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H: Onto the notation and interpretation of queueing theory-related markov chains
If in a server with probability $p$ one job arrives and independently with probability $q$ one job departs, could you please explain to me what is the quantity and let me know if I have understood? The queue is infinite.
Note that during ... |
H: Product of Lebesgue integrals is bounded
I'm attempting to show the following: If $f:[0,1] \to (0,\infty)$ is measurable, then $$ \int_0^1 f(x) \,dx \int_0^1 \frac{1}{f(x)} \,dx≥1$$
My first inclination is to use something like Fubini or Tonelli to turn the product into a single integral, but I'm not sure I can app... |
H: Is the absolute value of a definite integral equal to the definite integral of the absolute value of the integrand?
An interesting question occurred to me as I was reading some Physics: Is it true in general that $$\left|\int_a^b f(x) \, dx\right| = \int_a^b |f(x)| \, dx\, ?$$
If not, what properties must $f(x)$ sa... |
H: Why is $(A\times B)/(I\times J)=A/I\times B/J$?
Why is it true that $(A\times B)/(I\times J)=A/I\times B/I$? Here, $A$ and $B$ are (commutative) rings (with $1$), and $I$ and $J$ are respective ideals.
This looks similar to the chinese remainder theorem, but I'm not sure how this is (if it is at all) an application... |
H: Proving independence of the neighborhood axioms in topology
I have not found any information on this topic on StackExchange or through a few minutes of searching with Google.
This question is Exercise 8 in Section 2.1 on page 22 of Topology and Groupoids, by Brown. I am given the neighborhood axioms
If $N$ is a n... |
H: Inequality involving size of random variable
I’m following my professor’s notes, and I became confused by an inequality he used without justification, but I cannot see why it is true. Any help is appreciated.
Let $\{T_t\}_t$ be a sequence of iid, positive, integer-valued random variables. Suppose $$P\{T \ge x \} = ... |
H: How to show Spec$\frac{\Bbb C [x, y]}{(xy)}$ is connected in the Zariski topology
In exercise 3.6.E of Vakil's book, it is asked for the reader to find a ring for which its spec is connected and non-irreducible. Taking his hint, I thought of $A = \frac{\Bbb C [x, y]}{(xy)}$. It is not irreducible, because $V((x)) ... |
H: Are nimbers the largest field of characteristic 2?
Are nimbers the largest field* of characteristic 2?
The nimbers are defined as follows:
$$a+b=\operatorname{mex}(\{a+b'\}\cup\{a'+b\})$$
$$ab=\operatorname{mex}(\{ab'+a'b+a'b'\})$$
Where $a'<a,b'<b$, and $a$ and $b$ are arbitrary ordinals.
If yes, is there a proof?... |
H: Proof that $(1 + t)^n = \sum\limits_{k = 0}^{n} \binom{n}{k} t^k$
I'm not understanding this proof from Combinatorics: Topics, Techniques, Algorithms by Peter J. Cameron. Here is his proof of this theorem:
The theorem can be proven by induction on $n$. It is trivially true for $n = 0$. Assuming the result for $n$, ... |
H: Contest math application for Wilson's theorem
$$1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{23} = \frac{a}{23!}$$ Find the remainder when $a$ is divided by $13.$
I found this online and got stuck a bit. I approached the problem as such:
From the expression we get $$a=\frac{23!}{1}+\frac{23!}{2}+\dots+\frac{23!}{23!}... |
H: Math Fraction Problem
I am at the moment in the $5$th chapter of IGCSE mathematics and currently need help for this problem.
Joseph needs $6\frac12$ cups of cooked rice for a recipe of nasi goreng. If $2$ cups of uncooked rice with $2 \frac12$ cups of water make $4 \frac13$ cups of cooked rice, How many cups of unc... |
H: Making sense of a set notation
I'm trying to make sense of this notation I came across in a post here:
Let $\omega=\{(x,y)\in \mathbb{R}^2\,:\,0<x<y<2x<2\}$.
Putting $0<x<y<2x<2$ into WolframAlpha gives me the solution interval as:
$0<x<1$ and $x<y<2x$.
What I'm trying to understand is how do you resolve $0<x<y<2x<... |
H: Integrability of $\frac{1}{|z|^m}$
Suppose that we are in $\mathbb C^n$, and consider the unit ball $B_1(0)$ around the origin, I am interested in knowing the integrability of $\frac{1}{|z|^{2m}}$. Specifically, when is
$$\int_{B_1(0)} \frac{1}{|z|^{2m}} \mathrm d\operatorname{vol}< \infty?$$
For $n = 1$, $\int_{B_... |
H: Prove that exist $b \gt 0$, so that $f$ may be defined at $x=0$ and be continuous.
Given the function $$
f(x) =
\begin{cases}
(1 + 2^{\frac{3}{x}})^{bsin(x)} &\quad if \quad x\gt 0 \\
\\
\frac{arctan(9bx)}{x} &\quad if \quad x\lt 0 \\
\end{cas... |
H: How many ways are there to give cookies and candies to these kids?
The same kind of 5 cookies are given to 3 kids. At least one cookie is given to each
child. Also, the same kind of 5 candies are given to the kids who received only one
cookie. How many ways are there to give cookies and candies to these children?
M... |
H: How to find the Laurent series expansion of $\frac{2}{z^2-4z+8}$ by long division?
I'm trying to find the Laurent series expansion for
$$ \frac{2}{z^2-4z+8} $$
using polynomial long division.
However, I noticed that if I divide leading with the $8$ term, then I will only get positive power terms, namely
$$ \frac{1}... |
H: Break even points structure
I am creating a Matchmaking ranking structure where players have a value to determine their skill level. If I win I get points, say $X$ points and if I lose then I win $Y$ points.
I am playing around with the values in a spreadsheet to see long term where players will fit into.
How can I... |
H: Question about the elements of $\operatorname{Hom}_k(k,V)$
Let $V$ be a $\mathbb{K}$-vector space. I am trying to understand what the linear maps from $\operatorname{Hom}_\mathbb{K}(\mathbb{K},V)$ look like. Given some $v \in V$, the map $f_{v}:\mathbb{K} \rightarrow V$ given by $f_v(k)=kv$ is linear and so $f_v \i... |
H: Finding the associated unit eigenvector
Background
Find the eigenvalues $λ_1<λ_2$ and two associated unit eigenvectors of the symmetric matrix
$$A = \begin{bmatrix}-7&12\\12&11\end{bmatrix}$$
My work so far
$$A = \begin{bmatrix}-7-λ&12\\12&11-λ\end{bmatrix}=λ^2-4λ-221=(λ+13)(λ-17)$$
Thus
$$λ_1=-13$$
$$λ_2=17$$
To f... |
H: How do I integrate $\frac1{x^2+x+1}$?
I have tried this:
$$\frac1{x^2+x+1} = \frac1{\left( (x+\frac12)^2+\frac34\right)}$$
Now $u = x+\frac12$
$$\frac1{ u^2+\frac34 }$$ Now multiply by $ \frac34$
$$\frac1{ \frac43 u^2 + 1}$$
Now put the $\frac43$ outside the integral
$$\frac34 \int \frac1{u^2+1}\,du=\frac34\arctan... |
H: In triangle $\triangle ABC$, angle $\angle B$ is equal to $60^\circ$; bisectors $AD$ and $CE$ intersect at point $O$. Prove that $OD=OE$.
In triangle $\triangle ABC$, angle $\angle B$ is equal to $60^{\circ}$; bisectors $AD$ and $CE$ intersect at point $O$. Prove that $OD=OE$.
So I've already made a diagram(it is a... |
H: Prove that the function integer part is continuous for any topology in Z.
Prove that the function integer part $f:\mathbb{R}_l \to \mathbb{Z}, \quad f(x) =\lfloor x \rfloor $ is continuous for any topology in $\mathbb{Z}$.
where $\mathbb{R}_l$ is the topology of the lower limit in $\mathbb{R}$.
I know that if I pro... |
H: Proving that one can integrate a uniformly convergent series of functions term by term
I aim to understand the following proof from Serge Lang's Introduction to Complex Analysis at a graduate level
[
and I have the following definitions
My question is: What does the last paragraph of the theorem in question actua... |
H: Find all the integer solutions for: $3x^2+18x+95\equiv 0\pmod {143}$
I need help with the following question:
Find all the integer solutions for: $3x^2+18x+95\equiv 0\pmod {143}$
My solution: First I know that $143=11\cdot 13$ then because $\gcd (11,13) = 1$ then $3x^2+18x+95\equiv 0\pmod {143}$ if, and only if $... |
H: Proving $f$ is a real, uniformly continuous function on the bounded subset $E$ in $\mathbb{R}^1 \implies f$ is bounded on $E$.
(Baby Rudin Chapter 4 Exercise 8)
I am trying to prove:
$f$ is a real, uniformly continuous function on the bounded subset $E$ in $\mathbb{R}^1 \implies f$ is bounded on $E$.
My attempt:
... |
H: Is the braid group hyperbolic?
The braid groups satisfy a number of properties that one would expect of a hyperbolic group, liking having a solvable word problem, and having exponential growth. Are the braid groups hyperbolic groups? If not, is there any obvious property of hyperbolic groups showing that they are n... |
H: What is this notation $\mathbb{Z}_3[x]_{x^2+1}$?
What is this notation?
$$\mathbb{Z}_3[x]_{x^2+1}$$
I know $\mathbb{Z}_3[x]$ denotes the polynomials with coefficients in $\mathbb{Z}_3$. What does the $x^2+1$ bit denote?
AI: That is the notation used for the localization of the ring $\mathbb Z_3[x]$ at the multiplic... |
H: Estimating integral of $\frac{\log z}{z^2+a^2}$ over small upper-plane semi-circle
I'm trying to calculate $$\int^\infty_0{\frac{\log z}{z^2+a^2}\mathrm{d}z}$$
I was able to calculate this using residue calculus and different half-plane and full keyhole contours, but have not been able to nail down an approximation... |
H: Well defined function involving quotient spaces
I wanna know how to prove that a function involving a cartesian product of quotient space is well-defined.
Let's see this question: Bilinear form and quotient space
In that question, I guess that $g$ is well-defined if $(u_1+U_0,v_1+V_0) = (u_2+U_0, v_2+V_0)$ implies ... |
H: understand the proof of $\frac{2 n}{3} \sqrt{n}<\sum_{k=1}^{n} \sqrt{k}<\frac{4 n+3}{6} \sqrt{n}$
If $n \in \mathbb{N}^*$, prove that \begin{align*}\frac{2 n}{3} \sqrt{n}<\sum_{k=1}^{n} \sqrt{k}<\frac{4 n+3}{6} \sqrt{n}.\end{align*}
I am having trouble understanding the following proof of this problem:
Let
$a_{... |
H: How to solve $\log_2(x)+\log_{10}(x-7)=3$ using high-school math?
A question given in a grade 12 "advanced functions" class, asks to solve $\log_2(x)+\log_{10}(x-7)=3$ with a hint to change bases.
The given hint suggests the base of the second logarithm is 10, but when trying to massage the equation, how does one p... |
H: EGMO 2014/P3 : Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$
We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. ... |
H: Function how to show whether it is pointwise or uniform convergent?
My sequence is : $s_n=\frac{2}{3+n|x|}$ for real x and n $\in \mathbb{N}$
Now I have done this:
$\lim_\limits{n \to 0} \frac{2}{3+n|x|}= \frac{2}{3}$
$\lim_\limits{ n \to \infty} \frac{2}{3+n|x|}= 0$
Now it is pointwise convergent to these 2.
Now ... |
H: Proving that the $p$-adic topology is a neighborhood topology
This is (part of) Exercise 7 from Section 2.2 on page 22 of Topology and Groupoids, by Brown. I would appreciate any feedback regarding the quality of my proof.
Exercise:
Let $X = \mathbb{Z}$ and let $p$ be a fixed integer. A set $N
\subseteq \mathbb{Z}... |
H: Question about the proof of $(\forall a)[a \in \mathbb{F} \rightarrow -(-a) = a]$
The proof of the proposition $(\forall a)[a \in \mathbb{F} \rightarrow -(-a) \in \mathbb{F}]$ is given on page 6 of the following link. https://www.math.ucdavis.edu/~emsilvia/math127/chapter1.pdf
We have that $e$ denotes the additive ... |
H: Finite ordinal Exponentiation
I confused a little when i do arithmetic on ordinals especially multiplication
is what i wrote right? :
$(ω+1)$ = {$0,1,....ω$}
$(ω+1)(ω+1)$ = sup({lexicographic Order($(ω+1)×(ω+1)$)}) = $ω²+1$
$(ω+1)(ω+1)(ω+1)$ = sup({lexicographic Order($(ω+1)×(ω+1)×(ω+1)$)})= $ω³+1$
$(ω+1)ⁿ$ ... |
H: Prove or disprove that $\sum_{n=1}^{\infty} \frac{x^{3/2}\cos(nx)}{n^{5/2}}$ is differentiable on $(0, \infty)$
Let $f(x) = \sum_{n=1}^{\infty} \frac{x^{3/2}\cos(nx)}{n^{5/2}}$ on $(0, \infty)$.
(i) Is $f(x)$ differentiable on $(0, \infty)$?
(ii) Does the series uniformly converge to $f$ on $(0, \infty)$?
Any... |
H: Writing G as a product of groups
$G$ is a connected, locally compact group satisfying the second axiom of countability, and $C$ is a discrete central subgroup of $G$ such that $G/C$ is compact. In the book I am reading (Varadarajan, Lie Groups, Lie Algebras, and their Representations, Lemma 4.11.1) it says that one... |
H: connected component and path connected component
What are the connected components and path connected components of $\Bbb{Q}$ , $\Bbb{R}$ and $\Bbb{R}_\mathcal{l}$?
Definition: A component $C$ of a topological space $X$ is a maximal connected subspace.
I think every connected component of $\Bbb{R}_\mathcal{l}$ ... |
H: INMO : Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers
Let $a$ and $b$ be two positive rational numbers such that $\sqrt[3] {a} + \sqrt[3]{b}$ is also a rational number. Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers.
My first response was, isn't is obvious? ... |
H: Given that $x^2 + y^2 = 2x - 2y + 2$ , find the maximum value of $x^2 + y^2 + \sqrt{32}$ .
Given that $x^2 + y^2 = 2x - 2y + 2$ , find the maximum value of $x^2 + y^2 + \sqrt{32}$ .
What I Tried :- Since $x^2 + y^2 = 2x - 2y + 2$ , we have $2x - 2y + 2 + \sqrt{32}$
=> $2(x - y + 1 + 2√2)$ . From this step I am n... |
H: Showing that a function is Darboux Integrable using definition of integral
I have been given a function $f:[0, 4]→ \mathbb{R}$ such that $f(x) = 0$ for all $x \ne 2$ and $f(2) = 2$, and told to show that $f$ is Darboux integrable in $[0,4]$.
However, I don't understand how a function like this can be integrable, as... |
H: Standard symplectic form on a sphere is an area form in cylindrical coordinates.
Consider a symplectic form $\omega_x(\xi,\nu)=\langle x,\xi\times \nu\rangle$ on $S^2$ where $x\in S^2$ and $\xi,\nu\in T_x S^2$ and a parametrization $\phi:U\to S^2$ where $U=(0,2\pi)\times(-1,1)$ and $\phi(\theta,x_3)=(\sqrt{1-x_3^2}... |
H: Smaller binary relation on the set $\mathbb Q$
Given two binary relations $R$ and $S$ over sets $A$ and $B$,then $R$ is said to be contained in $S$ if $$\forall a,b: (a,b) \in R \implies (a,b) \in S$$
Moreover $R$ is considered to be smaller than $S$ if $R$ is contained in $S$,but $S$ is not contained in $R$,e.g.$$... |
H: A combinatorial lemma
The following combinatorial lemma is from Benson's polynomial invariants of finite groups, lemma 1.5.1 used to prove a generalisation of Noether's degree bound.
The polynomial in $n$ variables $x_1x_2\ldots x_n$ satisfies the identity$$
(-1)^nn!x_1x_2\ldots x_n = \sum_{I\subseteq\{1,\ldots,n\}... |
H: If $a_n=100a_{n-1}+134$, find least value of n for which $a_n$ is divisible by $99$
Let $a_{1}=24$ and form the sequence $a_{n}, n \geq 2$ by $a_{n}=100 a_{n-1}+134 .$ The first few terms are
$$
24,2534,253534,25353534, \ldots
$$
What is the least value of $n$ for which $a_{n}$ is divisible by $99 ?$
We have to f... |
H: Is $\zeta(x)>\frac{1}{x-1}$ when $1
I had originally found that $\lfloor\zeta(\zeta(n))\rfloor$, where $\zeta(n)$ is the Riemann Zeta Function, seemed to be relatively close to $\left\lfloor\frac{1}{\zeta(n)-1}\right\rceil$ for $n \in \mathbb{Z} $ such that $n \geq 2$. This led me to realize that for real values of... |
H: If the sum of a set of numbers is bigger, will the average of the set of numbers be bigger too?
If the sum of a set of real numbers $X$ is bigger than the sum of a set of numbers $Y$, will the average of $X$ be bigger than $Y$ and vice versa?
What is a proof either way?
AI: You are in fact asking if
$$a>b\implies \... |
H: What is the chance of getting a number less than $0.01$ when using Math.random()?
When using Math.random() in JavaScript in Google Firebase Functions I get $17$ decimal places like this:
$$0.35361536181287034$$
I wanted to see what the chance is of getting any number where the first two decimals are $0$ like this:
... |
H: Problem with understanding.
This is an example in the book of "Abstract algebra by Dummit & Foote " . I didn't understand the indicate part of this proof. Please anyone help.
How did they get class equation from that ?
AI: If an element $g\in G$ has order $p$, then it generates a cyclic subgroup $\langle g\rangle$ ... |
H: Trace of point begin circle under certain condition
Let $P$ be a point on a circle $(x-2)^2 + (y-2)^2 = 4$. Now let $Q$ be a point on the same line with $O$, the origin, $P$, and also on the first quadrant.
If $Q$ satisfies $\overline{OP}\times\overline{OQ}=6$, the trace of $Q$ is a circle, $(x-3)^2 + (y-3)^2 = 9$.... |
H: Property of a positive Lebesgue measure set in $\mathbb{R}^2$
Let $A\subset \mathbb{R}^2$ be a closed set of positive Lebesgue measure. Can we find positive Lebesgue measure sets $A_1,A_2\subset \mathbb{R}$ such that $A_1\times A_2\subseteq A$?
Note that the above is not true if $A$ is not assumed to be closed. For... |
H: $R$ is symmetric if and only if it is equal to its converse
Given a binary relation $R$ over set $A$ ,prove the following statement:
$R$ is symmetric if and only if it is equal to its converse.
$\implies$
$R$ symmetric iff $\forall a,b \in A$:
$$(a,b) \in R \iff (b,a) \in R$$
But how to show that it is equal to i... |
H: Is $\mbox{Rank}(A + A^2) \leq \mbox{Rank} (A)$?
Here, $A$ is an $n \times n$ matrix. I am not able to find any counterexample but not able to prove this as well. The examples I have tried so far shows me that $\mbox{Rank} (A + A^2) = \mbox{Rank} (A)$. I don't know how to show the inequality.
AI: $\DeclareMathOperat... |
H: A question about subring of Rational Numbers involving prime and maximal ideals
Edited :
I have this particular question in abstract algebra assignment given to me.
I have been studying algebra from Thomas Hungerford as a textbook.
Question : Let R be a subring of $\mathbb{Q}$ containing 1 . Then which 1 of follow... |
H: $x_1 + x_2 + x_3 + x_4 + x_5=5$ . Determine the maximum value of $x_1x_2+x_2x_3+x_3x_4+x_4x_5$.
Let $x_1 , x_2 , x_3 , x_4 , x_5$ be non-negative real numbers such that $x_1 + x_2 + x_3 + x_4 + x_5=5$ . Determine the maximum value of $x_1x_2+x_2x_3+x_3x_4+x_4x_5$.
Normally in such questions I use the fact that th... |
H: Question on determinant
Given $x \in \Bbb R$ and
$$P = \begin {bmatrix}1&1&1\\0&2&2\\0&0&3\end {bmatrix}, \qquad Q=\begin {bmatrix}2&x&x\\0&4&0\\x&x&6\end {bmatrix}, \qquad R=PQP^{-1}$$
show that
$$\det R = \det \begin {bmatrix}2&x&x\\0&4&0\\x&x&5\end {bmatrix}+8$$
for all $x \in \Bbb R$.
My attempt: $|R|=\frac{|P... |
H: Proof of a group (in the context of finite groups)
my task is the following:
Let $G$ be a finite set with an inner connection $\circ: G \times G \rightarrow G$, which is associative and for which a neutral element exists in $G$. In addition, for all $a,b,c \in G$, it applies that from $a \circ b = a \circ c$ also... |
H: Unimodularity: How are these notions related?
1. Definitions
We call a Hopf algebra $H$ unimodular if the space of left integrals $I_l(H)$ is equal to the space of right integrals $I_r(H)$.
We call a square integer matrix $M$ unimodular if $det(M)=\pm 1$.
Apparently, there exists a notion of unimodular group: "a l... |
H: Normal distribution sample
Since I'am beginner in statistics I'm stuck in simple exercise so will appreciate any help. I have mean, standard deviation and probability p(x) and need to get x. Here is the Exercise
The patient recovery time from a particular surgical procedure is
normally distributed with a mean of 5... |
H: Identity for sets
Identity for a set X:
The set X has an identity under the operation if there is
an element j in set X such that j * a = a * j = a for all elements a in set X.
According to my college book the counting numbers don' t have an identity.
But for the operation * there is number 1 (= j ) such that a *... |
H: Is AC necessary to prove that whenever $S$ is a set of non-empty sets, there exist a set $A$ which for each $s\in S$, $s\cap A\neq\varnothing$?
Is this theorem need Axiom of Choice ? or is it equal to that?
Let $S$ be set of non-empty sets, there exist a set $A$ which for each $s\in S$, $s\cap A\neq\varnothing$
I... |
H: Local representation of one map with respect to another map
Let $M$ be a smooth manifold of dimension $m$ and $(U,\phi)$, $(V,\psi)$ be two maps on $M$ such that $U\cap V\neq\emptyset$. I will write $\phi=(\phi^1,\dots,\phi^m)$ and $\psi=(\psi^1,\dots,\psi^m)$.
I want to prove that on $U\cap V$, we can write (using... |
H: Given $M\in\mathbb{R}^{m\times n}$ and $v_1,v_2\in\mathbb{R}^n,$ find $A\in\mathbb{R}^{m\times n}$ such that $Mv_1=Av_2$?
Suppose that we are given a matrix $M\in\mathbb{R}^{m\times n}$ and two vectors $v_1,v_2\in\mathbb{R}^n$. Under which conditions there exists a matrix $A\in\mathbb{R}^{m\times n}$ such that $Mv_... |
H: Complex integration around unit circle centresd at origin
I am trying masters entrance exam question papers and I am unable to solve this particular question, so I am asking it here.
Let C denote the unit circle centred at origin in $\mathbb{C} $ then find value of
$$\frac{1}{2\pi i}\int_{C} |1+z+z^2|^2 dz$$.
Attem... |
H: Series. Uniformly convergent on $\Bbb R$ vs. any interval $[-K,K]$
As the title indicates I am slightly unsure about a setup used for convergence of a series.
I was wondering if there is/what the difference is between showing that a series is uniformly convergent on $\Bbb R$ vs. uniformly convergent in any interval... |
H: Elements of a subgroup and explanation.
Find all of the elements in the subgroup K = $\langle (12)(34),(125)\rangle \leq S_5$.
I understand that you need to take the powers of the given elements and products, but I don't understand how for example $\beta^2=(125)(125)=(152)$. Where do you get $(152)$ from?
AI: $(1 \... |
H: The definition of the set of positive integers in "Topology 2nd Edition" by James R. Munkres.
I am reading "Topology 2nd Edition" by James R. Munkres.
In Ch. 1.4, Munkres defines the set of real numbers $\mathbb{R}$ with the field axioms (including completeness), and then defines $\mathbb{Z}_{+}$ as the smallest in... |
H: A deduction in 1st course of complex analysis if a particular series is given absolutely
I am trying assignment problems in complex analysis and I couldn't deduce the reasoning behind a particular Statement.
Suppose f is holomorphic in an open neighborhood of $z_{0} $ $\epsilon $
$\mathbb{C} $. Given that the seri... |
H: Why did we call a row operation "elementary"?
Why we called the three actions of row operation "elementary"? Is there a thing called "advanced" or "complicated" row operation?
I've seen the word "non-elementary" row operation is used to describe things like $R_1-R_2$, which is not written in the conventional $-1R_... |
H: Is weak connectivity sufficient for $0$ to be a simple eigenvalue of the weighted Laplacian?
$L$ is the weighted Laplacian of a weakly connected directed graph $G$,$$L=D-A$$ with the $L_{ij} \leq 0$ when $i \neq j$, $L_{ii} \geq 0$ and $\sum_{i=1}^{n}L_{ij}=0$.
My question is: Is $0$ a simple eigenvalue of $L$ ?
AI... |
H: Problem in Understanding the following steps in an integral
While doing quartic integral,I was unable to understand the step that leads to the answer.
Can somebody illustrates me how do we got from first integral to the next one?
AI: Just looking at the antideriavtive
$$I=\int\left(\frac{1-\cos (\theta )}{(1-a) \c... |
H: coordinates of focus of parabola
Find the coordinates of focus of parabola $$\left(y-x\right)^{2}=16\left(x+y\right)$$
rewriting:
$(\frac{x-y}{\sqrt{2}})^2=8\sqrt2(\frac{x+y}{\sqrt{2}})$
comparing with $Y^2=4aX$
$4a=8\sqrt2,a=2\sqrt2 $
$\Rightarrow$ coordinates of focus=2,2
Is this the correct approach?
AI: Using R... |
H: Is $\operatorname{div}(X)=\partial_i X^i$?
We know that for a vector field $X$, $\operatorname{div}(X)$ is defined as $\nabla_I X^I$. This is not the same as $\partial_i X^i$ right?
I would assume that $\nabla_I X^I=\partial_i X^i-X^j\Gamma_{ij}^i $.
AI: Yes, you are correct.
Another way to expand the divergence on... |
H: Prove that $\frac{1}{2} (x-1) x + y$ is a bijection. (on p.45 Munkres Topology 2nd Edition)
I am reading "Topology 2nd Edition" by James R. Munkres.
On p.45, Munkres leaves it to the readers to show that $g$ is bijection:
Show that $g(x, y) = \frac{1}{2} (x-1) x + y$ is a bijection from $\{(x, y) \in \mathbb{Z}_{+... |
H: Consequence of inequality
My question:
Let $\Omega$ in $\mathbb{R}^n$ bounded. For all $\varepsilon>0$, there exists a constant $C(\varepsilon)>0$ such that
$$
\label{lemma_gagliardo_nirenberg_2}
\|\varphi\|_{L^2(\Omega)}^2
\le \varepsilon \|\nabla \varphi\|_{L^2(\Omega)}^2 + C(\varepsilon) \|\varphi\|_{L^1(\O... |
H: Expected value with a die with 9 faces
You have a die with $9$ faces, which are numbered $1, 2, 3, \dots,9$. All the numbers have an equal chance of appearing. You roll the die repeatedly, write the digits one after another, and you stop when you obtain a multiple of $3$. For example, you could roll a $4$, then a ... |
H: A general summation of powers of roots of $x^2-x+q=0$
If $\alpha,\beta$ are the roots of the equation $x^2-x+q=0$ and $S_r=\alpha ^r + \beta ^r$, find $S_n$ in terms of $\sum a_iS_{n-i}$, where $a_i$ are constant terms for each $S_{n-1}$.
I tried to observe a pattern in the $S_r$:
$S_1=1, S_2=1-2q, S_3=1-3q,S_4=1-... |
H: Interpolation of three positive values at 0, 1 and 2 by a polynomial with non-negative coefficients
I am asking myself the question: Let $ y_0, y_1, y_2 $ be positive real numbers. Is there always a polynomial $ f $ with non-negative real numbers as coefficients which satisfies $ f( i ) = y_i $ for $ i = 0, 1, 2 $?... |
H: Showing the subset $\{(x_1,x_2) \in \mathbb{R}^2 : x_1 > x_2 \}$ is open
The metric is the typical Euclidean metric, $ \sqrt{ (x_1 - x_2)^2 + (y_1 - y_2)^2 } $.
I have solved this one, albeit with in my opinion quite excessive steps. I would love to know if there is a simpler way to do it. Below is my approach.
De... |
H: General proof for square–cube law
Can someone present a general (and easy) proof for square-cube law?
For similar objects 1 and 2,
$$
\frac{A_1}{A_2}=k^2 \ \mathrm{and} \ \frac{V_1}{V_2}=k^3,
$$
where k is the scale of objects 1 and 2.
AI: For a cube we have that
$A_1=l_1^2$
$V_1=l_1^3$
and
$A_2=l_2^2$
$V_2=l_2^... |
H: Evaluate $\sum_{r=1}^{m} \frac{(r-1)m^{r-1}}{r\cdot\binom{m}{r}}$
Evaluate:$$\sum_{r=1}^{m} \frac{(r-1)m^{r-1}}{r\cdot\binom{m}{r}}$$
Using the property:$$r\binom{m}{r}=m\binom{m-1}{r-1}$$
It is same as $$\sum_{r=2}^{m} \frac{(r-1)m^{r-1}}{m\cdot\binom{m-1}{r-1}}$$
How I do now?
AI: Let $$S=\sum_{r=1}^{m} \frac{(r... |
H: Trouble in understanding the proof of the existense of the HOMFLY polynomial
Following Page 252-254 of Christian Kassel's text "Quantum Groups" Chapter 10 very closely:
$Definition$ $4.1$
A triple $(L^+,L_-,L_0)$ of oriented links in $\mathbb{R}^3$ is a Conway triple if they can be represented by link Diagrams $D_... |
H: How many angles can be drawn using only a ruler and a compass?
So far I know that it’s possible to draw angles which are multiples of 15° (ex. 15°, 30°, 45° etc.).
Could anybody please tell me if it's possible to draw other angles which are not multiples of 15° using only a compass and a ruler.
AI: You can construc... |
H: Anti-canonical bundle of a bundle
Let $ F \to E \stackrel{\pi}{\to} B$ be a smooth fibre bundle, so that $F$, $E$, and $B$ are smooth manifolds.
I'm interested in what one can say about the anti-canonical bundle $K_E^*$ of the total space $E$, given the anti-canonical bundles of the fibre $F$ and base $B$. In parti... |
H: A question about whether an ideal is maximal or not
Edited New question : my earlier asked question was assosiated with this question:Consider the ideal $I=(x^2+1,y)$ in the polynomial ring $\mathbb{C}[x,y]$.Then which of the following is true
I already knew this question exists on MSE but here it's not answered ... |
H: Total combinations from number of unique combination?
How can you find count of the total combinations possible if you have the count of unique combinations?
Suppose I need to form a sum of $3$ using $\{1,2\}$
Unique combn. $(1+1+1)$,
$(1+2)$
hence '$2$' unique
but total is '$3$' i.e,
$(1+1+1)$,
$(1+2)$,
$(2+1)$
I ... |
H: sum of terms of series
If $$F(t)=\displaystyle\sum_{n=1}^t\frac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}$$ find $F(60)$.
I tried manipulating the general term(of sequence) in the form $V(n)-V(n-1)$ to calculate the sum by cancellation but went nowhere. I also tried using the fact that $$2n+\sqrt{4n^2-1}=\frac{... |
H: What is the integer part of the following fraction: $\frac{2012^{2013}+2013^{2014}}{2012^{2012}+2013^{2013}}$
What is the integer part of the following fraction: $\dfrac{2012^{2013}+2013^{2014}}{2012^{2012}+2013^{2013}}$.
This is a competition problem for 7th grade students.
The answer to this question is $2012$.... |
H: On hypergeometric square integral $\int_0^{\infty } \, _2F_1(a,b;c;-x){}^2 \, dx$
Preliminary. When $a,b>\frac12$, $c\not=0, -1,-2, ...$, one have (using Mellin transform)
$I=I(a,b,c)=\int_0^{\infty } \, _2F_1(a,b;c;-x){}^2 \, dx=\frac{\Gamma (c)^2 G_{4,4}^{3,3}\left(1\left|
\begin{array}{c}
0,1-a,1-b,c-1 \\
0,a... |
H: Checking whether $\mathbb C^n$ is semisimple or not
Consider $\mathbb C^n$ as a Banach algebra over $\mathbb C$. Is it semisimple?
A Banach algebra is semisimple if the intersection of all maximal ideals of it is ${0}$.
I can't figure out what are the maximal ideals of $\mathbb C^n$. How do I solve it?
AI: What ab... |
H: Does there exist some linear factor for every quadratic such that their product's $x^2$ and $x$ terms disappear
Can any quadratic with integer coefficients be multiplied by some linear factor, also with integer coefficients, such that the coefficients of the product's $x^2$ and $x$ terms are both zero and the coeff... |
H: Let the sum of the coefficients of the polynomial $(4x^2 - 4x + 3)^4(4 + 3x - 3x^2)^2$ be $S$ . Find $\frac{S}{16}$ .
Let the sum of the coefficients of the polynomial $(4x^2 - 4x + 3)^4(4 + 3x - 3x^2)^2$ be $S$ . Find $\frac{S}{16}$ .
Actually I have absolutely no other idea on how to find this without opening ... |
H: Ornstein - Uhlenbeck - Stochastic Differential Equation liminf and limsup
I came across an exercise which deals with a stochastic differential equation of the form
$$\mathrm dX(t)=-\theta[X(t)-\mu]\mathrm dt+\sigma\,\mathrm dW(t)$$
for $t>0$, where $\theta,\sigma>0$ and $\mu$ are fixed parameters. It requests to sh... |
H: List all the possible values for $\int_{\mathbb{R}}\sup_{k\in\mathbb{N}}f_k(x)dx$ under these conditions...
Question: Let $\{f_k(x)\}_{n=1}^\infty$ be a sequence of nonnegative functions on $\mathbb{R}$ such that $\sup_{x\in\mathbb{R}}f_k(x)=\frac{1}{k}$, and $\int_{\mathbb{R}}f_k(x)dx=1$. List all the possible va... |
H: Is there any other controller than PID Controller?
I have been given a Project to search for a controller having better transient response than PID Controller. I searched but I didn't find any research paper on it. All are talking about improving PID's transient response but I am unable to find any other Controller... |
H: A question based on analyticity of entire functions
This particular question was asked in quiz yesterday and I was unable to solve it.
Suppose f and g are entire functions and g(z) $\neq$0 for all z $\epsilon \mathbb{C} $ . If |f(z) |$\leq$ |g(z) | , the which one of them is true.
1.f is a constant function.
2.f(... |
H: What's wrong with this method of evaluating an integral?
I was trying to evalute the integral $$\int \frac{1}{x^2+1} \,dx$$ by partial fractions.
$$\frac{1}{x^2+1} = \frac{1}{2i}\left(\frac{1}{x-i} - \frac{1}{x+i}\right)$$
Therefore,
\begin{aligned} \int \frac{1}{x^2+1} \,dx &= \frac{1}{2i} \int \left(\frac{1}{x-i... |
H: Question 2 from Bredon's Topology and Geometry page 39
I got stuck on the following question from Bredon Topology and Geometry Chap 1 sec 12.
Suppose $X$ is paracompact. For any open subset $U$ of $X \times [0,\infty)$ which contains $X \times \{0\}$ show that there is a map $f:X\rightarrow (0,\infty)$ such that $(... |
H: Books: homotopy groups of inverse limits
Can anybody recommend a book, which contains comprehensive information about homotopy groups of inverse limits?
AI: I doubt that there is a book as you desire. There is only a loose relation between homotopy groups of the inverse limit and the homotopy groups of the spaces i... |
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