of $T$ denoted $R(T)$ is not $\ell^2$ since $e_1=(1,0,0...)\notin R(T)$.
Is $R(T)$ a closed subspace of $\ell^2$?
AI: It is not cl... |
H: Congruent numbers have congruent squarefree parts?
The problem is:
If $a\equiv b\pmod{p},$ then $Squarefree(a)\equiv Squarefree(b) \pmod{p}.$ Is this true?
I encountered such problem in homework I'm doing, where I need to check if the squarefree value of a polynomial is congruent to some numbers modulo $5$. Now I'm... |
H: $a,b\in\mathbb R^2$, is it ok to define $[a,b]$ to be the segment connecting $a$ and $b$?
How to denote higher dimensional segments?
For example, $a,b\in\mathbb R^2$, it is not necessarily that $a\leq b$. Is it ok to denote $[a,b]$ as the segment connecting $a$ and $b$?
AI: As long as you say that that's what you m... |
H: Probability of getting a specific sequence of length $4$ in $10$ coin tosses
This is more of a thinking question maybe, hope that's ok.
Suppose I toss a coin $10$ times. What is the probability that within these 10 tosses I get the sequence THHT.
My attempt:
If I have 10 coin tosses THHT can happen in only 1 way, s... |
H: Congruences: Solving $ax \equiv c \pmod m$,
This is from the Joseph Silverman book on Number Theory
Solving $ax \equiv c \pmod m$,
https://i.stack.imgur.com/H3QcK.png
I understand upto Step X in the above image. But how does he get from Step X to Step Y?
AI: Note that we will have infinitely many solutions such tha... |
H: $(a \cdot b)\circ c = (a \circ c) \cdot (b \circ c)$ and $(a \circ b)\cdot c = (a \cdot c) \circ (b \cdot c)$
Do $(a \cdot b)\circ c = (a \circ c) \cdot (b \circ c)$ and $(a \circ b)\cdot c = (a \cdot c) \circ (b \cdot c)$ lead to a contradiction?
Where $\cdot$ and $\circ$ are both binary operations who form gro... |
H: A question on application of derivatives
If the area of the circle increases at a uniform rate, show that the rate of increase of the circumference varies inversely as the radius.
My approach-
If area is increasing at a uniform rate then the area function should be $A(r)=pr+q$ where $p$ and $q$ are certain consta... |
H: Definition essential supremum
Consider the following fragment from Axler's "Measure, Integration & Real analysis":
Is the definition of $\Vert f \Vert_\infty$ unaffected when we change any of the strict inequalities into non-strict inequalities?
AI: If $\mu \{x:|f(x)| >t\}=0$ then $\mu \{x:|f(x)| \geq t+\epsilon\}... |
H: Calculating $ \lim_{n \to \infty} \left(\frac{n-3}{n}\right)^n $
While calculating the following limit:
$$ \lim_{n \to \infty} \left(\frac{n-3}{n}\right)^n $$
I have used the following procedure:
\begin{align}
\lim_{n \to \infty} \left(\frac{n-3}{n}\right)^n =
\lim_{n \to \infty} \left(\frac{\frac{1}{n}\cdot(n-3)}{... |
H: Extension of function beyond boundary of a closed set is continuous
Let $(X,\mathcal T)$ be a topological space and $A\subset X$ be a closed subset. Assume $g\in C(A,\mathbb C)$ is such that $g=0$ on $\partial A$. Define the extension $\tilde g$ by $\tilde g = g$ on $A$ and $\tilde g = 0$ on $A^c$. Prove that $\ti... |
H: Why is the Conway 'Look and Say' sequences constant defined by this polynom?
In his work on 'Look and Say' sequences,for instance beginning with $1$.
$$1//
11//
21//
1211//
111221//
312212$$
If $L_n$ is the length of the $n-th$ sequences, then it follows from Conway work that :
$$\lim_{n\to\infty} \
\frac{L_{n+1}}{... |
H: Find the linear function $f$ whose composition $f\circ f\circ f\circ\dots$ ($6$ times) is equal to $2x-1$ for all $x$
I am not able to start the solution, so can get a hint for this one?
Thanks
AI: If $f(x)=ax+b$ then $f(f(...(f(x)))))=a^{6}x+b(1+a+a^{2}+a^{3}+a^{4}+a^{5})$. Equate this to $2x-1$ and compare coeffi... |
H: How can I prove $\left|\frac{e^{it_p x_j}-1}{t_p}\right| \leq 2|x|$?
Background:
I am trying to understand why, if $X$ is a random variable with $\mathbb{E}\big[|X|\big]< +\infty$ and $\mu=\mathbb{P}^{X}$, then $$\frac{\partial}{\partial x_j}\overline{\mu}(u)=i\int e^{i \langle u,x \rangle} x_j \mu(dx).$$
My proble... |
H: Use of Hôpital's rules to calculate also the sequences
Hoping that my question is clear, I would like to understand because the L'Hôpital's rules are used in several questions on Math.SE (an answer for example) to calculate the sequences,
$$(a_n)_{n\in\Bbb N}, \quad \text {or} \quad \{a_n\}, \quad n\in\Bbb N$$
Duri... |
H: Is it possible to reach a monochromatic configuration only using 2x2 and 5x5 flips?
The following problem has been troubling me for quite a while now:
"The cells of a $10\times 10$ grid are either coloured blue or green. In a move you are allowed to select any $2\times 2$ or $5\times 5$ grid and reverse the colour ... |
H: Proving a limit using the $\epsilon$ - $\delta$ definition of limit.
Given $\lim _{x\to a}\left(f\left(x\right)\right)=\infty$ and $\lim _{x\to a}\left(g\left(x\right)\right)=c$ where $c \in R$, prove $\lim _{x\to a}\left[f\left(x\right)+g\left(x\right)\right]=\infty$.
My attempt:
Let for every $M>0$ exists $\delta... |
H: Proof of Cauchy-Schwarz in $\mathbb{R}^n$ using Law of Cosines
Let us consider the Cauchy-Schwarz Inequality for vectors in $\mathbb{R}^n$:
Let $u, v \in \mathbb{R}^n$. Then $|u \cdot v| \leq |u||v|$.
Wikipedia provides a proof for this setting by enforcing a condition on the discriminant of a polynomial.
What I do... |
H: Question regarding stochastic independence
Let $X,Y,Z$ be random variables and pairwise independent, i.e. $X$ independent from $Y$, $Y$ indepedent from $Z$, and $X$ independent from $Z$. I am interested in a rigorous argument, why $ (X,Y) $ is independent from $Z$?
AI: Counterexample: Let $\Omega$ be the sample spa... |
H: Rearranging basic algebraic expression
I am starting to relearn algebra as I am starting to use more linear algebra at work. I apologise for how simple this question is going to sound.
The formula that I am working with is:
$$
x_{1}+2\left(\frac{-1}{5}y_{1}+\frac{3}{5}y_{2}\right)=y_{2}
$$
and I want to rearrange i... |
H: If $\varphi$ is a linear functional, show that $\{\varphi\ge0\}^\circ=\{\varphi>0\}$
Let $E$ be a normed $\mathbb R$-vector space, $\varphi\in E'\setminus\{0\}$ and $H:=\{\varphi\ge0\}$.
I would like to show that $H^\circ=\{\varphi>0\}$ and $\partial H=\{\varphi=0\}$.
Should be trivial enough. It's clear that $H$ i... |
H: Logic expressions for an English verse.
Question:
This is an assignment question that I am having trouble with. The question goes like this.
What I did:
I made the predicate logic expression as follows.
$$\forall x\ LOVES(x, MYBABY)\ \wedge\ \forall x\ (x=ME \Longleftrightarrow LOVES(MYBABY, x))$$
Please tell me i... |
H: (Dummit and Foote) Group of order 105 with $n_3 = 1$ must be abelian
I was working on this problem: Let $G$ be a group of order $105 = 3\times 5\times 7$. Assume it has a unique normal Sylow 3-subgroup. Then prove that $G$ is abelian.
I worked out the following from Sylow's theorem:
$n_5 = 1$ or $n_5 = 21$
$n_7 = ... |
H: Formula for sequence of 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4 and so on.
I have been trying to figure out a formula for the sequence: $0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, ...$ and so on . It is not geometric and it is not arithmetic, I tried to apply these formulas, but of them are failing leading me to believe ... |
H: How do I solve $(\cos2x+1)^2=1/2$?
$x$ belongs to $$[0,\pi/2[$$
$$(\cos(2x+1))^2 = \frac{1}{2}$$
I tried to find $x$ using
$$2x+1=\frac{\pi}{4} +\frac{n\pi}{2}
\qquad\text{or}\qquad
2x+1=\frac{3\pi}{4} + n\pi$$
but I didn't find my answer in MCQ which is
$$\frac{3\pi-4}{8}
\qquad\text{and}\qquad
\frac{5\pi-4}{8}.$... |
H: Let $x_1,x_2,x_3,x_4$ denote the four roots of the equation $x^4 + kx^2 + 90x - 2009 = 0$. If $x_1x_2 = 49$, then find the value of $k$.
Let $x_1,x_2,x_3,x_4$ denote the four roots of the equation $$x^4 + kx^2 + 90x - 2009 = 0.$$ If $$x_1x_2 = 49,$$ then find the value of $k$.
What I tried :- From Vieta's Formula... |
H: Examples of finite-dimensional, semisimple, non-separable $k$-algebras?
1. Motivation
Let $k$ be a field.
Apparently, any separable $k$-algebra is finite-dimensional and semisimple.
Using Maschke's theorem for Hopf algebras, one can prove that for Hopf algebras the following stronger statement holds:
A $k$-Hopf al... |
H: proof that $\log(\det(XX^H))$ is not concave
Is there an elegant way to proof that $\log(\det(XX^H))$ is not concave, with respect to the complex-valued matrix elements of $X$?
AI: If $f:X\mapsto \log\det (X X^H)$ were concave you would have $f(0)\ge(f(X)+f(-X))/2$. This is violated by any $X$ of full rank, for whi... |
H: Poisson's equation Evan's book
I do not understand the line - ' naive differentiation through the integral near the sigularity is unjustified'. I am wondering if anyone could help explain that?
AI: If you are given a function of the form $$u(x) = \int_{{\mathbf R}^n} g(x,y) \, dy$$ it may be tempting to calculate... |
H: Prove that this continuous function takes the value $0$ for every $x$
Let $f : [0,1] \to \mathbb{R}$ be continuous, with $f(0) = f(1) = 0$. Suppose that for every $x ∈ (0,1)$ there exists $δ > 0$ such that both $x + δ$ and $x − δ$ belong to $(0,1)$ and $f(x) = (f(x −δ)+f(x+δ))/2$. Prove that $f(x) = 0$ for every $... |
H: Are finitely generated modules over a commutative ring always a direct sum of cyclic submodules?
Let's first motivate my question by looking at a finitely generated $k$-algebra $A$ over a field $k$.
Then $A$ in general does not have the form $k[a_1,a_2,\ldots,a_n]$ where $\{a_1,a_2,\ldots,a_n\}$ is a generating set... |
H: Uniform Boundedness and the Arzela-Ascoli Theorem in a Riemannian Manifold
Let $(M, g)$ be a complete Riemannian Manifold and let $(x_n)$ be a sequence of curves in $\Omega = \{x \in C^1([0, 1], \ M): \ x(0) = p, \quad x(1) = q, \quad \dot{x}(0) = v, \quad \dot{x}(1) = w \}$ such that $g(\dot{x}_n, \dot{x}_n)$ is u... |
H: Convergence Range of $\sum\limits_{n=1}^{\infty} \frac{x^{2n-1}}{2n-1}$
$$\sum\limits_{n=1}^{\infty} \frac{x^{2n-1}}{2n-1}$$
By the ratio test we need $\lim_{n \to \infty} | \frac{a_{n+1}}{a_n}| < 1 \rightarrow\lim_{n \to \infty} x^2 < 1$. Hence we get the range $-1 < x < 1$.
My question is if we sub $x = -1$ we ge... |
H: $\text{rank}(A - I) = 0$
If $A$ is a square matrix with real entries, does $\text{rank}(A-I)=0$ tell me that $A=I$? Since the zero matrix is the only matrix of rank zero? ($I$ is the corresponding identity matrix).
AI: Yes that's correct, indeed we have that $\forall x$, $x$ is in the null space of $(A-I)$ that is
... |
H: General real solution of a system of differential equations
Learning for an extra resit of a university exam I was trying to find my mistakes in the resit. However, I can't come to the given solution for the following question(the 4th answer is mine and the 3rd should be correct):
The question
When I calculated it ... |
H: An approximate solution of a second-order differential equation
In the book, Mathematical Methods for Students of Physics and Related Fields, Second Edition by Sadri Hassani, Page 667, the author has stated that, for the following differential equation
$y''(x) - x^2 y(x) \approx 0$,
where $x \to \infty$, one can ea... |
H: Stationary solution to a inhomogeneous differential equation.
Use the Transfer function to determine the stationary solution.
Given:
$$y'''(t) +4y''(t)+9y'+10y(t) = 2e^{-3t}\cos(t)$$
I have determined the transfer function to
$$H(s)=\frac{2}{s^{3}+4s^{2}+9s+10}$$
I have attempted to use the theory below to determin... |
H: Show that $S \leq \sup _{x \geq(1+\varepsilon) M_{n}} x^{-1 / r} \cdot x^{1 / r-1} \sum_{k=1}^{n} E|X_{k}|^{r} I(|X_{k}|>x^{1 / r}) $
Let $1<r<2$ and let $\left\{X_{n}, n \geq 1\right\}$ be a sequence of pairwise independent random tariables
with $E X_{n}=0$ and $E\left|X_{n}\right|^{r}<\infty$ for all $n \geq 1 .$... |
H: Convexity bound in Lieb and Loss.
I'm currently reading through Lieb and Loss's Analysis text. At the end of the proof of theorem 1.9 the authors prove the inequality
$$ \left( |a|+|b|\right)^{p}\leq(1-\lambda)^{1-p}|a|^{p}+\lambda^{1-p}|b|^{p} $$
where $a,b\in \mathbb{C} , p>1$ and $ 0<\lambda<1$. They cite the co... |
H: Find a limit of radicals
How to find a limit where $x\to\infty$ of the following expression? What is the steps?
My first guess is to apply this rule to numerator: $(a+b)(a-b) = a^2 - b^2$
AI: No, that won't help. As $x \to \infty$, you simply look at the dominating powers within each sum. For example, in the num... |
H: Solutions to the equation $xk = x^k$
The equation, $xk = x^k$ (where $x$ and $k$ are both integers).
Are there any solutions other than $\{ (1,1), (2,2) \}$ ?
AI: Well, $x^k=kx$ implies $x(x^{k-1}-k)=0$. Thus, $x=0$ works for all $k>0$ and $k=1$ works for all $x$. Aside from that, there are only solutions when $k... |
H: Prove that if f g and 2 are bounded, then f is also bounded
Let $f$ and $g$ be two functions from $\Bbb{R}$ to $\Bbb{R}$.
Prove that if $f-g$ and $2fg$ are bounded, then $f$ is also bounded.
So from what I understand is that $|f| \le M$ and $|g| \le N$. I know I can get rid of the absolute values by squaring both... |
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