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H: Any integer can be written as $x^2+4y^2$
If $n$ is a positive integer with $(n,8)=1$ and $-4$ is square $mod$ $n$ then $n$ can be written in this form:
$n=x^2+4y^2$.
I was using that there are x, y integers satisfying $x^2+4y^2=kn$ where $0<k<4$, let's consider $k=1,2,3$
If $k=1$ the result is completed.
If $k=2$,... |
H: What form of Leibniz rule is this (principal fiber bundle)?
Let $P(M,G)$ be a principal fiber bundle. Let $\sigma : U \subseteq M \rightarrow P$ be a smooth local section and $f : U \rightarrow G$ a smooth function. For $ a \in G$, $R_a : P \rightarrow P$ is the map $u \mapsto u \cdot a$. For $u \in P$, $L_u : G \m... |
H: Weak topology generated by identity maps
Let $(X, \tau_1)$ and $(X,\tau_2)$ be two topological spaces having the same underlying set X.
Let $\tau$ be the smallest topology on $X$ such that identity maps $I_1 : (X,\tau) \rightarrow (X,\tau_1)$ and $I_2 : (X,\tau) \rightarrow (X,\tau_2)$ are continuous.
Then if bot... |
H: How can $\left({1\over1}-{1\over2}\right)+\left({1\over3}-{1\over4}\right)+\cdots+\left({1\over2n-1}-{1\over2n}\right)+\cdots$ equal $0$?
How can $\left({1\over1}-{1\over2}\right)+\left({1\over3}-{1\over4}\right)+\cdots+\left({1\over2n-1}-{1\over2n}\right)+\cdots$ equal $0$?
Let
$$\begin{align*}x &= \frac{1}{1} +... |
H: proving that this function does not define a norm on $\mathbb R^2$ since the convexity
This problema use the previous part to conclude something, so I write all the parts.
First I have to prove that every norm in $\mathbb R^n$ is a convex function, I did it, it only requires the triangular inequality.
Then I have t... |
H: An identity involving the Beta function
I'm trying to show that
$$ \int _0^1 \frac{x^{a-1}(1-x)^{b-1}}{(x+c)^{a+b}}dx = \frac{B(a,b)}{(1+c)^ac^b}$$
Where $$B(a,b) := \int _0^1 x^{a-1}(1-x)^{b-1}dx $$ is the "Beta function". I am supposed to use a substitution but I'm pretty much stuck. I am familiar with the ba... |
H: uniform convergence of $\sum\limits_{n=0}^{\infty} \frac {(-1)^nx^{2n+1}}{(2n+1)!}$
Given the series $\displaystyle \sum_{n=0}^{\infty} \frac {(-1)^nx^{2n+1}}{(2n+1)!}$ does the series converge on $\mathbb R$? I found that the radius is $\infty$ and I know that for $\forall c\in [0,\infty)$ the sum converges unifor... |
H: Uncountably many equivalent Cauchy sequence?
RTP
There exists uncountably many Cauchy sequence of rationals that are equivalent.
I am trying to solve the above question, and my understanding is that $\Bbb R$ is a set of equivalent classes of Cauchy sequence of rationals. And two sequences are equivalent if they b... |
H: How does one prove that local diffeomorphism is submersion?
How does one prove that local diffeomorphism is submersion?
For a manifold, what does it being disconnected mean? I get what "disconnected" means for a graph, but not for a manifold.
AI: If $f:M\to N$ is a local diffeomorphism, then for any point $p\in M$... |
H: Number of days it took to climb the mountain (BdMO 2012 National Primary/Junior question)
From the Bangladesh Mathematical Olympiad 2012 National Secondary (Question 7, or ৭).
When Tanvir climbed the Tajingdong mountain, on his way to the top he saw it was raining $11$ times. At Tajindong, on a rainy day, it rains... |
H: What is the difference between isomorphism and homeomorphism?
I have some questions understanding isomorphism. Wikipedia said that
isomorphism is bijective homeomorphism
I kown that $F$ is a homeomorphism if $F$ and $F^{-1}$ are continuous. So my question is: If $F$ and its inverse are continuous, can it not be ... |
H: Find two closed subsets or real numbers such that $d(A,B)=0$ but $A\cap B=\varnothing$
Here is my problem:
Find two closed subsets or real numbers such that $d(A,B)=0$ but $A\cap B=\varnothing$.
I tried to use the definition of being close for subsets like intervals but I couldn't find any closed sets. Any hint?... |
H: Product and Quotient rule for Fréchet derivatives
Does anyone know whether the product/quotient rule for Fréchet derivatives still hold? For example, consider the evaluation operator:
$$\rho_x : (C[a,b],\|\cdot\|_\infty) \rightarrow (\mathbb{R},|\cdot|)$$
where $\|\cdot\|_\infty$ is the sup-norm and $|\cdot|$ the E... |
H: Quick question about proof of Levy's theorem
The following question arise from the proof of Levy's theorem in Richard Bass - Stochastic processes (can be seen via Google books, its on page 77).
So we have $(M_t)_{t\geq 0}$ a continuous local martingale, $M_0=0$ adapted to $\{\mathcal{F}_t\}$ s.t. $<M>_t = t$.
We le... |
H: Question about homomorphisms $f_{!}, f^{!}$.
Let $f: A \to B$ be a finite ring homomorphism and $N$ a $B$-module. $N$ can be considered as an $A$-module if we define $A \times N \to N$, $(a, n) \mapsto f(a)n$. Therefore we have a map $f_{!}: K(B) \to K(A)$.
Let $M$ be a $B$ module. $B$ can be considered as an $A$ m... |
H: Question about symmetric even polynomials
This might be an easy question but here goes. I am looking for a polynomial $P\in \mathbb{Q}[x,y,z]$ such that
$P$ is symmetric and homogenous.
$P$ is even in all three variables, i.e. $P\in \mathbb{Q}[x^2,y^2,z^2]$.
$P$ is divisible by $x+y+z$.
In two variables the equi... |
H: variance of two independent random variable
$X$ is normal with $E[X]=-1, Var(X)=4$, $Y$ is esponential with $E[Y]=1$, they are independent, if $T=pXY+q$ with $p, q \in R$, what is $Var(T)$,
I get $E[T]=q-p$ and $Var(T)=p^2(E[X^2]E[Y^2]-(E[X])^2(E[Y])^2)=$?
AI: Hint: For any random variable $Z$:
$$
{\rm E}[Z^2]=\mat... |
H: Relation between stirling numbers
Is there a relation between $$ \genfrac\{\}{0pt}{}{n}{n-2} $$
and $$ \genfrac\{\}{0pt}{}{n-1}{n-3} $$
Like the first one can be obtained from the second one by adding something?
AI: The Stirling numbers of the second kind satisfy a somewhat Pascal-like recurrence relation:
$${{n+1}... |
H: probability question is this true?
So my teacher was solving this exercise and she wrote $D (2x +6) = (2^2)\times D$
where $ D$ is the statistical dispersion.But wasnt she supposed to write $ D(2x+6)=(2^2)\times D+ 6$ ?
AI: If by Dispersion you specifically mean Variance, notice that:
$$Var[aX+b] = E[(aX+b)^2]-E^2... |
H: Spectrum of the sum of matrices
I have an $n$ by $n$ matrix $A$ such that:
$$A = J_n + (k-1)I_n$$
$I_n$ being the identity matrix and $J_n$ the all-$1$ matrix.
The spectra of those matrices are as follow:
$$Spec(J_n) = (k^2 -k +1)^1(0)^{n-1}$$
and
$$Spec((k-1)I_n) = (k-1)^n$$
I don't understand why I then have
$$Sp... |
H: Under what circumstances does this procedure terminate?
This earlier question (essentially) asked why the following loop will terminate. (This is Java code, so assume you're working with signed, 32-bit integers:)
final int initial = 2;
final int multiplier = 12381923;
for (int i = initial; i != 0; i += i * mul... |
H: Show $g^{-1} \wedge h^{-1}=(g\vee h)^{-1}$
Glass' Partially Ordered GroupsLemma 2.3.2 says:
Let G be a p.o. group and $g,h\in G$. If $g\vee h$ exists, then $g^{-1} \wedge h^{-1}=(g\vee h)^{-1}$
Proof: If $f\leq g^{-1},h^{-1}$ then $f^{-1}\geq g,h$. Thus $f^{-1}\geq g\vee h$. Hence $f\leq (g\vee h)^{-1}$.
I don't... |
H: Proving the relation $\det(I + xy^T ) = 1 + x^Ty$
Let $x$ and $y$ denote two length-$n$ column vectors. Prove that $$\det(I + xy^T ) = 1 + x^Ty$$
Is Sylvester's determinant theorem an extension of the problem? Is the approach the same?
AI: Here is an approach by upper triangularization for the sake of variety.
No... |
H: Some group theory interpretion problem
Very simple question I think, but I can't fully understand the following set:
We are given a group $G$ with a subgroup $H$. Then I have to answer some questions about the subgroup $$\bigcap_{g\in G} gHg^{-1}$$
Wat exactly are the elements of this group?
AI: That subgroup is ca... |
H: removing the remainder of a fraction
I would like to remove the remainder from a fraction if possible. I want a function
$$f(x,y) = x/y - remainder$$
for example
$$f(3,2) = 1$$
$$f(7,2) = 3$$
$$f(12,5) = 2$$
It seems so simple but its been bugging me for a while. Please help.
AI: You are looking for division with... |
H: separability of a space
When I have to show that some space $A$ IS NOT separable, does it always work if I find uncountable subset $B\subset A$, $|B|=2^{\aleph_0}$ and set C of disjoint open balls, $C=\{L(x,r): x\in B\}$.
If $A$ IS separable, then $\exists D\subset A$ such that $|D|\leq \aleph_0$ and $(\forall x... |
H: Number of solutions of a positive integral quadratic form is finite?
Is there an easy way to see the following:
Suppose Q is an integral quadratic form in $n$ variables that is positive definite, that is $Q(x) \geq 1$ for all $0 \neq x \in \mathbb{Z}^n$. Then the number of solutions to the equation $Q(x)=m$, for so... |
H: Find the volume using triple integrals
Using triple integrals and Cartesian coordinates, find the volume of the solid bounded by
$$ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 $$
and the coordinate planes $x=0, y=0,z=0$
My take
I have set the parameters to $$ 0\le x \le a$$ $$0\le y \le b\left( 1 - \frac{x}{a} \right... |
H: Need help solving the ODF $ f''(r) + \frac{1}{r}f'(r) = 0 $
I am currently taking complex analysis, and this homework question has a part that requires the solution to a differential equation. I took ODE over 4 years ago, so my skills are very rusty. The equation I derived is this:
$$ f''(r) + \frac{1}{r}f'(r) = 0 ... |
H: Convolution and integrating over G(t)
I'm struggling with the following expression in a statistics script:
$$H(x) = \int_{-\infty}^\infty F(x-t) dG(t)$$
What does the dG(t) mean exactly? I've never seen that notation before.
Background: Random variables X, Y with CDFs F and G, we want to calculate H which is the CD... |
H: Fourier series question
How do we know that a Fourier series expansion does exist for a given function $f(x)$? I mean, if $f(x)=x$ and we suppose that $x=a_1\sin(x)+a_2\sin(2x)$ with $-\pi\leq x \leq \pi$ the Fourier coefficients $a_1$ and $a_2$ still being the same as if we suppose that $x=a_1\sin(x)+a_2\sin(2x)+.... |
H: is prime spectrum $Spec(R)$ countable?
Let $R$ commutative ring with identity, given $Spec(R)=\{I|\text{$I$ prime ideal of $R$}\}$, does the set $Spec(R)$ countable? Also, if $\{\langle p^n \rangle\}$ is closed in $P_{-}Spec(R) = \{I| \text{$I$ primary ideal of $R$}\}$, does $\langle p^n \rangle$ maximal ideal? I t... |
H: Upper bound of binomial distribution
I'm studying a proof and I'm wondering which binomial approximation could have been use to establish the following bound:
$$\cfrac{1}{2} {n \choose r} {n-r \choose r} \le \cfrac{n^{2r}}{2(r!)^2}$$
I get that:
$$\cfrac{1}{2} {n \choose r} {n-r \choose r} = \cfrac{n!}{2(r!)^2 (n... |
H: How to show that T is a projection operator
For $x ∈ [0, 2π]$ let $G(x) = π^{−1}\cos x$, and define an operator $T$ on $L^2([0, 2π])$ as follows:
$$(Tf)(x) = ∫_0^{2π}G(x − x')f(x') \,dx'. $$
Show that $T$ is a projection operator.
I guess I must show that $T$ is selfadjoint and idempotent, in other words that:
$T=T... |
H: Why doesn't this series converge uniformly?
given series:
$$x^2+\frac{x^2}{(1+x^2)}+\frac{x^2}{(1+x^2)^2}+\frac{x^2}{(1+x^2)^3}+\frac{x^2}{(1+x^2)^4}+\ldots$$
in the interval $[-1,1].$
I tried to break the problem in two cases:1)$x=0$, in which case the series is zero.
2)$x\neq0$: I solved the G.P. and i got $(1+x^... |
H: Question about inverse limits.
I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. On Page 104, I have some questions about the proof that $\{A_n\}$ is surjective implies $d^A$ is surjective. We have to prove that given $(a_n) \in A = \varprojlim A_n$, there is $(x_n) \in\varprojlim A... |
H: When does $\varphi(n) = n/6$?
I am having trouble showing this. I was able to show when $\varphi(n) = n/2$, and $\varphi(n) = n/3$, but I don't know when $\varphi(n) = n/6$
AI: As $\frac{\phi(n)}{n}=\prod_{p|n} \frac{p-1}{p}$ you need to find all $n$ so that
$$\prod_{p|n} \frac{p-1}{p} =\frac{1}{6} $$
Let $p_1< ... |
H: How can a group of matrices form a manifold?
So for example, $GL(n,\mathbb{R})$ group. It is said that this group can be considered as manifold - but I do not get how this is possible. How does one then assign a neighborhood of a matrix, and talk of compactness? (of course manifold can be disconnected, but.)
AI: Ex... |
H: How to Find the First Moment of Area of a Circular Segment by Integration
Given a segment of circle symmetric about the $y$-axis, I'm wondering how to apply the integral $Q_x = \int y \, dA $ to find the first moment of area with respect to the $x$-axis. I'm having difficulties taking into account both the straight... |
H: How to integrate: $\int \frac{\mathrm dx}{\sin (x)-\sin(a)}$
How to integrate :
$$\int \frac{\mathrm dx}{\sin (x)-\sin(a)}$$
AI: Using Weierstrass substitution, $$\tan \frac x2=u$$
$$\implies \sin x=\frac{2u}{1+u^2}\text{ and } x=2\arctan u,dx=\frac{2du}{1+u^2}$$
$$I=\int\frac{dx}{\sin x-\sin \alpha} =\int\frac1{\f... |
H: Is there a contradiction in this definite integral computation?
EDIT:
This question is wrong. You don't need to waste your time trying to answer it. :D
I need help showing that:
$$ \int_a^b x f(x) dx = \frac {a+b} 2\int_a^bf(x)dx$$
My attempt.
$$ I = \int_a^bxf(x)dx = \int_a^b(b+a -x)f(b+a-x)dx $$
$$ = \int_a^b (... |
H: Why ring with only even numbers is not an integral domain?
Let $S$ be a set of all even integers. According to my text book, $(S,+,\cdot)$ is a ring which is not an integral domain. It is stated as a fact without an explanation and I fail to see the reason for this.
Why the ring from above is not an integral domai... |
H: equivalent measures, can be one finite and one not?
Let $\mu$ be a non-negative and Borel-finite measure on $\mathbb{R}$ and $\nu$ a non-negative measure on $\mathbb{R}$. If $\mu$ and $\nu$ are equivalent (one absolutely continuous with respect to the other) is it true that even $\nu$ is Borel-finite.
AI: The other... |
H: About the relation of rank(AB), rank(A), rank(B) and the zero matrix
Let $A$ be a $2 \times 4$ matrix and $B$ be a $4 \times 4$ matrix, prove that if $rank(A) = 2$ and $rank(B)=3$ then $AB \neq 0$.
I got stuck at $rank(AB) \leq 2 $
How do I continue from here?
AI: We know that,
$rank (AB)=rank (B)-\dim(Img(B)\cap K... |
H: Ring structure on subsets of the natural numbers
Let $$\mathcal{N}=\{\{k_1,\ldots,k_s\}:\ s>0,\ \mbox{and the}\ k_i\ \mbox{are non-negative and pairwise different integers}\}\cup\{\emptyset\}.$$ Note that there is a bijection with the naturals,
$$
\begin{array}{rccc}
B:&\mathcal{N}&\longrightarrow &\mathbb{N}\\
&... |
H: Is the endomorphism of $\mathbb{Z}_{p}$ induced by multiplication by $p^{n}$ surjective?
Let $p$ be a prime number. Is it true that $p^{n}\mathbb{Z}_{p}\cong\mathbb{Z}_{p}$ as additive groups for any natural number $n$ and if so, why? Here, $\mathbb{Z}_{p}$ denotes the ring of $p$-adic integers.
Any help would be v... |
H: Evaluate the maximum of: $A = \sin A\cdot\sin ^2 B\cdot \sin ^3 C$
Given a triangle ABC. Evaluate the maximum of:
$A = \sin A\cdot\sin ^2 B\cdot \sin ^3 C$
AI: let
$$f(A,B,C)=\ln{\sin{A}}+2\ln{\sin{B}}+3\ln{\sin{C}}+\lambda (A+B+C-\pi)$$
then
$$\begin{cases}
\dfrac{\partial f}{\partial A}=\cot{A}-\lambda=0\\
\dfr... |
H: Proving that: $-\frac{2 \;i\log(i^2)}{2} = \pi$
I'm trying to prove that: $$-\frac{2 \;i\log(i^2)}{2} = \pi$$
This is what I've tried:
$$-\frac{2 \;i\log(i^2)}{2} = -i \log(i^2) = -i (i \pi)\implies$$
$$-x\;(x y)=-x\;y\;x\implies$$
$$-i\;(i \pi) = -i\;i\;\pi = \pi \implies$$
$$-i\;i = (-i^2) = (-(-1)) = 1$$
Is thi... |
H: Henkin vs. "Full" Semantics for Second-order Logic and Multi-Sorted First Order Interpretations
In this paper by Jeff Ketland, he notes:
With Henkin semantics, the Completeness, Compactness and Löwenheim-Skolem Theorems all hold, because Henkin structures can be re-interpreted as many-sorted first-order structures... |
H: Show $A$ and $B$ have a common eigenvector if $A^2=B^2=I$
Let $n$ be a positive odd integer and let $A,B\in M_n(R)$ such that $A^2=B^2=I$. Prove that $A$ and $B$ have a common eigenvector.
AI: For $M=A$ or $B$, define
$$V_M^\pm=\{v\in\Bbb R^n:v\pm Mv\}.$$
By definition, $V_M^\pm$ are linear subspaces of $\Bbb R^n$.... |
H: Playing with a functional equation
I was playing with a functional equation and proved the result below:
Let $f$ be such that $$f(f(z))=z$$ If $f^{-1}$ exists then $$f(z)=f^{-1}(z)$$ If $f'$ exists then as $$(f^{-1}(z))'=\frac{1}{f'(f^{-1}(z))}=(f(z))'$$ then $(z)'=\frac{1}{f'(z)}$ so $f'(z)=1$ then $f(z)=z+C\right... |
H: Is the outer boundary of a connected compact subset of $\mathbb{R}^2$ an image of $S^{1}$?
A connected compact subset $C$ of $\mathbb{R}^2$ is contained in some closed ball $B$. Denote by $E$ the unique connected component of $\mathbb{R}^2-C$ which contains $\mathbb{R}^2-B$. The outer boundary $\partial C$ of $C$ i... |
H: Does $x^2>x^3+1 \implies x < -{1\over P}$?
How could one prove that:
$$x^2>x^3+1 \implies x < -{1\over P}$$
where $P$ denotes the plastic constant, the unique real root of $x^3-x-1=0$?
AI: The function $x^2 - x^3 - 1$ is monotone decreasing on $\{x < 0\}$ so we need to look at the (unique) $x$ where $$x^2 = x^3 + 1... |
H: Quadrilateral congruency theorem
Is there a congruence theorem that says that if three sides of two quadrilaterals are equal, then the two quadrilaterals are congruent?
I am grading some homework and a student appealed to such a theorem, but I cannot find it anywhere. I'd like to give them credit if it is the case.... |
H: Does the limit of a descending sequence of connected sets still connected?
Given a descending sequence of sets
$$
F_1\supset F_2\supset\cdots F_n\supset\cdots
$$
in which each $F_i$ is connected. I wonder if the limit set
$$
F=\bigcap_{i=1}^\infty F_i
$$
is still connected? I believe it is, but cannot make a pro... |
H: maximal ideal properly contains union of its square with the union of minimal prime ideals
One of the first theorems one encounters in the study of commutative algebra is that if $I$ is an ideal of a ring $A$ not contained in any of the prime ideals $P_1,\cdots,P_n$, then $I$ is not contained in $\cup_{i=1}^n P_i$.... |
H: Convergence - formula
There are sequences:
$\{x^n\}_{n\in N}$, where
$x^n=\langle x^n_1, x^n_2, x^n_3,...\rangle, n=1,2,...$
$x=\langle x_1,x_2,x_3,...\rangle$
How should I write that $x$ is limes of $x^n$? I use definition that
$lim_{n \rightarrow \infty}x_n=x$ iff $(\forall \epsilon >0)(\exists n_0\in N)(\foral... |
H: For $f$ an analytic function, what is
$f$ be analytic function, could any one tell me how to find the value of $$\int_{0}^{2\pi} f(e^{it})\cos t \,\mathrm dt$$
I am not able to apply any complex analysis result here, could any one give me hint?
AI: Make a change of variables, $z = e^{it}$, which maps the interval ... |
H: $\|x -y\|+\|y-z\|=\|x-z\|$ implies $y= a x + b z$ where $a +b =1$
$\|x -y\|+\|y-z\|=\|x-z\|$ implies $y= a x + b z$ where $a +b =1$
Hint: Take $m=x-y$ and $n= y-z$.
Does this follow from standard properties of inner product spaces (linearity, symmetry, and positive definiteness?) Or does Cauchy-Schwarz help? This... |
H: Understanding $\sum^n_{i=1}\int_{(i-1)\pi}^{i\pi}|\frac{\sin x}{x}|\,dx\ge\sum^n_{i=1}\frac{1}{i\pi}\int_{(i-1)\pi}^{i\pi}|\sin x|\,dx$
Could some please explain me the following inequality?
$$\sum^n_{i=1} \int_{(i-1)\pi}^{i\pi} \left|{\frac{\sin x}{x}}\right| \, dx\geq \sum^n_{i=1}\frac{1}{i\pi}\int_{(i-1)\pi}^{... |
H: “$f$ is a function from $A$ to $B$” vs. “$f $is a function from $A$ into $B$”?
When we say that
$f$ is a function from $A$ to $B$
is this different from saying
$f$ is a function from $A$ into $B$
I know what injective ("1-1"), surjective ("onto"), and bijective functions are, but is there such a thing as an "i... |
H: On chain complex morphisms
The following seems quite obvious to me. Nevertheless I would like to have another opinion.
Suppose $(A_\bullet,d_A)$ and $(B_\bullet,d_B)$ are chain cmplexes, such that
$d_A$ is the trivial differential (i.e $(d_A)_k(a)=0$ for all $k\in \mathbb{Z}$
and $a\in A_k$) and $d_B$ is not zero i... |
H: One Step in Proving the Gamma of $1 \over 2$
Good Afternoon All. There is one step in the proof that I never quite understood.
Let $I = \int_{0}^{\infty} e^{{-u}^2} du$. Then
$$I^2 = \int_{0}^{\infty} e^{{-u}^2} du \int_{0}^{\infty} e^{{-v}^2} dv$$
Now, if I say that since $\int_{0}^{\infty} e^{{-u}^2} du$ is no... |
H: Raising each side of a limit to a power?
If you have $\lim_{x\to \infty} f(x)=L$ $\quad$(or as $x \rightarrow c$ for that matter), is it correct to say that $\lim_{x\to \infty} f(x)^{g(x)}=L^{\lim_{x\to \infty}g(x)}$ as long as $L \not= 1$? I'm not looking for a very rigorous response, I am just looking for if this... |
H: Prove normalizing constant on normal CDF
So I know that the CDF of a standard normal will be:
$$ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{
\frac{-z^2}{2}} \, dz $$
How do I show that when I sub in mu and sigma, the equation will become:
$$ \frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{(y-\mu)^2}{2\sigma^2... |
H: How to calculate the order of commutator of two elements of a group $G$ in terms of their orders?
if $G$ is a group , $x,y \in G$ and $[x,y]$ is the commutator of $x$ and $y$
so , $[x,y]=x^{-1}y^{-1}xy$
is there a formula to compute $|[x,y]|$ in terms of $|x| , |y|$ ?
AI: No. If $|x|=|y|=2$ then $|[x,y]|$ can have... |
H: change of variables using a substitution
Let $D$ be the triangle with vertices $(0,0),(1,0)$ and $(0,1)$. Evaluate $$\iint_D \exp\left( \frac{y-x}{y+x} \right) \,dx\,dy$$ by making the substitution $u=y-x$ and $v=y+x$
My attempt
Finding the domain, $D$ $$ (x,y) \rightarrow(u,v) $$ $$ (0,0) \rightarrow(0,0) $$ $... |
H: If $G$ is a finite group, $H$ is a subgroup of $G$, and $H\cong Z(G)$, can we conclude that $H=Z(G) $?
If $G$ is a finite group, $H$ is a subgroup of $G$, and $H\cong Z(G)$, can we conclude that $H=Z(G) $?
I learnt that if two subgroups are isomorphic then it's not true that they act in the same way when this act... |
H: How to show that $ Ax \le b$ is convex?
For
$$ A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^m, c \in \mathbb{R} $$
one has to show that
$$ K:= \{ x \in \mathbb{R}^n: Ax \le b \}$$
is convex.
Now I'm aware that by definition, a set is convex $ \iff $ for all $x,y \in K, \lambda \in [0,1]$ any point $ \lambda x ... |
H: Find a subspace $W$ such that ... - am I right?
Let $U$ be a subspace of $\mathbb R^4$ spanned by $v_1=(1,-1,1,2)$ and $v_2=(3,1,2,1)$. Find a subspace $W$ of $\mathbb R^4$ such that $U ∩ W = \{{0}\}$ and $\dim(U) + \dim(W) = \dim(\mathbb{R^4})$.
Now obviously, $v_1,v_2$ form a basis for $U$ and thus $\dim(U)=2$. N... |
H: $G$ is a tree $\iff G$ contains no cycles, but if you add one edge you create exactly one cycle - is my proof correct?
$G$ is a tree $\iff G$ contains no cycles, but if you add one edge you create exactly one cycle
Could anyone please be so kind to check my proof? That would be very much appreciated. Thank you in ... |
H: Power of commutator formula
A few people remember a commutator formula of the form $[a,b]^n = (a^{-1} b^{-1})^n (ab)^n c$ where $c$ is a product of only a few commutators (say $n-1$) of them. Here $a,b$ are in a (free) group and $[a,b] := a^{-1} b^{-1} a b$.
Does anyone remember such a formula with proof?
Some suc... |
H: Connected, regular, bipartite graph and biconnected component
I'm learning for an exam and I can't solve this problem:
Prove that every undirected, connected, regular, bipartite graph has only one biconnected component. Give an example which shows that assumption of bipartition is necessary.
Can anyone help?
AI: ... |
H: Integrating by using change of variables and by making a substitution
Let $D$ be the region bounded by $x=0$, $y=0$, $x+y=1$ and $x+y=4$. Evaluate $$\iint_D \frac{dx\,dy}{x+y}$$ by making the change of variables $x=u-uv$, $y=uv$
My attempt
I understand I must first find the domain. $$x=u-uv, y=uv$$ $$x=u-y$$ $... |
H: Maximum of $ f(x,y) = 1 - (x^2 + y^2)^{2/3} $
For the function
$ f(x,y) = 1 - (x^2 + y^2)^{2/3} $
one has to find extrema and saddle points. Without applying much imagination, it is obvious that the global maximum is at $ (0,0)$.
To prove that, I set up the Jacobian as
$$ Df(x,y) = \left( -\frac{4}{3} x (x^2 y^2)^{... |
H: Prerequisites for understanding Borel determinacy
I have just learned that Gale-Stewart game is determined for open and closed sets, so naturally I'm interested to understand Borel determinacy which seems to be on a totally different level. What's the minimal set of knowledge to understand Borel determinacy without... |
H: Questions about cosets, conjugate classes etc
Some questions about subgroups, normal subgroups, conjugate classes etc, just to make sure I understand it :-)
The index of a subgroup $H$ in $G$, written as $[G:H]$ is defined as the number of left cosets of $H$ in $G$. I know that the a left coset of $H$ in $G$ is de... |
H: Is Gödel's incompleteness theorem provable without any model-theoretic notion?
The entry on Gödel's incompletenss theorem in Wikipedia says:
Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated ... |
H: Probability density/mass function
I am a bit confused as to the difference between the probability mass function and the probability density function for a distribution of discrete variables. I understand there would be no mass function for a continuous variable distribution, only a density function. But for discre... |
H: still trying to figure out details of Newton's method
I understand mostly what going on when using Newton's method to approximate some value but there are some details I am still hazy on (like how you come up with the correct $f(x)$ for the update rule). I would appreciate it if someone could help me fill in the de... |
H: Normal subgroups and factor groups
$\\$
A normal subgroup $N$ is a subgroup where the left cosets are the same as the right cosets. $N$ is normal $\iff $ $xnx^{-1} \in N, \forall x\in G$.
5.) Why is it that if $[G:H]=2 \implies $ $H$ is normal subgroup?
6.) Can we say that a factor group is just a group that has... |
H: Proving $\|e^A\|\le e^{\|A\|}$
I'm trying to prove this inequality:
$\|e^A\|\le e^{\|A\|}$, where $A$ is a matrix and $\|A\|:=\sup_{|x|=1}
|Ax|$.
My attempt of solution:
Since $e^A:=I+A+A^2/2!+A^3/3!+\ldots$
we have
$$\|e^A\|=\|I+A+A^2/2!+A^3/3!+\ldots\|=\sup_{|x|=1}\|(I+A+A^2/2!+A^3/3!+\ldots)x\|$$
$$=\sup_{|... |
H: Explanation of why $\frac{d}{dx} e^x=e^x$
I'm taught that all the way back when I'm in high school that $\dfrac{de^x}{dx}=e^x$ and $\int e^x dx=e^x$.
Can someone explain why this is the case?
AI: If you differentiate any exponential function, you can write something like this:
$$
\frac{d}{dx} 2^x = \lim_{h\to0}\fra... |
H: Distributing two distinct objects to identical boxes
Hiii,
I've been struck with a problem which deals with the distribution of two distinct objects such that p of one type and q of other type into three identical boxes.
As if it were only one object with q copies i'd have used integer partitioning, and if all obje... |
H: Calculate $e^{1/4}$ using Maclaurin series with accuracy of 0.001
I have to calculate $\sqrt[4]{e}$ with a deviation of less than $0.001$.
I was guided to use the Maclaurin series to solve this exercise.
So I've written down the series of $e$,
and now I don't have any idea how to proceed.
Any help? Thanks in adv... |
H: How many integer solutions to this 5 integers equation?
Ref to the question in Unusual 5th grade problem, how to solve it.
Find a positive integer solution $(x,y,z,a,b)$ for which
$$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$
Here is my question:
How many solutions for this questi... |
H: Complex Analysis Question from Stein
The question is #$14$ from Chapter $2$ in Stein and Shakarchi's text Complex Analysis:
Suppose that $f$ is holomorphic in an open set containing the closed unit disc, except for a pole at $z_0$ on the unit circle. Show that if $$\sum_{n=0}^\infty a_nz^n$$ denotes the power seri... |
H: If $H$, $K$ are characteristic subgroups of $G$, when is $\operatorname{Aut}(H\times K) = \operatorname{Aut}(H) \times \operatorname{Aut}(K)$ true?
If $H$, $K$ are characteristic subgroups of $G$, when is $\operatorname{Aut}(H\times K) = \operatorname{Aut}(H) \times \operatorname{Aut}(K)$ true?
What if $H$, $K$ are... |
H: Intersection of the $p$-sylow and $q$-sylow subgroups of group $G$
What can we say about the intersection of the $p$-sylow and $q$-sylow subgroups of group $G\;$?
It's not necessary that $p=q$.
Is there general statements about the intersections of sylows subgroups ?
You can give me restricted conditions, this is ... |
H: Why does this limit not exist?
Working through some limit exercises. The answer sheet says the limit below does not exist. Is this correct. Shouldn't it be $-\infty$?
$$\lim_{x \to 0^+} \left( \frac{1}{\sqrt{x^2+1}} - \frac{1}{x} \right)\ \ \ $$
AI: It is true that as $x$ approaches through positive values, $\frac{... |
H: Smooth retraction onto a differentiable manifold
Let $M\subset\mathbb{R}^n$ be a smooth k-dimensional differentiable manifold (by which I mean that it is locally diffeomorphic to an open set in $\mathbb{R}^k$). Let us suppose $M$ compact for simplicity.
How can one prove that there exists an open set $U\supset M$ ... |
H: Evaluating $\lim\limits_{x\to0}\frac{1-\cos(x)}{x}$
$$\lim_{x\to0}\frac{1-\cos(x)}{x}$$
Could someone help me with this trigonometric limit? I am trying to evaluate it without l'Hôpital's rule and derivation.
AI: $\displaystyle \lim_{x\to 0} \frac{1-\cos x}{x}= \lim_{x\to 0} \frac{(1-\cos x)(1+\cos x)}{
x(1+\cos x... |
H: Positive integers a and b such that $\sqrt{7-2\sqrt{a}}=\sqrt{5}-\sqrt{b}$
There exists positive integers a and b such that $\sqrt{7-2\sqrt{a}}=\sqrt{5}-\sqrt{b}$
How am I suppose to find these?
I squared both sides, turned out nasty though.
Help Appreciated!
AI: After squaring both sides you should get:
$$7-2\sqrt... |
H: If $m+n=5$ and $mn=3$, find $\sqrt{\frac{n+1}{m+1}} + \sqrt{\frac{m+1}{n+1}}$?
It is known that $m+n=5$ and $mn=3$. So what is the value of:
$$
\sqrt{\dfrac{n+1}{m+1}} + \sqrt{\dfrac{m+1}{n+1}}
$$
I think we're suppose to solve for the system of equations first, but I'm not getting any results that's useful.
AI: ... |
H: Proving that $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$
How am I suppose to prove that:
$$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$$
Do I use the way like how we count $1+2+ \cdots+100$ to estimate?
So $1/5050 \lt 20$, implying that it is indee... |
H: Simplify a recursive function in Maple
I have the following problem. Out of the runtime analysis of an divide and conquer algorithm I got the following formula for the necessary flops:
flops(n): = (89+1/3)*n^3 + 2 * flops(n/2)
and
flops(1):= 0
I want to sum it up and to remove the recursion with Maple. But I d... |
H: Question on a symmetric inequality
Let $a, b,c $ are positive real number satisfying $a^2+b^2+c^2+2abc=1$.
How can I prove that $a+b+c\ge \dfrac{3}{2}$ ?
AI: let $$a=\cos{A},b=\cos{B},c=\cos{C},A+B+C=\pi$$
because it is known
$$\cos^2{A}+\cos^2{B}+\cos^2{C}+2\cos{A}\cos{B}\cos{C}=1$$
then $$a+b+c=\cos{A}+\cos{B}+... |
H: closure of open set in topological space
How to prove : If $X$ is a topological space, $U$ is open in $X$, and $A$ is dense in $X$, then $\overline{U}=\overline{U \cap A}$
AI: Note that
$$
\overline U = \underset{ \mathcal F \text{ closed}}{ \bigcap_{U \subseteq \mathcal F} }\mathcal F, \quad \overline {U \cap A} ... |
H: Simple ODE question
Suppose i have an ODE $$f(y'',y',y,x) = 0.$$
Does the domain of $x$ where $f$ is defined have anything to do with the domain in which a solution $y = g(x)$ satisfies the equation ?
Knowing , a priori , the domain of $x$ where $f(y'',y',y,x)$ is defined, do I know in advance somet... |
H: How many even number in a sequence are there?
How many even numbers in the below numbers ?
$$\binom{k}{0},\binom{k}{1},\binom{k}{2},\ldots,\binom{k}{k}$$
Worng: Is it true that all of them are odd iff $k$ is odd, and if $k$ is even then $\binom{k}{2i}$ is odd, $\binom{k}{2i+1}$ is even. ?
AI: From the Wikipedia a... |
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