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H: How does the ring of smooth functions fail to be a field?
So if we have a differentiable manifold $M$, the set of smooth functions $C^{\infty}(M)$ on $M$ can be equipped with point-wise addition and multiplication.
Apparently it is important that this object will be a ring but not a field, however I am having trou... |
H: Increments in Derivatives
Say, I have an equation $y^2=x^3$.
So I can say $dy^2/dx=3x^2$.
So the very small increment in $y^2$ when $x$ becomes $dx$ is $dy^2$ which in this case is $3x^2dx$.
I also know that $dy^2/dy=2y$ so I can say $dy^2=2ydy$ and equate the increments but how do I know the increments $dy^2$ are... |
H: LR Test in Beta($\theta$, 1) with $H_0 = {\theta_0}$
I'm trying to obtain $\alpha$-level LR Test where $(X_1, ... X_n)$ are from Beta($\theta$, 1) with $H_0 = {\theta_0}$ and $H_0 \neq \theta_0$.
I'm looking for
$$
\lambda(X) = \frac{\sup_{\theta \in \Theta_0}l(\theta)}{\sup_{\theta \in \Theta}l(\theta)}
$$
Suppose... |
H: Is there a way to represent a Max Flow problem as a dynamic programming task?
I've recently started practising some graph theory problems, and I wanted to know if there is a method which would allow us to approach the Max Flow problem through dynamic programming. I cannot seem to find any resources where they outli... |
H: Which calculation for the eigenfunction and eigenvalue is wrong?
Given $\Omega = (0,1)$ and the eigenvalue problem $$-u''(x)+u(x) = \lambda u(x), \quad u'(0)=u(1)=0$$ or equivalently $-u''(x)=(\lambda -1)u(x).$
I first derived $u(x)=\cos(k\pi x)+1$ and $\lambda =k^2\pi^2+1$ which in my opinion solves the problem.
... |
H: Defining new function to solve the original function easier.
If I have some $f : \mathbb A \to \mathbb A$ that needs to be solve. Can I define $g : \mathbb B \to \mathbb B$ when $\mathbb A$ is a subset of $\mathbb B$ and solve for $g(x)$ then I conclude that $f(x) = g(x)$?
For example , if I have $f : \mathbb N \... |
H: Evaluation of line lintegral along the parabola $y=x^2$
Evaluation of $$\int_{C}xydx+(x+y)dy$$ aling the curve $y=x^2$ from $(-2,4)$ to $(1,1)$
What i try
Let $\vec{F}=\bigg<xy,x+y\bigg>$ and let $\vec{r}=\bigg<x,y\bigg>$
So we have to calculate $$\int_{C}\vec{F}\cdot \vec{dr}$$
Now let paramatrize the curve $y... |
H: Clarification of Baire category theorem: a (counter-?)example
I am trying to understand the statement of Baire's category theorem.
Why is $\bigcap_{n \in \mathbb{N}} A_n := \bigcap_{n \in \mathbb{N}} \big[(n, n+\frac{1}{2}) \cap \mathbb{Q}\big] = \emptyset$ dense in $\mathbb{R}$? As I understand $\mathbb{R}$ is a c... |
H: How to calculate the index $|\mathcal{O_K}/ \mathfrak{a}|$ in sage
Let $K=\mathbb{Q}(\sqrt{2})$. I want to calculate $|\mathcal{O_K}/ \mathfrak{a}|$ in sage with $\mathfrak{a}=3\mathbb{Z}$. This code:
sage: K.<sqrt(2)>=NumberField(x^2-2)
sage:01=K.order(sqrt(2))
OK=ring_ of _ integers()
sage: O1.index_in(OK)
d... |
H: H1N1 Probability Problem
Suppose that the probability of being affected by H1N1 flu virus is
0.02, the probability of those who regularly wash their hands among those affected by the H1N1 virus is known as 0.3, In general, if the
probability of people who washing their hands regularly between the
whole peopl... |
H: Is order matter when writing the roots of a quadratic equation?
Equation:
$x^2-x-6=0$
The two roots of this equation are $3$ and $-2$. When writing the answer can I also write it as $-2, 3$ or do I have to maintain a certain order?
AI: If the question offers the two choices you gave us in a comment, $-2, 3$ and ... |
H: Inverse trigonometric function of trigonometric function
This is a simple question.
We know that
$\sin( arcsin(2\pi) ) = 2\pi$
Because $arcsin$ is the inverse function of $sin$
So logically, the following should be true as well, since $sin$ is the inverse of $arcsin$
$\arcsin( sin(2\pi) ) = 2\pi$
but it actually eq... |
H: How much seconds of YouTube video will be played if I will press play/pause button at rational timesteps?
Suppose that I want to watch a continuous YouTube video. I start watching it when my clock shows $t_0=0$ seconds. Each time my clock shows rational amount of seconds I instantly press play/pause button. How muc... |
H: How do you compute the value of the right derivative of $f(x)= \sin (x)^{\cos (x)} +\cos (x)^{\sin (x)}$ when $x=0$.
How do you compute the value of the right derivative of $f(x)= \sin (x)^{\cos (x)} +\cos (x)^{\sin (x)}$ when $x=0$. I'm trying to learn calculus so some explanations wouldn't be so bad. I got stuck ... |
H: Question based on finer and coarser topological spaces
I am trying exercise 1.2 (question 12) of C. Wayne Patty Foundations of Topology.
I am struck on it.
Question is-> No. 12
I am unable to think how such a space in (1) must exist as union of topology is not always a topology. ( I could only think of Union o... |
H: Correct way to announce a theorem
Reading the following statement :
Theorem :
An operator $A$ has the property $P$ if $\underset{\lambda \rightarrow 0}{%
\lim }\left( A-\lambda \right) ^{-1}$ exists.
Does the reader understand that $\left( A-\lambda \right) ^{-1}$ is defined
on a neighborhood of $0$ ?
Or the corr... |
H: Testing Series $ \sum\limits_{n = 3}^{\infty} \frac{(-1)^n + 2\cos(\alpha n)}{n(\ln(n))^{\frac{3}{2}}} $
I have a problem which is related to testing the divergence or convergence for the sum of a series. For more details:
$$
\sum\limits_{n = 3}^{\infty} \frac{(-1)^n + 2\cos(\alpha n)}{n(\ln(n))^{\frac{3}{2}}}
$$
T... |
H: Convergence in Polish Spaces
If I have a point $x \in S$ where $S$ is some Polish Space with metric $d$, a closed set $F \subset S$, and $inf\;\{d(x,f)|f\in F\}=0$, then why can we say that there exists $f' \in F$ such that $d(x,f')=0$?
If we had some kind of compactness it would be easy but my topology class is a... |
H: Root of the derivative is unique in some intervals
Suppose that $p_n$ is the n-th (real) polynomial, and that it has $n$ simple real roots, let's call them for $x_{n,1}<x_{n,2}<\dots <x_{n,n}$. Then, by Rolle's theorem, $p_n'$ has a real root in each $(x_{n,1},x_{n,2})$, $(x_{n,2},x_{n,3})$, etc. How do you show th... |
H: Does $\lim\limits_{n\to\infty}|X_n-X|=0$ imply $\lim\limits_{n\to\infty}|X_n-X|^p=0\hspace{0.2cm}\forall p>1$?
Let $\left(X_n\right)_{n\geq1}$ be a sequence of random variables and $X$ a random variable as well. Does $\lim\limits_{n\to\infty}|X_n-X|=0$ imply that $\lim\limits_{n\to\infty}|X_n-X|^p=0\hspace{0.2cm}\f... |
H: Prove using bisection that if $f$ is continuous on $[a, b]$ and $f(a)<0
Let's define the following process:
1) If $f(\frac{a+b}{2})=0$, then we're done.
2) If $f(\frac{a+b}{2})\neq0$, then either $f$ changes sign on $[a, \frac{a+b}{2}]$, or on $[\frac{a+b}{2}, b]$. So consider next the interval where $f$ changes si... |
H: Let $X = \{ \sqrt{p} : p \text{ is prime} \}$, $Y \subseteq X$ and $\sqrt{p} \not\in Y$. Show that $[\mathbb{Q}(Y)(\sqrt{p}) : \mathbb{Q}(Y)] = 2$.
I am trying to solve Problem 22 from Chapter 5 of Patrick Morandi's Field and Galois Theory:
Let $K = \mathbb{Q}(X)$, where $X = \{ \sqrt{p} : p \text{ is prime} \}$.... |
H: Invertible Matrices and basis
Consider the $n\times n$ matrix $A$ and the basis $\{\vec{v_1}\ldots \vec{v_n}\}$ for $\mathbb{R}^n$. Prove if $\{A\vec{v_1} \ldots A\vec{v_n}\}$ is a basis for $\mathbb{R}^n$, then A is invertible.
If we let $B=\{A\vec{v_1} \ldots A\vec{v_n}\}$, does this mean the column vectors form ... |
H: Can the cross section of parallelepiped be a regular pentagon
Came across this question in a children's recreational mathematics book. Apparently, the cross section of a cube cannot be a regular pentagon. It could be a irregular pentagon though.
But if we generalize this problem, can the cross section of parallel... |
H: Prove that quadratic form is differentiable from the definition
I am trying to show that a quadratic form $Q: \mathbb{R}^n \to \mathbb{R}, Q(x)=x^T A x$ is differentiable from the definition of the differential.
I started by considering $Q(x+h)=(x+h)^T A (x+h)=x^T A x + x^T A h + h^T A x + h^T A h$.
We need to find... |
H: Transitive models of ZFC and power set
Let $M$ be a transitive model of ZFC.
From my understanding, if $x \in M$ then what $M$ believes to be its power set $\mathcal{P}(x)^M$ does not necessarily agree with the external power set $\mathcal{P}(x)$ (i.e. $\mathcal{P}(x)^M \neq \mathcal{P}(x)$), because $M$ might not ... |
H: Determining the differential of a map defined on a submanifold of $\Bbb R^n$
Let $M=\{(x,y,z,w)\in \Bbb R^4:x^3+y^3+z^3-3xyz=1\}$ and consider the function $f:M\to \Bbb R^2$ defined by $f(x,y,z,w)=(x+y+z+w,w^3+w)$. It is easily checked that $1$ is a regular value of the function $g:\Bbb R^4\to \Bbb R$, $(x,y,z,w)\m... |
H: how to prove φ(n) tends to infinity as $n$ grows?
I am wondering how can I prove $\lim\limits_{n \rightarrow \infty} {φ(n)=\infty}$
My attempt: For a prime number, we have φ(n)=n-1, so the equation above is proved. However, how can I prove it when n is a composite number?
What do you think about it? Could you ple... |
H: What does "x-y" denote in boolean logic?
I came across this equation in set theory :
x-y = y'-x'
where x and y are sets
If it was a "+" or "." , I could easily correlate it with OR and AND function.
But what does this "-" indicate in boolean logic? It indicates Set difference in set theory. Is it complement ? But c... |
H: A question involving faithful flatness, support of a module, and Spectrum of a ring
The following theorem is taken from Matsumura's Commutative Ring Theory [M] Theorem 7.3(i) and the paragraph before it. My questions only concern the proof of the Theorem below.
A ring homomorphism $f:A\longrightarrow B$ induces a m... |
H: What does it mean, “Nonindependent identically distributed and nonergodic static behavior.”
I have heard “independent identically distributed (iid) process” a lot.
It means mathematically that, for any $t$ and $t’$, $X(t)$ and $X(t’)$ follows the same distribution and independent each other.
Then, it can be also co... |
H: What is the meaning of the union of ascending chain $\bigcup_{k} N_{k}$
With $A\in \operatorname{End}(V)$ and $N(A^k)$ the nullspace of $A^k$, what does the 'union of ascending chain' mean, defined by:
$$\bigcup_{k} N(A^k)$$
I would assume that it means:
$$N(A)\cup N(A^2)\cup...\cup\ N(A^k)$$
Is this correct?
AI: C... |
H: Is $S_5=\left\{ \begin{pmatrix} x\\ y \end{pmatrix}\in \mathbb{C}^2 ;\, y=\bar{x}\, \right\}$ a subspace of $(\mathbb{C}^2,+,\bullet,\vec{0},1)$
Is $S_5=\left\{ \begin{pmatrix} x\\ y \end{pmatrix}\in \mathbb{C}^2 ;\, y=\bar{x}\, \right\}$ a subspace of $(\mathbb{C}^2,+,\bullet,\vec{0},1)$
With $\begin{pmatrix}x_1 \... |
H: Regular and irregular points of second order ODE
I want to find the regular and irregular singular points of this ODE
$$x \sin (x) y'' + 3y' + xy = 0$$
What can I do?
AI: To start of, I think you can see that when $x=0, x=n\pi$ and $x=\infty$, these might be singularity points. Divide the whole equation by $xsin(x)... |
H: Operation that returns a unique result for each unordered set of numbers
What operation $f$ can I apply to any two numbers $a$ & $b$, such that $$f(a,b) = f(b,a)$$
where $f(a,b)$ is unique for any combination of a & b in the set of whole numbers?
P.S. I'm really not sure what tag to use here, I'd appreciate someone... |
H: Showing that $\varnothing$ and $X$ are open sets in a metric space $(X, \rho)$.
Let $(X, \rho)$ be a metric space. Define the open ball with center $x_0 \in X$ and radius $r > 0$ by
$$B(x_0, r) = \{x \in X: \rho(x, x_0) < r\}$$
We say a subset $E$ of $X$ is an open set if for each $x \in E$, there is an $r > 0$ wit... |
H: Comparability graphs are perfect graphs (reference request )
By whom it was initially shown that the family of comparability graphs is a subclass of perfect graphs? I am a first year math student and i am working on the project with my group mates, at certain point of the "paper" we claim that comparability graphs ... |
H: Generating function of a Fibonacci series but with certain variation.
Let $f_n$ denote the $nth$ Fibonacci number then what is the generating function for the sequence$f_0,0,f_2,0,f_4,0,...$ $$\text{Attempt}$$Its known that the sequence has following two properties.\begin{align}
\smash[b]{\sum_{i=1}^n F_{2i-1}}&=F_... |
H: Finding The Tangent Line to $\sqrt{x} + \sqrt{y} = 1$
Hello everyone I have a function $\sqrt{x} +\sqrt{y} = 1$ and I have a tangent line to this function that cut
the axises at $A , B$.
How can I proof that $OA + OB = 1$.
$O = (0,0)$.
I tried to mark = $A = (a , 0) , B(0, b)$
and find the tangent line by A and B
... |
H: Subgroups of order 5 and 6 in a group $\mathbb{Z}_{10}$
According to my solution, we use Lagrange's Theorem and the fact, that all subgroups of a finite group have an order dividing the order of the group.
As a result, we can say that the orders of the subgroups of the group $\mathbb{Z}_{10}$ are $\{1,2,5,10\}$, w... |
H: Two candidates for definition of splitting field
Definition 1. [Bourbaki] A splitting field of $f\in \Bbbk[x]$ is an extension $\Bbbk\subset \mathbb K$ which splits $f$ (into possibly repeated linear factors) and satisfies $\mathbb K=\Bbbk(Z(f,\mathbb K))$ where $Z(f,\mathbb K)$ is the set of roots of $f$ in $\math... |
H: Determine unknown probability by observing results
Disclaimer: Sorry, i'm a self-taught non-native, apart from basics, i don't know the proper terms. I'm pretty sure there has to be a part of statistics that would help me deal with my problem, but somehow i'm unable to word my question well enough for google to be ... |
H: Notation question: $R(A)=\mathbb{C}^2$ with $A\in End(\mathbb{R}^2)$
I have a map, represented by the matrix $A=\left(\begin{array}{ll}1 & 0 \\1 & 1\end{array}\right)=End(\mathbb{R}^2)$.
My teacher wrote in his lecture notes that because it is invertible, we can conclude that $R(A)=\mathbb{C}^2$. To me it looks lik... |
H: Median (and consequently the mean) of an evenly-spaced list
Why is it the case you can find the median (and consequently the mean) of an evenly-spaced list by taking the mean of opposite terms? Where opposite refers to opposite positions (e.g. first and last).
What intuition allows you to most easily reconcile thi... |
H: How can we prove that $3t$ cannot be a perfect cube for any integer $t$ except 9?
If $t \in \mathbb{Z}$ then prove that $3t$ can never be a perfect cube except for $t=9$.
How can we prove things like these? I’m pretty new to Number-Theory and I find it difficult to prove things like these. Mathematical Induction ... |
H: linear algebra, given the dimension of the kernel of the transformation and finding k.
If the dimension of the kernel of the transformation
$$T \colon \mathbb{R}^3 \to \mathbb{R}^3,\ T((x, y, z)) = (2x + y, x + z, kx + 2y - z)$$
is $1$, find $k$.
I found,
$\operatorname{kernel} = \{1,-2,-1\}$
and $k$ is $3$.
is tha... |
H: Finding integral for this question-
$$F(x)=
\int_0^{\pi/2} \frac{1}{(\sqrt{sin(x)}+\sqrt{cos(x)} )^4}dx = $$
Any help would be highly appreciated.I first used an online integral finder but it only displayed me final answer which was =.33333 .
PS: I am still in high school so complex answers wouldn't be of any help... |
H: How to solve a fraction of imaginary numbers?
I have the following equation.
$a = \frac{(1/2) - (3/2)i}{(3/2) + (3/2)i}$
The solution says that $a^2 = 5/9$.
I don't know how I can perform the steps, could I get some feedback?
Thanks!
AI: Hint:
Multiply numerator and denominator by the complex conjugate of the lat... |
H: Understanding units mod $n$ are relatively prime to $n$
I am trying to prove that the only invertible elements in $\mathbb{Z}_n$ are those that are relatively prime to $n$.
The first half is straightforward. If $i$ is relatively prime to $n$, so $\gcd(i,n) = 1$, we have $\alpha i + \beta n = 1$, so $\alpha i - 1 =... |
H: Show that u is radially symmetric.
Consider $\Delta u=1$ on an annulus $a<r<b$ in $\Bbb{R}^2$, with $u$ vanishing on both the inner and outer circles. Here $0<a<b$, and $r=|x|=({x_1}^2+{x_2}^2)^{\frac{1}{2}}$ for $x=(x_1,x_2) \in \Bbb{R}^2$. Show that $u$ is radially symmetric.
I am not sure how to get to that, ... |
H: Independence of discrete random variable
Let $U_i$,$V_i,\ i=1,\dots,n$ be $2n$ real random variables i.i.d..
Are $(U_i-V_i)^2,\ i=1,\dots, n$ independent ?
I should check that
$$P((U_i-V_i)^2=k,(U_i-V_i)^2=p)=P((U_i-V_i)^2=k)P((U_j-V_j)^2=p)$$
is there any general theorem about such independence or should I use ge... |
H: Arithmetic Series Word problem
Taking a calculus class and trying to solve a series and sequence word problem but I am struggling on the easiest problem. Please someone talk me thru a baby step on this question please?
1) Xin has been given a 14 day training schedule by her coach. Xin will fun for A minutes on day... |
H: Prove that $v_{1} + W = v_{2} + W$ if and only if $v_{1} - v_{2}\in W$.
Let $W$ be a subspace of a vector space $V$ over a field $\textbf{F}$. For any $v\in V$ the set $\{v\} + W = \{v+w:w\in W\}$ is called the coset of $W$ containing $v$. It is customary to denote this coset by $v + W$ rather than $\{v\} + W$.
(a)... |
H: What does it mean for an ODE to be conservative?
What does it mean for an ODE to be conservative?
For example, I already read somewhere that the equation
$$w\cdot y''-y+y^{2k+1}=0,$$
with $w>0$ and $k\in \mathbb{N}$ constants fixeds, is conservative. In practice, what does this mean?
AI: Given the differential equa... |
H: Sigma Algebra property
Let $\mathcal{C}=\{(-\infty,x]:x\in\mathbb{R}\}$ and $\mathcal{C}_0=\{[a,b]:-\infty<a<b<\infty\}$. I want to show that $\sigma(\mathcal{C})=\sigma(\mathcal{C}_0)$. I think the general idea of the proof is to show that $A\subseteq \mathcal{C}\implies A\subseteq\sigma(\mathcal{C}_0)\implies\sig... |
H: Deriving the equation of a plane $Ax+By+Cz=D$.
Defining a plane as the span of two linearly independent vectors, I've been trying to derive the equation $$Ax+By+Cz=D$$ without much success. The equation seems to indictate that a vector
$$\vec{v}=\begin{bmatrix} x \\ y \\ z\end{bmatrix}$$
is in the plane if and only... |
H: Different boundaries of a set, its closure und interior in R
Denoting $\overline A$ as closure of A, $A°$ as interior of A.
I've proven that in a general topology (X,O), following is true:
$\partial\overline A \subseteq \partial A$ and $\partial A° \subseteq A$, where $A \subseteq X $ and $\partial A$ is defined a... |
H: Prove that if $A$ is open at $(X, d)$ and $B$ is any subset of $X$, then $\overline{(A\cap \overline{B})}=\overline{(A \cap B)}$
Hello friends could you help me with the following please:
Prove that if $A$ is open at $(X, d)$ and $B$ is any subset of $X$, then
$\overline{(A\cap \overline{B})}=\overline{(A \cap B)}$... |
H: Negating an alternate definition of a limit point
I know that one way to define $x$ as a limit point of set $A$ is to say that there is some sequence $\{a_n\}$ contained in $A$ which converges to $x$ and $a_n \neq x$ $\forall n \in \mathbb{N}$.
I'm trying to negate this definition to say $x$ is not a limit point of... |
H: Proper subsets and arbitrary subset of the containing set
Suppose there is a set $A$ and $P$ is a proper subset of $A$. Also, suppose that $B$ is any subset (not necessarily a proper subset) of $A$. Then, are we justified in writing
$$P \subseteq B \subseteq A$$
In other words, can we be sure that it is not the ca... |
H: Correct Interpretation of Notation
I was reading a parametrization and they used a peculiar way to write their equations which I am unfamiliar with as to how to properly interpret it.
K refers to Kelvins in this case and what I am particularly struggling with is with the symbolic meaning of the $min[1, max[0,f(x)]$... |
H: Laplace equation on unit disc
Let $u(x,y)$ be the solution of$$ u_{xx}+u_{yy}=64$$ in unit disc $\{(x,y): x^2+y^2<1\}$ and such that $u$ vanishes on the boundary of disc then find $u(\frac{1}{4},\frac{1}{√2})$
What I tried i know how to solve Laplace's equation so i make a transformation $ v = u -32x^2$
With this ... |
H: Maximum size of the automorphism group of a graph given some constraint?
Are there any results on the maximum size of the automorphism group of a graph with $n$ vertices and $m$ edges? What about for a graph with $n$ vertices and is $d$-regular?
AI: For your second question, Wormald, On the number of automorphisms ... |
H: Tensor product $E\otimes_{A} F$ of modules $E,F$ where $F$ has a basis
Let $A$ be a ring, $E$ a right $A$-module and $F$ a left $A$-module.
Let $(b_\mu)_{\mu\in M}$ be a basis of $F$. Then every element of
$E\otimes_AF$ can be written uniquely in the form $\sum_{\mu\in
M}(x_\mu\otimes b_\mu)$ where $x\in E^{... |
H: Find the remainder of the polynomial $f(x)$ divided by $(x-b)(x-a)$ given its remainder when divided by $x-a$ and $x-b$
I am hoping to get a feedback on my solution to the following problem and if there are better solutions I would love to take a look. Thank you for your time.
Let $f(x)$ be a polynomial with remain... |
H: Finding probability density - a result of an algoritm
I am supposed to find probability density of the random variable $Y$, which is an output of the following algorithm:
Gen $X \sim U(0,1)$
Gen $U \sim U(0,1)$
If $X<U$ then $Y:=X$ else $Y:=1-X$
return Y
In R:
X <- runif(10000, 0, 1)
U <- runif(10000, 0, 1)
res... |
H: Prove that $\lim_{s \to \infty} \sum_{x=1}^{2s} (-1)^x\sum_{n=1}^{x}\frac{1}{n!}=\cosh (1) -1$
How can we prove that $$\lim_{s \to \infty} \sum_{x=1}^{2s} (-1)^x\sum_{n=1}^{x}\frac{1}{n!}=\cosh (1) -1$$
It seems like this is some kind of telescopic series, but I don't know how to find the limit of this sum. Any hel... |
H: $x, y, z$ for dimensions
We tend to use $x$ for an arbitary first dimension, $y$ for one at right angles to it, $z$ for one at rigth angles to both of those ... what is the letter for $4\textrm{D}$?
AI: I think the letter $t$, because the fourth dimension is (usually) $\textrm{time}$ which is often symbolized with ... |
H: How can $\mathbb{Z} \times \mathbb{Z}$ be generated by $(1,1)$ and $(0, 1)$?
I read somewhere that $\mathbb{Z} \times \mathbb{Z}$ be generated by $(1,1)$ and $(0, 1)$ (taken together, I presume). I am trying to understand how this is true. I found this answer, based on which I developed the following argument:
The ... |
H: How to find number of edges in a graph?
Let G(V,E) be an undirected graph:
$$V={\{0,1\}}^n$$
E:
There is an edge between A and B iff, A and B differ in exactly one index
For example (when n=4 -which is the length of each world-):
There is an edge connecting 0000,0001 and another edge connecting 0100,0110 But, ther... |
H: How to prove that there exists a specific path?
Let G(V,E) be an undirected graph.
if there exists a vertex called u that its degree is not even then it is connected to another vertex v that its degree is also not even.
I tried to prove this by contradiction but reached a closed end, may I get help with this?
Note:... |
H: Linear transformation bounded iff its kernel is closed in infinite-dimensional Banach spaces
I am working on Problem 8, Chapter 6, in Luenberger's Optimization by Vector Space Methods. It states:
"Show that a linear transformation mapping one Banach space into another is bounded if and only if its nullspace is clos... |
H: Question about linear operator on polynomial space $L:\mathbb{R}[x,y]_{\leq 2} \rightarrow \mathbb{R}[x,y]_{\leq 2}$
I'm trying to find a a matrix of a linear operator defined mapping from the set of real polynomials of two variables of degree less than or equal 2 to itself by the following rule:
$L(f(x,y)) = (x^2 ... |
H: How to find the Moment Generating function of a function of random variables from their joint Moment Generating function?
Given the Joint Moment Generating function(MGF) of the random vector $X =(Y,Z)$
$$M_{Y,Z}(t_1,t_2) =e^{(t_1^2 + t_2^2 + t_1 t_2)/2} $$
How can I find the MGF of
$Y+Z$
$Y-Z$
Is there any gene... |
H: Symplectic structure on $\mathbb{S}^{2}$
This question has been asked several times but I cannot find a satisfactory answer. Consider $\mathbb{S}^{2} \subseteq \mathbb{R}^{3}$ and define, for every $p \in \mathbb{S}^{2}$ and every $u,v \in T_{p}\mathbb{S}^{2}$, the $\mathbb{R}$-bilinear form $\omega_{p}(u,v) := \la... |
H: Bochner integral in a direct sum of Banach spaces
Let $\mathcal{B} = \mathcal{B}_1\oplus\ldots\oplus \mathcal{B}_n$ be a direct sum of Banach spaces $\mathcal{B}_i$ each with norm $\|\cdot\|_{\mathcal{B}_i}$. The Banach space $\mathcal{B}$ has many equivalent norms. For instance, letting $v = (v_1,\ldots,v_n)\in\ma... |
H: Finding the order of $(1, 1) + \langle(2, 2)\rangle$ in the factor group $\mathbb{Z} \times \mathbb{Z} / \langle (2, 2)\rangle$
I read somewhere that the order of $(1, 1) + \langle(2, 2)\rangle$ in the factor group $\mathbb{Z} \times \mathbb{Z} / \langle (2, 2)\rangle$ is $2$. I am trying to understand how this is ... |
H: Compute integral using Cauchy Principal Value
Using the Cauchy Principal Value, I need to compute the following integral
$$\int_{-\infty}^\infty\frac{\cos(ax) - \cos(bx)}{x^2}dx$$
I have used the standard semi-circle contour with an indentation around the singularity at $x=0$. However integrating around the outer ... |
H: If $A,B$ are linear combinations based on common "underlying" random variables, can they still be independent?
Apologies if I am just having a mental block and missing something very obvious. Here is a conjecture that I think is obviously true, and yet I cannot prove it:
Let $X_1, X_2, \ldots, X_n, Y, Z$ be mutual... |
H: existence of negative root
I look if there exists a non-zero polynomial $p(x)= a_0 + a_1x + a_2x^2 + \cdots a_nx^n$ with positive integers coefficients : $\forall i, a_i\in \mathbb N $ such that $p(-\sqrt 2)=0$
AI: The minimal polynomial for $-\sqrt{2}$ over $\mathbb{Z}$ is $x^2-2$. Thus any such polynomial havi... |
H: $\int_\gamma F \cdot dr$, where $F=x^2i+y^2j+z^2k$
Given
$\int_\gamma F \cdot dr$
where $F=x^2i+y^2j+z^2k$ where $\gamma$ is the intersection of the sphere $x^2+y^2+z^2=a^2$ with the plane $y=z$.
I know it sounds quite silly and easy to compute this but the exercise make the statement about the intersection its... |
H: Is there a closed form to this summation?
I cannot get a closed form for
$\sum\limits_{r=0}^{m} \frac{(m+r) !}{(m-r)! (2 r)!}$ ‘
Does anyone have any idea on what it is?
AI: This sum is the Fibonacci number $F_{2 m+1}$. |
H: Exactly one root of $p_n$ between two consecutive roots of $p_{n+1}$
Let $p_n$ be a polynomial of exactly degree $n$, with positive leading coefficient, and suppose that it has $n$ simple real roots. Let $y_1<\dots <y_{n+1}$ be real (simple) roots of $p_{n+1}$. Assume that $p_n/p_{n+1}$ is decreasing in each interv... |
H: $\sigma$ algebra question
So I'm learning a little bit of measure theory currently, and I am a little in the dark as to the motivation for the definition of the $\sigma$ algebra. My biggest question is why the $\sigma$ algebra must be used rather than the power set (since a $\sigma$ algebra is a proper subset of th... |
H: Doubts about a proof of the existence of a lower bound on $\varphi(n)$
I recently attempted to solve the following question (exercise 0.2.10 from Dummit & Foote - Abstract Algebra):
Prove for any given positive integer $N$ there exist only finitely many integers $n$ with $\varphi(n) = N$, where $\varphi$ denotes Eu... |
H: Do all homogeneous systems with non-trivial solutions have columns of zeros?
I'm trying to think about this problem I'm faced with.
My peer stated that a non-trivial homogeneous system (which is square) has a column/row of zeros, but I'm trying to make sense of that. It's pretty mind-boggling at the moment. Can any... |
H: Proof of ∀(x,y)∈R[|x-y| ≥ |x| - |y|]
Is the following proof correct?
To prove: ∀(x,y)∈R[|x-y| ≥ |x| - |y|]; where R is the set of real numbers.
Proof:
Lemma: ∀(x,y)∈R[|x+y| ≤ |x| + |y|]
Since x and y are arbitrary real numbers we have,
∀(x,y)∈R[|x+(-y)| ≤ |x| + |-y|]
Since |y| = |-y|,
|x - y| ≤ |x| + |y| ⇔ -|x -... |
H: Does *smooth manifold* implies unique tangent space at each point?
The paper "A micro Lie theory
for state estimation in robotics" claims that
the space tangent to $M$ at $X$, which we note $T_X M$. The smoothness of the manifold, i.e., the absence of edges or spikes, implies the existence of a unique tangent spa... |
H: How to find this discrete limit?
While proving some discrete Hardy-type inequalities I tried to prove the following limit for non-negative sequence $a(n)$ and $p>1$
$$\lim_{p\rightarrow1}\frac{1}{p-1}\left[\sum_{n=1}^{r}a(n)^{p}-\left(\sum_{n=1}^{r}a(n)\right)^{p}\right]=\sum_{n=1}^{r}a(
n)\log \frac{a( n) }{\sum_{... |
H: A question about a localization of a graded ring
Let $R=\oplus_{i\in\mathbb{Z}} R_i$ be a (commutative) graded ring of type $\mathbb{Z}$. It can be shown that if $S$ is a multiplicative set consists of homogeneous elements, $R_S$ have a natural grading structure of type $\mathbb{Z}$.
My question is:
If $\mathfrak... |
H: A trivial question about continuity
$f:\mathbb{R}\longrightarrow\mathbb{R}:x\mapsto x^2$
This function is continuous as we all know.
Since for every point in the domain, we will always be able to draw a $\delta\epsilon-$rectangle,
for every $\epsilon$ which captures every point of $f(x)$ if it captures $x$.
As i fi... |
H: How to show function property holds for integers
I want to show that for all integers $x$ greater than 1,
$$f(x)=\left\lfloor{\frac{4x^2}{2x-1}-\left\lfloor{\frac{4x^2-4x}{2x-1}}\right\rfloor}\right\rfloor=3.$$
Upon graphing $f$, it's clear that this is probably true. I considered a monotonicity argument but I'm pr... |
H: Proving ${ \left\{\sum \left( ab+{b}^{2}+{c}^{2}+ac \right)\right\} }^{4}\geq 27\,{ \sum} ( ab+{b}^{2}+{c}^{2}+ac ) ^{3} ( c+a) ( a+b) $
For $a,b,c>0.$ Prove$:$ $$ \left\{ \sum\limits_{cyc} \left( ab+{b}^{2}+{c}^{2}+ac \right) \right\}^{4}\geq
27\,{ \sum\limits_{cyc}} \left( ab+{b}^{2}+{c}^{2}+ac \right) ^{3}
\le... |
H: Factor a quadratic when the leading coefficient is not equal to 1 and you can't factor by grouping?
If one has a quadratic, for example $5x^2-10x-2$, which has real roots which via the quadratic equation are $(5\pm \sqrt{35})/5$, can you find its factored form. As I understand it $5x^2-10x-2\ne(x-\text{root1})(x-\t... |
H: "Partition" without the disjoint condition
A partition of a set $A$ is defined as a set of pairwise disjoint sets whose union is $A$.
I'm interested in a related concept, where for a set $A$ you have $Q = \{A_1 \ldots A_n\}$ such that union of all $A_i$ is $A$ but $A_i$ needn't be pairwise disjoint.
I'm looking for... |
H: Differentiability of $\cos \lvert x\rvert$
I know that $f(x) = \cos\lvert x\rvert$ is differentiable at $x=0$ and I know what its graph looks like.
But if I differentiate $f(x)$ with respect to $x$ , I will have to apply the chain rule i.e,
$\frac {df(x)} {dx} = -sin\lvert x\rvert\cdot \frac {d\lvert x\rvert} {dx}$... |
H: Find the number of series with a certain condition
Q: Calculate the number of series: $a_1a_2a_3 \dots a_n$ of length $n$ that for all $a_i \in \{0,1,2,3\}$ and there is no occurrence of $3$ right of $0$.
Meaning: no $i,j \in \mathbb{N}$ exist so that $1 \leq i < j \leq n$ and $a_j = 3 , a_i = 0$
I tried to approa... |
H: Closed form solution to recurrence: $g(n)=(k-2)g(n-1)+(k-1)g(n-2)$
For the number of paths with exactly $n$ hops from one node to another in a $k$ node fully connected graph, we get the following recurrence:
$$g_n=(k-2)g_{n-1}+(k-1)g_{n-2}$$
With $g_1=1$, $g_0=0$ and $g_m=0 \;\; \forall \;\; m<0$.
Is there a way to... |
H: Combinatorics Problem : Additional Clause
A firm needs to obtain 5 van loads of mineral. The five vans available can
go to any of 11 places of mineral. The basic question
is how many possible ways this can be achieved. The 11 places all have different
kinds of elements in the mineral, so it is always important how ... |
H: Exist smooth function?
Let $f: \mathbb R \rightarrow [0,1] $ with $f(0)=1, f(1)=0$ a function differential such that exist $f^n(0)$, $f^n(1)$ for all $1 \leq n $ and they are equal to $0$ ($f^n$ is the $n$-th derivative ).
Can be $f([0,1])$ a smooth function?
I try moldering the functions like the $\sin, \cos$ or ... |
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