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H: Show that X1 and X2 are not independent
Let $X_1 \sim N(0,1)$ and $\xi$ be an independent symmetric random sign, i.e. $P(\xi = \pm1) = \frac{1}{2}$
Define $X_2= \xi X_1$. This random variable is normal, since for any $s\in\mathbb{R}$ (???)
Show that $X_1$ and $X_2$ are not independent.
I’ve no idea how to solve it,... |
H: Prove $a^k\equiv 1 \pmod k$ has no solution for infinitely many even integers k
Prove that there are infinitely many EVEN positive integers $k$ such that for each of
those $k$, the equation $\varphi(n) = k$ has no solution in positive integers $n$.
I believe there might be a way to approach this using Euler's Theor... |
H: Relation between generator polynomial and codeword in a cyclic code
I'm trying to solve the following exercise, but I can't use an hypothesis.
Let $F$ be a finite field and $a(x)$ a poly of degree $n$ over $F[x]$. Let $C$ the smallest cyclic code of length $n$ over $F$ with $a(x)$ as codeword, and let $g(x)$ the g... |
H: Interesting Partition Questions
There is a good question here.
My question is;
"x is a positive integer and
$\lfloor x\rfloor$ denote the largest integer smaller than or equal to $x$. Prove that $\lfloor n / 3\rfloor+1$ is the number of partitions of $n$ into distinct parts where each part is either a power of two ... |
H: How to find count of numbers in a range that satisfies both X mod N = A and X mod M = B?
For example how many numbers X are there from [0,100] that satisfies both X mod 8 = 2 and X mod 5 = 1.
My first approach was to find X such that X mod LCM(A,B) = (X mod A) + (X mod B). Because if a number is divisible by LCM(A,... |
H: Centering charts about a point in a euclidean space.
Let $M$ be a locally euclidean space. Let $p\in M$. Then there exists some chart $(V,\psi)$ such that $\psi(p)=0$.
Now, I first thought of breaking it into cases, but I didn't like that approach, so I was able to think of another way:
By definition of $M$, $p$ is... |
H: How to choose infinite number of different values from infinite set of infinite sets.
Let $ \aleph_{\alpha} $ be a cardinal and assume that $ \left\{ A_{\beta}:\beta<\aleph_{\alpha}\right\} $ is a set of sets, such that $ |A_{\beta}|=\aleph_{\alpha} $ for any $ \beta<\aleph_{\alpha} $.
Prove that exists set of se... |
H: Holomorphic function in complex analysis
For a holomorphic function $f$ we know that:
$f(x) \in R$, for every real number $x$
$f(1+i)=-1-i$
Determine $f(1-i).$
I'm really not seeing how this exercise can be solved. My best guess is the answer would be $f(1-i)=-1+i, $ but only because since the function sends ... |
H: Tangent Hyperplane $H$ to $X$ at $p \in X$ and hyperplane divisor $\operatorname{div}(H)$.
According to Rick Miranda (Algebraic Curves and Riemann Surfaces) we have the following Lemmas:
Lemma 3.7 (page 219): Suppose that $X \subset \mathbb{P}^n$ is a nondegenerate smooth curve (with $n \ge 2$). Then $X$ has only f... |
H: Let $T$ be the linear operator on $M_{n}(\textbf{R})$ defined by $T(A) = A^{t}$. Find a basis $\mathcal{B}$ s.t. $[T]_{\mathcal{B}}$ is diagonal.
Let $T$ be the linear operator on $M_{n\times n}(\textbf{R})$ defined by $T(A) = A^{t}$.
(a) Show that $\pm 1$ are the only eigenvalues of $T$.
(b) Describe the eigenvect... |
H: Finding a diagonalizable endomorphism $f : \mathbb{R}^4 \to \mathbb{R}^4$ such that $\text{ker}(f) = \text{im}(f)$
I've been struggling all day with this question. I tried to come up with a proof which shows that such an endomorphism does NOT exist, but I'm not sure it is correct.
Let $ B = (b_1, b_2, b_3, b_4)$ b... |
H: Is it true that $\sqrt{ab}\le \frac{a-b}{\ln a - \ln b}$ for any $a\neq b>0$?
Is it true that $\sqrt{ab}\le \frac{a-b}{\ln a - \ln b}$ for any $a\neq b>0$?
If so, any thoughts on how to prove this?
AI: The assertion is true. When $a>b>0$, it follows by taking $x:=\frac ab$ in the inequality
$$ \log x\le \sqrt x... |
H: Connection between cross product and determinant
When I calculate a cross product of two vectors in Cartesian coordinates, I calculate something that seems like the determinant of a 2x2 matrix.
Is there any connection between the determinant and the cross product?
AI: If $\vec{i},\vec{j},\vec{k}$ are the three basi... |
H: Convergence of $\sum_{n=1}^\infty \frac{1}{n}\frac{x^n}{1-x^n}$,$\sum_{n=1}^\infty \frac{(-1)^n}{x^2-n^2}$
I am having problems to find the values of x for which the following series converge. I understand the use of root test or ratio test to find the radius of convergence when the series is a power series, but in... |
H: Abbott's proof that any rearrangement of an absolutely convergent series converges to the same limit as the original
Here is his proof in full:
Assume $\sum\limits_{k = 1}^{\infty} a_k$ converges absolutely to $A$, and let $\sum\limits_{k = 1}^{\infty} b_k$ be a rearrangement of $\sum\limits_{k = 1}^{\infty} a_k$. ... |
H: Is this proof that $0x = 0$ correct?
I was wondering if this proof I wrote for $0x = 0$ (using only the field axioms) is correct. The proof is as follows:
$$0x = (1-1)x = x-x = 0$$
AI: There are a lot of steps you skipped. First, -1 is defined as the inverse of 1, so a rigorous notation is (-1).
$0 x = (1 + (-1))x ... |
H: Property of cyclic quadrilaterals
Suppose $ABCD$ is a cyclic quadrilateral and $P$ is the intersection of the lines determined by $AB$ and $CD$. Show that $PA·PB= PD·PC$
Could you help me please, I have no idea how to relate the property that he is cyclic, I have reviewed other posts of cyclic quadrilaterals and th... |
H: Angle between $n + 1$ equidistant unit vectors in $\mathbb{R}^n$
My question is basically as posed in the title. Suppose we are given $n + 1$ unit vectors in $\mathbb{R}^n$ so that the angle between any pair of them is the same. What is that angle? I have (unfounded) reason to believe that the angle is arccos$(\fra... |
H: Using negation of Uniqueness Quantifier to show a relation is not a function.
$A = \{1,2\}$
$B = \{2,3\}$
$R \subseteq A \times B$
$R = \{(1,2), (1,3), (2,3)\}$
I want to prove $R$ is not a function. That is, I want to show:
$$
\lnot \forall x(x \in A \to \exists y(y \in B \land (x,y) \in R \land \forall z((z \in... |
H: Rings with no zero divisors and an additional Hypothesis are commutative?
So, inspired by the question Finite integral domains are commutative?, i was wondering if the next generalization is also true:
Let $A$ be an integral domain (using the terminology of the linked question), such that every element $a \in A$ g... |
H: What is meant by "dot product between random variables?"
I was having a discussion with a colleague today about correlation coefficients, and I was told that correlation coefficient between 2 random variables $X$ and $Y$ is proportional to the dot product of the two random variables.
I asked him what he means by th... |
H: If there exists a positive $K$ such that $|f(x)| \leq K \int_a^x |f(t)|dt$ then $f(x) = 0$
Let $f$ is continuous on $[a,b]$. There exists positive K such that $|f(x)| \leq K \int_a^x |f(t)|dt$ then $f(x) = 0$.
I was trying to prove the statement above, by trying the smallest number c such that $f(x) = 0$ for any $x... |
H: Finding zeroeth coefficient of a Chebyshev polynomial expansion
Let $v_\theta = (\cos\theta,\sin\theta)$ be a unit vector in the plane. I have a kernel $p(\theta,\theta') = p(v_\theta\cdot v_{\theta'})$ that satisfies
$$\int_0^{2\pi} p(v_\theta\cdot v_{\theta'})\,d\theta' = 1\;\;\;(*)$$
for all $\theta\in [0,2\pi]$... |
H: How to prove that the second condition for Leibniz test is met for the series?
So, here is the series: $\sum^{\infty}_{2}\frac{k}{(k\ln x +x^2)^2}$. I need to show that $\frac{k+1}{((k+1)\ln x +x^2)^2} - \frac{k}{(k\ln x +x^2)^2} \ge 0 \ \ \ \forall x\in (0, \infty), \forall k \ge 2 $
How to do that?
AI: HINT:
As ... |
H: $\frac{z}{e^z - 1}$ power series at $z = 0$
I have a the following question:
Show that $f(z) = \frac{z}{e^z - 1}$ at has a removable singularity at $z = 0$ and that $f$ has power series expansion $\sum_{n=0}^\infty c_nz^n$. Calculate $c_0$ and $c_1$ and show that $c_{2n+1} = 0$ for $n \geq 1$. Find the radius of c... |
H: Tossing a fair coin 3 times
If a fair coin is tossed 3 times, what is the probability that it turn
up heads exactly twice?
Without having to list the coin like HHH, HHT, HTH, ect. to get to P=3/8. I would like to ask if there is any mathematical way to calculate this probability.
Please help, thank you!
AI: Since... |
H: Is it possible to build a $8×8×9$ block using $32$ bricks of dimensions $2×3×3$?
Is it possible to build a $8×8×9$ block using $32$ bricks of dimensions $2×3×3$?
I tried to show that $8×8×9$ block can't contain $32$ blocks of dimensions $2×3×3$ . For that I tried to colour $1×1×1$ cubes.
(It would give me something... |
H: Does real*real*real... = imaginary? $x\cdot x\cdot x\cdot x\cdot x\ ...\ =\ i, x \in \mathbb{R}$
Please be advised as is pointed out below, the video was incorrect and this:
$$ x=e^{\frac{\pi}{2}} \Rightarrow x^{x^{x^{x^{...}}}} = i$$Is completely false!
I recently watched the video by real^real^real^... = imagin... |
H: If $ |f(x)-f(y)| \leq 7|x-y|^{201} $ Then,
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that for any two real numbers $x$ and $y$
$$
|f(x)-f(y)| \leq 7|x-y|^{201}
$$
Then,
(A) $f(101)=f(202)+8$
(B) $f(101)=f(201)+1$
(C) $f(101)=f(200)+2$
(D) None of the above.
My approach:-
$$|(f(x)-f(y)... |
H: Applying chain rule in $f(x)=\sin(x)\cdot x\ln(x)$
Can we apply chain rule in function
$f(x)= \sin(x)\cdot x\ln(x)$
What i try:: Chain rule
$$\frac{d}{dx}\bigg(f(g(x)\bigg)=f'(g(x))\cdot g'(x)$$
So $$\frac{d}{dx}\bigg(\sin (x)\cdot x\ln(x)\bigg)=\sin(x)\cdot \frac{d}{dx}\bigg(x\ln(x)\bigg)+x\ln(x)\frac{d}{dx}(... |
H: Percent on 100 tries?
If i know that something happens 50% In 1 try.
What is percent of Something to Happen at least once in 100 tries?
My friend told me that chances are 50% but i dont understand how that is possible?
could you explain with more details please
AI: Let us consider tossing a fair coin (equivalent to... |
H: Meta cyclic p-group
While studying meta cyclic p groups, I came across an interesting class of meta cyclic groups which can be written as semi-direct product of two cyclic p-groups of order $p^m$ and $p^n$ respectively. These kind of groups are called split meta cyclic p-groups. I am trying to write their presentat... |
H: Polynomials question Part (a) and (b)
(a) In this multi-part problem, we will consider this system of simultaneous equations:
$$\begin{array}{r@{~}c@{~}l l}
3x+5y-6z &=&2, & \textrm{(i)} \\
5xy-10yz-6xz &=& -41, & \textrm{(ii)} \\
xyz&=&6. & \textrm{(iii)}
\end{array}$$
Let $a=3x$, $b=5y$, and $c=-6z$.
Determine th... |
H: Does convexity around a point imply the function is above the tangent at that point?
Let $\phi:\mathbb [0,\infty) \to [0,\infty)$ be a $C^2$ function, and let $c>0$ be a constant.
Suppose that for any $x_1,x_2>0, \alpha \in [0,1]$ satisfying $\alpha x_1 + (1- \alpha)x_2 =c$, we have
$$
\phi(c)=\phi\left(\alpha x_1 ... |
H: Linear transformation from complex to polynomial of degree 2
Is there a linear transformation $T:C^4 -> P_2(C)$ such that $im(T)= P_2(C)$?
AI: Something like
$$T(w_1,w_2,w_3,w_4)=w_1t^2+w_2t+w_3$$
can work. |
H: Does bounded $L^2$-norm of $r$-th derivative imply bounded $L^2$-norm of $(r-1)$-th derivative?
Let $f:[0,1]\to [0,\infty)$ such that $\int_0^1 \left|\frac{d^r}{d x^r} f(x)\right|^2 d x<\infty$ for some $1\leq r<\infty, r\in\mathbb{N}$. Does it hold that $\int_0^1 \left|\frac{d^{r-1}}{d x^{r-1}} f(x)\right|^2 d x<\... |
H: Determinant equal to zero, what does it mean?
If I understood correctly, a $determinant = 0$ means that the matrix has no area/volume/etc. But if a system of equations, say $Ax = b$ has a unique solution $x$, does that mean that the determinant can't be zero in that case?
AI: Yes, because if the determinant is zero... |
H: Having the dual base $\beta^*=$ {$\phi_1,\phi_2,\phi_3$} where $\phi_1(x,y,z)=x-y$, which is $\beta=$ { $v_1,v_2,v_3$ }?
Let's suppose that I have $\beta=$ { $v_1,v_2,v_3$ } a base of $\mathbb{R^3}$ and its dual base $\beta^*=$ {$\phi_1,\phi_2,\phi_3$} where $\phi_1(x,y,z)=x-y$.
Which is a base $\beta=$ { $v_1,v_2,... |
H: Finding All Solutions For $\sin(x) = x^2$
Hello everyone how can I find the count of the solution for $\sin(x) = x^2$?
I know there is a one solution in $x = 0$ and for the other solutions I tried to find the extreme point of the function: $y = x^2 - \sin(x)$ and $y'$ is:
$y' = 2x -\cos(x)$ but I don't know how to ... |
H: Check this proof: If two columns/rows of a matrix are the same, the determinant is $0$.
I have written this proof stating that if two rows or columns of a matrix are the same, then the determinant of the matrix is equal to 0. Is it correct?
Let us say we have an n x n matrix A, shown below:
For some $i,n \in \math... |
H: Distributing $n$ unique presents to $k$ kids
Question:
You have 9 presents to give to your 4 kids. How many ways can this be done if:
The presents are unique and each kid gets at least one present?
I know the solution is by using Principle of Inclusion and Exclusion.
$4^9 - [{4 \choose 1}3^9 - {4 \choose 2}2^9 +... |
H: Why can $\frac{{{x^2}}}{{\sqrt y }} \le t$ be prepresent as the the following equivalent second order cone constraint?
I am quite new to the field of convex optimization and in a research paper that I have read, some author represent this constraint $\frac{{{x^2}}}{{\sqrt y }} \le t$ as equivalent SOC constraint li... |
H: Curvature of Fernet curve on a sphere
The question is, how to prove that the curvature of any Frenet curve on a sphere with radius $R$ is bigger or equal to $1/R$.
I have managed to prove so far that the Gauss curvature of the sphere $x^2+y^2+z^2=R^2$ is $1/R^2$, but I don't know if this helps at all
AI: Suppose $\... |
H: Why do we say that probability of an individual event in a continuous distribution is 0?
So I understand that the probability a<x<b is the definite integral from a to b of tye probability density function and that makes sense. If we use that same definition to define the probability that x is equal to some value, ... |
H: Function of 'Max Width' of Crossing Rectangular Bars
(Looking at the Visual Example image below should greatly help with your understanding of my problem)
I'm trying to find the function for the increase of the 'maximum width' of two (identically sized, although their width is all that matters here) rectangular bar... |
H: Approximation in the integral to calculate the age of the universe
I'm computing the following integral: $$T=\frac{1}{H_{0}} \int^{1}_{0} \frac{da}{\sqrt{\frac{\Omega_{M}(t_{0})}{a}}\left( \sqrt{1+\frac{\Omega_{R}(t_0)}{\Omega_{M}(t_0) \,a}+\frac{\Omega_{\Lambda}(t_0) \,a^3}{\Omega_{M}(t_0) }}{} \right)}$$
with: $\... |
H: What's the kernel of $w\mapsto w \bullet \bullet$?
for $v\in K^n$, the dot product defines a linear transformation
$-\bullet v: K^n\to K, w\mapsto w\bullet v$. Let $e_i$ be the i-the basis vector of $K^n$. What's the kernel of $-\bullet e_i$?
I know that the dot product of two vectors is zero if they are orthogonal... |
H: A sufficient condition for a space not to be $T_1$ by a collection generating the topology.
I've thought about the following result, which I wanted to verify:
Let $X$ be a topological space where $\vert X\vert>1$ generated by a collection of subsets $\{ S_\alpha \}_{\alpha\in \Lambda}$, such that $\cup_{\alpha\in \... |
H: Is a homogeneous space $X$ of a compact Hausdorff group $G$ with closed stabilizer $G_x$ itself Hausdorff?
assume $G \times X \to X$ is a continuous group action of a compact Hausdorff topological group $G$ such that the action is transitive, i.e. $X$ is a homogeneous space. Let $G_x$ be the stabilizer subgroup of ... |
H: Linear Algebra - direction of ball after bounce off a plane
In an orthonormal system, a ball is thrown from a point =(2,6,5) towards a plane with
equation −=−2, in such a way that after it bounces off the plane, it passes through the point =(3,3,7). What is the ball's direction after the bounce.
This question has ... |
H: Well-defineness/existence of an integral
I am reading on page 2, it says
For a finite Borel measure $m$ on the real line $\mathbb{R}$, let us recall that its Cauchy transform $G_m$ is defined by
$$
G_m(z)=\int_{\mathbb{R}}\frac{1}{z-x}\mathrm{d}m(x),\qquad \textrm{for } z\in \mathbb{C}\setminus \mathbb{R}=\{z\in \... |
H: What is the conjugate of $[i + e^{iπt}]$?
What is the conjugate of the following complex number?
$$ Z(t) = i + e^{iπt} $$
Is it $Z(t) = i - e^{iπt}$ or $Z(t) = i - e^{-iπt}$? $t\in [0,1]$
AI: $$Z(t) = i + \cos(\pi t) + i\sin(\pi t)$$
$$Z(t) = \cos(\pi t) + i(1 + \sin(\pi t))$$
$$conjugate (Z(t)) = \bar{Z}(t) = \co... |
H: Show that $X$ must be degenerate at $n$.
For $X$ which is an integer-valued random variable
$$\mathbb{E}[X(X-1)(X-2)...(X-(k-1))] = \begin{cases}
k!\ {n \choose k},& \text{if } k \in \mathbb{N}\\
0, & \text{otherwise}
\end{cases}$$.
which can be rewritten as
$$\mathbb{E}\bigg[\frac{X!}{(X-k)!}\... |
H: Möbius transformation that carries the real axis to the unit circle
It is known that any (invertible) Möbius transformation carries lines and circles in the complex plane into lines and circles. Which Möbius transformations
$$
T(z)=\frac{az+b}{cz+d}
$$
carries the real axis into the unit circle? More generally, whi... |
H: What Goes Wrong in the Application of Lax-Milgram?
I have the following equation
$$\int\nabla u\nabla v =\int fv$$
So I want to find $f$ for which there is an unique $u$ so the above is satisfied for all $v$ in the space (say $H^1$).
So define bilinear map $B$ from $H^1$ to $\mathbb R$
$$B(u,v):=\int\nabla u \nabla... |
H: What is the value of $\int_0^1\pi e^{i\pi t}dt$?
What is the integration of the following equation?
$$\int_0^1Z(t)dt =\int_0^1\pi e^{i\pi t}dt$$
AI: The integral of $\int_0^1 \pi e^{i\pi t}\ dt$ Is by u substitution with $u = i\pi t$ and $du/dt= i\pi$ is equal to (also substituting the bounds) $\frac{\pi}{i\pi}\int... |
H: Is the function $f : \mathbb{R} \to \mathbb{R}$ such that $f(x) = x$ for $x \leq 0$ and $x+1$ for $x>0$ continuous?
Is the function $f : \mathbb{R} \to \mathbb{R}$ such that $f(x) = x$ for $x \leq 0$ and $x+1$ for $x>0$ continuous at $x = 0$?
If we consider the limit definition, then $\lim_{x \to 0+} f(x) = 1$ whil... |
H: Probabilities of Bivariate Normal Distribution
I have the following normal distribution that all of the parameter are known
$$\begin{pmatrix}
X_1\\
X_2\\
\end{pmatrix}
\sim N\left[\begin{pmatrix}
\mu_1 \\
\mu_2 \\
\end{pmatrix},\begin{pmatrix}\sigma^2 & \rh... |
H: Is a stochastic matrix always diagonalizable?
Let $A$ be a $n$-by-$n$ left stochastic matrix.
The followings are the properties I found so far:
Algebraic Property
$A$ has left eigenvector $[1,1,\cdots,1]$ ($n$ 1s) with corresponding eigenvalue $1$.
Geometric Property
Let $S$ be a $(n-1)$-simplex whose vertices are ... |
H: What are specific proofs of Jacobi Triple Product Identity?
I am looking for the Special Proofs.
Here is a reference from MSE.
Motivation for/history of Jacobi's triple product identity
I also know that a simple proof via Functional Equation from the book of An Invitation to q-series Hei-Chi-Chan. it has a very ni... |
H: Let $\langle x_n\rangle$ be a recursive relation. Find $\lim_{n\to\infty}\frac {x_n}{n^2}.$
Let $\langle x_n\rangle$ be a recursive relation given by $$x_{n+1}=x_n+a+\sqrt {b^2+4ax_n}, n\geq0, x_0 =0$$ and $a$ and $b$ are fixed positive integers. Find $$\lim_{n\to\infty}\frac {x_n}{n^2}.$$
AI: Clearly, $\lim_{n\to... |
H: Find the x-coordinate of the stationary point on the curve $\tan(x)\cos(2x)$ for $0 < x < \pi/2$
Can someone please show me how to find the x-coordinate for the stationary point for this curve?
$y=\tan(x)\cos(2x)$ for $0 < x < \pi/2$
This is what I've done so far:
$$\frac{dy}{dx}=\cos(2x)\sec^2(x)-2\tan(x)\sin(... |
H: Change the scale of sigmoid function to get a value between -0.5 to +.0.5?
Right now the sigmoid function usually gives a value between 0 to 1. I want to scale it down by 0.5 to it gives value between -0.5 to 0.5 ?
How can I do it?
AI: You said it yourself: you want to scale it down by 0.5. So take your
$$
f(x) = \... |
H: How do I evaluate $\lim_{n\to\infty} \,\sum_{k=1}^n\left(\frac{k}{n^2}\right)^{\frac{k}{n^2}+1}$?
I came across the following problem recently in a problem sheet aimed at high school students:
Evaluate $$\lim_{n\to\infty} \,\sum_{k=1}^n\left(\frac{k}{n^2}\right)^{\frac{k}{n^2}+1}.$$
I tried to rewrite the inner s... |
H: Graph homomorphism, but edges can be mapped to paths
Let $G$ and $H$ be graphs. Is there a name for a function $f$ which
Maps each vertex $x$ of $G$ to a vertex $f(x)$ of $H$
Maps each edge $e \in E(G)$ with endpoints $x$ and $y$ to a path $f(e)$ between $f(x)$ and $f(y)$
In other words, $f$ is like a graph homom... |
H: If $x+y+z=1$, prove that $9xyz+1\ge 4(xy+yz+zx)$
If $x+y+z=1$, prove that $9xyz+1\ge 4(xy+yz+zx)$ for $x,y,z\in \Bbb R^+$
I tried to solve this by splitting $9xyz$ as $3xyz+3xyz+3xyz$ and taking all the terms to the LHS before factoring, but I was unable to.
Also tried using Schur's inequality, but that didn't ... |
H: Amount of homomorphisms from $V$ to $S_4$
I am trying to count the amount of homomorphisms from the klein four group to $S_n$, so the homommorphisms $f: V_4\to S_n$.
I think I am almost there, but just wanted to let you guys know my way of reasoning, and if that is correct: the elements of $V_4$ all have order 2 (e... |
H: Equivalence of $\sigma$-field $\sigma(X,Y)$ and $\sigma (X+Y,X-Y)$
Consider any two random variable $X$ and $Y$. Is it correct to say that $\sigma$-field $\sigma(X,Y)$ and $\sigma (X+Y,X-Y)$ are equal?
In my logic this is correct, because,
$$
\sigma(X+Y,X-Y) = \sigma\{\omega:(X+Y, X-Y)(\omega)\in H\}
$$
Where $H\in... |
H: Proof that no integers satisfy $x^2+2y^2 = p$
Suppose $p$ is a prime such that $p\equiv 5,7 \ \pmod{8}$, then I want to show that there exist no integral solutions $(x,y)$ such that $x^2+2y^2=p$.
I did a simple approach of simply computing with $x,y=0,1,....7$. But I want to know a more technical approach with a go... |
H: Ratio of two infinite cardinal numbers
Suppose $G$ is the group of all functions between $[0,1]\to\mathbb{Z}$. Let $H$ be the subgroup defined as $H=\{f\in G: f(0)=0\}$. Then, what can be said about the cardinality of $H$ and its index in $G$?
I think the cardinality of $H=G=2^{c^2}$, where $c$ is the cardinality o... |
H: Modification of a reduced homology
Background:
In Hatcher's Algebraic Topology Chapter 2, reduced homology with coefficient $R$ of a space $X$ is defined as the homology groups of a chain complex
$$
...\to C_{2}(X)
\overset{\partial_{2}}\to C_{1}(X)
\overset{\partial_{1}}\to C_{0}(X)
\overset{\epsilon}\to R
\to 0
$... |
H: ()Most significant bit; bit of the greatest value
A binary number is number expressed in the binary numeral system. Let $n$ be the binary number. Each digit in a number as a bit. And the definition of the most significant bit is the following: The most significant bit is the bit position in a binary number with the... |
H: Solve the following EDO's
They ask me to solve
$$y' +2y + \int_{0}^{x} y(t)dt = f(x)$$ with $y(0)=0$ and
$$
f(x) =
\begin{cases}
0, & x < 5 \\
2, & x \geq 5
\end{cases}
$$
I don't know how to do it.
AI: Solve the following differential equation:
$$y^\prime +2y + \int_{0}^{x} y(t)dt = f(x)\tag{1}$$
where $y(0)=0$... |
H: How can i prove that if some set is a subset of every set in a family of sets, then it is a subset of the intersection of family too.
The question is stated as:
Prove: $(\forall B)(B \in F \Rightarrow C \subseteq B) \Rightarrow C \subseteq \bigcap_{A \in F}A$
Thats what i thinked in a textual way: If we assume that... |
H: Why can a deterministic first-mover always be exploited in 2-player zero-sum games?
I am reading a book about Bandit Algorithms in which the authors make the following observation. I am wondering if anyone could point me to an explanation of why the first-mover can always be exploited unless following a randomized ... |
H: How to prove using Zorn's lemma that if $X$ has at least two elements, then is a 1-1 function $H\colon X\to X$ such that $H(x)\neq x$ for all $x$?
This is the problem.
Let $X$ be an arbitrary set having at least two elements. Show that there is 1-1 function $H: X \to X$ such that for all $x \in X$, $H(x) \neq x$, b... |
H: Convergence of series problem
Consider the sequence given by $$x_n=\sqrt{-1+\sqrt{1+\frac{1}{n^{\alpha}}}}$$
For what values of $\alpha>0,$ $x_n \in l^1.$
I feel for $\alpha>4,$ $x_n\in l^1$ and for other values of $\alpha$ series $\sum |{x_n}|$ diverges.
How to prove it rigorously?
AI: As a hint : Maybe you know $... |
H: A number is selected from each of these sets,say $p$, $q$, $r$ respectively. The probability that $r=|p-q|$ is?
Let $P$, $Q$, $R$ be the sets of first $8$ natural numbers, first $12$ natural numbers,first $20$ natural numbers respectively.
A number is selected from each of these sets,say $p$, $q$, $r$ respectively.... |
H: How many elements of $M$ are similar to the following matrix?
Let k be the field with exactly $7$ elements. Let $M$ be the set of all $2\times 2$ matrices with entries in k. How many elements of $M$ are similar to the following matrix?
$ \begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}$
I know that two matrices in... |
H: If $ 3a+2b+c=7$ then find minimum value of $ a^2+b^2+c^2$
Question:- If $ 3a+2b+c=7$ then find the minimum value of $ a^2+b^2+c^2 $.
I used vectors to solve this problem.
Let $$α=3\hat{i}+2\hat{j}+\hat{k}$$
$$β=a\hat{i}+b\hat{j}+c\hat{k}$$
Using Cauchy-Schwarz inequality
we have, $|α.β|\le |α| |β|$
$=|3a+2b+c|\le\s... |
H: Number sequence with a given formula
$a_{n}$ is a sequence which is given by the formula: $a_{n+1}=\log_2(a_n)$, where $a_1=30$. What is a number of maximum elements of the sequence?
I tried to approximate each number but I think there must be more systematic way to solve this problem. Can you explain this to me? N... |
H: Is it a convex set?
I have the following set
$$A = \{ (x,y) \in \Bbb R^{2} : \log x + y^{2}\ge 1, x \ge 1, y \ge 0 \}$$
and I need to know if it's convex or not. I tried to have a look at this function $-\log x-y^{2}$, but the Hessian matrix is indefinite and I don't know what to do else.
AI: $(e,0)$ is in the set.... |
H: Does a continuous bijection from a compact, hausdorff space imply it is an homeomorphism?
Let $f:X \rightarrow Y$ be a bijective, continuous map between two toplogical spaces. Does X being compact and Hausdorff imply that $f$ must be a homeomorphism? I think it doesn't but I can not find an example.
AI: Let $X$ be ... |
H: Proof that Hadamard matrices of order $4k+2$ don't exist
It's known that Hadamard matrices can only exist for orders $1$, $2$ and $4k$. It's easy to show that there are no Hadamard matrices of order $2k+1$. But what is the proof that there are no Hadamard matrices of order $4k+2$?
AI: Assume the Hadamard matrix has... |
H: How many of the subsets of set $A$ are also subsets of set $B$?
$A=\left\{2,3,4,5,7,8\right\}$
$B=\left\{3,4,5,7,10\right\}$
How many of the subsets of set $A$ are also subsets of set $B$?
I find $2^4=16$. Because, $A∩B=\left\{3,4,5,7\right\}$.
But, the answer is $32$. But, why? Am I wrong?
AI: Here are your $2^{\#... |
H: Let $(G,\cdot)$ be a set with an associative operation. Show that the following two Axioms are equivalent
Let $(G,\cdot)$ be a set with an associative operation. Show that the following two Axioms are equivalent:
(a) : there exists a left-hand neutral element $e'$, so that $\forall a \in G: e'a=a$
(b): There exists... |
H: Missing piece for combinatorial proof
I am working on a combinatorial proof given that $n-k$ is divisible by $g$ (that is $mod(\frac{n-k}{g})=0$). I am missing a piece for this step:
$\frac{k-1}{n-1}{n+\frac{n-k}{g}-2 \choose n-2}+\frac{k+g}{n}{n+\frac{n-k}{g}-2 \choose n-1}=\frac{k}{n}{n+\frac{n-k}{g}-1 \choose n-... |
H: How can I prove $\int_{0}^{\infty}\frac{x^2+1}{(x^4+2ax^2+1)(x^s+1)} dx=\frac{π}{2\sqrt{2a+2}}$
Question:- Prove that $$\int_{0}^{\infty}\frac{x^2+1}{(x^4+2ax^2+1)(x^s+1)} dx=\frac{π}{2\sqrt{2a+2}}$$
I Recenty got stuck on evaluating this integral,the result is independent of $s$. with $s=2$ , I verified the result... |
H: Show one diagonal of $B D E C$ divides the other diagonal in the ratio?
Consider a triangle $A B C$. The sides $A B$ and $A C$ are extended to points $D$ and $E,$ respectively, such that $A D=3 A B$ and $A B=3 A C$. Then one diagonal of $B D E C$ divides the other diagonal in the ratio ?
My approach
I am trying to ... |
H: Example of matrix $A\neq I$ such that $A^3=I$
Let $A \neq I$ be a $3 \times 3$ matrix. I need to find an example of $A$ that satisfies $A^3 = I$.
Is there any "smart" way to do this? All I can think of is to either multiply $A$ for 3 times and then try to guess the factors or try to solve $A^2 = A^{-1}$. In either ... |
H: What is $P(\min\{X, Y\} = 1)$?
If $y=1,2,3$ and $x=0,1,2$ where $P(X=x, Y=y) = k(2x+3y)$
I need to find $P(\min\{X, Y\} = 1)$.
I thought I need to use that the CMF of the minimum is $1-P(X)$, and maybe to find k by doing derivative on the equation and to sum it up to 1?
would love any direction on this.
AI: Given t... |
H: If $X$ follows an $\operatorname{Exp}(\theta)$, does $1/X$ follow an $\operatorname{Exp}(1/ \theta)$?
I heard a teacher say that if $$X \sim \operatorname{Exp}(\theta)$$
then $$\frac{1}{X} \sim \operatorname{Exp}\left(\frac{1}{\theta}\right)$$
I don't trust this teacher because he has given us wrong answers before.... |
H: Find a vector that is as proportional as possible to a given vector under a set of linear constraints
Let $d\in \mathbb{R}^n, \ b\in \mathbb{R}^n, \ A \in \mathbb{R}^{m\times n},\ \lambda\in \mathbb{R}$.
Let $x=\lambda d+\varepsilon $, where $\varepsilon\in \mathbb{R}^n$.
Let $E_\lambda =\left \{\varepsilon\in \mat... |
H: inequality involving exponentials and square root function
I came across the following inequality from these notes. It states that
$\frac{e^{-\lambda}}{\sqrt{1-2\lambda}} \le e^{2\lambda^2}$ for $\lambda < \frac{1}{4}$.
Is there a way to show this using calculus? I tried setting $f(\lambda) = e^{2\lambda^2} -\frac{... |
H: How do I prove both arcs are equal?
As in the following image, the segments AD, DB, BE and EC make the same angle (x) relative to the diameter of the circle QP. How can I prove the arcs L1 (AB) and L2 (BC) are equal?
AI: This question is trivially answered by simply extending $AD$ and $CE$ and observing their inter... |
H: Condition for splitting the integrals
I wanted to ask what are the conditions for splitting a definite improper integral?
For example, is it true that $$ \int_{0}^{\infty}(f(x)+g(x)) ~dx=\int_{0}^{\infty}f(x)~dx+\int_{0}^{\infty}g(x)~dx $$
If both integrals are individually convergent and are continuous throughout ... |
H: Consider the unbounded region below $y=e^{2x}$ and $y =e^{-2x}$ , and above $y =0$ over interval $(−\infty, \infty)$. What is the area of this region?
Consider the unbounded region below $y=e^{2x}$ and $y =e^{-2x}$ , and above $y =0$ over
interval $(−\infty, \infty)$. What is the area of this region?
How can the... |
H: Find the area of the square $ABCD$ in terms of $u$ and $v$.
QUESTION: Given a square $ABCD$ with two consecutive vertices, say $A$ and $B$ on the positive $x$-axis and positive $y$-axis respectively. Suppose the other vertex $C$ lying in the first quadrant has coordinates $(u , v)$. Then find the area of the squar... |
H: Lebesgue points
Lebesgue points are defined by:
$$\lim_{r \to 0}\frac{1}{Leb(B(x,r))}\int_{B(x,r)}|f(y)-f(x)|dy=0 \text{ for a.e. } x \in \mathbb{R} \text{ w.r.t. } Leb$$
where $f\in L^1(\mathbb{R}), B(x,r)$ is the ball with center $x$ and radius $r$.
My question is: if we replace $Leb$ by general measure $\mu$? Do... |
H: A set as an algebraic structure
A set is collection of distinct objects: https://en.wikipedia.org/wiki/Set_(mathematics).
The word distinct implies the identity relation on each set: an element of a set is equal to itself, or $a = a$
(Does the word 'distinct' in the definition of Set implies an equivalence relation... |
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