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H: Let $A$ be $2\times 2$ Orthogonal matrix such that det$A= -1$. Show that $A$ represents reflection about the line in $R^2$.
Every matrix is corresponding to a linear transformation. Reflection is a linear transformation called $T$. So $T^2=I$. Also modulus of eigen values of $A$ is $1$.Since it has determinant $-1$... |
H: Why does this sequence generate all numbers in 2^n?
Here's a sequence, for fixed $n \in \mathbb{N}$:
$$ j_{i+1} = (5 j_i + 1) \pmod{ 2^n} $$
This update rule can be simplified to:
$$ j_{i} = \frac{5^i - 1}{4} \pmod{ 2^n}$$
One can easily verify the above via proof by induction.
Now when you start with $j_1 = 1$,... |
H: The series $\sum_{n=1}^\infty nc_n (x-a)^{n-1}$ converges uniformly to $f'$ on the interval $[a-r, a+r]$
(Tao Vol.2, P.77) Let $\sum_{n=0}^\infty c_n(x-a)^n$ be a power series, and let $R$ be the radius of convergence. Suppose that $R>0$ (the series converges at least one other point than $x= a$). Let $f : (a- R, ... |
H: Evaluating $\int\frac {dx}{1+x^2}$
Evaluate
$$\int\frac{dx}{1+x^2}$$
Please help me find my mistake.
I have integrated $\frac {1}{1+x^2}$ and gotten the correct result by making a mistake in the substitution.
I imagined a triangle, with $1 = \cos\theta$ and $x = \sin\theta$
I then integrated $d\theta$ and got $\a... |
H: Is the series $\frac{\sin (an)}{\ln(n+1)}$ convergent?
Is the series $\displaystyle \sum\frac{\sin (an)}{\ln{(n+1)}}$ convergent? ( $a\in \mathbb{R}$ )
I know that $\displaystyle \sum\frac{1}{\ln{n}}$ diverges. I am trying to use the comparison test but fails to come up with a good result.
AI: $\sum a_n \sin (nx)$ ... |
H: Why is the unique mapping from $\emptyset$ to $Y$ inclusion?
I am sorry, but I edited my question several times.
I am reading "Set Theory and General Topology" by Takeshi Saito(in Japanese).
The author wrote as follows:
When $X$ is a subset of $Y$, a mapping from $X$ to $Y$which maps $x \in X$ to $x \in Y$ is call... |
H: Solve the integral $\int_1^3\!\sqrt{x-\sqrt{x+\sqrt{x-...}}}\,\mathrm{d}x$
As an extension to my discussion in one of the answers to my previous question on simplifying the integrand, I'd like to evaluate the following integral: $$\int_1^3\!\sqrt{x-\sqrt{x+\sqrt{x-...}}}\,\mathrm{d}x$$
The above radical, when solve... |
H: How does WolframAlpha solve this recursion?
I have the following recursion:
$$x_n=\frac{n-1}{n}x_{n-1}+\frac{1}{n}\left\lfloor\frac{n}{2}\right\rfloor.$$
WolframAlpha gives a solution to this recursion as
$$x_n=\frac{C_1+\left\lceil\frac{1-n}{2}\right\rceil^2-\left\lceil\frac{1-n}{2}\right\rceil+\left\lfloor\frac{n... |
H: Definition of vertices of a polytop in $\mathbb{R}_{+}^d$
Having the following set:
$V=\left\{v\in{}\mathbb{R_{+}^{d}}\hspace{2pt}:\hspace{2pt}\sum_{i=1}^dv_i=K\hspace{4pt}\text{and}\hspace{4pt}\forall{}v_i\hspace{2pt},v_i\leq{}1\right\}$. That is, all the vectors with entries between 0 and 1 that sum up to $K\leq{... |
H: How to prove that $|f|\leqslant\|f\|_\infty$ almost everywhere?
Let $f\in\mathcal L^\infty(X,\Sigma,\mu)$, where $(X,\Sigma,\mu)$ is a measure space, and $\mathcal L^\infty(X,\Sigma,\mu)$ is the set of essentially bounded functions. That is,
$\|f\|_\infty:=\inf\{c>0:\mu\{x\in X:|f(x)|>c\}=0\}<\infty$.
How can we pr... |
H: Solving : $ \left(1+x^{2}\right) \frac{d y}{d x}+2 x y=4 x^{2} $
Solve the following ODE : $$
\left(1+x^{2}\right) \frac{d y}{d x}+2 x y=4 x^{2}
$$
After rearrangement I get : $$\frac{dy}{dx}=\frac{4x^{2}-2xy}{1+x^{2}}$$ please help me after this step.
AI: $\newcommand{\d}{\mathrm{d}}$
This is a linear DE. Rearrang... |
H: Probability of guessing everything wrong
I was doing some AP Biology practice, and noticed that instead of Bio, I was learning more about probability. Tests have A B C and D, say you could guess one, get an answer whether it was right or wrong, guess again, and again and again, until eventually you got it right.
Fo... |
H: $2310 $ points are evenly marked on circle.How many regular polygons can be drawn by joining some of these points.
Question:- $2310$ points are evenly marked on circle.How many regular polygons can be drawn by joining some of these points($N$ -sided polygon should be counted only once).
I came across this question ... |
H: Covering spaces and fibre bundles, using the of $SU(2)$ and $SO(3)$
I understand there are many questions on here that show give an explicit map to show that $SU(2)$ is a double cover of $SO(3)$ (seevia quaternions).
I am trying and use the fact that $SU(2)$ is a double cover of $SO(3)$ to write $SU(2)$ as some fib... |
H: Expected Utility Maximization
This is from Markowitz's Risk-Return Analysis: The Theory and Practice of Rational Investing (Volume One) Chapter 1.
Suppose, for example, that a decision maker can choose any probabilities $p_0$, $p_1$, $p_2$ that he or she wants for specified dollar outcomes
$D_0$ < $D_1$ < $D_2$
and... |
H: $b^* a^* ab \leq \Vert a\Vert^2 b^* b$ in a $C^*$-algebra.
Let $A$ be a $C^*$-algebra and $a,b \in A$. In a proof I'm reading the following is claimed:
$b^* a^* ab \leq \Vert a\Vert^2 b^* b$. I want to understand this:
Here is my reasoning: we view $A \subseteq \tilde{A}$ with $\tilde{A}$ the unitisation of $A$. Th... |
H: Jacobian of a system of equations including an ODE for Newton-Raphson
I want to use the Newton-Raphson method to solve a system of equations and in order to do so, I need to calculate the Jacobian. The system given describes the stages (called $k$) of an implicit numerical method, which I want to solve for using Ne... |
H: $N\subseteq G_x$ then is $N$ in the kernel of the group action
Let $G$ be a group that acts transitive on $X$. Show that if $N$ is a normal subgroup in $G$ and for a $x\in X$ holds $N\subseteq G_x$. Then lies $N$ in the kernel of the group action.
[This is one part of a task in a textbook, so that $G$ acts transiti... |
H: How to evaluate $ \:\sum _{n=3}^{\infty \:}\frac{4n^2-1}{n!}\:\: $?
I am trying to evaluate:
$$ \:\sum _{n=3}^{\infty \:}\:\:\frac{4n^2-1}{n!}\:\: \quad (1)$$
My attempt:
$$ \:\sum _{n=3}^{\infty \:}\:\:\frac{4n^2-1}{n!}\:\: \quad = 4\sum _{n=0}^{\infty \:} \frac{(n+3)^2}{(n+3)!} + \sum _{n=0}^{\infty \:} \frac{1... |
H: If $p(x)$ be the polynomial left as remainder when $x^{2019}-1$ is divided by $x^6+1$. What is the remainder left when $p(x)$ is divided by $x-3$?
If $p(x)$ be the polynomial left as remainder when $x^{2019}-1$ is divided by $x^6+1$. What is the remainder left when $p(x)$ is divided by $x-3$?
It is obvious that rem... |
H: How to interpret $P_{V^{\perp}}m$, where $P_{V^{\perp}}$ is an orthogonal projection and $m$ is a matrix?
How would you interpret $P_{V^{\perp}}m$ in the following extract?
Given is a vector $v\in \mathbb{R}^n$ and a symmetric tracefree $n \times n$-matrix $m$. Let $V^{\perp}$ be the orthogonal complement of the sp... |
H: How to find the number of terms in the expansion of $(a+b+c)^2$?
How to find the number of terms in the expansion of $(a+b+c)^2$?
Can anyone help me in finding its formula with proof?
AI: We basically have to divide the power of $2$ amongst the $ a , b, c$. Let the powers be $ x, y , z $.
$ x+y+z = 2$ where x , y,z... |
H: Simplifying $\sum_{k=0}^{24}\binom{100}{4k}.\binom{100}{4k+2}$
How to evaluate the following series: $$\sum_{k=0}^{24}\binom{100}{4k}\binom{100}{4k+2}$$
What I have tried : Considering expansion of $\displaystyle (1+x)^n=
\binom{n}0+ \binom n1 x + \binom n2 x^2+\cdots$
By this I can get easily the result :
$$\sum... |
H: Inequality like $AB \leq \frac{\epsilon}{2n}A^2 + \frac{n}{2\epsilon}B^2$
In this paper on page 18 there is an inequality that I cannot prove. I will only look at the term with $K$, since the term with $b$ follows analogously I think.
So we have, using $L_{\sigma}M>0$ to ease the notation:
$$
L_\sigma M \sum_{i=1}^... |
H: How to find the $n$-th term of the series 1, 22, 333, 4444, 55555..........?
How to find the $n$-th term of the series $1, 22, 333, 4444, 55555..........?$
Here the $i^{\text{th}}$ term is the concatenation of $i$ for $i$ times.
for ef $13$ the term = $13$ concatenated $13$ times = $13131313131313131313131313$
AI: ... |
H: Proof of the convergence of infinite series of positive terms.
I would like to prove the fact that the series of positive terms converges.
Precondition;
$\sum_{n=0}^{\infty} a_n$ is the series of positive terms, and
$\sum_{n=0}^{\infty} a_n$ converges.
Problem;
$\sum_{n=0}^{\infty} (a_n)^2$ converges.
I tried to pr... |
H: About modified rock-paper-scissors game
(The game that I am about to ask is not my original creation; it is from a Korean game program 'RunningMan'. Also, I am also Korean, and please understand my poor English.)
The game is about a modified rock-paper-scissors game.
8 people are playing this game, and each of th... |
H: How to use Bars and Stars method for equations with more than 1 non unity coefficients?
I know we can find the non negative integral solutions of the equation $x+y+z=24$ using Bars and Stars method.
The same can be extended to provide the solutions for equations like $2x+y+z = 24$.
But is there any way to find the ... |
H: For which odd number $k$ does $\ \varphi(n) \mid n-k \ $ has infinitely many solutions?
The Lehmer-totient problem: For a prime number $\ n\ $ we have $\ \varphi(n)=n-1\ $. In particular, we have $\ \varphi(n) \mid n-1\ $. Is there a composite number $\ n\ $ with $\ \varphi(n)\mid n-1\ $ ?
It is not known if there... |
H: Given $f$ is a Lebesgue measurable function and $\int_0^1 x^{2n}f = 0 ~~~ \forall n$ , then show that $f = 0$ a.e.
Given $f$ is a Lebesgue measurable function and $\int_0^1 x^{2n}f\,d\mu = 0 \quad \forall n$, then show that $f = 0$ a.e.
Of course, if it was given that $f \geq 0$ then this was pretty trivial.
My att... |
H: Zorn's Lemma - partial order, or preorder?
According to Wikipedia Zorn's Lemma says;
every partially ordered set containing upper bounds for every chain necessarily contains at least one maximal element.
According to Nlab Zorn's Lemma says something slightly different;
Every preorder in which every sub-total ord... |
H: Which matrices in $\operatorname{SL}_2(\mathbb R)$ have orthogonal squares?
This is a self answered question, which I post here since, it turned out to be quite nice, and wasn't trivial for me. Of course, I would be happy to see other approaches.
Question:
Characterize all the matrices $X \in \operatorname{SL}_2(\... |
H: Universal coefficients for an integral domain
Let $R$ be an integral domain with unit. In Burde-Zieschang's Knots it is claimed (page 219) that for a pair of spaces $(X,Y)$ $$H_i(X,Y;R) \cong H_i(X,Y;\mathbb{Z}) \otimes_{\mathbb{Z}} R.$$
Is it the case that $R$, as a $\mathbb{Z}$-module, is projective or free (so t... |
H: What do the double bars and the 2 2 mean in this equation?
I was trying to understand this equation which is used to find the Rotation and translation between a set of 3D points x, and y that brings them closest together. It says you look for the solution where it is minimized.
R I think is the 3x3 rotation matrix... |
H: How can one prove that $f(x) = 4x - \ln(x^2 + 1)$ is injective?
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x) = 4x - \ln(x^2 + 1)$. Prove that $f$ is injective.
How I thought to solve this problem was to use the derivative of the function and show that it is strictly positive and strictly i... |
H: Are Turing machines that move before assigning Turing complete?
A state in a Turing machine is an assignment and a move direction both depending on the current input.
But if we define Turing machines to be a move first and then an assignment both depending on the current input,
would this also be Turing complete?
e... |
H: Finding the joint distribution of two independent uniform random variables $X,Y$ given event $E$
If $E = \{\text{either} \ X < 1/3 \ \text{or} \ Y<1/3\}$ for $X\sim Unif(0,1)$ and $Y\sim Unif(0,2)$, does the joint distribution of $X,Y$ given event $E$ exist? I am assuming that $X$ and $Y$ are independent.
AI: Let $... |
H: A particular problem on series
Problem: Show that there exist $c>0$ such that for all $N\in \mathbb N$ we have $$
\sum_{n=N+1}^{\infty}\left(\sqrt{n+\frac{1}{n}}-\sqrt{n}\right)\le \frac{c}{\sqrt{N}}
$$
I have no clue how to solve this. All I know is this fact $$\int_0^1\left(\sum_{n\in \mathbb N}\frac{1}{
\sqrt{... |
H: Confused about linear independence of two vectors in $\mathbb R^3$
If I put the two vectors 1,0,0 and 0,1,0 next to each other
$$\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}$$
I can see that they are independent since I cannot write $(1,0,0)^t$ as $(0,1,0)^t$.
But I've heard that "If you have a row of $0$'... |
H: Determining convergence or Divergence of $\sum\limits_{n=1}^{\infty} (\frac{3n}{3n+1})^n$
$$\sum\limits_{n=1}^{\infty} \left(\frac{3n}{3n+1}\right)^n$$
A cauchy root test will not work since $\lim_{n \to \infty}\sqrt[n]{(\frac{3n}{3n+1})^n} = 1$. However, by inspection, as $n \to \infty$ the sum reduces to:
$$\sim^... |
H: Given a real world dataset of film budgets and their actual gross profit, find the optimal budget
So I am currently working on a project for a Data Science Bootcamp that I started about a month ago and we are examining the film industry to gleam certain insights that we determine. What I am looking to do is take th... |
H: simplified formula of expression having adjoint of a matrix
Well, the question is :
let A be a square matrix of order $n_n$ and adj(A) is the adjoint of matrix A
then is it possible to get a simplified formula for the expression
$$adj(adj(adj(...adj(A))))$$ here adj() has been written 'r' times.
The reason I am as... |
H: Big $O$ notation and limit superior
I'm reading the Wikipedia's link for big $O$ https://en.wikipedia.org/wiki/Big_O_notation and it says
[...] both of these definitions can be unified using the limit superior: $$f(x) = O(g(x)) \; \mbox{as} \; x \rightarrow a$$ if $$\limsup_{x \rightarrow a} \left|\frac{f(x)}{g(x)... |
H: Quotient of ring by radical ideal.
Let $R$ be a commutative ring with identity, and let $I$ be an ideal of $R$. Let $J$ be the radical of $I.$
What can we say about the quotient ring $R/J?$ Does it have any special properties? In particular, if $J$ is the nilradical of $R,$ what can we say about $R/J?$
I couldn't f... |
H: Real sequences and convergence almost everywhere.
Let $(X,\mu,\mathcal{A})$ be a finite measure space and $f_n$ measurable functions such that $f_n \to 0 $ almost everywhere.
Show that exists a sequence $a_n \to +\infty$ auch that $a_nf_n \to 0$ a.e.
I managed (by using the Borel-Cantelli lemma) to find a subse... |
H: What is a good algorithm to find the clique number of a vertex of a graph?
In this question, "graph" means a non-oriented, simple graph with no loop and no label on the edges or vertices.
A clique in a graph $G$ is a complete subgraph of $G$. The clique number $\omega_v(G)$ of a vertex $v$ of $G$ is the maximum of ... |
H: Is this Hankel matrix positive definite?
It's an exercise given in my book which says that
Question: Consider a matrix $A=(a_{ij})_{5×5}$, $1\leq i,j \leq 5$ such that $a_{ij}=\frac{1}{n_i+n_j+1}$, where $n_i,n_j\in\mathbb{N}$. Then in which of the following is $A$ positive definite?
(a) $n_i=i$, for all
$i=1,2,3,4... |
H: Normalizers of Sylow $p$-subgroups have restricted order
I saw an old post. It gives the following result.
Assume that $H$ is a subgroup of a finite group $G$, and that $P$ is a Sylow $p$-group of $H$. If $N_G(P) \subset H$ then $P$ is a Sylow p-subgroup of $G$.
My first question: I wonder if there should be “$m=... |
H: Let $X=\{f:\Bbb{R}\to\Bbb{R}\}$. Define $f\oplus g=h$ by $h(x)=f(x)+g(x)$. (i) Show $(X, \oplus)$ is a group. (ii) Show $\{f\in X\mid f(0)=0\}\le X$.
Let $X= \{f:\mathbb{R}\to\mathbb{R} \}$. Define an operation $\oplus$ such that $f \oplus g = h$, where $h(x) = f(x) + g(x)$ for all $x \in \mathbb{R}$ and $+$ is th... |
H: Only Two Isomorphism Classes of Groups of Order Four
So my textbook says that there are two classifications of groups of order four. Those two are:
$\mathbb{Z}_4\cong\{0,1,2,3\}$ under $+_4$, and
$\mathbb{K}_4\cong$ symmetry group of a (non-square) rectangle.
It also says if $P\cong Q$, and $P$ has $k$ elements of ... |
H: Gap between the Uniform and Uniform convergence on compacts topologies
Let $X$ and $Y$ be the set $C_0(\mathbb{R})$ (of functions vanishing at infinity) equipped with the topologies of compact-convergence and uniform convergence respectively. The second is clearly stronger than the first (as the closure of the comp... |
H: How can I express $(1+2+\dots+(k+1))^2$ using a $\sum$ instead?
Good morning from México, I am in my first semester of Mathematics and I started proving by induction that:
$$\sum_{i=0}^n i^3 = \left(\sum_{i=0}^n i\right)^2$$
This question has been answered before, three times actually, but not with the approach I a... |
H: Computing the pdf after clipping
Given a random variable X with uniform distribution on [-b,b], I want to compute the probability density function of Y = g(X) with
$$g(x) = \begin{cases} 0, ~~~x\in[-c,c]\\ x, ~~~\text{else} \end{cases}$$,
and $b>c$.
There is a discontinuity at $x = c$ and $x=-c$. Therefore, the dis... |
H: What does it mean for a complex function to be real-differentiable?
There is a proposition I read which claims
Let $U$ be an open set. If $f(z)$ is real-differentiable on $U$ and satisfies Cauchy Riemann Equation then it is complex differentiable on $U$.
I am now confused as to what does it mean for a complex fun... |
H: Base locus under a pull-back
Let $\pi: Y \to X$ be a surjective morphism of smooth projective varieties.
Question: Suppose the linear system $|D|$ of a divisor $D$ on $X$ is base-point free. Is the linear system $|\pi^*D|$ also base-point free?
Here $\pi^*$ is the pull-back. I have seen there are statements about ... |
H: Limit of $L^p$ norm is $L^\infty$ norm variation
I'm familiar with the result that $$\lim_{p \to \infty} ||f||_p=||f||_\infty$$ when $f \in L^p([0,1])$, but I've come a cross a variation of this fact that I'm having trouble showing.
The assertion is that given $f \in L^\infty(\mathbb{R})$ $$\lim_{n \to \infty}\left... |
H: Matrix whose square is in Jordan normal form
Let $$A = \begin{bmatrix}J_0^2 \\ & J_0^2 & \\ && J_{1/4}^3\end{bmatrix}\in M_7(\mathbb{Q})$$ Find, with proof, a matrix $B$ so that $B^2 = A$.
I'm not sure how to find this matrix. Clearly, $A$ is not invertible as it is contains rows solely consisting of zeros. I kn... |
H: Is there a single continuous function satisfy all these properties?
I'm finding one $C^2$, non-decreasing function $f$: $[0, 1] \to \mathbb{R}$ that
\begin{align}
f'(0) &= 0, \\
f'(0.5) &= \max f'(x) \text{ } \forall \text{ } x \in [0, 1] \\
f'(1) &= 0, \text{ and} \\
f''(0.5) &= 0. \\
\end{align}
A Gaussian func... |
H: Write the polynomial of degree $4$ with $x$ intercepts of $(\frac{1}{2},0), (6,0)$ and $(-2,0)$ and $y$ intercept of $(0,18)$.
Write the polynomial of degree $4$ with $x$ intercepts of $(\frac{1}{2},0), (6,0) $ and $ (-2,0)$ and $y$ intercept of $(0,18)$.
The root ($\frac{1}{2},0)$ has multiplicity $2$.
I am to wri... |
H: Showing $P\left(\bigcap_{n=1}^{\infty} B_n\right)=1$ if $P(B_n)=1$ for every $n$
Resnick - Probability path 2.11:
Let $\{B_n, n\geq 1\}$ be events with $P(B_n)=1$ for every $n$. Show that
$$P\left(\bigcap_{n=1}^{\infty} B_n\right)=1$$
I was thinking to use a sequence such that it is equal to $\bigcap\limits_{n=1}^{... |
H: What am I allowed to assume in questions like this?
Suppose $T \in \mathcal{L}(V, W)$, and $(w_1, \ldots, w_m)$ is a basis of $\operatorname{range}(T)$. Prove that there exists $(\varphi_1, \ldots, \varphi_m) \in \mathcal{L}(V, \mathbf{F})$ such that $$ T(v) = \varphi_1(v)w_1 + \cdots + \varphi_m(v)w_m $$ for ever... |
H: How do I evaluate $\sum_{m,n\geq 1}\frac{1}{m^2n+n^2m+2mn}$
I saw a problem here which state to evalute $$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{m^2n+n^2m+2mn}$$
My attempt
Let $$f(m,n)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{m^2n+n^2m+2mn}$$ and interchanging $m,n$ as $n,m$ we have $$f'(n,m) = \s... |
H: Subgroup of Plane Isometries Isomorphic to $O_2(\mathbb{R})$
Let $\mathcal{Isom}(\mathbf{E})$ be the group of the isometries of the euclidean plane $\mathbf{E}$, and for every point $P \in \mathbf{E}$, let $\mathcal{Isom}_P$ be the subgroup of all isometries of $\mathbf{E}$ that fix $P$. It is well known that $\mat... |
H: Proving that an open subgroup of $\mathbb{R}$ must be all of $\mathbb{R}$
Let $G$ be an open subset of $\mathbb{R}$ which is also a subgroup of the group $(\mathbb{R}, +)$. Show that $G = \mathbb{R}$.
(Hint: $0$ belongs to $G$ and hence $(-\epsilon,\epsilon)$ is a subset of $G$ for some $\epsilon > 0$. Use the fact... |
H: Proving Segal's Category $\Gamma \simeq \mathbf{FinSet}_*^{op}$
I'm trying to show that Segal's category $\Gamma$ is equivalent to $\mathbf{FinSet}_*^{op}$, the opposite of the category of finite pointed sets with basepoint preserving morphisms. I'm intuitively gathering that $\Gamma$ can be shown to be the skelet... |
H: Define a linear operator which has as Kernel the line $y=-x$ and as image the line $y=$x
So the question asks basically to determine a linear operator $F: \mathcal{R}^2 \rightarrow \mathcal{R}^2$ which has as Kernel the line $y=-x$ and as image the line $y=x$.
Here is what i tried: i supposed i could write the oper... |
H: Image of an open set under an injective immersion map is Borel measurable?
Let $M$ and $N$ be two smooth manifolds and $f:M\to N$ be an injective immersion. Let $U\subset M$ be an open subset, then is $f(U)$ necessarily a Borel measurable subset of $N$?
AI: You can just cover U with open subsets where f is an imbed... |
H: Simple exercise on a linear operator $T$
I'm given the following linear operator $$T(a,b)=(-2a+3b,-10a+9b)$$ on the vector space $V=\mathbb{R}^2$. I have to find the eigenvalues of $T$ and an ordered basis $\beta$ for $V$ such that $[T]_\beta$ is a diagonal matrix.
I've tried using the standard basis $\beta=\{(1,0)... |
H: Prove that $c_{m} \in[a, b],$ for all $m \geq 1, \lim _{m \rightarrow \infty} c_{m}$ exists and find its value.
Let $f:[0,1] \rightarrow[0, \infty)$ be a continuous function. Let
$$
a = \inf_{0 \leq x \leq 1} f(x) ~\text{ and }~ b = \sup_{0 \leq x \leq 1} f(x) .
$$
For every positive integer $m$, define
$$
c_{m}=\l... |
H: Decreasing Sequence of Open Sets in Tychonoff Pseudocompact Spaces has Nonempty Intersection of Closures
Let $(U_n)$ be a decreasing sequence of non-empty open sets in a Tychonoff pseudocompact space $X$. Then, show that $\cap \overline U_n \neq \phi$
This was part of a problem in Willard. I was able to do the re... |
H: One-sided limit of q norm on $[0,1]$
I'm trying to show that for $f \in L^q([0,1])$ for all $q$ with $1≤q≤p<\infty$, we have $$\lim_{q \to p^-}||f||_q=||f||_p$$
It's easy to show that $$\lim_{q \to p^-}||f||_q≤||f||_p$$
But the other direction is proving difficult. I've tried to use an epsilon argument similar to w... |
H: Understanding why the integral test is applicable in $\sum_{n=1}^{\infty} \frac{\ln n}{n^2}$?
Here is the statement of the integral test and question (e) that I want to solve:
My questions are:
1- I know that here $f(x) = \frac{\ln x}{x^2} $ as this is a question of MATH subject GRE test and $\log n$ means $\ln n.... |
H: Bilinear form and quotient space
Let $U,V$ be two finite dimensional vector spaces of a field $K$ and let $f:U \times V \to K$a bilinear form. The set $U_0 = \{u \in U: f(u,v) = 0,\forall v \in V \}$ is called the left kernel of $f$ and $V_0 = \{v \in V: f(u,v) =0, \forall u \in U\}$ is called the right kernel of $... |
H: If $\sum a_n^p$ converges, then $\sum a_n^q$ also converges? for $1 \leq p \leq q$ and $a_n > 0$, for all $n \in \mathbb{N}$.
I was able to give a counterexample to the reciprocal, but I'm having trouble proving this one (I even tried to look for a counterexample, but I couldn't). Any suggestions on how to do it, o... |
H: Dual of a $\mathbb{C}G$-module, can we get $g^{-1}$ outside $\theta(\cdot)$?
I am a bit lost with dual modules.
Let $G$ be a group, $\mathbb{C}$ be the complex numbers and let $V$ be a $\mathbb{C}G$-module. Then as I understand it, the dual is $V^*=\operatorname{Hom}_{\mathbb{C}G}(C,\mathbb{C}G)$. So if we have $\t... |
H: A problem involving sum of digits of integers
Question: The sum of the digits of a natural number $n$ is denoted by $S(n)$. Prove that $S(8n) \ge \frac{1}{8}S(n)$ for each $n \in \mathbb N$.
[source:Latvia 1995]
At first I thought this problem can be solved using induction on the number of digits. Say without the l... |
H: is there a type of number disjoint complex numbers
is there a type of number disjoint complex numbers?
Has anyone found a type of number or is there exists such number s.t. $\omega^*$ (or something else) that $\omega^*\not\in\mathbb{C}$ but grouped in a new type of number with its meanings such as $\omega^*\in\math... |
H: Find limits of $\lim _{x \to 0} \frac {x \cos x - \sin x} {x^2 \sin x}$ without l'Hopital's rule or Taylor Expansion.
Find the limit $\displaystyle \lim _{x \to 0} \frac {x \cos x - \sin x} {x^2 \sin x}$ without l'Hopital's rule or Taylor expansion.
My Try
$\displaystyle =\lim _{x \to 0} \frac {\cos x - \frac{\sin ... |
H: Find a positive-definite matrix $B$ such that $\langle A,B\rangle_F=0$
Let $A$ denotes an $n\times n$ Hermitian matrix. Is there any positive-definite matrix, $B$, such that $\langle A,B\rangle_F=0$?
AI: Such a positive definite $B$ exists if and only if $A$ is zero or indefinite.
If $B$ is positive definite and $A... |
H: How to find the series $\sum_{n=0}^{\infty} \frac{(-1)^n x^{4n}}{(4n!)}$
It is known that, $$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}=\cos(x)$$ and $$\sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}=\cosh(x)$$
From this follows that $$\sum_{n=0}^{\infty} \frac{x^{4n}}{(4n)!}=\frac{\cos(x)+\cosh(x)}{2}$$
How to find ... |
H: Evaluate the sum $\sum_{m,n\geq 1}\frac{1}{m^2n+n^2m+kmn} $
Motivated by the double summation posted here and recently here too. I came up with the general closed form for one parameter $k>0$ where $k$ being a positive integer.
$$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{m^2n+n^2m+kmn}=\frac{H^2_{k}-\psi_1... |
H: If $\sum_n \sqrt{a_n a_{n+1}}$ converges, then $\sum_n a_n$ might not converge.
This is the other direction of this question that reads
Let $a_n>0$ (!) for each $n\in\mathbb{N}$. Then $\sum_{n=1}^\infty a_n<\infty$ implies $\sum_{n=1}^\infty \sqrt{a_n a_{n+1}}<\infty$.
I need to prove that "the converse of the la... |
H: Can't solve quadratic function
I'm trying to get a simple quadratic function of the form $y=ax^2+bx+c$ which goes through the following points:
$$(0;0) \;(\frac d2;2)\; (d;0)$$
The solution I've calculated $3$ times on paper has always been $y=\frac8d- \frac{8x^2}{d^2}$, but this can't be true, for an arbitrary $d$... |
H: Show that a finite D-dimensional Hilbert Space has $D^2$ operators
I'm studying Group Theory and i've just arrived at this chapter about Hilbert Spaces and the author states that a this finite Hilbert Space has $D^2$ operators, one being the trivial one, by completeness:
\begin{equation}
\sum_{j=0}^D\,|\,j\,\rangl... |
H: Why isn't the Disjoint Union in Set a *product* in addition to being a coproduct?
So I'm understanding that in $Set$ that the cartesian product is a categorical product, and further get why the disjoint union is a categorical coproduct, but why is it also not a product? I want to understand where I'm going wrong in... |
H: Calculus of Variations: Looking for theorem that ensures that a given variational problem has maxima and minima
Is there a theorem that garuantees that a variational problem $I[y] = \int_a^bF(x,y,y')dx$ has local/ global maxima and minima?
Perhaps similar to the extreme value theorem for continuous functions on com... |
H: Is it true that $\mathbb{E}[X^n] > \mathbb{E}[X^{n-1}]\mathbb{E}[X]$ for all $n \geq 2$?
Let $X$ denote a random variable with a smooth distribution over $[0, 1]$. Is it true that
$$\mathbb{E}[X^n] > \mathbb{E}[X^{n-1}]\mathbb{E}[X]$$
for all integers $n = 2, 3, ...$? This seems to be a simple application of Jensen... |
H: What's the minimal positive integer solution to $13n^2+1 = m^2$
What's the minimal positive integer solution to $13n^2+1 = m^2$?
My first intuition was that $m$ has to be $13k+1$ or $13k-1$. But how do I proceed to find the minimum solutions?
The minimum I found through computer program is $(m,n) = (649,180)$
AI: R... |
H: Expected value and Lindeberg condition
The Lindeberg condition (classic) states that for a random variable $X$ with finite mean and variance, $\mu$ and $\sigma^2$, and for every $\varepsilon>0$
$$E(X^2 \boldsymbol{I}_{\{ X>\varepsilon\sqrt{n}\sigma\}}) \rightarrow 0, \quad \mbox{as} \quad n\rightarrow \infty,$$ whe... |
H: Convergence of a series with divergent odd and even sub-term series
If $\sum_{n=0}^{\infty} a_{2 n}$ and $\sum_{n=0}^{\infty} a_{2 n+1}$ both diverge, then prove/disprove $\sum_{n=0}^{\infty} a_{n}$ diverges as well.
Can we take the counters as
$$\sum_{n=1}^\infty a_n$$
Where $a_n=\left\{ \begin {array}{ll} \frac{1... |
H: How can I prove that $\sum_{n=0}^{\infty}\frac{\sin(7n)}{13^n}=\frac{13 \sin(7)}{170 - 26 \cos(7)}$?
$$\sum_{n=0}^{\infty}\frac{\sin(7n)}{13^n}=\frac{13 \sin(7)}{170 - 26 \cos(7)}$$
Have no clue how to prove it.
Possibly rewrite $\sin(7n)$ as $\frac{1}{2\sin(7)}\left(\cos(7n-7)-\cos(7n+7)\right)$.
But what next?
AI... |
H: Does $-6(x-\frac{1}{2})^2$ = $-\frac{3}{2}(2x-1)^2$?
I'm confused about a question I posted this morning.
I am trying to understand if $-6(x-\frac{1}{2})^2$ can be rewritten as $-\frac{3}{2}(2x-1)^2$?
I tried multiplying out the expression $-6(x-\frac{1}{2})^2$ to a polynomial form $36x^2-36x+9$ but that didn't tak... |
H: What is the normalised arc length measure on the unit circle?
Consider the following fragment from Murphy's '$C^*$-algebras and operator theory':
Can someone explain what the normalised arc length measure on $\Bbb{T}$ is? Is this some translation of Lebesgue measure on the circle?
AI: Are you familiar with Haar me... |
H: Sum of i.i.d random variables equals to infinity
Let $\{X_i\} $ be i.i.d random variables in $\mathbb{R}+$.
When $\sum_{i=1}^{\infty} X_i=\infty$?
AI: There exists $\epsilon > 0$ such that $\mathbb{P}(X_1 \geq \epsilon) > 0$. Then by the 2nd Borel-Cantelli Lemma,
$$\mathbb{P}(X_n \geq \epsilon \text{ i.o.}) = 1.$$
... |
H: How to calculate line integral using Green's theorem
I had this specific task in my math exam and didn't solve it correctly. Also, I, unfortunately, don't have any correct result. So I am asking you, if anyone can solve and explain it to me. I would be super grateful. Sorry for my bad English, it's my second langua... |
H: Probability of exactly $2$ sixes in $3$ dice rolls where $2$ dice have $6$ on $2$ faces?
Three dice are rolled. One is fair and the other two have 6 on two faces.
Find the probability of rolling exactly 2 sixes.
My textbook gives an answer of $\frac{20}{147}$ but I get an answer of: $$\frac{1}{6}\frac{2}{6}\frac{4}... |
H: Show that if $x \in \partial (A \cap B)$ and $x \not\in (A \cap \partial B)\cup (B \cap \partial A)$, then $x \in \partial A \cap \partial B$.
This question is for my exam prep. I want to solve the following example:
Show that if $x \in \partial (A \cap B)$ and $x \not\in (A \cap \partial B)\cup (B \cap \partial A)... |
H: Verify that $\mathcal{A}$ is a $\pi$-system and also determine $\lambda\langle\mathcal{A}\rangle$, the $\lambda$-system generated by $\mathcal{A}$.
Let $\Omega$ be a nonempty set and $\{A_{i}\}_{i\in\mathbb{N}}$ be a sequence of subsets of $\Omega$ such that $A_{i+1}\subset A_{i}$ for all $i\in\mathbb{N}$. Verify t... |
H: Prove that montonicity and continuity imply bijectivity.
Here's the question:
Let $f: [a,b] \to [f(a),f(b)]$ be monotonically increasing and continuous. Prove that $f$ is bijective.
Proof Attempt:
Let $f: [a,b] \to [f(a),f(b)]$ be monotonically increasing and continuous. Since it is monotonically increasing, it is... |
H: How can we show that each component of a 2-regular graph is a cycle?
I have done quite a few examples and this statement turns out to be true in each case. However I'm not sure how to formally prove it. Does there exist a theorem that can help with such type of proof?
AI: You need to show that a connected 2-regular... |
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