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H: Example of function where improper integral doesn't exist
Let $f \colon [1,\infty) \to \mathbb{R}$ be continuous, with $\lim_{x \to \infty}f(x) =0$. Does the integral \begin{equation*}\int_{1}^{\infty}\frac{f(x)}{x}\mathop{}\!\mathrm{d}x\end{equation*} necessarily converge?
I think the answer is "no", because from ... |
H: Proving $\pi=(27S-36)/(8\sqrt{3})$, where $S=\sum_{n=0}^\infty\frac{\left(\left\lfloor\frac{n}{2}\right\rfloor!\right)^2}{n!}$
I have to prove that:
$$\pi=\frac{27S-36}{8\sqrt{3}}$$
where I know that $$S=\sum_{n=0}^\infty\frac{\left(\left\lfloor\frac{n}{2}\right\rfloor!\right)^2}{n!}$$
Where do I get started?
AI: A... |
H: Excludе $x$ frоm this systеm оf equаtiоns.
How do you go about excluding $x$ from the system of equations?
\begin{cases}x^3-xy-y^3+y=0\\x^2+x-y^2=1\end{cases}
AI: The resultant of the two polynomials $x^3 - xy - y^3 + y$ and $x^2 + x - y^2 - 1$ with respect to $x$ is $5\,{y}^{5}-7\,{y}^{4}+6\,{y}^{3}-2\,{y}^{2}-y-1... |
H: Estimate definite integral by Maclaurin series with an error at most 10^-1
$\int_{0}^{1} \frac{\sinh x}{x}\mathrm{d}x$ with an error at most $10^{-1}$.
I tired to find expression of error in order to decide order n, but I can't express it because this Maclaurin series is not alternating series.
$$\frac{\sinh x}{x}=... |
H: Given $F(x)=\int\limits_x^{x^2}\frac{\sin t}{t}dt$, find $\lim_{x\rightarrow 0}F(x)$ and $\lim_{x\rightarrow 0}F'(x)$
Given $F(x)=\int\limits_x^{x^2}\frac{\sin t}{t}dt$, find $\lim_{x\rightarrow 0}F(x)$ and $\lim_{x\rightarrow 0}F'(x)$
Assume the options are ${-1, 0, 1}$
Intuitively I'm pretty sure the answer is ... |
H: Entropy of the Uniform Mixture of Discrete Probability Distribtuions
Consider the following inequality:
\begin{equation}
H\left(\frac{1}{3}p_{1} + \frac{1}{3}p_{2} + \frac{1}{3}p_{3}\right) \geq H(0.5p_{1} + 0.5p_{2})
\end{equation}
where H(.) denotes the Shannon entropy of the probability distribution in its argum... |
H: Why is the directional derivative not defined for $v_1=0$?
I have some trouble understanding how my prof. got to this conclusion.
I'm asked to find the directional derivative at point $\zeta=(0,0)$ of $$\left\{
\begin{array}{c} f(x,y)=\frac{x+xy}{\sqrt{x^2+y^2}} ,\forall(x,y)\neq(0,0) \\f(0,0)=0 \end{array}
\right... |
H: Show that if $\phi$ is an odd function on $(-l,l)$, its full Fourier series on $(-l,l)$ only has sine terms.
Show that if $\phi$ is an odd function on $(-l,l)$, its full Fourier series on $(-l,l)$ only has sine terms.
The full Fourier series is defined as $$\phi(x)=\frac{1}{2}A_0+\sum_{n=1}^{\infty}(A_n cos \frac... |
H: Niemytzki continious function
In Wiki, they state that this function, whose image is in [0,1] and that is defined on X with the Niemyetzki topology, is continuous:
Therefore, the preimages of sets of the form $(x-r, x+r) $ with $ x \in R $ and $r>0$ (that are exactly the open sets in R with the standard topology)... |
H: Proving that $f(n)=nlog(n)$ is a $b$-smooth function
First I start with the definition: a function $f:\mathbb{N} \rightarrow \mathbb{R}^{+}$ is b-smooth for an integer $b \geq 2$ if $f$ is eventually non decreasing and if
$$ \exists c \in \mathbb{R}^{+} \exists n_0 \in \mathbb{N} \forall n \geq n_0 \hspace{1cm} f(b... |
H: Why does this "gradient field test" not work on the spin field $S / r^2$? (from Strang's Calculus)
In section 15.2, Strang's Calculus explains that for any gradient field $\bf{F} = Mi + Nj$, ${\partial M \over \partial y} = {\partial N \over \partial x}$. (Strang calls this "test D" for identifying a vector field a... |
H: If $t<0$, what is $t\sum a_n$?
Let $t$ be a nonpositive real number (i.e. $t<0$) and $\{a_n\}$ be a nonnegative sequence if $$\sum a_n<\infty$$ then how do we prove or disprove that $$t\sum a_n<\infty?$$
AI: Since $a_n \geq 0$ by assumption, we have:
$$ \infty > \sum a_n \geq 0 $$
Now, if you multiply for some rea... |
H: Weak version of Hahn-Banach separation
Let $K\subset \mathbb{R}^n$ be a compact convex subset and $p\in \mathbb{R}^n$ be a point not in $K$. Then $p$ and $K$ can be strongly separated with a hyperplane $c_1 x_1+\cdots+c_nx_n=b$ (I mean "$c_1p_1+\cdots c_np_n <b$ and $c_1 k_1+\cdots c_nk_n > b$ forall $k\in K$").
I ... |
H: Solve $9x(1-x)y^{\prime\prime} - 12y^\prime + 4y = 0$ using power series
Let $y(x) = \sum_{n=0}^\infty c_nx^n \Rightarrow y^\prime(x) = \sum_{n=0}^\infty nc_nx^{n-1}, y^{\prime\prime}(x) = \sum_{n=0}^\infty n(n-1)c_nx^{n-2}$. Now:
$$9x(1-x)\sum_{n=0}^\infty n(n-1)c_nx^{n-2} - 12\sum_{n=0}^\infty nc_nx^{n-1} + 4\sum... |
H: Finding a Pivotal Quantity
Let $X_1, \dots, X_n$ be i.i.d with probability density function
$$f(x;\theta) = \theta \frac{8^\theta}{x^{\theta +1}}, \qquad x\geq 8,\qquad \theta > 0 $$
And given the statistic $W(X_1, \dots, X_n) = \sum_{i=1}^n ln\left(\frac{X_i}{8}\right)$, I want to find a function $g(\theta)$ s.t:... |
H: Having trouble working out how two vector expressions are equivalent
I'm doing some coursework on linear regression, and part of it requires finding a closed-form solution of the mean-squared error minimisation problem:
$$\min_{\bf w} \frac{1}{n} \sum_{i=1}^{n} (\mathbf{w}^T \mathbf{x}_i - y_i)^2$$
This is how I tr... |
H: Derive $P[D = 1|X]$ from $X = f(X,-1)P[D =-1|X] + f(X,1)P[D =1|X]$
Given the random variables $X: \Omega \to \mathbb R$ and $D: \Omega \to \{-1,1 \}$, and the (measurable) function $f: \mathbb R \times \{-1, 1 \} \to \mathbb R$.
Show that if
$$
X = f(X,-1)P[D =-1|X] + f(X,1)P[D =1|X],
$$
then
$$
\frac{X - f(X,... |
H: Are affine morphisms with coherent direct image finite?
Let $f:X \longrightarrow Y$ be a morphsim of Noetherian schemes. I was doing excersise 5.5 of Hartshorn Algebraic Geometry and in (c) i showed that finite morphisms preserve coherence (i.e. if $\mathscr{F}$ is coherent on $X$ then $f_*\mathscr{F}$ is coherent ... |
H: In a casino a player can win 1 euro with a probability of 18/28 and loses 1 euro with a probability of 20/38.
In a casino, for each bet on the wheel, a player wins 1 euro with the
probability of $\frac{18}{38}$ and loses 1 euro with a probability of $\frac{20}{38}$.
a) What is the average value won per game?
b)... |
H: ${\log}_{a}{x}\neq {\int}^{x}_{1}{\frac{1}{t}}dt$
In most calculus textbooks, $\ln{x}$ is defined to be ${\int}^{x}_{1}{\frac{1}{t}}dt$. Some textbooks validate this definition by demonstrating that this function $\int^{x}_{1}{\frac{1}{t}}dt$ has all the properties of a logarithmic function (I've included pictures ... |
H: Given that $n^4-4n^3+14n^2-20n+10$ is a perfect square, find all integers n that satisfy the condition
So, I tried solving that by
$$n^4-4n^3+14n^2-20n+10=x^2\\10=x^2-a^2, a^2=n^4-4n^3+14n^2-20n+10\\10=(x+a)(x-a)$$ but I couldn't find any integers when I solved it
AI: It always helps to form squares from the bigge... |
H: Bound on integral on function implies bound order of entire function
Let $f$ be an entire function such that $\int_{\mathbb{C}}|f(z)|^2e^{-|z|^2} <\infty$ (with Lebesgue measure on $\mathbb{C}$). I need to prove that $f(z)$ has order $\le 2$.
My ideas:
Try to find bounds on coefficients and derive information abo... |
H: $N(0,\sigma^2_n)$ and $\sigma^2_n\to\sigma^2$ imply $N(0,\sigma^2_n)\overset{d}{\to}N(0,\sigma^2)$?
The following result seems to be natural to me:
$$N(0,\sigma^2_n) \text{ and } \sigma^2_n\to\sigma^2 \implies N(0,\sigma^2_n)\overset{d}{\to}N(0,\sigma^2)$$
as $n\to\infty$, where $\overset{d}{\to}$ denotes convergen... |
H: How to calculate determinant of a (N-1) order Matrix?
For $n \geq 2,$ consider the following square matrix of order $(n-1)$
\begin{array}{cccccc}
3 & 1 & 1 & 1 & & 1 \\
1 & 4 & 1 & 1 & \dots & 1 \\
1 & 1 & 5 & 1 & \cdots & 1 \\
1 & 1 & 1 & 6 & & 1 \\
& & & & \ddots & \vdots \\
1 & 1 & 1 & 1 & \cdots & n+1
\end{arra... |
H: Finding an identity to simplify this combinatorics solution
Steve flips 499 coins and Marissa flips 500 coins. What is the probability that Marissa flips more heads than Steve does?
We use casework for each of the possible number of heads that Steve flips. The first is $$\frac{1}{2^{499}\cdot2^{500}}\left(\binom{... |
H: Definition of chain of tree
I am trouble understanding the definition of chain of tree at p15.
Here is a rooted tree. The root is "a".
abc is clearly chain.
However, I cannot understand whether bc is chain or not.
a
|
bーd
|
c
At first, I thought chain is only "abc","ab","a","abd".
But looking at the proof of Lem... |
H: Resolvent definition: bounded operator vs. unbounded operator
Maybe my question is obvious in some sense, but I ask here because I didn’t find a “satisfactory” answer on the web.
If we have a bounded or unbounded operator, the definition of resolvent changes? And, in general, why one should prefer to work with a bo... |
H: Radius of convergence of a power series summing from $- \infty$ to $0$.
How would one go about computing the radius of convergence of, say, the following power series:
$$\sum_{n=-\infty}^0 n \, 3^{-n} z^n.$$
It is tempting to directly apply the Cauchy-Hadamard theorem here, but the statement is true for power serie... |
H: What do $\min{[f,g]}$ and $\max{[f,g]}$ mean in my proof of continuity?
I must understand this proof of continuity. I understand the basics of continuity and algebra of continuity of limits. So I add the picture of the proof such that it is in the book of Real Analysis.I would really appreciate if someone could ... |
H: Computing a 3D rotation matrix aligning 1 orthonormal basis to another
I have 2 sets of 3 vectors ($\vec{u_1}$, $\vec{v_1}$, $\vec{w_1}$, $\vec{u_2}$, $\vec{v_2}$, $\vec{w_2}$) and the 3 vectors form an orthonormal basis. That is:
$$
|\vec{u}|=|\vec{v}|=|\vec{w}|=1 \\
\vec{u} \cdot \vec{v}=0 \\
\vec{u} \times \vec{... |
H: Is there a rigorous proof that $|G|=|\text{Ker}(f)||\text{Im}(f)|$, for some homomorphism $f\,:\,G\rightarrow G'$.
Is there a rigorous proof that $|G|=|\text{Ker}(f)||\text{Im}(f)|$, for some homomorphism $f\,:\,G\rightarrow G'$? Can anyone provide such a proof with explanations?
AI: Here is a proof in full detail.... |
H: Help with understanding how to find sum of series
Find the sum to $\sum_{n=0}^{\infty} \frac{(n+1)(n+2)}{3^n}x^n$
I do this:
$\sum_{n=0}^{\infty}x^n=\frac{1}{1-x}$ I then take the derivative twice and get
$\sum_{n=1}^{\infty}nx^{n-1}$
$\sum_{n=2}^{\infty}(n-1)nx^{n-2}$ And the change the sum index to n=0 and get an... |
H: Finding the area of the region defined in polar coordinates by $0\leq\theta\leq\pi$ and $0\leq r\leq\theta^3$
Find the area of the region defined in polar coordinates by $0 \leq \theta \leq \pi$ and $0 \leq r \leq {\theta}^3$.
I tried using the formula
$$A = \int_{0}^{\pi} \frac{1}{2} r(\theta)^2 d\theta$$
Howeve... |
H: Lipschitz continuity and boundedness of derivatives.
Suppose $f(x)$ is a function that has derivative in the domain of our concern. We all know that if the derivative is bounded then $f$ is Lipschitz continuous. I was wondering if the above statement is an if-and-only-if statement since if the derivative is unbound... |
H: $\sigma$-algebras and sample space
Does a $\sigma$-algebra always contain the sample space (or the full set over which it is defined) ?
I know the smallest $\sigma$-algebra over $\Omega$ can be defined as $G = \{\emptyset, A, A^{c}, \Omega\}$, and the largest by the powerset of $\mathscr{P}(\Omega)$, where
$A \sub... |
H: Let $h(Y)$ be a random variable such that $E|h(Y)| < \infty$ and $K(X)$ a random variable. Show that $E[h(Y)|X,K(X)] = E[h(Y)|X]$
I have an idea but I don't know how to write it formally. Well, if we are conditioning over $X$ and $K(X)$, this second random variable could be a transformation of $X$ but since we are ... |
H: Solve $\int_{|z|=2}\frac{e^{3z}}{(z-1)^3}dz$ using residue.
I'm trying to evaluate $$\int_{|z|=2}\frac{e^{3z}}{(z-1)^3}dz$$ using the residue theorem. I get a pole of order $3$ at 1 with a residue of $\frac{9}{2}e^3$. But since the absolute value of the residue (which in this case is exactly the residue) is bigger ... |
H: Number of full binary trees is Catalan, What is the number of Binary trees?
In exercise 12-4 of "Introduction to Algorithms" by Cormen et.al (third edition), they mention that the number of Binary trees with $n$ nodes is given by the Catalan numbers,
$$b_n = \frac{1}{n+1}{2n \choose n}$$
In the Wikipedia article on... |
H: What field do entries of eigenvectors belong in?
I have the following problem:
"Given the matrix $A = \begin{pmatrix} 1&i\\ i&1 \end{pmatrix}$, find the eigenspaces of the respective eigenvalues".
First I found the eigenvalues to be $\lambda_1 = 1+i$ and $\lambda_2 = 1-i$. Then, using that the eigenspace of an eig... |
H: Problem about inequality with symetric matrices and inner product
Let $A$ and $B$ be two matrices of order $n$ with entries in $\mathbb{R}$.
$\newcommand{\lg}{\langle}$ $\newcommand{\rg}{\rangle}$
a) If $A$ and $B$ are symmetric then
$$ \lg(A^{2} + B^{2})x, x \rg \geq \lg(AB+BA)x,x\rg $$
for any $x \in \mathbb{... |
H: Power series method to solve $(1-x)y^\prime + y = 1 + x, y(0) = 0$
Let $y(x) = \sum_{n=0}^\infty c_nx^n \Rightarrow y^\prime(x) = \sum_{n=0}^\infty nc_nx^{n-1}, y^{\prime\prime}(x) = \sum_{n=0}^\infty n(n-1)c_nx^{n-2}$. Now:
$$(1-x)\sum_{n=0}^\infty nc_nx^{n-1} + \sum_{n=0}^\infty c_nx^n = 1 + x$$
$$\sum_{n=0}^\inf... |
H: Proving the orthogonality of $\sin\frac{2\pi x}{\pi-e}$ and $\cos\frac{2\pi x}{\pi-e}$
I want to prove the orthogonality of the functions: $\sin\left(\dfrac{2\pi x}{b-a}\right)$ and $\cos\left(\dfrac{2\pi x}{b-a}\right)$, where $b=\pi$ and $a = e$
My work:
$$\begin{align}
\int^{\pi}_{e} \frac{1}{2} \sin\left(\fra... |
H: Birkhoff ergodic theorem on a lattice
Let $\mathbb{P}_0$ be a probability measure on $\mathbb{R}$. Let $\Omega = \mathbb{R}^{\mathbb{Z}^d}$ and $\mathbb{P} = (\mathbb{P}_0)^{\otimes \mathbb{Z}^d}$ so the the canonical process $X:\Omega \rightarrow \Omega$, defined by $X(\omega) =\omega$, can be regarded as iid rand... |
H: Schwarz space $S(\mathbb{R})$ is dense inside $L_p(\mathbb{R})$-spaces
I was wondering why the schwarz functions $S(\mathbb{R})$ are dense inside the $L_p(\mathbb{R})$ spaces and I was reading this answer, but I don't understand why the $g_t$ are in $S(\mathbb{R})$. Could someone explain this?
AI: Note the followin... |
H: Is the definition of the Riemann sum from Thomas' Calculus correct?
I'm having trouble with theoretical understanding of the Riemann sum with this explanation/definition from Thomas' Calculus. I checked Wikipedia and it seems to state virtually the same.:
On each subinterval we form the product $f(c_k)*∆x_k$. This... |
H: Find linear map depending on parameter t
I'm currently sitting on the following problem:
For what $t \in \mathbb{R}$ exists a linear map $\varphi_t: \mathbb{R}^{1x3}\rightarrow \mathbb{R}^{2}$
$\varphi_t:\left\{
\begin{array}{lcl}
\left( 0, -1, 0 \right) & \mapsto &
\left( \begin{array}{c} -1 \\ 1 \end{array}\r... |
H: Distance from a point to a complement of a set
Let $D=\{(x,y) \in \mathbb{R}^{2}: x^2+y^2 \leq 1\}$ be the unit disk and consider $U$ be a open subset of $\mathbb{R}^{2}$ such that $D \subset U$. Since $D$ is compact and $U^c$ is closed, $\operatorname{dist}(D,U^c)=r>0$.
Intuitively, it seems that $\|y\| \geq r+1$... |
H: Find $\lim\limits_{x \to \infty}{\mathrm{e}^{-x}\int_{0}^{x}{f\left(y\right)\mathrm{e}^{y}\,\mathrm{d}y}}$
Given $f(x)$ is a continuous function defined in $(0,\: \infty)$ such that $$\lim_{x \to \infty}f(x)=1$$ Then Find $$L=\lim_{x \to \infty}{\mathrm{e}^{-x}\int_{0}^{x}{f\left(y\right)\mathrm{e}^{y}\,\mathrm{d}y... |
H: What is the $\|f\|_{L^{\infty}}=$?
For
$$f(x):=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$
what is the $\|f\|_{L^{\infty}}=$?
Is it just $1$?
AI: We have $f(x)=\tanh{x}$ which is continuous on the whole real line. Since $f^{\prime}(x)=1/\cosh^{2}{x}>0$ for all $x\in \mathbb{R}$, then $f$ is strictly increasing on $\math... |
H: Prove universal morphism is unique up to unique isomorphism.
I'm following along Wikipedia page(https://en.wikipedia.org/wiki/Universal_property) on universal property, and this seems it should be trivial, but I couldn't finish the proof.
The definition I am working with:
The problem
So I want to prove that if $(... |
H: Maximize area of isosceles triangle with given median
Given icoscales triangle with sides $a, b, c, a = b$; median performed to the side of triangle, say, to b, denoted as $m_b.$
Note: I need to maximize area of triangle. I need to solve it using inequalities, not Lagrange multipliers etc.
Firstly i tried to solve ... |
H: Show that if $A$ and $B$ are compact subsets of $(\mathbb{R}^m,||.||_2)$ not empty and disjointed, then $\inf\{||a-b||_2:a\in A,b\in B\} > 0$
Show that if $A$ and $B$ are compact subsets of $(\mathbb{R}^m,||.||_2)$ not empty and disjointed, then $$\inf\{||a-b||_2:a\in A,b\in B\} > 0$$
I know the definitions and I b... |
H: Find a constant $c$, such that $f(x)\leq cx^2$ for every $x\geq 0$?
For
$$f(x):=\log(e^x+e^{-x})$$ and
$$f'(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$
Can we find a constant $c$, such that $f(x)\leq cx^2$ for every $x\geq 0$?
Clearly, $f'(0)=0$. It seems we need to have a bound on $f''$.
Based on the some answers,... |
H: Can $\frac{\mathrm d}{\mathrm dx}$ increase the support of a function?
Let "support" mean the closed support.
Can $\frac{\mathrm d}{\mathrm dx}$ increase the support of a function? That is, is there any $f=f(x)$ in one variable with $\operatorname{supp}(f)$ completely contained in $\operatorname{supp}(f')$? Can we ... |
H: What's the advantages of writing standard calculus into lie differentiation?
Say $\dot{x} = f(x)$, $x\in\mathcal{M}$, $\phi: \mathcal{M} \mapsto \mathcal{M}$.
Then $\dot{\phi} = f \cdot \nabla_x \phi $.
However, one can define a lie differentiation and write
$f \cdot \nabla_x \phi = \mathcal{L}_f \phi$.
QUESTION:
... |
H: Axiomatizability of a relative complement
Given a fixed first order lexicon $\mathcal{L}$, suppose $\mathcal{K}$ is an axiomatizable class such that $\mathcal{K}\subseteq Mod(\varphi)$ for some sentence $\varphi$. If $Mod(\varphi)-\mathcal{K}$ is axiomatizable, Does $\mathcal{K}$ necessarily to be finitely axiomati... |
H: Find value of t then the series is convergent?
$$\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 n } } { ( n + 2 ) ^ { 2 n + t } ( n + t ) ^ { n + 2 t } }$$
I use the ratio test, than I saw the series failed.
Right or wrong ?
AI: Using root test, I find:
$$\sqrt[n]{\frac { n ^ { 3 n } } { ( n + 2 ) ^ { 2 n + t } ( n ... |
H: How to prove that there is an open simply connected subspace containing a simple closed curve
Let $\Omega$ be an open set in $\mathbb C$. Let $\gamma$ be a simple closed curve (i.e., $\gamma$ is homeomorphic to $S^1$) in $\mathbb C$. Let $W$ be the bounded component of $\mathbb C\setminus\gamma$. Suppose $\gamma\cu... |
H: Does this sequence of functions converges uniformly?
¿Converges uniformly? $$f_n (x)= n^3 x^n (1-x)^4$$ for $$ x\in\mathcal [0,1]$$
I have this, it's clear that $$\lim_{x\to \infty}f_n (x)= n^3 x^n (1-x)^4 =0$$
Then
$$ |f(x)-f_n (x)|=|f_n (x)| $$
So, note that
$$n^3 x^n (1-x)^4 \leq n^3 x^n \leq n^3 < \epsilon $$ ... |
H: Convergence $\sum \frac {1} {k^2}$
Let $a_n=\frac{1}{1^2} + \frac{1}{2^2} + .....+\frac{1}{n^2}\;\forall n \in \mathbb{Z_+}$. Prove that $a_n \leq 2 - \frac{1}{n} \;\forall n \in \mathbb{Z_+}$.Deduce the convergence of ${a_n}$.
I have proved the inequality using mathematical induction. But I'm stucked at proving ... |
H: mapping between sets
If $f$ is a mapping between two sets $A$ and $B$ and if $ a\in A $ and $f(a)=a$ what is that called? I looked though some Abstract Algebra books but couldn't recall.
AI: If $f(a) = a$ that's called a "fixed point".
It is usually assumed that $f: X\to X$ but that isn't necessary. It is necessar... |
H: Nonhomogenous examples in Fraïssé limit
On wiki page on Fraïssé limit, it says that neither $⟨\Bbb{N}, < ⟩$ nor $⟨\Bbb{Z}, < ⟩$ are the Fraïssé limit of FCh (Fraïssé class) because although both of them are countable and have FCh as their age (the class of all finitely generated substructures), neither one is homo... |
H: Original Fraïssé's paper and texts on Fraïssé theory
I wonder where I can find the Fraïssé's paper "Sur l’extension aux relations de quelques propriet es desordres", appeared in Annales Scientifiques de l'Ecole Normale Superieure. Troisieme Śerie 71 (1954), 363–388." (or an English translation).
Also I'd li... |
H: $(x,y)$ pairs in lattice $Z^2$ that are co-prime with euclidean-norm at most $k$
Let $B(k) = \{(x,y)\in Z^2 ~|~ x^2+y^2\leq k^2\}$, where $Z$ is the set of integers.
It is quite straight forward to show that $|B(k)|$ is $\Theta(k^2)$.
My question is whether the number of co-prime pairs $(x,y)$ in $B(k)$ also $\Thet... |
H: Infinite Cartesian Product: Understanding
I'm having a bit of trouble understanding the definition of the infinite cartesian product, particularly with the intuition behind it.
According to my textbook, Enderton's Elements of Set Theory, the infinite cartesian product takes the cartesian product of each set $X_i$ f... |
H: System of equations involving 3 variables , whether it it solvable for real values of k
Can anyone confirm if I am correct for this question, thank you.
There are positive real numbers $x$ and $y$ which solve the equations $2x + ky = 4, \;x + y = k$,for
(a) all values of $k$
(b) no values of $k$
(c) $k = 2$ only
(... |
H: There is a function $f:X \to \Bbb{R}$ such that $a$ is a limit point of $X$, $f$ does not have limit at $a$, but $|f(x)|$ has a limit at $a$.
Is this true? There is a function $f:X \to \Bbb{R}$, $X \subseteq \Bbb{R}$, such that $a$ is a limit point of $X$, $f$ does not have limit at $a$, but $|f(x)|$ has a limit at... |
H: Uniform Continuity of Characteristic Function
I am trying to understand the concept of uniform continuity as it pertains to characteristic functions.
First my understanding of uniform continuity:
Def:
$$\forall x_0, \forall \epsilon>0, \exists \delta>0,\hspace{4mm} \text{if}\hspace{4mm} |x-x_0|<\delta \hspace{... |
H: how to prove that this problem?
Is there partial laplace derivative equations? I am so confused.
Show that the function provides the equation.
\begin{equation}
\label{simple_equation0}
u = {\varphi }(xy)+\sqrt{xy}{\psi}(\frac{y}{x})
\end{equation}
\begin{equation}
\label{simple_equation1}
x^{2}\frac{\part... |
H: Prove that 2 is not a primitive root of any prime of the form $3\cdot 2^n+1$ for $p>13$
I am really struggling with this proof. This doesn't seem like it should be that hard. All I have been trying to do is find a $k<3.2^n$ such that $2^k\equiv 1($mod $ 3\cdot 2^n+1)$, but it turns out there are a lot of numbers be... |
H: The binomial coefficient $\left(\begin{array}{l}99 \\ 19\end{array}\right)$ is $ 107,196,674,080,761,936, x y z $ , Find $x y z$
The binomial coefficient $\left(\begin{array}{l}99 \\ 19\end{array}\right)$ is a 21 -digit number:
$
107,196,674,080,761,936, x y z
$
Find the three-digit number $x y z$
I showed th... |
H: Finding eigenvalues and eigenvectors of a certain matrix
Find eigenvalues and eigenvectors of a matrix $A_{n\times n}$ where elements $a_{ij} $ of $A_{n\times n}$ are given as
\begin{cases}
\alpha, & \text{if }i=j \\[2ex]
1, & \text{if }|i-j|=1\\[2ex]
0 & \text{otherwise}
\end{cases}
where $\alpha$ is a constant... |
H: Calculate $6^{1866}$ in $\mathbb{Z}_{23}$
Calculate $6^{1866}$ in $\mathbb{Z}_{23}$
$Solution:$ Note that $1866=22\cdot 84 + 18$ then by Fermat's theorem
$$[6^{1866}]=[6^{22}]^{84}[6^{18}]=[1]^{84}[6^{18}]=[6^{18}]$$
Then $6^6=46656=2028\cdot 23 + 12$ It is true that $$[6^{1866}]=[6^6]^3=[-11]^3=[121][-11]=[72]=[3]... |
H: continuous and inverse function problem
Show that $f:\Bbb{R^n} \rightarrow \Bbb{R^m}$ is continuous if and only if for each subset $E \subseteq \Bbb{R^m}$ we have $$f^{-1}(E^\circ) \subseteq [f^{-1}(E)]^{\circ}$$, where $E^\circ$ denotes the interior of the set E.
Theorem: A function $f:\Bbb{R^n} \rightarrow \Bbb... |
H: Confusion on a directional derivative expression
I'm studying "A Visual Introduction to Differential Forms and Calculus on Manifolds" and came across a confusing part on intro to directional derivatives. First the definition
The directional derivative of $f:\mathbb{R}^2\to\mathbb{R}$ at $(x_0,y_0)$ in the directio... |
H: A problem regarding a maximal ideal in a polynomial ring in several variables
$\mathbf {The \ Problem \ is}:$ Is the ideal $I =\langle x^2-2,y^2+1,z\rangle$ maximal in the polynomial ring $R =\mathbb Q[x,y,z]$ ?
$\mathbf {My \ approach} :$ Actually, by $3rd$ isomorphism theorem of rings, quotenting both $R$ and $I$... |
H: Finding a bijective correspondence between $X^{\omega}$ and $\mathcal{P}(\mathbb{Z}_+)$
Let $X = \{ 0,1 \}$ and let $\mathcal{P} (\mathbb{Z}_+) $. Find a
bijective correspondence between $\mathcal{P} (\mathbb{Z}_+) $ and the
cartesian product $X^{\omega} $ or ${\bf show}$ there isn't one
Attempt to solution:
I cl... |
H: If $a_n$ converges to $a$, can we say $a_n^c$ converges to $a^c$?
I'm doing a lot of practice problems with sequences, and I've noticed a number of problems ask about the convergence of the sequence raised to a positive power. It seems like in all the examples that I've tried, if $a_n$ converges to $a$, then $a_n^c... |
H: Method to solve factored quadratic diophantine equations?
Is there a method that can solve all quadratic diophantine equations of the following type
$$X (X + a) = Y (Y + b)$$
where $a,b$ are given integers?
AI: $X (X + a) = Y (Y + b) \implies (2 X + a)^2 - (2 Y + b)^2 = a^2 - b^2$
Get finite set solutions of differ... |
H: Another series involving $\log (3)$
I will show that
$$\sum_{n = 0}^\infty \left (\frac{1}{6n + 1} + \frac{1}{6n + 3} + \frac{1}{6n + 5} - \frac{1}{2n + 1} \right ) = \frac{1}{2} \log (3).$$
My question is can this result be shown more simply then the approach given below? Perhaps using Riemann sums?
Denote the s... |
H: Asymptotically, can supercomputers easily solve the Travelling Salesman Problem, why or why not?
I just want to know if supercomputers could easily solve the TSMP or would still take a lot of time, as it does now?
AI: You must understand that no classical computer in the world nor in the universe will ever solve a ... |
H: show $I(a,1) + I(-a,1)\ge2$
$I(a,b)$= $\int_1^e x^a\ln^bx \,dx, b > 0$
I need to show that $I(a,1) + I(-a,1)\ge2$
I took both integrals. For the first one I get:
$I(a,1)$= $\int_1^e x^a\ln x \,dx$ = $\frac1{(a+1)^2}$($ae^{a+1} + 1)$
The second one should be the same but $-a$ will take place for $a$.
Is there any e... |
H: Number of homomorphisms from direct products of $\mathbb{Z}_n$ to $\mathbb{Z}_{18}$
How many homomorphisms are there from $\mathbb Z_3\times \mathbb Z_4\times\mathbb Z_9$ to $\mathbb Z_{18}$.
I tried to find possible kernals. The answer is $54$ but I'm getting something else. Can anyone show me some easy way to co... |
H: Isomorphism of cohomology related to Kunneth formula
Let $S$ be a smooth complex (rational) algebraic surface, and $\mathcal{F}$ be a quasi-coherent sheaf such that $H^2(S, \mathcal{F}) = 0$.
Then, by Kunneth formula, we have $H^2(S \times S, \mathcal{F} \boxtimes \mathcal{F} ) \simeq H^1(S, \mathcal{F}) \otimes H^... |
H: Proving a system of equations has only one solution
I want to find all critic point of $f(x,y)=xe^y-ye^x$. So I calculated $\nabla f(x,y)=(e^y-ye^x, xe^y-e^x)$ and tried to solve $$\begin{cases}e^y-ye^x=0 \\ xe^y-e^x=0 \end{cases}\qquad \Rightarrow \qquad \begin{cases}(1-xy)=0 \\ xe^y=e^x \end{cases}$$
A trivial so... |
H: prove that $\int_{0}^{\cos^2{x}}\arccos\sqrt{t}\ dt + \int_{0}^{\sin^2{x}}\arcsin\sqrt{t}\ dt =\frac{\pi}{4}$
I have to prove the following problem:
$$\int_{0}^{\cos^2{x}}\arccos\sqrt{t}\ dt + \int_{0}^{\sin^2{x}}\arcsin\sqrt{t}\ dt =\frac{\pi}{4}$$
I know that $\arccos(x)+\arcsin(x)=\frac{\pi}{2}$ but now, I don'... |
H: Which is the spectrum of this operator?
Let $T : \ell_2 \to \ell_2$, $T(x_1,x_2,x_3,...,x_n,...) = (0,\frac{x_1}{1},\frac{x_2}{2},\frac{x_3}{3},..., \frac{x_n}{n},...)$. Which is the spectrum of this operator?
AI: It is easy to find the spectrum if you realize that this is a compact operator. To show that it is com... |
H: What does the notation $A\in\mathscr{B}(H_1, H_2)$ mean?
I am sorry for the trivial question, but I am a little bit confused about this notation in literature. Let $H_1$ and $H_2$ be two Hilbert spaces. I am interested in understanding what means that an operator $A$ is bounded from $H_1$ to $H_2$, i.e. $A\in\maths... |
H: The Axiom of Choice: Proof Validity
Synopsis
In Enderton's Element's of Set Theory, he introduces several forms of the Axiom of Choice. Currently, I've gotten through the first and second forms. Mainly:
(1) For any relation $R$, there is a function $H \subseteq R$ with dom $H$ = dom $R$
(2) For any set $I$ and any... |
H: Riemann Integration vs Lebesgue Integration
If a function is not Riemann integrable, what does it mean geometrically?
Is it that we can't use integration to find out area under the curve?
So basically when a function is Riemann integrable then it can tell us about area.
I am asking this because I saw a function ... |
H: infinite-dimensional inner product space
I've been asked to get an example of an infinite-dimensional inner product space, so I wrote this
As an example is there anything wrong with what I wrote?
AI: Example: infinite dimensional Hilbert spaces are inner product spaces and they have infinite dimension. $l^2$ is ... |
H: Show that $\sum_{i=1}^\infty\frac{42}{(i+1)^3}$ converges.
I am having some difficulties proving this statement. Any help would be appreciatted. I have proved that $\left (\frac{1}{2^i} \right)_{i \in \mathbb{N}}$ is summable, however, I couldn't prove it for this one.
P.S. I am not sure if the formula is readable... |
H: How to solve $\tanh(x-y)=\frac{y}{2t}?$
How to solve the following equation about variable $y$:
$$\tanh(x-y)=\frac{y}{2t}?$$
where $x$ is fixed and for small $t>0$ and large $t<\infty$, there would be different cases.
AI: With suitable change of variables/constants, the equation can be written in the cleaner form
... |
H: how does knowing the indeterminate form of a limit help in solving that limit?
i know what indeterminate forms are, but fail to find their use while solving questions on limits. I know that 0/0 and infinity/infinity forms indicate use of l'Hopital's rule, but i dont know what other indeterminate forms lead us to, f... |
H: Can I always change the order of integration in an ordered multidimensional integral?
Imagine I have an integral of the following form:
$$I = \int_{-\infty}^{\infty} d\tau_1 \int_{\tau_1}^\infty d\tau_2 \int_{\tau_2}^\infty d\tau_3\ f(\tau_1,\tau_2,\tau_3) \tag{1}$$
Can I always commute the integrals, by changing t... |
H: Prove that if $a_n \to 1$ then $\sqrt[n]{a_n} \to 1$ if $n \to \infty$
Prove that if $a_n \to 1$ then $\sqrt[n]{a_n} \to 1$ if $n \to \infty$.
What could be the way to prove that in that case also $\sqrt[n]{a_n} \rightarrow 1$?
AI: You are confused. The question has nothing to do with any series. It is question ab... |
H: Calculate the following integral $\int_{|z|=1}\frac{z^m}{(z-a)^n}dz$
Given $n,m\in\mathbb{N},|a|\neq1$
Calculate the following integral $\int_{|z|=1}\frac{z^m}{(z-a)^n}dz$
I thought maybe using Cauchy's integral formula and I'm not sure what happens when $a$ is outside the domain
AI: For $|a| <1$ it is $2\pi i/{(n... |
H: Parties and distributions
It is known that the distribution of the people coming to the party is Poisson with a rate of 0.9. The DJ is going to play only if there are people. What is the probability that the DJ is going to play in front of one person exactly when it is known that no more than two people came to the... |
H: Modular calculation high exponent?
I want to show, that $5^{96}\equiv -1 \pmod{193}$, without using the formula for quadratic residue.
So far I have :
$5^{96}\equiv 5^{4\cdot24} \equiv 625^{24}\equiv 46^{24}\equiv 186^{12}\equiv -7^{12}\equiv 7^{12}\equiv 7^{3\cdot4}\equiv 150^4\equiv -43^4\equiv 43^4\equiv 112^2\... |
H: Infinite dimensional separable Hilbert spaces having an open countable dense subset
While working on infinite dimensional Hilbert spaces, I came up with the following question: under what conditions is the existence of an open dense countable subset assured? If we assume that the space is separable, we are certain ... |
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