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H: How to find the constant value for this differential equation
Given
$$
\dot{y} = \frac{1}{4}y(1-\frac{1}{20}y), \quad y(0)=1
$$
using separable approach, we get
$$
4\ln\left( \frac{|y|}{|y-20|} \right) = t+C
$$
Is it possible to find the value of $C$? The natural logarithm is undefined at negative values.
AI: Wheth... |
H: Estimating $\int_{0}^{1}\sqrt {1 + \frac{1}{3x}} \ dx$.
I'm trying to solve this:
Which of the following is the closest to the value of this integral?
$$\int_{0}^{1}\sqrt {1 + \frac{1}{3x}} \ dx$$
(A) 1
(B) 1.2
(C) 1.6
(D) 2
(E) The integral doesn't converge.
I've found a lower bound by manually calculating $\int... |
H: Solving the differential equations as shown below
I recently came across a question in which we had to to solve the set of differential equations:
$10dx/dt+x+y/2=0 $ and $6d(x-y)/dt= y$
I tried a lot to solve these equations but I was unable to do so. I tried adding them eliminating $t$, but I couldn't even solv... |
H: Locus of a point with constant distance ratio $e$ to two circles.
Please help to obtain.. in as elegant a form as possible.. the locus of point $P$ equations if its distances to two circles:
$$ (x-h)^2 + y^2 = a^2;\;(x+h)^2 + y^2 = b^2 ;\;$$
are in a constant ratio $e,$ or
$$\dfrac{ \sqrt{(x + h)^2 + y^2} - a} { \s... |
H: How do you draw a $K_{m.n}$ graph?
$K_{3,3}$ is a complete bipartite graph with $6$ nodes split into $2$ groups of $3$ nodes. All of nodes in one group are connected to all of the nodes in the other groups, but not with nodes in the same group. Here's what it looks like:
However, what is a graph of $K_{m,n}$ suppo... |
H: Calculate Distance (Not Squared) between two vectors using Inner Product
I'm stuck on a Inner Product question:
Calculate the distance (non squared) between $x=[4 2 1]$ and $y=[0 1 0]$
using inner product defined as
Can someone kindly help with the solution?
AI: Hint: One you have computed the vector $v = x - y$, ... |
H: Proving $f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + ... $ is continuous for a fixed $x_0 \in (-1,1)$ by using the Weierstrass M-test
I am trying to prove that $f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + ...$ is continuous for a fixed $x_0 \in (-1,1)$ by using the Weierstrass M-test. ... |
H: How $N(2\zeta^{2n})=2^{p-1}$?
Let, $\zeta$ be the $p$ th root of unity and $\mathbb{Z}[\zeta])$ be the number ring generated by $\zeta$, and $N$ is the norm function.
Why or how $N(2\zeta^{2n})=2^{p-1}$?
The source of the problem is -
AI: The norm is a homomorphism from the multiplicative group $\mathbb Q[\zeta]^\... |
H: If each element in $A$ is greater than all elements in a unique subset of $B$, is the average of $A$ greater than average of $B$?
Suppose I have two sets $A = \{a_1, ..., a_n\}$ and $B = \{ b_1, ..., b_m\}$ of non-negative real numbers and where $a_1\geq a_2 \geq ... \geq a_n$, and $|A| < |B|$.
Now, suppose $B$ can... |
H: If a 'distance function' does not possess triangle inequality property, would the limit of a converging sequence still be unique?
Let $X$ be a set and $d$ be a function such that $d:X\times X\to \mathbb{R}$ such that it satisfies positivity, that is, $d(x,y)\geq 0$ and $d(x,y)=0 \iff x=y.$ Moreover suppose it satis... |
H: How do I evaluate the line integral $\int _c \mathbf{F}\cdot\mathrm{d}\mathbf{r}$
How do I evaluate the line integral $$\int _c \mathbf{F}\cdot\mathrm{d}\mathbf{r}$$ where $\mathbf{F} = x^2\mathbf{i} + 2y^2\mathbf{j}$ and $c$ is the curve given by $\mathbf{r}(t)=t^2\mathbf{i} + t\mathbf{j}$ for $t \in [0,1]$.
I hav... |
H: Find the sum: $\sum_{n=1}^{20}\frac{(n^2-1/2)}{(n^4+1/4)}$
Hint: this is a telescoping series sum (I have no prior knowledge of partial fraction decomposition)
Attempt: I tried to complete the square but the numerator had an unsimplifiable term. So I couldn't find a pattern. I just need a hint on how to convert thi... |
H: Why every field is a local ring, but the ring of integers $\mathbb{Z}$ not?
It is said, that $\mathbb{Z}$ is trivially not a local ring, because the sum of any two non-units must be a non-unit in a local ring and for example $-2+3=1$. But why every field is said to be local ring? Isn't the violation of exact this r... |
H: trace of $(I_m + AA^T)^{-1}$ and $(I_n + A^TA)^{-1}$ for real matrix A
Let $A$ be $m \times n$ real matrix.
(1) Show that $X=I_m + AA^T$ and $Y=I_n+A^TA$ are invertible.
(2) Find the value of $tr(X^{-1}) - tr(Y^{-1}) $
attempt for (1):
$AA^T$ is a real symmetric matrix, therefore can be diagonalized. Let $\lambda$... |
H: Maximum value of the expression $E=\sin\theta+\cos\theta+\sin2\theta$.
Find the maximum value of the expression $E=\sin\theta+\cos\theta+\sin2\theta$.
My approach is as follow ,let $E=\sin\theta+\cos\theta+\sin2\theta$, solving we get
$E^2=1+\sin^22\theta+\sin2\theta+2\sin2\theta(\sin\theta+\cos\theta)$ not able t... |
H: Classify all groups of a given order
I have to classify all groups of order 87 and 121. How can I do that? I saw other posts but there is no unified approach that helped me so far...
AI: $\gcd(87, \varphi(87)) = 1$ so there is only one group of order 87: $C_{87}$.
$121 = 11^2$ so there is exactly two groups of this... |
H: What is the definition of nodal singularity of an algebraic curve?
What is the definition of nodal singularity of an algebraic curve ?
I got the following definition from here:
A nodal singularity of an algebraic curve is one of the forms parameterized by the equation $xy=0$. A nodal curve is a curve with a nodal... |
H: Is the commutator subgroup of a subgroup the same as the commutator subgroup of the group intersected with that subgroup?
I might be overthinking this, but anyway:
Let $G$ be a group and $H$ a subgroup. Let $K'$ be the commutator subgroup of $H$, i.e. $K' = \langle [x, y] \mid x, y, \in H \rangle$. Is it true that ... |
H: I want to know radius of convergence. $\sum ^{\infty }_{n=0}\left\{ 3+\left( -1\right) ^{n}\right\} x^{n}$
$$\sum ^{\infty }_{n=0}\left\{ 3+\left( -1\right) ^{n}\right\} x^{n}$$
I used ratio test to find radius of convergence.
$$\lim _{n\rightarrow \infty }\left| \dfrac {\left\{ 3+\left( -1\right) ^{n+1}\right\} x... |
H: Determine that a series is rational
Determine whether $$\sum_{n=1}^\infty 1/10^{n!} $$ is rational. I have tried thinking about decimal representations such as that of $1/11$, and the fact that this sum is equal to $0.1100010....1...........1$ etc, but I don't know if the distance between $1$'s increases fast enoug... |
H: Inverse of block anti-diagonal matrix
Let $A \in \mathbb R^{n\times n}$ be an invertible block anti-diagonal matrix (with $d$ blocks), i.e.
$$
A = \begin{pmatrix} & & & A_1 \\ & & A_2 & \\ & \cdot^{\textstyle \cdot^{\textstyle \cdot}} & & \\ A_d\end{pmatrix},
$$
with all square blocks $A_1, \ldots, A_d$ invertible... |
H: Is this space subspace of $[0,1]^{\mathbb{N}}$ Polish?
Let $D := \{ x \in [0, 1]^{\mathbb{N}} \mid \forall n: x_n = 1 \Rightarrow x_{n+1} = 1 \}$
be the set of sequences in $[0, 1]$ such that if $x_n = 1$ for some $n$ then $x_{n+m} = 1$ for all $m \geq 0$.
I know that $[0, 1]^{\mathbb{N}}$ is Polish. Is $D$ with t... |
H: Simplify fraction $4x/(x-1)$ to $ 4+(4/(x-1))$
I have put the fraction into Symbolab which gives some step-by-step explanation on why this os correct, but I am unable to grasp how this is possible.
AI: Adding and $+4$ and $-4$ to the numerator, if $x\in \Bbb R \setminus \{1\}$, you have:
$$\frac{4x}{x-1}=\frac{(4x-... |
H: Compute mass function of $U=X+2Y$
Let be $X$ and $Y$ random variables and let be the joint density $f_{X,Y}(x,y)=\frac{xy}{96}I_{R}(x,y)$
where $R:=\{(x,y):0<x<4,1<y<5\}$. Let be $U:=X+2Y$, its distribution function is
$$F_U(u)=P(U\leq u) = \int\int_{\{(x,y):x+2y\leq u\}} f_{X,Y}(x,y) \,dx\,dy =$$
$$\int\int_{\{(x,... |
H: Understanding what is wrong in a limit development
I have the following limit:
$$\lim_{x\to -\infty} \frac{\sqrt{4x^2-1}}{x}$$
I know that the result is $-2$ and I know how to achieve it. However on the first try I made the following development and I still can't see what I am doing wrong:
$$\mathbf1)\lim_{x\to-\in... |
H: IMO $2001$ problem $2$
Let $a,b,c \in \mathbb{R}_+^*$. Prove that $$\frac{a}{\sqrt{a^2+8bc}} + \frac{b}{\sqrt{b^2+8ca}}+ \frac{c}{\sqrt{c^2+8ab}} \geqslant 1.$$
I tried to follow the proposed solution for this which depended on Hölder's inequality, but I'm a bit confused about how they came up with the expression... |
H: Projection of space curve shortens
Let $C$ be a rectifiable, open curve in $\mathbb{R}^3$,
and let $|C|$ be its length.
Orthogonally project $C$ to a plane $\Pi$ (e.g., the $xy$-plane).
Call the projected curve $C_{\perp}$, and its length $|C_{\perp}|$.
I would like to claim $|C_{\perp}| \le |C|$.
I would appreciat... |
H: Block matrix of tensor product
$K$ is a field, $K^n$ is a vector space with $(e_1, \ldots, e_n)$. The tensor product $K^n \otimes K^n$ has the basis $\mathcal {B} = (e_1 \otimes e_1, \ldots, e_1 \otimes e_n, e_2 \otimes e_1,, \ldots, e_2 \otimes e_n, e_3 \otimes e_1, \ldots, e_n \otimes e_n)\,.$
Look at the matrice... |
H: Construct the joint probability mass function of $X$ and $Y$
Two fair dice are thrown. Let $X$ be the random variable that represents the
maximum obtained on any of the two dice and $Y$ the one which denotes the sum of what was obtained in both dice. Construct the joint probability mass function of $X$ and $Y$.
A... |
H: Problem with the parametrisation of this surface integral
I am facing troubles in understanding (read: "guessing") the correct way to parametrise this integral:
$$\int_{\Sigma} \dfrac{1}{\sqrt{1 + x^2 + y^2}}\ \text{d}\sigma$$
Where $\Sigma = \{(x, y, z)\in\mathbb{R}^3; x^2 + y^2 \leq 1; z = \sin^2(x^2+y^2)\}$
Is t... |
H: How do you prove that the derivative $\tan^{-1}(x)$ is equal to $\frac{1}{1+x^2}$ geometrically
How do you prove that the derivative of $\tan^{-1}(x)$ is equal to $\frac{1}{1+x^2}$ geometrically?
I figured it out by working it out using implicit differentiation.
I also found how to plot a semi-circle using $\cos^2(... |
H: Is every factorial totient?
A positive integer $\ n\ $ is called totient , if there is a positive integer $\ m\ $ such that $\ \varphi(m)=n\ $ holds , where $\ \varphi(m)\ $ is the totient function.
Is $\ k!\ $ totient for every positive integer $\ k\ $?
For $\ 2\le k\le 200\ $ I could find positive integers $\ a... |
H: Increasing sequence of sigma-algebras
On a non-empty set $E$, let $(\mathcal{E}_n)$ be an increasing sequence of sigma-algebras, i.e. such that, for every $n \leq m$, $\mathcal{E}_n \subseteq \mathcal{E}_m$. Let us denote by $\mathcal{E}$ its limit, i.e.
$$
\mathcal{E} = \sigma\left(\bigcup_{n\geq 0} \mathcal{E}_n\... |
H: When can we say that a sequence is bigger or smaller than another sequence
Let's say there are two sequences $\{a_n\}$, $\{b_n\}$. I often see the inequality
$$\{a_n\} > \{b_n\}.$$
I don't understand how can we make such comparison to determine if a sequence is bigger or smaller than another one.
AI: Usually we wil... |
H: How to evaluate $\int_{0}^{\pi} \sin ^{n}(\eta) d \eta$?
I have encountered the following integral:
$$\int_{0}^{\pi} \sin ^{n}(\eta) d \eta=\underbrace{\left[\left(\sin ^{n-1}(\eta)\right)(-\cos (\eta))\right]_{\eta=0}^{\pi}}_{=0} -\int_{0}^{\pi}\left((n-1) \sin ^{n-2}(\eta) \cos (\eta)\right)(-\cos (\eta)) d \eta$... |
H: Finite Summations can be Interchanged
In the proof of associativity of matrix multiplication, the reason for one of the steps is given as -Finite summations can be interchanged. What is meant by this statement?
AI: In other words, they mean the following: if $c_{ij}$ is a number for every $i = 1,\dots,m$ and $j = 1... |
H: Identicality, equality and linearity of i.i.d random variables. What are some examples if iid r.vs are unequal among themselves?
I need to understand some underlying concepts and facts regarding i.i.d random variables.
The problem is $$\textrm{Suppose } \mathrm{X_1, X_2, X_3 } \textrm{ are i.i.d positive valued r.... |
H: Interested in a closed form for this recursive sequence.
Consider the following game: you start with $ n $ coins. You flip all of your coins. Any coins that come up heads you "remove" from the game, while any coins that come up tails you keep in the game. You continue this process until you have removed all coin... |
H: Plane, two lines and distance problem
I have been working on this exercise and am kinda struggling with it. This is the exercise and what I have done so far. Any tips would be greatly appreciated!
Given the plane $x+y=0$ and two lines:
$p_1: \frac{x}{3} = \frac{y+1}{1} = \frac{z-3}{-2}$ and $p_2$ (form with 2 plan... |
H: Clarification on the use of subsequences to prove that in a metric space a sequence in a compact subset admits a convergent subsequence in the subset
Lemma: Let $(X,\tau)$ be a topological space and let $K \subseteq X$ compact. If $E\subseteq K$ is infinite then $Der(E)\neq \emptyset$ , where $Der(E)$ is the set of... |
H: Show that $(\sum a_{n}^{3} \sin n)$ converges given $\sum{a_n}$ converges
Given that $\sum a_{n}$ converges $\left(a_{n}>0\right) ;$ Then $(\sum a_{n}^{3} \sin n)$ is
My approach:
Since, $\sum a_{n}$ converges, we have $\lim _{n \rightarrow \infty} n \cdot a_{n}$ converges.
i.e. $\left|n \cdot a_{n}\right| \leq 1$... |
H: Linear programming word problem
A sports event for a school has 300 tickets. They'll sell tickets to students for $5$ dollars and to teachers for $6$ dollars. School rules say that there must be at least $1$ teacher for every $5$ students on the trip. The school also wants to have at least twice as many students as... |
H: Determining a Laurent series with trigonometric functions
I could use some help with the Laurent series around $z_{0}=0$ for all $z\in\mathbb{C}\backslash\{n\cdot\pi;\;n\in\mathbb{N}\} $ of the following function:
$$f(z)=\frac{e^{sin(z)}-cos(z)-z}{sin^{2}(z)} $$
In particular, I'm having problems trying to figure o... |
H: What is the need to include the "additive identity exists" axiom in the set of vector space axioms?
A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold, according to S. Axler's Linear Algebra Done Right:
Commutativity
Associativity
Existenc... |
H: Sanity check: is this simple formula for pseudoinverse of $[\mathbf{U} \cdots \mathbf{U}]$ correct?
Let $\mathbf{U}$ be some matrix, and then consider the "block row vector"
$$ \underbrace{[\mathbf{U} \cdots \mathbf{U}]}_{N \text{ times}} \,. $$
Claim: The pseudoinverse of this is the "block column vector"
$$ \fr... |
H: Understanding one step in the Neyman-Pearson lemma proof
In Georgii's book they state:
Given $(\chi,\mathcal{F},P_0,P_1)$ with simple hypothesis and alternative and $0<\alpha<1\ $ a given significance level. Then:
$(a)$ There exists a Neyman-Pearson-test $\phi$ with $E_0(\phi)=\alpha$.
They begin the proof like thi... |
H: Power of orthogonal matrix
Suppose $U$ is an orthogonal matrix, and $D$ is a diagonal matrix. Let $I$ denote the identity matrix. Let $k$ be a positive integer.
I think the following holds:
$$(I - UDU^T)^k = U(I - D)^kU^T$$
But I got a little lost while writing out the steps
\begin{align*}
(I - UDU^T)^k &= (UIU^T -... |
H: How do we solve the ODE $y''= \frac{1}{ \cosh (y')}$?
I want to solve
$$y''= \frac{1}{ \cosh (y')} $$
$y(0)=1, y'(0)=0 $
Can I do it by substituting $ y'=z $, $ y''=z' $
and solving
$$ z'= \frac{1}{ \cosh z}$$
$$ \Leftrightarrow \frac{dz}{dx}= \frac1{ \cosh z } $$
$$ \Leftrightarrow \int \cosh z dz = \int 1 dx $$
$... |
H: Family vs. Child when a girl is chosen, what is the probability that the second child is a girl, textbook clarification?
In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
I was able to understand the difference between selecting a child and a family... |
H: If $X_1, \ldots, X_t$ is a sequence of iid r.v.'s, what is the expected $T=t$ such that $X_t>X_1$?
If $X_1, \ldots, X_t$ is a sequence of iid r.v.'s that are say, indexed by time, what is the expected $T=t$ such that $X_t>X_1$? Is it enough to know they are i.i.d or do we need a distribution on them?
AI: (Here I am... |
H: Find the volume between $z=\sqrt{x^{2}+y^{2}}$ and $x^2+y^2+z^2=2$ in spherical cordinates
I am asking to find the volume of the volume trap above the cone $z=\sqrt{x^{2}+y^{2}}$ and below the sphere $x^2+y^2+z^2=2$
When I checked the solution I noticed that it was writen as $$V=\int_{0}^{2 \pi} \int_{0}^{\frac{\... |
H: "Pedantic" derivation of geodesic equation using pullback bundles
I'm trying to get more comfortable with manipulations involving connections and vector fields so I've tried to derive the geodesic equations without having to resort to any familiarities using standard calculus, everything computed "properly" from th... |
H: limit of subsequence where $X_n - X_{n-1}\rightarrow 0$.
suppose $X_{n}$ is a sequence of real numbers such that $X_{n} - X_{n-1} \rightarrow 0$.
prove that the limit of subsequence is empty or single point set or interval.
.
I know the limit of subsequence is the set of limits of subsequences of {$P_{n}$} n=1,2,..... |
H: Computing the quantity $ \frac{{x}\cdot{y}}{{\|x\|\|y\|}}$ in terms of $a$ where $x=(1,0)$ and $y=(a,-2)$
Let $x=(1,0)$ and $y=(a,-2)$ be two vectors $ℝ^2$, where $a$ is a real number. Then compute the quantity
$$
\frac{{x}\cdot{y}}{{\|x\|\|y\|}}$$
in terms of $a$.
My work so far:
$$x\cdot y=1\cdot a+0\cdot(-2)=... |
H: The neighborhood $U$ of the identity generates a connected Lie group $G$. Proof check.
I want to check if my proof is correct.
Consider the subgroup $H$ of $G$ which generated by $U$. Then it's enough to show that $H=G$. But, if $H\neq G$, then $G$ can be written as a disjoint union of cosets $gH$ i.e. $G=\cup_{g\i... |
H: Limit Problem Involving Number Sets
Let $\mathbb{N} = \left\{ 1, 2, 3, ... \right\}$.
For each $n \in \mathbb{N}$, let $A_n$ be a finite set of real numbers.
Assume $\forall m, n \in \mathbb{N}, ~m \neq n \Rightarrow A_m \cap A_n = \emptyset~$.
Assume $\forall \varepsilon > 0, \exists x \in \mathbb{R}, (\exists n \... |
H: Password counting: number of $8$-character alphanumeric passwords in which at least two characters are digits
I had an exam yesterday and wanted to confirm if I am correct, one of the questions was:
Suppose there is a string that has 8 characters composed of alphanumeric characters (A-Z, 0-10) how many combinations... |
H: If $f,g\in\mathcal C^1[0,1],\,f$ monotone, and $g(x)>g(1)=g(0)$ on $(0,1)$, then $\int_0^1 f(x)g'(x)\,dx=0$ if and only if $f$ is constant
The Problem: Let $f,g$ be continuously differentiable on $[0,1],\,f$ monotone, and $g(x)>g(0)=g(1)$ on $(0,1).$ Prove that
$$\int_0^1 f(x)g'(x)\,dx=0\quad\text{if and only if }f... |
H: What does it mean to say data points in a complementary cumulative distribution plot are correlated?
While studying, I came across the following quote:
"A more serious disadvantage is that successive points on a cumulative distribution plot are correlated — the cumulative distribution function in general only chang... |
H: Non Abelian Normal Field Extension with Abelian Subextensions
It is known that a subextion $L/F/K$ of an abelian (Galois) field extension $L/K$ is also abelian. The converse is not true: even when assuming that $L/K$ is Galois and $L/F$ and $F/K$ are abelian, $L/K$ might not be abelian.
I am looking for an explicit... |
H: Give a function uniformly continuous with respect to one metric and not with respect to another, while both induce the same topology
I would very much appreciate an example to the above question or some hints to construct one.
Such a function should not exist in normed vector spaces: If the topologies induced by tw... |
H: Constructing a locally integrable function
Let $\epsilon\in(0,1)$ and $F^{\epsilon}:\mathbb{R}^2\to\mathbb{R}$ defined by
$$F^{\epsilon}(x)=\log(|x|^2+\epsilon^2)$$
How can I construct a $g \in L^1_{loc}({\mathbb{R}}^2)$ such that
$$|F^{\epsilon}(x)|\leq g(x), \ \ \forall x\in \mathbb{R}^2, \ \ \epsilon\in(0,1). $... |
H: General solution to eigenvalue problem when $\lambda $ is negative.
I have a simple one for you guys.
So I was reading this PDEs book which regularly discusses the eigenvalue problem
$F''(x)+\lambda F(x)=0$.
For $\lambda =-\mu^2 $, i.e negative eigenvalues, that is if
$F''(x)-\mu^2F(x)=0$
then the general solution ... |
H: Tournament of 32 teams, highest rank always wins
$32$ teams, ranked $1$ through $32$, enter a basketball tournament that works as follows: the teams are randomly paired and in each pair, the team that loses is out of the competition. The remaining $16$ teams are randomly paired, and so on, until there is a winner.... |
H: Finding the angle between vectors $\mathbf x$ and $\mathbf y$ in radians
Two unit vectors $\mathbf{x}$ and $\mathbf{y}$ in $\Bbb R^n$ satisfy $\mathbf{x}\cdot\mathbf{y}=\frac{\sqrt{2}}{2}$ in radians. How would I go about finding the angle between $\mathbf{x}$ and $\mathbf{y}$?
As I don't know the $\mathbf{x}$ and ... |
H: Convergence and Comparison of Topology
Let $(X,\mathcal{T}_1)$ and $(X,\mathcal{T}_2)$ be a topological space endowed with two different topologies. If any convergent net $\{x_v\}$ in $(X,\mathcal{T}_1)$ is convergent in $(X,\mathcal{T}_2)$, does it imply that $\mathcal{T}_1\supseteq\mathcal{T}_2$?
AI: The question... |
H: Can the functions be chosen so that they increase/decrease monotonically?
Assume we are in the interval [a,b] and we have a function $f\in L^1([a,b])$. Then since the continuous functions are dense in $L^1([a,b])$ we can choose a sequence $f_n$ of continuous functions such that they converge to $f$ in $L^1([a,b])$.... |
H: Evaluating $\lim\limits_{x\to \infty}\left(\frac{20^x-1}{19x}\right)^{\frac{1}{x}}$
Evaluate the limit $\lim\limits_{x\to \infty}\left(\dfrac{20^x-1}{19x}\right)^{\frac{1}{x}}$.
My Attempt
$$\lim_{x\to \infty}\left(\frac{20^x-1}{19x}\right)^{\frac{1}{x}}=\lim_{x\to \infty}\left(\frac{(1+19)^x-1}{19x}\right)^{\frac{... |
H: If $ a-b \mid ax-by$, then $\gcd(x,y) \ne1$?
So is this true for positive integers $a,b,x,y>1$ with $a>b$ and $x>y$?
AI: Well $a-b|ax - bx$ and $a-b|ax-by$ so $a-b|(ax-bx) - (ax-by)=b(y-x)$ and ... why not?
So we can have $a-b |b$, for example $a=8;b=6$ and $x,y$ can be anything....$y=57, x = 56$ for example. $8... |
H: Proving finite-stabilizers of a tensor group action
Let $G$ be a group with subgroup $H$ of finite index. Let $X$ be a $G$-set, (ie. G acts on $X$), then we can define the tensor $G$-set $$G \otimes_H X := (G \times X) / \simeq $$ where the equivalence relation is defined as $(gh,x) \simeq (g,hx)$ for all $g \in G,... |
H: Is a simple and solvable group cyclic?
Is not enough with the simple hypothesis?
If $g\neq e$ then $\{e\}\neq \left\langle g \right\rangle$, also $\left\langle g \right\rangle$ is a normal subgroups of $G$. If $G$ is a simple group, and $\left\langle g \right\rangle$ is not the identity therefor $G=\left\langle g \... |
H: Inqualitiy with exponent
I was trying to prove this inequality $\exp(\frac{1}{\pi})+\exp(\frac{1}{e})\geq 2 \exp(\frac{1}{3})$ My attempt was using AM-GM mean $\exp(\frac{1}{\pi})+\exp(\frac{1}{e})\geq2 \exp(\frac{1}{2\pi e})$.
AI: Since $$\pi+e<6,$$ by AM-GM and C-S and we obtain:
$$e^{\frac{1}{\pi}}+e^{\frac{1}{... |
H: evaluation of infinite series expansion
$(1)$ How can i find $\displaystyle \sum^{\infty}_{n=0}b_{n}x^n,$ If
$$\sum^{\infty}_{n=0}b_{n}x^n=\bigg(\sum^{\infty}_{n=0}x^{n}\bigg)^2$$
$(2)$ How can i find $\displaystyle \sum^{\infty}_{n=0}c_{n}x^n,$ If
$$\bigg(\sum^{\infty}_{n=0}c_{n}x^n\bigg)\cdot \bigg(\sum^{\inf... |
H: Eigenvalues of $n^2 \times n^2$ matrix with $(n-1)^2$ along diagonal and $1$ or $1-n$ elsewhere depending on adjacencies.
I reduced a confounding and challenging problem to the task of proving an unweildy inequality. Luckily, I managed to reduce the inequality to proving that a certain quadratic form is positive se... |
H: Factor Theorem in $\mathbb{Z}_m[x]$
All the numbers I mentioned below are integers.
Question:
In $\mathbb{Z}_m[x]$, if $f(c_1) = 0$ and $f(c_2) = 0$, it does not always follow that $(x - c_1)(x - c_2) \mid f(x).$ What hypothesis on $c_1, c_2$ is needed to make that true?
I think when $c_1$ and $c_2$ are in diff... |
H: Question about the radius of convergence in complex analysis
This is related to this question. Tell me if this should be in the original question.
Is it possible for a complex function to have both conditions below?
Let $0<r<R$ and $a>0$ be a real number that satisfies $a+r<R$.
Condition 1: Power series expansion a... |
H: Upper bound of line integral along simple closed curve.
Let $U$ be an open set in $\Bbb C$ and $f\in H(U)$.
Fix a point $z\in U$.
Consider the line integral $$\displaystyle \oint_{\partial D(z,\varepsilon)} \frac{f(\zeta)-f(z)}{\zeta-z}$$
Since $\frac{f(\zeta)-f(z)}{\zeta-z}$ is continuous on $U-\{z\}$ and $[0,2\pi... |
H: Least squares method to get the fit formula
This is my first post here. For one of my projects I need to do a temperature compensation according to the distance, browsing I found an article called "High precision infrared temperature measurement system based on dsitance compensation" that does exactly what I need. ... |
H: On properties of quotients in the abelian category
Suppose A is an Abelian group, and B a subgroup. Moreover, suppose A/B = C is a Free Abelian group.Then, we have A is isomorphic to: $$B \oplus C$$. Is there an elementary proof of this without the use of category theory? Thanks.
AI: If we think of abelian groups ... |
H: Evaluate the Improper Integral(help)
I encountered the following integral while solving a log-normal distribution question. Initially, I thought since its a odd function, it evaluates to zero. But I think, since its a improper integral, we cannot to do simply. Upon further inspection, I found that neither does the ... |
H: Prove that four vectors of the three dimensional Euclidean space are always linearly dependant.
Could anyone check my proof?
Statement: Prove that four vectors of the three dimensional Euclidean space are always linearly dependant.
Proof: A group of vectors are linear dependent if their determinant is zero. Suppose... |
H: Given that $f\leq g$ a.e then how to show that Essential sup $f\leq $ Essential sup $g$?
Given that $f\leq g$ a.e then how to show that Essential sup $f\leq $ Essential sup $g$?
$$\text{ess} \sup f=\inf\{b\in \mathbb R\mid \mu(\{x:f(x)>b\})=0\}$$
From the definition, it is clear that inequality holds. But how to ... |
H: Prove that a directed graph with two vertices can only reach each other if their strongly connected components can as well
Hi I am struggling to prove the following question regarding strongly connected components in a directed graph. Any help would be appreciated, thanks in advance.
Prove that a directed graph wit... |
H: Locus of midpoint of line with endpoints always on x and y axis.
I came across the following question:
A line segment of length 6 moves in such a way that its endpoints remain on the x-axis and y-axis. What is the equation of the locus of its midpoint?
And I proceeded with the following:
Let (x,y) be the midpoint... |
H: Does every infinite graph contain a maximal clique?
The original problem is stated in terms of the tolerance relation (reflexive and symmetric, but not necessarily transitive): Is every tolerance subset contained in a maximal tolerance subset?
For a set $X$ with a tolerance relation $r \subset X \times X$, a subset... |
H: If a matrix has linearly independent columns, does it automatically have a left inverse?
If a matrix has linearly independent columns, does it automatically have a left inverse?
So I know the opposite is true. That is, if a matrix has a left inverse, that means that the columns of the matrix are linearly indepen... |
H: $A = (a_{ij})$ in the matrix definition
I have the following matrix definition
An m × n (read “m by n”) matrix A over a set S is a rectangular array
of elements of S arranged into m rows and n columns: (an mn matrix
shown)
We write $A = (a_{ij})$.
What is the meaning of $A = (a_{ij})$? $a_{ij}$ is an elements in ... |
H: What is the probability to form a triangle with the three pieces of the stick?
On a stick $1$ meter long is casually marked a point $X \sim U[0,1]$. Let $X=x$, is also marked a second point $Y\sim U[x,1]$.
1) Find the density of $(X,Y)$ showing the domain.
$$\rightarrow \quad f_{XY}(x,y)=\frac{1}{1-x}\mathbb{I}... |
H: Distribution of XY with X and Y Bernoulli distributed
I have a Problem with this exercise:
$X,Y:\Omega \to \{0,1\}$ are random variables with $X$~Bernoulli($\frac{1}{2}$) and $Y$~Bernoulli($\frac{3}{4}$). We also know that $P(X=Y=0)=\frac{1}{4}$
I already showed that $X$ and $Y$ are not independet. Now I want to de... |
H: Find the minimum of the set $A=\left\{\int_0^1(t^2 - at-b)^2 dt\, : \,a,b \in \mathbb{R}\right\}$.
Let $$A=\left\{\int_0^1(t^2 - at-b)^2 dt\, : \,a,b \in \mathbb{R}\right\}\,.$$ Find the minimum of $A$.
$\textbf{My attempt:}$
Well, we have
$ 0 \leq\int_0^1(t^2 - at-b)^2 dt = \frac{1}{5} - \frac{a}{2} + \frac{a^... |
H: Detail of Proof of Theorem 6.17 in Probability Theory (A. Klenke)
There is a part of the proof of Theorem 6.17 that I don't understand.
Definition 6.16. A family $\mathcal{F} \in \mathcal{L}^1(\mu)$ is called uniformly integrable if
$$ \inf_{0 \leq g \in \mathcal{L}^1(\mu)} \sup_{f \in \mathcal{F}} \int (|f| - g)^+... |
H: Hard Differential Equation
Can anyone help me to find the solution of this ODE : $$4(y')^2-y^2+4=0.$$
I've tried to find it's solution by putting $y = e^{at}$ (for null solution) and $y = 2$ (for particular solution). My final solution is $$y = c_1 e^{0.5t} + c_2 e^{-0.5t} + 2,$$
but didn't match with the solution ... |
H: The operator norm $\|L\|$
Let $C_0([0, 1])$be a subspace of $C([0, 1])$, a functional space consisting of real-value continuous functions over the interval $[0, 1]$, such that
$C_0 ([0, 1]) = \left\{ f \in C([0, 1]) \mid \int_0^1 f(t) dt = 0 \right\}$
, and define the norm as $\| f \|_\infty = \sup_{x \in [0, 1]} |... |
H: Matrix norm inequality $\| Bx\| \geq |\lambda| \| x \|$ for a real symmetric $B$
For a symmetric invertible matrix $B \in \mathbb{R}^{n \times n}$ with eigenvalues $\lambda_1, ..., \lambda_n \in \mathbb{R}$, it holds that for all $x \in \mathbb{R}^{n}$ and for any $\lambda \in \lambda_1, ..., \lambda_n$,
$$\|Bx\| \... |
H: Brownian Motion at hitting time defined as an infimum
I'm reading a book on Brownian Motion, and they define the hitting time as
$$T_x = \inf\{t > 0 : B(t) = x \}$$
Later on they state that $B(T_x)=x$.
Why would they use inf instead of min? With inf, if we have infinite amount of crossings/hits at x in finite time,... |
H: Find the range of the function $f(x)=\sqrt {\log (\sin^{-1}x +\frac 23 \cos ^{-1} x)}$
For the inner function
$$\sin^{-1} s +\frac 23 (\frac{\pi}{2}-\sin^{-1} x)$$
$$\frac 13 (\pi +\sin^{-1}x)$$
$$\frac{-\sin^{-1} x}{3}$$
Since it is inside a log function which is inside a square root
$$-\frac{\sin^{-1} x}{3} \ge 1... |
H: Calculate line integral $\int_{\ell} y \cos x d \ell$
I am asking to calculate the integral $$\int_{\ell} y \cos x d \ell$$ while $\ell$ is the graph of the function $\phi(x)=sin(x)$ in the domain $x \in [0,\frac{\pi}{2}]$ .
So what I understand is that $\frac{d\ell}{dx}=cos(x)$ so I can substitude it in my integra... |
H: Direct sum of eigenspaces
This is a problem in Chapter 4, Algebra, Michael Artin, 2nd.
Let $T$ be a linear operator on a finite dimensional vector space $V$, such that $T^2=I$. Prove that for any vector $v$ in $V$, $v-Tv$ is either an eigenvector with eigenvalue $-1$ or the zero vector. Prove that $V$ is the direct... |
H: Geometric multiplicity of eigenvalues in a diagonal block matrix
I'm trying to prove that the geometric multiplicity of an eigenvalue in a diagonal block matrix is the sum of the geometric multiplicities of the eigenvalue with respect to every block. I know that if I have a diagonal block matrix with $k$ blocks and... |
H: Finding real $(x,y)$ solutions that satisfies a system of equation.
I was given:
$x + y^2 = y^3 ...(i) \\ y + x^2 = x^3...(ii)$
And was asked to find real $(x,y)$ solutions that satisfy the equation.
I substracted $(i)$ by $(ii)$:
$x^3 - y^3 + y^2 - x^2 + x - y = 0$
Then factored it out so I have:
$(x-y)(x^2 + xy +... |
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