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H: Integral and series convergence intuition
I have this problem I ran into during my studies to the upcoming exam: I don't feel I have the intuition of whether a series or an integral converges or not. What are the things I should look for when looking at a closed form expression, and thinking whether it converges or... |
H: Restrictions of automorphisms to elementary substructures
Suppose that I have structures $M \preceq M'$ (in some first-order language). I have a set $A$, with $M \subseteq A \subseteq M'$, and an automorphism $f$ of $M'$. Is it is always possible to find an $M''$, with $M \preceq M'' \preceq M'$, and $A \subseteq... |
H: Determining the structure of the $\mathbb{Z}$-module $\mathbb{Z}^3/K$, with $K=\langle (2,1,-3),(1,-1,2)\rangle$
Well, this is the exercise:
Determine the structure of $\mathbb{Z}^{(3)}/K$ where $K$ is generated by $f_1=(2,1,-3)$, $f_2=(1,-1,2)$.
Looking at the proof of the fundamental structure theorem for fini... |
H: Closure Operator and Set Operations
In Engelking, General Topology stand the following exercise:
Show that for any sequence $A_1, A_2, \ldots$ of subsets of a topological space we have
$$
\overline{\bigcup_{i=1}^{\infty} A_i} = \bigcup_{i=1}^{\infty} \overline{A_i} \cup \bigcap_{i=1}^{\infty} \overline{\bigcup_{j=... |
H: $\sum \limits_{n=1}^{\infty}{a_n^2}$ converges $\implies \sum \limits_{n=1}^{\infty}{\frac{a_n}{n}}$
I need some help to solve the following problem:
*Show that if $\sum \limits_{n=1}^{\infty}{a_n^2}$ converges, then $\sum \limits_{n=1}^{\infty}{\dfrac{a_n}{n}}$ converges.
I tried to solve it by using the Ratio Tes... |
H: Uniform convergence of $\sin\left ( {\frac{1}{n^{3}x}} \right )$
I ran into this question and im not sure how to solve it:
Check uniform convergence of:
$$f_{n}(x)=\sin\left ( {\frac{1}{n^{3}x}} \right )\quad \;x\in (0,1]$$
I tried finding the supremum of the function and then take the limit as $n \to\infty$. I got... |
H: Laplace transform of sin(at)
Given $f(t)= \sin (at)$ I want to calculate the Laplace transform of $f(t)$.
I have determined by using integration by parts twice, that the answer should be $$F(s)= \frac{a}{s^2+a^2}$$
Now I want to recalculate it by using just that $$\sin (at)= \frac{1}{2i} \left( {e^{ati}-e^{-ati}}\r... |
H: The principle of duality for sets
The Wikipedia article on the algebra of sets briefly mentions the following:
These are examples of an extremely important and powerful property of set algebra, namely, the principle of duality for sets, which asserts that for any true statement about sets, the dual statement obtai... |
H: How to prove that normal matrix with property $A^2=A$ is Hermitian?
I am given a matrix $A\in M(n\times n, \mathbb{C})$ normal (in matrix form $AA^*=A^*A$) and $A^2=A$. The task is to prove that the matrix is Hermitian.
But when I try something like $A^*=\,\,...$ , then I can't reach $A$, because I can't "get rid o... |
H: I do not understand this integral,please help...
$$\int_0^{\infty} P(y > z) \, dz = \int_0^{\infty} \int_z^{\infty} h(y) \, dy \, dz = \int_0^{\infty} \int_0^y \, dz \, h(y) \, dy$$
Why do we have the last equality? I used Fubini and derived the following:
$$ \int_z^{\infty} \int_0^{\infty} h(y) \, dz \, dy.$$
Not ... |
H: Can you solve this problem with functions?
We are given $f(x)=ax^2+2x+b$, a is not $0$, $Df=R$ and $f\circ g=g\circ f$ where $g(x)=x$ has only solution $x_0$. Then we have to show that $ab\leq 1/4$.
AI: If $x_0$ is a solution for $g(x)=x$, then we also have
$$g(f(x_0))=f(g(x_0))=f(x_0)$$
so $f(x_0)$ is also a solu... |
H: Can one apply the classifying space functor $B$ more than once?
For a topological monoid $M$, the classifying space $BM$ is at least a pointed topological space as far as I know.
From where to where is the construction $B$ a functor actually? Can I plug in an $A_\infty$ space $M$ or even a $H$-space $M$? What do I... |
H: How do I create a sigmoid-esque function with the following properties?
For a range of $x$ values between $A$ and $B$ I would like $f(x) \rightarrow x$. For values less than $A$ I would like $f(x)$ to exhibit a sigmoid-esque convergence to $A'$ where $A'$ is $A - \delta$ for some small $\delta$. Similarly, for ... |
H: Which axioms of ZFC or PA are known to not be derivable from the others?
Which, if any, axioms of ZFC are known to not be derivable from the other axioms?
Which, if any, axioms of PA are known to not be derivable from the other axioms?
AI: There are several interesting issues here.
The first is that there are diff... |
H: Lebesgue Measure of a k-cell
Working through Rudin's RCA construction (Theorem 2.20, p. 53) of the Lebesgue measure using the Riesz Representation Theorem. Rudin constructs a linear functional $\Lambda$ on $\operatorname{C}_c(\mathbb{R}^k)$ such that
$$\Lambda f := \lim\limits_{n \to \infty} 2^{-nk} \sum\limits_{x ... |
H: local diffeomorphism on $\mathbb{R}$ and on manifolds.
I find the proof of diffoemorphism in Guillemin & Pallock's Differential Topology 1.3.3 is more or less independent of the fact that the manifold happen to be $\mathbb{R}$, and therefore are the same.
Then I am asking if my two proofs (primarily the latter one)... |
H: domain of square root
What is the domain and range of square $\sqrt{3-t} - \sqrt{2+t}$? I consider the domain the two separate domain of each square root.
My domain is $[-2,3]$. Is it right? Are there methods on how to figure out the domain and range in this kind of problem?
AI: You are right about the domain. As t... |
H: Examples of types of mathematical models
I am a student currently doing a course on modelling and simulation. I came across the classifications of mathematical models and studied that they can classified as static or dynamic, deterministic or stochastic, and as discrete or continuous. This means any mathematical ... |
H: Where to find Rudin's references.
Principles of mathematical analysis(Rudin) comes with various references to proves and results that have appeared in several magazines such as A.m.s and Monthly Math, but I have entered these pages and I actually don't know how to look for the articles that the book quotes.
AI: If ... |
H: Derivative of the maximum of two random variables
For any two real numbers $a$ and $b$ and any two random variables (with no mass points in their distributions) $x$ and $y$, why is it that the derivative of $E[\max\{a+x,b+y\}]$ with respect to $a$ is $\Pr(a+x>b+y)$?
AI: Warning: The statement is not true as you hav... |
H: Convergence of $\int_0^\infty x \sin e^x \, dx$
I'm trying to demonstrate the convergence/divergence of a couple of integrals. They are: $\int_0^\infty x \sin e^x\,dx$$\int_0^{\pi/2} \sin(\sec x)\,dx$
There was a previous exercise similar to the first one, $\int_0^\infty e^x \sin e^x \, dx$. I concluded that this ... |
H: $\overline{A}\cap B\neq\emptyset$ or $\overline{B}\cap A\neq\emptyset$ implies $\operatorname{dist}(A,B)=0$
$\overline{A}\cap B\neq\emptyset$ or $\overline{B}\cap A\neq\emptyset$ implies $\operatorname{dist}(A,B)=0$. I have tried to prove this but was unable to find a decent method. Any help will be appreciated
AI:... |
H: Solving Differential equation with partial fraction decomposition
I am a little rusty with some calculus and need some help with the follow equation:
\begin{equation}
\int\dfrac{f'(x)}{f(x)}dx = \int\dfrac{1}{-x+x^2y}dx
\end{equation}
Where $y$ is a constant. My idea is to use some kind of $U$ substitution as I kno... |
H: Verifying statements for non-zero matrix
Let $N$ be non-zero $3 \times 3$ matrix with the property $N^2=0$. Which of the following is true?
(A) $N$ is not similar to a diagonal matrix.
(B) $N$ is similar to diagonal matrix.
(C) $N$ has one non-zero eigenvector.
(D) $N$ has three linearly independent eigenvectors.
A... |
H: Combinatorially showing $\lim_{n\to \infty}{\frac{2n\choose n}{4^n}}=0$
I am trying to show that $\lim_{n\to \infty}{\frac{2n\choose n}{4^n}}=0$. I found that using stirling's approximation, I can get:
$$
\lim_{n\to \infty}{\frac{2n\choose n}{4^n}}=
\lim_{n\to \infty}{\frac{(2n)!}{n!^2*2^{2n}}}=
\lim_{n\to \infty}{... |
H: I want to prove an inequality between limits.
I want to prove that if $f:\mathbb R \rightarrow \mathbb R$ is continuous and $x_n$ a bounded sequence, then $\liminf_{n\rightarrow \infty}f(x_n) < f(\liminf_{n\rightarrow \infty}x_n)$.
Suppose $\liminf_{n\rightarrow \infty}x_n=a$
$$f\text{ continuous }\Rightarrow \fora... |
H: Help with graphing a piecewise function
What would be the graph and domain of this function?
My domain is $(- \infty, \infty)$. I am stuck on graphing $-2x$.
$$g(x)=\begin{cases}
x+9 & \text{if }x<-3,\\
-2x & \text{if }|x|\leq 3,\\
-6 & \text{if }x>3.
\end{cases}$$
AI: $-2x$ is a straight line. Your domain for this... |
H: Prove that $(x+1)^{\frac{1}{x+1}}+x^{-\frac{1}{x}}>2$ for $x > 0$
Let $x>0$. Show that
$$(x+1)^{\frac{1}{x+1}}+x^{-\frac{1}{x}}>2.$$
Do you have any nice method?
My idea $F(x)=(x+1)^{\frac{1}{x+1}}+x^{-\frac{1}{x}}$
then we hvae $F'(x)=\cdots$ But it's ugly.
can you have nice methods? Thank you
by this I have see... |
H: If an arithmetic progression starts with 4, what is the common difference if the sum of the first 12 terms is twice the sum of the first 8 terms?
An arithmetic progression (AP) has 4 as its first term. What is the common difference if the sum of the first 12 terms is 2 times the sum of the first 8 terms?
AI: Let $d... |
H: Are the strong limit cardinals precisely those of the form $\beth_\lambda$, where $\lambda$ is a limit ordinal or $0$?
I know that $\aleph_\lambda$ is a weak limit cardinal iff $\lambda$ is a limit ordinal or $0$. In the absence of GCH, can we similarly prove that $\kappa$ is a strong limit cardinal iff $\kappa=\be... |
H: gauss map takes geodesics to geodesics
Let $S$ be a regular surface, and let's consider $\gamma: I \to S$ be a geodesic. Let $ N: S \to S^2 $ be the gauss map. Then $ \beta(s) = N(\alpha(s))$ is a curve $\beta : I \to S^2$ (where $S^2$ denotes the unit sphere). I want to prove that $\beta$ it's also a geodesic, and... |
H: Given a day of the week and the day of the month, what is the range of time within which it will uniquely specify a single date?
In other words, given "2 Tues", (e.g. today, 2 July 2013), for how long must I wait until it is Tuesday on a 2nd of the month again?
How does this interval change for each week? Is it co... |
H: Why do we care about specifying events in a probability space?
Why aren't probability spaces just defined as $(\Omega, p)$ pairs with $\Omega$ as the sample space, $\sum_{\omega \in \Omega}p(\omega) = 1$, and for a subset $A \subseteq \Omega$, $\Pr(A) := \sum_{\omega \in A}p(\omega)$?
Said another way, why aren't a... |
H: Probability (arranging around a circle)
What is the probability that when arranging n people around a circle, two people with the same birthday (assume no leap years) will be adjacent to each other?
AI: Your probability model is not quite clear. If you have $n$ given people and arrange them randomly, the probabilit... |
H: Is there a transitive set that is non empty and doesn't contain the empty set?
Sorry for being so naive and because this is maybe a silly question but all the examples I can think of contains the empty set, and it's not clear to me whether this makes sense at least intuitively.
For example, let's supose that we ha... |
H: Proofs: the running in the sun conjecture (I made it up - explained below). Is it true and how can it be proven?
The running in the sun conjecture (I made it up). Is it true and how can it be proven?
This is just for fun. It might actually turn out to be easy - or it might be hard - I'm not sure. I'm an engineer, n... |
H: How to integrate these integrals
$$\int^{\frac {\pi}2}_0 \frac {dx}{1+ \cos x}$$
$$\int^{\frac {\pi}2}_0 \frac {dx}{1+ \sin x}$$
It seems that substitutions make things worse:
$$\int \frac {dx}{1+ \cos x} ; t = 1 + \cos x; dt = -\sin x dx ; \sin x = \sqrt{1 - \cos^2 x} = \sqrt{1 - (t-1)^2} $$
$$ \Rightarrow
\int \f... |
H: Which of these statements regarding metric spaces are true?
The following are a few statements in various metric spaces mcqs that I couldn't figure if they are true or false. Please offer some help to get answer them
Let $(X,d)$ be a metric space
1) If $ A,B \subseteq X $ and $ A,B $ are bounded
$\mathrm{dist}(A,B... |
H: How to generate sequence like this?
Can you tell what algorithm can generate sequence $x_1, x_2, x_3, x_4, ...$ satisfying:
$x_n$ is real, and always $0<x_n<1$.
Every change between $x_n$ and $x_{n+1}$, such as increase or decrease and their amount, can be controlled by a variable with values, say, $+ε$ or $-ε$ wh... |
H: Why does $\oint\mathbf{E}\cdot{d}\boldsymbol\ell=0$ imply $\nabla\times\mathbf{E}=\mathbf{0}$?
I am looking at Griffith's Electrodynamics textbook and on page 76 he is discussing the curl of electric field in electrostatics. He claims that since
$$\oint_C\mathbf{E}\cdot{d}\boldsymbol\ell=0$$
then
$$\nabla\times\mat... |
H: Ways of enumerating a countable set
My questions are about enumerating countable sets.
Say we have a countable set $X$.
How many ways are there of enumerating X?
It seems to me, just based on the examples i've seen in maths so far, that most of the time, it doesn't matter too much how you enumerate a set if it's... |
H: Show that $\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx =-\frac{\pi^2 \sqrt{2}}{16}$
I could prove it using the residues but I'm interested to have it in a different way (for example using Gamma/Beta or any other functions) to show that
$$
\int_{0}^{\infty}\frac{\ln\left(x\right)}{x^{4} + 1}\,{\rm d}x
=-\frac{\,\pi^{... |
H: A problem on countabiliy and families of sets
Let $X$ be a non-empty set and $(A_{\lambda})_{\lambda\in \Delta } $ be a family of subsets of $X$.
a) $ \Delta $ is countable and $(A_{\lambda}$ is countable for each
$\lambda\in \Delta) \implies \prod_{\lambda\in \Delta} A_{\lambda} $
is countable
b) $ \Delta $... |
H: Did I solve this System of differential equations right?
My Problem is this given System of differential equations.
$$y_{1}^{\prime}=5y_{1}+2y_{2} \\
y_{2}^{\prime}=-2y_{1}+y_{2}$$
I am looking for the solution. According to one of my earlier Questions, I tried the method on my own. Now i fear the solution could be... |
H: Trace, Kronecker and vec relations
I'm reading a paper and got stuck on one of the simplifications that was done without any elaboration. I've taken a course on Linear Algebra, but this is a little out of reach for me...
The simplification done is this (note that $T$ is not the transpose but a constant) where it's ... |
H: Determine the sets on which $f$ is continuous and discontinuous.
Let $f:\mathbb R\to\mathbb R$ be defined by $$f(x):=\begin{cases} x, &\text{if } x\in\mathbb Q\;;\\ -x, &\text{if } x\in\mathbb R\setminus\mathbb Q.\end{cases}$$
Determine the sets on which $f$ is continuous and discontinuous. Prove your answer.
I k... |
H: Characterization of positive elements in unital C*-algebra
Let $\mathcal{A}$ be a unital C*-algebra (not necessarily commutative) and let $A^*=A\in \mathcal{A}$ be a self-adjoint element with $\vert\vert A \vert\vert \leq 2$?
I want to show that $\vert\vert \mathbb{1}-A \vert\vert \leq 1 \Leftrightarrow \sigma(A)\s... |
H: Question on different definitions of upper (hemi)semicontinuity for set-valued maps
In this thesis(page $8-10$), it is asserted, two definitions are equivalent, if the set-valued map $f$ maps to a compact space.
Definition $1$:$f : X \to 2^Y$ is upper semicontinuous if:
$f(x)$ is compact for all $x \in X$, and
f... |
H: How can we prove $\pi =1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\cdots\,$?
I saw a beautiful result in Wikipedia which was proved by Euler; but I do not know how it can be proved:
$$\pi =1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}... |
H: Unitary linear map
So my professor gave me this question :
Let $V$ be a vector space. Let $e_{1},\ldots,e_{n}$ be an orthonormal basis for $V$, $T\colon V\to V$ be a linear map and $\forall 1\leq i \leq n$ $\|T(e_{i})\|=1$. Is $T$ unitary ?
So I know that there is a counterexample. But I have a proof and I would... |
H: Derivative of tan(x) with product and chain rules instead of quotient rule
So I usually just use the product and chain rules for quotient functions, because I can never remember which product to substract from which in the numerator. But somehow I'm doing it wrong for $\tan(x)$. Say $a$ and $b$ are functions of $x$... |
H: find the radius of convergence of $(1+(-1)^{n}2^{1+n})x^{n}$
I ran into this question and I don't really know how to find the radius of convergence.
the power series is:
$$\sum_{n=0}^\infty(1+(-1)^{n}2^{1+n})x^{n}$$
Thanks in advance.
AI: HINT:
$$(1+(-1)^n2^{1+n})x^n=x^n+2(-2x)^n$$
Using this, $\sum_{0\le n<\infty}... |
H: exercise on uniform integrability
I cannot figure out the following exercise:
Let $F$ be the family of functions $f$ on $[0,1]$, each of which is (Lebesgue) integrable over $[0,1]$ and has $\int_a^b|f|\le b-a$ for all $[a,b]\subseteq[0,1]$. Is $F$ uniformly integrable over $[0,1]$?
AI: The answer is 'yes'.
To prov... |
H: Expressing polynomial roots expression in terms of coefficients
This is my first question on MSE. Apologies in advance for any textual or LaTeX errors.
I'm stuck with this problem:
Given $x^3 - bx^2 + cx - d = 0$ has roots $\alpha$, $\beta$, $\gamma$, find an expression in terms of $b$, $c$ and $d$ for:
(i) $\a... |
H: If a sequence of natural numbers satisfies $\gcd(a_{i+1},a_{i})>a_{i-1}$, then $a_{n}>2^n$
Given a sequence $\{a_{n}\}$ in $\mathbb{N}$ such that $\gcd(a_{i+1},a_{i})>a_{i-1},$ for any $i\ge 2$, show that $a_{n}>2^{n-1}$.
Thank you everyone, my friend asked me about this problem, and I feel this problem is really... |
H: Sufficient condition for self-adjoint subset of bounded linear operators on a Hilbert space being irreducible
Let $H$ be a Hilbert space and denote as $B(H)$ the bounded linear operators on $H$. Let $M$ be a subset of $B(H)$, s.t. for $A \in M$, also $A^* \in M$.
How can one show that if the commutant has the form... |
H: How to check if a normal vector of a plane points towards or away from a certain point
The problem is as follows: I have a plane defined by three points. I also have a fourth point. I now want to calculate the normalvector of the plane defined by the first three points, but i want the Normalvector to point towards ... |
H: Statements regarding relations in R
Suppose $\rho$ is a relation on $R$. I want to verify whether the following statements are true. Looks simple but proving them seems to be difficult for me.
$\rho\circ\rho$ is a subset of $\rho$
$\rho\circ\rho=\rho$ implies $\rho=i_{D(\rho)}$ ($D(\rho)$ being the domain of $\rho... |
H: Prove that graph without independent set is not $4$-colorable
Let $G$ be an undirected graph with $n$ vertices so that $n\geq35$ and there isn't an independent set (IS) of size $4$ in $G$.
(If $S$ is a group of vertices from graph $G$, $S$ is an independent set if there isn't an edge in graph $G$ between each two v... |
H: Harmonic Series is $\theta(\ln n)$
How does one prove that $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = H_n = \theta(\ln n)$ by using Riemann Sums? I have seen in the MIT OCW 6.042 that if $f$ is continuous and increasing then $$f(1) + \int_1^n f(x) \, dx < f(1) + f(2) + \cdots + f(n) < f(n) + \int_1^n ... |
H: Lang $SL_2$: fin-dim irreducible subspace for abelian group has dim < 2
Lang $SL_2(\mathbb R)$ p. 24, Theorem 2 : Let $\pi$ be an irreducible representation of G on a Banach space H. Let $H_n$ be the subspace of vectors v s.t. $$\pi(r(\theta))v = e^{in\theta}v.$$ If dim $H_n$ is finite, then dim $H_n$ = 0 or 1. Th... |
H: Confused about harmonic series and Euler product
So Euler argued that
$$1 + \frac{1}{2} + \frac{1}{3} + \frac {1}{4} + \cdots = \frac {2 \cdot 3 \cdot 5 \cdot 7 \cdots} {1 \cdot 2 \cdot 4 \cdot 6 \cdots} $$
which you can rearrange to
$$ \left( \frac {1 \cdot 2 \cdot 4 \cdot 6 \cdots} {2 \cdot 3 \cdot 5 \cdot 7 \cdo... |
H: Conditional or absolute convergence of an integral $\int_{0}^{\infty}\frac{\sin x}{1+x^{2}}dx$
I ran into a few problems where i had to check absolute\conditional convergence of a few integrals.
I'm sure theres a method to check this, i just can't find the trick.
I wan't help with one of the problems so that i can ... |
H: Asymptotic solutions for inequalities
How do I determine the order (big o) of $\omega$ in $e^{-\omega/\epsilon}\leq10^{-9}$ and $e^{-\omega/\epsilon}\leq\epsilon$, where $\epsilon$ is a small parameter.
AI: In the first case, taking logs yields
$$-\omega/\epsilon \leq -9 \ln(10)\\
\omega \geq 9 \epsilon \ln(10)\\
\... |
H: Forcing and antichains
What would be a good way to show:
If $p \Vdash (\exists\alpha) \phi(\alpha)$, then there is an antichain $A$ maximal below $p$, and a set of ordinal $\{\gamma_{q} | q \in A\}$ s.t. $(\forall q \in A$) $q \Vdash \phi(\gamma_{q})$
AI: Use the mixing lemma to conclude that there exists a name $... |
H: What does $f: A \times A \to A$ mean?
What does $f: A \times A \to A$ mean?
Can you give some examples please?
AI: Normally this means that $f$ is a function with domain $A\times A$ and codomain $A$. One way to represent a function is with ordered pairs. For this example, it would be a set of ordered pairs $((a,b... |
H: A problem on an open sets in $\mathbb R$ and expressing in pairwise disjoint intervals
Let $G$ be an open subset of $\Bbb R$. Define the relation $x \sim y $ on $G$ such that $x \sim y$ iff there exists an open interval $I$ such that $x,y \in I$ and $I \subseteq G $
Verify that '$\sim$' is an equivalence relation ... |
H: Quesion on a detail of the proof of Schauder-Tychonoff fixed point theorem
I'm trying to understand the proof of Schauder-Tychonoff fixed point theorem on page $96-97$, in Fixed Point Theory and Applications, Ravi P. Agarwal,Maria Meehan,Donal O'Regan, which can be found here in googlebooks.
Let $F: C \to E$ be a c... |
H: If $a_n$ is divergent, then $f(a_n)$ is divergent
Suppose $f(x)$ is strictly increasing on $\mathbb R$ and $a_n$ is divergent. Then sequence $f(a_n)$ is also divergent.
How to prove that this is false? Actually, a counterexample will suffice.
AI: If $a_n$ is increasing, then $f(a_n)$ is increasing thus convergent i... |
H: Hilbert space on line bundle
Suppose that $L$ is a complex line bundle on a manifold $M$ with measure $\mu$, How can we prove, $L^2(M,L,\mu)$ is Hilbert space?
AI: You must be leaving something out of the question here - you need an inner product in order to say that something is a Hilbert space.
Probably the situa... |
H: Translation-invariant operator
Let $T$ be a translation invariant bounded linear operator $L^p(\mathbb{R}^d)\rightarrow L^q(\mathbb{R}^d)$, i.e. $T(\tau_c f)=T f$ where $\tau_cf(x)=f(x+c)$ for $c\in\mathbb{R}^d$. Then I have read in this marvellous post by Tao that necessarily $q\ge p$ ("the larger exponents are al... |
H: Jensen inequality
Does Jensen inequality, which is $\mathbb{E}(g(x)) \geq g(\mathbb{E}X)$ if $g$ is convex, assume that $\mathbb{E}X$ (expected value of random variable $X$) must belong to $R(X)$ (range of random variable $X$)?
I THANK YOU.
AI: The answer is negative: consider the easy example $X:\{\omega_1,\omega_... |
H: Prove by using the integral test question
Prove by using the integral test the correctness of these results:
$$\sum_{n=1}^{\infty} \frac{1}{n^p} \rightarrow P>1 : \text{Convergent} , P\leq1 : \text{Diverging}$$
$$\sum_{n=1}^{\infty} \frac{1}{n\ln^q(n)} \rightarrow q>1 : \text{Convergent}, q\leq1 : \text{Diverging}... |
H: Matrix $I + 2 A A^T$ is nonsingular for any A
Suppose A is $m\times n$ matrix with real entries. Could you prove that $\det (I + 2 A A^T) \neq 0$
AI: Suppose $S$ is symmetric and positive semi-definite. Then $S=U\Lambda U^T$ for some orthogonal $U$ and diagonal $\Lambda$. Then $I+S = I+U \Lambda U^T= U (I+\Lambda) ... |
H: Name for grid system
Is there a name for a type of grid you might find in Battleship? Where coordinates don't relate to points on a grid but rather the squares themselves?
AI: "Grid" is as good a name as any: See Regular Grid in Wikipedia: In particular, see the "related" grid: the Cartesian Grid:
"A Cartesian gr... |
H: Is the multiplication of two complex numbers with $|z|=1$ a complex number with modulus 1?
If we have two complex numbers $a, b \in \mathbb{C}$ such that $|a|=1$ and $|b|=1$ is $|a\cdot b|=1$ as well?
I am trying to determine if the set $\left(\{z\in\mathbb{C}:|z|=1\},\cdot\right)$ is a group.
I am not sure if it... |
H: What is the definition of a flat morphism?
When we say that a morphism $f: E \rightarrow M $
between two algebraic varieties (over $\mathbb{C}$)
is a flat morphism, what does it mean? Does it mean that that the
"dimension" of every fiber $f^{-1}(x)$ is the same for all $x$?
Or do we also have to check some comp... |
H: graph of complicated equation
Graph of the equation $(x+y) (x^2 + y^2 -1) = 0$ is just the line $y=-x$ and the circle $x^2 + y^2 = 1$.
Is it generally true that the graph of $f(x,y) \cdot g(x,y) = 0$ may be drawn as union of graphs of $f(x,y) =0 $ and $g(x,y) = 0$?
AI: Note that the equation you've written is an eq... |
H: Steinhausen set in $[0,1]$
Does every Steinhausen set have positive lebesgue measure? Steinhausen set is a set $A\in [0,1]:0\in \operatorname{int}(A-A),\mu(A)\geq 0$.
AI: No. The Cantor set has measure zero, yet $0$ is an interior point of its difference set (in fact, the difference set is $[-1,1]$). A proof of th... |
H: how to solve this partial differential equation
while in its equilibrium position a uniform string stretched between the points (0,0) and (ℓ,0) (hint cn=0 since equilibruim)
AI: I take it you mean the wave equation should be solved here, which it more or less has. Your job is to match the boundary condition given.... |
H: Do equivalent norms preserve dual spaces?
Suppose that $X^*$ is the dual space of a normed space $X$. If we renorm the space $X^*$ with a new norm equivalent to the first one, is this new normed space the dual of $X$ as well?
(I think it suffices to prove that a functional $f$ is continuous with a norm1 if and only... |
H: A function/distribution which satisfies an integral equation. (sounds bizzare)
I think I need a function, $f(x)$ such that $\large{\int_{x_1}^{x_2}f(x)\,d{x} = \frac{1}{(x_2-x_1)}}$ $\forall x_2>x_1>0$. Wonder such a function been used or studied by someone, or is it just more than insane to ask for such a function... |
H: Quick question about binary strings
Determine the unambigious expression which generates every string in this set.
The set of all binary strings which contains 001111 as a substring.
I am thinking that the answer should be
{0,1}$^*${001111}{0,1}$^*$
But the answer says that it should be
{0,1}$^*$\{1}$^*$({00}{0}$^... |
H: Transforms to orthogonal basis
Given a function $f(x)$ we can use the Fourier Transform to find $F(w)$, which represents how we can build up the signal by $\sin$/$\cos$ as we find the coefficients, frequencies and phases. (I realise this is very simplified I just wanted to fit it in a sentence)
My question is could... |
H: Convergence to $0$ of Jacobi theta function
I'm trying to prove that a function $$f(y) = \sum_{k=-\infty}^{+\infty}{(-1)^ke^{-k^2y}}$$ is $O(y)$ while $y$ tends to $+0$. I have observed that
$f(y) = \vartheta(0.5,\frac{iy}{\pi})$ where $\vartheta$ is a Jacobi theta function. It seems that these functions are very ... |
H: Finding $\lim\limits_{x\to0}x^2\ln (x)$ without L'Hospital
I am preparing a resit for calculus and I encountered a limit problem.
The problem is the following: $\lim\limits_{x\to0}x^2\ln (x)$
I am not allowed to use L'Hospital.
Please help me, I am stuck for almost an hour now.
AI: Note that we should rather consid... |
H: How and what to teach on a first year elementary number theory course?
In the late 80’s and early 90’s there was the idea of ‘calculus reform’ and some emphasis and syllabus changed. The order of doing things in calculus also changed with the advantage of technology.
Similarly in linear algebra there was a linear ... |
H: Trig substitution $\int x^3 \sqrt{1-x^2} dx$
$$\int x^3 \sqrt{1-x^2} dx$$
$x = \sin \theta $
$dx = \cos \theta d \theta$
$$\int \sin^3 \theta d \theta$$
$$\int (1 - \cos^2 \theta) \sin \theta d \theta$$
$u = \cos \theta$
$du = -\sin\theta d \theta$
$$-\int u^2 du$$
$$\frac{-u^3}{3} $$
$$\frac{\cos^3 \theta}{3}$... |
H: Difference between two definitions of Manifold
I've been studying Differential Geometry on Spivak's Differential Geometry book. Since Spivak just works with notions of metric spaces and analysis, I'm doing fine. The point is that Spivak presents the following definition of a manifold:
A manifold $M$ is a metric sp... |
H: Complexity of all nearest neighbours problem
Given a set $P$ of points $P=\left\{p_1,p_2,\dots p_n\right\}\subset\mathbb{R}^2$
What I want to show is that every correct algorithm which finds for every $p_i$ the nearest neightbour,
i.e. the point $p_j$ such that $dist(p_i,p_j)\le dist(p_i,p_k)$ for every $k\neq i$,... |
H: Proving there is no solutions in diophantine equations
I recently saw a question that I couldn´t answer, so I decided to do a little C code to test if some solutions were possible but I got nothing. The problem is:
If $m,n, p \in \mathbb{Z}^+$ give the number of solutions of
$$4mn-m-n=p^2$$
I couldn't find any answ... |
H: Integration trig substitution $\int \frac{dx}{x\sqrt{x^2 + 16}}$
$$\int \frac{dx}{x\sqrt{x^2 + 16}}$$
With some magic I get down to $$\frac{1}{4} \int\frac{1}{\sin\theta} d\theta$$
Now is where I am lost. How do I do this? I tried integration by parts but it doesn't work.
AI: HINT:
$$\int\frac{d\theta}{\sin\theta}=... |
H: How to check if the given lines are coplanar?
I have two lines:
$$\frac{x-1}{3}=\frac{y+2}{-2}=\frac{z}{1} = L_1$$
$$ \frac{x+1}{4} = \frac{y-3}{1}=\frac{z}{\alpha} =L_2$$
How can I find the value of $\alpha$ for which these two lines lie on the same plane?
Just notice that this is not a homework question. I took i... |
H: how to solve this first-order nonlinear ode
how to solve this differential equation:
$A\cdot(dT(x)/dx)(1873.382+2.2111\cdot T(x))=90457.5-2.149\cdot10^{-10 }-10\cdot T(x)^4$
where A is a constant
Thank you
AI: It looks like your equation takes the form
$$A \frac{dT}{dx} (B + C T) = D - E T^4$$
where the constants a... |
H: Solving an integral of form $ F(x)=\int (2x-1)e^{2x}\ dx $
I have this integral in a worksheet please help me solve it
$$ F(x)=\int (2x-1)e^{2x}\ dx $$
AI: Let $u = (2x - 1)$ and $dv = e^{2x}dx$. Now continue with integration by parts. |
H: When to begin a new paragraph?
When writing a mathematical article, when should someone begin a new paragraph? Is there some specific rule or convention?
And more generally, what rules are there about articles-writing?
AI: Mathematical writing, like all writing, is best when it reads smoothly. A good way to test t... |
H: How do you show that isometry is an equivalence relation among metric spaces?
Ok, to start I am new to metric spaces. I have studied equivalence relations in Algebra, but unfamiliar with the e.r. in metric spaces.
Here is my question:
We say that metric spaces (X,$d_X$) and (Y,$d_Y$) are isometric if there is an is... |
H: Physical meaning of "probability density"
Is there some way of describing the co-domain of probability density functions? Does it relate in some way to something physically meaningful? I was given that question today - and I was at a loss. Density for me, is the co-domain of pdfs - a scalar dimension with values fr... |
H: Differential equation two solutions, how so?
I tried to solve $7x^3y'=4*\sqrt{y}$ with $y(1)=1$ now I thought that Picard Lindelöf would tell me that there is a (at least in a local area for x=1) unique solution unfortunately I found two:
$y(x)=(-\frac{1}{7x^2}+\frac{8}{7})^2, y(x)=(-\frac{1}{7x^2}-\frac{6}{7})^2$... |
H: Show that the set of functions under composition is isomorphic to $S_3$
Show that the set $\{f_1, f_2, f_3, f_4, f_5, f_6\}$ of functions $\mathbb{R}-\{0, 1\}\rightarrow \mathbb{R}-\{0,1\}$ under composition is isomorphic to $S_3$, where $$f_1(x)= x\\ f_2(x) = 1-x\\f_3(x)=1/x\\f_4(x)=1-1/x\\f_5(x)=1/(1-x)\\f_6(x)=x... |
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