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H: Conditions that a function is analytic in the complex plane of its independent variable?
I have no a mathematic undergraduate background, so I am very sorry if this question is too naive.
Consider a simple example: $f(x)=\vert x \vert^3$ and $g(x)=x^3$ where $x\in \mathbb{C}$. Why $f(x)$ is not analytic in the comp... |
H: Smooth function with non-zero derivative having bounded number of zeros in a compact interval?
Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function. Suppose $f^{(n)}(a) \neq 0$ for some $n \geq 2$.
I was wondering is it possible to prove that there exists $\epsilon > 0$ such that
$f$ only has finitely many zer... |
H: Does the integral of a function exist at a sharp point in the function?
This is a pretty basic and easy question to answer, but I am not certain about its answer (I am still studying at highschool).
Let's say we have a function with some sharp point and we want to find its improper integral, such as: f(x)=|1-x| (th... |
H: Addition-related property of the determinant of a $2 \times 2$ block matrix
For $2 \times 2$ matrices,
$$\det \begin{bmatrix}
a&b \\
c+e&d
\end{bmatrix} =\det\begin{bmatrix}
a&b \\
c&d
\end{bmatrix}+ \det \begin{bmatrix}
a&b \\
e&0
\end{bmatrix}$$
If $A$ is $m \times m$ matrix and $D$ is $n\times n$ mat... |
H: On the use of Weierstrass' M-test for uniform convergence of series including unbounded terms
Let $A$ be a subset of $\mathbb{R}$ and for each integer $k\in\mathbb{N}$ consider a sequence of functions $\{f_k(x)\}_{k=1}^\infty$ defined on the set $A$. Suppose that there is an integer $n^*$ such that $\sup_{x\in A}|f... |
H: About differential equations
I do not understand the concept of differential enter image description here for example
What have both the sides been differentiated by to get this result.I understand that dz^2/dz gives 2z,where has the extra dz come from also what has the side with theta been differentiated by to get... |
H: What is the answer to this question?
A cylinder with closed ends has a total surface area ; the radius of the base is and the height is . Find an expression for in terms of and .
The expression I get cancels down to $√a^2=a$ which seems obvious.
AI: HINT: The total surface area $S$ of a closed cylinder with radi... |
H: Confidence interval for $\theta$ when $X_i$'s are i.i.d $N(\theta,\theta)$
Let $X_i$ be i.i.d. r.v. with $N(\theta,\theta)$
I calculated $$E[\bar{X_n}] = \theta$$ $$Var[\bar{X_n}] = \theta/n$$
And want to construct a confidence interval $I_{\theta}$ that is centered around $\bar{X_n}$ such that $\theta \in P(I_{\th... |
H: Bearings Question involving cosine rule
Ship A is 120 nautical miles from lighthouse L on a bearing of 072°T, while ship B is 180 nautical miles from L on a bearing of 136°T. Calculate the distance between the two ships to the nearest nautical mile.
I've stuck on this question for a while.I have tried using the c... |
H: Equivalence in strong convergence of operators
I am trying to see if I have that $(T_n)\in L(X)$ bounded and that $T_nx$ converges to $Tx$ for every $x$ in a dense subset of $X$ a Banach space, then $T_n$ converges strongly to $T$.
Let suppose that $D$ is the dense subset, I was able to see that if $x\in cl D$ the... |
H: Bijection between $\mathbb{Z}/ m\mathbb{Z}$ and the set of elements coprime to all divisors of $m$
Let $m \in \mathbb{N}$. I want to show that there's a bijection between the sets
$$A = \{(q,a) \mid q\in \mathbb{N} \text{ divides }m, \text{ and }a\in \mathbb{Z}/q\mathbb{Z} \text{ is such that }\gcd{(a,q)}= 1\}, \t... |
H: Sum of one open and one closed set in $\mathbb{R}^n$ is open or closed or none? NBHM 2012 PhD question.
A and B are subsets of $R^n$ where A is open and B is closed. Define A+B as $$A+B= \{a+b: a\in A, b\in B \}.$$
is A+B open or closed or none of them?
I tried to consider n=1 and work with $\mathbb{R}$ only. By ta... |
H: Prove that $\int\sum k\chi_{f^{-1}}<\infty$
For a Lebesgue measurable sets and functions problem, I need to prove this statement:
Being $A\subset\mathbb{R}$ a measurable set with $m(A)<\infty$, and $f:A\rightarrow[0,\infty)$ a Lebesgue measurable function:
$$\sum_{n=0}^\infty(m(\{x\in A:f(x)\geq n\}))<\infty\Lon... |
H: Conceptual meaning of a differential
When we find the derivative of $z^2$ with respect to $z$ it means the slope of the graph,Which comes out to be $2z$.
$$ \frac{dz^2}{dz}=2z $$if we take $dz$
on the other side it becomes $dz^2=2zdz$ which is known as the differnetial of $z^2$. I am not sure what this means, does... |
H: Prove that $\frac{1}{\sqrt[3]2}=\sqrt{\frac 5{\sqrt[3]4}-1}-\sqrt{(3-\sqrt[3]2)(\sqrt[3]2-1)}$
Playing around with denesting radicals, I arrived at the following formula which appears to be correct.
$$\frac 1{\sqrt[3]2}=\sqrt{\frac 5{\sqrt[3]4}-1}-\sqrt{(3-\sqrt[3]2)(\sqrt[3]2-1)}$$
If one were to prove this stri... |
H: Finding the adjoint of linear map with respect to an inner product
Consider the map $A:\mathbb{R}^2\mapsto\mathbb{R}^n$ given by $A\left(x_{1}, x_{2}\right)=\left(2 x_{1}, x_{1}-x_{2}\right)$. I want to find the adjoint with respect to the inner product on $\mathbb{R}^2$.
My attempt:
$$\langle A x, y\rangle=\left\l... |
H: Proof for Division Algorithm, from the book *Contemporary Abstract Algebra* by Joseph A. Gallian
This is the same question as Q2 of Need help with understanding the proof for Division Algorithm, from the book *Contemporary Abstract Algebra* by Joseph A. Gallian
"Q2: I don't understand how 0∉S implies a≠0. We could ... |
H: Is my proof that nonnegative polynomials on $[0,1]$ form a convex set correct?
I want to prove that the set $K = \{c \in R^n\mid c_{1} + c_{1}t +\dotsb+ c_{n}t^{n-1} ≥ 0 \forall t \in [0,1]\}$ is a cone i.e that for $x \in K$ and $\theta \ge 0$, $\theta x\in K$.
Is the follow attempt a correct proof?
Let's conside... |
H: Unbiased estimator for median (lognormal distribution)
Assume that
$Y \sim N(\mu,\sigma^2)$
$X = e^Y$
Then X is lognormal distributed with parameters $\sigma$ and $\mu$.
I know that
$E(X) = E(e^Y) = \eta e^{\frac{\sigma^2}{2}} $
The median in the lognormal distrubution is $\eta = e^\mu$, and that $\eta* = e^\wi... |
H: Prove $(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$
If $a, b, c$ are distinct real numbers, prove that
$(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$
I thought of using AM-GM but that is surely not getting me anywhere ( Maybe some step before i can do AM-GM ?)
I thought of applying... |
H: Can every convergent series uniformly converge in some way?
I found myself thinking on this idea for a long time, I really not understand the intuition of the difference between uniform convergence and convergence. But when I thought on the definition deeply, I figured that
If I have convergence of $f_n$ so I know ... |
H: Is the value of $\mathbb E_{\forall i=1,2,\cdots,n,x_i\sim N(0,1)}\,\Vert\vec x\Vert_2=\sqrt{n}$?
I simulated the result with computer and my guess seems to be correct. Assuming that $\mathbb E_{\forall i=1,2,\cdots,n,x_i\sim N(0,1)}\,\Vert\vec x\Vert_2=\sqrt{\mathbb E_{\forall i=1,2,\cdots,n,x_i\sim N(0,1)}\,\sum_... |
H: How to prove $\sum_{i=1}^n i.i! = (n+1)!-1$ with mathematical induction?
I'm trying to prove
$$
\sum_{i=1}^n i.i! = (n+1)!-1
$$
with mathematical induction. The first step I did after prove it for 1 was:
$$
\sum_{i=1}^{n+1} i.i! = (n+2)!-1= (n+2).(n+1)!-1
$$
but I can't do anything more.
AI: The equality is trivia... |
H: A question related to discrete and subspace topology
I am unable to think how to prove this question.
Question is - Let A be a subgroup of Real Line under Addition. Show that either A is dense in Real Line Or else the subspace topology of A is discrete topology.
I tried by assuming A is not dense in Real Line b... |
H: Equality of expectations if identically distributed
Let $X,Y,Z$ be independent random variables on some probability space $(\Omega, P)$ with values in $\mathbb R$ such that $X \sim Y$, i.e. $X$ and $Y$ have the same distribution. Let $f \colon \mathbb R^2 \to \mathbb R$ be a measurable function. Is it always true t... |
H: $f(x)=(\sin(\tan^{-1}x)+\sin(\cot^{-1}x))^2-1, |x|>1$
Let $f(x)=(\sin(\tan^{-1}x)+\sin(\cot^{-1}x))^2-1, |x|>1$. If $\frac{dy}{dx}=\frac12\frac d{dx}(\sin^{-1}(f(x)))$ and $y(\sqrt3)=\frac{\pi}{6}$, then $y(-\sqrt3)=?$
$$f(x)=(\frac{x}{\sqrt{x^2+1}}+\frac{1}{\sqrt{x^2+1}})^2-1=\frac{2x}{1+x^2}$$
$$\frac{dy}{dx}=\... |
H: Prove $P= 7\,{c}^{4}-2\,ab{c}^{2}-2\,ab \left( a+b \right) c+ \left( a+b \right) ^{2} \left( {a}^{2}+{b}^{2} \right) \geqq 0$
For $a,b,c$ are reals$.$ Prove$:$ $$P= 7\,{c}^{4}-2\,ab{c}^{2}-2\,ab \left( a+b \right) c+ \left( a+b \right) ^{2} \left( {a}^{2}+{b}^{2} \right) \geqq 0$$
I found this from Michael Rozenber... |
H: Checking the uniform convergence of sequence of functions
I have been trying some questions on uniform convergence.Got stuck in one of those questions which says that
For a positive real number p, let (f$_n$) is a sequence of functions defined on [$0,1$] by
$$f_n(x) =
\begin{cases}
n^{p+1}x, \text{if 0 $\le$ $x$ $... |
H: Let, $V$ be a vector subspace of $\Bbb{R}^n$. Prove that, $V$ is a closed set in $\Bbb{R}^n$ with respect to usual metric.
Let, $\{u_1,u_2,\ldots,u_k\}$ be a basis for $V$. Let, $\{v_n\}$ be sequence of vectors in $V$ such that $v_j\to v$ where $v\in\Bbb{R}^n$. My target is to prove $v\in V$.
Now for each $n\in\Bbb... |
H: The cross product of three vectors
Is the cross product of three vectors associative? If not, then how do I determine $A\times B\times C$? Is it a vague statement? What I did was $A\times B$ then $(A\times B)\times C$.
AI: The cross product is not associative; you have to use brackets to disambiguate. Normally one ... |
H: How to evaluate $\lim_{n\to+\infty}\frac{(n+1)\ln(n+1)}{n\ln(n)}$?
How to evaluate the limit $$\lim_{n\to+\infty}\frac{(n+1)\ln(n+1)}{n\ln(n)}\,?$$
I tried to search it in the forum but I didn't find, this limit is pretty famous and I know it is equal to 1, but I not sure I understand the technique there
AI: Using... |
H: Arithmetic Sequences on Prime numbers
It is known that $a_1, a_2,$...$, a_{50}$ is an arithmetic sequence with common difference $d$, and $a_i (i=1, 2, ..., 50)$ are primes. If $a_1>50$, prove that $d>600,000,000,000,000,000.$
Honestly I have completely no idea how to solve this question. I don't even know how $6... |
H: integral of fractional part $\int_0^1\left\{\frac 1x\right\}dx$ convergent?
$$I=\int_0^1\left\{\frac 1x\right\}dx=\int_1^\infty\frac{\{u\}}{u^2}du=\sum_{k=1}^\infty\int_0^1\frac{\{v+k\}}{(v+k)^2}dv=\sum_{k=1}^\infty\int_0^1\frac{v}{(v+k)^2}dv=\sum_{k=1}^\infty\ln\left(\frac{k+1}k\right)+\frac k{k+1}-1$$
and I belie... |
H: $\sin x = \cos y, \sin y = \cos z, \sin z = \cos x$
For real numbers $x,y,z$ solve the system of equations:
$$\begin{align} \sin x = \cos y,\\
\sin y = \cos z,\\
\sin z = \cos x\end{align}$$
Source: high school olympiads, from a collection of problems for systems of equations, no unusual tricks involved.
So far I ... |
H: Disprove that $V$ is a linear subspace of $R^3$
Disprove that $V$ is a linear subspace of $R^3$, where $V$ = {$(x, y, z)$ ∈ $R^3$ : $x + 2y = 0$ or $5x − z = 0$}.
So here $dim$ = 2, $basis$ = { $(1, 2, 0), (5, 0, -1)$ } if I understand it correctly.
So it is a subspace of $R^3$.
Or am I mistaken here?
AI: Firstly, ... |
H: write the following complex numbers in the form a+bi
I need to write the following complex numbers in the form a+bi, where a and b are real numbers and i=√I
(3-4i)+(1+ √1)
(2 + 2i)(2 - 3i)
AI: $$x = (3+4i) + (1+\sqrt{-1}) = 3+1+4i + i = 4+5i\,,$$
assuming the convention $\sqrt{-1}=i$.
$$y=(2+2i)(2-3i) = 4+4i-6i-6i^... |
H: Doubt regarding groups formation under matrix multiplication
When considering the set of matrices:
$$Sp(n) = \{ S \in \text{GL} (2n, \mathbb{R}) \hspace{2mm} \text{s.t.} \hspace{2mm} S^T \Omega S= \Omega\} \tag{1}$$
where
$$\tag{2} \Omega = \begin{pmatrix} 0 & \mathbb{I}_{n} \\ - \mathbb{I}_{n} & 0\end{pmatrix}$$
w... |
H: Understanding double asymptotics
Suppose I have a double sequence $x_{i,t}$ of numbers where $i=1,\dots,n$ and $t=1,\dots,T$ and a function $f(n,T)$ of these numbers, e.g. $f(n,T)=\sum_{t=1}^{T} \sum_{i=1}^{n} x_{i,t} / nT$.
The paper am reading is interested in the quantity $f(n,T)$ as $n,T\to \infty$ with $n=O(T)... |
H: Find the GCD of $S = \{ n^{13} - n \mid n \in \mathbb{Z} \}.$
Determine the greatest common divisor of the elements of the set $S = \{ n^{13} - n \mid n \in \mathbb{Z} \}.$
I put in a few values of $n$ and I know that $10$ is a common factor, but I'm not sure if that's the greatest common factor.
AI: By Fermat's L... |
H: sum upto n terms where rth term is $r(r+1)2^r$
sum upto n terms where rth term is $r(r+1)2^r$.
I tried to make a telescoping series but failed.It seems like i have to subtract and add something from r(r+1) such that power of 2 also change.Is there a systematic approach?
AI: Hint:
$$S=\sum_{r=1}^n r(r+1)x^r$$
$$... |
H: Proving the inequality that $\dfrac{x^2 + x^{-2}}{x-x^{-1}} \geq 2 \sqrt{2}$ for $x > 1$
Question: Show that $$\dfrac{x^2 + x^{-2}}{x-x^{-1}} \geq 2 \sqrt{2}$$ for $x > 1$.
My attempts: After spending some time trying to prove it by $AM-GM$ and with algebraic manipulation, I tried to use trigonometric substitutions... |
H: Eigenvector of a complete graph Laplacian
Can somebody help me prove why $v=\begin{bmatrix} 1 \dots 1\end{bmatrix}^T$ is the eigenvector of every complete graph Laplacian matrix?
Thanks!
AI: Recall that the definition of a graph Laplacian is $L=D-A$, where $D$ is the degree matrix and $A$ is the adjacency matrix.
... |
H: self-adjoint operator and symmetric operator
we recently learned about self-adjoined operator with the formal definition $ ⟨Tv, w⟩ = ⟨v, Tw⟩$ for every $v, w$ in $V.$
In the other side we talked that self-adjoined can be represented as a symmetric operator (or matrix).
can you explain the geometric interoperation o... |
H: $A^{-1}XB = I$ Solve for X matrix equation
$A^{-1}XB = I$, $A$ and $B$ are given and they are square matrixes.
If I want to solve this matrix equation for $X$, I need to change it to the form like this $X = A×B×I$?
AI: If $B$ is invertible, i.e. $B^{-1}$ exists, then, multiplying by $A$ from the left and by $B^{-1}... |
H: Density of $g(Y)=\frac{1}{2}\mathbb{E}[X|Y]$
Let $(X,Y)$ a random variable with density $f(x,y)=cx(y-x)e^{-y}$ for $0 \leq x \leq y <\infty$. Find:
1) the value of $c$.
$\rightarrow c=1$
2) the density of $X|Y=y$.
$\rightarrow f_{X|Y}(x,y):=\frac{f_{XY}(x,y)}{f_Y(y)}=\frac{fx(y-x)}{y^3}$
3) the density of the ... |
H: Variance of $X$, an uniformly random sum from a finite set $S$.
This is from my class. Can I have an explanation of what going on in the last equality (i.e. $\operatorname{Var}\left(\varepsilon_{i}\right) s_{i}^{2}=\frac{1}{4} \sum_{i=1}^{k} s_{i}^{2}$)?
Let $S=\left\{s_{1}, s_{2}, \ldots, s_{k}\right\} \subsete... |
H: Where is the error? Application of FTC
Please, I need a help to see the error on this argument:$$\int_0^tf(a)g(t-a)da=\int_0^tf(t-a)g(a)da\implies$$
$$\dfrac{d}{dt}\int_0^tf(a)g(t-a)da=\dfrac{d}{dt}\int_0^tf(t-a)g(a)da\implies$$
$$f(t)g(0)=f(0)g(t)$$
If $f(0)=g(0)=k\neq 0$, então
$$f(t)=g(t)\forall t\in\Bbb R.$$
I... |
H: Let $f(x) = x + 2x^2 + 3x^3 + 4x^4 + 5x^5 + 6x^6$, and $S = [f(6)]^5 + [f(10)]^3 + [f(15)]^2$. Find the remainder when $S$ is divided by 30.
Let $f(x) = x + 2x^2 + 3x^3 + 4x^4 + 5x^5 + 6x^6$, and let $S = [f(6)]^5 + [f(10)]^3 + [f(15)]^2$. Compute the remainder when $S$ is divided by 30.
I don't really know how to ... |
H: On finding domain of a trigonometric function
Given, f(theta)= 11cos^2 (theta) - 9sin^2 (theta) + [15 sin (theta) . Cos (theta)].
Find the Range of the function give above and express the above function in the form of {a cos(2theta+ alpha) + b},where a,b, alpha are real numbers.
I tried using completing the sqaure... |
H: If you can reach a point in $R^4$ does that automatically mean that your set of vectors must be Linearly Independent in $R^4$?
I am working on part c) and given that we can reach a point $(1,1,1,1)$ does this mean we are linearly independent in $R^4$?
We can formulate this as a 4x6 matrix and as such the rank must... |
H: product of matrices, and its norm
I have a product of matrices $\prod\limits_{i=1}^{n} a_i$, If $b$ is an eigenvalue of $a_i$ for any $i$, then $|b|<1$.
(1) Under what norm or condition, $\|\prod\limits_{i=1}^{n} a_i\|<r<1$
(2) If I had infinite such matrices, $\|\prod\limits_{i=1}^{\infty} a_i\|<r<1$ true for som... |
H: Verification of Discrete metric space
When we define a metric on $\mathbb{N}$ by $d(m,n)=|\frac{1}{n}-\frac{1}{m}|$. I need to show this metric gives discrete topology on $\mathbb{N}$. For this i need to show singletons are open or in other words for any $n\in \mathbb{N}$ there exists $r>0$ such that $B_{d}(n,r)=\{... |
H: Does the converse of if $x$ is a transitive set then $\bigcup (x^{+})=x$ hold?
I am studying Set Theory from the book Set Theory: A First Course by Daniel W. Cunningham. This proves that If $x$ is a transitive set, $\bigcup x^+ = x$, where $x^+ = x\cup\{x\}$. I really wonder if the converse is true. I tried proving... |
H: Optimality proof for the coin-change problem of 1, 2, 5 and 10
I have four types of coins: 1, 2, 5 and 10. When I am given a number $k \in \mathbb{N}^{+}$, I have to return the least number of coins to reach that number. Using a greedy algorithm I can simply return all the possible 10 coins, and from the remaining,... |
H: Calculating the characteristic polynomial of a 3x3 matrix
I had to calculate the eigenvalues of the following matrix.
$$H=h\begin{pmatrix}A+\frac{1}{2}(B+C) & = & \frac{1}{2}(B-C) \\ 0 & B+C & 0 \\ \frac{1}{2}(B-C) & 0 & A+\frac{1}{2}(B+C)\end{pmatrix}$$
for that, I calculated the characteristic polynomial
$$ \text... |
H: How to find a solution for a matrix with 1 equation and 3 unknown variables?
The task is to find all solutions for $A_1 x = 0$ with $x\in \mathbb R^3$
$$A_1 = \begin{pmatrix}
6 & 3 & -9 \\
2 & 1 & -3 \\
-4 & -2 & 6 \\
\end{pmatrix}
$$
The given solution is as follows:
$$L_1 = \{ \lambda \begin{pmatrix}
1 \\
1 \\
1 ... |
H: Definition of topological space & open sets
I am just getting into topology, and I have a doubt regarding open sets.
Let $(X, \mathcal{T})$ be a topological space. Given an open set of $X$, $A$, and subset of $X$, $B$ such that
$$A\cap B \in \mathcal{T}$$
$$A\cup B \in \mathcal{T}$$
Can I conclude that $B$ is also... |
H: Find out whether linearity for the functions $f$ and $g$ persists
Given $$f: \mathbb{C}^3 \rightarrow \mathbb{C}^2, \begin{bmatrix}
a\\
b\\
c
\end{bmatrix} \mapsto \begin{bmatrix}
ia+b\\
c
\end{bmatrix}, \,\,\,\,\,\,\,g:
\mathbb{C}^3 \rightarrow \mathbb{C}^2, \begin{bmatrix}
a\\
b\\
c
\end{bmatrix} \mapsto \b... |
H: Determine a total cost of producing x units
marginal cost is $C'(x) = 5 + \frac{10}{\sqrt{x}}$, it is known that producing 100 units costs 950$, how much would it be to produce 400 units?
from that I can calculate total cost function which is $C(x) = 5x + 20\sqrt{x}$
Is it enough to just plug in 400? Or should I pl... |
H: When is $\sqrt{k}\operatorname{Var}\left(\sum_{i=1}^{k}X_{i}\right) \leq \sum_{i=1}^{k}\operatorname{Var}(X_{i}) $ true?
Assume we have $k$ dependent random variables $X_{1}, \dots, X_{k}$ with $\operatorname{Var}(X_{i}) < \infty$.
In which case
$$\sqrt{k}\operatorname{Var}\left(\sum_{i=1}^{k}X_{i}\right) \leq \s... |
H: How to prove that some specific group is not isomorphic to any member of any of the 5 families of groups?
For example, let's say we have a group Q8 (Quaternion group).
How to prove that this group is not isomorphic to any member of any of the families of groups (Cyclic, Abelian, Dihedral, Symmetric, Alternating)? ... |
H: Integrability of $\frac{x_1}{|x|^{n}}$ over the unit ball
Is $\frac{x_1}{|x|^{n}}$ integrable over the unit ball of $B_1(\mathbb{R}^n)$? That is, is
$$\int_{B_1(\mathbb{R}^n)} \frac{|x_1|}{|x|^{n}}<\infty?$$
I know that $|x|^{-a}$ is integrable over the ball if $a<n$, but what if $a<n$ in just one dimension? My he... |
H: Existence of $f$ such that $f(x,|x|^2)f(y,|y|^2)=0$ whenever $x \cdot y=0$
Does there exists a non trivial continuous function (other than $f=0$) with the following :
$f:R^4 \to [0, \infty)$
Let a $x,y \in R^3$ and their respective Euclidean norm squared $|x|^2$ and $|y|^2$ and their dot product $x \cdot y$
$f(x,|... |
H: I have to find the conditional pmf's $f_{X|Y}(x|y)$ and $f_{Y|X}(y|x)$
$f(x, y)$ = ($\frac{1}{x+y−1}$$+$ $\frac{1}{x+y+1}$ $−$ $\frac{2}{x+y}$)
I have to find the conditional pmf's $f_{X|Y}(x|y)$ and $f_{Y|X}(y|x)$
I know we can use the following formula, but I do not know how to apply it:
$f_{X|Y}(x|y)$ = $P(X=x|Y... |
H: Multivariate Analysis: formula with unknown origin
A formula to study the stability of a multivariant variable was given to me.
The formula is introduced below:
$$ \sum_{i=1}^k (p_{i,2}-p_{i,1})\log \bigl(\frac{p_{i,2}}{p_{i,1}}\bigl) $$
Where:
$p_{i,j}$ is the relative frequency of the observed value $i$ in the ... |
H: Can we represent an improper integral as $\int_{-\infty}^{\infty} f(x)\,dx = \lim_{a \to \infty} \int_{-a}^a f(x)\,dx$?
I was reading on improper integrals, and came across :
$$\int_{-\infty}^{\infty} f(x)\,dx = \lim_{A \to -\infty}
\int_A^Cf(x)\,dx + \lim_{B \to \infty} \int_C^B f(x)\,dx$$
My question is a rathe... |
H: Expected value of $1$'s in a matrix product defined over $\mathbb{Z}_2$
Let $\mathbf{A}$, $\mathbf{B}$ be random boolean matrices of $n \times n$ size, such that the matrix entry is $1$ with probability $p$ and $0$ otherwise. All entries are independent. How many $1$'s on average will be in the product of matrices,... |
H: generating set of $\mathbb{Z}$
I have some troubles with identification of the generating set in the next group:
If I want to create a group $\mathbb{Z}$ from commutative monoid $\mathbb{N}$ I should take $\mathbb{N}^2$ and factorize it by $(n_1,m_1) = (n_2,m_2)$ if $n_1+m_2 = n_2+m_1$. After that, the operation $... |
H: Product $PN$ of normal subgroups is abelian
I am trying to show that every non-abelian group $G$ of order $6$ has a non-normal subgroup of order $2$ using Sylow theory.
First, Sylow's Theorem says the number of Sylow $2$-subgroups $n_2$ is either $1$ or $3$. Assume that $n_2=1$. Then $G$ has a normal subgroup $P$ o... |
H: If$ f(x)=x^{10000}-x^{5000}+x^{1000}+x^{100}+x^{50}+x^{10}+1$, what is the number of rational roots of $f(x)=0$?
The question is:
If$$ f(x)=x^{10000}-x^{5000}+x^{1000}+x^{100}+x^{50}+x^{10}+1$$ what is the number of rational roots of $f(x)=0$?
I used descrates rule.
As number of time sign change is two therefore po... |
H: Quadratic with missing Linear Coefficient
Let $x^2-mx+24$ be a quadratic with roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible?
I'm assuming we can use Vieta's Formula.
We can say $x_1+x_2=m,$ and $x_1\cdot x_2=24.$
$16$ values satisfy both of these conditions,... |
H: Probability of being "close" to a lattice point
Let $X = \{1, \cdots, n\}$, and let $T$ be the set of $t$-tuples over $X$.
Now choose a random point $x$ from $[1, n]^t$ (note that $x$ is a tuple of real numbers, not necessarily a lattice point), and define $\epsilon_1, \cdots, \epsilon_{|T|}$ to be the distances (... |
H: Inverse of the $y=x^x$ in implicit form?
I want to find the inverse of the function $y=x^x$ in implicit form and not by using Lambert W function. Can you tell me how to find it?
Thanks.
AI: The Lambert $W$ function was precisely introduced to solve the equation
$$y=xe^x$$ and those that can reduce to that form, suc... |
H: Convergence of integral $\int_1^2 \frac{\sqrt{x}} {\ln(x)} \,dx $
I want to determine whether or not the integral $$\int_1^2 \frac{\sqrt{x}} {\ln(x)} \,dx $$ converges.
I have tried things like $$\int_1^2 \frac{\sqrt{x}} {\ln(x)} \,dx \leq \int_1^2 \frac{2} {\ln(x)} dx ,$$ but I find myself unable to evaluate the l... |
H: Uniform continuity implies continuity in topological vector spaces
Let $E$ and $F$ be topological vector spaces and $A \subset E$. I want to prove that: if $f: A \longrightarrow F$ is a uniformly continuous function, then $f$ is continuous.
I want that, by definition of uniformly continuous we have for all $V \in \... |
H: Automorphism of vector space $V$ such that $\varphi(S_1)=S_2$
Let $V$ be a finite dimensional $K$-vector space. Let $S_1,S_2\subset V$ be subspaces such that $\dim S_1=\dim S_2=n$. Show there is an automorphism $\varphi:V\to V$ such that $\varphi(S_1)=S_2$.
Let $\{a_1,\ldots,a_n\}$ be a basis for $S_1$, and let $... |
H: Is a collection of pairwise disjoint closed intervals countable?
Attempt: If we argue the same way as we did for the case of open intervals: Since the closed intervals are disjoint, we can identify each closed interval with a rational number in that interval and since the rationals are countable, their subsets are ... |
H: Contradiction problem from $P(x)=a_nx^n+a_{n-1}x^{n-1}+ \dots+ a_0$
Prove that there is no polynomial $$P(x)=a_nx^n+a_{n-1}x^{n-1}+ \dots+ a_0$$ with integer coefficients and of degree at least $1$ with the property that $P(0), P(1), P(2), \dots$ are all prime numbers.
How should one approach this? Contradiction ... |
H: Compute $\lim_{n \rightarrow \infty} \lim_{R \rightarrow \infty} \int_0^R \sin{(x/n)} \sin{(e^x)}dx$.
Another practice preliminary question for you all. This time, a double limit of an integral.
Problem Compute $\lim_{n \rightarrow \infty} \lim_{R \rightarrow \infty} \int_0^R \sin{(x/n)} \sin{(e^x)}dx$. Hint: I... |
H: Find three vectors in ${R^3}$ such that the angle between all of them is pi/3?
Is there a simple way to do this? I have found that ${(a.b)/|a||b|}$ must be equal to ${1/2}$ but from there I am stuck how to proceed. Any help?
P.s. This is from the MIT 2016 Linear Algebra course and is not homework.
AI: Diagonals of ... |
H: Evaluate $f_n(\alpha,\beta)=\int_0^{\infty}\mathrm{e}^{-x^n}\sin(\alpha x)\cos(\beta x)\,dx$
I was just looking at the following function but couldn’t understand how to form a way of integrating this and how does it depend upon $n, \alpha$ and $\beta$ can anyone please help
$$f_n(\alpha,\beta)=\int_0^{\infty}\mathr... |
H: Transitive actions of $\mathbb{Z}_6$ on itself
Two actions of $\mathbb{Z}_6$ on itself that we naturally might consider are $\overline{m} \cdot \overline{n} = \overline{m}+\overline{n}$, and $\overline{m} \cdot \overline{n} = \overline{n}-\overline{m}$. In fact these actions are isomorphic. On the other hand, we ca... |
H: Uniform convergence of a sequence of functions which is integral of another sequence
I was going through some questions on pointwise and uniform convergence. Got stuck in one of those which says:
Let $g_n(x) = \sin^2(x+\frac{1}{n})$ be defined on $[0,\infty).$
and $f_n(x) = \int_0^xg_n(t)\,dt.$
I am supposed to dis... |
H: Simplifying $a = \dfrac{\sqrt{x}}{x+3} $
Solving equations involving terms of the form $ \dfrac{3x}{6x^2} $ is easy. You can cancel the $x$ in the numerator and end up with: $ \dfrac{3}{6x} $.
However, I am presented with an equation of the following form:
$$a = \dfrac{\sqrt{x}}{x+3} $$
Where, $a$ is a constant.
Tr... |
H: How can I solve the differential equation $r'=r(1-r)$
I seperated the variables and decomposed the fraction to get $r(1-r)=ce^t$, but I don't know where to go from here.
AI: $$\dfrac {dr}{r(1-r)}=dt$$
$$ \left({\dfrac 1 r- \dfrac 1{(r-1)}}\right)dr=dt$$
Integrate:
$$\ln r -\ln (r-1)=t+c$$
$$\dfrac r {r-1}=ke^t$$
$$... |
H: What is the error of a sine function defined using a unit polygon instead of a unit circle?
I have made up a geometric definition for a function $\mathrm{polysin}(n, θ)$:
Construct a regular polygon of $n$ sides. Place the first vertex at
$(1,0)$ and place the rest going counterclockwise.
Let a line through the or... |
H: How can I prove this statement about mean and variance?
How can I prove that:
$$E(a) = a\, \text{ and }\, V(a) = 0?$$
AI: Suppose $X$ discrete random variable with constant value $a$. So distribution looks like$$a\space a \space ...a\\p_1\space p_2 \space... p_n$$
Then
$$EX = \sum_{i=1}^{n} a \cdot p_i = a \cdot \s... |
H: Does removing the "heaviest" edge of all cycles in an (unweighted) graph result in a minimum spanning tree?
Background:
A graph is connected if there is a path between all pairs of vertices.
A graph has a cycle if there exists two vertices with an edge between them and a path between them that doesn’t use that edge... |
H: Changing reduced partial sum into a multiplicative function
I have a partial sum in the form of $$\sum_{\substack{n \leq x\\k|n}} f(n)$$ for a fixed $k \in \mathbb{Z}$ where $f(n)$ is a multiplicative function. Is there a way to reduce this partial sum into another sum such that I can exploit the multiplicative pr... |
H: Is there a way to determine a function that could model the transformation of one function to another?
Let's say I have a function centered at the origin, say $f(x)= x^2$, at an initial time. After some time has passed, the initial function $f(x)$ has transformed into a different function, say $g(x)=6x^7$. Is it ma... |
H: If original set of vectors have zero mean, will the orthogonally projections of the vectors onto another vector have zero mean?
Consider vectors $x_1, \cdots, x_n \in \mathbb{R}^m$. Define the vector $\mu \in \mathbb{R}^m$ to be the mean of the vectors:
$$
\mu = \frac{1}{n}\sum_{i=1}^n x_i
$$
Assume that $\mu = 0$,... |
H: Necessary and sufficient condition for $f_n$($x$) = $b_{n}x$+$c_{n}x^2$ to uniformly converge to zero
Have been trying some questions on uniform and point-wise convergence of sequence of functions. Got stuck in this. I have to prove the following:-
Let ($b$$_n$) and ($c$$_n$) be sequences of real numbers then $\su... |
H: Algebra/number theory solution check, number of 0's at end of integer
As part of a larger problem, I wish to calculate the value of $\frac{1993^2+1993}{2} \pmod {2000}$. The top reduces to $42$. However, $\gcd(2,2000)>1$, so the solution is not $21$, and carrying out the division would require changing the modulus.... |
H: A Map is continuous on the inverse image of the set $(-\infty,r]$. Does this inverse image a closed set?
Let $U$ be a topological space and a map $g:U\to \mathbb{R}$. For a given $r\in\mathbb{R}$, define $E:= \{x\in U: g(x)\leq r\}$. If $g$ is continuous at every point of $E$, then Is it true that $E$ is closed set... |
H: Multiobjective optimization
I need some clarification on multi objective optimization. I would like to know if a problem has three objectives with completely different variables, should such a problem be solved as three independent single objective optimization problem or could the problem be solved using a mult... |
H: If a function $f$ is $L$-periodic then $f'$ has $2$ zeros in $[0,L)$?
Let $f: \mathbb{R} \longrightarrow \mathbb{R} $ be a differentiable and odd function. If $f$ is periodic and the (minimal) period $L>0$, then $f'$ has $2$ zeros in $[0,L)$?
For example, this occurs if we consider $f(x)=\sin(x)$, for all $x \in ... |
H: Is $\forall x ((A = \{a | P(a)\} \wedge x \in A ) \rightarrow P(x))$ an axiom of some system?
In section 1.3 of Vellemans's 'How to Prove it', the author states the following:
"In general, the statement $y \in {x | P(x)}$ means the same thing as $P(y)$,..."
I couldn't find a proof of this, and wondered if $\forall... |
H: How to calculate average growth when it's negative?
We have annual reports for company's revenue and can calculate annual growth as
$yg = {y_{i+1} \over y_i}$.
And then we can calculate the average monthly growth as $mg = ({y_{i+1} \over y_i})^{1 \over 12}$.
So for reports 2000-12 $1m and 2001-12 $2m the average m... |
H: Find the sequence $a_n$ so that $\sum_{n=1}^{\infty} a_nsin(nx) = f(x)$ where $f(x)$ is a piecewise function.
Trying to solve a problem I reached a point where I know that $$\sum_{n=1}^{\infty} a_nsin(nx) = f(x) \text{, where }f(x) = \begin{cases}
x & 0 \leq x \leq \frac\pi2 \\[5pt]
\pi - x & \frac\pi2 < x \leq \pi... |
H: How do I find the sum of a power series $\underset{n=3}{\overset{\infty}{\sum}}\frac{x^n}{(n+1)!n\,3^{n-2}}$?
I have found the area of convergence to be $ x \in (-\infty, \infty)$, and this is how far I had gotten before getting stuck:
$$
\begin{aligned}
\sum_{n=3}^{\infty} \frac{x^{n}}{(n+1) ! n 3^{n-2}} &=\sum_{k... |
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