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I learned that the sufficient and necessary condition for a finite state Markov chain to have a unique stationary distribution is there's only one closed communication class. For example from this tutorial
Every Markov Chain with a finite state space has a unique stationary distribution unless the chain has two or more... |
EDIT: Here is a related post which concern quadratic vector fields rather than Van der pol equation. In this linked post we see that the convexity of limit cycle play a crucial role. On the other hand the unique limit cycle of Van der pol equation is convex. So there is a Riemanian metric on $\mathbb{R}^2 \setminus C$ ... |
Problem
Let $\{A_\alpha\}$ be a collection of subsets of $X$; let $X=\bigcup_{\alpha}A_\alpha$. Let $f:X\rightarrow Y$;suppose that $f\vert_{A_\alpha}$ is continuous for each $\alpha$.
An index family of sets $\{A_\alpha\}$ is defined to be locally finite if each point $x$ has a neighborhood that intersects $A_\alpha$ ... |
If I were to cover the Fibonacci Sequence in introducing sequences to Calculus students, I would probably avoid some of the more obscure properties. I would focus on what early Calculus students should know: for example, how to prove that a sequence converges, and if it does, then how to find what it converges to.
(Not... |
However, each reaction at a different temperature has a different equilibrium constant. With that said, the amount of each molecule will change which also changes the concentration so the amount of NH3 would not necessarily be 2.
This is true. I have not watched the video, but if he is suggesting that the stoichiometri... |
Bit of a strange question, but what is it? My physics teacher said it was kind of like a "push" that pushes electrons around the circuit. Can I have a more complex explanation? Any help is much appreciated.
Your teacher was right.
Current is electric charges (usually electrons) moving. They don't do that by themselves ... |
Let's say I have a solid cylinder of uniform mass density, radius $R$ and height $h$. I know that the moment of inertia of this cylinder rotating about the axis parallel to the height and passing through the center of mass is $\frac{MR^2}{2}$. How would the Moment of Inertia (about the same axis) change if I were to cu... |
I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of logarthmic blow ups too, possibly as a corollary).
Consider a BV function $f(t)$ in $L^2(\mathbb{R})$ such that $f(t) =... |
A somewhat similar question to what I'm going to ask is this one. The problem is basically that one has the heat equation $c^2\nabla^2u = u_t$ in which initial and boundary conditions are given. But these boundary conditions do not match continuously with the initial conditions. In the question I referred to in the fir... |
Answer to Question #2602 in Calculus for Alex T Simon
Question #2602
Use polar coordinate to find the volume of solid inside the cylinder and inside the ellipsoid.
Expert's answer
1) Suppose that one of the base D of the cylinder is a disk of radius R in the plane xy and centered at the origin. Let also its height is e... |
Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature
Department of Mathematics, Hunan Normal University, Changsha 410081, Hunan, China
$u_{t}(x,~t)-\int_{0}^{t}[\beta_{1}(t-s)\,u_{xx}(x,~s) - \beta_{2}(t-s)\,u_{xxxx}(x,~s)]ds = f(x,~t),$
$ 0<x<1,~ 0... |
Both devices are listed to work with I2C clock frequencies up to 400 kHz. The two devices are the only devices on the I2C bus.
However, working out the calculations for the pull-up resistor bounds gives some rather odd values.
Calculating the I2C minimum pullup resistor value:
\begin{equation} R_{min} = \frac{Vcc - 0.4... |
Is there a finite group such that, if you pick one element from each conjugacy class, these don't necessarily generate the entire group?
No, this is impossible. This is a standard lemma, but I'm finding it easier to give a proof than a reference: Let $G$ be your finite group. Suppose that $H$ were a proper subgroup, in... |
How to Implement a Point Source with the Weak Form
Today we continue our discussion on the weak formulation by looking at how to implement a point source with the weak form. A point source is a useful tool for idealizing the situation where a source is concentrated in a very small region of the modeling domain. We will... |
Publications Influence
Claim Your Author Page
Ensure your research is discoverable on Semantic Scholar. Claiming your author page allows you to personalize the information displayed and manage publications.
C. P. Shen, C. Z. Yuan, Y. Ban, H. Aihara, D. M. Asner, I. Badhrees, 27 A. M. Bakich, E. Barberio, P. Behera, V. ... |
DCDS
We make two observations concerning the generalised Korteweg de Vries
equation $u_t + $u
xxx$ = \mu ( |u|^{p-1} u )_x$. Firstly we give a scaling argument that shows, roughly speaking,
that any quantitative scattering result for $L^2$-critical equation ($p=5$) automatically implies an analogous scattering
result f... |
Continuing the answer given here: How to determine if this series converges absolutely/conditionally or diverges?
wrt to this series: $$ \sum_{n=2}^{\infty} \ln \left(1+\frac{(-1)^n}{n}\right) $$ will you please help me understand why is it legitimate to split this series into odd and even terms? i.e.- if the series in... |
This answer focuses on identifying families of solutions to the problem described in the question.
I've made two provisional conjectures in order to make progress with the problem:
The result can be stated for three $2n$-gons rather than two $n$-gons and one $2n$-gon.
Solutions have mirror symmetry. Or equivalently, in... |
1,409 Works
In 1971 Baumgartner showed it is consistent that any two $\aleph_1$-dense subsets of the real line are order isomorphic. This was important both for the methods of the proof and for consequences of the result. We introduce methods that lead to an analogous result for $\aleph_2$-dense sets. Keywords : forcin... |
Question:Give an example of a 1-dimensional subspace U of R3.
Answer: Any line through the origin is a one-dimensional subspace of R3.
How do i write this as a set? i mean U=?
thanks
You should know this from analytic geometry: $\displaystyle U=\{t\cdot (a,b,c)\,\mid\,\,t\in\mathbb{R}\,,\,(a,b,c)\neq (0,0,0)\}$ is a st... |
Should we assume that the 58% chance of winning is fixed and that points are independent?
I believe that Whuber's answer is a good one, and beautifully written and explained, when the consideration is that every point is independent from the next one. However I believe that, in practice it is only an interesting starti... |
Possible Duplicate: Long underscore in LaTeX
I'm writing a document in which I encourage the reader to "fill in the blanks" in order to be actively engaged with the proof.
Is there any way to write a clean, long underscore? Currently I'm using
\underline{\hspace*{2cm}}, but it doesn't look so great...
Here's a sample o... |
Learn how to add your own Javascript libraries to Kotobee Author.
Adding your own Javascript libraries can help perform certain functions, such as rendering Mathematical formulae (e.g. KaTeX) or displaying a fancy tooltip (e.g. Tippy). Since Kotobee Author provides you with access to the internal files of the ebook pro... |
I just watched the Blender Guru video How to Make a Beer in Blender and stopped at the point where it advises us to make the liquid slightly larger in diameter than the inner diameter of the glass - approximately half way between the inner and outer walls.
I then turned to Fluid in a Glass(and why you’ve been doing it ... |
Here is a beautiful result from numerical analysis. Given any nonsingular $n\times n$ system of linear equations $Ax=b$, an optimal Krylov subspace method like GMRES must necessarily terminate with the exact solution $x=A^{-1}b$ in no more than $n$ iterations (assuming exact arithmetic).
The Cayley-Hamilton theorem pro... |
I was wondering when a stochastic process defined via a SDE is Markovian? The SDE may involved Ito integral, Lebesgue integral, jump component, and any other things. The reason I ask this question is that I don't understand when we can and cannot discuss properties of a Markov process, such as Kolmogorov equations, for... |
Difference between revisions of "LaTeX:Symbols"
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|<math>\upl... |
Singularity of controls in a simple model of acquired chemotherapy resistance
1.
Inter-Faculty Individual Doctoral Studies in Natural Sciences and Mathematics, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland
2.
Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Sloneczna 54... |
Basically 2 strings, $a>b$, which go into the first box and do division to output $b,r$ such that $a = bq + r$ and $r<b$, then you have to check for $r=0$ which returns $b$ if we are done, otherwise inputs $r,q$ into the division box..
There was a guy at my university who was convinced he had proven the Collatz Conject... |
People say that the probability density function of the continuity equation for the Dirac equation is definite positive. I wanted to see it myself and followed the same path as Bjorken & Drell's did in their RQM book. They say "We left-multiply the Dirac equation with the Hermitian conjugate wave function, then we take... |
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
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Now showing items 1-10 of 24
Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
Given a semilattice $S$, a subset $E$, and a positive integer $n$, let $E^{[n]}$ be the set of all products of $n$-tuples in $E$. Thus $\bigcup_{n\geq 1} E^{[n]}$ is nothing but the subsemigroup of $S$ generated by $E$, which I'll denote by $\langle E\rangle$.
The following definition arose in some work I am writing up... |
To construct the DFA for a cross-section of the languages (string must be accepted by both DFAs) You can work as follows:
Make sure the transition function for the input DFAs is complete. The new set of states for the DFA is the cartesian product of the states of the 2 DFAs $Q' = Q_1 \times Q_2$. The transition functio... |
Let $Q$ be a finite set of states, $\Sigma$ a finite alphabet, $q_0\in Q$ the start state and $F\subseteq Q$ the set of accepting sets. Let $\{\delta_k:Q\times\Sigma\rightarrow Q\}_{k=1}^n$ be a set of $n$ possible transition functions and $m$ a fixed natural number. Consider the following problem: find a word $w\in\Si... |
The common understanding is that, setting air resistance aside, all objects dropped to Earth fall at the same rate. This is often demonstrated through the thought experiment of cutting a large object in half. The halves clearly can't then fall more slowly just by being sliced in two.
However, I believe the answer is th... |
Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. There are two types: Riddler Express for those of you who want something bite-size and Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,
1 ... |
Roughly speaking there are two ways for a series to converge: As in the case of \(\sum 1/n^2\), the individual terms get small very quickly, so that the sum of all of them stays finite, or, as in the case of \( \sum (-1)^{n-1}/n\), the terms do not get small fast enough (\(\sum 1/n\) diverges), but a mixture of positiv... |
If you want the expected value, one answer is $n E[S_{(m)}]$, where $S_{(m)}$ is the $m$th order statistic of a sample of $n$ gamma$(k,1)$ random variables. While this expression may not have a simple closed form, you may be able to get a decent-sized approximate answer from the literature on moments of order statistic... |
I will start with an example. Consider a symmetry breaking pattern like $SU(4)\rightarrow Sp(4)$. We know that in $SU(4)$ there is the Standard Model (SM) symmetry $SU(2)_L\times U(1)_Y$ but depending on which vacuum we use to break this symmetry, in a case you can totally break the SM symmetry, with the vacuum : $$\Si... |
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C
Carbides and grain defect formation in directionally solidified nickel-base superalloys. Advanced Technologies for Superalloy Affordability as held at the 2000 TMS Annual Meeting. :2000.. 2000.
Carburization of W-and R... |
Let $m\geq4$ be an even integer, $V\subset\mathbb{C}^{m-1}$ be the solution set of the following polynomial equations: \begin{cases} &\sum\limits_{s=1}^{2t-1}z_sz_{2t-s}+\sum\limits_{s=2t+1}^{m-1}z_sz_{m+2t-s}=0,\quad t=1,\dots,m/2-1,\\ &z_sz_{\frac{m}{2}+s}=0,\quad s=1,\dots,m/2-1,\\ &z_sz_{m-s}=0,\quad s=1,\dots,m/2.... |
Let $X$ be a Banach space and $Y$ a closed subspace of $X$. If $\varphi\in Y^*$, then Hahn-Banach allows us to extend $\varphi$ to a $\tilde\varphi\in X^*$, such that $\|\tilde\varphi\|=\|\varphi\|$. This extension can happen in infinitely many ways, in general.
My question is the following:
Can we guarantee the existe... |
Adventure Begins
The game Pokenom Go has just been released. Pokenom trainers can now travel the world, capture Pokenom in the wild and battle each other! Bash — the Pokenom trainer — has decided to drop out of his university to pursue his childhood dream of becoming the best Pokenom trainer!
However, Linux — Bash’s un... |
There are many explanations to be found about shot noise in optics, but the answers I find are incompatible. There are three ways shot noise in optics is explained.
(Note that according to Wikipedia, in general, shot noise is a type of noise which can be modeled by a Poisson process.)
It is the noise purely arising for... |
11 0
Hello!!
I did this problem and have gotten it wrong-- but my math is right. Maybe I'm missing something? Here's the question:
We want to support a thin hoop by a horizontal nail and have the hoop make one complete small-angle oscillation each 2.0 s. What must the hoop's radius be?
So this is a physical pendulum, a... |
Just as a single integral has a domain of one-dimension (a line) and a double integral a domain of two-dimension (an area), a triple integral has a domain of three-dimension (a volume). Furthermore, as a single integral produces a value of 2D and a double integral a value of 3D, a triple integral produces a value of hi... |
Here's an asymptotic bound - hopefully it's tight.
We throw $N$ little balls of diameter $D$ randomly (uniformly) inside the big sphere of radius $R$.
UPDATE: The new version (lower half) is better than what follows.
Neglecting border effects (reasonable if $N$ is large) the probability that the ball $i$ is "free" (no ... |
I would like to calculate the distribution function from the characteristic function. There is a formula given as
$$F_X(x)=\frac{1}{2}+\frac{1}{2\pi}\int_{0}^\infty \frac{e^{i w x}\phi(-w)-e^{-i w x}\phi(w)}{i w}\mbox{d}w$$ Actually, I want to evaluate $F_X(x)$ at some real number $b<0$. I wrote some codes in Mathemati... |
I want to evaluate the integral: $$\int_{-\infty}^{0}\frac{2x^2-1}{x^4+1}\,dx$$
using contour integration.
I re-wrote it as: $\displaystyle \int_{0}^{\infty}\frac{2x^2-1}{x^4+1}\,dx$. I am considering of integrating on a semicircle contour with center at the origin. I considered the function $\displaystyle f(z)=\frac{2... |
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1. J/ψ production cross section and its dependence on charged-particle multiplicity in p + p collisions at s=200 GeV
Physics Letters B, ISSN 0370-2693, 11/2018, Volume 78... |
We solve the following problem that arises for example in sparse signal reconstruction problems such as compressed sensing: \[ \mbox{minimize } ||x||_1 \mbox{ ($L_1$) }\\ \mbox{subject to } Ax = b \]
with \(x\in R^n\), \(A \in R^{m \times n}\) and \(m\leq n.\) Reformulate the problem expressing the \(L_1\) norm of \(x\... |
Skills to Develop
Solve equations with fraction coefficients Solve equations with decimal coefficients
be prepared!
Before you get started, take this readiness quiz.
Solve Equations with Fraction Coefficients
Let’s use the General Strategy for Solving Linear Equations introduced earlier to solve the equation \(\frac{1}... |
Monstrous Moonshine
Undergraduate math student at NYU
Member for 4 years
1 profile view
Last seen Apr 18 at 17:41 Communities (29) Top network posts 18 Why do people study curve shortening flows? 13 Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli? 12 Can a non-local ring have only two pr... |
News for cp3
Louvain-La-Neuve
September 24, 2019
New result on BR(K^+ \to \pi^+ \nu \bar{\nu}) from NA62...
A way to investigate the presence of new physics at very high energy, not accessible to LHC, is searching for deviations from the SM of quantities predicted very precisely within the SM, even at small energies. O... |
However, there's an interesting 3-sigma anomaly in an otherwise obscure search so let me tell you what it is.
It appears in the following preprint:
Search for Type III Seesaw Model Heavy Fermions in Events with Four Charged Leptons using \(5.8\,{\rm fb}^{-1}\) of \(\sqrt{s} = 8\TeV\) data with the ATLAS Detector (ATLAS... |
It's hard to say just from the sheet music; not having an actual keyboard here. The first line seems difficult, I would guess that second and third are playable. But you would have to ask somebody more experienced.
Having a few experienced users here, do you think that limsup could be an useful tag? I think there are a... |
Answer : B.
$L_1 = \{ a^pb^q | p + q \geq 10^6 \}$
$L_1$ is Regular. Moreover, $10^6$ is nothing but a large constant number. If it had been $10^7 \,\,or\,\,10^2$ instead of $10^6$, It wouldn't have mattered, Right? So, Let's get rid of such big number and replace it with, say, $1.$
So, $L_1' = \{ a^pb^q | p + q \geq 1... |
In general matching with dependent types can be quite subtle! You'll note that in the Coq documentation that the
extended pattern-matching syntax is
match t as x in T1 return T2 with
| C1 a1 ... an
...
In particular, ommiting any of the
as,
in or
return clauses can prevent type inference of the statement.
Intuitively, ... |
Let $X_1, X_2, ...$ be independent random variables.
Define $$\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, \ldots)$$ and $$\mathscr{T} = \bigcap_{n} \mathscr{T}_n,$$ the tail σ-algebra of $(X_1, X_2, \ldots)$.
Are $\sigma(X_1), \sigma(X_2), ...$ independent of $\mathscr{T}$?
If so, why?
If not, why, and what about $$\sigma... |
A Brief Introduction to the Weak Form
This is an introduction to the weak form for those of us who didn’t grow up using finite element analysis and vector calculus in our daily lives, but are nevertheless interested in learning about the weak form, with the help of some physical intuition and basic calculus.
About the ... |
fskilnik wrote:
GMATH
practice exercise (Quant Class 15)
What is the value of \(k\) ?
(1) \(x = 2\)
(2) \(y^2 = 3\)
\(? = k\,\,\,\left[ {{\rm{degrees}}} \right]\)
\(\left( {1 + 2} \right)\,\,\left\{ \matrix{
\,\Delta \,{\rm{below}}\,\,{\rm{unique}}\,\,\left( {{{30}^ \circ },{{60}^ \circ },{{90}^ \circ } + \,\,y\,\,{\rm... |
Search
Now showing items 1-10 of 15
A free-floating planet candidate from the OGLE and KMTNet surveys
(2017)
Current microlensing surveys are sensitive to free-floating planets down to Earth-mass objects. All published microlensing events attributed to unbound planets were identified based on their short timescale (bel... |
Consider the function $f: (0,1) \rightarrow \mathbb{R}$.
\
$f(n) =\left\{\begin{array}{ll} \dfrac{1}{x} - 2 & \text{if}\ 0 < x \leq \dfrac{1}{2}\\ \dfrac{1}{x-1} + 2 & \text{if} \ \dfrac{1}{2} < x < 1\end{array}\right.$
\
We claim that $f$ is a bijective function between the open unit interval $(0,1)$ and $\mathbb{R}$ ... |
This is actually parts of an exercise I found in W.Hodges'
shorter model theory (P.147):
Let $\mathcal{L}$ be a first-order language and T a theory in $\mathcal{L}$. Also let $\mathcal{A}$ and $\mathcal{B}$ are models of T and $\mathcal{C}$ is an $\mathcal{L}$-structure such that $\mathcal{A} \subseteq \mathcal{C} \sub... |
The general theorem is: for all odd, distinct primes $p, q$, the following holds: $$\left( \frac{p}{q} \right) \left( \frac{q}{p} \right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$$
I've discovered the following proof for the case $q=3$: Consider the Möbius transformation $f(x) = \frac{1}{1-x}$, defined on $F_{p} \cup {\inft... |
Answer
(b) is the sketch that better represents.
Work Step by Step
1. Considering a pure solution: $[H_3O^+] = [A^-] = x$ Therefore: $Ka = \frac{[H_3O^+][A^-]}{[HA]} = \frac{x^2}{[HA]}$ 2. If the molarity of the solution is doubled: $Ka = \frac{y^2}{2[HA]}$ ** The ka is constant, but the $[H_3O^+]$ changes, so we consi... |
I am currently reading Atkins and Friedman's "Molecular Quantum Mechanics" (4th ed), looking at the Rayleigh-Ritz variation method. Starting from the Schrödinger equation $\hat{H}\psi = E \psi$, we get the "Rayleigh ratio"
$$ E = \frac{\int \psi^*\hat{H}\psi d\tau}{\int \psi^*\psi d\tau} $$
Setting a trial function to ... |
Is there any
(i.e., to be found in probability books) metric for the distance between two probability measures, defined only on a subset of their support? official
Take, for example, the total variation distance: $$TV(\mu,\nu)=\sup_{A\in\mathcal{F}}|\mu(A)-\nu(A)|.$$
If $X$ and $Y$ are two real positive continuous rand... |
[EDIT]: After getting a very nice answer by Kevin Buzzard I realize that my question was a little bit too vague and I try to restate it more precisely.
As the title says, I would like to understand an isomorphism of Hida from a more geometric perspective than what I normally read. What bothers me is that there are two ... |
Suppose a group \(G\) has a finite index subgroup that maps onto the free group of rank 2. Show that every countable group can be embedded in one of the quotient groups of \(G\).
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Suppose a group \(G\) has a finite index subgroup that maps onto the free group of rank 2. Show that every countabl... |
Your confusion is justified -- the author's definitions abuse notation, at least in my opinion, leading to unnecessary ambiguity.
$I$ is the index set, this means it is the set of labels which are assigned to each event in the collection of events. Of course, each event is assigned at most one label, and each label is ... |
Instability of bound states for 2D nonlinear Schrödinger equations
1.
Department of Mathematical Sciences, Yokohama City University, Seto 22-2, 236-0027, Japan
$iu_t+\Delta u+|u|^{p-1}u=0\quad$ for $x\in \mathbb R^2$
and $t>0$, where $(r,\theta)$ are polar coordinates and $m\in\mathbb N$. Using the Evans function, we p... |
In quantum computation, what is the equivalent model of a Turing machine? It is quite clear to me how quantum
circuits can be constructed out of quantum gates, but how can we define a quantum Turing machine (QTM) that can actually benefit from quantum effects, namely, perform on high-dimensional systems?
In quantum com... |
Results from the kaonic hydrogen X-ray measurement at DAFNE and outlook to future experiments 43 Downloads Citations Abstract
The \(\overline{K}N\) system at rest plays a key role for the understanding of strong interaction of hadrons with strangeness involved. The experiment SIDDHARTA used X-ray spectroscopy of kaonic... |
I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More specific the following:
Let $(M,\omega, J)$ be a compact Kähler manifold with Kähler form $\omega$ and complex structure $J$... |
How can you derive the Carnot efficiency using only properties of reversible cycles?
closed as unclear what you're asking by Whit3rd, Gert, stafusa, Jon Custer, Kyle Kanos Nov 22 '17 at 11:15
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it... |
I think that everybody will agree that \({(\sqrt{9+\sqrt{80}}+\sqrt{9-\sqrt{80}})^2}\) is much easier to understand than (sqr root of (9 +sqr root of 80)+sqr root of (9 - sqr root 80))^2.
So, in order to help you with the questions you post more efficiently please use the following guide to write math formulas.
Square ... |
Give an example of a perfect set in $\mathbb R^n$ that does not contain any of the rationals.
(Or prove that it does not exist).
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this comm... |
Under what conditions is a specific sorting algorithm actually the fastest one?
1) When implemented in a parallel way in hardware, does it need to have a reasonably low latency while requiring as few gates as possible?Yes -> use a bitonic sorter or Batcher odd-even mergesort, latency is $\Theta(\log(n)^2)$ and the numb... |
Consider a variant of the traditional coupon collector's problem. There're $n$ kinds of coupons and there's a $1 \times n$ grid. Each grid corresponds to one kind of coupon. Once picking a coupon, we color the corresponding grid. I'd like to ask what's the expected number of coupons to pick such that the maximum number... |
Let $a \in \Bbb{R}$. Let $\Bbb{Z}$ act on $S^1$ via $(n,z) \mapsto ze^{2 \pi i \cdot an}$. Claim: The is action is not free if and only if $a \Bbb{Q}$. Here's an attempt at the forward direction: If the action is not free, there is some nonzero $n$ and $z \in S^1$ such that $ze^{2 \pi i \cdot an} = 1$. Note $z = e^{2 \... |
Start from the opposite task.
If $\displaystyle \int x^x \, dx=F(x)$ then $\displaystyle F'(x)=x^x$
First we need to find asymptotic evaluation of the integral. Let us take it in the form
$$F(x)=x^xg(x)$$
So it has to be:
$$F'(x)=x^x((1+\ln(x))g(x)+g'(x))=x^x$$
From there it is sufficient to take $g(x) \sim \frac{1}{1+... |
Yes, and no. There is a "categorically comprehensive" reason for this trace map to exist, but not necessarily to construct it. And to prove that this reason is valid, one does require categorical machinery. The elevator speech answer is:
THH is an algebra in some symmetric monoidal category. K is the unit in this symme... |
Analytical problem :
what you are expecting is
positive diffusion : you want the $T_i$ values to spread over your domain as time passes to eventually reach $T_i(t\rightarrow \infty) = cte$
if $\alpha$ was a negative number, you would have what is called
negative diffusion : you'll have exactly the opposite i.e. the gra... |
2018-08-25 06:58
Recent developments of the CERN RD50 collaboration / Menichelli, David (U. Florence (main) ; INFN, Florence)/CERN RD50 The objective of the RD50 collaboration is to develop radiation hard semiconductor detectors for very high luminosity colliders, particularly to face the requirements of the possible u... |
Can I be a pedant and say that if the question states that $\langle \alpha \vert A \vert \alpha \rangle = 0$ for every vector $\lvert \alpha \rangle$, that means that $A$ is everywhere defined, so there are no domain issues?
Gravitational optics is very different from quantum optics, if by the latter you mean the quant... |
Interesting recursive functions -
et R={i∣∃j:f(j)=i} be the set of distinct values that...can sum1 explain clearly ???wat is d meaning of dis
R={i∣∃j:f(j)=i} be the set of distinct values that f takes
R={i∣∃j:f(j)=i} be the set of distinct values that f takes
Someone please explain why is it written "takes" ? R contain... |
Learning Outcomes
Solve equations that include square roots.
Square roots occur frequently in a statistics course, especially when dealing with standard deviations and sample sizes. In this section we will learn how to solve for a variable when that variable lies under the square root sign. The key thing to remember is... |
Is there a standard example of two abelian varieties $A$, $B$ over some number field $k$ which are $k_v$-isomorphic for every place $v$ of $k$ but not $k$-isomorphic ?
(If you upvote this answer, please consider upvoting the answers by Felipe Voloch and David Speyer too, since this answer builds on their ideas.)
The sm... |
Let $f\colon X\to Y$ be a flat morphism of irreducible projective algebraic varieties over $\mathbb{C}$ (or any other algebraically closed field of characteristic 0). Assume that $Y$ is smooth, and the generic fiber of $f$ is smooth (these two assumptions seem to be important).
Let $U\subset Y$ be a Zariski open non-em... |
Is Bekenstein entropy limit inconsistent with universal continuity?
Yes, but that doesn't mean the Bekenstein bound is correct and everything is fine. Entropy can be considered as "sameness", related to available energy. And the expression $S \leq \frac{2 \pi k R E}{\hbar c}$ has a c in it, but the coordinate speed of ... |
The following equation describes the motion of a rigid body rotation, such as a gyroscope:
$$ \frac{d\textbf{L}}{dt} ={\bf{\tau}}= \textbf{r}\times m\textbf{g}= {\omega}\times \textbf{L}$$
where $L$ is the angular momentum, $\tau$ is the torque, $r$ is the position vector, $g$ is gravity, and $\omega$ is the angular ve... |
The task is to find the period of small oscillations in the potential
$$U=U_0\tan^2{\Big(\frac{x^2}{a^2}\Big)}.$$
I started with finding the stable equilibrium points:
$\frac{dU}{dx}=0$
$2U_{0}\tan{\Big(\frac{x^2}{a^2}\Big)}\frac{1}{\cos^{2}{\Big(\frac{x^2}{a^2}\Big)}}\frac{2x}{a^{2}}=0$
The solutions of this equation ... |
With which notation do you feel uncomfortable?
closed as not constructive by Loop Space, Chris Schommer-Pries, Qiaochu Yuan, Scott Morrison♦ Mar 19 '10 at 6:10
As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this quest... |
The formulation of an ARIMA model with exogenous regressors is not generally the same as a linear regression model with lagged dependent variables. To my knowledge, the formulation in software packages for the ARIMA model with exogenous regressors is the following:
$$\left[ y(t) - \beta_0 - \beta_3 \hbox{levelshift}(t)... |
Data on the mean multiplicity of strange hadrons produced in minimum bias proton--proton and central nucleus--nucleus collisions at momenta between 2.8 and 400 GeV/c per nucleon have been compiled. The multiplicities for nucleon--nucleon interactions were constructed. The ratios of strange particle multiplicity to part... |
Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$.
Consider a $G$-invariant (w.r.t. the coadjoint action) embedded submanifold $U \subset \mathfrak{g}^*$, such that the maximal-d... |
I am still not sure about this problem and I was hoping someone can help me put it to rest once and for all.
Suppose $X=[0,1]$ and $m$ is the $\sigma$ algebra of Lebesgue measurable sets on $X$ and $\mu$ is Lebesgue measure on $X$, consider the function $g_1=2\chi_{[0,1/2)}, g_2=4\chi_{[1/2,3/4)}, g_3=8\chi_{[3/4,7/8)}... |
Here, I will demonstrate an astonishing fact (and thus an astonishing lack of understanding):
It is well known from the uncertainty principle that an electron cannot be at rest.
However, consider the relativistic limit, in which we treat the electron as a spinor. Then, by solving the free equation of motion, we get \be... |
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