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H: Equivalent metrics give the same topology
Definition
If $d_1$ and $d_2$ are distance in $X$ then we say that they are equivalent iff and only if there exist $\alpha,\beta>0$ such that
$$
\alpha\cdot d_1(x,y)\le d_2(x,y)\le\beta\cdot d_1(x,y)
$$
for any $x,y\in X$.
Theorem
Two metric are equivalent if and only if t... |
H: Why doesn't $\sum_{n=0}^\infty2^n=-1$?
Now of course I'm not stranger to the fact that adding finite (and in many cases - infinite) amount of positive numbers always yeilds a positive number, but in many cases, often the finite limit isn't equivalent to the strange nature of infinity. It seems that mathematics tend... |
H: Is this series $\sum_{n=0}^{\infty} 2^{(-1)^n - n} = 3 $ or $ \approx 3$
Is this series $\sum_{n=0}^{\infty} 2^{(-1)^n - n} = 3 $ or $ \approx 3$
Hello all.
In Ross' elementary Analysis Chapter 14 there is an example of the above series.
It simply says it converges.
I have computed that it converges to 3.
The way I... |
H: Convergence of $\sqrt{1-\sqrt{{1}-\sqrt{{1}-\sqrt{{1-\sqrt{{1}}}...}}}}$
Recently, as is evident from many of my recent questions, I have been very interested in nested radicals. Recently I attempted to investigate the following infinite nested radical and arrived at a strange, counter-intuitive result.
$$\sqrt{1-\... |
H: How can I prove that the following are happening: $\ln\Big(1+\frac{1}{x}\Big)=\frac{1}{x}+o\Big(\frac{1}{x}\Big)$?
How can I prove that the following are happening ($x\to\infty$): $\ln\Big(1+\frac{1}{x}\Big)=\frac{1}{x}+o\Big(\frac{1}{x}\Big)$ and $\Big(1+\frac{1}{x}\Big)^{p}=1+\frac{p}{x}+o\Big(\frac{1}{x}\Big)$, ... |
H: Does the Riemannian distance function obey the Leibniz rule?
Let $M$ be a Riemannian manifold, and let $
\alpha,\beta:[0,\delta) \to M$ be two smooth paths satisfying $\alpha(t) \neq \beta(t)$ for every $t$.
Suppose that for every $t$ there is a unique length-minimizing geodesic from $\alpha(t)$ to $\beta(t)$. This... |
H: Is it possible to calculate $x$ and $y$?
I am struggling with this.
If $p=\frac{(x+y)}{2}$ and $q=\frac{y}{x}$ and you know the values for $p$ and $q$, can you calculate what $x$ and $y$ are?
I tried $p=3.5$ and $q=2.5$ as an example. Given this $p$ and $q$, is the only possible solution $x=2$ and $y=5$? Or are the... |
H: dimension of intersection of subspaces, one of which of dimension $n-1$
Let $H$ be a linear subspace of dimension $n-1$ of a linear space $V$ of dimension $n$.
Let $W$ be some subspace of $V$.
Show one of the following holds:
$W \subseteq H$ or
$\dim(W \cap H) = \dim(W) - 1$
This makes a lot of sense to me, as if... |
H: Understanding notation and meaning of uniform convergence of a power series
I am trying to understand correctly the idea of uniform convergence in power series, and in general the notation for series of functions.
When we define a series of functions, we are defining a 'new' object, in which the terms of the series... |
H: If an abelian group has subgroups of orders $m$ and $n$, respectively, then it has a subgroup whose order is $\operatorname{lcm}(m,n)$.
I'm solving Exercise 2.5.26 from Topics in Algebra by Herstein. Could you please verify if my attempt is fine or contains logical mistakes?
Let $G$ be an abelian group and $H,K$ i... |
H: Is there a general form for the relation between $df(x)$ and $dx$?
Let's say that $f(x)$ is a function and $dx$ is a slight change in the input, $x$. It has been used here as standard calculus notation and hence, $dx \rightarrow 0$. $df$, or $df(x)$ is again, by standard calculus notation, the change in the value o... |
H: proof of uniform convergence of sequence of functions
Let be $f_n$ a sequence of functions with $f_n:\mathbb{R}\supset[\alpha, \beta] \to \mathbb{R}$ and $f_n$ continuous and bounded for all $n\in \mathbb{N}$. Further, $f_n$ converges point-wisely: $\lim\limits_{n\to\infty}f_n(x) = f(x)$, where $f(x)$ is also cont... |
H: Finding the Constant Term of a Polynomial
How the coefficient of $t^{m+q}$ in the product $g'h'$ is
$$g_0h_{m+q}+g_1h_{m+q-1}+ \cdots +g_mh_q+\cdots g_{m+q}h_0 ?$$
For example, if $g_2=x^2+1, h_3=x^3+x^2+1$, then what is the value of $g_0h_{m+q}$? Plz explain reason in general for other terms.
The source of the pro... |
H: Why doesn’t this work: proof for the lower bound of the size $\sigma(k)$ of the smallest tournament with the Schütte property $S_k$?
Definitions:
Tournament and Dominating set
In a tournament $T$, for $u,v \in V(T)$, an edge “$u \rightarrow v$” means $u$ defeats $v$.
We say $T$ has the Schütte property $S_{k}$ if... |
H: Group actions, faithful, transitive
I have some questions on group actions, which occured from this problem:
We have the group $G=\operatorname{GL}_m(\mathbb{R})\times \operatorname{GL}_n(\mathbb{R})$ and the set $X=\operatorname{Mat}_{m\times n}(\mathbb{R})$
The group action is given as follows
$G\times X\to X$,... |
H: Spiral equation
Considering concentric arcs, of equal developed length, whose start point is aligned:
I am looking for the equation of the spiral passing through the end points.
Some help to solve this problem will be welcome!
Edit: The result
AI: In polar coordinates, every arc starts at $\theta=0$ and ends at ... |
H: Given $\frac{z_1}{2z_2}+\frac{2z_2}{z_1} = i$ and $0, z_1, z_2$ form two triangles with $A, B$ the least angles of each. Find $\cot A +\cot B$
Question: If $z_1$ and $z_2$ are two complex numbers satisfying
$\frac{z_1}{2z_2}+\frac{2z_2}{z_1} = i$
and $0, z_1, z_2$ form two non-similar triangles. $A, B$ are the leas... |
H: Notation for "and"
Consider the real numbers $a$ and $b$. I would like to express in math notation, that $c = 2$ if $a > 0$ and $b > 0$. Can I use the wedge symbol here like
$c = 2 \quad \text{ if }a>0 \wedge b>0$
or is this wrong usage of the wedge symbol? Is there a better way to express this?
AI: It depends on y... |
H: Proving that, given a surjective linear map, a set is a generator of a vector space
I know how to prove that, given $L: V \rightarrow W$ to be an injective linear map and $\{v_1, v_2, ..., v_n\}$ to be a linearly independent set in $V$, $\{L(v_1), L(v_2),...,L(v_n)\}$ is a linearly independent set in $W$
Proof:
We'... |
H: Nature of the series $\sum_{n=1}^\infty u_n$ where $u_{n+1} = \int_{0}^{u_{n}} \cos(x)^{n}dx$
Can you find the nature of the series $\sum_{n=1}^\infty$ given by $u_{n+1} = \int_{0}^{u_{n}} \cos^n(x)dx$?
You can show $u_{n}$ is convergent and the limit is 0. However it seems more difficult to find an equivalent to $... |
H: If a complex analytic function is injective on a dense subset of an open connected domain, is it injective on the whole domain?
Let $D$ be an open connected domain inside $\mathbb C$ and let $g: D\rightarrow \mathbb C$ be an analytic function. Suppose there is a dense subset of $D$ on which $g$ is injective. Does $... |
H: Is $R[x]/(g) \cong R[x]^{n-1}$ as groups, where $g$ is monic of degree $n$?
I am trying a problem that asks if it is true that $R[x]/(g) \cong R[x]^{(n-1)}$ (as groups under addition), when $g$ is a monic polynomial of degree $n$, and where $(g)$ is the ideal generated by $g$. Note, $R[x]^{(n)} = \{f \in R[x] : \de... |
H: Proof of $\exists!A \ \forall B \ (A \cup B = B) $
This is example 3.6.2 of How to Prove It by Velleman, and the proof he gives is the following:
proof
Existence: Clearly $∀ B ( \varnothing ∪ B = B )$, so ∅ has the required property. Uniqueness: Suppose $∀ B ( C ∪ B = B )$ and $∀ B ( D ∪ B = B )$. Applying the firs... |
H: Does ${f(x)=\ln(e^{x^2})}$ reduce to ${x^2\ln(e)}$ or ${2x\ln(e)}$?
I'm confused with the expression ${f(x) = \ln(e^{x^2})}$.I know the rule ${\log_a(x^p) = p\log_a(x)}$. So does the given expression reduce to ${x^2\ln(e)}$ or ${2x\ln(e)}$?
AI: It depends whether by ${e^{x^2}}$ you mean
$${e^{(x^2)}}$$
or
$${(e^{... |
H: Let $G$ be a finite nilpotent group and $G'$ its commutator subgroup. Show that if $G/G'$ is cyclic then $G$ is cyclic.
So I thought the cleanest way to do this was to simply prove $G' = 1$ since if $G$ is cyclic $G' = 1$ and then $G \cong G/G'$, but I got no where with this.
My next idea was since $G$ is nilpotent... |
H: Exists $t^*\in \mathbb{R}$ such that $y(t^*)=-1$?
$y'=y^2-3y+2, y(0) = \frac{3}{2}$
Exists $t^*\in \mathbb{R}$ such that $y(t^*)=-1$?.
How to prove without solving the ode? Any hint?
AI: Your equation admits two constant solutions, $y=1$ and $y=2$. By uniqueness, any other solution cannot intersect these constant s... |
H: Topological proof for the unsolvability of the quintic
Sorry, but in order to ask the question, you will have to view this video
http://drorbn.net/dbnvp/AKT-140314.php.
Here a topological proof for the unsolvability of the quintic is given, based on ideas of Vladimir Arnold. It's amazing because it does not require... |
H: Assert the range of a binomial coefficient divided by power of a number
I found this question in a previous year post graduate entrance exam for mathematics.The question was
What is the range of
\begin{equation*}
\frac{200 \choose 100}{4^{100}}
\end{equation*}
The choices were
\begin{align*}
[\frac{3}{4}, 1) && \... |
H: Linearized system for $ \begin{cases} \frac{d}{dt} x_1 = -x_1 + x_2 \\ \frac{d}{dt} x_2 = x_1 - x_2^3 \end{cases} $ is not resting at rest point?
Assume there is the dynamical system
$$
\begin{align}
\frac{d}{dt} x_1 &= -x_1 + x_2 \\
\frac{d}{dt} x_2 &= x_1 - x_2^3
\end{align}
$$
The system is at rest at the point... |
H: Is it possible to construct a continuous and bijective map from $\mathbb{R}^n$ to $[0,1]$?
Let $U$ be a non-trivial finite-dimensional vector space over $\mathbb R.$ I am trying to use a bijective and continuous map $f: U \to [0,1]$ and $d(x,y)=|f(x)-f(y)|$ to prove that there exist a metric on $U$ that makes $U$ c... |
H: What is the Fourier transform of $|x|$?
I am trying to find the Fourier transform of $|x|$ in the sense of distributions in its simplest form. Here is what I have done so far:
Let
$$f(x)=|x|=\lim_{a\rightarrow 0}\frac{1-e^{-a|x|}}{a},$$
then the Fourier transform is given by
$$\begin{aligned}
\hat{f}(\xi)&=\int_{-\... |
H: How to graph the following type of functions and discuss its differentiability:
I am having trouble in graphing a particular type of functions where a function is divided piecewise, and for some pieces we have to be draw maximum part and for some we have to draw minimum part, for example,
Graph $g(x)$: $$
\text { L... |
H: A finite group $G$ is called an $N$-group if the normalizer $N_G(P)$ of every non-identity p-subgroup $P$ of $G$ is solvable.
Prove that if $G$ is an $N$-group, then either (i) $G$ is solvable, or (ii) $G$ has a unique minimal normal subgroup $K$, the factor group $G/K$ is solvable, and $K$ is simple.
Suppose that ... |
H: Right adjoint to the forgetful functor $\text{Ob}$
Let $\text{Ob}:\textbf{Cat}\rightarrow\textbf{Set}$ be the forgetful functor mapping a small category to its set of objects. Consider the functor $R:\mathbf{Set}\rightarrow\textbf{Cat}$ mapping a set $X$ to the category having $X$ as its set of objects and a single... |
H: Let $\lambda$ be a real eigenvalue of matrix $AB$. Prove that $|\lambda| > 1$.
Let $A$ and $B$ be real symmetric matrices with all eigenvalues strictly greater than 1. Let
$\lambda$ be a real eigenvalue of matrix $AB$. Prove that $|\lambda| > 1$.
My solution:
Let $a$ and $b$ be eigenvalues of $A$ and $B$ corre... |
H: Evaluating an improper integral - issues taking the cubic root of a negative number
Problem:
Evaluate the following integral.
$$ \int_{-1}^{-1} \frac{dx}{x^\frac{2}{3}} $$
Answer:
This integral includes the point $x = 0$ which results in a division by $0$. To get around this difficulty, we break the integral into t... |
H: Integral of a product of Bessel functions of the first kind
I want to do this integral $H(\rho)=\int_{0}^{\infty} J_1(2 \pi Lr)J_0(2\pi \rho r)dr$, where $J_1$ and $J_0$ are Bessel functions of the first kind and $L\in \mathbb{R}$ is a constant, so I tried to do this in the Mathematica, but he failed. When I tried ... |
H: Connectives in George Tourlakis' Mathematical Logic
In page 10 of Mathematical Logic, Tourlakis says that "Readers who have done some elementary course in logic, or in the context of a programming course, may have learned that ¬, ∨ are the only connectives one really needs since the rest can be expressed in terms o... |
H: Inverting product of non-square matrices?
I am working on an Optimization problem and I need to show that
$$AB(B^TAB)^{-1} B^T = I_n$$
where $A$ is $n \times n$ and invertible, $B$ is $n \times k$ with rank $k$, and $k \le n$.
If $B$ is square, then this is a simple calcuation. For $B$ non-square, I have tried the ... |
H: Let $L\in End(V)$ with $L(V)=W$. Then $Tr(L)=Tr(L|_W)$
Let $V$ be a finite dimensional vector space and $W$ a subspace. Let $L$ be an endomorphism with image is $W$. Then $Tr(L)=Tr(L|_W)$ where $L|_W$ denotes the restriction of $L$ to $W$.
I am unsure what definition of trace I should use to prove this, it seems "... |
H: Connected and Hausdorff topological space whose topology is stable under countable intersection,
We know that the converging sequences of a discrete space are the stationary sequences.
I am looking for two examples for spaces (not empty or reduced to a singleton)
connected and Hausdorff topological space where the... |
H: let $\{a_n\} \downarrow 0$, show that $\sum_1^\infty a_n$ converge if and only if $\sum_1^\infty 2^na_{2^n}$ converge
let $\{a_n\} \downarrow 0$, show that $\sum_1^\infty a_n$ converge if and only if $\sum_1^\infty 2^na_{2^n}$ converge.
If $\{a_n\}$ is a non-increasing sequence, then this is Cauchy condensation te... |
H: Solution to autonomous differential equation with locally lipscitz function
As I was learning about the following theorem and its proof from the book Nonlinear Systems by H. K. Khalil, I encountered a difficulty in grasping some parts of the proof.
Theorem: Consider the scalar autonomous differential equation
\begi... |
H: Evaluating $\sum\limits_{i=\lceil \frac{n}{2}\rceil}^\infty\binom{2i}{n}\frac{1}{2^i}$
Let $a_n = \sum\limits_{i=\lceil \frac{n}{2}\rceil}^\infty\binom{2i}{n}\frac{1}{2^i}$. Prove that $a_{n+1} = 2a_n + a_{n-1}$.
I have tried considering the derivatives of $\frac{1}{1-x^2}=1+x^2+x^4+...$, and although this may work... |
H: Prove/disprove that $ \mathbb{R}^2 / $~ is hausdorff, when: $(x_1,x_2) $~$(y_1,y_2)$ if there is $t>0$ such that $x_2 = tx_1 $ and $ty_2 = y_1$
We look at the following equivalence relation on $\mathbb{R}^2$:
$(x_1,x_2) $~$(y_1,y_2)$ iff there exists $t>0$ such that $x_2 = tx_1 $ and $ty_2 = y_1$
The task is to pro... |
H: Prove a metric space is totally bounded
Let $X = 2^{\mathbb{Z+}}$ be the space of binary sequences $(x_k)_{k\ge1}$ with each $x_k \in \{0,1\}.$ Define a metric on $X$ by $d(x,y) = \sum^\infty_{k=1} |x_k−y_k|/2^k.$ I am trying to use the definition of a totally bounded space but I haven't found the $\varepsilon$-net... |
H: Definition of ring of dual numbers
In an exercise from Vakil's algebraic geometry notes, he asks us to describe the set $\rm Spec\space k[\epsilon]/(\epsilon^2)$, where $k$ is a field. A comment from this question gives a solution, but it's under the assumption that $\epsilon$ is transcendental, and so the assumpti... |
H: The Grothendieck-Serre Correspondence : Obstructions to the construction of the cyclic group of order 8 as a Galois group?
I am reading this note called as The Grothendieck-Serre Correspondence by Leila Schneps where this quote occurs:
the author still recalls Serre’s unexpected reaction of spontaneous delight upo... |
H: Are chain complexes chain equivalent to free ones?
Given a chain complex $A_\bullet\in\mathrm{Ch(\mathbf{Ab})}$, are there exist some chain complex $A'_\bullet\in\mathrm{Ch(\mathbf{Ab})}$ which is chain equivalent to $A_\bullet$ such that $A'_p$ are all free abelian groups?
AI: I'm answering my own question.
No. A ... |
H: Question about step in Ahlfors' proof of Cauchy's inequality in complex analysis.
I'm reading Ahlfors' Complex Analysis. In this book, he provides the following proof of Cauchy's inequality. Using $|a -b|^2 = |a|^2 + |b|^2 - 2 \Re\left(a\overline{b}\right)$ he establishes the following
$$0\le \sum_{k=1}^n \bigr\lv... |
H: Is the determinant a tensor?
I was reading Schutz's book on General Relativity. In it, he says that a(n) $M \choose N$ tensor is a linear function of $M$ one-forms and $N$ vectors into the real numbers.
So does that mean the determinant of an $n \times n$ matrix is a $0 \choose n$ tensor because it is a function th... |
H: Injection from $(K\otimes_{\mathbb{Z}}\hat{\mathbb{Z}})^*\to \prod_p(K\otimes_{\mathbb{Z}}\mathbb{Z}_p)^*$?
Suppose $K$ is a number field. The projections $\hat{\Bbb{Z}} = \prod_p\Bbb{Z}_p\to\Bbb{Z}_p$ give a map
$$(K\otimes_{\Bbb{Z}}\hat{\Bbb{Z}})^*\to \prod_p(K\otimes_{\Bbb{Z}}\Bbb{Z}_p)^*.$$
I want to show that ... |
H: What are some ("small") Riesel numbers without a covering set?
In the thread Does every Sierpinski number have a finite congruence covering? some examples of proven Sierpiński numbers that seem to have no full covering sets, are given. So it seems natural to ask if the same type of examples have been found for Ries... |
H: Understanding the $\alpha$ existence in the definition of the boundary map in case of simplicial homology and its absence in singular homology.
Here is the boundary map in case of simplicial homology(AT pg.105):
And here is the boundary map in case of singular homology(AT pg.108):
My question is:
I know that ... |
H: Is this a valid strategy to find group new group automorphisms given you already know some?
It is well known that Aut($G$), the group of all automorphisms of a group $G$, is a group under function composition. Lets say that you know $n$ of the automorphisms (plus the identity isomorphism) on $G$, hence you have
$$H... |
H: Why does this not satisfy the conditions of a metric?
Suppose we would like to define a metric on New York City, let t : NYC × NYC → R+ is
a function that measures the time it takes to travel between two points in New York City.
Why doesn’t t satisfy the criteria of being a metric?
I know that a metric must satisfy... |
H: $X_{1},X_{2},X_{3}\overset{i.i.d.}{\sim}N(0,1)$, find m.g.f. of $Y=X_{1}X_{2}+X_{1}X_{3}+X_{2}X_{3}$
I tried this
$X_{1}X_{2}+X_{1}X_{3}+X_{2}X_{3}=X_{1}(X_{2}+X_{3})+\frac{1}{4}(X_{2}+X_{3})^{2}-\frac{1}{4}(X_{2}-X_{3})^{2}$
$U=X_{2}+X_{3}\sim N(0,2)$
$\psi_{X_{1}(X_{2}+X_{3})}(t)=\psi_{X_{1}U}(t)=\frac{1}{\sqrt{2... |
H: Why random variables is a function? It seems that it violates the definition of function.
I am watching Lecture 5 of MIT 6.041 Probabilistic Systems Analysis and Applied Probability. In the lecture, Professor said Random Variable is a function that maps elements from the sample space to a real number. For example, ... |
H: Finding the volume when a parabola is rotated about the line $y = 4$.
Problem:
Find the volume generated by revolving the region bounded below by the
parabola $y = 3x^2 + 1$ and above by the line $y = 4$ about the line $y = 4$.
Answer:
Let $V$ be the volume we are trying to find. We want to find the points where th... |
H: $f:[0,1]\rightarrow[0,1]$, measurable, and $\int_{[0,1]}f(x)dx=y\implies m\{x:f(x)>\frac{y}{2}\}\geq\frac{y}{2}$.
Question: Suppose $f:[0,1]\rightarrow[0,1]$ is a measurable function such that $\int_0^1f(x)dx=y$. Prove that $m\{x:f(x)>\frac{y}{2}\}\geq\frac{y}{2}$.
My thoughts: Since we have $\int_0^1f(x)dx=y\impl... |
H: Showing that $\|f\|_{\infty}\leq \liminf_{p\to \infty}\|f\|_p$.
i'm trying to prove the next problem, and I wanted to know if my answer is correct.
Problem:
Let $(\Omega,\mathcal{F},\mu)$ be a $\sigma$-finite measurable space.
If $f\in L^p$ for all $p\in [1,\infty)$, show that
$$\|f\|_{\infty} \leq \liminf_{p\to \i... |
H: Smoothly varying finite dimensional vector subspaces tracing out infinite dimensional locus
Is it possible that a smoothly varying finite dimensional vector subspaces tracing out infinite dimensional locus?
More precisely, Let $E$ be a vertor bundle on a (compact) smooth manifold $M$ and $T_t:\Gamma(M,E)\to \Gamma(... |
H: Find minimum value of $\frac{\sec^4 \alpha}{\tan^2 \beta}+\frac{\sec^4 \beta}{\tan^2 \alpha}$
Find minimum value of $\frac{\sec^4 \alpha}{\tan^2 \beta}+\frac{\sec^4 \beta}{\tan^2 \alpha}$
I know this question has already answered here Then minimum value of $\frac{\sec^4 \alpha}{\tan^2 \beta}+\frac{\sec^4 \beta}{\... |
H: On Hypergeometric Series and OEIS Sequence
I have been searching an integer sequence in OEIS. The sequence is the following: OEIS A321234 (https://oeis.org/A321234) . So far, so good. However, this sequence is the denominator of a Hypergeometric Series, the following one:
$${}_3 F_2([1/2, 1, 1], [3/2, 3/2], x).$$
T... |
H: Fundamental Group of $\mathbb{RP}^n$
I was tring to culculate the fundamental group of $\mathbb{RP}^n$ with VAN KAMPEN to have a better understanding on how to use this theorem.
$\left(\mathbb{RP}^n:=S^n/ \sim \left((x_0,\cdots,x_n) \sim(-x_0,\cdots,-x_n)\right)\right)$
By consider the $A=\left\{[(x_0,\cdots, x_n)]... |
H: the definition of invertible sheaf on a functorial scheme in category theory
We define a functorial scheme as in "Two functorial definitions of schemes".A invertible sheaf is important and I'm interested in category theory, so I hope to define invertible sheaf in category theory like a scheme. However, the definiti... |
H: A question on locally integrable function on $\mathbb{R}^n$
I am currently doing some practice problems for the analysis qual. I have some thoughts on the following problem, and it would be great if someone could see if I am on the right track:
Problem:
Let $O\subset \mathbb{R}^n$ be an open set, and let $f\in L_{l... |
H: Why does $\frac{n!n^x}{(x+1)_n}=\left(\frac{n}{n+1}\right)^x\prod_{j=1}^{n}\left(1+\frac{x}{j}\right)^{-1}\left(1+\frac{1}{j}\right)^x$
Why does $$\frac{n!n^x}{(x+1)_n}=\left(\frac{n}{n+1}\right)^x\prod_{j=1}^{n}\left(1+\frac{x}{j}\right)^{-1}\left(1+\frac{1}{j}\right)^x$$ where the subscript n is the rising factor... |
H: Discrete Probability: alice and mary take a math exam
So I have a problem here, I know the answer but I have no idea how to solve it. This is one of a sample problem given to us by our prof. Can someone please help me out how to figure this problem out?
Alice and Mary take a math exam. The probability of passing t... |
H: Finding $E \in \mathcal A \otimes \mathcal B$ such that $E \neq E^y \times E_x,$ for some $x \in X, y \in Y.$
Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be two measure spaces. What has been stated in my book is that $\mathcal A \times \mathcal B$ may not necessarily be a $\sigma$-algebra of subsets of th... |
H: Problem with $A.M.\geq G.M.$ Inequality
Question:
If $x^2+y^2=1$, prove that $-{\sqrt2}\leq x+y \leq\sqrt2$
My approach:
$$\frac{x^2+y^2}{2}\geq\sqrt{x^2y^2}$$
$$ \frac12\geq xy$$
$$\frac{-1}{\sqrt2}\leq\sqrt{xy}\leq \frac{1}{\sqrt2} $$
Now how do I proceed from here?
AI: Proceeding from your approach, you had
$$... |
H: Yes/ No Is $X$ is homeomorphics to $Y$?
$X=\{(x,y) \in \mathbb{R^2}: x^2+y^2=1 \}$ and $Y=\{(x,y) \in \mathbb{R^2}: x^2+y^2=1 \} \cup \{(x,y) \in \mathbb{R^2}: (x-2)^2+y^2=1 \} $ be the subspaces of $\mathbb{R}^2$
Now my question is that
Is $X$ is homeomorphics to $Y$ ?
My attempt : Yes
Both $X$and $Y$ are connec... |
H: Proving that $\mathbb{Z}[i]/\langle 2+3i\rangle $ is a finite field
Prove that $\mathbb{Z}[i]/\langle 2+3i\rangle $ is a finite field.
Hi. I can't try a few steps in the next solution
$$\mathbb{Z}[i]/\langle 2+3i\rangle \simeq \mathbb{Z}[x]/\langle 1+x^2,2+3x\rangle$$ and $9(1+x^2)+(2-3x)(2+3x)=13$ then $13\in \lan... |
H: What is $T(p(a x^2+b))$ when $T(p(x))=x^2p'(x)$
Let $T$ be linear transformation on $\mathbb{P} \rightarrow \mathbb{P}$ that $T(p(x))=x^2p'(x)$.
1. Then, what is the value of $T(p(ax^2+b))$?
I think it could be $x^2p'(ax^2+b)$ or $2ax^3p'(ax^2+b)$ or even something else.
2. What would be the result when $T$ is on $... |
H: A question for a measurable function $g$ on a finite measure space such that $fg\in L^p$ for all $f\in L^p$
Let $\mu$ be a finite positive measure on $X$ and let $1\leq p<\infty$. Suppose that $g:X\to \Bbb R$ satisfies $fg\in L^p$ whenever $f\in L^p(\mu)$. I want to show that $||g||_\infty =\sup \{||fg||_p:f\in L^p... |
H: Maximum of function abs
Let function $f(x)=|2x^3-15x+m-5|+9x$ for $x\in\left[0,3\right]$ and $m\in R$. Given that $\max f(x) =60$ with $x\in\left[0,3\right]$, find $m$.
I know how to solve this kind of problem for $g(x)=|2x^3−15x+m−5|$. However, the $+9x$ is confusing me.
AI: Case 1:
Let $2x^3-15x+m-5\gt0$ at the p... |
H: $\lim\limits_{R\to0^+}\int\limits_{x^2+y^2\le R^2}e^{-x^2}\cos(y)dxdy=?$
$$\lim\limits_{R\to0^+}\int\limits_{x^2+y^2\le R^2}e^{-x^2}\cos(y)dxdy=?$$
First I want to show $f(x,y)=e^{-x^2}\cos(y)$ doesn't go crazy at $(0,0)$ otherwise it is already clearly continuous and bounded.
So $$|e^{-x^2}\cos(y)|\le e^{-x^2}\to ... |
H: Does the spiral Theta = L/R have a name?
Note: I intentionally left the equation in the title in plain text instead of MathJax, so it is searchable.
Here is a spiral's equation in polar coordinates:
$$\theta=L/r,$$
and in Cartesian coordinates:
$$(x,y) = \left(r\cdot\cos\frac Lr,\, r\cdot\sin\frac Lr\right)$$
for $... |
H: Every $T_1$, $C_2$, regular space is normal
By my attempt, I know that if $X$ is $T_1$, $C_2$, and regular, then it is metrizable. So is every metrizable space also normal? this seems correct but not sure how to see it.
AI: Is every metrizable space normal? Yes.
Let $(X,d)$ be a metric space and $P$, $Q$ be two non... |
H: What is the difference between ${3 \choose 2}$ and ${3 \choose 1}{2 \choose 1}$?
While choosing 2 from 3, we do ${3 \choose 2}$. But what would happen if we do ${3 \choose 1}{2 \choose 1}$? [Choosing 1 from 3 and again 1 from the remaining two)?
I know the latter is incorrect but can anyone give me conceptual view ... |
H: Probability question about picking $2$ types of balls out of $3$
I need help with this question:
A bag contains $2$ red balls, $6$ blue balls and $7$ green balls. Victoria draws $2$ balls out of
the bag. What is the probability that she gets a red ball and a blue ball?
I can figure out the probability of picking $1... |
H: Two conditional expectations equal almost everywhere
Suppose $X$ is a continuous random variable. If $\mathbb{E}[X\,|\,\mathcal{F}]=\mathbb{E}[X\,|\,\mathcal{G}]$ almost everywhere for two sub-sigma algebra $\mathcal{F}$ and $\mathcal{G}$, does this imply $\mathcal{F}$ and $\mathcal{G}$ are set theoretically identi... |
H: Evaluate $ \cos a \cos 2 a \cos 3 a \cdots \cos 999 a $ where $a=\frac{2 \pi}{1999}$
Evaluate
$
\cos a \cos 2 a \cos 3 a \cdots \cos 999 a
$
where $a=\frac{2 \pi}{1999}$
I know this question has already been answered many times but my doubt is different
Solution: Let $P$ denote the desired product, and let
$
Q=\s... |
H: Positive semi-definite real matrix with unit diagonal
Give an example of a $n\times n$ positive semi-definite real matrix $M\in \mathbb{R}^{n \times n}$, such that the following two conditions hold:
the eigenvalues $\lambda_1, \dots, \lambda_n$ of $M$ are $\lambda_i \leq 1$ for all $i\in [n]$;
the diagonal entrie... |
H: Is $\sin(\frac{1}{|z|})$ holomorphic on $\Bbb C-\{0\}$?
$f(z)=\sin(\frac{1}{|z|})$, $z\in \Bbb C-\{0\}$.
$$\frac{\partial f}{\partial\bar{z}}=\frac{\partial \sin(\frac{1}{|z|})}{\partial\bar{z}}=\frac{\partial}{\partial \bar{z}}\sin(\frac{1}{(z\bar z)^{1/2}})=\cos(\frac{1}{|z|})(2^{-1}z)(z\bar{z})^{\frac{-3}{2}} \n... |
H: Checking compactness of $[0,1]$ using the definition
By definition a set is compact if every open covering has a finite subcover.
I made an open covering of $[0,1]$ by taking a fixed $\epsilon= \dfrac1{10,00000}$ (or even more small) radius neighborhood around each point in $[0,1$]. But I could not find its finite ... |
H: Closed-form expression for $\prod_{n=0}^{\infty}\frac{(4n+3)^{1/(4n+3)}}{(4n+5)^{1/(4n+5)}}$?
I have recently come across this infinite product, and I was wondering what methods I could use to express the product in closed-form (if it is even possible):
$$\prod_{n=0}^{\infty}\dfrac{(4n+3)^{1/(4n+3)}}{(4n+5)^{1/(4n+... |
H: Are homology groups of a chain complex isomorphic to that of free chain complex?
Given a chain complex $A_\bullet\in\mathrm{Ch(\mathbf{Ab})}$, are there exist some chain complex $A'_\bullet\in\mathrm{Ch(\mathbf{Ab})}$ whose homology groups are all isomorphic to that of $A_\bullet$ such that $A'_p$ are all free abel... |
H: Can one obtain from the following diagram that the map $f_3$ is injective?
Let $A_i$ and $B_i$ be $R$-modules $(i=1,2,3)$.
If in the diagram
each map is $R$-linear, the rows are exact, both squares commute, and $f_1, f_2, \alpha_1, \beta_1$ are injective, is it possible to prove that $f_3$ is injective?
AI: Let $A... |
H: Karush-Kuhn-Tucker in Quadratic Program
I read an paper on Quadratic Programming:
Paul A. Jensen, Jonathan F. Bard: Operations Research Models and Methods Nonlinear Programming Methods.S2 Quadratic Programming Available here: https://www.me.utexas.edu/~jensen/ORMM/supplements/methods/nlpmethod/S2_quadratic.pdf
It d... |
H: Solve $\lfloor \ln x \rfloor \gt \ln \lfloor x\rfloor$
The question requires finding all real values of $x$ for which $$\lfloor \ln x\rfloor \gt \ln\lfloor x\rfloor $$ To start off, one could note that $$\lfloor \ln x \rfloor =\begin{cases} 0,& x\in[1,e) \\ 1,& x\in[e,e^2) \\ 2, &x\in [e^2,e^3) \\ 3,& x\in [e^3,e^4... |
H: Showing that for some group it is abelian iff $x • (y • x ^{−1} ) = y$
I have to show that for some group it is abelian iff $x • (y • x^{−1}) = y$.
This is what I did:
Starting with the given statement $x • (y • x^{−1}) = y$ which implies $x • (y • x^{−1}) • x= y • x$. Since it is a group associativity can be used... |
H: If $x$ is a local minimum of $f$ restricted to an open subset, why can we conclude it is a minimum of $f$ as well?
Let $(E,\tau)$ be a topological space and $f:E\to\mathbb R$. We say that $x\in E$ is a local minimum of $f$ if $$f(x)\le f(y)\;\;\;\text{for all }y\in N\tag1$$ for some open neighborhood $N$ of $x$.
Le... |
H: Probability one random variable is less than another random variable but higher than the same other random variable with a factor
Consider the two independent random variables $X$ and $Y$. Assume both are uniformly distributed on $[0,1]$. I want to calculate the probability that $X$ lies between some "low" linear t... |
H: What is the function $E(x)$?
When reading Problems in Calculus of One Variable (a translated Russian book), I came across unfamiliar notation "$E(x)$". It is neither expected value nor $\exp(x)$. Here is a picture of the function used in context, which I hope someone can deduce what it means from
$$\int\limits_0^x ... |
H: Show that $\lim_n a_n = 0$ implies $\lim_n \frac{\sum_{m=1}^n m a_m}{\sum_{m=1}^n m} = 0$
Let ${a_n}$ be a sequence in $\mathbb{R}$. I want to show that
$$
\lim_n a_n = 0 \implies \lim_n \frac{\sum_{m=1}^n m a_m}{\sum_{m=1}^n m} = 0.
$$
My attempt is like this:
Fix any $\varepsilon>0$. There exists $N$ such that $... |
H: Solve the following multiple integral
Let
$I = \int_{1}^{2}\int_{1}^{2}\int_{1}^{2}\int_{1}^{2} \frac{x_1 + x_2 + x_3 - x_4}{x_1 + x_2 + x_3 + x_4} dx_1 dx_2 dx_3 dx_4 $
Then $I$ equals
$ (a)\ \frac{1}{2} \\
(b)\ \frac{1}{3} \\
(c)\ \frac{1}{4} \\
(d)\ 1 $
I've tried to solve this and I also could do the whol... |
H: Raising Indices of a Conformally Transformed Metric
This is a bit of a silly question, but if a conformally transformed metric is given by
$g_{ij} = A^4 h_{ij}$,
where $A$ is a function of the spatial and time coordinates, and if one raises the indices does one accordingly have
$g^{ij} = A^4 h^{ij}$?
AI: $g^{ij}$ i... |
H: Why is the spectrum of a shift operator the closed unit disk?
Consider the following text from Murphy's: "$C^*$-algebras and operator theory":
In example 2.3.2, why is $\sigma(u) = \Bbb{D}$ (= the closed unit disk)?
I can see that $\sigma(u) \subseteq \Bbb{D}$ and $\sigma(u^*) = \Bbb{D}.$
Thanks in advance!
AI: $$... |
H: Evaluate the limit $\lim_{x\to 0} \frac{\ln |1+x^3|}{\sin^3 x}$
I tried this with L’Hopital and got $1$ as the answer. I don’t know if it can be applied considering the presence of the modulus function, but the answer is right. But I want a solution without using that rule, and I don’t know how to start this. Can I... |
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