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TIFR 2014 Problem 28 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects. The image is a front cover of a book named Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert. This book is very useful for the preparation of TIFR Entrance.
Also Visit: College Mathematics Program of Cheenta
Problem:
Let \(X\) be a topological space such that every function \(f: X \to \mathbb{R}\) is continuous. Then
A. \(X\) has the discrete topology.
B. \(X\) has the indiscrete topology.
C. \(X\) is compact.
D. \(X\) is not connected.
Discussion:
We know that if \(Y\) is a discrete space then any function \(g: Y \to Z \) is continuous.
Option A asks whether the converse to this is true in the case that \(Z= \mathbb{R} \).
To prove/disprove whether \(X\) has the discrete topology or not it is enough to prove whether every singleton set is open or not.
If we can show that for every \(x\in X\) there exists a function \(f_x :X \to \mathbb{R}\) such that \(f_x^{-1} (-1,1) = \{x\} \) then we are done. Because we are given that \(f_x\) if exists must be continuous, and since \((-1,1)\) is open in \(\mathbb{R}\) we will have the inverse image of it open in \(X\), so \(\{x\} \) will be open in \(X\).
Now, this target is easy to handle. We define for each \(x\in X\)
\(f_x (x) = 0 \) and \(f_x (y) =2\) for \(y \neq x \).
This \(f_x\) satisfies our desired property. So \(X\) is discrete.
Taking \(X= \mathbb{Z}\) (for example) shows that \(X\) does not need to be indiscrete nor does it have to be compact.
Taking \(X= \{0\} \) shows that \(X\) may be connected. Ofcourse if \(X\) has cardinality more than 1, it is not connected.
Helpdesk What is this topic:Real Analysis What are some of the associated concept:Continuity, Discrete Space Book Suggestions:Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert |
Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$
1.
School of Mathematics and Statistics, Shandong University of Technology Zibo 255049, China
2.
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
3.
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
4.
School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, Texas 78539, USA
$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + u + \lambda K\left( x \right)\phi u = a\left( x \right){{\left| u \right|}^{p - 2}}u}&{{\text{in }}{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2}}&{{\text{in }}{\mathbb{R}^3},} \end{array}} \right.$
$\lambda >0$
$2 < p \le 4$
$K\left( x\right) $
$a\left( x\right) $
$\mathbb{R}^{3}$
$% \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) = K_{\infty }\geq 0$
$\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) = a_{\infty }>0$
$\lambda$
$p$
$3.18 \approx \frac{{1 + \sqrt {73} }}{3} < P \le 4$ Keywords:Positive solution, Schrödinger-Poisson system, variational method, ground state, fibering maps, Sobolev embedding theorem, concentration-compactness principle. Mathematics Subject Classification:Primary: 35J20, 35J61, 35A01; Secondary: 35B40. Citation:Juntao Sun, Tsung-Fang Wu, Zhaosheng Feng. Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1889-1933. doi: 10.3934/dcds.2018077
References:
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K. J. Brown and Y. Zhang,
The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,
[10] [11]
C. Y. Chen, Y. C. Kuo and T. F. Wu,
Existence and multiplicity of positive solutions for the nonlinear Schrödinger-Poisson equations,
[12] [13] [14]
P. Drábek and S. I. Pohozaev,
Positive solutions for the $p$ -Laplacian: Application of the fibering method,
[15] [16] [17]
I. Ianni and G. Vaira,
Non-radial sign-changing solutions for the Schrödinger-Poisson problem in the semiclassical limit,
[18] [19]
P. L. Lions,
The concentration-compactness principle in the calculus of variations,
[20]
P. L. Lions,
The concentration-compactness principle in the calculus of variations,
[21]
A. Mao, L. Yang, A. Qian and S. Luan,
Existence and concentration of solutions of Schrödinger-Poisson system,
[22] [23] [24] [25] [26] [27]
J. Sun, H. Chen and J. J. Nieto,
On ground state solutions for some non-autonomous Schrödinger-Poisson systems,
[28] [29]
J. Sun, T. F. Wu and Z. Feng,
Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system,
[30]
G. Tarantello,
On nonhomogeneous elliptic equations involving critical Sobolev exponent,
[31] [32]
Z. Wang and H. Zhou,
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^{3}$,
[33]
L. Zhao, H. Liu and F. Zhao,
Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential,
[34]
show all references
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[1] [2] [3] [4] [5] [6] [7] [8] [9]
K. J. Brown and Y. Zhang,
The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,
[10] [11]
C. Y. Chen, Y. C. Kuo and T. F. Wu,
Existence and multiplicity of positive solutions for the nonlinear Schrödinger-Poisson equations,
[12] [13] [14]
P. Drábek and S. I. Pohozaev,
Positive solutions for the $p$ -Laplacian: Application of the fibering method,
[15] [16] [17]
I. Ianni and G. Vaira,
Non-radial sign-changing solutions for the Schrödinger-Poisson problem in the semiclassical limit,
[18] [19]
P. L. Lions,
The concentration-compactness principle in the calculus of variations,
[20]
P. L. Lions,
The concentration-compactness principle in the calculus of variations,
[21]
A. Mao, L. Yang, A. Qian and S. Luan,
Existence and concentration of solutions of Schrödinger-Poisson system,
[22] [23] [24] [25] [26] [27]
J. Sun, H. Chen and J. J. Nieto,
On ground state solutions for some non-autonomous Schrödinger-Poisson systems,
[28] [29]
J. Sun, T. F. Wu and Z. Feng,
Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system,
[30]
G. Tarantello,
On nonhomogeneous elliptic equations involving critical Sobolev exponent,
[31] [32]
Z. Wang and H. Zhou,
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^{3}$,
[33]
L. Zhao, H. Liu and F. Zhao,
Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential,
[34]
[1]
Sitong Chen, Xianhua Tang.
Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system.
[2]
Sitong Chen, Junping Shi, Xianhua Tang.
Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity.
[3]
Zhengping Wang, Huan-Song Zhou.
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$.
[4]
Xu Zhang, Shiwang Ma, Qilin Xie.
Bound state solutions of Schrödinger-Poisson system with critical exponent.
[5] [6]
Xianhua Tang, Sitong Chen.
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials.
[7]
Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang.
Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent.
[8]
Zhanping Liang, Yuanmin Song, Fuyi Li.
Positive ground state solutions of a quadratically coupled schrödinger system.
[9]
Yi He, Lu Lu, Wei Shuai.
Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents.
[10]
Lun Guo, Wentao Huang, Huifang Jia.
Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $.
[11]
Claudianor O. Alves, Minbo Yang.
Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$.
[12]
Yongpeng Chen, Yuxia Guo, Zhongwei Tang.
Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents.
[13]
Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang.
Standing waves for Schrödinger-Poisson system with general nonlinearity.
[14]
Kaimin Teng, Xiumei He.
Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent.
[15]
Miao-Miao Li, Chun-Lei Tang.
Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent.
[16] [17] [18]
Mingzheng Sun, Jiabao Su, Leiga Zhao.
Infinitely many solutions for a Schrödinger-Poisson system with
concave and convex nonlinearities.
[19]
Margherita Nolasco.
Breathing modes for the Schrödinger-Poisson system with a multiple--well external potential.
[20]
2018 Impact Factor: 1.143
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Rate of turn is dependent on the following two items:
The horizontal component of lift (centripetal force) The tangential velocity of the aircraft (true airspeed)
The rate or turn is directly proportional to the horizontal component of lift and inversely proportional to the tangential velocity of the aircraft.
For a given angle of bank, the vertical and horizontal components of lift will be the same, regardless of airspeed in level flight.
Consequently the airplane will experience the same centripetal acceleration, regardless of airspeed.
Since the tangential velocity is slower, any kind of centripetal force will produce a greater rate of turn for a slower flying aircraft as opposed to a faster moving aircraft and this can be shown by the centripetal acceleration equation
$$a_c = \frac{v^2}{r}$$
so both slow flying airplane with a true airspeed $v_s = 100$ knots and fast flying airplane with a true airspeed $v_f = 200$ knots experience the same centripetal acceleration.
$$\dfrac{v_s^2}{r_s} = \dfrac{v_f^2}{r_f} = 4\ \dfrac{v_s^2}{r_f}$$
or, $$\dfrac{1}{r_s} = \dfrac{4}{r_f}$$
Consequently $r_s < r_f$; in this case $r_f = 4\ r_s$
Since the angular velocity is equal to the tangential speed divided by the radius.
$$\omega = v/r$$
the angular speed of the slower aircraft will be greater than the faster aircraft.
$$\omega_s = v_s/r_s$$
and
$$\omega_f = \dfrac{v_f}{r_f} = \dfrac{2 \ v_s}{4 \ r_s} = \frac{1}{2}w_s$$
So our twice as slow airplane turns twice as fast as the faster one does under these conditions. |
I have some confusion about the "Reversed effective force" as it appears in the derivation of D'Alembert's principle. In Goldstein d'Alembert's principle is given as:
$(F-\dot{p}) \cdot \delta r = 0$
First I have sources that seem to be contradictory.
This book, on page 8, indicates that "reversed effective forces" are not real forces. However, this book indicates otherwise.
http://books.google.com/books?id=4wkLl4NvmWAC&pg=PA345&lpg=PA345&dq=what+is+a+reversed+effective+force&source=bl&ots=obL2JFMwAw&sig=lpf5LvJJM_iNvyHDhXAV0gIaDO0&hl=en&sa=X&ei=FkMsVeG_C8ahNv3vgPAE&ved=0CCYQ6AEwAQ#v=onepage&q=what%20is%20a%20reversed%20effective%20force&f=false
On page 345, this book indicates "this is the force exerted by the moving body to resist the change in its state.
I think that the former source is probably correct, but I lack good intuition of D'Alembert's principle. For reference the reversed effective force is represented by $-ma$ or $-\dot{p}$. My best guess as to what the reversed effective force is that it it the force that is required to appear if you view the system from the perspective of the accelerating body. Is this correct? If it is then how is d'Alembert's principle applicable to inertial systems? |
Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" link, and I am particularly interested in the question of factorizing volume-preserving diffeomorphisms on a compact smooth and connected $n$-dimensional manifold $M$ which are isotopic to the identity.
Now, chapter 5 of the book in particular deals with this infinite dimensional Lie group, and at page 129 they take into consideration exactly this question, and they provide the factorization specifically for maps $h$ isotopic to the identity, under the constraint that they belong to the kernel of the so-called flux homomorphism $S_\omega$.
Its definition is as follows:
assume $\omega$ to be a volume form on $M$ and let $\varphi_t\in\textrm{Diff}^\infty_\omega(M)$ be a smooth isotopy to the identity. Let $i(\cdot)$ denote the interior product and define $$I_{\varphi_t}(\omega)=\int_0^1(\varphi_t^*i(\dot\varphi_t)\omega)dt.$$
The cohomology class $[I_{\varphi_t}(\omega)]\in H^{n-1}(M,\mathbb R)$ depends only on the homotopy class relatively to fixed ends of the isotopy $\varphi_t\in\textrm{Diff}^\infty_\omega(M)$.
Let $G_\omega(M)$ be the group consisting of homotpy classes $[\varphi_t]$ of isotopies $\varphi_t\in\textrm{Diff}^\infty_\omega(M)$, relatively to fixed ends. The mapping $$S_\omega:G_\omega(M)\ni[\varphi_t]\mapsto I_{\varphi_t}(\omega)\in H^{n-1}(M,\mathbb R)$$ is the so-called flux homomorphism. Therefore $[\varphi_t]\in\ker S_\omega\Leftrightarrow I_{\varphi_t}(\omega)$ is an exact $n-1$ form. You can read more on $S_\omega$ in chapter 3 of the book I mentioned above.
My question now is as follows: let $\textrm{Diff}^\infty_\omega(M)_0$ be the subgroup of $\textrm{Diff}^\infty_\omega(M)$ of isotopic to the identity volume-preserving diffeomorphisms on $M$. Is it possible to find a sufficiently small neighborhood of the identity map $O\subset \textrm{Diff}^\infty_\omega(M)_0$ such that $f\in\ker S_\omega$ for any $f\in O$? I am pretty confident this is possible if $M$ is also simply connected, but in the general setting I explained at the beginning? I was wondering this could also be the case if all the maps in $O$ had common support within a contractible set $U\subset M$ but is this a full neighborhood of the identity map in $\textrm{Diff}^\infty_\omega(M)_0$?
Thanks in advance to all those who will reply and help me clarify this point.
-Guido- |
Answer
: 3
Squaring both sides
Let $x=\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{...}}}}}$
$\begin{align}x^2&=\left(\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{...}}}}}\right)^2\\
&=3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{...}}}}\\
&=3+2x\end{align}$
So $x^2-2x-3=0\Rightarrow (x-3)(x+1)=0$, so $x=3$ or $x=-1$.
$x$ positive since so $x=3$.
Check: $\sqrt{3+2\times(3)}=3$
'Working it out' using a sequence
Let $x=\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{...}}}}}$
We can approximate $x$ by imagining that we are going to find that large square root as a calculation.
We can start by finding $\sqrt{3}$, and then double it, add $3$, and find the square root of that, and so on.
So if we let $a=\sqrt{3}$, and then $b=\sqrt{3+2\times a}=\sqrt{3+2\sqrt{3}}$, then $c=\sqrt{3+2\times b}=\sqrt{3+2\times\sqrt{3+2\times\sqrt{3}}}$, and $d=\sqrt{3+2\times c}=\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{3}}}}$, then the sequence $a, b, c, d, ...$ will get closer and closer to $x$.
Using a calculator, or a spreadsheet, $a=1.73205$, $b=2.54246$, $c=2.84340$, $d=2.94734$, $e=2.98239$, $f=2.99413$, ...
It looks like they are getting closer and closer to $3$, so it looks like $x=3$. If we got to $3$ in our sequence, then the next number in the sequence would be $\sqrt{3+2\times 3}$, which is $\sqrt{3+6}=\sqrt{9}=3$. So if the sequence ever gets to $3$, it will stay at $3$ forever, which strongly suggests that $x=3$. |
Understand the problem
Consider the following subsets of the plane: $$ \displaystyle { C_1 = \{ (x, y) : x > 0, y = \frac{1}{x} \} } $$ and $$ \displaystyle { C_2 = \{ (x, y) : x < 0, y = -1 + \frac{1}{x} \} } $$
Given any two points P = (x, y) and Q = (u, v) of the plane, their distance d(P, Q) is defined by $$ \displaystyle { d(P, Q) = \sqrt{(x-u)^2 + (y-v)^2} } $$
Show that there exists a unique choice of points \( P_0 \in C_1 \) and \( Q_0 \in C_2 \) such that $$ d(P_0, Q_0) \leq d(P, Q) $$ for all \( P \in C_1 \) and \( Q \in C_2 \).
Source of the problem
I.S.I. (Indian Statistical Institute, B.Stat, B.Math) Entrance. Subjective Problem 8 from 2019
Topic
Differential Calculus
Difficulty Level
8 out 10
Suggested Book
Problems in Calculus by I.A. Maron
Start with hints
Do you really need a hint? Try it first!
Shortest distance from a point to a line is the length of the perpendicular drawn from the point to the line.
This is the key idea.
Suppose f(t) and g(s) are two (smooth) curves. Then the shortest distance between them is along the common
normal (perpendicular) on the two curves. How to draw perpendiculars to curves? Pick a point on the curve (suppose A in the picture).
Next, draw a tangent at A (since the curve is smooth, that infinitely differentiable at every point, we can do this).
Locally (near A), this tangent is the approximation of the curve.
Finally draw a perpendicular to this tangent line, at the point A. This is regarded as perpendicular to the curve (
normal to the curve) at A. How to find the shortest path between two (smooth curves)?
Draw all possible normals to both curves. That draws all ‘perpendiculars’ to both curves erected at all points on both curves.
If any normal (perpendicular) is common between the two curves then that is possibly the shortest path. (You still need to check some other details. But that is part of a calculus course, not this discussion).
Can you show there is a common normal between f(x) = 1/x (x > 0) and \( g(x) = -1 + \frac{1}{x}, (x < 0) \)?
Consider \( f(x) = \frac{1}{x}, (x > 0) \).
The slope of the tangent line at any point is the derivative of the function.
\(\frac {d} {dx} f(x) = \frac {-1}{x^2} \)
Slope of normal which is perpendicular to the tangent is negative reciprocal of it. Hence it is \( x^2 \).
Parametrize the curve \( f(x) = \frac{1}{ x} \) as \( (t, \frac{1}{ t}) \)
Hence the equation of the normal through a point \( (t, \frac{1}{ t}) \) is $$ y – \frac{1}{t} = t^2 (x – t) $$
Similarly, parametrize the other curve and find the equation of the normal.
Use some other variable for this.
Consider \( g(x) = -1 + \frac{1}{x}, (x > 0) \).
The slope of the tangent line at any point is the derivative of the function.
\(\frac {d} {dx} g(x) = \frac {-1}{x^2} \)
Slope of normal which is perpendicular to the tangent is negative reciprocal of it. Hence it is \( x^2 \).
Parametrize the curve \( g(x) = -1 + \frac{1}{ x} \) as \( (r, -1 + \frac{1}{ r}) \)
Hence the equation of the normal through a point \( (r, -1 + \frac{1}{ r}) \) is $$ y + 1 – \frac{1}{r} = r^2 (x – r) $$
We have generic equations for family of normal for each curve:
\( y + 1 – \frac{1}{r} = r^2 (x – r) \)
\( y – \frac{1}{t} = t^2 (x – t) \)
Can you find (or show the existence of) r and t such that these two equations are equal? This would show that there is a common normal and hence prove the existence of shortest path.
Let us write the equations in the slope-intercept form:
\( y = r^2 x + \frac{1}{r} – r^3 – 1 \\ y = t^2 x + \frac{1}{t} – t^3 \)
Comparing the coefficients we have:
\( r^2 = t^2 \) hence r = t or r = -t
We also have $$ \frac {1}{r} – r^3 – 1 = \frac {1}{t} – t^3 $$
Plugging in r = t we get -1 = 0 from second equation which is contradiction.
Plug in r = – t
$$ -\frac {1}{t} + t^3 – 1 = \frac {1}{t} – t^3 $$
(Assume t is not 0 (why?)) Simplifying we have
$$ 2t^4 – t – 2 = 0 $$
We want to show that all the roots of this equation are not complex. (If we have a real root, then we get a common normal, hence shortest part; this common normal will not be the longest path because the length of the longest path between these two curves easily goes to infinity).
It is enough to show the existence of real roots.
Consider the function \( 2t^4 – t – 2 = h(t) \)
Notice that h(0) = -2 < 0 and h(2) > 0
Hence by intermediate value property theorem there is at least one real root between 0 and 2.
Hence we are done. Watch the video (coming up soon) Connected Program at Cheenta
Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are: B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.
The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances. |
Problem: To prove the $\textsf{NP-Completeness}$ of the problem of "Packing Squares (with different side length) into A Rectangle", $\textsf{3-Partition}$ is reduced to it, as shown in the following figure.
In the $\textsf{3-Partition}$ instance, there are $n$ elements $(a_1, \cdots, a_i, \cdots, a_n)$. The target sum $t$ is $t = \frac{\sum a_i}{n/3}$.
In the reduction,
$B$ is a huge (constant) number and each $a_i$ is represented by a $(B + a_i) \times (B + a_i)$ square. The blank in the rectangle will be filled by unit ($1 \times 1$) squares. Questions:I don't quite understand the trick of "adding a huge number $B$" in the reduction. I guess it is used to force that any packing scheme will give a solution to $\textsf{3-Partition}$. But how?
Question 1: What is the trick of "adding a huge number" for in the reduction from $\textsf{3-Partition}$? Specifically, why does this reduction work? Why is this trick necessary, i.e., why wouldn't the reduction work if we left out $B$ (set $B=0$)?
I tried to identify the flaw of the proof of "any packing gives a 3-partition" but could not get the key point.
Actually I have also seen other reductions from $\textsf{3-Partition}$ that also use this trick. So,
Question 2: What is the general purpose of this trick of "adding a huge number" in the reductions from $\textsf{3-Partition}$ (
if there is)? Note: This problem is from the video lecture (from 01:15:15) by Prof. Erik Demaine. I should have first checked the original paper "Packing squares into a square". However, it is not accessible to me on the Internet. If you have a copy and would like to share, you can find my mailbox in my profile. Thanks in advance. |
According to the WP definition of MAP:
In Bayesian statistics, a maximum a posteriori probability (MAP) estimate is an estimate of an unknown quantity, that
equals the mode of the posterior distribution.
(emphasis added) which, given a posterior $f(\theta \mid x)$, can be defined as:
$$\hat{\theta}_{\mathrm{MAP}}(x) = \underset{\theta}{\operatorname{arg\,max}} \ f(\theta \mid x) $$
As far as I understand it, the mode of a distribution
depends on how I construct its histogram (or KDE). This looks to me to be in contradiction with the above definition, where the MAP is the $\theta$ value found for the maximum of the sampled $f(\theta \mid x)$ and does not depend on anything else.
What am I missing? |
Under the auspices of the Computational Complexity Foundation (CCF)
A typical obstacle one faces when constructing pseudorandom objects is undesired correlations between random variables. Identifying this obstacle and constructing certain types of "correlation breakers" was central for recent exciting advances in the construction of multi-source and non-malleable extractors. One instantiation of correlation breakers is correlation breakers with advice. These are algorithms that break the correlation a "bad" random variable $Y'$ has with a "good" random variable $Y$ using an "advice" - a fixed string $\alpha$ that is associated with $Y$ which is guaranteed to be distinct from the corresponding string $\alpha'$ associated with $Y'$. Prior to this work, explicit constructions of correlation breakers with advice require the entropy of the involved random variables to depend linearly on the advice length.
In this work, building on independence-preserving mergers, a pseudorandom primitive that was recently introduced by Cohen and Schulman, we devise a new construction of correlation breakers with advice that has optimal, logarithmic, dependence on the advice length. This enables us to obtain the following results.
* We construct an extractor for $5$ independent $n$-bit sources with min-entropy $(\log{n})^{1+o(1)}$. This result puts us tantalizingly close to the goal of constructing extractors for $2$ sources with min-entropy $O(\log{n})$, which would have exciting implications to Ramsey theory.
* We construct non-malleable extractors with error guarantee $\epsilon$ for $n$-bit sources, with seed length $d = O(\log{n}) + (\log(1/\epsilon))^{1+o(1)}$ for any min-entropy $k = \Omega(d)$. Prior to this work, all constructions require either very high min-entropy or otherwise have seed length $\omega(\log{n})$ for any $\epsilon$. Further, our extractor has near-optimal output length. Prior constructions that achieve comparable output length work only for very high min-entropy $k \approx n/2$.
* By instantiating the Dodis-Wichs framework with our non-malleable extractor, we obtain near-optimal privacy amplification protocols against active adversaries, improving upon all (incomparable) known protocols. |
Consider the one-dimensional Ising model with constant magnetic field and node-dependent interaction on a finite lattice, given by
$$H(\sigma) = -\sum_{i = 1}^N J_i\sigma_i\sigma_{i + 1} - h\sum_{i = 1}^N\sigma_i$$
where $\sigma = \{\sigma_i\}_{i = 1,\dots, N}\in\Omega := \{\pm 1\}^N$, $\{J_i\}_{i = 1,\dots, N}$ are nearest neighbor interaction strength couplings, and $h \in \mathbb{R}$ is the magnetic field. Let's consider the ferromagnetic case, that is, $J_i \geq 0$ for $i = 1, \dots, N$, and for the sake of simplicity (though this doesn't matter in the thermodynamic limit), take periodic boundary conditions. Neither in the finite volume, nor in the thermodynamic limit does this model exhibit critical behavior for finite temperatures.
On the other hand, as soon as we allow $h$ to be complex (and fix the temperature), even in the finite volume $N$, the partition function has zeros as a function of $h$. In the thermodynamic limit these zeros accumulate on some set on the unit circle in the complex plane (Lee-Yang circle theorem).
Now the question: let's consider information geometry of the Ising model, as described above, when $h$ is real. In this case the induced metric is defined and the curvature does not develop singularities (obviously, since there are no phase transitions). Now, what about information geometry of the Ising model when $h$ is complex? This is a bit puzzling to me, since then the partition function attains zeros in the complex plane, so that the logarithm of the partition function is not defined everywhere on the complex plane, and the definition of metric doesn't extend directly to this case (the metric involves the log of the partition function), let alone curvature.
Is anyone aware of any literature in this direction? I thought it would be a good idea to ask before I try to develop suitable methods from scratch. |
We can think of the Rubik's cube Cayley graph $\Gamma=(V,E)$ with each (colored) edge $E$ being one of the Singmaster moves $\langle U,U^{2},U^{3}=U^{-1},D,D^{2},D^{3},\cdots\rangle$ and each vertex $V$ being one of the $43252003274489856000\approx 4.3e{19}$ different configurations of the $3\times 3\times 3$ cubes.
The diameter of a graph is the longest shortest path in the graph. Classical algorithm for determining the diameter is polynomial in $\vert V \vert$; see, e.g., this answer from a sister site.
As mentioned above, God's number is (related to) this diameter; to know the longest shortest path between to vertices for a Cayley graph on a group, it suffices to know how many steps away from the solved state one is. We know, thanks to Rokicki, Kociemba, Davidson, and Dethridge among others, that God's number is $20$. The algorithms they executed was polynomial in $\vert V\vert$, e.g. polynomial in $4.3e{19}$.
Heiligman's quantum algorithm for graph diameter, mentioned in the comments, achieves a Grover speedup over Djikstra's algorithms, with "a total quantum cost of $O(|V|^{9/4})$." However, I believe Heiligman encodes the graph much as a classical algorithm would; e.g. with $O(|V|)$ qubits. Clearly if $|V|=4.3e{19}$ then this would not help.
Instead, another way to encode a Rubik's cube, as hinted in the other questions, is of course to prepare a uniform superposition over all $4.3e{19}$ states. This only takes $\log 4.3e{19}$ qubits.
Quantum algorithms are good at talking about "eigenvalues" and "eigenvectors" and "eigenstates." Applying all Singmaster moves to a uniform superposition of all $4.3e{19}$ states does not change the state; i.e. the uniform superposition is an eigenstate of the Markov chain on the Cayley graph.
There are relations between the
diameter of a graph and the eigenvalues/eigenvectors of the corresponding adjacency/Laplacian matrix, especially the spectral gap, the distance between the two largest eigenvalues ($\lambda_1-\lambda_2$). A quick Google search of "diameter eigenvalue" produces this; I recommend exploring similar Google searches.
Spectral gaps are
exactly what limits the adiabatic algorithm. Thus, perhaps by knowing how fast an adiabatic algorithm needs to run to evolve from the uniform superpositon to the solved state for various subgroups/subspaces of the Rubik's cube group, one could estimate the spectral gap, and use this to bound God's number. But I'm quickly out of my league here and I doubt any sense of accuracy is achievable. |
Given two vector spaces $V_1$ and $V_2$, we can define their tensor product $V_1 \otimes V_2$, which is a new vector space. Product states in this vector space can be written
$$ v_1 \otimes v_2 $$
where $v_1 \in V_1$ and $v_2 \in V_2$. Not every element of $V_1 \otimes V_2$ can be written like this as a separable product state, but you can form a basis of $V_1 \otimes V_2$ using the product states, and then the most general state is some linear combination of those product states. (This is the idea of entanglement: an entangled state is one that cannot be written as a product state.)
Furthermore, given two linear operators $T_1: V_1 \rightarrow W_1$ and $T_2: V_2 \rightarrow W_2$, we can define their tensor product as a new operator
$$T_1 \otimes T_2: V_1 \otimes V_2 \rightarrow W_1 \otimes W_2 $$
by
$$ T_1 \otimes T_2 (v_1 \otimes v_2) = T_1(v_1) \otimes T_2(v_2). $$
This definition of $T_1 \otimes T_2$ can be extended for non-product states by linearity (remember that $V_1 \otimes V_2$ has a basis of product states).
Beware that the symbol $\otimes$ is being used in three distinct ways! It can denote a tensor product of vector spaces, of vectors, or of operators. Physicists often omit the $\otimes$ notation entirely and just use juxtaposition, but hopefully including the tensor products explicitly will answer your questions.
Also note that mathematicians have more sophisticated definitions of tensor products, but for the our purposes this should be sufficient.
Now, if we have two electrons, we have two spin spaces. We consider their tensor product, whose basis states might look like
$$ \left| \chi_1 \right> \otimes \left| \chi_2 \right> $$
where $\left| \chi_1 \right>$ is the state of one electron and $\left| \chi_2 \right>$ the state of the other electron. Then we define the spin operator on this space by
$$ S = S_1 \otimes I + I \otimes S_2 $$
where $I$ is the identity operator. Using the definition of the tensor product of operators, it follows that
$$ S \left( \left| \chi_1 \right> \otimes \left| \chi_2 \right> \right) = S_1 \left| \chi_1 \right> \otimes \left| \chi_2 \right> + \left| \chi_1 \right> \otimes S_2 \left| \chi_2 \right>. $$
With this notation it should be clear why $S_1$ only acts on $\left| \chi_1 \right>$ and $S_2$ only acts on $\left| \chi_2 \right>$. For example, if $\left| \chi_1 \right> = \left| \uparrow \right>$ and $\left| \chi_2 \right> = \left| \uparrow \right>$, then
\begin{align}S_z \left( \left| \uparrow \right> \otimes \left| \uparrow \right> \right) &= S_{1z} \left| \uparrow \right> \otimes \left| \uparrow \right> + \left| \uparrow \right> \otimes S_{2z} \left| \uparrow \right> \\&= \frac{\hbar}{2} \left| \uparrow \right> \otimes \left| \uparrow \right> + \left| \uparrow \right> \otimes \frac{\hbar}{2} \left| \uparrow \right> \\&= \hbar \left( \left| \uparrow \right> \otimes \left| \uparrow \right> \right).\end{align}
We see that our definition of $S$ seems to be sensible. |
Theorem:For any closed 3-manifold $M$ and any rational number $4.5 < r < 6$ there is a triangulation $T$ of $M$ for which the average edge-degree $\mu(T)$ is $r$. Here the degreeof an edge $e$ is the number of 3-simplices having $e$ as a face.
Now, suppose $T$ is a triangulation of a closed $n$-manifold $M$. (NOTE: By triangulation, I mean a combinatorial $n$-manifold, but results using a more relaxed definition would be great.) Let us extend the notion of average edge-degree to something I like to call the
average bone-degree
$$\mu(T)=\frac{1}{f_{n-2}}\sum_{\tau^{n-2}\in T} \mbox{deg}(\tau^{n-2}).$$
Here $f_k$ is the number of $k$-simplices in $T$, the sum runs over all codimension-2 simplices (called the
bones of $T$), and deg($\tau^{n-2}$) is the number of $n$-simplices in $T$ with $\tau^{n-2}$ as a face.
Note that by simple double-counting arguments we may alternately write this as
$$\mu(T) = \frac{n(n+1)}{2}\frac{f_n}{f_{n-2}} = n \frac{f_{n-1}}{f_{n-2}}.$$
Question:Does anyone know of results similar to the theorem above but for $n\geq 4$? That is, have the possible values for $\mu(T)$ been characterized when $n\geq 4$? Results for any manifold $M$ would be great.
I seem to remember stumbling across something like this for $n=4$ once, but I can no longer find it. I know that there is a large body of research on the "f-vectors" of manifolds, $\mathbf{f}(T) = \left( f_0, f_1, \ldots, f_n \right)$, so perhaps someone familiar with this work can help.
Thanks for any assistance pointing me in the correct direction! |
Let
$T>0$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\in[0,\:T]}$ be a right-continuous filtration on $(\Omega,\mathcal A)$ $B$ be a Brownian motion on $(\Omega,\mathcal A,\mathcal F,\operatorname P)$
Note that $$\mathcal E_0\left\{\Phi\in\mathcal L^2(\operatorname P\otimes\left.\lambda^1\right|_{[0,\:T]}):\Phi\text{ is }\mathcal F\text{-progressively measurable}\right\}$$ is a closed subspace of $\mathcal L^2(\operatorname P\otimes\left.\lambda^1\right|_{[0,\:T]})$, $$\mathcal E:=\left\{\Phi\in\mathcal E_0:\Phi=\sum_{i=1}^n\zeta_{i-1}1_{(t_{i-1},\:t_i]}\text{ for some }n\in\mathbb N\text{, }0\le t_0<\cdots<t_n\le T\text{ and }\zeta_0,\ldots,\zeta_{n-1}\in\mathcal L^0(\operatorname P)\right\}$$ is a dense closed subspace of $\mathcal E_0$, $$\mathcal M^2:=\left\{X\subseteq\mathcal L^2(\operatorname P):X\text{ is an almost surely right-continuous }\mathcal F\text{-martingale with }\operatorname P\left[X_0=0\right]=1\right\}$$ equipped with $$\left\|X\right\|_{\mathcal M^2}^2:=\sup_{t\in[0,\:T]}\left\|X_t\right\|_{\mathcal L^2(\operatorname P)}^2=\left\|X_T\right\|_{\mathcal L^2(\operatorname P)}^2$$ is a complete semi-normed space and $$\mathcal M^2_c:=\left\{X\in\mathcal M^2:X\text{ is almost surely continuous}\right\}$$ is a closed subspace of $\mathcal M^2$. Let $$(\Phi\cdot B)_t:=\sum_{i=1}^n\zeta_{i-1}\left(B_{t_i\:\wedge\:t}-B_{t_{i-1}\:\wedge\:t}\right)\;\;\;\text{for }t\in[0,\:T]$$ for $\Phi\in\mathcal E$ of the form as in the definition of $\mathcal E$. Then $\Phi\cdot B\in\mathcal M^2_c$ and $$\mathcal E\to\mathcal M^2_c\;,\;\;\;\Phi\mapsto\Phi\cdot B\tag 1$$ is a linear isometry. Thus, $(1)$ can be uniquely extended to a linear isometry $\mathcal E_0\to\mathcal M^2_c$.
Question:$\;\;\;$Let $\Phi\in\overline{\mathcal E}$ and $$\Phi^\varsigma:=\sum_{i=1}^n\Phi_{t_{i-1}}1_{(t_{i-1},\:t_i]}$$ for some $n\in\mathbb N$, $0=t_0<\cdots<t_n=T$ and $\varsigma:=\left\{t_0,\ldots,t_n\right\}$. Can we show that $$\left\|\Phi^\varsigma-\Phi\right\|_{\mathcal E}\xrightarrow{\left|\varsigma\right|\to 0+}0\tag 2$$ where $$\left|\varsigma\right|:=\max_{1\le i\le n}\left(t_i-t_{i-1}\right)$$ such that we could conclude that $$\left\|\left(\Phi^\varsigma\cdot B\right)_T-\left(\Phi\cdot B\right)_T\right\|_{\mathcal L^2(\operatorname P)}=\left\|\Phi^\varsigma\cdot B-\Phi\cdot B\right\|_{\mathcal M^2}\xrightarrow{\left|\varsigma\right|\to 0+}0\tag 3$$ by definition of $\Phi\cdot B$?
It's easy to show that $(2)$ holds, if $\Phi$ is continuous in mean-square, i.e. $$\Phi_t\in\mathcal L^2(\operatorname P)\;\;\;\text{for all }t\in[0,T]\tag 4$$ and $$\left(t\mapsto\Phi_t\right)\in C^0\left([0,T],\mathcal L^2(\operatorname P)\right)\;,\tag 5$$ but I don't think that $(2)$ holds in general.
However, in the book
Numerical Solution of Stochastic Differential Equations, the authors state (they use a different notation, but that's exactly what they state) that $(3)$ holds in general (i.e. for all $\Phi\in\overline{\mathcal E}$).
That's weird cause they've defined $\Phi\cdot B$ (again, using a different notation) in the same way as I did. Moreover, they've stated that $(2)$ holds, if $\Phi$ is continuous in mean-square, in their proof of density of $\mathcal E$ in $\overline{\mathcal E}$. If $\Phi$ is not continuous in mean-square, they show that it can be approximated by processes which are continuous in mean-square. However, that doesn't mean that the $\Phi^\varsigma$ form an approximating sequence, does it? So, what am I missing? |
Strongly-correlated metals often become insulators due to the repulsive Coulomb interaction, and the basic model here is the Mott-Hubbard Model:
$$H=-t\sum(\hat{c}_{i,\sigma}^{\dagger}\hat{c}_{j,\sigma}+\hat{c}_{j,\sigma}^{\dagger}\hat{c}_{i,\sigma})+U\sum\hat{n}_i^{\uparrow}\hat{n}_i^{\downarrow}$$
Where $U$ represents the Coulomb energy cost of having two electrons on the same site/state.
A very influential paper by Jaan, Allen, Sawatzky makes a distinction between the Mott insulator and the Charge-transfer insulator (J Zaanen, GA Sawatzky, JW Allen - Physical Review Letters, 1985).
For the charge transfer insulator, charges can move between individual sites within a unit cell (i.e. there are at least 2 orbital states for each unit cell $i$) with an energy cost $\Delta$. The charge transfer gap then represents the cost of moving an electron between the anion and cation within the unit cell. I assume this introduces another term in the Hubbard Hamiltonian that looks like: $$H_{CT}\propto\Delta\sum(\hat{c}_{C}^{\dagger}\hat{c}_{A}+\mathrm{h.c.})$$ Where $C$ denotes the cation, and $A$ the anion.
Often phase diagrams of $U$ and $\Delta$ are drawn like the one at the bottom of this post.
My question:
Why is the differentiation between the Charge transfer insulator and Mott insulator important? Sure, the physical origin of the gap $U$ and $\Delta$ require two different orbitals, but what difference does it make with regards to superconductivity, antiferromagnetism, etc.?
In other words, the Mott and Charge-transfer insulators are microscopically different, but who cares and why? |
This question already has an answer here:
Using the Master Method, I need to prove (True, or False and why) the following (3) recurrences:
$T(n) = 3T(\frac{n}{2}) + n = \Theta(n\ln(n))$
$T(n) = T(\sqrt{n}) + 1 = \Theta(n^2)$
$T(n) = 2T(\frac{n}{3} + 1) + n = \Theta(n\ln(n))$
I understand the Master Method (I think, largely from: https://stackoverflow.com/questions/13430256/understanding-master-theorem) and from my CLRS Text.
Problem 1: Would be case 1, but I don't understand how to use it in this case? What would my ɛ be in this case to perform the logarithmic function? I know that ultimately, my time complexity will be $\Theta(n^{1.58})$, but I don't know how to prove that without knowing ɛ
Problem 2: I have no idea.
Problem 3: I believe this is false, because this can be rewritten as $T(n) = 2T(\frac{4n}{3}) + n$, which would be Case 2, and therefore give me a time complexity of $\Theta(ln(n))$.
This is my first time taking anything this granular with algorithms, and despite reading CLRS and looking at videos online, I am a bit lost, and would appreciate any help. Thanks in advance. |
I am trying to find a control model for the system of a balancing robot. The purpose of this project is control $\theta_2$ by the 2 motors in the wheels i.e. through the torque $τ$ I started with the dynamic equations and went to find the transfer function.
Then I will find the PID gains that will control the robot and keep it balanced with the most optimum response. For the time being I am only interested in finding the transfer function for the dynamic model only.
Here is an example: https://www.youtube.com/watch?v=FDSh_N2yJZk
However, I am not sure of my result.Here are the free body diagrams for the wheels and the inverted pendulum (robot body) and calculations below:
Dynamic Equations:
$$ \begin{array}{lcr} m_1 \ddot{x}_1 = F_r - F_{12} & \rightarrow & (1)& \\ m_2 \ddot{x}_2 = F_{12} & \rightarrow & (2) &\\ J_1 \ddot{\theta}_1 = F_r r - \tau & \rightarrow & (3) &\\ J_2 \ddot{\theta}_2 = \tau - mgl\theta & \rightarrow & (4) & \mbox{(linearized pendulum)}\\ \end{array} $$
Kinematics:
$$ x_1 = r\theta_1 \\ x_2 = r\theta_1 + l\theta_2 \\ $$
Equating (1) and (3): $$ m_1 \ddot{x}_1 + F_{12} = F_r \\ \frac{J_1 \ddot{\theta}_1}{r} + \frac{\tau}{r} = F_r $$
Yields:
$$ \frac{J_1 \ddot{\theta}_1}{r} - m_1 \ddot{x}_1 + \frac{\tau}{r} = F_{12} \rightarrow (5) $$
Equating (5) with (2):
$$ \frac{J_1 \ddot{\theta}_1}{r} - m_1 \ddot{x}_1 + \frac{\tau}{r} - m_2 \ddot{x}_2 = 0 \rightarrow (6) \\ $$
Using Kinematic equations on (6):
$$ (J_1 - m_1 r^2 - m_2 r^2) \ddot{\theta}_1 + m_2 l r \ddot{\theta}_2 = -\tau \rightarrow (7) \\ $$
Equating (7) with (4):
$$ \begin{array}{ccc} \underbrace{(J_1 - m_1 r^2 - m_2 r^2) }\ddot{\theta}_1 &+& \underbrace{(m_2 l r + J_2 ) }\ddot{\theta}_2 &+& \underbrace{m_2 gl}\theta &= 0 \rightarrow (8) \\ A & &B & & C & \\ \end{array} $$
Using Laplace transform and finding the transfer function:
$$ \frac{\theta_1}{\theta_2} = -\frac{Bs^2 + C}{As^2} \\ $$
Substituting transfer function into equation (7):
$$ (J_1 - m_1 r^2 - m_2 r^2) \frac{\theta_1}{\theta_2}\theta_2 s^2 + m_2 lr\theta_2 s^2 = -\tau \\ $$
Yields: $$ \frac{θ_2}{τ} = \frac{-1}{(mlr-B) s^2+C} $$
Simplifying: $$ \frac{θ_2}{τ}= \frac{1}{J_2 s^2-m_2 gl} $$
Comments:
-This only expresses the pendulum without the wheel i.e. dependent only on the pendulums properties.
-Poles are real and does verify instability. |
Difference between revisions of "Kakeya problem"
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:<math>k_n\gtrsim 3^{(n+1)/2}.</math>
:<math>k_n\gtrsim 3^{(n+1)/2}.</math>
−
One can derive essentially the same conclusion using the "bush" argument, as follows. Let <math>E\subset{\mathbb F}_3^n</math> be a Kakeya set, considered as a union of <math>N := (3^n-1)/2</math> lines in all different directions. Let <math>\mu</math> be the largest number of lines that are concurrent at a point of <math>E</math>. The number of point-line incidences is at most <math>|E|\mu</math> and at least <math>3N</math>, whence <math>|E|\ge 3N/\mu</math>. On the other hand, by considering only those points on the "bush" of lines emanating from a point with multiplicity <math>\mu</math>, we see that <math>|E|\ge 2\mu+1</math>.
+
One can derive essentially the same conclusion using the "bush" argument, as follows. Let <math>E\subset{\mathbb F}_3^n</math> be a Kakeya set, considered as a union of <math>N := (3^n-1)/2</math> lines in all different directions. Let <math>\mu</math> be the largest number of lines that are concurrent at a point of <math>E</math>. The number of point-line incidences is at most <math>|E|\mu</math> and at least <math>3N</math>, whence <math>|E|\ge 3N/\mu</math>. On the other hand, by considering only those points on the "bush" of lines emanating from a point with multiplicity <math>\mu</math>, we see that <math>|E|\ge 2\mu+1</math>. one obtains
<math>|E|\gtrsim\sqrt{6N} \approx 3^{(n+1)/2}</math>.
<math>|E|\gtrsim\sqrt{6N} \approx 3^{(n+1)/2}</math>.
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since the set of all vectors in <math>{\mathbb F}_3^n</math> such that at least one of the numbers <math>1</math> and <math>2</math> is missing among their coordinates is a Kakeya set.
since the set of all vectors in <math>{\mathbb F}_3^n</math> such that at least one of the numbers <math>1</math> and <math>2</math> is missing among their coordinates is a Kakeya set.
−
This estimate can be improved using an idea due to Ruzsa. Namely, let <math>E:=A\cup B</math>, where <math>A</math> is the set of all those vectors with <math>r/3+O(\sqrt r)</math> coordinates equal to <math>1</math> and the rest equal to <math>0</math>, and <math>B</math> is the set of all those vectors with <math>2r/3+O(\sqrt r)</math> coordinates equal to <math>2</math> and the rest equal to <math>0</math>. Then <math>E</math>, being of size just about <math>(27/4)^{r/3}</math> (which is not difficult to verify using [[Stirling's formula]]) contains lines in positive proportion of directions. Now one can use the random rotations trick to get the rest of the directions in <math>E</math> (losing a polynomial factor in <math>n</math>).
+
This estimate can be improved using an idea due to Ruzsa. Namely, let <math>E:=A\cup B</math>, where <math>A</math> is the set of all those vectors with <math>r/3+O(\sqrt r)</math> coordinates equal to <math>1</math> and the rest equal to <math>0</math>, and <math>B</math> is the set of all those vectors with <math>2r/3+O(\sqrt r)</math> coordinates equal to <math>2</math> and the rest equal to <math>0</math>. Then <math>E</math>, being of size just about <math>(27/4)^{r/3}</math> (which is not difficult to verify using [[Stirling's formula]])contains lines in positive proportion of directions. Now one can use the random rotations trick to get the rest of the directions in <math>E</math> (losing a polynomial factor in <math>n</math>).
Putting all this together, we seem to have
Putting all this together, we seem to have
Revision as of 04:46, 19 March 2009
Define a
Kakeya set to be a subset [math]A\subset{\mathbb F}_3^n[/math] that contains an algebraic line in every direction; that is, for every [math]d\in{\mathbb F}_3^n[/math], there exists [math]a\in{\mathbb F}_3^n[/math] such that [math]a,a+d,a+2d[/math] all lie in [math]A[/math]. Let [math]k_n[/math] be the smallest size of a Kakeya set in [math]{\mathbb F}_3^n[/math].
Clearly, we have [math]k_1=3[/math], and it is easy to see that [math]k_2=7[/math]. Using a computer, it is not difficult to find that [math]k_3=13[/math] and [math]k_4\le 27[/math]. Indeed, it seems likely that [math]k_4=27[/math] holds, meaning that in [math]{\mathbb F}_3^4[/math] one cannot get away with just [math]26[/math] elements.
General lower bounds
Trivially,
[math]k_n\le k_{n+1}\le 3k_n[/math].
Since the Cartesian product of two Kakeya sets is another Kakeya set, we have
[math]k_{n+m} \leq k_m k_n[/math];
this implies that [math]k_n^{1/n}[/math] converges to a limit as [math]n[/math] goes to infinity.
From a paper of Dvir, Kopparty, Saraf, and Sudan it follows that [math]k_n \geq 3^n / 2^n[/math], but this is superseded by the estimates given below.
To each of the [math](3^n-1)/2[/math] directions in [math]{\mathbb F}_3^n[/math] there correspond at least three pairs of elements in a Kakeya set, etermining this direction. Therefore, [math]\binom{k_n}{2}\ge 3\cdot(3^n-1)/2[/math], and hence
[math]k_n\gtrsim 3^{(n+1)/2}.[/math]
One can derive essentially the same conclusion using the "bush" argument, as follows. Let [math]E\subset{\mathbb F}_3^n[/math] be a Kakeya set, considered as a union of [math]N := (3^n-1)/2[/math] lines in all different directions. Let [math]\mu[/math] be the largest number of lines that are concurrent at a point of [math]E[/math]. The number of point-line incidences is at most [math]|E|\mu[/math] and at least [math]3N[/math], whence [math]|E|\ge 3N/\mu[/math]. On the other hand, by considering only those points on the "bush" of lines emanating from a point with multiplicity [math]\mu[/math], we see that [math]|E|\ge 2\mu+1[/math]. Comparing the two last bounds one obtains [math]|E|\gtrsim\sqrt{6N} \approx 3^{(n+1)/2}[/math].
A better bound follows by using the "slices argument". Let [math]A,B,C\subset{\mathbb F}_3^{n-1}[/math] be the three slices of a Kakeya set [math]E\subset{\mathbb F}_3^n[/math]. Form a bipartite graph [math]G[/math] with the partite sets [math]A[/math] and [math]B[/math] by connecting [math]a[/math] and [math]b[/math] by an edge if there is a line in [math]E[/math] through [math]a[/math] and [math]b[/math]. The restricted sumset [math]\{a+b\colon (a,b)\in G\}[/math] is contained in the set [math]-C[/math], while the difference set [math]\{a-b\colon (a-b)\in G\}[/math] is all of [math]{\mathbb F}_3^{n-1}[/math]. Using an estimate from a paper of Katz-Tao, we conclude that [math]3^{n-1}\le\max(|A|,|B|,|C|)^{11/6}[/math], leading to [math]|E|\ge 3^{6(n-1)/11}[/math]. Thus,
[math]k_n \ge 3^{6(n-1)/11}.[/math] General upper bounds
We have
[math]k_n\le 2^{n+1}-1[/math]
since the set of all vectors in [math]{\mathbb F}_3^n[/math] such that at least one of the numbers [math]1[/math] and [math]2[/math] is missing among their coordinates is a Kakeya set.
This estimate can be improved using an idea due to Ruzsa. Namely, let [math]E:=A\cup B[/math], where [math]A[/math] is the set of all those vectors with [math]r/3+O(\sqrt r)[/math] coordinates equal to [math]1[/math] and the rest equal to [math]0[/math], and [math]B[/math] is the set of all those vectors with [math]2r/3+O(\sqrt r)[/math] coordinates equal to [math]2[/math] and the rest equal to [math]0[/math]. Then [math]E[/math], being of size just about [math](27/4)^{r/3}[/math] (which is not difficult to verify using Stirling's formula), contains lines in a positive proportion of directions. Now one can use the random rotations trick to get the rest of the directions in [math]E[/math] (losing a polynomial factor in [math]n[/math]).
Putting all this together, we seem to have
[math](3^{6/11} + o(1))^n \le k_n \le ( (27/4)^{1/3} + o(1))^n[/math]
or
[math](1.8207\ldots+o(1))^n \le k_n \le (1.88988+o(1))^n.[/math] |
Difference between revisions of "De Bruijn-Newman constant"
(→t>0)
Line 57: Line 57:
In [KKL2009, Theorem 1.4], it is shown that for any fixed <math>t>0</math>, the number <math>N_t(T)</math> of zeroes of <math>H_t</math> with real part between 0 and T obeys the asymptotic
In [KKL2009, Theorem 1.4], it is shown that for any fixed <math>t>0</math>, the number <math>N_t(T)</math> of zeroes of <math>H_t</math> with real part between 0 and T obeys the asymptotic
−
:<math>N_t(T) = \frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + t \log T + O(1) </math>
+
:<math>N_t(T) = \frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + t\log T + O(1) </math>
as <math>T \to \infty</math> (caution: the error term here is not uniform in t). Also, the zeroes behave like an arithmetic progression in the sense that
as <math>T \to \infty</math> (caution: the error term here is not uniform in t). Also, the zeroes behave like an arithmetic progression in the sense that
Revision as of 16:45, 27 January 2018
For each real number [math]t[/math], define the entire function [math]H_t: {\mathbf C} \to {\mathbf C}[/math] by the formula
[math]\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du[/math]
where [math]\Phi[/math] is the super-exponentially decaying function
[math]\displaystyle \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3 \pi n^2 e^{5u}) \exp(-\pi n^2 e^{4u}).[/math]
It is known that [math]\Phi[/math] is even, and that [math]H_t[/math] is even, real on the real axis, and obeys the functional equation [math]H_t(\overline{z}) = \overline{H_t(z)}[/math]. In particular, the zeroes of [math]H_t[/math] are symmetric about both the real and imaginary axes. One can also express [math]H_t[/math] in a number of different forms, such as
[math]\displaystyle H_t(z) = \frac{1}{2} \int_{\bf R} e^{tu^2} \Phi(u) e^{izu}\ du[/math]
or
[math]\displaystyle H_t(z) = \frac{1}{2} \int_0^\infty e^{t\log^2 x} \Phi(\log x) e^{iz \log x}\ \frac{dx}{x}.[/math]
In the notation of [KKL2009], one has
[math]\displaystyle H_t(z) = \frac{1}{8} \Xi_{t/4}(z/2).[/math]
De Bruijn [B1950] and Newman [N1976] showed that there existed a constant, the
de Bruijn-Newman constant [math]\Lambda[/math], such that [math]H_t[/math] has all zeroes real precisely when [math]t \geq \Lambda[/math]. The Riemann hypothesis is equivalent to the claim that [math]\Lambda \leq 0[/math]. Currently it is known that [math]0 \leq \Lambda \lt 1/2[/math] (lower bound in [RT2018], upper bound in [KKL2009]). Contents [math]t=0[/math]
When [math]t=0[/math], one has
[math]\displaystyle H_0(z) = \frac{1}{8} \xi( \frac{1}{2} + \frac{iz}{2} ) [/math]
where
[math]\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{s/2} \Gamma(s/2) \zeta(s)[/math]
is the Riemann xi function. In particular, [math]z[/math] is a zero of [math]H_0[/math] if and only if [math]\frac{1}{2} + \frac{iz}{2}[/math] is a non-trivial zero of the Riemann zeta function. Thus, for instance, the Riemann hypothesis is equivalent to all the zeroes of [math]H_0[/math] being real, and Riemann-von Mangoldt formula (in the explicit form given by Backlund) gives
[math]\displaystyle N_0(T) - (\frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} - \frac{7}{8})| \lt 0.137 \log (T/2) + 0.443 \log\log(T/2) + 4.350 [/math]
for any [math]T \gt 4[/math], where [math]N_0(T)[/math] denotes the number of zeroes of [math]H_0[/math] with real part between 0 and T.
The first [math]10^{13}[/math] zeroes of [math]H_0[/math] (to the right of the origin) are real [G2004]. This numerical computation uses the Odlyzko-Schonhage algorithm.
[math]t\gt0[/math]
For any [math]t\gt0[/math], it is known that all but finitely many of the zeroes of [math]H_t[/math] are real and simple [KKL2009, Theorem 1.3]. In fact, assuming the Riemann hypothesis,
all of the zeroes of [math]H_t[/math] are real and simple [CSV1994, Corollary 2].
Let [math]\sigma_{max}(t)[/math] denote the largest imaginary part of a zero of [math]H_t[/math], thus [math]\sigma_{max}(t)=0[/math] if and only if [math]t \geq \Lambda[/math]. It is known that the quantity [math]\frac{1}{2} \sigma_{max}(t)^2 + t[/math] is non-decreasing in time whenever [math]\sigma_{max}(t)\gt0[/math] (see [KKL2009, Proposition A]. In particular we have
[math]\displaystyle \Lambda \leq t + \frac{1}{2} \sigma_{max}(t)^2[/math]
for any [math]t[/math].
The zeroes [math]z_j(t)[/math] of [math]H_t[/math] (formally, at least) obey the system of ODE
[math]\partial_t z_j(t) = - \sum_{k \neq j} \frac{2}{z_k(t) - z_j(t)}[/math]
where the sum may have to be interpreted in a principal value sense. (See for instance [CSV1994, Lemma 2.4]. This lemma assumes that [math]t \gt \Lambda[/math], but it is likely that one can extend to other [math]t \geq 0[/math] as well.)
In [KKL2009, Theorem 1.4], it is shown that for any fixed [math]t\gt0[/math], the number [math]N_t(T)[/math] of zeroes of [math]H_t[/math] with real part between 0 and T obeys the asymptotic
[math]N_t(T) = \frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + \frac{t}{16} \log T + O(1) [/math]
as [math]T \to \infty[/math] (caution: the error term here is not uniform in t). Also, the zeroes behave like an arithmetic progression in the sense that
[math] z_{k+1}(t) - z_k(t) = (1+o(1)) \frac{4\pi}{\log |z_k|(t)} = (1+o(1)) \frac{4\pi}{\log k} [/math]
as [math]k \to +\infty[/math].
Threads Polymath proposal: upper bounding the de Bruijn-Newman constant, Terence Tao, Jan 24, 2018. Other blog posts and online discussion Heat flow and zeroes of polynomials, Terence Tao, Oct 17, 2017. The de Bruijn-Newman constant is non-negative, Terence Tao, Jan 19, 2018. Lehmer pairs and GUE, Terence Tao, Jan 20, 2018. A new polymath proposal (related to the Riemann hypothesis) over Tao's blog, Gil Kalai, Jan 26, 2018. Code and data Wikipedia and other references Bibliography [B1950] N. C. de Bruijn, The roots of trigonometric integrals, Duke J. Math. 17 (1950), 197–226. [CSV1994] G. Csordas, W. Smith, R. S. Varga, Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis, Constr. Approx. 10 (1994), no. 1, 107–129. [G2004] Gourdon, Xavier (2004), The [math]10^{13}[/math] first zeros of the Riemann Zeta function, and zeros computation at very large height [KKL2009] H. Ki, Y. O. Kim, and J. Lee, On the de Bruijn-Newman constant, Advances in Mathematics, 22 (2009), 281–306. Citeseer [N1976] C. M. Newman, Fourier transforms with only real zeroes, Proc. Amer. Math. Soc. 61 (1976), 246–251. [RT2018] B. Rodgers, T. Tao, The de Bruijn-Newman constant is non-negative, preprint. arXiv:1801.05914 |
I have figured it out.
The way I included degeneracy was correct but I made some subtle mistakes during substitution of some parameters.
The formula for $\ln(\Omega)$ when degeneracy $g_j$ is taken into account is:$$\ln(\Omega)= N \cdot \ln(N) - N - \sum^n_{j=1}[n_j \cdot \ln(n_j) - n_j] + \sum^n_{j=1} [\ln(g_j) \cdot n_j]$$Substituting $n_j = g_j \cdot \frac{N}{Z} \cdot e^{\beta E_j}$ (I erroneously left out the $g_j$ during this substitution) along with rewriting, splitting the summations and simplifying eventually gives me:$$\ln(\Omega) = N \cdot \ln(Z) - \beta U$$Which is the exact equation as when degeneracy is
not taken into account, and thus I get the same value for $\beta$ when taking the derivative of $S = k_B \cdot \ln(\Omega)$ and putting it next to the equation of entropy $dS = \frac{dU}{T}$. Details Derivation
Formula for $\ln(\Omega)$ when taking degeneracy into account$$\ln(\Omega)= N \cdot \ln(N) - N - \sum^n_{j=1}[n_j \cdot \ln(n_j) - n_j] + \sum^n_{j=1} [\ln(g_j) \cdot n_j]$$
According to Boltzmann Statistics $n_j = g_j \frac{N}{Z} e^{\beta E_j}$. Furthermore, $\ln(n_j) = \ln\big(g_j \frac{N}{Z}\big) + \beta E_j$. Substituting these parameters:
$$\ln(\Omega)= N \cdot \ln(N) - N - \sum^n_{j=1}\bigg[g_j \frac{N}{Z} e^{\beta E_j}\cdot \bigg(\ln\big(g_j \frac{N}{Z}\big) + \beta E_j\bigg) - g_j \frac{N}{Z} e^{\beta E_j}\bigg] + \sum^n_{j=1} \bigg[\ln(g_j) \cdot g_j \frac{N}{Z} e^{\beta E_j}\bigg]$$
Splitting the 1st summation into 3 summations between the + and – signs and removing the brackets that appear after splitting:$$\ln(\Omega)= N \cdot \ln(N) - N - \sum^n_{j=1}\bigg[g_j \frac{N}{Z} e^{\beta E_j}\cdot \ln\big(g_j \frac{N}{Z}\big)\bigg] - \beta\sum^n_{j=1}\bigg[g_j \frac{N}{Z} e^{\beta E_j} \cdot E_j)\bigg] + \sum^n_{j=1}\bigg[g_j \frac{N}{Z} e^{\beta E_j}\bigg] + \sum^n_{j=1} \bigg[\ln(g_j) \cdot g_j \frac{N}{Z} e^{\beta E_j}\bigg]$$The 2nd summation is equal to the total energy $U$, and the 3rd summation is equal to the total number of particles $N$ which cancels the $-N$ term.$$\ln(\Omega)= N \cdot \ln(N) - \sum^n_{j=1}\bigg[g_j \frac{N}{Z} e^{\beta E_j}\cdot \ln\big(g_j \frac{N}{Z}\big)\bigg] - \beta U + \sum^n_{j=1} \bigg[\ln(g_j) \cdot g_j \frac{N}{Z} e^{\beta E_j}\bigg]$$In the 1st summation term, substituting $\ln\bigg(g_j \frac{N}{Z}\bigg) = \ln(g_j) + \ln\big(\frac{N}{Z}\big)$ and then splitting that summation between the newly created + sign as well and removing the appearing brackets:$$\ln(\Omega)= N \cdot \ln(N) - \sum^n_{j=1}\bigg[g_j \frac{N}{Z} e^{\beta E_j}\cdot \ln(g_j)\bigg] - \ln\big(\frac{N}{Z}\big) \sum^n_{j=1}\bigg[g_j \frac{N}{Z} e^{\beta E_j}\bigg] -\beta U + \sum^n_{j=1} \bigg[\ln(g_j) \cdot g_j \frac{N}{Z} e^{\beta E_j}\bigg]$$The first and 3rd summations cancel each other out. The 2nd summation is equal to $N$, giving:$$\ln(\Omega)= N \bigg(\ln(N) - \ln\big(\frac{N}{Z}\big)\bigg) - \beta U$$Since, $\ln(N) - \ln\big(\frac{N}{Z}\big) = \ln(Z)$, this gives:$$\ln(\Omega) = N\cdot \ln(Z) - \beta U$$The $Z$ and $U$ are functions of $\beta$. Equation for entropy is $S = k_B \cdot \ln(\Omega)$. Thus deriving $\frac{dS}{d\beta}$:$$\frac{dS}{d\beta} = k_B \bigg(\frac{N}{Z} \cdot \frac{dZ}{d\beta} - \bigg(U + \beta \frac{dU}{d\beta}\bigg)\bigg)$$Since $\frac{dZ}{d\beta} = \frac{UZ}{N}$ this gives:$$\frac{dS}{d\beta} = -k_B \cdot \beta \frac{dU}{d\beta}$$Knowing that $dS = \frac{dU}{T}$ (when a fixed volume is assumed), this yields:$$\beta = - \frac{1}{k_B T}$$ |
Newspace parameters
Level: \( N \) = \( 128 = 2^{7} \) Weight: \( k \) = \( 2 \) Character orbit: \([\chi]\) = 128.b (of order \(2\) and degree \(1\)) Newform invariants
Self dual: No Analytic conductor: \(1.02208514587\) Analytic rank: \(0\) Dimension: \(2\) Coefficient field: \(\Q(\sqrt{-2}) \) Coefficient ring: \(\Z[a_1, a_2, a_3]\) Coefficient ring index: \( 2 \) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.
Character Values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).
\(n\) \(5\) \(127\) \(\chi(n)\) \(-1\) \(1\)
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \) 65.1
0 − 2.82843 i 0 0 0 0 0 −5.00000 0 65.2 0 2.82843 i 0 0 0 0 0 −5.00000 0
Char. orbit Parity Mult. Self Twist Proved 1.a Even 1 trivial yes 8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes 4.b Odd 1 yes 8.b Even 1 yes
This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut +\mathstrut 8 \) acting on \(S_{2}^{\mathrm{new}}(128, [\chi])\). |
Given corr(A,B) and corr(A,C) you can obtain bounds on corr(B,C) (and similar such calculations involving more variables), but the bounds are in general quite wide. Indeed, typically such calculations aren't very informative at all.
Specifically, by looking at the relationship between the ordinary pairwise correlation and the partial correlation:
$$\rho_{BC\cdot A } = \frac{\rho_{BC} - \rho_{AB}\rho_{AC}} {\sqrt{1-\rho_{AB}^2} \sqrt{1-\rho_{AC}^2}}$$
you can rearrange the formula to back out bounds for $\rho_{BC}$:
$$\rho_{BC}=\rho_{AB}\rho_{AC}+\rho_{BC\cdot A } {\sqrt{1-\rho_{AB}^2} \sqrt{1-\rho_{AC}^2}}$$
and noting that the partial correlation must lie between -1 and 1, this implies that $\rho_{BC}$ is bounded to lie in
$$\rho_{AB}\rho_{AC}\pm {\sqrt{1-\rho_{AB}^2} \sqrt{1-\rho_{AC}^2}}\,.$$
e.g. Let's say $\rho_{AB}=0.8$ and $\rho_{AC}=0.6$.
Then $\rho_{BC}= 0.6 \times 0.8 \pm \sqrt{(1-.64)(1-.36)}=0.48\pm 0.48 = (0,0.96)$
With more variables the situation becomes more complex; in some situations it's easier to work with Cholesky decompositions.
If you impose additional structure on the problem then in some situations those bounds might reduce.
Additional details may help. |
This is a tricky problem. Can anyone help me with the procedure and answer?
Evaluate $$ \lim_{h\to 0} \left( \frac{f(x+hx)}{f(x)}\right)^{1/h}, \text{for }f(x)=x. $$
Note that you are given $f(x)=x$. With this piece of information, we can replace the functions in the given equation with their respective algebraic representations. For instance, the denominator would simply be $x$. What would the numerator look like? Now, as $h\rightarrow 0$, consider what the value inside the parenthesis tends towards, and similarly for the exponent.
Hint Taking the logarithm, you have to calculate$$\lim_{h\to 0} \frac{\ln(f(x+hx))- \ln(f(x))}{h}$$
That is the definition of the derivative of $\ln( f(x))$. |
A) \[3-i\sqrt{6},\pm \sqrt{\frac{3}{2}}\]
B) \[3-i\sqrt{6},\pm \frac{3}{\sqrt{2}}\]
C) \[3-i\sqrt{6},\pm \frac{\sqrt{3}}{2}\]
D) None of these
Correct Answer:
C
\[{{x}^{2}}-6x+15=0\]is quadratic corresponding to roots \[3\pm i\sqrt{6}\] and dividing the given equation by this, we get \[4{{x}^{2}}-3=0\]Þ \[x=\pm \frac{\sqrt{3}}{2}\].
Solution :
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Consider $U$ a open and bounded subset of $R^n$, with smooth boundary. A weak solution for the problem :
$$ \Delta^2 u = f \ \in \Omega \ and \ u=\frac{\partial u}{\partial\nu} = 0 \text{ in } \partial U$$
where $f \in L^{2}(U)$ is a function(say u )with $u \in H^{2}_{0}(U)$ with
$$ \int_{U} \Delta u \Delta v dx = \int_{U}fv dx, \forall \ v \ \in H^{2}_{0}(U)$$.
I believe this definition of weak solution is obtained taking a smooth function $u$ satisfing pointwise the two conditions of the problem and before integrate by parts the first equation and using the second condition of the problem I obtain the integral identity of the weak solution for test functions. Am I right ? I trying to do this calculus but i am getting anywhere...
Someone can help me ?
Thanks in advance |
I'm working through Jeevanjee's "An Introduction to Tensors and Group Theory for Physicists", and while trying to prove that the matrix representation $M(A^\dagger)$ of a Hermitian adjoint $A^\dagger$ is the conjugate-transpose of the matrix $M(A)$, I cannot figure out how to move past the gap between dual vectors and so-called 'metric dual' vectors.
Given an orthonormal basis $\{e_i\}_{i=1,\ldots,n}$ of vector space $V$, he defines a
dual basis $\{e^i\}_{i=1,\ldots,n} \in V^*$ such that $e^i(e_j)=\delta_{ij}$.
He then defines the
metric dual of a vector $v \in V$ as $\tilde{v} \in V^*$ with $\tilde{v}(w) = (v|w)$, where $(\cdot|\cdot)$ is a non-degenerate Hermitian form and $w \in V$.
In proofs of $M(A^\dagger)=\overline{M(A)}^T$, it seems necessary to use the fact that $(\cdot|\cdot)$ is Hermitian (i.e. $(v|w)=\overline{(w|v)}$), but I don't see how to move from $M(A^\dagger)_j^i = e^i(A^\dagger e_j)$ to $M(A^\dagger)_j^i \stackrel{?}{=} \tilde{e_i}(A^\dagger e_j) = (e_i| A^\dagger e_j)$ in the general case where it the metric is not necessarily Euclidean. That is to say, $(e_i|e_j)$ is not necessarily $\delta_{ij}$, and equivalently $\tilde{e_i}$ is not necessarily $e^i$.
Of course, since the map $e_i \mapsto \tilde{e_i}$ is bijective, $\{\tilde{e_i}\}_{i=1,\ldots,n}$ is a basis of $V^*$, and we can write $e^i = \sum_k a_{ik} \tilde{e_k}$ for scalars $a_{ik}$, giving
$$\begin{align} M(A^\dagger)_j^i &= e^i(A^\dagger e_j) \\ &= \sum_k a_{ik} \tilde{e_k} (A^\dagger e_j) \\ &= \sum_k a_{ik} (e_k | A^\dagger e_j) \\ &= \sum_k a_{ik} (A^\dagger e_j | e_k )^* \\ &= \sum_k a_{ik} (e_j | A e_k )^*. \end{align}$$
Then taking the dual of the dual to be itself, $\sum_k a_{ik} e_k = \tilde{e^i}$ and $$M(A^\dagger)_j^i = (e_j|A\tilde{e^i})^*.$$
But is not $\tilde{e^i} = e_i$ true only for the case of the Euclidean metric where $(e_i|e_j) = \delta_{ij}$? |
In plasmas, the collision rate among ions or electrons is much larger than the collision rate between ions and electrons. Why is that so?
There are lots of different types of plasmas.
In a thermal plasma the electrons and ions will have the same temperature.
In a non-thermal plasma the discharge is driven by some external power supply e.g. capacitatively coupled RF, inductively coupled, pulsed DC E field etc.
In a non-thermal plasma the electrons generally have a higher temperature than the ions because the energy from the RF or E field couples with the electrons more efficiently. The electrons transfer energy to the gas to sustain the plasma.
Strictly speaking non-thermal plasmas are not at equilibrium and we cannot necessarily define a temperature, but temperature is generally a useful concept to use.
Collision rates are generally lower for electrons than ions, but not always (e.g. very low energy electrons and SF6 has huge collision rate).
The reason for the temperature difference is due partly to the driving energy being mostly coupled to the electrons and the partly because the energy is not rapidly transfered by collisions to the neutral gas and ions.
Definitions
Let us first start with some definitions of parameters, in no particular order. I will be describing elastic collisions, assuming a quasi-neutral (i.e., $n_{e} = \sum_{s} \ n_{s} \ Z_{s}$) plasma. Thus, the collisions involve long-range forces and are called Coulomb collisions.
Constants Particle/Plasma Paramters $B_{o}$ = the magnitude of the quasi-static magnetic field $s$ = the letter used to identify particle species (e.g., $e$ for electrons, $i$ for ions) $n_{s}$ = the particle number density of species $s$ $q_{s} = e \ Z_{s}$, or the charge of species $s$ $m_{s}$ = the mass of species $s$ $T_{s}$ = the average temperature of species $s$ (in a non-equilibrium ionized gas like a plasma, I am referring to expression shown at https://physics.stackexchange.com/a/218643/59023) $\lambda_{De} = \sqrt{ \tfrac{ \varepsilon_{o} \ k_{B} \ T_{e} }{ n_{e} \ q_{e}^{2} } }$, or the electron Debye length $V_{Ts} = \sqrt{ \tfrac{ 2 \ k_{B} \ T_{s} }{ m_{s} } }$, or the thermal speed of species $s$ ( Note:it's specifically the most probable speedhere) $\omega_{ps} = \sqrt{ \tfrac{ n_{s} \ q_{s}^{2} }{ \varepsilon_{o} \ m_{s} } }$, or the plasma frequency of species $s$ $\Omega_{ps} = \tfrac{ q_{s} \ B_{o} }{ \gamma \ m_{s} }$, or the gyrofrequency (or cyclotron frequency) of species $s$, and $\gamma$ is the relativistic Lorentz factor Collision-Specific Parameters $\mu_{ss'} = \tfrac{ m_{s} \ m_{s'} }{ m_{s} + m_{s'} }$, or the reduced mass of species $s$ and $s'$ $\lambda_{D} = \lambda_{De} \ \sqrt{ \sum_{s} \left( \tfrac{ n_{s} \ Z_{s}^{2} }{ n_{e} } \right) \ \left( \tfrac{ T_{e} }{ T_{s} } \right) }$, or a generalized Debye length $V_{Tss'}^{2} = V_{Ts}^{2} + V_{Ts'}^{2}$, or the average thermal speed of the species $s$ and $s'$ $b_{min} \simeq \tfrac{ q_{s} \ q_{s'} }{ \left( 4 \ \pi \ \varepsilon_{o} \right) \ \mu_{ss'} \ V_{Tss'}^{2} }$, or the minimum impact parameter, where we have assumed the classical limit $\Lambda \simeq \tfrac{ 12 \ \pi \ n_{e} \ \lambda_{De}^{3} }{ Z_{i} }$, which is the plasma parameter (or Coulomb logarithm) $\Lambda_{ss'}$ = the generalized impact parameter integral between test particle species $s$ and moving background species $s'$ Coulomb Collisions
If we assume only $90^{\circ}$ scattering (i.e., no small-angle collisions), then one can define a
reference collision frequency between test particle species $s$ and moving background species $s'$, given by:$$\nu_{ o,ss' }\left( v \right) \simeq \frac{ 4 \ \pi \ n_{s} \ q_{s}^{2} \ q_{s'}^{2} }{ \left( 4 \ \pi \ \varepsilon_{o} \right)^{2} \ m_{s}^{2} \ v^{3} } \ \ln \Lambda_{ss'} \tag{1}$$where $v$ is the speed of moving background species $s'$ relative to test particle species $s$ and $\Lambda_{ss'}$ is given by:$$\begin{align} \ln \Lambda_{ss'} & = \int_{b_{min}}^{b_{max}} \ \frac{ db }{ b } = \ln \left( \frac{ \lambda_{D} }{ b_{min} } \right) \tag{2a} \\ \Lambda_{ss'} & \simeq \frac{ \left( 4 \pi \varepsilon_{o} \right) \mu_{ss'} \ V_{Tss'}^{2} }{ \sqrt{2} \ Z_{s} \ Z_{s'} \ e^{2} } \left[ \left( \frac{ \omega_{ps} }{ V_{Ts} } \right)^{2} + \left( \frac{ \omega_{ps'} }{ V_{Ts'} } \right)^{2} \right]^{-1/2} \tag{2b}\end{align}$$where Equation 2b is the classical limit of the last expression in Equation 2a.
The collision rate in Equation 1 is more often expressed as Maxwellian-averages in the Lorentz collision model for electron-proton, proton-proton, and electron-electron collisions [e.g., see
Wilson et al., 2018 and references therein], given by:$$\begin{align} \nu_{ep} & = \frac{ 2 \ \sqrt{ 4 \pi } \ n_{p} \ e^{4} }{ 3 \left( 4 \pi \varepsilon_{o} \right)^{2} \ \mu_{ep}^{2} \ V_{Tep}^{3} } \ \ln \Lambda_{ep} \tag{3a} \\ \nu_{pp} & = \frac{ 4 \ \sqrt{ \pi } \ n_{p} \ e^{4} }{ 3 \left( 4 \pi \varepsilon_{o} \right)^{2} \ m_{p}^{2} \ V_{Tp}^{3} } \ \ln \Lambda_{pp} \tag{3b} \\ \nu_{ee} & = \frac{ 4 \ \sqrt{ \pi } \ n_{e} \ e^{4} }{ 3 \left( 4 \pi \varepsilon_{o} \right)^{2} \ m_{e}^{2} \ V_{Te}^{3} } \ \ln \Lambda_{ee} \tag{3c}\end{align}$$ Answers
In plasma, collision rate among ions or electrons is much larger than the collision rate between ions and electrons...
No, I think you have it backwards. If the only
collisions are $90^{\circ}$ Coulomb collisions, then the collision rate between electrons and ions is higher than both the single species collision rates, as can be seen from Equations 3a--3c above.
...why is that so?
This can be understood by looking at the expression for $b_{min}$. For electron-proton collisions with $T_{e} \sim T_{i}$, then we have $\mu_{ei} \sim m_{e} \ll m_{i}$ and $V_{Tei} \sim V_{Te} \gg V_{Ti}$, which results in $b_{min,ei} \rightarrow \tfrac{ e^{2} }{ \left( 4 \ \pi \ \varepsilon_{o} \right) \ m_{e} \ V_{Te}^{2} } \sim b_{min,ee}$. For proton-proton collisions, $\mu_{ei} \sim m_{i}$ and $V_{Tei} \sim V_{Ti}$, which results in $b_{min,ii} \rightarrow \tfrac{ e^{2} }{ \left( 4 \ \pi \ \varepsilon_{o} \right) \ m_{i} \ V_{Ti}^{2} } \sim \left( \tfrac{ T_{e} }{ T_{i} } \right) b_{min,ei}$.
The larger(smaller) the minimum impact parameter, the smaller(larger) the $\Lambda_{ss'}$ contribution to the collision rate. The ions are also much slower on average than the electrons. This means that the probability of interacting with another ion in a given unit time is smaller as well, since it takes them longer to move the same distance as the much lighter electrons. Thus, the collision rates are effectively controlled by the higher mobility of the electrons.
why ions and electrons are at different temperatures in plasma?
Generally the Coulomb collision rates are much slower than many other rates except for regions like the solar chromosphere or in some lab plasmas. For instance, if we use the following typical solar wind parameters $n_{e}$ ~ 6 $cm^{-3}$, $T_{e}$ ~ 15 eV, and $Z_{i} = 1$, then $\nu_{ei} \sim 8 \times 10^{-6} \# \ s^{-1}$ or an effective collision time scale, $\tau_{ei} \sim 1.25 \times 10^{5} s$. The corresponding time scale for effective collisions with a magnetically turbulent spectrum (e.g., Kolmogorov magnetic power spectrum is $\propto k^{5/3}$) is ~1-10 seconds or ~4-5 orders of magnitude faster. The effective collision rates between particles and electromagnetic waves is even faster, with up to 7 orders of magnitude more collisions per unit time than equivalent electron-ion Coulomb collisions (e.g., http://adsabs.harvard.edu/abs/2007PhRvL..99d1101W).
For the ions and electrons to equilibriate (i.e., $T_{e} \sim T_{i}$), the Coulomb collision rates would need to dominate to allow for an equipartition of energy among the various species. Most plasmas are considered
weakly collisional or collisionless, which means that the Coulomb collision rates are negligible compared to other processes that would affect the velocity distributions.
Thus, it is generally the case that $T_{e} \neq T_{i}$ in most plasmas unless there are significant collisions.
Updates
[I removed some extraneous equations from the background section and added references.]
why ions and electrons are at different temperatures in plasma?
A recent statistical study of 10 years of measurements [e.g.,
Wilson et al., 2018] found that the electron-to-proton temperature ratio is ~1.64 on average (median value is ~1.27) in the solar wind near Earth. They also noted that the effective collision rates introduced by electromagnetic and electrostatic fluctuations can be upwards of seven orders of magnitude higher than any particle-particle Coulomb collision rate. Thus, while Coulomb collisions may slowly act to try and relax a plasma to thermal equilibrium, other, faster processes can act to drive it away. References R. Hernandez and E. Marsch, J. Geophys. Res. 90, pp. 11062, doi:10.1029/JA090iA11p11062, 1985. R.W. Schunk, Planet. Space Sci. 23, pp. 437--485, doi:10.1016/0032-0633(75)90118-X, 1975. R.W. Schunk, Rev. Geophys. Space Phys. 15, pp. 429--445, doi:10.1029/RG015i004p00429, 1977. L.B. Wilson III et al., Astrophys. J. Suppl. 236(2), pp. 41, doi:10.3847/1538-4365/aab71c, 2018.
Consider a plasma that just's been formed and then left alone. Being far from equilibrium, the plasma will evolve towards an equilibrium state. At this stage, it's not very useful to characterize the plasma with a temperature because the velocity distribution would bear little resemblance to a Boltzmann distribution, or really any kind of distribution function with meaningful moments.
Let's assume that the plasma is being driven towards equilibrium by collision events. The particles will lose energy during each inelastic collision event, with some being more inelastic than others. The degree of inelasticity will depend significantly on the mass ratio: ion-ion collisions and electron-electron collisions will be inelastic compared to ion-electron collisions.
Since the $i-i$ and $e-e$ collisions are relatively inelastic, it often occurs that the ions and electrons quickly develop their own "temperatures". The ion velocity distribution rapidly approaches a Boltzmann distribution with temperature, and the electrons also rapidly approach a Boltzmann distribution with some different temperature. If you were to superpose the two, you'd get a funny-looking distribution with a tall top and wide tails, i.e. not global equilibrium.
The "centeredness" or "tailedness" (basically kurtosis) of the total distribution tells you how far from equilibrium the plasma is. The further the system is from equilibrium, the faster it will try to equilibrate via collisions. So if the temperatures are very different, the $i-e$ collision frequency will be high. If the temperatures are not too different, the $i-e$ will be lower, but still higher than the $e-e$ and $i-i$ systems that have already approximately equilibrated with themselves.
Note: In laboratory experiments, it's common to produce plasmas whose ions and electrons do not settle to a Boltzmann distribution. In that case, the kurtosis argument flies out the window. Another major complication arises if the electrons and ions have different flow speeds, where so-called "two-stream" or "bump-on-tail" instabilities become important in the equilibration physics. |
If you want to do a project in one of the following areas, I would be glad to supervise it. If you have some of your own ideas, particularly in a mathematical area, then talk to me and we'll see if its OK.
Mostly these ideas are suitable for mathematical investigation. These are particularly suitable for MEng Mathematical Computation students. Key point: YOU choose your own project (provided someone agrees to supervise it).
Relation Algebra
A relation algebra is a bit like a boolean algebra but it has extra operators designed to handle binary relations. A relation algebra comes with a constant called the
identity (standing for the equality relation), a unary operator called converse (that reverses a binary relation) and a binary operator called composition. A relation algebra is defined by a set of about 8 axioms. Although every boolean algebra can be represented as a field of sets, not every relation algebra can be represented as a field of binary relations. Possible projects in this area include the following. Design a program that generates all possible relation algebras of a given size. Design a program that tries to test whether a relation algebra is representable (in fact this is undecidable, in general). Design a program that tests whether a relation algebra has a finite representation (it is not known whether this problem is decidable or not). Consider algebras over the signature with identity, <= and * only. A finite algebra A (with an identity element, an associative * operator and a partial order) is representable if it is isomorphic to a set of binary relations where the identity is the equality relation, <= is inclusion of sets and * is composition of binary relations. Note that the base set for the binary relations might be infinite. Write a program to generate structures of this type and test whether they can be represented over finite base sets. Design an efficient program that tests whether a given set of constraints over a given relation algebra can be satisfied by points in a representation (see www.cs.ucl.ac.uk/fileadmin/UCL-CS/staff/Robin_Hirsch/Papers/matteo.pdf). There is a mistake in Lemma 15 of www.cs.ucl.ac.uk/fileadmin/UCL-CS/staff/Robin_Hirsch/Papers/matteo.pdf Write a program to construct a set of constraints over this algebra, where every triangle is consistent but there is no global solution, or prove that this cannot happen. Write a program to find the smallest possible relation algebra that is weakly representable but not actually representable (the smallest known has 7 atoms, but smaller ones might exist).
Consider the following signature for algebras of binary relations: an ordering <= to denote that one binary relation is included in another, a converse operator that returns the converse of a binary relation, a composition operator that returns the composition of two binary relations and an identity constant. A representation of an algebra in this signature is an isomorphism to a set of genuine binary relations over some base, where all the operators have their natural meaning. We do not know whether a finite representable algebra necessarily has a representation over a finite base set. This needs investigating. You might want to implement a program that searches for finite representations for given algebras.
Graph Theory
Various projects are possible here.
0. The
spectrum of a graph is the multiset of eigenvalues of the adjacency matrix (see https://en.wikipedia.org/wiki/Spectral_graph_theory). In many (but not all) cases the spectrum of a graph determines the graph uniquely, up to isomorphism. It is not known whether the spectrum determines the graph for ALMOST ALL graphs. In this project you implement an isomorphism checker (preferably a fairly efficient one) and investigate whether the spectrum determines the graph.
1. Consider the following two-player, "forth game" \Gamma(G, H) played over two directed graphs G, H. Each player has two colours: red and blue. In the initial round the first player (A) picks a set of nodes of G and colours them red, the second player (E) picks a set of nodes from H and colours them red. The next round is similar: A picks another set of nodes of G and colours them blue, E does the same in H. In subsequent rounds, A picks a set of nodes of G and a colour (red or blue) and resets the nodes of that colour to the chosen set, E does the same in H. The game goes on, possibly forever.
At each stage, the two colours partition the nodes of G into four parts: not coloured, red but not blue, blue but not red, and both red and blue; the nodes of H are similarly partitioned into four. If at any stage one of these four parts is empty in G but non-empty in H, or the other way round, then A wins and the game is over. Also, if there are two of the four parts with an edge connecting some node in the first part to a node in the second part, but no edge between the corresponding parts of H, or the other way round, then again A wins and the game is over.
If A does not win at any finite stage then E is the winner.
The problem is to find two finite non-isomorphic graphs G, H that cannot be distinguished by this game, i.e. E has a winning strategy in \Gamma(G, H) and also a winning strategy in \Gamma(H, G).
The project might involve writing a computer program to construct suitable graphs and test them to see if they solve the problem.
2. A third problem is to give bounds to the so-called Ramsey numbers (see http://en.wikipedia.org/wiki/Ramsey_theory ). There are lots of problems to do with the 0-1 laws and random graphs (ask me for more details).
3. Solve the 17x17 grid colouring problem (and win a prize of £289) http://blog.computationalcomplexity.org/2009/11/17x17-challenge-worth-28900-this-is-not.html
Definition: The n x m grid is c-colorable if there is a way to c-color the vertices of the n x m grid so that there is no rectangle with all four corners the same color.
4. The following problem would help us understand the relation between the complexity classes P and NP.
Write a program that constructs a graph G with the following properties. (i) G is partitioned into n sets V_0, V_1, ..., V_{n-1} each with three nodes. (ii) For each i<n there are no edges within V_i. (iii) There is no clique of G with n nodes (a clique is a set of nodes where all edges between these nodes are edges of G). (iv) The following algorithm produces a non-empty list of triangles (a triangle is a clique of 3 nodes)
Make a list H of all triangles of G;
While (H changes)
For each triangle T in H, for each i<n if there is no vertex v\in V_i such that every triangle from T \cup {v} belongs to H then delete T from H;
End While
Return (H);
Clustering Problem
Let G be an undirected graph. A k-clustering S_0, S_1, ..., S_{k-1} is a partition of the set of nodes of G into k disjoint sets. A positive error in a k-clustering is a pair of nodes in different clusters connected by an edge, a negative errors is a pair of nodes in the same cluster not connected by an edge. The total error of a k-clustering is the number of positive and negative errors. We aim to minimise the total error. There are two optimisation problems to consider. First, let k be fixed and find the optimal k-clustering for G. Second consider the general optimization problem where k is not fixed and you have to choose k and a k-clustering so as to minimise the total error.
Based on the MinCut theorem, I would conjecture that the 2 clustering problem can be solved in polynomial time. I conjecture that for fixed ki>2 the optimization problem is NP-complete, also that the general optimization problem is NP-complete. You might attempt to prove some of these conjectures. Also, you might construct algorithms that find near optimal clusterings.
Cannibals
Have a look at this paper about a game called Cannibals. Implement this game and consider playing the game over other graphs (not necessarily just a grid as in the paper). Try to work out good intelligent strategies for such games.
Music Harmoniser
Design and implement a program to create harmonies for a tune. Various projects could be possible. Design "fugue completers" to automatically compute a four part fugue.
Temporal Reasoning and Games
Consider a set of temporal constraints between intervals of time. So, for example, if I, J, K represent intervals of time, we might be told that I is properly contained in J (it starts after J starts and it ends before J ends), J is either strictly before K or strictly after K and I either meets K (the end of I is the start of J) or it equals K. We want to know whether these constraints have a solution: is there a way of giving I, J, K definite intervals of time so that the constraints are satisfied?
Conventional algorithms for this are not complete: they sometimes miss inconsistencies in a set of constraints and return "yes" when the answer should be "no". For this project you need to study how a two player game is capable of detecting inconsistencies and implementing this as a computer algorithm. Temporal Databases
Consider a database that changes over time. How do we handle temporal changes? Logical aspects: what language is best to describe this? First-order logic, with one time variable (c.f. Kowalski's event calculus); a temporal language with temporal operators (e.g. F - future and P - past); in a non-monotonic framework?
Planning
Design your own artificial planner. Example: the blocks world. Given blocks stacked on a table the problem is to move them (probably one at a time) till they are in the correct position. Only your program has to work it out for itself. Possibility of exploring mathematical aspects e.g. you can use a temporal reasoning system based on intervals of time to do this.
Genetic Algorithms/Programming
Genetic algorithms evolve solutions to problems, but the solutions need not a plain data. If the solution is in fact a code that represents a strategy for some game, then a genetic algorithm could be used to evolve knowledge of how to play the game. This is a very interesting idea and one that has, to date, not been investigated very much. The main problem is how to find simple and correct ways to encode the idea of a strategy as data that can be combined by the genetic algorithm. You may have some ideas about this yourself. At each "iteration" all the current solutions would play each other to determine how good they were. The best would breed to produce, you hope, at least some that are even better. Of course what solutions evolve as good depends on what the existing competition is like and it may be necessary to "salt" the population with known good strategies to get started. I feel sure that the game would have to be a very simple one like noughts and crosses (also called tick-tac-toe) and you would need a lot of imagination to take this project on (and some luck) but I think it could be fascinating. Also, try to use GP for simple two-player games which are not zero sum. The most famous of these is the prisoners dilemma, but there are various others. Try three-player games and n-player games which involve co-operation as well as competition between players.
Genetic Algorithms and Intractability.
Most interesting computer science problems are intractable and may even be undecidable. For the latter there can be no correct algorithmic solution and for the former we know that any solution cannot run in p-time. Nevertheless we may be able to find probabilistic solutions, i.e. algorithms that are not always correct but have a high probability of being correct. One approach of this type is by using genetic algorithms to find probabilistic solutions to intractable problems. A good example of an intractable problem is PSAT. Examples of undecidable problems are tiling problems and the satisfiability problem for first-order logic. The following links include databases of formulas to test your programs on
http://www.intellektik.informatik.tu-darmstadt.de/SATLIB/benchm.html ftp://dimacs.rutgers.edu/pub/challenge/satisfiability/benchmarks/cnf/ http://www.nlsde.buaa.edu.cn/~kexu/benchmarks/benchmarks.htm http://www.satlive.org/bytype.jsp?reftypefrom=-3 [thanks to Marcin Sadowski for these links].
Higher dimensional geometry kit
Graphic displays for simple 4 dimensional shapes.
Logic theorem prover
Implement a theorem prover for first-order logic, or for various modal and temporal logics. Could use the tableau method. Investigate formulas such that the tableau method will terminate, i.e. find a set of first-order formulas which are
decidable.
Non-linear dynamics and fractal geometry
Investigate chaotic behaviour of iterative systems. Construct some fractal pictures. Consider the outcome of iterating different, non-linear functions. Study Julia sets, strange attractors etc.
Combinatorics
Lots of problems here. See http://www.math.uiuc.edu/~west/openp/ for example.
Some of Robin Hirsch's project ideas |
Exactly 3 months ago, I discussed an interesting article by Kevin Hartnett in the Quanta Magazine that described an exciting story of Mr/Ms Ewin Tang, an ex-student of Aaronson's in Austin who is now a grad student at University of Washington. Tang was ordered to prove a proposition, basically a miniversion of \(P\neq NP\), as if it were a fact, except that he was finally led to prove the converse. Needless to say, lots of people had previously wasted their time with efforts to prove something that couldn't have been proven – and the activities done in order to prove X are often substantially different from those needed to prove non(X) which is why most of the mental energy was completely incorrectly allocated.
Now, the same Kevin Hartnett wrote another story with a similar lesson – in the absence of a proof, the mathematicians' belief in a certain conclusion may very well be a prejudice that is gonna be reversed. His text
2016 paper by Melanie Wooden-Machete Trump and her 3 pals (OK, fair enough, I wanted to increase the number of views of their preprint page).
First, let me answer the question from that title. If the questions are of a purely qualitative, binary type, e.g. the question "whether the supremum of a set of ranks is finite or infinite", then
no amount of "evidence"that is short of a proof is enough! If we can't complete a proof, we should really say that no other comments are truly relevant so the amount of evidence is zero.
OK, what's going on? We may write down curves such as\[
y^2 = x^3 - 4x + 1
\] i.e. elliptic curves and search for the list of all possible rational solutions \((x,y)\in {\mathbb Q}^2\) to this equation. Well, there are infinitely many.
Here, I need to warn you: the elliptic curve doesn't mean that it is an ellipse. An ellipse would only have at most the second powers of the coordinates, my example has the third power as well. Instead, what is "elliptic" about it is that if you extend \(x,y\) to complex numbers \(\CC\), the curve will have a complex dimension 1 and the topology of a torus – which has one extra "handle" on top of the topology of a sphere. The sphere ends up being a rational curve and because the "handle" needs some square-root branch cuts to be created from the sphere, it is called the next-to-rational i.e. elliptic curve. OK, if you want to know why it's really called "elliptic", it comes from the elliptic integrals.
(The torus, a complex elliptic curve, is the most rudimentary part of the complex algebraic geometry you need to learn if you want to study F-theory compactifications in string theory.)
OK, let's return to the list of rational solutions. Imagine that you find one point \((x,y)\in{\mathbb Q}^2\) on that curve. From that single point, you can find other points: draw any line through that point whose slope \(dy/dx\) is rational. That line will intersect the elliptic curve at some other points, and you can prove that their coordinates will be rational, too. The "offspring" may be created to further generations, and you find many solutions. The animation at the top sketches this proliferation of the descendants.
Amusingly, you get a significant percentage of the rational solutions in this way. If you think about the construction of the new rational solutions, you will see that the set of rational solutions actually forms an Abelian group of a certain kind. The only "hard" part of the construction of all solutions is the "starting point".
If you had a circle instead of an elliptic curve, one rational starting point would actually be enough to find all rational solutions – through the process of picking the rational slope and intersections, using the points that you already know. So we say that the rank of the circle is one. This statement can be proven. Circles are simple.
But if you already think about all the possible elliptic curves, the ranks may be higher than one. Hartnett describes another story of "group think" whose general sociological skeleton is something that you have heard of many times. There is some "lore" that everyone is obliged to believe although it's not really proven. The "lore" has some particular influential people who force it on others. In this particular story, it is especially Noam Elkies – at his time, the youngest mathematician who got tenure at Harvard.
The simplest way to see that sociologically, the "lore" is being pushed everywhere, is to look at Wikipedia that states:
A common conjecture is that there is no bound on the largest possible rank for an elliptic curve. In 2006, Noam Elkies discovered an elliptic curve with a rank of at least 28: [an elliptic curve with two terrible googol-like integer coefficients]"A common conjecture" really means nothing else than "rationally unjustified group think" but "common conjecture" probably sounds more intimidating which is what the writers wanted. The reference for the "common conjecture" is a Croatian pop math page which actually calls it a "folklore conjecture", not a "common conjecture", but certain people never hesitate to "improve" a statement because the propagation of an answer to this not yet settled question is something like a religion with its (biased) missionaries.
OK, this Dr Elkies, and less importantly, a few others (in Hartnett's story, the number of people who actually "guarantee" the group think is incredibly small – it seems that there has been a silent majority that simply allowed the loud minority to determine the group think – like in discussions about political issues), have been saying it's almost certain that the ranks are unbounded – because he could produce some ranks above 20, namely 19 and 28 etc., and it's almost infinity. But is it true? Well, one comment we could make is that Elkies isn't quite impartial – he has a clash of interests because his name has been rather linked to these statements about the unbounded ranks. If you study the atmosphere in that field, you will see that people are pushed to pay lip service to this
faith. Those who didn't were considered heretics or idiots by a network of folks who enforce the group think. But there's no proof.
In 2016, Melanie Wooden-Machete Trump et al. have presented the heuristic paper I mentioned. Well, it's called a heuristic paper because it doesn't contain any rigorous proof of the most important propositions. But it's still over 40 pages of heavy professional mathematics that I would be
visually unableto distinguish from fourty pages with a very finely constructed, complete, perfectionist proof. When we say that the paper is a heuristic, it simply doesn't mean at all that the paper looks like Leo Vuyk's Strawberry Universes. ;-)
They didn't attack the rank-of-the-elliptic-curve problem itself but a similar problem with "alternating integer matrices" and they discussed some properties of primes etc. But it "looks like" the possible values of the ranks are analogous for their "matrix model" as it is for the original problem involving the elliptic curves.
Some parts of their models are rigorously proven – I don't quite know whether "everything aside from the equivalence to the elliptic curve problem" has been proven rigorously. There's obviously a lot of things that I don't know here.
But the conclusion is: The rank is bounded, thus the religion is incorrect. In fact, the last step before they declare that the rank is bounded proves a much stronger and interesting statement: the number of elliptic curves (well, elliptic curves' "caricatures" in their model) with the rank exceeding 21 is finite. If there are finitely many above 21, the allowed ranks can't be unbounded because you can pick the maximum rank among the finitely many ranks above 21.
Great. So this heuristic paper doesn't rigorously settle the problem, either. But it gives a strong argument in the opposite direction than what the lore used to say. Now, Elkies offers a plausible counter-argument: maybe Melanie has only studied some "generic" elliptic curves that are close enough to the mean of a distribution in some parameter space – while those hypothetical curves that have ever larger ranks are increasingly far from the mean and non-generic in some way.
Maybe. But at this level, it becomes clear that he's just guessing. Even if his counter-argument were relevant, there could be another counter-counter-argument that could weaken his counter-argument and have the last laugh, after all. He's proposing something that may be relevant and reverse their conclusions – or not. The real point is that there is no proof in one way or another but there are always "incomplete arguments to make you think in one way or another". A funny thing about incomplete arguments is that they may always be cherry-picked or artificially fabricated. So if you wish, you may collect a large number of arguments that support your case and claim that all arguments that you could find agree and point in a certain direction.
However, that "uniformity" is purely due to your bias, prejudice, and/or dishonesty. Some other people could have cherry-picked (and helped to artificially construct) the arguments pointing in the opposite direction, too!
Because of Melanie's model and a previous model that's been around for years, it seems rather clear that "something special" is changing when the rank reaches the critical value of 21. But that has moral consequences. If your car – let's call it a Tesla (how do you call a Tesla on a hill? A miracle) – half-breaks after 21 miles, it is rather easy to believe that the car will completely break after a finite number of miles. The maximum survival of such a Tesla car may be as small as 28 miles – 28 is the maximum known rank of an elliptic curve. It may still be infinite but if it could survive infinite usage, why would it half-break after 21 miles?
My feeling is that the argument by Melanie et al. is "finer" because they actually found an important and actual finite value of the rank where things start to change a lot, 21. On the other hand, Elkies says that the supremum of the ranks is \(\infty\) because, apparently, his \(28\approx \infty\): it is the relatively largeness of the number \(28\) that should impress you and make you believe that he can also get to infinity. His rough argument is equivalent to \(1/28\approx 0\) while Melanie et al. are able to see \(1/21\neq 0\) (because something that is possible below 21 isn't possible above 21, i.e. because \(28\approx 21 = O(1)\)) which means – in another approximation \(21\approx 28\) – that they probably have a better resolution! And when we want to see whether there is one star or two stars, \(0\) and \(1/{\rm rank}_{\rm max}\), it is better to have a finer resolution. Do you get my point? I am half-joking but the qualitative lesson I want to convey is meant totally seriously. Melanie et al. have found a
characteristic finite scalein the possible values of the rank. They have disproven some kind of a "scaling invariance" for ranks!
(Analogously, it's plausible that sometime in the future, a bright algorithmic complexity theorist will find something special happening for some perhaps incomplete calculations of \(NP\) problems that last \(N^{21}\) steps – e.g. that it seems to be much easier or more likely to find a solution if the exponent is above 21. This would analogously suggest that polynomial-time algorithms exist because we would see that "the exponent around 21 already starts to be enough". Maybe all \(NP\) problems could be solved in \(CN^{28}\) steps.)
There are clearly lots of "spectra" in mathematics that are bounded. Take all simple compact Lie groups and study their ranks and dimensions. The dimension of \(SO(N)\) is \(N(N-1)/2\), isn't it? So it scales like \(N^2\). To make the behavior uniform for large \(N\), let's divide the dimension of a Lie group by the squared rank and call the ratio Lie-Motl Rank. The group \(SO(N)\) has the rank \(N/2\) for an even \(N\) so the dimension over the squared rank goes to \(2\) from below. A similar Lie-Motl rank will be found for symplectic groups and odd \(N\) orthogonal ones. For unitary groups, \(SU(N)\), rank \(N-1\), and \(N^2-1\) dimension, the Lie-Motl Rank will be around \(1\).
Are there simple compact Lie groups with the Lie-Motl Rank above \(2\)? You bet. Just look at the five exceptional groups. \(E_8\) will be found to maximize the Lie-Motl Rank, at \(248/8^2 = 31/8\) – it is almost four.
My model is obviously much less "directly relevant" for the rank of the elliptic curves (unless you find some shocking equivalence) but the general lesson is very similar. Infinitely many simple compact Lie groups have the Lie-Motl Ranks around \(1\) or \(2\) but well above \(2\), or perhaps some \(2+\epsilon\) to eliminate the problem of ranks going to two from above, the number of simple compact Lie groups that may get this high in the rank is limited – it is the exceptional groups. And the Lie-Motl Ranks simply must be at most \(31/8\) of the \(E_8\) group.
These distribution-like issues
may be analogousfor the ranks of the elliptic curves. You may see that I am not proving anything. I am just inventing excuses. I am rationalizingone possible answer, making it look more plausible. But I don't really have any strong faith that the rank must be bounded. Instead, I want to say that it's important to be open-minded in the absence of the proof – because the other side's arguments are just excuses and rationalizations, too.
And rationalizations aren't proofs. In fact, in pure mathematics dealing with discrete qualitative propositions (those that you need to assume to be 50-50
a priori), I think it is correct to say that a rationalization isn't even evidence. It doesn't mean that I am never using "rationalizations". But I only talk about them as "evidence" if I believe that they are actual sketches of a proofthat has a certain chance to be completed. If that interpretation is impossible, we've been just wasting our time because the rationalization doesn't really imply anything.
More typically, I am defending intense research into something – e.g. string theory – even in the absence of certainty that it is correct (or the right theory of the Universe, in this example). Why? Because I am not taking any resources away from any meaningful alternative research. You can't get any interesting results just from the assumption that "string theory isn't the right theory of the Universe". We can
provethat string theory leads to vastly more convincing and interesting results than the knownalternatives which is a sufficient justification for a frantic research into string theory. String theory may only compete with ideas that are already out there, not with some non-existent, future, or hypothetical competitors. But in the case of the mathematical propositions, there could exist interesting and heavily unequivalent strategies to do research that assume both possibletruth values of the conjectures. |
Mandal, Sukhendu and Natarajan, Srinivasan (2005)
Inorganic–organic hybrid structure: Synthesis, structure and magnetic properties of a cobalt phosphite–oxalate, $[C_4N_2H_{12}][Co_4(HPO_3)_2(C_2O_4)_3]$. In: Journal of Solid State Chemistry, 178 (7). pp. 2376-2382.
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Abstract
A hydrothermal reaction of a mixture of cobalt (II) oxalate, phosphorous acid, piperazine and water at $150^oC$ for 96 h followed by heating at $180^oC$ for 24 h gave rise to a new inorganic–organic hybrid solid, $[C_4N_2H_{12}][Co_4(HPO_3)_2(C_2O_4)_3]$, I. The structure consists of edge-shared $CoO_6$ octahedra forming a $[Co_2O_{10}]$ dimers that are connected by $HPO_3$ and $C_2O_4$ units forming a three-dimensional structure with one-dimensional channels. The amine molecules are positioned within these channels. The oxalate units have a dual role of connecting within the plane of the layer as well as out of the plane. Magnetic susceptibility measurement shows the compound orders antiferromagnetically at low temperature $(T_N=22K)$. Crystal data: I, monoclinic, space group = $P2_1/_c$ (No. 14). a=7.614(15), b=7.514(14), c=17.750(3)\AA , $\beta=97.351(3)^o$, $V=1007.30(3)\AA^3$, Z=2, $\rho_{calc}=2.466g/cm^3$, $\mu(MoK\alpha)=3.496mm^{-1}, R_1=0.0310$ and $wR_2=0.0807$ data $[I>2\sigma(I)]$.
Item Type: Journal Article Additional Information: Copyright of this article belongs to Elsevier. Keywords: Inorganic–organic hybrid; Cobalt; Channels structure; Antiferromagnetic Department/Centre: Division of Chemical Sciences > Solid State & Structural Chemistry Unit Depositing User: Ramya Krishna Date Deposited: 15 Mar 2007 Last Modified: 19 Sep 2010 04:35 URI: http://eprints.iisc.ac.in/id/eprint/9742 Actions (login required)
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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 Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ... |
Remember that in the analogous case $\nabla \times \nabla f = 0$, some intuition for the result can be attained by integration: by Green's theorem this is equivalent to $\int \nabla f \cdot ds = 0$ around every closed loop, which is true because $\int_{\gamma} \nabla f \cdot ds = f(\gamma(1)) - f(\gamma(0)).$ Thus our intuition is that curl measures circulation, and $\nabla f$ cannot circulate because this would introduce a discontinuity in $f$ around a loop.
Let's try the same thing: by the divergence theorem, it suffices to show that $\int_\Sigma (\nabla \times V) \cdot \hat n\ dA = 0$ for every closed surface $\Sigma$. By Stokes' theorem we know $$\int_\Sigma (\nabla \times V) \cdot \hat n\ dA = \int_{\partial \Sigma}V\cdot ds,$$which vanishes because $\Sigma$ is closed (i.e. $\partial \Sigma = \emptyset$).
In more intuitive terms, the divergence measures flux through a small cube; but the flux of a curl through a closed surface must be zero because there is no boundary curve for the circulation to accumulate upon.
As Nameless alluded to in his comment, you can get a more unified understanding of what's going on here by studying differential forms. All these geometric differential operators $\nabla, \nabla \times, \nabla \cdot$ are exterior derivatives $d_0,d_1,d_2$, and the identity $d_{k+1} \circ d_k = 0$ can be seen either by expanding out the partial derivative expression and noting that everything cancels (what I assume you've done in your proof), but also by applying the general Stokes theorem twice and noting that the boundary of a boundary is always empty. |
ISSN:
1937-1632
eISSN:
1937-1179
All Issues
Discrete & Continuous Dynamical Systems - S
August 2014 , Volume 7 , Issue 4
Issue on nonlinear elliptic and parabolic partial differential equations
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Abstract:
The study of elliptic and parabolic nonlinear partial differential equation has manifold aspects and can be seen from a number of different perspectives and points of view. The mathematical tools involved can, similarly, be quite different since they may include, for example, functional analysis, calculus of variations, topological techniques, geometric analysis, semigroup theory, numerical methods.
It would therefore be hopeless to give a complete picture of the research going on nowadays on these topics. Nevertheless, we tried to collect in the present volume several contributions of some leading scholars in these fields, hoping that they could give some hints of some of the main research lines in the field, identifying in particular some setting in which the elliptic theory is an important clue to analyze the parabolic situation, and viceversa.
In the papers collected here, all of which have been anonymously refereed as requested by the high standards of this Journal, several rapidly developing and important topics are discussed and developed.
Among them, we mention the following ones: two-phases free boundary problems; qualitative properties of solutions of Lane-Emden-Fowler equations; Hopf fibration and singularly perturbed elliptic equations; biharmonic elliptic boundary value problems; positivity preserving issues in models for clamped plates; Liouville theorems for Hardy-Littlewood-Sobolev systems; regularity of solutions of degenerate elliptic equations; decay properties for degenerate parabolic problems; porous media and fast diffusion equations driven by fractional Laplacians; fine asymptotics of solutions to the fast diffusion equation; well-posedness of nonlinear integral equations with general kernels and asymptotics of the corresponding solutions; local lower and upper bounds for solutions to doubly nonlinear singular parabolic equations; nonlinear initial value problems that model evolution and selection in living systems in connection with kinetic theory; relationships between optimal inequalities and nonlinear flows; generalized solutions, via non-archimedean fields, to equations which may not have solutions in distributional sense; models for oscillations in suspension bridges.
The success of this collection depends on the quality of the papers and of the high reputation of the Authors: we are grateful to all the contributors of the present volume.
Abstract:
This paper deals with a new kind of generalized functions, called ``ultrafunctions", which have been introduced recently in [5] and developed in [10] and [11]. Their peculiarity is that they are based on a Non Archimedean field, namely on a field which contains infinite and infinitesimal numbers. Ultrafunctions have been introduced to provide generalized solutions to equations which do not have any solutions, not even among the distributions. Some applications of this kind will be presented in the second part of this paper.
Abstract:
We study the existence of solutions $u $ belonging to $L^1(0,T; W_0^{1,1}(\Omega)) \cap L^{\infty}(0,T;L^2(\Omega))$ of a class of nonlinear problems whose prototype is the following \begin{equation} \label{prob1} \left\{ \begin{array}{lll} \displaystyle u_t - {\rm div} \left( \frac{\nabla u}{(1+|u|)^2} \right) = 0, & \hbox{in} & \Omega_T; \\ u=0, & \hbox{on} & \partial\Omega \times (0,T); & & & & \hbox{(1)}\\ u(x,0)= u_0(x) \in L^2(\Omega), & \hbox{ in} & \Omega. \end{array} \right. \end{equation} We investigate also the asymptotic estimates satisfied by distributional solutions that we find and the uniqueness.
Abstract:
We consider a family of degenerate elliptic equations of the form div $(\nabla F(\nabla u)) = f$, where $F\in C^{1,1}$ is a convex function which is elliptic outside a ball. We prove an excess-decay estimate at points where $\nabla u$ is close to a nondegenerate value for $F$. This result applies to degenerate equations arising in traffic congestion, where we obtain continuity of $\nabla u$ outside the degeneracy, and to anisotropic versions of the $p$-laplacian, where we get Hölder regularity of $\nabla u$.
Abstract:
We prove some Liouville theorems for systems of integral equations and inequalities related to weighted Hardy-Littlewood-Sobolev inequality type on $R^N$ . Some semilinear singular or degenerate higher order elliptic inequalities associated to polyharmonic operators are considered. Special cases include the Hénon-Lane-Emden system.
Abstract:
We present some recent progress on the analysis of two-phase free boundary problems governed by elliptic operators, with non-zero right hand side. We also discuss on several open questions, object of future investigations.
Abstract:
This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.
Abstract:
We find a continuum of extinction rates of solutions of the Cauchy problem for the fast diffusion equation $u_\tau=\nabla\cdot(u^{m-1}\,\nabla u)$ with $m=m_*:=(n-4)/(n-2)$, here $n>2$ is the space-dimension. The extinction rates depend explicitly on the spatial decay rates of initial data and contain a logarithmic term.
Abstract:
In this paper we show some properties regarding the local behaviour of local weak solutions to a class of doubly nonlinear singular parabolic equations.
Abstract:
It is known that the Dirichlet bilaplace boundary value problem, which is used as a model for a clamped plate, is not sign preserving on general domains. It is also known that the corresponding first eigenfunction may change sign. In this note we will show that even a constant right hand side may result in a sign-changing solution.
Abstract:
We study the existence and the large time behavior of global-in-time solutions of a nonlinear integral equation with a generalized heat kernel \begin{eqnarray*} & & u(x,t)=\int_{{\mathbb R}^N}G(x-y,t)\varphi(y)dy\\ & & \qquad\quad +\int_0^t\int_{{\mathbb R}^N}G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds, \end{eqnarray*} where $\varphi\in W^{\ell,\infty}({\mathbb R}^N)$ and $\ell\in\{0,1,\dots\}$. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.
Abstract:
We first review some history prior to the failure of the Tacoma Narrows suspension bridge. Then we consider some popular accounts of this in the popular physics literature, and the scientific and scholarly basis for these accounts and point out some failings. Later, we give a quick introduction to three different models, one single particle, one a continuum model, and two systems with two degrees of freedom.
Abstract:
We study the asymptotic behaviour as $p\rightarrow \infty$ of the nodal radial solutions $u_p$ of the problem \begin{equation*} \left\{ \begin{array}{rlll} -\Delta u&=&|u|^{p-1}u& \text{in }\Omega \\ u&=&0& \text{on }\partial\Omega, \end{array} \right. \end{equation*} where $\Omega$ is an annulus in $\mathbb{R}^N$, $N\geq 2$. We also analyze the spectrum of the linearized operator associated to $u_p$ in the case when $u_p$ has only two nodal regions. In particular, we prove that the Morse index of $u_p$ tends to $\infty$ as $p$ goes to $\infty$.
Abstract:
This paper is devoted to the qualitative analysis of a new broad class of nonlinear initial value problems that model evolution and selection in living systems derived by the mathematical tools of the kinetic theory of active particles. The paper is divided into two parts. The first shows how to obtain the nonlinear equations with proliferative/distructive nonlinear terms. The latter presents a detailed analysis of the related initial value problem. In particular, it is proved that the corresponding initial value problem admits a unique non--negative maximal solution. However, the solution cannot be in general global in time, due to the possibility of blow--up. The blow--up occurs when the biological life system is
globally proliferative, see Theorem 3.3. Abstract:
In this article we show how the Hopf fibration can be used to generate special solutions of singularly perturbed elliptic equations on annuli. Indeed, by the Hopf fibration the equation can be reduced to a lower dimensional problem, to which known results on single (or multiple point) concentration can be applied. Reversing the reduction process, one obtains solutions concentrating on circles and spheres, which are given as the fibres of the Hopf fibration.
Abstract:
We address the question, for which $\lambda \in \mathbb{R}$ is the boundary value problem \begin{equation*} \left\{ \begin{array}{cc} \Delta ^{2}u+\lambda u=f & \text{in }\Omega , \\ u=\Delta u=0 & \text{on }\partial \Omega , \end{array} \right. \end{equation*} positivity preserving, that is, $f\geq 0$ implies $u\geq 0$. Moreover, we consider what happens, when $\lambda $ passes the maximal value for which positivity is preserved.
Abstract:
We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous medium equation, $\partial_t u +(-\Delta)^{s}(u^m)=0$, and some of its generalizations. Contrary to usual porous medium flows, the fractional version has infinite speed of propagation for all exponents $0 < s < 1$ and $m > 0$; on the other hand, it also generates an $L^1$-contraction semigroup which depends continuously on the exponent of fractional differentiation and the exponent of the nonlinearity.
After establishing the general existence and uniqueness theory, the main properties are described: positivity, regularity, continuous dependence, a priori estimates, Schwarz symmetrization, among others. Self-similar solutions are constructed (fractional Barenblatt solutions) and they explain the asymptotic behaviour of a large class of solutions. In the fast diffusion range we study extinction in finite time and we find suitable special solutions. We discuss KPP type propagation. We also examine some related equations that extend the model and briefly comment on current work.
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Answer
Brian needs $8\frac{1}{3}$ cups of tomato sauce to make $2\frac{1}{2}$ times the usual amount of spaghetti sauce.
Work Step by Step
Let $x$ = the number of cups of tomato sauce Brian needs Brian use $3\frac{1}{3}$ cups of tomato sauce for 1 spaghetti sauce recipe. Brian needs to use $x$ cups of tomato sauce for $2\frac{1}{2}$ times the usual amount of his spaghetti sauce recipe. Ratio and proportion can be used to solve this problem. Setting up the ratio and proportion gives $$\dfrac{3\frac{1}{3}}{1} = \dfrac{x}{2\frac{1}{2}}$$ Converting the mixed numbers to improper fractions give: $$\dfrac{\frac{10}{3}}{1}=\dfrac{x}{\frac{5}{2}}$$ Cross-multiply to obtain: $$1(x) = \frac{10}{3} \cdot \frac{5}{2} x=\frac{50}{6} \\x=8\frac{2}{6} \\x=8\frac{1}{3}$$ Thus, Brian needs $8\frac{1}{3}$ cups of tomato sauce to make $2\frac{1}{2}$ times the usual amount of spaghetti sauce. |
Job cards for GiBUU
We use so-called "job cards" for passing input parameters to GiBUU. A job card is just a simple plain-text file which contains all these parameters, determining the setup of the simulation being performed (e.g. type of projectile and target, beam energy, number of runs, which analysis to perform, which output to generate, etc).
Before you go on studying the example job cards given below, please note the following:
Each job card consists of several "namelists". Each namelist contains input parameters for one specific module within GiBUU. Text after an exclamation mark (!) is considered as a comment in a namelist Some of the example files given below are not strictly conform with the Fortran 2003 standard (we use "$end" as ending and not "/", and we use "$" instead of "&"). So code compiled with some compilers might complain about our convention (if this is the case then please replace "$end" by "/" and "$" by "&"). Unless indicated otherwise, distances are given in fm, Energies and momenta in GeVand hadronic cross sections in mb.
For detailed information on almost all possible namelists and its switches we refer to our online namelist documentation.
Examples TAPS: \( \gamma A\rightarrow\omega X \) @ 0.9-2.2 GeV Pion or proton induced reaction on nuclei in the energy regime of 3 to 50 GeV, as e.g. possible within the HARP and NA61/Shine experiments. Jobcard switches for Deuterium targets FOPI: \( \pi^-A \) @ 1.7 GeV ANKE: \( pA \) @ 2.83 GeV CLAS/g7: \( \gamma A\rightarrow e^+e^-X \) @ 1-5 GeV High energy photon induced: Using HiLepton in the limit \( Q^2\rightarrow 0 \) HADES: External particle input neutrino: T2K 0pi on water, preprint arXiv:1712.07134 [hep-ex]
More jobcards can be found here. |
In Dodelson's textbook modern cosmology ch4.5, I wonder if anyone knows how he did the integration by part over the phasespace on the distribution function of Eq(4.72) and Eq(4.78).
Here is part of Eq(4.72) $$\int \frac{d^3p}{(2\pi)^3}p\frac{\partial f_{dm}}{\partial p} = \frac{4\pi}{(2\pi)^3}\int_0^\infty dpp^3\frac{\partial f_{dm}}{\partial p}$$ $$= -3 \frac{4\pi}{(2\pi)^3}\int_0^{\infty}dpp^2f_{dm}$$
I don't understand why he assumes this in the last step. $$p^3f_{dm}|_0^{\infty} = 0.$$
Here is part of Eq(4.78) $$\int_0^\infty dp \dfrac{p^4}{E}\dfrac{\partial f_{dm}}{\partial p}=\int_0^\infty dp f_{dm} \left( \dfrac{p^5}{E^3} - \dfrac{4p^3}{E}\right)$$
Similarly, he also used integration by part but why is this correct. $$\frac{p^4}{E} f_{dm}|_0^{\infty} = \dfrac{p^5}{E^3}$$
Dodelson mentions that we don't need an explicit expression of the zero-order distribution function for dark matter, $f_{dm}$, during the derivation, but then I could not follow his integration in this case.
Any help is appreciated. |
2019-09-04 12:06
Soft QCD and Central Exclusive Production at LHCb / Kucharczyk, Marcin (Polish Academy of Sciences (PL)) The LHCb detector, owing to its unique acceptance coverage $(2 < \eta < 5)$ and a precise track and vertex reconstruction, is a universal tool allowing the study of various aspects of electroweak and QCD processes, such as particle correlations or Central Exclusive Production. The recent results on the measurement of the inelastic cross section at $ \sqrt s = 13 \ \rm{TeV}$ as well as the Bose-Einstein correlations of same-sign pions and kinematic correlations for pairs of beauty hadrons performed using large samples of proton-proton collision data accumulated with the LHCb detector at $\sqrt s = 7\ \rm{and} \ 8 \ \rm{TeV}$, are summarized in the present proceedings, together with the studies of Central Exclusive Production at $ \sqrt s = 13 \ \rm{TeV}$ exploiting new forward shower counters installed upstream and downstream of the LHCb detector. [...] LHCb-PROC-2019-008; CERN-LHCb-PROC-2019-008.- Geneva : CERN, 2019 - 6. Fulltext: PDF; In : The XXVII International Workshop on Deep Inelastic Scattering and Related Subjects, Turin, Italy, 8 - 12 Apr 2019 Registre complet - Registres semblants 2019-08-15 17:39
LHCb Upgrades / Steinkamp, Olaf (Universitaet Zuerich (CH)) During the LHC long shutdown 2, in 2019/2020, the LHCb collaboration is going to perform a major upgrade of the experiment. The upgraded detector is designed to operate at a five times higher instantaneous luminosity than in Run II and can be read out at the full bunch-crossing frequency of the LHC, abolishing the need for a hardware trigger [...] LHCb-PROC-2019-007; CERN-LHCb-PROC-2019-007.- Geneva : CERN, 2019 - mult.p. In : Kruger2018, Hazyview, South Africa, 3 - 7 Dec 2018 Registre complet - Registres semblants 2019-08-15 17:36
Tests of Lepton Flavour Universality at LHCb / Mueller, Katharina (Universitaet Zuerich (CH)) In the Standard Model of particle physics the three charged leptons are identical copies of each other, apart from mass differences, and the electroweak coupling of the gauge bosons to leptons is independent of the lepton flavour. This prediction is called lepton flavour universality (LFU) and is well tested. [...] LHCb-PROC-2019-006; CERN-LHCb-PROC-2019-006.- Geneva : CERN, 2019 - mult.p. In : Kruger2018, Hazyview, South Africa, 3 - 7 Dec 2018 Registre complet - Registres semblants 2019-05-15 16:57 Registre complet - Registres semblants 2019-02-12 14:01
XYZ states at LHCb / Kucharczyk, Marcin (Polish Academy of Sciences (PL)) The latest years have observed a resurrection of interest in searches for exotic states motivated by precision spectroscopy studies of beauty and charm hadrons providing the observation of several exotic states. The latest results on spectroscopy of exotic hadrons are reviewed, using the proton-proton collision data collected by the LHCb experiment. [...] LHCb-PROC-2019-004; CERN-LHCb-PROC-2019-004.- Geneva : CERN, 2019 - 6. Fulltext: PDF; In : 15th International Workshop on Meson Physics, Kraków, Poland, 7 - 12 Jun 2018 Registre complet - Registres semblants 2019-01-21 09:59
Mixing and indirect $CP$ violation in two-body Charm decays at LHCb / Pajero, Tommaso (Universita & INFN Pisa (IT)) The copious number of $D^0$ decays collected by the LHCb experiment during 2011--2016 allows the test of the violation of the $CP$ symmetry in the decay of charm quarks with unprecedented precision, approaching for the first time the expectations of the Standard Model. We present the latest measurements of LHCb of mixing and indirect $CP$ violation in the decay of $D^0$ mesons into two charged hadrons [...] LHCb-PROC-2019-003; CERN-LHCb-PROC-2019-003.- Geneva : CERN, 2019 - 10. Fulltext: PDF; In : 10th International Workshop on the CKM Unitarity Triangle, Heidelberg, Germany, 17 - 21 Sep 2018 Registre complet - Registres semblants 2019-01-15 14:22
Experimental status of LNU in B decays in LHCb / Benson, Sean (Nikhef National institute for subatomic physics (NL)) In the Standard Model, the three charged leptons are identical copies of each other, apart from mass differences. Experimental tests of this feature in semileptonic decays of b-hadrons are highly sensitive to New Physics particles which preferentially couple to the 2nd and 3rd generations of leptons. [...] LHCb-PROC-2019-002; CERN-LHCb-PROC-2019-002.- Geneva : CERN, 2019 - 7. Fulltext: PDF; In : The 15th International Workshop on Tau Lepton Physics, Amsterdam, Netherlands, 24 - 28 Sep 2018 Registre complet - Registres semblants 2019-01-10 15:54 Registre complet - Registres semblants 2018-12-20 16:31
Simultaneous usage of the LHCb HLT farm for Online and Offline processing workflows LHCb is one of the 4 LHC experiments and continues to revolutionise data acquisition and analysis techniques. Already two years ago the concepts of “online” and “offline” analysis were unified: the calibration and alignment processes take place automatically in real time and are used in the triggering process such that Online data are immediately available offline for physics analysis (Turbo analysis), the computing capacity of the HLT farm has been used simultaneously for different workflows : synchronous first level trigger, asynchronous second level trigger, and Monte-Carlo simulation. [...] LHCb-PROC-2018-031; CERN-LHCb-PROC-2018-031.- Geneva : CERN, 2018 - 7. Fulltext: PDF; In : 23rd International Conference on Computing in High Energy and Nuclear Physics, CHEP 2018, Sofia, Bulgaria, 9 - 13 Jul 2018 Registre complet - Registres semblants 2018-12-14 16:02
The Timepix3 Telescope andSensor R&D for the LHCb VELO Upgrade / Dall'Occo, Elena (Nikhef National institute for subatomic physics (NL)) The VErtex LOcator (VELO) of the LHCb detector is going to be replaced in the context of a major upgrade of the experiment planned for 2019-2020. The upgraded VELO is a silicon pixel detector, designed to with stand a radiation dose up to $8 \times 10^{15} 1 ~\text {MeV} ~\eta_{eq} ~ \text{cm}^{−2}$, with the additional challenge of a highly non uniform radiation exposure. [...] LHCb-PROC-2018-030; CERN-LHCb-PROC-2018-030.- Geneva : CERN, 2018 - 8. Registre complet - Registres semblants |
Could anyone kindly help me evaluate the period of the following function:
$$f(x)=\sin\left(2\left(\tan^{-1}\left(\frac{\tan(x)}k\right)\right)\right)$$
where $k$ is some positive real constant.
Had it been only $\sin2(\tan^{-1}(\tan(x)))$, I could have found its period but this constant $k$ is proving to be a challenge for me.
I guess, for certain values of $k$, f may not be periodic at all, can we find under what condition this function will be periodic?
Thanks for your time.
PS: I am in electrical engineering and I used to find the period of function ten years back, but now I have really forgotten the concepts and formulas and I don't have any book at my disposal either. |
I'm self-studying Analysis on Manifolds by Munkres. I understood the theory of the Riemann integral over bounded rectangles and more general rectifiable sets in $\mathbb{R}^n$. In the part on improper integrals, I am becoming confused.
Munkres defines the extended or improper integral of a continuous function $f$ over an open set $A \subset \mathbf{R}^n$. He chooses any sequence $C_N$ of compact rectifiable sets such that $A = \bigcup_N C_N$ and $C_N \subset \text{Int }C_{N+1}$ for all $N$ and states that the extended integral exists if and only if the sequence $\int_{C_N} |f|$ is bounded and
$$\int_Af = \lim_{N \to \infty} \int_{C_N} f.$$
He states "...if the ordinary integral exists, then so does the extended integral and the two integrals are equal", but then " ...the extended integral may exist when the ordinary integral does not."
This makes sense if you think about integrals over intervals in $\mathbf {R}$. If a function is unbounded or the interval is unbounded then the Riemann integral does not exist but the improper integral can. But Munkres claims this even if $A$ is a bounded, open set and $f:A \to \mathbf{R}$ is a bounded, continuous function.
How is this possible? |
ISSN:
1531-3492
eISSN:
1553-524X
All Issues
Discrete & Continuous Dynamical Systems - B
March 2015 , Volume 20 , Issue 2
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Abstract:
We consider an immersed finite element method for solving one dimensional Pennes bioheat transfer equation with discontinuous coefficients and nonhomogenous flux jump condition. Convergence properties of the semidiscrete and fully discrete schemes are investigated in the $L^{2}$ and energy norms. By using the computed solution from the immerse finite element method, an inexpensive and effective flux recovery technique is employed to approximate flux over the whole domain. Optimal order convergence is proved for the immersed finite element approximation and its flux. Results of the simulation confirm the convergence analysis.
Abstract:
The subject matter of this paper concerns the paraxial approximation for the transport of charged particles. We focus on the magnetic confinement properties of charged particle beams. The collisions between particles are taken into account through the Boltzmann kernel. We derive the magnetic high field limit and we emphasize the main properties of the averaged Boltzmann collision kernel, together with its equilibria.
Abstract:
This paper is devoted to investigate the problem of controlling chaos for a pendulum system with parametric and external excitations. By using Melnikov methods, the criteria of controlling chaos are obtained. Numerical simulations are given to illustrate the effect of the chaos control for this system, suppression of homoclinic chaos is more effective than suppression of heteroclinic chaos, and the chaotic motions can be suppressed to period-motions by adjusting parameters of chaos-suppressing excitation. Finally, we calculate the maximum Lyapunov exponents (LE) in parameter-plane and observe the frequency of chaos-suppressing excitation also play an important role in the process of chaos control.
Abstract:
In this paper, we study a reaction-diffusion population model with time delay. We establish a comparison principle for coupled upper/lower solutions and prove the existence/uniqueness result for the model. We then show the global asymptotic behavior of the model.
Abstract:
In this work we analyze a system of nonlinear evolution partial differential equations modeling the fluid-structure interaction associated to the dynamics of an elastic vesicle immersed in a moving incompressible viscous fluid. This system of equations couples an equation for a phase field variable, used to determine the position of vesicle membrane deformed by the action of the fluid, to the $\alpha$-Navier- Stokes equations with an extra nonlinear interaction term. We prove global in time existence and uniqueness of solutions for this system in suitable functional spaces even in the three-dimensional case.
Abstract:
The objective of this paper is to study systematically the dynamical properties of a predator-prey model with nonlinear predator harvesting. We show the different types of system behaviors for various parameter values. The results developed in this article reveal far richer dynamics compared to the model without harvesting. The occurrence of change of structure or bifurcation in a system with parameters is a way to predict global dynamics of the system. It has been observed that the model has at most two interior equilibria and can exhibit numerous kinds of bifurcations (e.g. saddle-node, transcritical, Hopf-Andronov and Bogdanov-Takens bifurcation). The stability (direction) of the Hopf-bifurcating periodic solutions has been obtained by computing the first Lyapunov number. The emergence of homoclinic loop has been shown through numerical simulation when the limit cycle arising though Hopf-bifurcation collides with a saddle point. Numerical simulations using MATLAB are carried out as supporting evidences of our analytical findings. The main purpose of the present work is to offer a complete mathematical analysis for the model.
Abstract:
The McKendrick/Von Foerster equation is a transport equation with a non-local boundary condition that appears frequently in structured population models. A variant of this equation with a size structure has been proposed as a metastatic growth model by Iwata
et al.
Here we will show how a family of metastatic models with 1D or 2D structuring variables, based on the Iwata model, can be reformulated into an integral equation counterpart, a Volterra equation of convolution type, for which a rich numerical and analytical theory exists. Furthermore, we will point out the potential of this reformulation by addressing questions coming up in the modelling of metastatic tumour growth. We will show how this approach permits to reduce the computational cost of the numerical resolution and to prove structural identifiability.
Abstract:
To describe population dynamics, it is crucial to take into account jointly evolution mechanisms and spatial motion. However, the models which include these both aspects, are not still well-understood. Can we extend the existing results on type structured populations, to models of populations structured by type and space, considering diffusion and nonlocal competition between individuals?
We study a nonlocal competitive Lotka-Volterra type system, describing a spatially structured population which can be either monomorphic or dimorphic. Considering spatial diffusion, intrinsic death and birth rates, together with death rates due to intraspecific and interspecific competition between the individuals, leading to some integral terms, we analyze the long time behavior of the solutions. We first prove existence of steady states and next determine the long time limits, depending on the competition rates and the principal eigenvalues of some operators, corresponding somehow to the strength of traits. Numerical computations illustrate that the introduction of a new mutant population can lead to the long time evolution of the spatial niche.
Abstract:
In this article, using the well-known Superconvergent Patch Recovery (SPR) method, we present a gradient superconvergence post-processing scheme for the tensor-product quadratic pentahedral finite element approximation to the solution of a general second-order elliptic boundary value problem in three dimensions over fully uniform meshes. The supercloseness property of the gradients between the finite element solution $u_h$ and the tensor-product quadratic interpolation $\Pi u$ is first given. Then we show that the gradient recovered from the finite element solution by using the SPR method is superconvergent to $\nabla u$ at interior vertices.
Abstract:
This paper is concerned with a three dimensional diffusive Lotka-Volterra system which is combined with cooperative-competitive interactions between the three species. By using the method of super-sub solutions and comparison principle with cross iteration, some results on the asymptotic spreading speed of the system are established under certain assumptions on the parameters appearing in the system.
Abstract:
We investigate the solution structure and stability of a semiconductor laser receiving time-delayed and frequency-filtered optical feedback from two external filters. This system is referred to as the 2FOF laser, and it has been used as pump laser in optical telecommunication and as light source in sensor applications. The underlying idea is that the two filter loops provide a means of stabilizing and controling the laser output. The mathematical model takes the form of delay differential equations for the (real-valued) population inversion of the laser active medium and for the (complex-valued) electric fields of the laser cavity and of the two filters. There are two time delays, which are the travel times of the light from the laser to each of the filters and back.
Our analysis of the 2FOF laser focuses on the basic solutions, known as continuous waves or external filtered modes (EFMs), which correspond to laser output with steady amplitude and frequency. Specifically, we consider the EFM-surface in the $(\omega_s,\,N_s,\,dC_p)$-space of steady frequency $\omega_s$, the corresponding steady population inversion $N_s$, and the feedback phase difference $dC_p$. This surface emerges as the natural object for the study of the 2FOF laser because it conveniently catalogues information about available frequency ranges of the EFMs. We identify five transitions, through four different singularities and a cubic tangency, which change the type of the EFM-surface locally and determine the EFM-surface bifurcation diagram in the $(\Delta_1,\,\Delta_2)$-plane. In this way, we classify the possible types of the EFM-surface, which consist of a combination of bands (covering the entire $dC_p$-range) and islands (covering only a finite range of $dC_p$).
We also investigate the stability of the EFMs, where we focus on saddle-node and Hopf bifurcation curves that bound regions of stable EFMs on the EFM-surface. It is shown how these stability regions evolve when parameters are changed along a chosen path in the $(\Delta_1,\,\Delta_2)$-plane. From a viewpoint of practical interests, we find various bands and islands of stability on the EFM-surface that may be accessible experimentally.
Beyond their relevance for the 2FOF laser system, the results presented here also showcase how advanced tools from bifurcation theory and singularity theory can be employed to uncover and represent the complex solution structure of a delay differential equation model that depends on a considerable number of input parameters, including two time delays.
Abstract:
We study a nonlocal reaction-diffusion-advection equation arising from the study of a single phytoplankton species competing for light in a poorly mixed water column. When the diffusion coefficient is very small, the phytoplankton population concentrates around certain zeros of the advection function. The corresponding phytoplankton distribution approaches a $\delta$-like function centered at those zeros.
Abstract:
In this paper, we investigate two problems in porous thermoelasticity where the heat conduction is given by Cattaneo's law and prove exponential decay results in the presence of both macro- and micro-dissipations.
Abstract:
We address in this work the minimization of the $L^q$-norm $(q>2)$ of semidiscrete controls for parabolic equation. As shown in [15], under the main approximation assumptions that the discretized semigroup is uniformly analytic and that the degree of unboundedness of control operator is lower than 1/2, uniform controllability is achieved in $L^2$ for semidiscrete approximations for the parabolic systems. The main goal of this paper is to overcome the limitation of [15] about the order 1/2 of unboundedness of the control operator. Namely, we show that the uniform controllability property also holds in $L^q \ (q>2)$ even in the case of a degree of unboundedness greater than 1/2. Moreover, a minimization procedure to compute the approximation controls in $L^q\ (q>2)$ is provided. An example of application is implemented for the one-dimensional heat equation with Dirichlet boundary control.
Abstract:
The generalized, two-component Hunter-Saxton system comprises several well-known models of fluid dynamics and serves as a tool for the study of one-dimensional fluid convection and stretching. In this article a general representation formula for periodic solutions to the system, which is valid for arbitrary values of parameters $(\lambda,\kappa) \in \mathbb{R} \times \mathbb{R}$, is derived. This allows us to examine in great detail qualitative properties of blow-up as well as the asymptotic behaviour of solutions, including convergence to steady states in finite or infinite time.
Abstract:
In this paper, the solutions of Navier-Stokes equations governing 2-D incompressible flows with the Dirichlet boundary condition are analyzed. We derive a condition for boundary layer separation, and the condition is determined by initial values and external forces. More importantly, the condition can predict when and where the boundary layer separation occurs directly. In addition, we also get an algebraic equation for the separation point and the separation time. The algebraic equation can tell us where the boundary layer separation does not occur in a short period of time. The main technical tool is the geometric theory of incompressible flows developed by T. Ma and S. Wang in [15].
Abstract:
Optimization problem for a stochastic N-dimensional competitive Lotka-Volterra system is studied in this paper. The considered system is driven by both white noise and jumping noise, and the jumping noise is modeled by a stochastic integral with respect to a Poisson counting measure generated by a Poisson point process. For two types of objective functions, namely, time-averaged yield and sustained yield, the optimal harvesting efforts as well as the corresponding maximum yields are given respectively. Moreover, almost sure equivalence between these two objective functions is proved by ergodic method. This paper provides us a new idea to study the stochastic optimal harvesting problem with sustained yield, and this idea can be popularized to other stochastic systems.
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ISSN:
1531-3492
eISSN:
1553-524X
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Discrete & Continuous Dynamical Systems - B
May 2015 , Volume 20 , Issue 3
Special issue on the asymptotic dynamics of non-autonomous systems
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Abstract:
We were very pleased to be given the opportunity by Prof. Peter Kloeden to edit this special issue of
Discrete and Continuous Dynamical Systems - Series Bon the asymptotic dynamics of non-autonomous systems.
For more information please click the “Full Text” above.
Abstract:
This review paper treats the dynamics of non-autonomous dynamical systems. To study the forwards asymptotic behaviour of a non-autonomous differential equation we need to analyse the asymptotic configurations of the non-autonomous terms present in the equations. This fact leads to the definition of concepts such as skew-products and cocycles and their associated global, uniform, and cocycle attractors. All of them are closely related to the study of the pullback asymptotic limits of the dynamical system, from which naturally emerges the concept of a pullback attractor. In the first part of this paper we want to clarify these different dynamical scenarios and the relations between their corresponding attractors.
If the global attractor of an autonomous dynamical system is given as the union of a finite number of unstable manifolds of equilibria, a detailed understanding of the continuity of the local dynamics under perturbation leads to important results on the lower-semicontinuity and topological structural stability for the pullback attractors of evolution processes that arise from small non-autonomous perturbations, with respect to the limit regime. Finally, continuity with respect to global dynamics under non-autonomous perturbation is also studied, for which appropriate concepts for Morse decomposition of attractors and non-autonomous Morse--Smale systems are introduced. All of these results will also be considered for uniform attractors. As a consequence, this paper also makes connections between different approaches to the qualitative theory of non-autonomous differential equations, which are often treated independently.
Abstract:
We give an abstract framework for studying nonautonomous PDEs, called a generalized evolutionary system. In this setting, we define the notion of a pullback attractor. Moreover, we show that the pullback attractor, in the weak sense, must always exist. We then study the structure of these attractors and the existence of a strong pullback attractor. We then apply our framework to both autonomous and nonautonomous evolutionary systems as they first appeared in earlier works by Cheskidov, Foias, and Lu. In this con- text, we compare the pullback attractor to both the global attractor (in the autonomous case) and the uniform attractor (in the nonautonomous case). Finally, we apply our results to the nonautonomous 3D Navier-Stokes equations on a periodic domain with a translationally bounded force. We show that the Leray-Hopf weak solutions form a generalized evolutionary system and must then have a weak pullback attractor.
Abstract:
We give a comprehensive study of strong uniform attractors of nonautonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces that are not translation compact, but nevertheless allow the attraction in a strong topology of the phase space to be verified and discuss in a more detailed way the class of so-called normal external forces introduced before. We also develop a unified approach to verify the asymptotic compactness for such systems based on the energy method and apply it to a number of equations of mathematical physics including the Navier-Stokes equations, damped wave equations and reaction-diffusing equations in unbounded domains.
Abstract:
We construct the trajectory attractor $\mathfrak{A}_{\Sigma }$ for the non-autonomous dissipative 2d Euler systems with periodic boundary conditions that contain time dependent dissipation terms $-r(t)u$ such that $0<\alpha \le r(t)\le \beta$, for $t\ge 0$. External forces $g(x,t),x\in \mathbb{T}^{2},t\ge 0,$ also depend on time. The corresponding non-autonomous dissipative 2d Navier--Stokes systems with the same terms $-r(t)u$ and $g(x,t)$ and with viscosity $\nu >0$ also have the trajectory attractor $\mathfrak{A}_{\Sigma }^{\nu }.$ Such systems model large-scale geophysical processes in atmosphere and ocean. We prove that $\mathfrak{A}_{\Sigma }^{\nu }\rightarrow \mathfrak{A}_{\Sigma }$ as viscosity $\nu \rightarrow 0+$ in the corresponding metric space. Moreover, we establish the existence of the minimal limit $\mathfrak{A}_{\Sigma }^{\min }\subseteq \mathfrak{A}_{\Sigma }$ of the trajectory attractors $\mathfrak{A}_{\Sigma }^{\nu }$ as $\nu \rightarrow 0+.$ Every set $\mathfrak{A}_{\Sigma }^{\nu }$ is connected. We prove that $\mathfrak{A}_{\Sigma }^{\min }$ is a connected invariant subset of $\mathfrak{A}_{\Sigma }.$ The problem of the connectedness of the trajectory attractor $\mathfrak{A}_{\Sigma }$ itself remains open.
Abstract:
We consider a stochastically perturbed coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a random pullback attractor of finite fractal dimension. Our argument is based on some modification of the method of quasi-stability estimates recently developed for deterministic systems.
Abstract:
As a direct consequence of well-established proof techniques, we establish that the invariant projectors of exponential dichotomies for parameter-dependent nonautonomous difference equations are as smooth as their right-hand sides. For instance, this guarantees that the saddle-point structure in the vicinity of hyperbolic solutions inherits its differentiability properties from the particular given equation.
Abstract:
The multiplicative ergodic theorem by Oseledets on Lyapunov spectrum and Oseledets subspaces is extended to linear random difference equations with random delay. In contrast to the general multiplicative ergodic theorem by Lian and Lu, we can prove that a random dynamical system generated by a difference equation with random delay cannot have infinitely many Lyapunov exponents.
Abstract:
The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.
Abstract:
We study the concept of dissipativity in the sense of Willems for nonautonomous linear-quadratic (LQ) control systems. A nonautonomous system of Hamiltonian ODEs is associated with such an LQ system by way of the Pontryagin Maximum Principle. We relate the concepts of exponential dichotomy and weak disconjugacy for this Hamiltonian ODE to that of dissipativity for the LQ system.
Abstract:
This paper investigates relevant dynamical properties of nonautonomous linear cooperative families of ODEs and FDEs based on the existence of a continuous separation. It provides numerical algorithms for the computation of the dominant one-dimensional subbundle of the continuous separation and the upper Lyapunov exponent of the semiflow. The extension of the theory to general linear cooperative families of ODEs and FDEs without strong monotonicity is also given. Finally these methods and results are applied in the study of nonlinear families of neural networks of Hopfield type with sigmoidal activation function.
Abstract:
The aim of this paper is study the problem of global asymptotic stability of solutions for $\mathbb C$-analytical dynamical systems (both with continuous and discrete time). In particular we present some new results for the $C$-analytical version of Belitskii--Lyubich conjecture. Some applications of these results for periodic $\mathbb C$-analytical differential/difference equations and the equations with impulse are given.
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Now showing items 1-10 of 155
J/Ψ production and nuclear effects in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-02)
Inclusive J/ψ production has been studied with the ALICE detector in p-Pb collisions at the nucleon–nucleon center of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement is performed in the center of mass rapidity ...
Measurement of electrons from beauty hadron decays in pp collisions at root √s=7 TeV
(Elsevier, 2013-04-10)
The production cross section of electrons from semileptonic decays of beauty hadrons was measured at mid-rapidity (|y| < 0.8) in the transverse momentum range 1 < pT <8 GeV/c with the ALICE experiment at the CERN LHC in ...
Suppression of ψ(2S) production in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-12)
The ALICE Collaboration has studied the inclusive production of the charmonium state ψ(2S) in proton-lead (p-Pb) collisions at the nucleon-nucleon centre of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement was ...
Event-by-event mean pT fluctuations in pp and Pb–Pb collisions at the LHC
(Springer, 2014-10)
Event-by-event fluctuations of the mean transverse momentum of charged particles produced in pp collisions at s√ = 0.9, 2.76 and 7 TeV, and Pb–Pb collisions at √sNN = 2.76 TeV are studied as a function of the ...
Kaon femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV
(Elsevier, 2017-12-21)
We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Femtoscopy is used to measure the space-time ...
Anomalous evolution of the near-side jet peak shape in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 TeV
(American Physical Society, 2017-09-08)
The measurement of two-particle angular correlations is a powerful tool to study jet quenching in a $p_{\mathrm{T}}$ region inaccessible by direct jet identification. In these measurements pseudorapidity ($\Delta\eta$) and ...
Online data compression in the ALICE O$^2$ facility
(IOP, 2017)
The ALICE Collaboration and the ALICE O2 project have carried out detailed studies for a new online computing facility planned to be deployed for Run 3 of the Large Hadron Collider (LHC) at CERN. Some of the main aspects ...
Evolution of the longitudinal and azimuthal structure of the near-side peak in Pb–Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV
(American Physical Society, 2017-09-08)
In two-particle angular correlation measurements, jets give rise to a near-side peak, formed by particles associated to a higher $p_{\mathrm{T}}$ trigger particle. Measurements of these correlations as a function of ...
J/$\psi$ elliptic flow in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV
(American Physical Society, 2017-12-15)
We report a precise measurement of the J/$\psi$ elliptic flow in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV with the ALICE detector at the LHC. The J/$\psi$ mesons are reconstructed at mid-rapidity ($|y| < 0.9$) ...
Enhanced production of multi-strange hadrons in high-multiplicity proton-proton collisions
(Nature Publishing Group, 2017)
At sufficiently high temperature and energy density, nuclear matter undergoes a transition to a phase in which quarks and gluons are not confined: the quark–gluon plasma (QGP)1. Such an exotic state of strongly interacting ... |
In a paper by Joos and Zeh, Z Phys B 59 (1985) 223, they say:This 'coming into being of classical properties' appears related to what Heisenberg may have meant by his famous remark [7]: 'Die "Bahn" entsteht erst dadurch, dass wir sie beobachten.'Google Translate says this means something ...
@EmilioPisanty Tough call. It's technical language, so you wouldn't expect every German speaker to be able to provide a correct interpretation—it calls for someone who know how German is used in talking about quantum mechanics.
Litmus are a London-based space rock band formed in 2000 by Martin (bass guitar/vocals), Simon (guitar/vocals) and Ben (drums), joined the following year by Andy Thompson (keyboards, 2001–2007) and Anton (synths). Matt Thompson joined on synth (2002–2004), while Marek replaced Ben in 2003. Oli Mayne (keyboards) joined in 2008, then left in 2010, along with Anton. As of November 2012 the line-up is Martin Litmus (bass/vocals), Simon Fiddler (guitar/vocals), Marek Bublik (drums) and James Hodkinson (keyboards/effects). They are influenced by mid-1970s Hawkwind and Black Sabbath, amongst others.They...
@JohnRennie Well, they repeatedly stressed their model is "trust work time" where there are no fixed hours you have to be there, but unless the rest of my team are night owls like I am I will have to adapt ;)
I think u can get a rough estimate, COVFEFE is 7 characters, probability of a 7-character length string being exactly that is $(1/26)^7\approx 1.2\times 10^{-10}$ so I guess you would have to type approx a billion characters to start getting a good chance that COVFEFE appears.
@ooolb Consider the hyperbolic space $H^n$ with the standard metric. Compute $$\inf\left\{\left(\int u^{2n/(n-2)}\right)^{-(n-2)/n}\left(4\frac{n-1}{n-2}\int|\nabla u|^2+\int Ru^2\right): u\in C^\infty_c\setminus\{0\}, u\ge0\right\}$$
@BalarkaSen sorry if you were in our discord you would know
@ooolb It's unlikely to be $-\infty$ since $H^n$ has bounded geometry so Sobolev embedding works as expected. Construct a metric that blows up near infinity (incomplete is probably necessary) so that the inf is in fact $-\infty$.
@Sid Eating glamorous and expensive food on a regular basis and not as a necessity would mean you're embracing consumer fetish and capitalism, yes. That doesn't inherently prevent you from being a communism, but it does have an ironic implication.
@Sid Eh. I think there's plenty of room between "I think capitalism is a detrimental regime and think we could be better" and "I hate capitalism and will never go near anything associated with it", yet the former is still conceivably communist.
Then we can end up with people arguing is favor "Communism" who distance themselves from, say the USSR and red China, and people who arguing in favor of "Capitalism" who distance themselves from, say the US and the Europe Union.
since I come from a rock n' roll background, the first thing is that I prefer a tonal continuity. I don't like beats as much as I like a riff or something atmospheric (that's mostly why I don't like a lot of rap)
I think I liked Madvillany because it had nonstandard rhyming styles and Madlib's composition
Why is the graviton spin 2, beyond hand-waiving, sense is, you do the gravitational waves thing of reducing $R_{00} = 0$ to $g^{\mu \nu} g_{\rho \sigma,\mu \nu} = 0$ for a weak gravitational field in harmonic coordinates, with solution $g_{\mu \nu} = \varepsilon_{\mu \nu} e^{ikx} + \varepsilon_{\mu \nu}^* e^{-ikx}$, then magic? |
M. Guru Prem Prasad
Guwahati - 781 039, Assam, India
Department of Mathematics, Indian Institute of Technology, Guwahati
Publications:
Guru Prem Prasad M.
Chaotic burst in the dynamics of $f_\lambda (z) = \lambda \frac{\sinh (z)}{z}$
2005, vol. 10, no. 1, pp. 71-80
Abstract
In this paper, a one-parameter family of non-critically finite entire functions $\mathscr{F} \equiv \{f_\lambda(z)=\lambda f(z): \lambda \in \mathbb{R} \setminus \{0\}\}$ with $f(z) = \frac{\sinh z}{z}$ is considered and the dynamics of the entire transcendental functions $f_\lambda \in \mathscr{F}$ is studied in detail. It is shown that there exists a parameter value $\lambda^* > 0$ such that the Julia set of $f_\lambda (z)$ is nowhere dense subset for $0 < |\lambda| \leqslant \lambda^* (\approx 1.104)$. For $|\lambda| > \lambda^*$ the set explodes and becomes equal to the extended complex plane. This phenomenon is referred to as a chaotic burst in the dynamics of the functions $f_\lambda$ in the one-parameter family $\mathscr{F}$. |
Hi, Can someone provide me some self reading material for Condensed matter theory? I've done QFT previously for which I could happily read Peskin supplemented with David Tong. Can you please suggest some references along those lines? Thanks
@skullpatrol The second one was in my MSc and covered considerably less than my first and (I felt) didn't do it in any particularly great way, so distinctly average. The third was pretty decent - I liked the way he did things and was essentially a more mathematically detailed version of the first :)
2. A weird particle or state that is made of a superposition of a torus region with clockwise momentum and anticlockwise momentum, resulting in one that has no momentum along the major circumference of the torus but still nonzero momentum in directions that are not pointing along the torus
Same thought as you, however I think the major challenge of such simulator is the computational cost. GR calculations with its highly nonlinear nature, might be more costy than a computation of a protein.
However I can see some ways approaching it. Recall how Slereah was building some kind of spaceitme database, that could be the first step. Next, one might be looking for machine learning techniques to help on the simulation by using the classifications of spacetimes as machines are known to perform very well on sign problems as a recent paper has shown
Since GR equations are ultimately a system of 10 nonlinear PDEs, it might be possible the solution strategy has some relation with the class of spacetime that is under consideration, thus that might help heavily reduce the parameters need to consider to simulate them
I just mean this: The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge fixing degrees of freedom, which correspond to the freedom to choose a coordinate system.
@ooolb Even if that is really possible (I always can talk about things in a non joking perspective), the issue is that 1) Unlike other people, I cannot incubate my dreams for a certain topic due to Mechanism 1 (consicous desires have reduced probability of appearing in dreams), and 2) For 6 years, my dream still yet to show any sign of revisiting the exact same idea, and there are no known instance of either sequel dreams nor recurrence dreams
@0celo7 I felt this aspect can be helped by machine learning. You can train a neural network with some PDEs of a known class with some known constraints, and let it figure out the best solution for some new PDE after say training it on 1000 different PDEs
Actually that makes me wonder, are the space of all coordinate choices more than all possible moves of Go?
enumaris: From what I understood from the dream, the warp drive showed here may be some variation of the alcuberrie metric with a global topology that has 4 holes in it whereas the original alcuberrie drive, if I recall, don't have holes
orbit stabilizer: h bar is my home chat, because this is the first SE chat I joined. Maths chat is the 2nd one I joined, followed by periodic table, biosphere, factory floor and many others
Btw, since gravity is nonlinear, do we expect if we have a region where spacetime is frame dragged in the clockwise direction being superimposed on a spacetime that is frame dragged in the anticlockwise direction will result in a spacetime with no frame drag? (one possible physical scenario that I can envision such can occur may be when two massive rotating objects with opposite angular velocity are on the course of merging)
Well. I'm a begginer in the study of General Relativity ok? My knowledge about the subject is based on books like Schutz, Hartle,Carroll and introductory papers. About quantum mechanics I have a poor knowledge yet.
So, what I meant about "Gravitational Double slit experiment" is: There's and gravitational analogue of the Double slit experiment, for gravitational waves?
@JackClerk the double slits experiment is just interference of two coherent sources, where we get the two sources from a single light beam using the two slits. But gravitational waves interact so weakly with matter that it's hard to see how we could screen a gravitational wave to get two coherent GW sources.
But if we could figure out a way to do it then yes GWs would interfere just like light wave.
Thank you @Secret and @JohnRennie . But for conclude the discussion, I want to put a "silly picture" here: Imagine a huge double slit plate in space close to a strong source of gravitational waves. Then like water waves, and light, we will see the pattern?
So, if the source (like a Black Hole binary) are sufficent away, then in the regions of destructive interference, space-time would have a flat geometry and then with we put a spherical object in this region the metric will become schwarzschild-like.
if**
Pardon, I just spend some naive-phylosophy time here with these discussions**
The situation was even more dire for Calculus and I managed!
This is a neat strategy I have found-revision becomes more bearable when I have The h Bar open on the side.
In all honesty, I actually prefer exam season! At all other times-as I have observed in this semester, at least-there is nothing exciting to do. This system of tortuous panic, followed by a reward is obviously very satisfying.
My opinion is that I need you kaumudi to decrease the probabilty of h bar having software system infrastructure conversations, which confuse me like hell and is why I take refugee in the maths chat a few weeks ago
(Not that I have questions to ask or anything; like I said, it is a little relieving to be with friends while I am panicked. I think it is possible to gauge how much of a social recluse I am from this, because I spend some of my free time hanging out with you lot, even though I am literally inside a hostel teeming with hundreds of my peers)
that's true. though back in high school ,regardless of code, our teacher taught us to always indent your code to allow easy reading and troubleshooting. We are also taught the 4 spacebar indentation convention
@JohnRennie I wish I can just tab because I am also lazy, but sometimes tab insert 4 spaces while other times it inserts 5-6 spaces, thus screwing up a block of if then conditions in my code, which is why I had no choice
I currently automate almost everything from job submission to data extraction, and later on, with the help of the machine learning group in my uni, we might be able to automate a GUI library search thingy
I can do all tasks related to my work without leaving the text editor (of course, such text editor is emacs). The only inconvenience is that some websites don't render in a optimal way (but most of the work-related ones do)
Hi to all. Does anyone know where I could write matlab code online(for free)? Apparently another one of my institutions great inspirations is to have a matlab-oriented computational physics course without having matlab on the universities pcs. Thanks.
@Kaumudi.H Hacky way: 1st thing is that $\psi\left(x, y, z, t\right) = \psi\left(x, y, t\right)$, so no propagation in $z$-direction. Now, in '$1$ unit' of time, it travels $\frac{\sqrt{3}}{2}$ units in the $y$-direction and $\frac{1}{2}$ units in the $x$-direction. Use this to form a triangle and you'll get the answer with simple trig :)
@Kaumudi.H Ah, it was okayish. It was mostly memory based. Each small question was of 10-15 marks. No idea what they expect me to write for questions like "Describe acoustic and optic phonons" for 15 marks!! I only wrote two small paragraphs...meh. I don't like this subject much :P (physical electronics). Hope to do better in the upcoming tests so that there isn't a huge effect on the gpa.
@Blue Ok, thanks. I found a way by connecting to the servers of the university( the program isn't installed on the pcs on the computer room, but if I connect to the server of the university- which means running remotely another environment, i found an older version of matlab). But thanks again.
@user685252 No; I am saying that it has no bearing on how good you actually are at the subject - it has no bearing on how good you are at applying knowledge; it doesn't test problem solving skills; it doesn't take into account that, if I'm sitting in the office having forgotten the difference between different types of matrix decomposition or something, I can just search the internet (or a textbook), so it doesn't say how good someone is at research in that subject;
it doesn't test how good you are at deriving anything - someone can write down a definition without any understanding, while someone who can derive it, but has forgotten it probably won't have time in an exam situation. In short, testing memory is not the same as testing understanding
If you really want to test someone's understanding, give them a few problems in that area that they've never seen before and give them a reasonable amount of time to do it, with access to textbooks etc. |
I’ve added a new library to Incanter called
incanter.latex that adds the ability to include LaTeX formatted equations as annotations and subtitles in charts. The library is based on the fantastically useful JLaTeXMath library.
The following examples require Incanter version 1.2.2-SNAPSHOT or greater. Add the following dependency to your project.clj file:
[incanter "1.2.2-SNAPSHOT"]
Load the necessary libraries.
(use '(incanter core stats charts latex))
Define the latex-formatted equation; I’ll use the
str function so I can break the equation across multiple lines. Notice that I have to use two backslashes where I would only need one if I were were working directly in LaTeX; this is because the backslash is an escape character in Clojure/Java strings. (def eq (str "f(x)=\\frac{1}{\\sqrt{2\\pi \\sigma^2}}" "e^{\\frac{-(x - \\mu)^2}{2 \\sigma^2}}"))
The equation can be rendered as an image with the
latex function. The rendered equation can then be viewed in a window or saved as a png file with the view and save functions respectively. (view (latex eq)) (save (latex eq) filename)
Use the
add-latex function to add an annotation to a chart. The following example adds the above equation to a function-plot of the Normal PDF. (doto (function-plot pdf-normal -3 3) (add-latex 0 0.1 eq) view)
Use the
add-latex-subtitle function to add a rendered LaTeX equation as a subtitle to the chart (this particular chart does not have a main title). (doto (function-plot pdf-normal -3 3) (add-latex-subtitle eq) view)
The complete code for the above examples can be found here. |
The question itself is very short and sweet, and requires no background.
Find a solution to the following system:$$\left\{\begin{array}{l}a,b,c,d,e,f>0\\\\a+b+c+d+e+f=1\\\\\displaystyle\frac{c}{a+c}\geqslant .999\\\\\displaystyle\frac{d}{b+d}\leqslant .001\\\\\displaystyle\frac{c}{c+d+e}\leqslant .001\\\\\displaystyle\frac{d}{c+d+e}\geqslant .999\end{array}\right.$$I don't need to find the entire solution set. I literally just need one particular solution. It's basic algebra but it's giving me a headache so maybe someone has some software they can use to find it. Why am I asking?
Philosopher Robin Collins wrote a paper where he claims that if $H$ and $K$ are competing hypotheses and $E$ is observed evidence with $P(E|H)\gg P(E|K)$, then $E$ is "strong evidence" for $H$ over $K$. This is an ambiguous claim, but my guess is that he means something like the following:
(LP) If $H\cap K=\emptyset$ and $P(E|H)\gg P(E|K)$, then $P(H|E)\not\ll P(K|E)$.
I believe that (LP) is false, but I'm struggling to produce a counterexample. In particular, I need a counterexample where $P(H\cap K)\neq 1$, $.99\leqslant P(E|H),P(K|E)<1$, and $0<P(E|K),P(H|E)\leqslant .01$. I then assign variables based on the following Venn diagram:
Finding a solution to the system will disprove (LP). |
A nice solution is given by the natural embedding of $S_n$ into $\mathrm{GL}_n(\mathbb{F}_2)$.Here $S_n$ is the symmetric group of $n$ elements. Every element of $S_n$ permutes the canonical basis of $\mathbb{F}_2^n$, hence gives an $\mathbb{F}_2$-linear automorphism of $\mathbb{F}_2^n$. This embeds $S_n$ as a subgroup of $\mathrm{GL}_n(\mathbb{F}_2)$....
If $b^2 + ba + 1$ divides $a^2 + ab+1$, then it also divides $(a^2+ab+1)-(b^2+ba+1) = a^2-b^2$, which factors as $(a+b)(a-b)$.However, it can't share any divisors with the first factor: $\gcd(b^2+ba+1, a+b) = \gcd(b^2+ba+1 - b(a+b), a+b) = \gcd(1, a+b) = 1$.Therefore $b^2+ba+1$ also divides $\frac{a^2-b^2}{a+b} = a-b$.However, $b^2+ba+1 > a > a-b$...
Suppose it is, then also $$13\mid n^2+5n+16$$ so $$13\mid (n^2+5n+16)-13n=n^2-8n+16$$so $$ 13\mid n-4\implies 169\mid (n-4)^2 = n^2-8n+16$$So $$169 \mid (n^2+5n+16)-(n^2-8n+16)= 13n\implies 13\mid n$$ But then from 1.st relation we get $$13\mid 16$$ a contradiction!
If $b^2+ba+1$ divides $a^2+ab+1$ then $b^2+ba+1 \le a^2+ab+1$ and so $b \le a$.Suppose $b<a$. Then $b^2+ba+1$ divides the difference $(a^2+ab+1) - (b^2+ba+1) = a^2-b^2$.But $b^2+ba+1 = b(a+b)+1 = 1 \mod a+b$ and $a^2-b^2=(a-b)(a+b) = 0 \mod a+b$.So either $a+b=1$, in which case $b=0, a=1$; or $b \not \lt a$ in which case $a=b$.
Suppose $p$ is prime.If $p$ divides $2n^7+1$ & $3n^3+2$then $p$ divides $2(2n^7+1)-(3n^3+2)=n^3(4n^4-3)$then $p$ divides $4n^4-3$ ( See Footnote )then $p$ divides $2(4n^4-3)+3(3n^3+2)= n^3(8n+9)$then $p$ divides $8n+9$ (See Footnote)then $p$ divides $9(3n^3+2)-2(8n+9)=n(27n^2-16)$then $p$ divides $27n^2-16$ (See Footnote)then $p$ divides ...
I have a solution, but I'm sure there's a better way to do this. The greatest common divisor $g$ of of $2n^7+1$ and $3n^3+2$ must also divide $$3(2n^7+1)-2n^4(3n^3+2)=3-4n^4$$ then $g$ must also divide $$4n(3n^3+2)-3(4n^4-3)=8n+9$$ Then $g$ must divide $$ 3n^4(8n+9)-8(3n^3+2)=27n^2-16$$Continuing in this manner, we eventually find that $g$ must divide $...
Let $d = \gcd(2n^7+1, 3n^3+2)$. Then since $2n^7+1 \ | \ 2^3n^{21}+1$ and $3n^3 + 2 \ | \ 3^7n^{21}+2^7$, we must have $$d \ | \ 3^7(2^3n^{21}+1) - 2^3(3^7n^{21}+2^7) \quad\Rightarrow\quad d \ | \ 1163.$$Since $1163$ is a prime, if the fraction is reducible, $1163 \ | \ 3n^3 + 2$. Since $1163 \equiv 2 \pmod 3$, $n^3 \equiv -2\cdot 3^{-1} \equiv 587 \pmod{...
Let us prove that for every $\varepsilon > 0$ there exist infinitely many integers $n > 0$ such that $P(n^2 + 1) < \varepsilon n$.Obviously, it is enough to prove that there exists one such $n$. As you noticed, if $n = 2m^2$, for some integer $m > 0$, then $n^2 + 1 = (2m^2 - 2m + 1)(2m^2 + 2m + 1)$. Hence, it is enough to prove that we can ...
This gcd can be computed purely mechanically by a slight generalization of the Euclidean algorithm which allows us to scale by integers $\,c\,$ coprime to the gcd during the modular reduction step, i.e.$$(a,b)\, = \,(a,\,cb\bmod a)\ \ \ {\rm if}\ \ \ (a,c) = 1\qquad\qquad\ \, $$which is true since $\,(a,c)= 1\,\Rightarrow\, (a,\,cb\bmod a) = (a,\,b\bmod ...
Hint:Term $n-1$ in $S_2$ has the value $n$. If term $n-1$ in $S_n$ still has the value $n$ then this means this value $n$ was not changed at any point during the construction of $S_3, S_4, \dots S_n$ (because once a value in the sequence is changed, it can only increase - it can never go back to its original value).But if $n$ was divisible by some $m$ in ...
Let's try to complete the square. Equivalences below are $\bmod 169$.$x^2+5x+16\equiv 0$$4x^2+20x+64\equiv 0$$4x^2+20x+25=(2x+5)^2\equiv 25-64=-39$We need to find a quantity whose square is $\equiv -39$. Unfortunately this is a multiple of $13$ and the only square multiples of $13$ are also multiples of $169$ --therefore $\equiv 0\not\equiv -39$. ...
Let $n\in \Bbb Z$ be so that $n^2+5n+16$ is divisible by $13$. Then working modulo $13$ we have$$\begin{aligned}n^2+5n+16&\equiv n^2 + 18n + 81 \\&= (n+9)^2 \qquad\text{ modulo }13\ .\end{aligned}$$So $n$ is of the form $n=4+13k$, we substitute and get (computation in $\Bbb Z$):$$\begin{aligned}n^2+5n+16&= (13k +4)^2 + 5(13k+4) + 16 \\...
I don't see how your observation of the divisors of powers of primes can be applied here. The question says $a,b,c,d$ are relatively prime, which means no $2$ of them have any factor in common, but none of them are necessarily prime and, in fact, one or more may be $1$, not to mention possibly being composite numbers.Instead, first note $a,b,c,d$ are each ... |
Problem 1 from the EGMO2017 What Is It About? Problem
Solution 1
The main device for the solution is angle chasing and the main configuration to which the angle chasing will be applied is the right triangle:
Since in any right triangle the circumcenter is the midpoint of the hypotenuse, $\angle YOZ=2\angle YXZ,\,$ as the central and inscribed angles subtended by the same arc.
To return to the problem, assume $\angle ADC\gt\angle ABC.\,$ (The case where $\angle ADC\lt\angle ABC\,$ is entirely analogous; in case $\angle ADC=\angle ABC\,$ $ABCD\,$ is a rectangle, the circle degenerates into a straight line and the four points $A,\,$ $M\,$ $N,\,$ and $C\,$ are collinear.)
Now, in $\Delta ABD,\,$ $\angle AMD=2\angle ABD\,$ and, in $\Delta BCD,\,$ $\angle CMD=2\angle CBD,\,$ such that
$\angle AMC=2\angle ABC.$
In $\Delta PAS,\,$ $\angle ANS=2\angle APN\,$ and, in $\Delta QCR,\,$ $\angle CNR=2\angle CQR$ such that
$\begin{align}\angle ANC&=2(\angle APN+\angle CQR)\\ &=2(180^{\circ}-\angle PBQ)=2\angle ABC=\angle AMC. \end{align}$
So that, finally, $\angle ANC=\angle AMC,\,$ meaning they are inscribed into the same circle and, thus points $A,\,$ $M\,$ $N,\,$ and $C\,$ are indeed concyclic.
Solution 2
Let $\hat{x}$, $\hat{y}$, $\hat{u}$, and $\hat{v}$ be unit vectors along $\vec{AP}$, $\vec{AS}$, $\vec{CR}$ and $\vec{CQ}$, respectively. $(\hat{u},\hat{v})$ and $(\hat{x},\hat{y})$ are two fixed orthogonal systems and can thus be related by an orthogonal transformation:
$\displaystyle\begin{align} \hat{u}&= \cos\xi\hat{x}+\sin\xi\hat{y} \\ \hat{v}&= -\sin\xi\hat{x}+\cos\xi\hat{y}. \end{align}$
Here, $\xi$ is the angle between $\hat{x}$ and $\hat{u}$.
Let the radius of the circumcircles of $CRQ$ and $APS$ be $r$ and $R$, respectively. Let $\angle CNR = 2\alpha$ and $\angle ANS = 2\beta$. Thus,
$\displaystyle\begin{align} \vec{CR}&= 2r\sin\alpha\hat{u} \\ \vec{CQ}&= 2r\cos\alpha\hat{v} \\ \vec{CN}&= r\sin\alpha\hat{u}+r\cos\alpha\hat{v} \\ \vec{AS}&= 2R\sin\beta\hat{y} \\ \vec{AP}&= 2R\cos\beta\hat{x} \\ \vec{AN}&= R\cos\beta\hat{x}+R\sin\beta\hat{y} \end{align}$
and
$\displaystyle\begin{align} \vec{NS}&= \vec{AS}-\vec{AN}=-R\cos\beta\hat{x}+R\sin\beta\hat{y} \\ \vec{NR}&= \vec{CR}-\vec{CN}=r\sin\alpha\hat{u}-r\cos\alpha\hat{v} \\ &= (r\sin\alpha\cos\xi+r\cos\alpha\sin\xi)\hat{x} + (r\sin\alpha\sin\xi-r\cos\alpha\cos\xi)\hat{y} \\ &= r\sin(\alpha+\xi)\hat{x}-r\cos(\alpha+\xi)\hat{y} \end{align}$
$\vec{NR}$ and $\vec{NS}$ are parallel. Thus,
$\displaystyle\begin{align} \cos\beta\cos(\alpha+\xi)-\sin\beta\sin(\alpha+\xi)&= 0 \\ \cos(\alpha+\beta+\xi)&= 0 \end{align}$
Thus, $\alpha+\beta=(2K+1)\frac{\pi}{2}-\xi$ for integral $K$. The above analysis holds if $M$ is chosen as a special case of $N$. In this case, $R$ ans $S$ coalesce into $D$ and $Q$ and $P$ intp $B$. Note that as $N$ is continuously transformed into $M$, the integral value of $K$ has to remain fixed otherwise, $\angle CNA = 2(\alpha+\beta)$ will have to make step jumps of multiples of $\pi$. This also means that $\angle CNA = \angle CMA$. Thus, $CNMA$ are concyclic.
Acknowledgment
I am grateful to Vincent Pantaloni for bringing this problem (Problem 1 from the European Girls' Mathematical Olympiad 2017) to my attention over the twitter and to Vaggelis Stamatiadis for the solution (Solution 1) at the Οι Ρομαντικοι της Γεωμετριας (Romantics of Geometry) facebook group. Solution 2 is by Amit Itagi
65620555 |
Advances in Differential Equations Adv. Differential Equations Volume 8, Number 7 (2003), 843-872. Ultracontractivity and convergence to equilibrium for supercritical quasilinear parabolic equations on Riemannian manifolds Abstract
Let $(M,g)$ be a compact Riemannian manifold without boundary and dimension $d\ge3$. Let $u(t)$ be a solution to the problem $\dot u=\triangle_pu$, $u(0)=u_0$, $\triangle_p$ being the Riemannian $p$--Laplacian with $p>d$. Let also $\overline{u}$ be the (time--independent) mean of $u(t)$. We will prove ultracontractive estimates of the type $\Vert u(t)-\overline{u}\Vert_\infty\le C\Vert u(0)-\overline{u}\Vert_q^\gamma/t^\beta$. The constant $C$ depends only on $p$ and $q$, on geometric quantities of $M$ and on the dimension of the manifold, while the exponents $\beta$ and $\gamma$ depend only on $p$ and $q$ and differ according to the regimes $t\to0$ and $t\to+\infty$. Similar bounds hold when $\triangle_p$ is replaced by the subelliptic $p$--Laplacian associated to a collection of Hörmander vector fields. We also prove the L$^q$--L$^\infty$ Höolder continuity of the solutions, and apply similar methods to study the same questions for evolution equations on manifolds with boundary. The bounds are sharp in several of the above cases. The method relies on the theory of nonlinear Markov semigroups (\cite{CG2}) and on the connection between nonlinear ultracontractivity and logarithmic Sobolev inequality for the $p$--energy functional.
Article information Source Adv. Differential Equations, Volume 8, Number 7 (2003), 843-872. Dates First available in Project Euclid: 19 December 2012 Permanent link to this document https://projecteuclid.org/euclid.ade/1355926814 Mathematical Reviews number (MathSciNet) MR1988681 Zentralblatt MATH identifier 1031.58004 Subjects Primary: 58D07: Groups and semigroups of nonlinear operators [See also 17B65, 47H20] Secondary: 35K55: Nonlinear parabolic equations 35K65: Degenerate parabolic equations 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07] 58J35: Heat and other parabolic equation methods Citation
Bonforte, Matteo; Cipriani, Fabio; Grillo, Gabriele. Ultracontractivity and convergence to equilibrium for supercritical quasilinear parabolic equations on Riemannian manifolds. Adv. Differential Equations 8 (2003), no. 7, 843--872. https://projecteuclid.org/euclid.ade/1355926814 |
I'm looking to model a repeating sinc pulse in LTSPICE, however I can only get it to fire once. I'm looking to delay it by x many ms, and trigger the pulse the again.
Is this possible with LTSPICE ?
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Just similar to what I wrote in this answer you can do almost anything repetitive by using the time variable and a modulus. So let us set this up step by step. First just a sinc function for convenience:
.function sinc(x) { sin(x)/x }
Using this together with the
time variable and a little scaling we can get this b source:
V=sinc(time*100)
to produce this here:
This isn't repetetive so let us create a modulo function and use that to bring back the parameter into range:
.function mod(x,y) { (x/y)-int(x/y) }V=sinc(mod(time,1)*100)
Better, but not quite nice as the "pulse" starts at 0 of the sinc again. It would be better if it started a bit lower, so lets shift and align it a little bit nicer:
V=sinc(mod(time,pi/3)*100-pi)
You should now be able to take this as a starting point to calculate a waveform that matches your needs.
I'm not sure if this is helpful in the context of your simulation, but you can produce an approximation to a series of sinc pulses with a slight negative DC bias by simply summing together a set of cosine waves. Start with one at the pulse repetition frequency, and add integer harmonics of that frequency, all at the same amplitude:
$$pulse(t) = \sum_{i=1}^N cos (2\pi i f t)$$
The value you select for N will determine the width of the individual pulses; a higher value will create narrower pulses. Also, as N increases, the peak amplitude of the pulses increases linearly as well; scale by \$\frac{1}{N}\$ if desired.
See an example on Wolfram Alpha |
A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated.
Let g a Hecke-Maass form for SL3(Z) which do not come from a symmetric square lift, and f be a Hecke-Maass cusp form for SL2(Z) of level $q$. The Rankin-Selberg L-function is defined by $$L( s,g\times f)=\sum_{m,n\ge 1}\frac{A(m,n)a(n)}{(m^2n)^s}.$$
My question is
how to prove subconvexity bound on level aspect
$$L(\frac{1}{2},g\times f)\ll q^{3/4-\epsilon},\quad \text{some constant }\epsilon>0 \hskip2em ?$$ And who has studied it? please show me their works.
Remark: Suppose $f_1,f_2$ be Hecke-Maass cusp forms on SL2(Z). In many papers, the upperbound for $L(\frac{1}{2},\text{sym}^2(f_1)\times f_2)$ has greatly improved. However there was few literature involving in L-function with general GL(3) Hecke-Maass form twisted by a GL(2) cusp form, i.e. $L(\frac{1}{2},g\times f)$.
So far I know that Rizwanur Khan ( link his paper here) prove a conditional result, he proved that suppose $f$ be holomorphic, and $\sum_{n<L}a(n)^2\gg L^{1-\epsilon} \text{ for} L>q^{\frac{1}{4}+\frac{1}{2001}}$, then $L(\frac{1}{2},g\times f)\ll q^{3/4-1/2001}.$ If there are other literature studying $L(\frac{1}{2},g\times f)$, please guide me their names or papers.
Another stupid little question is let $g$ be a a Hecke-Maass form for SL3(Z) which is self-dual, I'm not sure whether $g$ must come from a symmetric square lift or not? |
This is several years late, but it may be helpful nonetheless.
As alluded to by Buschi, Olivia has given an explicit answer to this in Theorem 2.1.29 of her monograph
Theories, Sites and Toposes:
Let $\mathbb{T}$ be a geometric theory, $\mathcal{E}$ a Grothendieck
topos and $M$ a model of $\mathbb{T}$ in $\mathcal{E}$. Then
$\mathcal{E}$ is a classifying topos of $\mathbb{T}$ and $M$ is a
universal model (i.e. generic model) of $\mathbb{T}$ iff the following
conditions are satisfied:
The family $F$ of objects which can be built from the interpretations in $M$ of the sorts, function symbols and relation
symbols over the signature of $\mathbb{T}$ by using geometric logic
constructions (i.e. the objects given by the domains of the
interpretations in $M$ of geometric formulae over the signature of
$\mathbb{T}$) is separating for $\mathcal{E}$.
The model $M$ is conservative for $\mathbb{T}$, that is for any geometric sequent $\sigma$ over the signature of $\mathbb{T}$, $\sigma$ is valid in $M$ if and only if it is provable in
$\mathbb{T}$.
Any arrow $k$ in $\mathcal{E}$ between objects $A$ and $B$ in the family $F$ of condition (1) is definable; that is, if $A$ (resp. $B$)
is equal to the interpretation of a geometric formula $\phi(\vec{x})$
(resp. $\psi(\vec{y})$) over the signature of $\mathbb{T}$, there
exists a $\mathbb{T}$-provably functional formula $\theta$ from
$\phi(\vec{x})$ to $\psi(\vec{y})$ such that the interpretation of
$\theta$ in $M$ is equal to the graph of $k$.
I'm not sure if you were looking necessary and sufficient conditions on $M$ and $\mathcal{E}$, or just merely sufficient conditions, but since this Theorem gives an 'iff' result, one might try and prove the sufficiency of certain (perhaps more intuitive) critiera on $M$ and $\mathcal{E}$ by checking against the conditions listed in this theorem, i.e. by proving results of the flavour: 'If $M$ and $\mathcal{E}$ satisfy condition $X$, then they satisfy the 3 conditions of this theorem.'
Extending this thought, I am curious to see how these conditions relate to the special case mentioned in Dylan's comment. In particular, how does weak contractibility relate to the conditions spelt out by Olivia? This is not obvious to me, but I haven't taken the time to properly work through the details. |
In general, (.*)stationary orbits are simply the altitude where a circular orbit has a period equal to the rotational period of the central body.
In order to find this altitude, we can transform Kepler's third law to solve for R:
Kepler's third law:$\frac{T^2}{R^3} = \frac{4\pi^2}{G M_{central}}$
Solved for $R$: $R = \sqrt[3]{\frac{T^2 G M_{central}}{4\pi^2}}$
Using $M_{earth}$ and $T_{earth}$:
$R = \sqrt[3]{\frac{(86164)^2 * (6.67e-11) * (5.97e24)}{4\pi^2}} = \sqrt[3]{\frac{2.97e24}{39.5}} = 4.22e7m = 42,200km$,
where 86164 seconds is the approximate length of Earth's sidereal day (~23h, 56m, 4s).
Subtracting the radius of Earth which is ~6370km, this is an altitude of ~35,800km. This can be confirmed by google.
Using $M_{mars}$ and $T_{mars}$ (which is 40 minutes longer than an Earth day):
$R = \sqrt[3]{\frac{(88643)^2 * (6.67e-11) * (6.39e23)}{4\pi^2}} = \sqrt[3]{\frac{3.36e23}{39.5}} = 2.04e7m = 20,400km$,
where 88643 seconds is the approximate length of a Martian sidereal day (~24h, 37m, 23s).
Subtracting the radius of Mars which is ~3390km, this gives an altitude of about 17,000km.
You can easily look up and substitute the numbers of any celestial body for this! |
Recall that in the adversarial contextual $K$-action bandit problem, at the beginning of each round $t$ a context $c_t\in \Ctx$ is observed. The idea is that the context $c_t$ may help the learner to choose a better action. This led Continue Reading
In most bandit problems there is likely to be some additional information available at the beginning of rounds and often this information can potentially help with the action choices. For example, in a web article recommendation system, where the goal Continue Reading
In the post on adversarial bandits we proved two high probability upper bounds on the regret of Exp-IX. Specifically, we showed: Theorem: There exists a policy $\pi$ such that for all $\delta \in (0,1)$ for any adversarial environment $\nu\in [0,1]^{nK}$, Continue Reading
A stochastic bandit with $K$ actions is completely determined by the distributions of rewards, $P_1,\dots,P_K$, of the respective actions. In particular, in round $t$, the distribution of the reward $X_t$ received by a learner choosing action $A_t\in [K]$ is $P_{A_t}$, Continue Reading
In the last post we showed that under mild assumptions ($n = \Omega(K)$ and Gaussian noise), the regret in the worst case is at least $\Omega(\sqrt{Kn})$. More precisely, we showed that for every policy $\pi$ and $n\ge K-1$ there exists Continue Reading
Continuing the previous post, we prove the claimed minimax lower bound. We start with a useful result that quantifies the difficulty of identifying whether or not an observation is drawn from similar distributions $P$ and $Q$ defined over the same Continue Reading
In this post we introduce the concept of minimax regret and reformulate our previous result on the upper bound on the worst-case regret of UCB in terms of the minimax regret. We briefly discuss the strengths and weaknesses of using Continue Reading
We now describe the celebrated Upper Confidence Bound (UCB) algorithm that overcomes all of the limitations of strategies based on exploration followed by commitment, including the need to know the horizon and sub-optimality gaps. The algorithm has many different forms, Continue Reading
With most of the background material out of the way, we are almost ready to start designing algorithms for finite-armed stochastic bandits and analyzing their properties, and especially the regret. The last thing we need is an introduction to concentration Continue Reading
On Monday last week we did not have a lecture, so the lectures spilled over to this week’s Monday. This week was devoted to building up foundations, and this post will summarize how far we got. The post is pretty Continue Reading |
I was reading Landau's
Foundations of Analysis. He starts his construction of number systems by stating five axioms. My question is related to the fifth, the axiom of induction: Let there be given a set $\mathscr{N}$, of natural numbers with the following properties:
I)
1 belongs to $\mathscr{N}$
II)
If $x$ belongs to $\mathscr{N}$ then so does $x'$ Then $\mathscr{N}$ contains all the natural numbers
(Here $x'$ is the successor of $x$ )
Now consider the following proposition:
Let $(A,≤)$ be an ordered set. If $S$ is a finite subset of $A$ then $S$ contained a maximum and a minimum.
This can be proved by induction on the cardinal of $S$.
Now my (first) question is, for the induction to "work" is it necessary to first prove that each $S$ has an unique cardinal? That is, that there is an unique $n\in \Bbb{N}$ such that there exists a bijection between $S$ and $\left \{ 1, \cdots ,n \right \}$.
In accordance with the axiom of induction we would be forming the set: $$\mathscr{N}=\left \{ n \in \Bbb{N} : S\subset A \ \wedge \ \#(S)=n \Rightarrow S \ \text{contains a maximum and a minimum} \right \}$$
Is this correct?
Now, for my second (third) question, consider Bernoulli's inequality:
$$\prod_{i=1}^{n}(1+a_i)≥1+\sum_{i=1}^{n}a_i \, \quad \text{if $a_i≥-1$, $i=1,...n$}$$
If I wanted to construct the set $\mathscr{N}$ cited in the axiom, how would I do it? Furthermore, should any care be taken when defining the set, analogous to proving that the cardinal is unique in the previous question? |
I am working on a problem where I have successfully reduced a version of Black Scholes to the Heat Equation and then shown the solution to be:
$$u(x,t)=\frac{1}{2\sqrt{t\pi}}\int_{-\infty}^\infty{f(\xi)e^{-\frac{(x-\xi)^2}{4t}}}d\xi$$
I now need to show that if $f(x)$ is continuous then $$\lim_{t\rightarrow 0+}u(x,t)=f(x)$$
Further, there is a tip that a change of variables of $p=\frac{(\xi-x)}{2\sqrt{t}}$ may help.
I think that I need to do some integration by parts, show that some part of the integration go to zero as $t\rightarrow0+$ which will cancel out and then by linearity I can say that as u(x,t) is a solution and then $f(x)$ must also be a solution. I am just missing the first step and hoped someone can give me a nudge!
Thanks for any help! |
28A75 Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] Refine
We show that the intersection local times \(\mu_p\) on the intersection of \(p\) independent planar Brownian paths have an average density of order three with respect to the gauge function \(r^2\pi\cdot (log(1/r)/\pi)^p\), more precisely, almost surely, \[ \lim\limits_{\varepsilon\downarrow 0} \frac{1}{log |log\ \varepsilon|} \int_\varepsilon^{1/e} \frac{\mu_p(B(x,r))}{r^2\pi\cdot (log(1/r)/\pi)^p} \frac{dr}{r\ log (1/r)} = 2^p \mbox{ at $\mu_p$-almost every $x$.} \] We also show that the lacunarity distributions of \(\mu_p\), at \(\mu_p\)-almost every point, is given as the distribution of the product of \(p\) independent gamma(2)-distributed random variables. The main tools of the proof are a Palm distribution associated with the intersection local time and an approximation theorem of Le Gall.
Symmetry properties of average densities and tangent measure distributions of measures on the line (1995)
Answering a question by Bedford and Fisher we show that for every Radon measure on the line with positive and finite lower and upper densities the one-sided average densities always agree with one half of the circular average densities at almost every point. We infer this result from a more general formula, which involves the notion of a tangent measure distribution introduced by Bandt and Graf. This formula shows that the tangent measure distributions are Palm distributions and define self-similar random measures in the sense of U. Zähle.
Tangent measure distributions were introduced by Bandt and Graf as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by contractive mappings, which are not similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models of Bedford and Fisher.
Tangent measure distributions are a natural tool to describe the local geometry of arbitrary measures of any dimension. We show that for every measure on a Euclidean space and every s, at almost every point, all s-dimensional tangent measure distributions define statistically self-similar random measures. Consequently, the local geometry of general measures is not different from the local geometry of self-similar sets. We illustrate the strength of this result by showing how it can be used to improve recently proved relations between ordinary and average densities.
We show that the occupation measure on the path of a planar Brownian motion run for an arbitrary finite time intervalhas an average density of order three with respect to thegauge function t^2 log(1/t). This is a surprising resultas it seems to be the first instance where gauge functions other than t^s and average densities of order higher than two appear naturally. We also show that the average densityof order two fails to exist and prove that the density distributions, or lacunarity distributions, of order threeof the occupation measure of a planar Brownian motion are gamma distributions with parameter 2. |
I posted this on Math Stack Exchange one month ago, but did not receive any responses. The original question (in a simplified form) can be found here.
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be an $L^{\infty}$ function on $[0,2\pi)^d$, whose Fourier coefficients are real and are given by $\hat{f}$, and in particular $\hat{f}(0) = 0$. Suppose also that $\Gamma \subset \mathbb{R}^d$ is a rational lattice of full rank. I would like to compute the $L^4$-norm over $[0,2\pi)^d$ by using its Fourier series. Explicitly:
$$\displaystyle \int_{[0,2\pi)^d} |f(k)|^4 \mathrm{d}k = \int_{[0,2\pi)^d}\left( \sum_{a \in \Gamma} \hat{f}(a)e^{iak} \right)\left( \sum_{b \in \Gamma} \hat{f}(b)e^{-ibk} \right) \left( \sum_{c \in \Gamma} \hat{f}(c)e^{ick} \right)\left( \sum_{d \in \Gamma} \hat{f}(d)e^{-idk} \right)\mathrm{d}k,$$
where I have used the relation $|f|^4$ = $(|f|^2)^2$ = $f^2 \cdot \bar{f}^2$. Could anyone tell me what the resulting sum looks like, inside the integral? I'm not familiar with how to take Cauchy products of series whose indices range over a rational lattice. Essentially what I am trying to do is a repeat of the argument here, except with four copies of the Fourier series (up to conjugation) rather than two -- and then use the fact that
$$\displaystyle \int_{0}^{2\pi} e^{ink}\mathrm{d}k = \begin{cases} 0, & n\neq 0, \\ 2\pi, & n = 0, \end{cases}$$
to impose a condition on the resulting sum. If $d = 1$, and we take the lattice $\Gamma$ to be $\mathbb{Z}$, and let the indices range from $0$ to $\infty$ then I've ended up with:
$$\displaystyle \int_{0}^{2\pi} \sum_{a = 0}^{\infty}\sum_{b=0}^{a}\sum_{c=0}^{b}\sum_{d=0}^{c} \hat{f}(d)\hat{f}(c-d)\hat{f}(b-c)\hat{f}(a-b)e^{i(2d - 2c + 2b - a)k}\mathrm{d}k,$$
from which we can impose the condition $2d - 2c + 2b - a = 0$ to get rid of the integral and introduce a factor of $2\pi$. But where can we go from here? I know that the Fourier coefficients for general dimension $d$ are:
$$\displaystyle \hat{f}(n) = (2\pi)^{d/2}\rho^{d/2}\|n\|^{-d/2}J_{d/2}(\rho \|n\|),$$
where $\| \cdot \|$ denotes the Euclidean norm, $\rho > 0$ is a constant, and $J_{\nu}$ is the Bessel function of the first kind. How can we simplify the sum from here?
For background information on where this problem comes from, see this paper (in particular, pages 10-11, 15-16). The paper considers $\sigma_{p}$ for $p = 1,2$, and this is my attempt to generalise that work to the case $p = 4$ and for general $p.$ The goal is to bound the $L^4$-norm. |
Permanent link: https://www.ias.ac.in/article/fulltext/pram/078/04/0613-0623
A sequential three-dimensional (3D) particle-in-cell simulation code PICPSI-3D with a user friendly graphical user interface (GUI) has been developed and used to study the interaction of plasma with ultrahigh intensity laser radiation. A case study of laser–plasma-based electron acceleration has been carried out to assess the performance of this code. Simulations have been performed for a Gaussian laser beam of peak intensity $5 \times 10^{19}$ W/cm
2 propagating through an underdense plasma of uniform density $1 \times 10^{19}$ cm -3, and for a Gaussian laser beam of peak intensity $1.5 \times 10^{19}$ W/cm 2 propagating through an underdense plasma of uniform density $3.5 \times 10^{19}$ cm -3. The electron energy spectrum has been evaluated at different time-steps during the propagation of the laser beam. When the plasma density is $1 \times 10^{19}$ cm -3, simulations show that the electron energy spectrum forms a monoenergetic peak at $\sim 14$ MeV, with an energy spread of $\pm 7$ MeV. On the other hand, when the plasma density is $3.5 \times 10^{19}$ cm -3, simulations show that the electron energy spectrum forms a monoenergetic peak at $\sim 23$ MeV, with an energy spread of $\pm 7.5$ MeV.
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Volume 93 | Issue 6 December 2019
Click here for Editorial Note on CAP Mode |
This is slightly contrived, but consider a situation where you have two balls, of mass $M$ and $m$, where $M=16\times100^N\times m$ for some integer $N$. The balls are placed against a wall as shown:
We push the heavy ball towards the lighter one and the wall. The balls are assumed to collide elastically with the wall and with each other. The smaller ball bounces off the larger ball, hits the wall and bounces back. At this point there are two possible solutions: the balls collide with each other infinitely many times until the larger ball reaches the wall (assume they have no size), or the collisions from the smaller ball eventually cause the larger ball to turn around and start heading in the other direction - away from the wall.
In fact, it is the second scenario which occurs: the larger ball eventually heads away from the wall. Denote by $p(N)$ the number of collisions between the two balls before the larger one changes direction, and gaze in astonishment at the values of $p(N)$ for various $N$:
\begin{align}p(0)&=3\\p(1)&=31\\p(2)&=314\\p(3)&=3141\\p(4)&=31415\\p(5)&=314159\\\end{align}
and so on. $p(N)$
is the first $N+1$ digits of $\pi$!
This can be made to work in other bases in the obvious way.
See 'Playing Pool with $\pi$' by Gregory Galperin. |
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3. Search for dark matter and other new phenomena in events with an energetic jet and large missing transverse momentum using the ATLAS detector
Journal of High Energy Physics, ISSN 1126-6708, 1/2018, Volume 2018, Issue 1, pp. 1 - 53
Results of a search for new phenomena in final states with an energetic jet and large missing transverse momentum are reported. The search uses proton-proton...
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4. Combination of inclusive and differential $ \mathrm{t}\overline{\mathrm{t}} $ charge asymmetry measurements using ATLAS and CMS data at $ \sqrt{s}=7 $ and 8 TeV
Journal of High Energy Physics (Online), ISSN 1029-8479, 04/2018, Volume 2018, Issue 4
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5. Search for Higgs boson decays to beyond-the-Standard-Model light bosons in four-lepton events with the ATLAS detector at $\sqrt{s}=13$ TeV
Journal of High Energy Physics (Online), ISSN 1029-8479, 02/2018, Volume 2018, Issue 6
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Phys. Rev. Lett. 120 (2018) 202007 A search for the narrow structure, $X(5568)$, reported by the D0 Collaboration in the decay sequence $X \to B^0_s \pi^\pm$,...
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7. Measurement of the cross section for isolated-photon plus jet production in $pp$ collisions at $\sqrt s=13$ TeV using the ATLAS detector
12/2017
Phys. Lett. B 780 (2018) 578 The dynamics of isolated-photon production in association with a jet in proton-proton collisions at a centre-of-mass energy of 13...
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The European Physical Journal C: Particles and Fields, ISSN 1434-6052, 2017, Volume 77, Issue 5, pp. 1 - 53
During 2015 the ATLAS experiment recorded $$3.8\,{\mathrm{fb}}^{-1}$$ 3.8 fb - 1 of proton–proton collision data at a centre-of-mass energy of...
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European Physical Journal. C, Particles and Fields, ISSN 1434-6044, 10/2017, Volume 78, Issue 1
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European Physical Journal C, ISSN 1434-6044, 2017, Volume 77, Issue 7, pp. 1 - 73
The reconstruction of the signal from hadrons and jets emerging from the proton–proton collisions at the Large Hadron Collider (LHC) and entering the ATLAS...
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12. Measurement of Higgs boson production in the diphoton decay channel in pp collisions at center-of-mass energies of 7 and 8 TeV with the ATLAS detector
Physical Review D, ISSN 1550-7998, 2014, Volume 90, Issue 11, p. 112015
A measurement of the production processes of the recently discovered Higgs boson is performed in the two-photon final state using 4.5 fb(-1) of proton-proton...
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13. Constraints on new phenomena via Higgs boson couplings and invisible decays with the ATLAS detector
Journal of High Energy Physics, ISSN 1126-6708, 2015, Volume 2015, Issue 11, pp. 1 - 52
The ATLAS experiment at the LHC has measured the Higgs boson couplings and mass, and searched for invisible Higgs boson decays, using multiple production and...
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14. Measurement of the production cross-section of a single top quark in association with a Z boson in proton–proton collisions at 13 TeV with the ATLAS detector
Physics Letters B, ISSN 0370-2693, 05/2018, Volume 780, Issue C, pp. 557 - 577
The production of a top quark in association with a boson is investigated. The proton–proton collision data collected by the ATLAS experiment at the LHC in...
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15. Search for heavy particles decaying into top-quark pairs using lepton-plus-jets events in proton–proton collisions at $\sqrt{s} = 13$ $\text {TeV}$ with the ATLAS detector
European Physical Journal. C, Particles and Fields, ISSN 1434-6044, 07/2018, Volume 78, Issue 7
Here, a search for new heavy particles that decay into top-quark pairs is performed using data collected from proton–proton collisions at a centre-of-mass...
PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
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16. Search for pair production of up-type vector-like quarks and for four-top-quark events in final states with multiple b-jets with the ATLAS detector
Journal of High Energy Physics, ISSN 1126-6708, 7/2018, Volume 2018, Issue 7, pp. 1 - 68
A search for pair production of up-type vector-like quarks (T ) with a significant branching ratio into a top quark and either a Standard Model Higgs boson or...
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Journal Article |
Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(4963\) \(x\mathstrut +\mathstrut \) \(96223\):
\(\beta_{0}\) \(=\) \( 1 \) \(\beta_{1}\) \(=\) \( 64 \nu^{2} + 640 \nu - 211989 \) \(\beta_{2}\) \(=\) \( 8896 \nu^{2} + 662400 \nu - 29657664 \)
\(1\) \(=\) \(\beta_0\) \(\nu\) \(=\) \((\)\(\beta_{2}\mathstrut -\mathstrut \) \(139\) \(\beta_{1}\mathstrut +\mathstrut \) \(191193\)\()/573440\) \(\nu^{2}\) \(=\) \((\)\(-\)\(\beta_{2}\mathstrut +\mathstrut \) \(1035\) \(\beta_{1}\mathstrut +\mathstrut \) \(189750951\)\()/57344\)
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
This newform does not admit any (nontrivial) inner twists.
\( p \) Sign \(2\) \(-1\)
This newform can be constructed as the kernel of the linear operator \(T_{3}^{3} \) \(\mathstrut -\mathstrut 96764 T_{3}^{2} \) \(\mathstrut -\mathstrut 20431242576 T_{3} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\( \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(8))\). |
Rate of turn is dependent on the following two items:
The horizontal component of lift (centripetal force) The tangential velocity of the aircraft (true airspeed)
The rate or turn is directly proportional to the horizontal component of lift and inversely proportional to the tangential velocity of the aircraft.
For a given angle of bank, the vertical and horizontal components of lift will be the same, regardless of airspeed in level flight.
Consequently the airplane will experience the same centripetal acceleration, regardless of airspeed.
Since the tangential velocity is slower, any kind of centripetal force will produce a greater rate of turn for a slower flying aircraft as opposed to a faster moving aircraft and this can be shown by the centripetal acceleration equation
$$a_c = \frac{v^2}{r}$$
so both slow flying airplane with a true airspeed $v_s = 100$ knots and fast flying airplane with a true airspeed $v_f = 200$ knots experience the same centripetal acceleration.
$$\dfrac{v_s^2}{r_s} = \dfrac{v_f^2}{r_f} = 4\ \dfrac{v_s^2}{r_f}$$
or, $$\dfrac{1}{r_s} = \dfrac{4}{r_f}$$
Consequently $r_s < r_f$; in this case $r_f = 4\ r_s$
Since the angular velocity is equal to the tangential speed divided by the radius.
$$\omega = v/r$$
the angular speed of the slower aircraft will be greater than the faster aircraft.
$$\omega_s = v_s/r_s$$
and
$$\omega_f = \dfrac{v_f}{r_f} = \dfrac{2 \ v_s}{4 \ r_s} = \frac{1}{2}w_s$$
So our twice as slow airplane turns twice as fast as the faster one does under these conditions. |
Full disclosure, I've asked a very similar question before which didn't have much success as it was too specific. However I want to try again by restricting the scope of the question.
Scenario
I'm performing measurements of a transfer function of a DAQ (NI USB-6251). I have placed 2 MΩ in series between the input and output of the DAQ:
The output/input impedances in the above schematic come from the DAQ datasheet.
Due to the 100 pF at the input to the DAQ I expect to see a low pass filter with cutoff frequency at $$ \frac{1}{2\pi (2\times 10^6)(100\times10^{-12})} = 795\ \text{Hz} $$
Measured results
Fitting an RC model to the measurements I find the capacitance to be around 57 pF (1396 Hz 3 dB point). This then fits quite well to the measurements, up to about 10 kHz.
The question
Why at high frequencies do the measurements deviate from the model? Has anyone seen this behaviour before?
Edit
Below is the measurement with no R2, i.e. no 2 M resistor.
There still seems to be a deviation in amplitude and phase from the anticipated flat response. The amplitude response fits what is specified in the datasheet in Figure 2. Edit 2
I have updated my model: it now actually includes the 0.2 Ω output resistance, and now includes a parasitic capacitance across the resistor. Hand-tuning using the amplitude response, I found 650 fF to be a good fit, illustrated in the figure below. While the phase is over-compensated I am beginning to see that this is likely in higher-order parasitic effects as was mentioned by Daniel Turizo in my previous question. It does not look like it will be easy to get a perfect fit at higher frequencies, particularly because I am working on a breadboard... |
To get some feeling for the problem, we make the following simplifyingassumptions:
$B$ is independent of $x$, say $B \in \mathcal S(\overline{\mathbb R}_+)$.
$A$ admits a (holomorphic) functional calculus.
There is a (holomorphic) function $h\colon \sigma(A)\to\overline{\mathbb H}_-$ such that $\bigl(i\tau - \lambda\,\hat{B}(\tau)\bigr)\bigr|_{\,\tau=h(\lambda)} =0$ for $\lambda\in\sigma(A)$, where $\hat{B}(\tau)=\int_0^\infty e^{-it\tau}B(t)\,dt$ is the Fourier transform of $B$.
Then it is readily seen that $$ u(t) = e^{ith(A)} u_0$$is a solution of the original problem. Notice that, in general,
one can only solve backwards in time.
There are several difficult points with this approach, the two obvious ones being:
The range of $i\tau\,/\,\hat{B}(\tau)$ might miss parts of the spectrum of $A$. This puts restrictions on the initial value $u_0$.
There might be several choices for the function $h$. This leadsto non-uniqueness of the solution $u$.
Here is an
example: Take $B(t) = H(t)\,e^{-t}$. Then $\hat{B}(\tau)= \frac{1}{1+i\tau}$. Choose $A\in \operatorname{Op}S^2$ with principal symbol $|\xi|^2$ and $\sigma(A)\subset\{\lambda\in{\mathbb C}\mid \Re \lambda>0\}$. Then $h(\lambda) = \frac{i}2\left(1-\sqrt{1+4\lambda}\right)$. It follows that $h(A)\in \operatorname{Op}S^1$ with principal symbol $-i\,|\xi|$. This implies that the operators $e^{ith(A)}$ for $t<0$ belong to $\operatorname{Op}S^{-\infty}$, i.e., they are regularizing, and there is no propagation of singularities.
ADDED: I'm not aware of any holomorphic functional calculus for pseudodifferential operators - with $m>0$ and not necessarily having $A$ be normal (i.e., $AA^*=A^*A$ as unbounded operators in $L^2$) - which does what is required here.
Indeed, $\hat B$ extends to a $C^\infty$ function in $\overline{\mathbb H}_-$ which is holomorphic in $\mathbb H_-$, with$$ \hat B(\tau) \sim \sum_{k\geq 0} \frac{B^{(k)}(+0)}{(i\tau)^{k+1}} \quad \text{as $\tau\to \infty$ in $\overline{\mathbb H}_-$}$$(and this asymptotic expansion can be differentiated any number of times). Therefore, assuming $B(+0)\neq0$, one has$$ \frac{i\tau}{\hat{B}(\tau)} = -\,\frac{\tau^2}{B(+0)} + O(\tau) \quad \text{as $\tau\to\infty$ in $\overline{\mathbb H}_-$.}$$So, one would expect $h(A)$ to belong to $\operatorname{Op}S^{m/2}$ and to have principal symbol$$ -i\sqrt{B(+0)} \ a_m^{1/2}(x,\xi),$$where $a_m(x,\xi)$ denotes the principal symbol of $A$.
Of course,
$h(\lambda)$ need not be an algebraic function of $\lambda$ - as was the case in the example above - and then in order to prove such a result one cannot directly appeal to known facts about complex powers of (hypoelliptic) pseudodifferential operators.
Still, as $$ h(\lambda) \sim \sum_{l\geq0} c_l\lambda^{1/2-l} \quad\text{as $\lambda\to\infty$}$$in a suitable sector depending on the choice of $B$ (again, this asymptotic expansion can be differentiated any number of times), with $c_0 = -i\sqrt{B(+0)}$ as just seen and $c_l$ for $l\geq0$ computable in terms of $B(+0),\dots, B^{(l)}(+0)$, there is a good chance that such a result holds. See also this previous post.
ADDED: Suppose that $e^{at}B\in \mathcal S(\overline{\mathbb R}_+)$ holds for some $a>0$. Then $\hat{B}$ extends to a $C^\infty$ function on $ia+\overline{\mathbb H}_-=\{\tau\in \mathbb B\mid\Im\tau\leq a\}$ which is holomorphic on $ia+\mathbb H_-$ (and still has an asymptotic expansion as $\tau\to\infty$, as above).
Here is another
example using this observation: Let $B(t)=-\,H(t)\,e^{-t}$ and $A\in\operatorname{Op}S^2$ with principal symbol $|\xi|^2$. Suppose that $\sigma(A)\subset \{\lambda\in{\mathbb C}\mid (\Im \lambda)^2<\Re\lambda\}$. Then $i\tau/\hat{B}(\tau)=\tau^2-i\tau$ and there are the two choices $h_\pm(\lambda)= \frac12\left(i\pm \sqrt{4\lambda-1}\right)$ for $h$. Furthermore, $h_\pm(A) \in \operatorname{Op}S^1$ with principal symbol $\pm|\xi|$. To specify a unique solution $u$ requires a second initial condition $u_t(0)=u_1$. Contrary to the situation considered before, the solution$$ u(t) = e^{-t/2}\left(\cos\left(t\sqrt{A-1/4}\right)u_0 + \frac{\sin\left(t\sqrt{A-1/4}\right)}{\sqrt{A-1/4}}\left(\frac{u_0}2+u_1\right)\right)$$is defined for all $t\in\mathbb R$. Moreover, now one has propagation of singularities. |
Gradient blowup rate for a semilinear parabolic equation
1.
College of Science, Xi’an Jiaotong University, Xi’an, 710049, China
2.
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
xx$ +x^m |u_x|^p, p> 0, m\geq 0$, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We show that the spatial derivative of solutions is globally bounded in the case $p\leq m+2$ while blowup occurs at the boundary when $p>m+2$. Blowup rate is also found for some range of $p$. Mathematics Subject Classification:Primary: 35K55, 35B4. Citation:Zhengce Zhang, Bei Hu. Gradient blowup rate for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 767-779. doi: 10.3934/dcds.2010.26.767
[1]
Jong-Shenq Guo, Bei Hu.
Blowup rate estimates for the heat equation with a nonlinear gradient source term.
[2]
Zhengce Zhang, Yanyan Li.
Gradient blowup solutions of a semilinear parabolic equation with exponential source.
[3]
Zhengce Zhang, Yan Li.
Global existence and gradient blowup
of solutions for a semilinear parabolic equation with exponential source.
[4] [5]
Thierry Cazenave, Yvan Martel, Lifeng Zhao.
Finite-time blowup for a Schrödinger equation with nonlinear source term.
[6]
Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang.
Blowup rate estimate for a system of semilinear parabolic equations.
[7]
Chunlai Mu, Zhaoyin Xiang.
Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux.
[8]
Zaihui Gan, Boling Guo, Jian Zhang.
Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials.
[9]
Masahoto Ohta, Grozdena Todorova.
Remarks on global existence and blowup
for damped nonlinear Schrödinger equations.
[10]
Bertram Düring, Daniel Matthes, Josipa Pina Milišić.
A gradient flow scheme for nonlinear fourth order equations.
[11]
Xiaohong Li, Fengquan Li.
Nonexistence of solutions for nonlinear
differential inequalities with gradient nonlinearities.
[12] [13] [14]
Paolo Baiti, Helge Kristian Jenssen.
Blowup in $\mathbf{L^{\infty}}$ for a class of genuinely nonlinear hyperbolic systems of conservation laws.
[15]
Congming Peng, Dun Zhao.
Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation.
[16]
Evgeny Galakhov, Olga Salieva.
Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets.
[17]
Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta.
Gradient estimates and comparison principle
for some nonlinear elliptic equations.
[18]
El-Sayed M.E. Mostafa.
A nonlinear conjugate gradient method for a special class of matrix optimization problems.
[19]
Wataru Nakamura, Yasushi Narushima, Hiroshi Yabe.
Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization.
[20]
Gaohang Yu, Shanzhou Niu, Jianhua Ma.
Multivariate spectral gradient projection method
for nonlinear monotone equations with convex constraints.
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Continuing the previous post, we prove the claimed minimax lower bound. We start with a useful result that quantifies the difficulty of identifying whether or not an observation is drawn from similar distributions $P$ and $Q$ defined over the same Continue Reading
In this post we introduce the concept of minimax regret and reformulate our previous result on the upper bound on the worst-case regret of UCB in terms of the minimax regret. We briefly discuss the strengths and weaknesses of using Continue Reading
We now describe the celebrated Upper Confidence Bound (UCB) algorithm that overcomes all of the limitations of strategies based on exploration followed by commitment, including the need to know the horizon and sub-optimality gaps. The algorithm has many different forms, Continue Reading
With most of the background material out of the way, we are almost ready to start designing algorithms for finite-armed stochastic bandits and analyzing their properties, and especially the regret. The last thing we need is an introduction to concentration Continue Reading
On Monday last week we did not have a lecture, so the lectures spilled over to this week’s Monday. This week was devoted to building up foundations, and this post will summarize how far we got. The post is pretty Continue Reading
Dear Interested Reader, Together with Tor, we have worked a lot on bandit problems in the past and developed a true passion for them. At the pressure of some friends and students (and a potential publisher), and also just to Continue Reading |
Effective mass of electrons in silicon¶
In this tutorial, you will learn how to compute the electron effective mass for silicon.
Introduction¶
It is well known that DFT methods, or to be more specific the LDA and GGA exchange-correlation functionals, are not very adept at predicing the band gap of semiconductors. They do, however, in many cases give rather accurate curvatures of the bands. We can use this to compute the effective mass of holes and electrons by fitting a parabola to the minimum/maximum of the conduction/valence bands.
The effective mass is a parameter that depends on several assumptions and definitions, so it will be important to properly understand the underlying concept, instead of just using it as a black box. For other materials, these assumptions may be different!
Background¶
The lowest conduction band of Si has 6 equivalent minima, located along (101) (called \(\Delta\)) and its permuted equivalent directions in reciprocal space. The minimum is located at (x,0,x) where x is about 0.425, or 85% of the distance to the first Brillouin zone boundary at X=(1/2,0,1/2). At this point, the energy isosurfaces are ellipsoids, thus you get different values for the effective mass depending on if you look in the longitudinal (along ) or transverse (perpendicular to \(\Delta\)) directions. See Refs. [1] and [2] for more details.
By (1/2,0,1/2) we mean a k-vector \(K=(\mathbf{G}_A+\mathbf{G}_C)/2\), where \(\mathbf{G}_A\), \(\mathbf{G}_B\), \(\mathbf{G}_C\) are the three unit vectors that span the reciprocal unit cell. If we express this point in Cartesian coordinates (still in reciprocal space) we find that the \(\Delta\) direction is parallel to \(\mathbf{K}_Y\). Obviously then, \(\mathbf{K}_X\) and \(\mathbf{K}_Z\), which are parallel to \(\mathbf{G}_B+\mathbf{G}_C\) and \(\mathbf{G}_A+\mathbf{G}_B\), respectively, are perpendicular to \(\mathbf{K}_Y\). Thus the transverse directions are (011) and (110), and the longitudinal one is (101) - see the figure below illustrating the Brillouin zone of silicon.
What we now need to do is:
generate a sequence of points \(k\) around a specified point \(k_0\), in a given direction; compute the energy eigenvalues \(E(k)\) for the lowest conduction band for these \(k\)-points; perform a numerical (finite difference) second order derivative of the band, \(d = \partial^2 E(k)/\partial k|_{k=k_0}\); compute the effective mass as \(m^* = \hbar^2/(2d)\,m_e\) where \(m_e\) is the free electron mass.
If we do this for the longtitudinal (L) and transverse (T) directions, we will obtain three values, \(m^*_L\), \(m^*_{T1}\), and \(m^*_{T2}\) (actually, for Si \(m^*_{T1} = m^*_{T2} = m^*_{T}\)), which we can insert in the well-known expressions for the conductivity and density of states effective masses:
where the factor 6 which enters the expression for the conductivity mass is the number of equivalent \(\Delta\) valleys.
Set up the calculation¶
The first step is to do a self-consistent DFT calculation for Si.
The details involved in such a task are described in the Calculate the band structure of a crystal.There, the extended Hückel method was used, so you should modify the calculator to use
ATK-DFT instead.Note that by default the LDA exchange correlation functional is used and you just need to specify a proper k-point mesh, i.e. 15x15x15.
Before saving and running your DFT calculation, you can set up the effective mass analysis directly from the Scripter. Addanalysis after the calculator:
Double-click on EffectiveMass to set up the effective mass calculation. Here, you have several options and you can find all the details for each of these options in the
EffectiveMass.
In order to calculate the effective mass at the conduction mass minimum enter the following parameters:
Change the k-point to “Other (Fractional)” and use the [0.425, 0, 0.425] coordinates. Select one band above the Fermi level (conduction band only) and one band below the Fermi level. However, feel free to select as many bands as you wish! As you will see later they will be conveniently listed in the LabFloor. Only remember that the effective masses calculated for all these bands correspond to the specified [0.425, 0, 0.425] point. Set the Direction to Fractionaland set a=1, b=0, c=1, corresponding to the longitudinal direction, c.f. Background. Leave all the other default parameters. Add a second EffectiveMass analysis object with the same settings, except the direction which should be a=1, b=1, c=0, corresponding to one of the transverse directions.
Finally, set the output filename,
silicon_lda.hdf5 for example, save and run the script.The job will take just a few seconds.
Analyze the results¶ Bandstructure¶
You will now find the Bandstructure and EffectiveMass objects in the LabFloor. Let’s first check the bandstructure: double-click on it and use the Bandstructure Analyzer tool to plot it.
Inspecting the bandstructure it’s clear that the band gap is severly underestimated, but the shape is essentially correct; the gap is indirect and we do have a conduction band minimum at about 85% of the distance to X.
Effective mass 1¶
Select one of the
EffectiveMass object from the LabFoor and use the Effective mass analyzer to inspect it:
The
Effective mass analyzer window will then open:
Here, the effective mass corresponding to the specified band and direction is reported. From the two
EffectiveMass objects you will obtain: Longitudinal effective mass, [1,0,1] direction, \(m^*=0.90\,m_e\). Transverse effective mass, [1,1,0] direction, \(m^*=0.187\,m_e\).
With these values we can now compute the density of states mass \(m^*_{DOS}=1.05\) and \(m^*_C=0.26\), very close indeed to the reference values 1.08 and 0.26 (see e.g. http://ecee.colorado.edu/~bart/book/effmass.htm ). Thus DFT actually does reproduce the effective mass of the electrons in the \(\Delta\) valley very well.
Effective mass 2¶
You may have noticed at the bottom right corner of the
Bandstructure analyzer window the Effective mass button.You do have indeed a second interactive tool at your disposal to calculate the effective masses.
This is a very convenient tool since you can select the k-point for which the effective mass will be calculated directly from the plot:
zoom in in the desired region of the plot and left-click to the desired point
Click on “Effective mass...” to open the effective mass tool:
Here, all the relevant parameters such as bandindex and k-point coordinates corresponding to the clicked point will be automatically set. Note that you can change and refine these parameters.
Press “Calculate” to calculate the effective mass.
Check on the LabFloor where the Effective Mass object is stored and follow the steps above to get the effective mass.
The effective mass tools contain a couple of additional parameters which we have not mentioned.
Stencil order: it is possible to select the number of k-points that will be used to generate the band structure for which we perform the numerical derivative. Delta: this parameter controls how far away from the chosen k-point we go. The default is typically fine; if the band structure is strongly non-parabolic, the value can be decreased a bit perhaps. Going further¶
You can now also try the same calculation with the extended Hückel method.
If you followed the Calculate the band structure of a crystal you already have the necessary HDF5 file. The band minimum is not at x=0.452 this time, but closer to x=0.445. The results are very similar, however (\(m^*_L=1.00\) and \(m^*_T=0.17\)).
Hole masses in Si¶
For Si, we compute a light hole mass (at the Γ point [0,0,0]) \(m^*_{LH}=-0.16\) (band index=1), and for the two degenerate heavy hole bands the result is \(m^*_{HH}=-0.25\) (band indices 2 and 3, respectively). These values are obtained in the directions of the Cartesian axes (i.e. if we use the same directions as for the electron, along \(\Delta\) from \(\Gamma\) to X). If, however, we look in the \(\Lambda\) direction from \(\Gamma\) to L (by setting the direction in the Analyzer to [1,1,1]), we find \(m^*_{HH}=-0.64\) and \(m^*_{LH}=-0.09\), a result of fact that the constant-energy surfaces for holes are strongly warped in Si, especially for the heavy hole bands [1] [2].
To get proper hole masses that can be compared to experiments one should however also include spin-orbit interaction. |
We owe Paul Dirac two excellent mathematical jokes. I have amended them with a few lesser known variations.
A.
Square root of the Laplacian: we want $\Delta$ to be $D^2$ for some first order differential operator (for example, because it is easier to solve first order partial differential equations than second order PDEs). Writing it out,
$$\sum_{k=1}^n \frac{\partial^2}{\partial x_k^2}=\left(\sum_{i=1}^n \gamma_i \frac{\partial}{\partial x_i}\right)\left(\sum_{j=1}^n \gamma_j \frac{\partial}{\partial x_j}\right) = \sum_{i,j}\gamma_i\gamma_j \frac{\partial^2}{\partial x_i x_j},$$
and equating the coefficients, we get that this is indeed true if
$$D=\sum_{i=1}^n \gamma_i \frac{\partial}{\partial x_i}\quad\text{and}\quad \gamma_i\gamma_j+\gamma_j\gamma_i=\delta_{ij}.$$
It remains to come up with the right $\gamma_i$'s. Dirac realized how to accomplish it with $4\times 4$ matrices when $n=4$; but a neat follow-up joke is to simply define them to be the elements $\gamma_1,\ldots,\gamma_n$ of
$$\mathbb{R}\langle\gamma_1,\ldots,\gamma_n\rangle/(\gamma_i\gamma_j+\gamma_j\gamma_i - \delta_{ij}).$$
Using symmetry considerations, it is easy to conclude that the commutator of the $n$-dimensional Laplace operator $\Delta$ and the multiplication by $r^2=x_1^2+\cdots+x_n^2$ is equal to $aE+b$, where $$E=x_1\frac{\partial}{\partial x_1}+\cdots+x_n\frac{\partial}{\partial x_n}$$ is the Euler vector field. A boring way to confirm this and to determine the coefficients $a$ and $b$ is to expand $[\Delta,r^2]$ and simplify using the commutation relations between $x$'s and $\partial$'s. A more exciting way is to act on $x_1^\lambda$, where $\lambda$ is a formal variable:
$$[\Delta,r^2]x_1^{\lambda}=((\lambda+2)(\lambda+1)+2(n-1)-\lambda(\lambda-1))x_1^{\lambda}=(4\lambda+2n)x_1^{\lambda}.$$
Since $x_1^{\lambda}$ is an eigenvector of the Euler operator $E$ with eigenvalue $\lambda$, we conclude that
$$[\Delta,r^2]=4E+2n.$$
B.
Dirac delta function: if we can write
$$g(x)=\int g(y)\delta(x-y)dy$$
then instead of solving an inhomogeneous linear differential equation $Lf=g$ for each $g$, we can solve the equations $Lf=\delta(x-y)$ for each real $y$, where a linear differential operator $L$ acts on the variable $x,$ and combine the answers with different $y$ weighted by $g(y)$. Clearly, there are fewer real numbers than functions, and if $L$ has constant coefficients, using translation invariance the set of right hand sides is further reduced to just one, $\delta(x)$. In this form, the joke goes back to Laplace and Poisson.
What happens if instead of the ordinary geometric series we consider a doubly infinite one? Since
$$z(\cdots + z^{-n-1} + z^{-n} + \cdots + 1 + \cdots + z^n + \cdots)= \cdots + z^{-n} + z^{-n+1} + \cdots + z + \cdots + z^{n+1} + \cdots,$$
the expression in the parenthesis is annihilated by the multiplication by $z-1$, hence it is equal to $\delta(z-1)$. Homogenizing, we get
$$\sum_{n\in\mathbb{Z}}\left(\frac{z}{w}\right)^n=\delta(z-w)$$
This identity plays an important role in conformal field theory and the theory of vertex operator algebras.
Pushing infinite geometric series in a different direction,
$$\cdots + z^{-n-1} + z^{-n} + \cdots + 1=-\frac{z}{1-z} \quad\text{and}\quad 1 + z + \cdots + z^n + \cdots = \frac{1}{1-z},$$
which add up to $1$. This time, the sum of doubly infinite geometric series is zero!Thus the point $0\in\mathbb{Z}$ is the sum of all lattice points on the non-negative half-line and all points on the positive half-line:
$$0=[\ldots,-2,-1,0] + [0,1,2,\ldots] $$
A vast generalization is given by Brion's formula for the generating function for the lattice points in a convex lattice polytope $\Delta\subset\mathbb{R}^N$ with vertices $v\in{\mathbb{Z}}^N$ and closed inner vertex cones $C_v\subset\mathbb{R}^N$:
$$\sum_{P\in \Delta\cap{\mathbb{Z}}^N} z^P = \sum_v\left(\sum_{Q\in C_v\cap{\mathbb{Z}}^N} z^Q\right),$$
where the inner sums in the right hand side need to be interpreted as rational functions in $z_1,\ldots,z_N$.
Another great joke based on infinite series is the Eilenberg swindle, but I am too exhausted by fighting the math preview to do it justice. |
For a binomial tree, everywhere in Hull and other literature, we have found the formulas for
$$u = \exp(\sigma \sqrt{h})$$
but for binomial trees based on forward prices, we get a different formula
$$u=\exp((r−\delta)h+\sigma\sqrt{h})$$
Could anyone please provide an explanation of why there is this extra term of $\exp(r-\delta)$ multiplied here?
I understand that $\delta$ is for the constant dividend yield but why is there a difference in formulas for $u$ when binomial tress are constructed using forward prices? |
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 Conjecture even tho several lecturers had told him otherwise, and he sent his paper (written on Microsoft Word) to some journal citing the names of various lecturers at the university
Here is one part of the Peter-Weyl theorem: Let $\rho$ be a unitary representation of a compact $G$ on a complex Hilbert space $H$. Then $H$ splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of $G$.
What exactly does it mean for $\rho$ to split into finite dimensional unitary representations? Does it mean that $\rho = \oplus_{i \in I} \rho_i$, where $\rho_i$ is a finite dimensional unitary representation?
Sometimes my hint to my students used to be: "Hint: You're making this way too hard." Sometimes you overthink. Other times it's a truly challenging result and it takes a while to discover the right approach.
Once the $x$ is in there, you must put the $dx$ ... or else, nine chances out of ten, you'll mess up integrals by substitution. Indeed, if you read my blue book, you discover that it really only makes sense to integrate forms in the first place :P
Using the recursive definition of the determinant (cofactors), and letting $\operatorname{det}(A) = \sum_{j=1}^n \operatorname{cof}_{1j} A$, how do I prove that the determinant is independent of the choice of the line?
Let $M$ and $N$ be $\mathbb{Z}$-module and $H$ be a subset of $N$. Is it possible that $M \otimes_\mathbb{Z} H$ to be a submodule of $M\otimes_\mathbb{Z} N$ even if $H$ is not a subgroup of $N$ but $M\otimes_\mathbb{Z} H$ is additive subgroup of $M\otimes_\mathbb{Z} N$ and $rt \in M\otimes_\mathbb{Z} H$ for all $r\in\mathbb{Z}$ and $t \in M\otimes_\mathbb{Z} H$?
Well, assuming that the paper is all correct (or at least to a reasonable point). I guess what I'm asking would really be "how much does 'motivated by real world application' affect whether people would be interested in the contents of the paper?"
@Rithaniel $2 + 2 = 4$ is a true statement. Would you publish that in a paper? Maybe... On the surface it seems dumb, but if you can convince me the proof is actually hard... then maybe I would reconsider.
Although not the only route, can you tell me something contrary to what I expect?
It's a formula. There's no question of well-definedness.
I'm making the claim that there's a unique function with the 4 multilinear properties. If you prove that your formula satisfies those (with any row), then it follows that they all give the same answer.
It's old-fashioned, but I've used Ahlfors. I tried Stein/Stakarchi and disliked it a lot. I was going to use Gamelin's book, but I ended up with cancer and didn't teach the grad complex course that time.
Lang's book actually has some good things. I like things in Narasimhan's book, but it's pretty sophisticated.
You define the residue to be $1/(2\pi i)$ times the integral around any suitably small smooth curve around the singularity. Of course, then you can calculate $\text{res}_0\big(\sum a_nz^n\,dz\big) = a_{-1}$ and check this is independent of coordinate system.
@A.Hendry: It looks pretty sophisticated, so I don't know the answer(s) off-hand. The things on $u$ at endpoints look like the dual boundary conditions. I vaguely remember this from teaching the material 30+ years ago.
@Eric: If you go eastward, we'll never cook! :(
I'm also making a spinach soufflé tomorrow — I don't think I've done that in 30+ years. Crazy ridiculous.
@TedShifrin Thanks for the help! Dual boundary conditions, eh? I'll look that up. I'm mostly concerned about $u(a)=0$ in the term $u'/u$ appearing in $h'-\frac{u'}{u}h$ (and also for $w=-\frac{u'}{u}P$)
@TedShifrin It seems to me like $u$ can't be zero, or else $w$ would be infinite.
@TedShifrin I know the Jacobi accessory equation is a type of Sturm-Liouville problem, from which Fox demonstrates in his book that $u$ and $u'$ cannot simultaneously be zero, but that doesn't stop $w$ from blowing up when $u(a)=0$ in the denominator |
Is there a "simple" mathematical proof that is fully understandable by a 1st year university student that impressed you because it is beautiful?
closed as primarily opinion-based by Daniel W. Farlow, Najib Idrissi, user91500, LutzL, Jonas Meyer Apr 7 '15 at 3:40
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
Here's a cute and lovely theorem.
There exist two irrational numbers $x,y$ such that $x^y$ is rational.
Proof. If $x=y=\sqrt2$ is an example, then we are done; otherwise $\sqrt2^{\sqrt2}$ is irrational, in which case taking $x=\sqrt2^{\sqrt2}$ and $y=\sqrt2$ gives us: $$\left(\sqrt2^{\sqrt2}\right)^{\sqrt2}=\sqrt2^{\sqrt2\sqrt2}=\sqrt2^2=2.\qquad\square$$
(Nowadays, using the Gelfond–Schneider theorem we know that $\sqrt2^{\sqrt2}$ is irrational, and in fact transcendental. But the above proof, of course, doesn't care for that.)
How about the proof that
$$1^3+2^3+\cdots+n^3=\left(1+2+\cdots+n\right)^2$$
I remember being impressed by this identity and the proof can be given in a picture:
Edit: Substituted $\frac{n(n+1)}{2}=1+2+\cdots+n$ in response to comments.
Cantor's Diagonalization Argument, proof that there are infinite sets that can't be put one to one with the set of natural numbers, is frequently cited as a beautifully simple but powerful proof. Essentially, with a list of infinite sequences, a sequence formed from taking the diagonal numbers will not be in the list.
I would personally argue that the proof that $\sqrt 2$ is irrational is simple enough for a university student (probably simple enough for a high school student) and very pretty in its use of proof by contradiction!
Prove that if $n$ and $m$ can each be written as a sum of two perfect squares, so can their product $nm$.
Proof: Let $n = a^2+b^2$ and $m=c^2+d^2$ ($a, b, c, d \in\mathbb Z$). Then, there exists some $x,y\in\mathbb Z$ such that
$$x+iy = (a+ib)(c+id)$$
Taking the magnitudes of both sides are squaring gives
$$x^2+y^2 = (a^2+b^2)(c^2+d^2) = nm$$
I would go for the proof by contradiction of an infinite number of primes, which is fairly simple:
Assume that there is a finite number of primes. Let $G$ be the set of allprimes $P_1,P_2,...,P_n$. Compute $K = P_1 \times P_2 \times ... \times P_n + 1$. If $K$ is prime, then it is obviously notin $G$. Otherwise, noneof its prime factors are in $G$. Conclusion: $G$ is notthe set of allprimes.
I think I learned that both in high-school and at 1st year, so it might be a little too simple...
By the concavity of the $\sin$ function on the interval $\left[0,\frac{\pi}2\right]$ we deduce these inequalities: $$\frac{2}\pi x\le \sin x\le x,\quad \forall x\in\left[0,\frac\pi2\right].$$
The first player in Hex has a winning strategy.
There are no draws in hex, so one player must have a winning strategy. If player two has a winning strategy, player one can steal that strategy by placing the first stone in the center (additional pieces on the board never hurt your position) then using player two's strategy.
You cannot have two dice (with numbers $1$ to $6$) biased so that when you throw both, the sum is uniformly distributed in $\{2,3,\dots,12\}$.
For easier notation, we use the equivalent formulation "You cannot have two dice (with numbers $0$ to $5$) biased such that when you throw both, the sum is uniformly distributed in $\{0,1,\dots,10\}$."
Proof:Assume that such dice exist. Let $p_i$ be the probability that the first die gives an $i$ and $q_i$ be the probability that the second die gives an $i$. Let $p(x)=\sum_{i=0}^5 p_i x^i$ and $q(x)=\sum_{i=0}^5 q_i x^i$.
Let $r(x)=p(x)q(x) = \sum_{i=0}^{10} r_i x^i$. We find that $r_i = \sum_{j+k=i}p_jq_k$. But hey, this is also the probability that the sum of the two dice is $i$. Therefore, $$ r(x)=\frac{1}{11}(1+x+\dots+x^{10}). $$ Now $r(1)=1\neq0$, and for $x\neq1$, $$ r(x)=\frac{(x^{11}-1)}{11(x-1)}, $$ which clearly is nonzero when $x\neq 1$. Therefore $r$ does not have any real zeros.
But because $p$ and $q$ are $5$th degree polynomials, they must have zeros. Therefore, $r(x)=p(x)q(x)$ has a zero. A contradiction.
Given a square consisting of $2n \times 2n$ tiles, it is possible to cover this square with pieces that each cover $2$ adjacent tiles (like domino bricks). Now imagine, you remove two tiles, from two opposite corners of the original square. Prove that is is now no longer possible to cover the remaining area with domino bricks.
Proof:
Imagine that the square is a checkerboard. Each domino brick will cover two tiles of different colors. When you remove tiles from two opposite corners, you will remove two tiles with the
samecolor. Thus, it can no longer be possible to cover the remaining area.
(Well, it may be
too "simple." But you did not state that it had to be a university student of mathematics. This one might even work for liberal arts majors...)
One little-known gem at the intersection of geometry and number theory is Aubry's reflective generation of primitive Pythagorean triples, i.e. coprime naturals $\,(x,y,z)\,$with $\,x^2 + y^2 = z^2.\,$ Dividing by $\,z^2$ yields $\,(x/z)^2+(y/z)^2 = 1,\,$ so each triple corresponds to a rational point $(x/z,\,y/z)$ on the unit circle. Aubry showed that we can generate all such triples by a very simple geometrical process. Start with the trivial point $(0,-1)$. Draw a line to the point $\,P = (1,1).\,$ It intersects the circle in the
rational point $\,A = (4/5,3/5)\,$ yielding the triple $\,(3,4,5).\,$ Next reflect the point $\,A\,$ into the other quadrants by taking all possible signs of each component, i.e. $\,(\pm4/5,\pm3/5),\,$ yielding the inscribed rectangle below. As before, the line through $\,A_B = (-4/5,-3/5)\,$ and $P$ intersects the circle in $\,B = (12/13, 5/13),\,$ yielding the triple $\,(12,5,13).\,$ Similarly the points $\,A_C,\, A_D\,$ yield the triples $\,(20,21,29)\,$ and $\,(8,15,17),\,$ We can iterate this process with the new points $\,B,C,D\,$ doing the same we did for $\,A,\,$ obtaining further triples. Iterating this process generates the primitive triples as a ternary tree
$\qquad\qquad$
Descent in the tree is given by the formula
$$\begin{eqnarray} (x,y,z)\,\mapsto &&(x,y,z)-2(x\!+\!y\!-\!z)\,(1,1,1)\\ = &&(-x-2y+2z,\,-2x-y+2z,\,-2x-2y+3z)\end{eqnarray}$$
e.g. $\ (12,5,13)\mapsto (12,5,13)-8(1,1,1) = (-3,4,5),\ $ yielding $\,(4/5,3/5)\,$ when reflected into the first quadrant.
Ascent in the tree by inverting this map, combined with trivial sign-changing reflections:
$\quad\quad (-3,+4,5) \mapsto (-3,+4,5) - 2 \; (-3+4-5) \; (1,1,1) = ( 5,12,13)$
$\quad\quad (-3,-4,5) \mapsto (-3,-4,5) - 2 \; (-3-4-5) \; (1,1,1) = (21,20,29)$
$\quad\quad (+3,-4,5) \mapsto (+3,-4,5) - 2 \; (+3-4-5) \; (1,1,1) = (15,8,17)$
See my MathOverflow post for further discussion, including generalizations and references.
I like the proof that there are infinitely many Pythagorean triples.
Theorem:There are infinitely many integers $ x, y, z$ such that $$ x^2+y^2=z^2 $$ Proof:$$ (2ab)^2 + ( a^2-b^2)^2= ( a^2+b^2)^2 $$
One cannot cover a disk of diameter 100 with 99 strips of length 100 and width 1.
Proof: project the disk and the strips on a semi-sphere on top of the disk. The projection of each strip would have area at most 1/100th of the area of the semi-sphere.
If you have any set of 51 integers between $1$ and $100$, the set must contain some pair of integers where one number in the pair is a multiple of the other.
Proof: Suppose you have a set of $51$ integers between $1$ and $100$. If an integer is between $1$ and $100$, its largest odd divisor is one of the odd numbers between $1$ and $99$. There are only $50$ odd numbers between $1$ and $99$, so your $51$ integers can’t all have different largest odd divisors — there are only $50$ possibilities. So two of your integers (possibly more) have the same largest odd divisor. Call that odd number $d$. You can factor those two integers into prime factors, and each will factor as (some $2$’s)$\cdot d$. This is because if $d$ is the largest divisor of a number, the rest of its factorization can’t include any more odd numbers. Of your two numbers with largest odd factor $d$, the one with more $2$’s in its factorization is a multiple of the other one. (In fact, the multiple is a power of $2$.)
In general, let $S$ be the set of integers from $1$ up to some even number $2n$. If a subset of $S$ contains more than half the elements in $S$, the set must contain a pair of numbers where one is a multiple of the other. The proof is the same, but it’s easier to follow if you see it for a specific $n$ first.
The proof that an isosceles triangle ABC (with AC and AB having equal length) has equal angles ABC and BCA is quite nice:
Triangles ABC and ACB are (mirrored) congruent (since AB = AC, BC = CB, and CA = BA), so the corresponding angles ABC and (mirrored) ACB are equal.
This congruency argument is nicer than that of cutting the triangle up into two right-angled triangles.
Parity of sine and cosine functions using Euler's forumla:
$e^{-i\theta} = cos\ (-\theta) + i\ sin\ (-\theta)$
$e^{-i\theta} = \frac 1 {e^{i\theta}} = \frac 1 {cos\ \theta \ + \ i\ sin\ \theta} = \frac {cos\ \theta\ -\ i\ sin\ \theta} {cos^2\ \theta\ +\ sin^2\ \theta} = cos\ \theta\ -\ i\ sin\ \theta$
$cos\ (-\theta) +\ i\ sin\ (-\theta) = cos\ \theta\ +i\ (-sin\ \theta)$
Thus
$cos\ (-\theta) = cos\ \theta$
$sin\ (-\theta) = -\ sin\ \theta$
$\blacksquare$
The proof is actually just the first two lines.
I believe Gauss was tasked with finding the sum of all the integers from $1$ to $100$ in his very early schooling years. He tackled it quicker than his peers or his teacher could, $$\sum_{n=1}^{100}n=1+2+3+4 +\dots+100$$ $$=100+99+98+\dots+1$$ $$\therefore 2 \sum_{n=1}^{100}n=(100+1)+(99+2)+\dots+(1+100)$$ $$=\underbrace{101+101+101+\dots+101}_{100 \space times}$$ $$=101\cdot 100$$ $$\therefore \sum_{n=1}^{100}n=\frac{101\cdot 100}{2}$$ $$=5050.$$ Hence he showed that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}.$$
If $H$ is a subgroup of $(\mathbb{R},+)$ and $H\bigcap [-1,1]$ is finite and contains a positive element. Then, $H$ is cyclic.
Fermat's little theorem from noting that modulo a prime p we have for $a\neq 0$:
$$1\times2\times3\times\cdots\times (p-1) = (1\times a)\times(2\times a)\times(3\times a)\times\cdots\times \left((p-1)\times a\right)$$
Proposition (No universal set): There does not exists a set which contain all the sets (even itself)
Proof: Suppose to the contrary that exists such set exists. Let $X$ be the universal set, then one can construct by the axiom schema of specification the set
$$C=\{A\in X: A \notin A\}$$
of all sets in the universe which did not contain themselves. As $X$ is universal, clearly $C\in X$. But then $C\in C \iff C\notin C$, a contradiction.
Edit: Assuming that one is working in ZF (as almost everywhere :P)
(In particular this proof really impressed me too much the first time and also is very simple)
Most proofs concerning the Cantor Set are simple but amazing.
The total number of intervals in the set is zero.
It is uncountable.
Every number in the set can be represented in ternary using just
0 and
2. No number with a
1 in it (in ternary) appears in the set.
The Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval of nonzero length. The irrational numbers have the same property, but the Cantor set has the additional property of being closed, so it is not even dense in any interval, unlike the irrational numbers which are dense in every interval.
The Menger sponge which is a 3d extension of the Cantor set
simultaneously exhibits an infinite surface area and encloses zero volume.
The derivation of first principle of differentiation is so amazing, easy, useful and simply outstanding in all aspects. I put it here:
Suppose we have a quantity $y$ whose value depends upon a single variable $x$, and is expressed by an equation defining $y$ as some specific function of $x$. This is represented as:
$y=f(x)$
This relationship can be visualized by drawing a graph of function $y = f (x)$ regarding $y$ and $x$ as Cartesian coordinates, as shown in Figure(a).
Consider the point $P$ on the curve $y = f (x)$ whose coordinates are $(x, y)$ and another point $Q$ where coordinates are $(x + Δx, y + Δy)$.
The slope of the line joining $P$ and $Q$ is given by:
$tanθ = \frac{Δy}{Δx} = \frac{(y + Δy ) − y}{Δx}$
Suppose now that the point $Q$ moves along the curve towards $P$.
In this process, $Δy$ and $Δx$ decrease and approach zero; though their ratio $\frac{Δy}{Δx}$ will not necessarily vanish.
What happens to the line $PQ$ as $Δy→0$, $Δx→0$? You can see that this line becomes a tangent to the curve at point $P$ as shown in Figure(b). This means that $tan θ$ approaches the slope of the tangent at $P$, denoted by $m$:
$m=lim_{Δx→0} \frac{Δy}{Δx} = lim_{Δx→0} \frac{(y+Δy)-y}{Δx}$
The limit of the ratio $Δy/Δx$ as $Δx$ approaches zero is called the derivative of $y$ with respect to $x$ and is written as $dy/dx$.
It represents the slope of the tangent line to the curve $y=f(x)$ at the point $(x, y)$.
Since $y = f (x)$ and $y + Δy = f (x + Δx)$, we can write the definition of the derivative as:
$\frac{dy}{dx}=\frac{d{f(x)}}{dx} = lim_{x→0} [\frac{f(x+Δx)-f(x)}{Δx}]$,
which is the required formula.
This proof that $n^{1/n} \to 1$ as integral $n \to \infty$:
By Bernoulli's inequality (which is $(1+x)^n \ge 1+nx$), $(1+n^{-1/2})^n \ge 1+n^{1/2} > n^{1/2} $. Raising both sides to the $2/n$ power, $n^{1/n} <(1+n^{-1/2})^2 = 1+2n^{-1/2}+n^{-1} < 1+3n^{-1/2} $.
Can a Chess Knight starting at any corner then move to touch every space on the board exactly once, ending in the opposite corner?
The solution turns out to be childishly simple. Every time the Knight moves (up two, over one), it will hop from a black space to a white space, or vice versa. Assuming the Knight starts on a black corner of the board, it will need to touch 63 other squares, 32 white and 31 black. To touch all of the squares, it would need to end on a white square, but the opposite corner is also black, making it impossible.
The Eigenvalues of a skew-Hermitian matrix are purely imaginary.
The Eigenvalue equation is $A\vec x = \lambda\vec x$, and forming the vector norm gives $$\lambda \|\vec x\| = \lambda\left<\vec x, \vec x\right> = \left<\lambda \vec x,\vec x\right> = \left<A\vec x,\vec x\right> = \left<\vec x, A^{T*}\vec x\right> = \left<\vec x, -A\vec x\right> = -\lambda^* \|\vec x\|$$ and since $\|\vec x\| > 0$, we can divide it from left and right side. The second to last step uses the definition of skew-Hermitian. Using the definition for Hermitian or Unitarian matrices instead yields corresponding statements about the Eigenvalues of those matrices.
I like the proof that not every real number can be written in the form $a e + b \pi$ for some integers $a$ and $b$. I know it's almost trivial in one way; but in another way it is kind of deep. |
Consider the mode expansion of a (chiral) scalar field confined to a disc with circumference L: $$ \phi(x) = \phi_{0} + p_{\phi} \frac{2\pi}{L} x + \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} e^{-(k_{n}a)/2} \left(e^{-ik_{n}x} \ b_{n}^{\dagger}+e^{ik_{n}x} \ b_{n}\right) $$ with $k_{n}=\frac{2\pi n}{L}$ , $\phi_{0}$ some "zero-mode", $p_{\phi}$ some "conjugate momentum" and $a$ some short-distance cut-off. The operators fulfill the following bosonic commutation relations $$ \left[b_{n}^{\dagger} , b_{n'}\right]=\delta_{n,n'} \quad\text{and}\quad \left[\phi_{0},p_{\phi}\right]=i $$
(Fermionic) Vertex operators are defined by $$ V_{\alpha}(x)=:e^{i\alpha\phi(x)}: $$ with $: \ ... \ :$ denoting normal ordering. Inserting the mode expansion of $\phi(x)$ into the definition of the vertex operator yields to lowest order in $\frac{a}{L}$:
$$ :e^{i\alpha\phi(x)}: = \left(\frac{L}{2\pi a}\right)^{\Delta(\alpha)} e^{i\alpha\phi(x)} $$ with the "scaling dimension" $\Delta(\alpha)=\frac{\alpha^{2}}{2}$. The pre factor on right side in front of the exponential is sometimes called "Klein factor".
Now here are my questions (They may really be "Newbie"-CFT-questions;) ) :
Since the right hand side is only an approximation of of $:e^{i\alpha\phi(x)}:$ to lowest order I am wondering whether the left hand-side reproduces the correct (say) fermonic commutators in all cases and whether hand side only partially reproduces the correct fermonic commutators?
If the right-handside only indeed only partially reproduces the correct commutation relations how can we say that a certain product of fermionic operators (say a product of 3 fermionic operators) indeed obeys the correct sermonic commutators when written in the "bosonized language"?
What is the importance of the higher-order terms in $\frac{a}{L}$ in the "expansion" of the vertex operator?
Is all this a more general construction in CFT?
I am looking forward to your responses! |
Difference between revisions of "Fourier series"
(→top: clean up & uniformity)
(Added complex Fourier series definition)
Line 3: Line 3:
'''Fourier series''' express a piecewise continuous, bounded, periodic function as a linear combination of [[orthogonal]] [[sine]] and [[cosine]] functions. The seeds of the modern theory were developed by [[Joseph Fourier]].
'''Fourier series''' express a piecewise continuous, bounded, periodic function as a linear combination of [[orthogonal]] [[sine]] and [[cosine]] functions. The seeds of the modern theory were developed by [[Joseph Fourier]].
−
The Fourier [[series (mathematics)|series]] of a function
+
The Fourier [[series (mathematics)|series]] of a function f(t)is of the form:
−
:<math> f(t) = \frac{
+ −
where,
+
:<math> f(t) = \frac{}{2} + \sum_{n=1}^{\infty} [a_n \cos(\omega_n t) + b_n \sin(\omega_n t) ] </math>
+ +
where, nis an integerand
+
:<math> \omega_n = n\frac{2\pi}{T}</math>
:<math> \omega_n = n\frac{2\pi}{T}</math>
:<math>a_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt</math>
:<math>a_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt</math>
:<math>b_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt </math>
:<math>b_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt </math>
−
Fourier series can be generalized to [[Fourier transformation]]s for other classes of functions, for instance the space <math>L^2(\mathbb R^n)</math> of
+
Fourier series can be generalized to [[Fourier transformation]]s for other classes of functions, for instance the space <math>L^2(\mathbb R^n)</math> of square integrable . Even more generally, it is possible to carry out Fourier analysis in the setting of [[compact space|locally compact]] [[abelian]] [[topology|topological]] [[group]]s, where the fundamental observation of Pontryagin duality provides the necessary theoretical underpinnings.
+ + + + + + + + +
[[Category:Calculus]]
[[Category:Calculus]]
Latest revision as of 16:41, 23 November 2016
This article/section deals with mathematical concepts appropriate for late high school or early college. Fourier series express a piecewise continuous, bounded, periodic function as a linear combination of orthogonal sine and cosine functions. The seeds of the modern theory were developed by Joseph Fourier.
The Fourier series of a function is of the form:
where, is an integer and
Fourier series can be generalized to Fourier transformations for other classes of functions, for instance the space of square integrable functions. Even more generally, it is possible to carry out Fourier analysis in the setting of locally compact abelian topological groups, where the fundamental observation of Pontryagin duality provides the necessary theoretical underpinnings.
The fourier series can also be described using complex numbers. The complex Fourier series is:
where |
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Now showing items 1-10 of 26
Production of light nuclei and anti-nuclei in $pp$ and Pb-Pb collisions at energies available at the CERN Large Hadron Collider
(American Physical Society, 2016-02)
The production of (anti-)deuteron and (anti-)$^{3}$He nuclei in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has been studied using the ALICE detector at the LHC. The spectra exhibit a significant hardening with ...
Forward-central two-particle correlations in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV
(Elsevier, 2016-02)
Two-particle angular correlations between trigger particles in the forward pseudorapidity range ($2.5 < |\eta| < 4.0$) and associated particles in the central range ($|\eta| < 1.0$) are measured with the ALICE detector in ...
Measurement of D-meson production versus multiplicity in p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV
(Springer, 2016-08)
The measurement of prompt D-meson production as a function of multiplicity in p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV with the ALICE detector at the LHC is reported. D$^0$, D$^+$ and D$^{*+}$ mesons are reconstructed ...
Measurement of electrons from heavy-flavour hadron decays in p–Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV
(Elsevier, 2016-03)
The production of electrons from heavy-flavour hadron decays was measured as a function of transverse momentum ($p_{\rm T}$) in minimum-bias p–Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV with ALICE at the LHC for $0.5 ...
Direct photon production in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV
(Elsevier, 2016-03)
Direct photon production at mid-rapidity in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 2.76$ TeV was studied in the transverse momentum range $0.9 < p_{\rm T} < 14$ GeV/$c$. Photons were detected via conversions in the ALICE ...
Multi-strange baryon production in p-Pb collisions at $\sqrt{s_\mathbf{NN}}=5.02$ TeV
(Elsevier, 2016-07)
The multi-strange baryon yields in Pb--Pb collisions have been shown to exhibit an enhancement relative to pp reactions. In this work, $\Xi$ and $\Omega$ production rates have been measured with the ALICE experiment as a ...
$^{3}_{\Lambda}\mathrm H$ and $^{3}_{\bar{\Lambda}} \overline{\mathrm H}$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(Elsevier, 2016-03)
The production of the hypertriton nuclei $^{3}_{\Lambda}\mathrm H$ and $^{3}_{\bar{\Lambda}} \overline{\mathrm H}$ has been measured for the first time in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV with the ALICE ...
Multiplicity dependence of charged pion, kaon, and (anti)proton production at large transverse momentum in p-Pb collisions at $\sqrt{s_{\rm NN}}$= 5.02 TeV
(Elsevier, 2016-09)
The production of charged pions, kaons and (anti)protons has been measured at mid-rapidity ($-0.5 < y < 0$) in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV using the ALICE detector at the LHC. Exploiting particle ...
Jet-like correlations with neutral pion triggers in pp and central Pb–Pb collisions at 2.76 TeV
(Elsevier, 2016-12)
We present measurements of two-particle correlations with neutral pion trigger particles of transverse momenta $8 < p_{\mathrm{T}}^{\rm trig} < 16 \mathrm{GeV}/c$ and associated charged particles of $0.5 < p_{\mathrm{T}}^{\rm ...
Centrality dependence of charged jet production in p-Pb collisions at $\sqrt{s_\mathrm{NN}}$ = 5.02 TeV
(Springer, 2016-05)
Measurements of charged jet production as a function of centrality are presented for p-Pb collisions recorded at $\sqrt{s_{\rm NN}} = 5.02$ TeV with the ALICE detector. Centrality classes are determined via the energy ... |
I think there’s a rule somewhere that says “You can’t call yourself a data scientist until you’ve used a Naive Bayes classifier”. It’s extremely useful, yet beautifully simplistic. This article is my attempt at laying the groundwork for Naive Bayes in a practical and intuitive fashion.
Motivating Problem
Let’s start with a problem to motivate our formulation of Naive Bayes. (Feel free to follow along using the Python script or R script found here.)
Suppose we own a professional networking site similar to LinkedIn. Users sign up, type some information about themselves, and then roam the network looking for jobs/connections/etc. Until recently, we only required users to enter their current job title, but now we’re asking them what industry they work in. New users are supplying this info as they sign up, but old users aren’t bothering to update their information. So, we need to build a text classification model to do it for them. Our data looks like this
job_title job_category underwriter manager finance mortgage data analyst finance junior underwriter finance sales manager sales junior medical sales associate sales senior sales manager sales sales manager sales senior data analyst technology data analyst technology data manager technology data analyst manager NA junior data analyst NA
Our goal - to estimate the probability those two unlabeled job titles should be categorized as Technology, Sales, or Finance, at which point we can make our best guess for the user. Formalizing this a bit, we want to find \(p(C_k \vert \textrm{job_title})\) where \(C_1, C_2, \text{and} C_3\) are the classes Technology, Sales, and Finance. (Note: this type of problem is called Document Classification.)
How about that first unlabeled title, “data analyst manager”? We should probably label it as Technology, but how do we train a computer to figure that out? If we had trillions of training samples we might be able to estimate \(p(C_k \vert \textrm{job_title}=\textrm{"data analyst manager"})\) empirically (i.e. by measuring the relative frequency of each class for samples where job_title = “data analyst manager”). Unfortunately we only have 10 training samples (none of which have the title “data analyst manager”) so we’ll have to be a little more creative in our approach.
The word “data” seems pretty important. It occurs in all of the Technology samples, none of the Sales samples and only one of the Finance samples. On the other hand the word “manager” appears in every single category, so it’s probably not as useful. The big takeaway here is that we can use word occurrences to build a probabilistic model. Let’s start tracking words then…
Finance
job_title analyst associate data junior manager medical mortgage sales senior underwriter underwriter manager 0 0 0 0 1 0 0 0 0 1 mortgage data analyst 1 0 1 0 0 0 1 0 0 0 junior underwriter 0 0 0 1 0 0 0 0 0 1 Sales
job_title analyst associate data junior manager medical mortgage sales senior underwriter sales manager 0 0 0 0 1 0 0 1 0 0 junior medical sales associate 0 1 0 1 0 1 0 1 0 0 senior sales manager 0 0 0 0 1 0 0 1 1 0 sales manager 0 0 0 0 1 0 0 1 0 0 Technology
job_title analyst associate data junior manager medical mortgage sales senior underwriter senior data analyst 1 0 1 0 0 0 0 0 1 0 data analyst 1 0 1 0 0 0 0 0 0 0 data manager 0 0 1 0 1 0 0 0 0 0 Unlabeled
job_title analyst associate data junior manager medical mortgage sales senior underwriter data analyst manager 1 0 1 0 1 0 0 0 0 0 junior data analyst 1 0 1 1 0 0 0 0 0 0
Updating our model a bit, we want to find \(p(C_k \vert \mathbf{x})\) where \(\mathbf{x}\) is our feature vector of word occurrences. In the case of “data analyst manager” \(\mathbf{x} = (x_1, x_2, ... x_{10}) = (1,0,1,0,1,0,0,0,0,0)\).
Bayes’ Theorem
Before we continue, let’s review some probability rules and Bayes’ Theorem
\[ \begin{aligned} P(A\mid B) &= \frac{P(A \cap B)}{P(B)} \Rightarrow P(A \cap B) = P(A\mid B)\, P(B)\\ P(B\mid A) &= \frac{P(A \cap B)}{P(A)} \Rightarrow P(A \cap B) = P(B\mid A)\, P(A)\\ \Rightarrow P(A\mid B) &= \frac{P(B\mid A)\,P(A)}{P(B)}, \text{ if } P(B) \neq 0. \text{ (Bayes' Theorem)} \end{aligned} \]
For us this means
\[ \boxed{p(C_k \vert \mathbf{x}) = \frac{p(\mathbf{x} \vert C_k)p(C_k)}{p(\mathbf{x})}} \]
Let’s break down those pieces.
\(p(C_k)\) = frequency of class \(C_k\) / total number of samples \(p(\mathbf{x} \vert C_k)\) = (frequency of \(\mathbf{x}\) / number of samples) where class = \(C_k\) \(p(\mathbf{x})\) = frequency of \(\mathbf{x}\) / total number of samples \(p(C_k)\)
This is the easy part. To calculate \(p(C_k)\) we can use empirical relative frequencies given by our training data.
\[ \begin{aligned} p(\text{Technology}) &= \frac{3}{10} = 0.3\\ p(\text{Sales}) &= \frac{4}{10} = 0.4\\ p(\text{Finance}) &= \frac{3}{10} = 0.3 \end{aligned} \]
These probabilities are called
priors. Our method of estimating them using the training data is common, but not necessary. Suppose we have reason to believe that the true priors for Technology, Sales, and Finance are 0.2, 0.4, and 0.4 respectively. With a Naive Bayes model, we can just plug those babies in for \(p(C_k)\) and our model will adjust accordingly. By contrast, using priors for tree based models like Random Forest is not nearly as easy. \(p(\mathbf{x} \vert C_k)\)
Now let’s consider \(p(\mathbf{x} \vert C_k)\). Using our training data we would calculate \(p(\mathbf{x} \vert \text{Technology})\) to be 0. (Remember \(p(\mathbf{x} \vert \text{Technology})\) represents the probability that only and all of the words “data”, “analyst”, and “manager” appear in a random job title given that it’s in the Technology class. In our training samples this never occurred, so our empirical probability estimate is 0.) This is a problem. We know \(p(\mathbf{x} \vert \text{Technology})\) should be greater than 0, but we don’t have enough (or in this case, any) samples to accurrately estimate it. The way we’ll get around this problem is to make a
naive assumption - we’ll assume that the features (i.e. the occurrence of words in a job title) are independent variables. Obviously this is not true. When the word “data” appears in a job title, it immediately increases the probability that the word “analyst” appears in the title. However let’s assume the assumption is valid (or close to being valid). Then
\(p(\mathbf{x} \vert C_k) = p(x_1 | C_k)p(x_2 | C_k) \ldots p(x_{10} | C_k)\)
For our example “data analyst manager” this means
\({p(\mathbf{x}|\text{Technology}) = p(x_1=1|\text{Tech})p(x_2=0|\text{Tech}) \ldots p(x_{10}=0|\text{Tech})=} \frac{2}{3}\frac{3}{3} \ldots \frac{3}{3} = 0.1481\)
Similarly, we can calculate \(p(\mathbf{x}|\text{Sales})=0\) and \(p(\mathbf{x}|\text{Finance})=0.005487\)
\(p(\mathbf{x})\)
We run into the same issue as before when we try to estimate \(p(\mathbf{x})\). According to our training data \(p(\mathbf{x}) = 0\), but the
true value of \(p(\mathbf{x})\) should obviously be > 0. However, consider our end-goal which is to determine \(p(\text{Technology} \vert \mathbf{x})\), \(p(\text{Sales} \vert \mathbf{x})\), and \(p(\text{Finance} \vert \mathbf{x})\). Per Bayes’ Theorem, these probabilities are equivalent to
\[ \begin{aligned} p(\text{Technology} \vert \mathbf{x}) &= \frac{p(\mathbf{x} \vert \text{Technology})p(\text{Technology})}{p(\mathbf{x})}\\ p(\text{Sales} \vert \mathbf{x}) &= \frac{p(\mathbf{x} \vert \text{Sales})p(\text{Sales})}{p(\mathbf{x})}\\ p(\text{Finance} \vert \mathbf{x}) &= \frac{p(\mathbf{x} \vert \text{Finance})p(\text{Finance})}{p(\mathbf{x})} \end{aligned} \]
Notice that \(p(\mathbf{x})\) is just a scaling factor. It affects the values of \(p(\text{Technology} \vert \mathbf{x})\), \(p(\text{Sales} \vert \mathbf{x})\), and \(p(\text{Finance} \vert \mathbf{x})\) equally, And since these probabilities must sum to 1 we can solve for \(p(\mathbf{x})\):
\[ \begin{aligned} 1 &= p(\text{Technology} \vert \mathbf{x}) + p(\text{Sales} \vert \mathbf{x}) + p(\text{Finance} \vert \mathbf{x}) \\ \Rightarrow 1 &= \frac{p(\text{Technology})p(\mathbf{x} \vert \text{Technology})}{p(\mathbf{x})} + \frac{p(\text{Sales})p(\mathbf{x} \vert \text{Sales})}{p(\mathbf{x})} + \frac{p(\text{Finance})p(\mathbf{x} \vert \text{Finance})}{p(\mathbf{x})} \\ \Rightarrow p(\mathbf{x}) &= p(\text{Technology})p(\mathbf{x} \vert \text{Technology}) + p(\text{Sales})p(\mathbf{x} \vert \text{Sales}) + p(\text{Finance})p(\mathbf{x} \vert \text{Finance}) \\ \end{aligned} \]
So, in our example we can calculate \(p(\mathbf{x})\) to be 0.04609.
Putting it all together
we can generate probability estimates
\[ \begin{aligned} p(\text{Technology} \vert \mathbf{x}) &= \frac{p(\mathbf{x} \vert \text{Technology})p(\text{Technology})}{p(\mathbf{x})} = \frac{0.3 \times 0.1481}{0.04609} = 0.9643\\ p(\text{Sales} \vert \mathbf{x}) &= \frac{p(\mathbf{x} \vert \text{Sales})p(\text{Sales})}{p(\mathbf{x})} = \frac{0.4 \times 0}{0.04609} = 0\\ p(\text{Finance} \vert \mathbf{x}) &= \frac{p(\mathbf{x} \vert \text{Finance})p(\text{Finance})}{p(\mathbf{x})} = \frac{0.3 \times 0.005487}{0.04609} = 0.03571 \end{aligned} \]
Finally, if we want to turn this model into a classifier we just need to use a decision rule like
Label the sample using the class with the highest probability.
Awesome! But there’s still a nagging issue… Suppose we try to classify the title “junior data analyst”. With our current model, we’d get
\[ \begin{aligned} p(\mathbf{x}|\text{Technology}) &= p(x_1=1|\text{Tech})p(x_2=0|\text{Tech}) \ldots p(x_{4}=1|\text{Tech}) \ldots = \frac{2}{3}\frac{3}{3} \ldots \frac{0}{3} \ldots = 0\\ p(\mathbf{x}|\text{Sales}) &= p(x_1=1|\text{Sales})p(x_2=0|\text{Sales}) \ldots p(x_{4}=1|\text{Sales}) \ldots = \frac{2}{4}\frac{3}{4} \ldots \frac{1}{4} \ldots = 0\\ p(\mathbf{x}|\text{Finance}) &= p(x_1=1|\text{Finance})p(x_2=0|\text{Finance}) \ldots p(x_{4}=1|\text{Finance}) \ldots = \frac{2}{3}\frac{3}{3} \ldots \frac{1}{3} \ldots = 0.005487 \end{aligned} \]
Our intuition tells us that “junior data analyst” is more likely to be a Technology job than a Finance job, but since the word “junior” never appeared in any of the Technology training samples, \(p(x_{4}=1|\text{Technology}) = 0\) which will ultimately cause \(p(\text{Technology} \vert \mathbf{x})\) to be 0. In order to deal with this issue, we’ll introduce something called Additive Smoothing which will ensure that \(p(x_i|C_k)\) is never 0 by making \(p(x_i|C_k) = \frac{frequency(x_i|C_k) + \alpha}{frequency(C_k) + \alpha n_i}\) where \(n_i\) is the number of possible values for \(x_i\) (in our example, 2). If \(\alpha = 1\) this is known as Laplace Smoothing. In our example, Laplace Smoothing produces the following:
\[ \begin{aligned} p(\mathbf{x}|\text{Technology}) &= \frac{2+1}{3+1 \cdot 2}\frac{3+1}{3+1 \cdot 2} \ldots \frac{0+1}{3+1 \cdot 2} \ldots = 0.01132 \Rightarrow p(\text{Technology}|\mathbf{x}) = 0.7431\\ p(\mathbf{x}|\text{Sales}) &= \frac{2+1}{4+1 \cdot 2}\frac{3+1}{4+1 \cdot 2} \ldots \frac{1+1}{4+1 \cdot 2} \ldots = 0.0001058 \Rightarrow p(\text{Sales}|\mathbf{x}) = 0.00926\\ p(\mathbf{x}|\text{Finance}) &= \frac{2+1}{3+1 \cdot 2}\frac{3+1}{3+1 \cdot 2} \ldots \frac{1+1}{3+1 \cdot 2} \ldots = 0.003775 \Rightarrow p(\text{Finance}|\mathbf{x}) = 0.2477 \end{aligned} \]
Generalizing
Recall our model which is a direct translation of Bayes’ Theorem
\[ p(C_k \vert \mathbf{x}) = \frac{p(\mathbf{x} \vert C_k)p(C_k)}{p(\mathbf{x})} \]
From our naive assumption, we transformed \(p(\mathbf{x} \vert C_k)\) into the product \(\prod_{i=1}^n p(x_i \vert C_k)\). Then we calculated \(p(x_i \vert C_k)\) using relative frequencies. The underlying argument for using relative frequencies is that \(x_i\) was a Bernoulli Random Variable (because \(x_i\) is either 0 or 1 for our example). However, in the general case \(x_i\) can be from any distribution and our Naive Bayes model will still work as long as we can closely estimate \(p(x_i \vert C_k)\). This includes multinomial random variables (commonly used in document classification problems like ours, but where word-repeats can occur) as well as continuous random variables. That means we can include things like a user’s bio (word frequencies) and age as additional features in our model.
Questions or comments? Drop me a line – bgorman@GormAnalysis.com Enjoyed this article?Show your support and buy some GormAnalysis merch. |
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Now showing items 1-10 of 108
Measurement of electrons from beauty hadron decays in pp collisions at root √s=7 TeV
(Elsevier, 2013-04-10)
The production cross section of electrons from semileptonic decays of beauty hadrons was measured at mid-rapidity (|y| < 0.8) in the transverse momentum range 1 < pT <8 GeV/c with the ALICE experiment at the CERN LHC in ...
Event-by-event mean pT fluctuations in pp and Pb–Pb collisions at the LHC
(Springer, 2014-10)
Event-by-event fluctuations of the mean transverse momentum of charged particles produced in pp collisions at s√ = 0.9, 2.76 and 7 TeV, and Pb–Pb collisions at √sNN = 2.76 TeV are studied as a function of the ...
Anomalous evolution of the near-side jet peak shape in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 TeV
(American Physical Society, 2017-09-08)
The measurement of two-particle angular correlations is a powerful tool to study jet quenching in a $p_{\mathrm{T}}$ region inaccessible by direct jet identification. In these measurements pseudorapidity ($\Delta\eta$) and ...
Enhanced production of multi-strange hadrons in high-multiplicity proton-proton collisions
(Nature Publishing Group, 2017)
At sufficiently high temperature and energy density, nuclear matter undergoes a transition to a phase in which quarks and gluons are not confined: the quark–gluon plasma (QGP)1. Such an exotic state of strongly interacting ...
Multiplicity dependence of the average transverse momentum in pp, p-Pb, and Pb-Pb collisions at the LHC
(Elsevier, 2013-12)
The average transverse momentum <$p_T$> versus the charged-particle multiplicity $N_{ch}$ was measured in p-Pb collisions at a collision energy per nucleon-nucleon pair $\sqrt{s_{NN}}$ = 5.02 TeV and in pp collisions at ...
Production of light nuclei and anti-nuclei in $pp$ and Pb-Pb collisions at energies available at the CERN Large Hadron Collider
(American Physical Society, 2016-02)
The production of (anti-)deuteron and (anti-)$^{3}$He nuclei in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has been studied using the ALICE detector at the LHC. The spectra exhibit a significant hardening with ...
Forward-central two-particle correlations in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV
(Elsevier, 2016-02)
Two-particle angular correlations between trigger particles in the forward pseudorapidity range ($2.5 < |\eta| < 4.0$) and associated particles in the central range ($|\eta| < 1.0$) are measured with the ALICE detector in ...
K$^{*}(892)^{0}$ and $\phi(1020)$ meson production at high transverse momentum in pp and Pb-Pb collisions at $\sqrt{s_\mathrm{NN}}$ = 2.76 TeV
(American Physical Society, 2017-06)
The production of K$^{*}(892)^{0}$ and $\phi(1020)$ mesons in proton-proton (pp) and lead-lead (Pb-Pb) collisions at $\sqrt{s_\mathrm{NN}} =$ 2.76 TeV has been analyzed using a high luminosity data sample accumulated in ...
Directed flow of charged particles at mid-rapidity relative to the spectator plane in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV
(American Physical Society, 2013-12)
The directed flow of charged particles at midrapidity is measured in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV relative to the collision plane defined by the spectator nucleons. Both, the rapidity odd ($v_1^{odd}$) and ...
Measurement of D-meson production versus multiplicity in p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV
(Springer, 2016-08)
The measurement of prompt D-meson production as a function of multiplicity in p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV with the ALICE detector at the LHC is reported. D$^0$, D$^+$ and D$^{*+}$ mesons are reconstructed ... |
Under the auspices of the Computational Complexity Foundation (CCF)
We prove a model-independent non-linear time lower bound for a slight generalization of the quantified Boolean formula problem (QBF). In particular, we give a reduction from arbitrary languages in alternating time t(n) to QBFs describable in O(t(n)) bits by a reasonable (polynomially) succinct encoding. The reduction works for many reasonable ... more >>>
Over the years, proof systems for propositional satisfiability (SAT)
have been extensively studied. Recently, proof systems for quantified Boolean formulas (QBFs) have also been gaining attention. Q-resolution is a calculus enabling producing proofs from DPLL-based QBF solvers. While DPLL has become a dominating technique for SAT, QBF has been tackled ... more >>>
We revisit the complexity of the satisfiability problem for quantified Boolean formulas. We show that satisfiability
of quantified CNFs of size $\poly(n)$ on $n$ variables with $O(1)$ quantifier blocks can be solved in time $2^{n-n^{\Omega(1)}}$ by zero-error randomized algorithms. This is the first known improvement over brute force search in ... more >>>
In sharp contrast to classical proof complexity we are currently short of lower bound techniques for QBF proof systems. In this paper we establish the feasible interpolation technique for all resolution-based QBF systems, whether modelling CDCL or expansion-based solving. This both provides the first general lower bound method for QBF ... more >>>
We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit $C(x_1,\ldots,x_n)$ of size $s$ and degree $d$ over a field ${\mathbb F}$, and any inputs $a_1,\ldots,a_K \in {\mathbb F}^n$,
$\bullet$ the Prover sends the Verifier the values $C(a_1), \ldots, C(a_K) \in {\mathbb F}$ and ... more >>>
We devise a new technique to prove lower bounds for the proof size in resolution-type calculi for quantified Boolean formulas (QBF). The new technique applies to the strong expansion system IR-calc and thereby also to the most studied QBF system Q-Resolution.
Our technique exploits a clear semantic paradigm, showing the ... more >>>
We define a cutting planes system CP+$\forall$red for quantified Boolean formulas (QBF) and analyse the proof-theoretic strength of this new calculus. While in the propositional case, Cutting Planes is of intermediate strength between resolution and Frege, our findings here show that the situation in QBF is slightly more complex: while ... more >>>
We aim to understand inherent reasons for lower bounds for QBF proof systems and revisit and compare two previous approaches in this direction.
The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff & Pich, LICS'16). Here we ... more >>>
As a natural extension of the SAT problem, different proof systems for quantified Boolean formulas (QBF) have been proposed. Many of these extend a propositional system to handle universal quantifiers.
By formalising the construction of the QBF proof system from a propositional proof system, by the addition of the ... more >>>
We give an analogue of the Riis Complexity Gap Theorem for Quanti fied Boolean Formulas (QBFs). Every fi rst-order sentence $\phi$ without finite models gives rise to a sequence of QBFs whose minimal refutations in tree-like Q-Resolution are either of polynomial size (if $\phi$ has no models) or at least ... more >>>
In their influential paper `Short proofs are narrow -- resolution made simple', Ben-Sasson and Wigderson introduced a crucial tool for proving lower bounds on the lengths of proofs in the resolution calculus. Over a decade later their technique for showing lower bounds on the size of proofs, by examining the ... more >>>
For quantified Boolean formulas (QBF), a resolution system with a symmetry rule was recently introduced by Kauers and Seidl (Inf. Process. Lett. 2018). In this system, many formulas hard for QBF resolution admit short proofs.
Kauers and Seidl apply the symmetry rule on symmetries of the original formula. Here we ... more >>> |
in definition of $\alpha$-inaccessible cardinals on Wikipedia, we can read :
For example, denote by $ψ_0(λ)$ the λth inaccessible cardinal, then the fixed points of $ψ_0$ are the 1-inaccessible cardinals.
But, the function $\psi_0$ is not a normal function (as it would implies that $cf(\psi_0(\omega))=\omega$, and, not so inaccessible !). So what (smallest, if any) hypothesis do we need to prove that $\psi_0$ has a fixed point and how to ?
same question for $\alpha$-hyper-inaccessible.
Is it sufficient for a cardinal to be inaccessible, hyper-inaccessible, hyper-hyper-inaccessible,... and so on, to be Mahlo ? |
Let's consider this system
Transmitter --> SWR meter --> Antenna
A (V)SWR of 1.3 means that the magnitude of your reflection coefficient $\Gamma$ between your transmitter and your antenna is
$$|\Gamma| = \frac{SWR-1}{SWR+1} = \frac{0.3}{2.3}\approx 0.13$$
which is pretty good, actually, meaning that but 13% of the field strength your transmitter inserts into the antenna is reflected by the antenna, or that 87% actually reach the antenna!
You can convert that into a relation of power inserted and reflected, simply by squaring it – then things look like this: The reflection dampening factor (i.e. how much of your energy is reflected is $$a=\frac{1}{|\Gamma|^2}\approx59\approx 17dB$$.
Now, for a power amplifier to reach maximum power output, it must be connected to a device that matches its source impedance $Z_s$. In that case, 0% of power is reflected. In all other situations (e.g. $Z_s=50\,\Omega$, $Z_a\ne50\, \Omega$, output power is below the maximum output. I don't actually know how your device displays forward power – that's something non-trivial to measure – but the fact that we don't have a unity SWR indicates your PA can't sink it's full output.
It's known that
$$\begin{align}\Gamma &= \frac{Z_s-Z_a}{Z_s+Z_a}\\\implies\\0.13 &= \frac{|Z_s-Z_a|}{Z_s+Z_a}\\0.13(Z_s+Z_a) &= |Z_s-Z_a|\\0.13Z_s+0.13Z_a &= |Z_s-Z_a|\\0.13Z_s+0.13Z_a &= Z_s - Z_a &\vee&& 0.13Z_s+0.13Z_a &= Z_a - Z_s\\-0.87Z_s&= - 1.13 Z_a &\vee&& 1.13 Z_s &= 0.87 Z_a \\Z_a&\approx 0.77 Z_s &\vee&& Z_a &\approx 1.30 Z_s\text{ .}\end{align}$$I'm going to assum $Z_s = 50\,\Omega$, so your antenna impedance is either $38.5$ or $65.0\,\Omega$.
That doesn't really explain the power loss your device is displaying, but it's common for HF measurement devices to not be on identical terms. |
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Higher harmonic flow coefficients of identified hadrons in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(Springer, 2016-09)
The elliptic, triangular, quadrangular and pentagonal anisotropic flow coefficients for $\pi^{\pm}$, $\mathrm{K}^{\pm}$ and p+$\overline{\mathrm{p}}$ in Pb-Pb collisions at $\sqrt{s_\mathrm{{NN}}} = 2.76$ TeV were measured ... |
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 $z = e^{2 \pi i \theta}$ for some $\theta \in [0,1)$.
Then the equation becomes $e^{2 \pi i(\theta + an)} = 1$, which holds if and only if $2\pi (\theta + an) = 2 \pi k$ for some $k \in \Bbb{Z}$. Solving for $a$ gives $a = \frac{k-\theta}{n}$...
What if $\theta$ is irrational...what did I do wrong?
'cause I understand that second one but I'm having a hard time explaining it in words
(Re: the first one: a matrix transpose "looks" like the equation $Ax\cdot y=x\cdot A^\top y$. Which implies several things, like how $A^\top x$ is perpendicular to $A^{-1}x^\top$ where $x^\top$ is the vector space perpendicular to $x$.)
DogAteMy: I looked at the link. You're writing garbage with regard to the transpose stuff. Why should a linear map from $\Bbb R^n$ to $\Bbb R^m$ have an inverse in the first place? And for goodness sake don't use $x^\top$ to mean the orthogonal complement when it already means something.
he based much of his success on principles like this I cant believe ive forgotten it
it's basically saying that it's a waste of time to throw a parade for a scholar or win he or she over with compliments and awards etc but this is the biggest source of sense of purpose in the non scholar
yeah there is this thing called the internet and well yes there are better books than others you can study from provided they are not stolen from you by drug dealers you should buy a text book that they base university courses on if you can save for one
I was working from "Problems in Analytic Number Theory" Second Edition, by M.Ram Murty prior to the idiots robbing me and taking that with them which was a fantastic book to self learn from one of the best ive had actually
Yeah I wasn't happy about it either it was more than $200 usd actually well look if you want my honest opinion self study doesn't exist, you are still being taught something by Euclid if you read his works despite him having died a few thousand years ago but he is as much a teacher as you'll get, and if you don't plan on reading the works of others, to maintain some sort of purity in the word self study, well, no you have failed in life and should give up entirely. but that is a very good book
regardless of you attending Princeton university or not
yeah me neither you are the only one I remember talking to on it but I have been well and truly banned from this IP address for that forum now, which, which was as you might have guessed for being too polite and sensitive to delicate religious sensibilities
but no it's not my forum I just remembered it was one of the first I started talking math on, and it was a long road for someone like me being receptive to constructive criticism, especially from a kid a third my age which according to your profile at the time you were
i have a chronological disability that prevents me from accurately recalling exactly when this was, don't worry about it
well yeah it said you were 10, so it was a troubling thought to be getting advice from a ten year old at the time i think i was still holding on to some sort of hopes of a career in non stupidity related fields which was at some point abandoned
@TedShifrin thanks for that in bookmarking all of these under 3500, is there a 101 i should start with and find my way into four digits? what level of expertise is required for all of these is a more clear way of asking
Well, there are various math sources all over the web, including Khan Academy, etc. My particular course was intended for people seriously interested in mathematics (i.e., proofs as well as computations and applications). The students in there were about half first-year students who had taken BC AP calculus in high school and gotten the top score, about half second-year students who'd taken various first-year calculus paths in college.
long time ago tho even the credits have expired not the student debt though so i think they are trying to hint i should go back a start from first year and double said debt but im a terrible student it really wasn't worth while the first time round considering my rate of attendance then and how unlikely that would be different going back now
@BalarkaSen yeah from the number theory i got into in my most recent years it's bizarre how i almost became allergic to calculus i loved it back then and for some reason not quite so when i began focusing on prime numbers
What do you all think of this theorem: The number of ways to write $n$ as a sum of four squares is equal to $8$ times the sum of divisors of $n$ if $n$ is odd and $24$ times sum of odd divisors of $n$ if $n$ is even
A proof of this uses (basically) Fourier analysis
Even though it looks rather innocuous albeit surprising result in pure number theory
@BalarkaSen well because it was what Wikipedia deemed my interests to be categorized as i have simply told myself that is what i am studying, it really starting with me horsing around not even knowing what category of math you call it. actually, ill show you the exact subject you and i discussed on mmf that reminds me you were actually right, i don't know if i would have taken it well at the time tho
yeah looks like i deleted the stack exchange question on it anyway i had found a discrete Fourier transform for $\lfloor \frac{n}{m} \rfloor$ and you attempted to explain to me that is what it was that's all i remember lol @BalarkaSen
oh and when it comes to transcripts involving me on the internet, don't worry, the younger version of you most definitely will be seen in a positive light, and just contemplating all the possibilities of things said by someone as insane as me, agree that pulling up said past conversations isn't productive
absolutely me too but would we have it any other way? i mean i know im like a dog chasing a car as far as any real "purpose" in learning is concerned i think id be terrified if something didnt unfold into a myriad of new things I'm clueless about
@Daminark They key thing if I remember correctly was that if you look at the subgroup $\Gamma$ of $\text{PSL}_2(\Bbb Z)$ generated by (1, 2|0, 1) and (0, -1|1, 0), then any holomorphic function $f : \Bbb H^2 \to \Bbb C$ invariant under $\Gamma$ (in the sense that $f(z + 2) = f(z)$ and $f(-1/z) = z^{2k} f(z)$, $2k$ is called the weight) such that the Fourier expansion of $f$ at infinity and $-1$ having no constant coefficients is called a cusp form (on $\Bbb H^2/\Gamma$).
The $r_4(n)$ thing follows as an immediate corollary of the fact that the only weight $2$ cusp form is identically zero.
I can try to recall more if you're interested.
It's insightful to look at the picture of $\Bbb H^2/\Gamma$... it's like, take the line $\Re[z] = 1$, the semicircle $|z| = 1, z > 0$, and the line $\Re[z] = -1$. This gives a certain region in the upper half plane
Paste those two lines, and paste half of the semicircle (from -1 to i, and then from i to 1) to the other half by folding along i
Yup, that $E_4$ and $E_6$ generates the space of modular forms, that type of things
I think in general if you start thinking about modular forms as eigenfunctions of a Laplacian, the space generated by the Eisenstein series are orthogonal to the space of cusp forms - there's a general story I don't quite know
Cusp forms vanish at the cusp (those are the $-1$ and $\infty$ points in the quotient $\Bbb H^2/\Gamma$ picture I described above, where the hyperbolic metric gets coned off), whereas given any values on the cusps you can make a linear combination of Eisenstein series which takes those specific values on the cusps
So it sort of makes sense
Regarding that particular result, saying it's a weight 2 cusp form is like specifying a strong decay rate of the cusp form towards the cusp. Indeed, one basically argues like the maximum value theorem in complex analysis
@BalarkaSen no you didn't come across as pretentious at all, i can only imagine being so young and having the mind you have would have resulted in many accusing you of such, but really, my experience in life is diverse to say the least, and I've met know it all types that are in everyway detestable, you shouldn't be so hard on your character you are very humble considering your calibre
You probably don't realise how low the bar drops when it comes to integrity of character is concerned, trust me, you wouldn't have come as far as you clearly have if you were a know it all
it was actually the best thing for me to have met a 10 year old at the age of 30 that was well beyond what ill ever realistically become as far as math is concerned someone like you is going to be accused of arrogance simply because you intimidate many ignore the good majority of that mate |
Ok, I admit it. I'm confused. I'm a physics student attempting to learn some group theory and topology in my spare time. I was reading about group representations. For example I get that the set of spherical harmonics $Y_{lm}(\theta,\phi)$ form a set of irreducible representations of $SO(3)$. What I don't get is their dimension. For example here (page 144 as it reads on the paper heading) it is stated:
The $Y_{lm}(\theta,\phi)$ form a $(2l+1)$ -dimensional representation of $SO(3)$.
Now, in utilizing the spherical harmonics in physics, I know that I'm working in a three dimensional space. I further know that I can represent any “well behaved” function $f(\theta ,\phi)$ on the unit sphere in $R^3$ in terms of a series of these spherical harmonics (properly weighted with coefficients) like so:
$$f(\theta,\phi)=\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}a_{lm}Y_{lm}(\theta,\phi)$$
It's not lost on me that the dimensionality for a given representation is the same as the number of $m$ values (ie the second summation). I know the function "lives" in a $2$-dimensional space (the unit sphere). So what is going on? Is there a mapping or reference to another space I'm missing? is this a physicist's notational/dictionarial clash with the mathematician's?? Thank you in advance. |
By "configuration spaces of $\mathbb{R}^n$" I mean ordered configuration spaces:$$\operatorname{Conf}_k(\mathbb{R}^n) = \{ (x_1,\dots,x_k) \in (\mathbb{R}^n)^k \mid x_i \neq x_j, \, \forall i \neq j \}.$$ These spaces are "formal" over the rationals, meaning that their cohomology ring completely encodes their rational homotopy type.
For $n = 2$ there is no need to involve operads, because there is a direct map $H^*(\operatorname{Conf}_k(\mathbb{C})) \xrightarrow{\sim} \Omega^*(\operatorname{Conf}_k(\mathbb{C}))$, sending the usual generators of the cohomology to $d\log(z_i - z_j)$ (Arnold, 1969). For $n \ge 3$ this doesn't work because the Arnold relations do not hold "on the nose" in $\Omega^*(\operatorname{Conf}_k(\mathbb{R}^n))$.
The proofs I know for $n \ge 3$ all involve operads. More precisely, the little disks operads, whose components are homotopy equivalent to the configuration spaces above. The theorem is that these operads are formal. Hence the configuration spaces are formal. See the results of Kontsevich (and Guillén Santos–Navarro–Pascual–Roig to descend to $\mathbb{Q}$), Tamarkin (for $n = 2$ actually), Lambrechts–Volić, Petersen, Fresse–Willwacher...
I am curious, is there a proof somewhere in the literature that has nothing to do with operads? I have not found any, but, well, I cannot be 100% sure... |
@JosephWright Well, we still need table notes etc. But just being able to selectably switch off parts of the parsing one does not need... For example, if a user specifies format 2.4, does the parser even need to look for e syntax, or ()'s?
@daleif What I am doing to speed things up is to store the data in a dedicated format rather than a property list. The latter makes sense for units (open ended) but not so much for numbers (rigid format).
@JosephWright I want to know about either the bibliography environment or \DeclareFieldFormat. From the documentation I see no reason not to treat these commands as usual, though they seem to behave in a slightly different way than I anticipated it. I have an example here which globally sets a box, which is typeset outside of the bibliography environment afterwards. This doesn't seem to typeset anything. :-( So I'm confused about the inner workings of biblatex (even though the source seems....
well, the source seems to reinforce my thought that biblatex simply doesn't do anything fancy). Judging from the source the package just has a lot of options, and that's about the only reason for the large amount of lines in biblatex1.sty...
Consider the following MWE to be previewed in the build in PDF previewer in Firefox\documentclass[handout]{beamer}\usepackage{pgfpages}\pgfpagesuselayout{8 on 1}[a4paper,border shrink=4mm]\begin{document}\begin{frame}\[\bigcup_n \sum_n\]\[\underbrace{aaaaaa}_{bbb}\]\end{frame}\end{d...
@Paulo Finally there's a good synth/keyboard that knows what organ stops are! youtube.com/watch?v=jv9JLTMsOCE Now I only need to see if I stay here or move elsewhere. If I move, I'll buy this there almost for sure.
@JosephWright most likely that I'm for a full str module ... but I need a little more reading and backlog clearing first ... and have my last day at HP tomorrow so need to clean out a lot of stuff today .. and that does have a deadline now
@yo' that's not the issue. with the laptop I lose access to the company network and anythign I need from there during the next two months, such as email address of payroll etc etc needs to be 100% collected first
@yo' I'm sorry I explain too bad in english :) I mean, if the rule was use \tl_use:N to retrieve the content's of a token list (so it's not optional, which is actually seen in many places). And then we wouldn't have to \noexpand them in such contexts.
@JosephWright \foo:V \l_some_tl or \exp_args:NV \foo \l_some_tl isn't that confusing.
@Manuel As I say, you'd still have a difference between say \exp_after:wN \foo \dim_use:N \l_my_dim and \exp_after:wN \foo \tl_use:N \l_my_tl: only the first case would work
@Manuel I've wondered if one would use registers at all if you were starting today: with \numexpr, etc., you could do everything with macros and avoid any need for \<thing>_new:N (i.e. soft typing). There are then performance questions, termination issues and primitive cases to worry about, but I suspect in principle it's doable.
@Manuel Like I say, one can speculate for a long time on these things. @FrankMittelbach and @DavidCarlisle can I am sure tell you lots of other good/interesting ideas that have been explored/mentioned/imagined over time.
@Manuel The big issue for me is delivery: we have to make some decisions and go forward even if we therefore cut off interesting other things
@Manuel Perhaps I should knock up a set of data structures using just macros, for a bit of fun [and a set that are all protected :-)]
@JosephWright I'm just exploring things myself “for fun”. I don't mean as serious suggestions, and as you say you already thought of everything. It's just that I'm getting at those points myself so I ask for opinions :)
@Manuel I guess I'd favour (slightly) the current set up even if starting today as it's normally \exp_not:V that applies in an expansion context when using tl data. That would be true whether they are protected or not. Certainly there is no big technical reason either way in my mind: it's primarily historical (expl3 pre-dates LaTeX2e and so e-TeX!)
@JosephWright tex being a macro language means macros expand without being prefixed by \tl_use. \protected would affect expansion contexts but not use "in the wild" I don't see any way of having a macro that by default doesn't expand.
@JosephWright it has series of footnotes for different types of footnotey thing, quick eye over the code I think by default it has 10 of them but duplicates for minipages as latex footnotes do the mpfoot... ones don't need to be real inserts but it probably simplifies the code if they are. So that's 20 inserts and more if the user declares a new footnote series
@JosephWright I was thinking while writing the mail so not tried it yet that given that the new \newinsert takes from the float list I could define \reserveinserts to add that number of "classic" insert registers to the float list where later \newinsert will find them, would need a few checks but should only be a line or two of code.
@PauloCereda But what about the for loop from the command line? I guess that's more what I was asking about. Say that I wanted to call arara from inside of a for loop on the command line and pass the index of the for loop to arara as the jobname. Is there a way of doing that? |
My question is straightforward:
Do fermionic operators associated to different species commute or anticommute? Even if these operators have different quantum numbers? How can one prove this fact in a general QFT?
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I want to give another perspective. Bosons vs. fermions are usually introduced by considering the many-body wavefunction
$$|\psi(x_1,\ldots,x_N)\rangle$$
in the position basis, where $x_1,\ldots,x_N$ are the positions of $N$ particles. If these $N$ particles are distinguishable, then permuting the $x_j$ should result in a physically equivalent wavefunction, which differs from the original one by a phase. These phases have to be consistently defined, meaning they give a homomorphism from the permutation group $S_N \to U(1)$. It turns out there are two such homomorphisms, a trivial one and one which is $-1$ to the number of swaps. These correspond to bosonic or fermionic particles, respectively.
Now we consider a Hilbert space that includes $N$-particle states for all $N \ge 0$, with a state $|0\rangle$ we call the vacuum. We define the creation operator $a^\dagger(x)$ which creates a particle at $x$ (all particles are indistinguishable still). Then we define
$$|\psi(x_1,\ldots,x_N)\rangle = a^\dagger(x_N)\cdots a^\dagger(x_1)|0\rangle.$$
We see that our permutation action is thus equivalent to commutativity or anticommutativity of the creation operators in the bosonic or fermionic case, respectively.
This permutation symmetry imposes real constraints on the allowed $|\psi(x_1,\ldots,x_N)\rangle$, such as the Pauli exclusion principle for fermions.
However, permutation symmetry is not as impressive with multiple particle types. Suppose we had another species of particles at positions $y_1,\ldots,y_M$ and a joint wavefunction
$$|\psi(x_1,\ldots,x_N,y_1,\ldots,y_M)\rangle.$$
This wavefunction is fixed by $S_N \times S_M$, up to phases, which determine whether each species is bosonic or fermionic. There is no way to swap an $x$ particle with a $y$ particle, however.
We can likewise define another set of creation operators $b^\dagger(y)$ which create a $y$ particle at position $y$. And for which
$$|\psi(x_1,\ldots,x_N,y_1,\ldots,y_M)\rangle = b^\dagger(y_M) \cdots b^\dagger(y_1)a^\dagger(x_N)\cdots a^\dagger(x_1) |0\rangle.$$
We see that the permutation representations of $S_N$ and $S_M$ determine the commutation relations of the two species of creation operators with themselves, but do they determine the commutation relations of them with each other? Actually contrary to several other answers
they do not.
To see why, let's study two distinct species of fermion. Distinct physically means that the individual particle numbers, which are schematically
$$N = \int dx a^\dagger(x) a(x) \qquad M = \int dx b^\dagger(x) b(x)$$
are both conserved. This is key to the argument. Note these are Hemitian.
Suppose we now do the usual quantization, where $a$ and $b$ anti-commute. Let us define
$$b' = (-1)^N b.$$
We observe that the $b'$ anti-commute with each other but
commute with the $a$'s. Furthermore, we can define the many-body states using the $b'$'s and they only differ by what we had above by a sign. Finally, the time evolution of the $b'$ is equivalent to the time evolution of $b$ because $N$ is conserved.
The reason why some other answers got it wrong thinking about fancy things like $\mathbb{Z}_2$-graded algebras and super-Poisson brackets is that when there are multiple species, there are multiple gradings: in this case a $\mathbb{Z}_2 \times \mathbb{Z}_2$ grading.
In a geometric point of view, which is relevant for bosonization, we would say that neutral fermions all couple to the same spin structure. However, when there are global symmetries, then we can attach charge operators to our fermions for these global symmetries (as we did above), effectively creating multiple spin structures seen by different fermion species.
For more, see this paper: https://arxiv.org/abs/1312.0831
Fermionic creation and annihilation operators always satisfy commutation relations with bosonic (or more generally even) operators and anticommutation relations with the fermionic creation and annihilation operators (or more generally odd operators). This follows from the properties of super Poisson brackets. See Poisson Superalgebra
In particular, the creation operators for distinct, orthogonal modes always anticommute.
Fermions $f_i,\,f_j$ with respective momenta $\pi_i,\,\pi_j$
satisfy the equal-time canonical anticommutation relations $$\left\{\ f_i,\,f_j \right\} = \left\{\ \pi_i,\,\pi_j \right\} = 0,\,\left\{\ f_i\left(t,\,\mathbf{x}\right),\,\pi_j \left(t,\,\mathbf{x'}\right)\right\} = i\hbar \delta_{ij} \delta \left(\mathbf{x},\,\mathbf{x'}\right),$$where the second $\delta$ is a Dirac delta. The $i=j$ special case is a generalisation of a theory of a single fermon $f$ of momentum $\pi$. Why do the $i\neq j$ cases use anticommutators instead of commutators? Because we want our rules to be invariant under $f_i \to \sum_j M_{ij} f_j,\,\pi_i \to \sum_j \left(M^{-1}\right)_{ji} \pi_j$ for invertible choices of the matrix $M$. There's no consistent way to achieve this by using commutators sometimes. A similar explanation is available in terms of the ladder operators.
I think the existing answers to the question are incomplete and confusing, so let me elaborate.
The Hilbert space of two
independent fermions of different species (think electron and proton) is the tensor product of the respective spaces, and the field operators for the two fermions act independently on the two factors in the tensor product ($\psi_1\otimes I_2$ and $I_1\otimes \psi_2$ with $I_{1,2}$ being the identity operators on the two small Hilbert spaces). Operators like this commute just by definition, since they concern completely independent degrees of freedom.
This applies to absolutely any quantum fields: as long as they create/annihilate independent degrees of freedom, they commute. Two fermions, two bosons, a fermion and a boson, you name it. The only time anti-commutation comes up is when you write down the relations for one fermion with itself.
However, sometimes you have several "flavors" of fermions that "mix" under the action of some group (think quarks under $SU(3)$). In this case they have to anti-commute for the relations to remain invariant under the action of the mixing group. This is a sign that the better way to think about flavors is as mere components of one larger fermionic field with an extra index, which now doesn't just enumerate the fermions but turns them into a single element of a representation of a group.
To summarize, the "species" of any two fields commute, and the "flavors" of one fermionic field anti-commute (because they are really just components of one bigger fermion). |
By definition, Hadamard transformation (acting on a qubit) maps the unit vector in the $Y$ axis direction of the Bloch Sphere ($S^2$) to its negative, equivalent to a rotation of $\pi$ rad around $X+Z$ axis. I understand it pictorially.
I have trouble showing this explicitly using the matrix representation of Hadamard transformation $H= \frac{1}{\sqrt{2}}\begin{bmatrix} {1}&{1}\\ {1}&{-1} \end{bmatrix}$for a qubit $\hat y$. This qubit $\hat y$ (the unit vector in the $Y$ axis direction of the Bloch Sphere) has $\theta=\frac{\pi}{2}$ and $\phi=\frac{\pi}{2}$, where these angles are the ones that define the state of a qubit($\lvert \psi\rangle=cos\,(\frac{\theta}{2})\lvert 0\rangle+e^{i\phi}\,sin\,(\frac{\theta}{2})\lvert 1\rangle$). Therefore, $\hat y = \frac{1}{\sqrt2}\lvert 0\rangle + e^{i\frac{\pi}{2}}\frac{1}{\sqrt2}\lvert 1\rangle= \frac{1}{\sqrt2}\lvert 0\rangle + \frac{i}{\sqrt2} \lvert 1\rangle$. Now acting the Hadamard transformation matrix on it we have: $\begin{equation} H\cdot\hat y=\frac{1}{\sqrt{2}}\begin{bmatrix} {1}&{1}\\ {1}&{-1} \end{bmatrix}\begin{bmatrix} \frac{1}{\sqrt2}\\\frac{i}{\sqrt2}\end{bmatrix}=\frac{1}{2}\begin{bmatrix}1+i\\1-i\end{bmatrix}\end{equation}$. In contrast, we expected the result of $H\cdot\hat y$ be instead $-\hat y$ by the pictorial definition of Hadamard transformation. What am I missing? Any help regarding the mistake that I'm making here is appreciated.
By definition, Hadamard transformation (acting on a qubit) maps the unit vector in the $Y$ axis direction of the Bloch Sphere ($S^2$) to its negative, equivalent to a rotation of $\pi$ rad around $X+Z$ axis. I understand it pictorially.
You have to take out a factor $\frac{1+i}{\sqrt{2}}$ -- the action on the Bloch sphere is only up to a phase. Also, not that the qubit vector corresponding to $-\hat y$ is $(|0\rangle-i|1\rangle)/\sqrt{2}$. |
Seeing that in the Chomsky Hierarchy Type 3 languages can be recognised by a DFA (which has no stacks), Type 2 by a DFA with one stack (i.e. a push-down automaton) and Type 0 by a DFA with two stacks (i.e. with one queue, i.e. with a tape, i.e. by a Turing Machine), how do Type 1 languages fit in...
Considering this pseudo-code of an bubblesort:FOR i := 0 TO arraylength(list) STEP 1switched := falseFOR j := 0 TO arraylength(list)-(i+1) STEP 1IF list[j] > list[j + 1] THENswitch(list,j,j+1)switched := trueENDIFNEXTIF switch...
Let's consider a memory segment (whose size can grow or shrink, like a file, when needed) on which you can perform two basic memory allocation operations involving fixed size blocks:allocation of one blockfreeing a previously allocated block which is not used anymore.Also, as a requiremen...
Rice's theorem tell us that the only semantic properties of Turing Machines (i.e. the properties of the function computed by the machine) that we can decide are the two trivial properties (i.e. always true and always false).But there are other properties of Turing Machines that are not decidabl...
People often say that LR(k) parsers are more powerful than LL(k) parsers. These statements are vague most of the time; in particular, should we compare the classes for a fixed $k$ or the union over all $k$? So how is the situation really? In particular, I am interested in how LL(*) fits in.As f...
Since the current FAQs say this site is for students as well as professionals, what will the policy on homework be?What are the guidelines that a homework question should follow if it is to be asked? I know on math.se they loosely require that the student make an attempt to solve the question a...
I really like the new beta theme, I guess it is much more attractive to newcomers than the sketchy one (which I also liked). Thanks a lot!However I'm slightly embarrassed because I can't read what I type, both in the title and in the body of a post. I never encountered the problem on other Stac...
This discussion started in my other question "Will Homework Questions Be Allowed?".Should we allow the tag? It seems that some of our sister sites (Programmers, stackoverflow) have not allowed the tag as it isn't constructive to their sites. But other sites (Physics, Mathematics) do allow the s...
There have been many questions on CST that were either closed, or just not answered because they weren't considered research level. May those questions (as long as they are of good quality) be reposted or moved here?I have a particular example question in mind: http://cstheory.stackexchange.com...
Ok, so in most introductory Algorithm classes, either BigO or BigTheta notation are introduced, and a student would typically learn to use one of these to find the time complexity.However, there are other notations, such as BigOmega and SmallOmega. Are there any specific scenarios where one not...
Many textbooks cover intersection types in the lambda-calculus. The typing rules for intersection can be defined as follows (on top of the simply typed lambda-calculus with subtyping):$$\dfrac{\Gamma \vdash M : T_1 \quad \Gamma \vdash M : T_2}{\Gamma \vdash M : T_1 \wedge T_2}...
I expect to see pseudo code and maybe even HPL code on regular basis. I think syntax highlighting would be a great thing to have.On Stackoverflow, code is highlighted nicely; the schema used is inferred from the respective question's tags. This won't work for us, I think, because we probably wo...
Sudoku generation is hard enough. It is much harder when you have to make an application that makes a completely random Sudoku.The goal is to make a completely random Sudoku in Objective-C (C is welcome). This Sudoku generator must be easily modified, and must support the standard 9x9 Sudoku, a...
I have observed that there are two different types of states in branch prediction.In superscalar execution, where the branch prediction is very important, and it is mainly in execution delay rather than fetch delay.In the instruction pipeline, where the fetching is more problem since the inst...
Is there any evidence suggesting that time spent on writing up, or thinking about the requirements will have any effect on the development time? Study done by Standish (1995) suggests that incomplete requirements partially (13.1%) contributed to the failure of the projects. Are there any studies ...
NPI is the class of NP problems without any polynomial time algorithms and not known to be NP-hard. I'm interested in problems such that a candidate problem in NPI is reducible to it but it is not known to be NP-hard and there is no known reduction from it to the NPI problem. Are there any known ...
I am reading Mining Significant Graph Patterns by Leap Search (Yan et al., 2008), and I am unclear on how their technique translates to the unlabeled setting, since $p$ and $q$ (the frequency functions for positive and negative examples, respectively) are omnipresent.On page 436 however, the au...
This is my first time to be involved in a site beta, and I would like to gauge the community's opinion on this subject.On StackOverflow (and possibly Math.SE), questions on introductory formal language and automata theory pop up... questions along the lines of "How do I show language L is/isn't...
This is my first time to be involved in a site beta, and I would like to gauge the community's opinion on this subject.Certainly, there are many kinds of questions which we could expect to (eventually) be asked on CS.SE; lots of candidates were proposed during the lead-up to the Beta, and a few...
This is somewhat related to this discussion, but different enough to deserve its own thread, I think.What would be the site policy regarding questions that are generally considered "easy", but may be asked during the first semester of studying computer science. Example:"How do I get the symme...
EPAL, the language of even palindromes, is defined as the language generated by the following unambiguous context-free grammar:$S \rightarrow a a$$S \rightarrow b b$$S \rightarrow a S a$$S \rightarrow b S b$EPAL is the 'bane' of many parsing algorithms: I have yet to enc...
Assume a computer has a precise clock which is not initialized. That is, the time on the computer's clock is the real time plus some constant offset. The computer has a network connection and we want to use that connection to determine the constant offset $B$.The simple method is that the compu...
Consider an inductive type which has some recursive occurrences in a nested, but strictly positive location. For example, trees with finite branching with nodes using a generic list data structure to store the children.Inductive LTree : Set := Node : list LTree -> LTree.The naive way of d...
I was editing a question and I was about to tag it bubblesort, but it occurred to me that tag might be too specific. I almost tagged it sorting but its only connection to sorting is that the algorithm happens to be a type of sort, it's not about sorting per se.So should we tag questions on a pa...
To what extent are questions about proof assistants on-topic?I see four main classes of questions:Modeling a problem in a formal setting; going from the object of study to the definitions and theorems.Proving theorems in a way that can be automated in the chosen formal setting.Writing a co...
Should topics in applied CS be on topic? These are not really considered part of TCS, examples include:Computer architecture (Operating system, Compiler design, Programming language design)Software engineeringArtificial intelligenceComputer graphicsComputer securitySource: http://en.wik...
I asked one of my current homework questions as a test to see what the site as a whole is looking for in a homework question. It's not a difficult question, but I imagine this is what some of our homework questions will look like.
I have an assignment for my data structures class. I need to create an algorithm to see if a binary tree is a binary search tree as well as count how many complete branches are there (a parent node with both left and right children nodes) with an assumed global counting variable.So far I have...
It's a known fact that every LTL formula can be expressed by a Buchi $\omega$-automata. But, apparently, Buchi automata is a more powerful, expressive model. I've heard somewhere that Buchi automata are equivalent to linear-time $\mu$-calculus (that is, $\mu$-calculus with usual fixpoints and onl...
Let us call a context-free language deterministic if and only if it can be accepted by a deterministic push-down automaton, and nondeterministic otherwise.Let us call a context-free language inherently ambiguous if and only if all context-free grammars which generate the language are ambiguous,...
One can imagine using a variety of data structures for storing information for use by state machines. For instance, push-down automata store information in a stack, and Turing machines use a tape. State machines using queues, and ones using two multiple stacks or tapes, have been shown to be equi...
Though in the future it would probably a good idea to more thoroughly explain your thinking behind your algorithm and where exactly you're stuck. Because as you can probably tell from the answers, people seem to be unsure on where exactly you need directions in this case.
What will the policy on providing code be?In my question it was commented that it might not be on topic as it seemes like I was asking for working code. I wrote my algorithm in pseudo-code because my problem didnt ask for working C++ or whatever language.Should we only allow pseudo-code here?... |
@HarryGindi So the $n$-simplices of $N(D^{op})$ are $Hom_{sCat}(\mathfrak{C}[n],D^{op})$. Are you using the fact that the whole simplicial set is the mapping simplicial object between cosimplicial simplicial categories, and taking the constant cosimplicial simplicial category in the right coordinate?
I guess I'm just very confused about how you're saying anything about the entire simplicial set if you're not producing it, in one go, as the mapping space between two cosimplicial objects. But whatever, I dunno. I'm having a very bad day with this junk lol.
It just seems like this argument is all about the sets of n-simplices. Which is the trivial part.
lol no i mean, i'm following it by context actually
so for the record i really do think that the simplicial set you're getting can be written as coming from the simplicial enrichment on cosimplicial objects, where you take a constant cosimplicial simplicial category on one side
@user1732 haha thanks! we had no idea if that'd actually find its way to the internet...
@JonathanBeardsley any quillen equivalence determines an adjoint equivalence of quasicategories. (and any equivalence can be upgraded to an adjoint (equivalence)). i'm not sure what you mean by "Quillen equivalences induce equivalences after (co)fibrant replacement" though, i feel like that statement is mixing category-levels
@JonathanBeardsley if nothing else, this follows from the fact that \frakC is a left quillen equivalence so creates weak equivalences among cofibrant objects (and all objects are cofibrant, in particular quasicategories are). i guess also you need to know the fact (proved in HTT) that the three definitions of "hom-sset" introduced in chapter 1 are all weakly equivalent to the one you get via \frakC
@IlaRossi i would imagine that this is in goerss--jardine? ultimately, this is just coming from the fact that homotopy groups are defined to be maps in (from spheres), and you only are "supposed" to map into things that are fibrant -- which in this case means kan complexes
@JonathanBeardsley earlier than this, i'm pretty sure it was proved by dwyer--kan in one of their papers around '80 and '81
@HarryGindi i don't know if i would say that "most" relative categories are fibrant. it was proved by lennart meier that model categories are Barwick--Kan fibrant (iirc without any further adjectives necessary)
@JonathanBeardsley what?! i really liked that picture! i wonder why they removed it
@HarryGindi i don't know about general PDEs, but certainly D-modules are relevant in the homotopical world
@HarryGindi oh interesting, thomason-fibrancy of W is a necessary condition for BK-fibrancy of (R,W)?
i also find the thomason model structure mysterious. i set up a less mysterious (and pretty straightforward) analog for $\infty$-categories in the fappendix here: arxiv.org/pdf/1510.03525.pdf
as for the grothendieck construction computing hocolims, i think the more fundamental thing is that the grothendieck construction itself is a lax colimit. combining this with the fact that ($\infty$-)groupoid completion is a left adjoint, you immediately get that $|Gr(F)|$ is the colimit of $B \xrightarrow{F} Cat \xrightarrow{|-|} Spaces$
@JonathanBeardsley If you want to go that route, I guess you still have to prove that ^op_s and ^op_Delta both lie in the unique nonidentity component of Aut(N(Qcat)) and Aut(N(sCat)) whatever nerve you mean in this particular case (the B-K relative nerve has the advantage here bc sCat is not a simplicial model cat)
I think the direct proof has a lot of advantages here, since it gives a point-set on-the-nose isomorphism
Yeah, definitely, but I'd like to stay and work with Cisinski on the Ph.D if possible, but I'm trying to keep options open
not put all my eggs in one basket, as it were
I mean, I'm open to coming back to the US too, but I don't have any ideas for advisors here who are interested in higher straightening/higher Yoneda, which I am convinced is the big open problem for infinity, n-cats
Gaitsgory and Rozenblyum, I guess, but I think they're more interested in applications of those ideas vs actually getting a hold of them in full generality
@JonathanBeardsley Don't sweat it. As it was mentioned I have now mod superpowers, so s/he can do very little to upset me. Since you're the room owner, let me know if I can be of any assistance here with the moderation (moderators on SE have network-wide chat moderating powers, but this is not my turf, so to speak).
There are two "opposite" functors:$$ op_\Delta\colon sSet\to sSet$$and$$op_s\colon sCat\to sCat.$$The first takes a simplicial set to its opposite simplicial set by precomposing with the opposite of a functor $\Delta\to \Delta$ which is the identity on objects and takes a morphism $\langle k...
@JonathanBeardsley Yeah, I worked out a little proof sketch of the lemma on a notepad
It's enough to show everything works for generating cofaces and codegeneracies
the codegeneracies are free, the 0 and nth cofaces are free
all of those can be done treating frak{C} as a black box
the only slightly complicated thing is keeping track of the inner generated cofaces, but if you use my description of frak{C} or the one Joyal uses in the quasicategories vs simplicial categories paper, the combinatorics are completely explicit for codimension 1 face inclusions
the maps on vertices are obvious, and the maps on homs are just appropriate inclusions of cubes on the {0} face of the cube wrt the axis corresponding to the omitted inner vertex
In general, each Δ[1] factor in Hom(i,j) corresponds exactly to a vertex k with i<k<j, so omitting k gives inclusion onto the 'bottom' face wrt that axis, i.e. Δ[1]^{k-i-1} x {0} x Δ[j-k-1] (I'd call this the top, but I seem to draw my cubical diagrams in the reversed orientation).
> Thus, using appropriate tags one can increase ones chances that users competent to answer the question, or just interested in it, will notice the question in the first place. Conversely, using only very specialized tags (which likely almost nobody specifically favorited, subscribed to, etc) or worse just newly created tags, one might miss a chance to give visibility to ones question.
I am not sure to which extent this effect is noticeable on smaller sites (such as MathOverflow) but probably it's good to follow the recommendations given in the FAQ. (And MO is likely to grow a bit more in the future, so then it can become more important.) And also some smaller tags have enough followers.
You are asking posts far away from areas I am familiar with, so I am not really sure which top-level tags would be a good fit for your questions - otherwise I would edit/retag the posts myself. (Other than possibility to ping you somewhere in chat, the reason why I posted this in this room is that users of this room are likely more familiar with the topics you're interested in and probably they would be able to suggest suitable tags.)
I just wanted to mention this, in case it helps you when asking question here. (Although it seems that you're doing fine.)
@MartinSleziak even I was not sure what other tags are appropriate to add.. I will see other questions similar to this, see what tags they have added and will add if I get to see any relevant tags.. thanks for your suggestion.. it is very reasonable,.
You don't need to put only one tag, you can put up to five. In general it is recommended to put a very general tag (usually an "arxiv" tag) to indicate broadly which sector of math your question is in, and then more specific tags
I would say that the topics of the US Talbot, as with the European Talbot, are heavily influenced by the organizers. If you look at who the organizers were/are for the US Talbot I think you will find many homotopy theorists among them. |
Under the auspices of the Computational Complexity Foundation (CCF)
In this paper we prove two results about $AC^0[\oplus]$ circuits.
We show that for $d(N) = o(\sqrt{\log N/\log \log N})$ and $N \leq s(N) \leq 2^{dN^{1/d^2}}$ there is an explicit family of functions $\{f_N:\{0,1\}^N\rightarrow \{0,1\}\}$ such that
$f_N$ has uniform $AC^0$ formulas of depth $d$ and size at ... more >>> |
Let $X$ denote the flag variety of a semi-simple group $G$ (in characteristic 0) and let $T_X$ denote its tangent bundle. I would like to ask the following question(s):
1) Is it true that for any $n\geq 0$ we have $H^i(X,T_X^{\otimes n})=0$ for $i>0$?
2) More generally, let $\lambda$ be a dominant weight of $G$ and let $\mathcal O(\lambda)$ be the corresponding line bundle on $X$. Is it true that $H^i(X, T_X^{\otimes n}\otimes \mathcal O(\lambda))=0$ for $i>0$?
When tensor powers of $T_X$ are replaced by symmetric powers, this is known to be true (for example it is proved in a paper of Kumar, Lauritzen and Thomsen). |
Multiplying Whole Numbers
Let's start multiplying whole numbers.
Suppose 5 of your friends gave you 8 apples. How many apples do you have? It is easily to calculate it using addition: 8+8+8+8+8=40.
But it is very long and time consuming to make such calculations. Morevore, suppose, you have 100 friends that give you 5 apples. You need to add five 100 times!
For this we use multiplication.
Multiplication of numbers `a` and `b` is $$$\color{purple}{a \cdot b=\underbrace{a+a+a+a+...+a}_{b}}$$$`.
For example, `3*4=3+3+3+3`, `6*7=6+6+6+6+6+6+6=42`.
Multiplication tells us how many times to use number in addtion.
Each number being multiplied is called
factor and the result is called product.
So, in `3*4=12` both 3 and 4 are factors and 12 is product.
There is another notation for multiplication: `times`, so `a*b` and `a times b` are equivalent.
A nice property of multiplication is that order doesn't matter.
For example `3 times 4=3+3+3+3=12` and `4 times 3=4+4+4=12`. As can be seen result is same.
For any numbers `a` and `b` we have that `color(blue)(a times b=b times a)`.
Another interesting facts:
Zero multiplied by any number is zero. For example, `25 times 0=0`, `0 times 1568=0`. Any number multiplied by 1 is number itself. For example, `67 times 1=67`, `1 times 8445=8445`.
For result of multiplication of some numbers see multiplication table.
Now, let's see how to multiply any whole numbers.
Example 1 . Find `34 times 12`.
Let's write numbers one under another:
$$$\begin{array}{l@{\,}l@{\,}l} \ & 3&4 \\ \times & \color{blue}{1}&\color{green}{2} \\ \hline & & \\ \end{array}$$$
Start with the right (green) digit. Multiply it with all digits of first number:
$$$ \begin{array}{l@{\,}l@{\,}l} \ & \color{brown}{3}&\color{purple}{4} \\ \times & \color{blue}{1}&\color{green}{2} \\ \hline & \color{green}{2 \times \color{brown}{3}}&\color{green}{2 \times \color{purple}{4}} \\ \end{array}$$$
$$$\begin{array}{l@{\,}l@{\,}l} \ & \color{brown}{3}&\color{purple}{4} \\ \times & \color{blue}{1}&\color{green}{2} \\ \hline & 6&8 \\ \end{array}$$$
Now, multiply 1 (blue digit) with all digits of first number but start writing result from the position of blue digit:
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l} & & \color{brown}{3}&\color{purple}{4} \\ \times & & \color{blue}{1}&\color{green}{2} \\ \hline & & 6 &8 \\ &\color{blue}{1 \times \color{brown}{3}}&\color{blue}{1 \times \color{purple}{4}}&\end{array}$$$
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l} & & \color{brown}{3}&\color{purple}{4} \\ \times & & \color{blue}{1}&\color{green}{2} \\ \hline & & 6 &8 \\ &3&4&\end{array}$$$
Now you need to add these two resulting numbers. But they are shifted, so just imagine that there are zeros, on places where digit is missing:
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l} & & \color{brown}{3}&\color{purple}{4} \\ \times & & \color{blue}{1}&\color{green}{2} \\ \hline & \color{red}{0}& 6 &8 \\ &3&4&\color{red}{0}\end{array}$$$
Now, use addition technique to add these two numbers:
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l} & & \color{brown}{3}&\color{purple}{4} \\ \times & & \color{blue}{1}&\color{green}{2} \\ \hline & 0& 6 &8 \\+ &3&4&0 \\ \hline & 4&0 &8\end{array}$$$
So,
`34 times 12=408` .
Let's do a harder example.
Example 2. Multiply 852 by 697.
First, we write number one under another:
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l} & 8 & 5 & 2 \\ \times & 6 & 9 &7 \\ \hline & & & \\\end{array}$$$
Multiply 7 with all digits of first number.
For this multiply 7 and 2. Result is 14. We need only 1 digit, so we take 4 and remember 1.
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l} & 8 & \overset{\color{green}{+1}}{5} & \color{green}{2} \\ \times & 6 & 9 &\color{green}{7} \\ \hline & & & \color{green}{4}\\\end{array}$$$
Next, multiply 7 and 5. Result is 35. Add remembered 1. Result is 36. Again we have two digits. So take 6 and remember 3.
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l} & \overset{\color{blue}{+3}}{8} & {\color{blue}{5}} & 2 \\ \times & 6 & 9 &\color{blue}{7} \\ \hline & & \color{blue}{6}& 4\\\end{array}$$$
Now, multiply 7 and 8. Result is 56. Add remembered 3. Result is 59. Again we have two digits, but we are done with last digit of second number, i.e. 7, so just write 59.
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l} & & \color{red}{8} & 5 & 2 \\ \times & & 6 & 9 &\color{red}{7} \\ \hline & \color{red}{5} & \color{red}{9}& 6& 4\\\end{array}$$$
Next, work with 9. Multiply 9 by 2. Result is 18. Take 8 and remember 1.
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l} & & 8 & \overset{\color{green}{+1}}{5} & \color{green}{2} \\ \times & & 6 & \color{green}{9} &7 \\ \hline & 5 & 9& 6& 4\\ & & & \color{green}{8}&\\\end{array}$$$
Multiply 9 by 5. Result is 45. Add remembered 1. Result is 46. Take 6 and remember 4.
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l} & & \overset{\color{blue}{+4}}{8} & \color{blue}{5} & 2 \\ \times & & 6 & \color{blue}{9} &7 \\ \hline & 5 & 9& 6& 4\\ & & \color{blue}{6}& 8&\\\end{array}$$$
Multiply 9 by 8. Result is 72. Add remembered 4. Result is 76. Since we are done with 9 then write 76.
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l@{\,}l} & & & \color{red}{8} & 5 & 2 \\ \times & & & 6 & \color{red}{9} &7 \\ \hline & & 5 & 9& 6& 4\\ & \color{red}{7}& \color{red}{6}& 6& 8&\\\end{array}$$$
Finally, deal with 6. Multiply 6 and 2. Result is 12. Take 2 and remember 1.
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l@{\,}l} & & & 8 & \overset{\color{green}{+1}}{5} & \color{green}{2} \\ \times & & & \color{green}{6} & 9 &7 \\ \hline & & 5 & 9& 6& 4\\ & 7& 6& 6& 8&\\ & & &\color{green}{2} & &\\\end{array}$$$
Multiply 6 by 5. Result is 30. Add remembered 1. Result is 31. Take 1 and remember 3.
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l@{\,}l} & & & \overset{\color{blue}{+3}}{8} & \color{blue}{5} & 2 \\ \times & & & \color{blue}{6} & 9 &7 \\ \hline & & 5 & 9& 6& 4\\ & 7& 6& 6& 8&\\ & & \color{blue}{1}&2 & &\\\end{array}$$$
At last multiply 6 by 8. Result is 48. Add remembered 3. Result is 51. We are done with 6, so write 51.
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l@{\,}l@{\,}l} & & & & \color{red}{8} & 5 & 2 \\ \times & & & & \color{red}{6} & 9 &7 \\ \hline & & & 5 & 9& 6& 4\\ & & 7& 6& 6& 8&\\ & \color{red}{5}&\color{red}{1} & 1&2 & &\\\end{array}$$$
Now, imagine zeros on places of missing digits:
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l@{\,}l@{\,}l} & & & & 8 & 5 & 2 \\ \times & & & & 6 & 9 &7 \\ \hline &\color{red}{0} &\color{red}{0} &5 & 9& 6& 4\\ &\color{red}{0} & 7& 6& 6& 8&\color{red}{0}\\ & 5&1 & 1&2 &\color{red}{0} &\color{red}{0}\\\end{array}$$$
Finally, add these numbers:
$$$\begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l@{\,}l@{\,}l} & & & & 8 & 5 & 2 \\ \times & & & & 6 & 9 &7 \\ \hline &0 &0 &5 & 9& 6& 4\\+ &0 & 7& 6& 6& 8&0\\ +& 5&1 & 1&2 &0&0\\ \hline&5&9&3&8&4&4\end{array}$$$
So,
`852 times 697=593844` .
Next, arises question how to multiply numbers that have different number of digits? Actually the same way, just write number with smaller number of digits under the number with larger number of digits.
Example 3. Multiply 23 by 233.
Write 23 under 233 since 23 has smaller number of digits:
$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} &2& 3&3 \\ \times & &2&3 \\ \hline & & & \\ \end{array}$$$
We will do this example faster:
$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l} & &\color{red}{2}& \color{brown}{3}&\color{purple}{3} \\ \times & & &\color{blue}{2}&\color{green}{3} \\ \hline & & \color{green}{3 \times \color{red}{2}}& \color{green}{3 \times \color{brown}{3}}&\color{green}{3 \times \color{purple}{3}} \\ & \color{blue}{2 \times \color{red}{2}}& \color{blue}{2 \times \color{brown}{3}}&\color{blue}{2 \times \color{purple}{3}}& \\\end{array}$$$
Or
$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l} & &2& 3&3 \\ \times & & &2&3 \\ \hline & &6& 9&9 \\ & 4& 6&6& \\\end{array}$$$
Finally, use addition:
$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l@{\,}l} & &2& 3&3 \\ \times & & &2&3 \\ \hline & 0&6& 9&9 \\ +& 4& 6&6& 0\\ \hline &5&3&5&9 \end{array}$$$
So,
`23 times 233=5359` .
Finally, lat's see how to multiply more than two numbers.
It is easy, just requires more work.
For example, suppose you need to multiply 25, 31 and 54. Take 25 and 31. Multiply them. Result is 775 (verify!). Now multiply 775 and 31. Result is 24025 (verify!).
So, `25 times 31 times 54=24025`.
It is also worth noting that order of numbers is not important. You could easily take 31 and 54, multiply them and then multiply result by 25.
Now, it is your turn. Take pen and paper and solve following problems:
Exercise 1. Find `32 times 31`. Answer: 992. Exercise 2. Find `58 times 12`. Answer: 696. Exercise 3. Find `523 times 86`. Answer: 44987. Exercise 4. Find `567 times 228`. Answer: 129276. Exercise 5. Find `745 times 51 times 99times 4581`. Answer: 17231454405. |
Let $f: A\rightarrow B$ be a function and consider a subset $Y\subseteq B$.
,is $f(f^{-1}(Y)) = Y$ always true?
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Yes, there are counterexamples. Let $A = B = \mathbb{R}$ and let $f(x) = x^2$. Let $Y = \{x \in \mathbb{R} : x \leq 0\}$. Then $$f(f^{-1}(Y))= f(\{0\}) = \{0\} \subsetneq Y.$$
It's always true that $f(f^{-1}(Y)) \subset Y$, as if $x \in f^{-1}(Y)$ then by definition $f(x) \in Y$.
If $Y \subset f(A)$, then it is true, as if $y \in Y$, then since $Y \subset f(A)$ there exists some $x \in A$ such that $f(x) = y$, thus $y = f(x)\in f(f^{-1}(Y))$. Therefore $$Y \subset f(f^{-1}(Y)).$$
No. For example, set $A=B=Y=\mathbb{N}$, and $f:x\mapsto 2x$. Then $f^{-1}(Y)=\mathbb{N}$, but $$\mbox{$f(\mathbb{N})=\{$evens$\}\subsetneq Y$.}$$ |
Advances in Operator Theory Adv. Oper. Theory Volume 4, Number 3 (2019), 604-624. Atomic characterizations of Hardy spaces associated to Schrödinger type operators Abstract
In this article, the authors consider the Schrödinger type operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$, where the matrix $A$ is symmetric and satisfies the uniformly elliptic condition and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_q(\mathbb{R}^n)$ with $q\in(n/2,\,\infty)$. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$ be a variable exponent function satisfying the globally $\log$-Hölder continuous condition. The authors introduce the variable Hardy space $H_L^{p(\cdot)}(\mathbb{R}^n)$ associated to $L$ and establish its atomic characterization. The atoms here are closer to the atoms of variable Hardy space $H^{p(\cdot)}(\mathbb{R}^n)$ in spirit, which further implies that $H^{p(\cdot)}(\mathbb{R}^n)$ is continuously embedded in $H_L^{p(\cdot)}(\mathbb{R}^n)$.
Article information Source Adv. Oper. Theory, Volume 4, Number 3 (2019), 604-624. Dates Received: 27 November 2018 Accepted: 19 December 2018 First available in Project Euclid: 2 March 2019 Permanent link to this document https://projecteuclid.org/euclid.aot/1551495623 Digital Object Identifier doi:10.15352/aot.1811-1440 Mathematical Reviews number (MathSciNet) MR3919034 Zentralblatt MATH identifier 07056788 Citation
Zhang, Junqiang; Liu, Zongguang. Atomic characterizations of Hardy spaces associated to Schrödinger type operators. Adv. Oper. Theory 4 (2019), no. 3, 604--624. doi:10.15352/aot.1811-1440. https://projecteuclid.org/euclid.aot/1551495623 |
PhD Seminar Seminar programme Seminar details
In the first part of this talk, I would like to introduce my research group at my home university. The research in Software Engineering Group of Nanjing University focuses on formal modeling and verification, program analysis and testing, and the research work also covers areas such as aspect-oriented programming, model-driven architecture, service-oriented computing, and so on. Some selected research results and tools will be briefly introduced in this part of the talk.
Web services bridge the gap between heterogeneous systems. By providing formal support to web services, we can use existing research results to improve the understanding and increase the confidence to the services. In our work, we give a Petri-net based model to BPEL web services. Using this model, we perform verification on BPEL services against scenario-based specifications (such as message sequence charts and UML Sequence Diagrams). In addition to verification, we also focus on web service behavior extraction. Third party web services cannot always fit the customers' requirement, some critical behavior may not be guaranteed to appear in each of the service invocation. The behavior extraction is targeted to provide a wrapper service to ensure that the required behavior occurs in every execution of the target service. In the second part of the talk, the idea and approach of the behavior verification and extractions will be presented.
In (universal) algebra, G. Birkhoff proved his famous variety and completeness theorems in the 1930's which are results about equational logic. In the 1990's, a theory of coalgebras was developed which allows to reason abstractly about "systems", for example various kinds of transition systems and automata. Soon Birkhoff's theorems could be dualised, and one obtains a coequational logic which is about properties of such systems instead of properties of algebras.
In this talk we first review the notion of a cofree coalgebra (on the category of sets and functions); then we see how coequations and coequational logic are defined. In the main part, we present simple deduction systems for coequations that are sound and complete for a wide range of system types: we start with systems of polynomial type and extend to a subclass of accessible types using a representation technique for coequations.
A recurrent challenge that appears in enterprises is the need to enhance the functionality of their software eco-system by making some of the existing applications to interoperate with others. In the literature, this problem is known as Enterprise Application Integration (EAI) and is all about making two or more existing applications, that belongs to the same enterprise, to synchronize their data or to create new functionalities on top of them; in either case, the software that implements the integration is called the integration solution. This seminar will give an introduction on a DSL called Guaraná to design EAI solutions.
Local search algorithms represent a traditional heuristic approach to intractable computational problems. We addressed the inconvenience that a typical local search is not concerned with having a global vision over its own evolution, over its trajectory through the search space. We present two position-guided algorithms that work on top of a local search process - i.e. Tabu Search (TS). The goal is to guide the local search toward certain targeted regions of the search space. These techniques are focused on problems for which one can define a distance measure between candidate solutions.
The first algorithm (TS-Div) records its own trajectory and “pays attention” not to visit the same search space regions (spheres) repeatedly. The second algorithm (TS-Int) makes deep investigations in a “limited perimeter” around a given candidate solution. TS-Int employs a breath-first-search routine to enumerate all spheres from this “limited perimeter,” and each of these spheres is thoroughly explored by numerous independent TS processes. We experimentally observed that if such a “limited perimeter” contains a global optimum, TS-Int does not fail in eventually finding it. TS-Div ensures diversity, TS-Int enforces intensification, and together they reached very good results on a competitive problem (graph colouring). Details on the numerical experiments are available in a recent paper to be published by Computers & Operations Research.
The Two Dimensional Range Minimum Query (\mbox{2D-RMQ}) problem is to preprocess a static two dimensional array~$A$ of size $m\times n$, where~$m\le n$ such that subsequent queries asking for the index of the minimum element in a rectangular range within~$A$ can be answered efficiently. We show that every algorithm enabled to access~$A$ during the query and using~$O(N/c)$ bits additional space, requires query time~$\Omega(c)$ for $N=m\cdot n$ and any value of~\mbox{$1 \le c \le N$}. In particular this lower bound holds for the 1D-RMQ problem. We complement this lower bound with an algorithm that with~$O(N)$ preprocessing time and~$O(N/c)$ bits additional space achieves~$O(c\log2 c)$ query time. For~$c=1$, this is the first optimal algorithm using $O(N)$ bits additional space with $O(1)$ query time. We also consider the problem in the Encoding model where the query algorithms use an encoding data structure to solve the problem without utilizing~$A$. For this model, we present an~$O(1)$ query time algorithm using~$O(mn\log n)$ bits for the encoding data structure. We also give an alternative proof for the formerly known~$\Omega(mn\log m)$ bits for the size of the encoding data structure. |
Elastic scattering, mean free path, mobility: Impurity scattering in a silicon nanowire¶ Version: 2017.0
In this tutorial you will study elastic scattering by dopant atoms in a silicon nanowire. Based on calculated transmission spectra you will learn how to estimate the scattering mean free path and mobility, taking into account both the effect of doping and the explicit scattering due to the dopant atoms.
Introduction¶
In this tutorial you will learn how the transmission spectra can be used to estimate the elastic scattering mean free path and the doping dependent mobility.
The main focus of the tutorial is not on the actual calculations of the transmission spectra, but rather on the subsequent analysis. As some of the calculations are rather time-consuming, data-files with the results are provided as downloads, so you can focus on the analysis.
The methods used in the tutorial are based on Ref. [MRJB07], where it was shown that the transmission function of a very long silicon nanowires with several structural defects can be accurately reproduced using a statistical average of the transmissions calculated for relatively small nanowires with a single defect.
Theoretical background¶
At low temperatures, the conductance of a system can be approximated as \(G = G_0 \cdot T(E_F)\), where \(G_0=2e^2/h\) is the conductance quantum and \(T(E_F)\) is the transmission at the Fermi energy. The resistance of the system is \(R=1/G\).
These considerations can be generalized to define an energy dependent and unitless resistance \(R(E)=1/T(E)\), which is just the inverse of the transmission. To get a physical resistance, \(R(E)\) should be multiplied with \(1/G_0\).
For a one-dimensional system without any defects (pristine),the transmission always takes integer values, \(T_0(E)=N(E)\).We shall denote the corresponding resistance \(R_c(E)=1/T_0(E)\)as the contact resistance.If a structural defect (labeled as
1 in the following) is introduced to the system,the transmission will always be lower than the transmission of the pristine system,\(T_1(E)<T_0(E)\).We will now write the resistance of the system with the defect as a sum of two contributions:
which defines the scattering resistance \(R_{1,s}(E)\) of defect
1.If multiple different defects are present – this could be dopant atoms in different positions – we can calculate an average transmission\(\langle T(E) \rangle\).Using the same notation as in the equation above,we will define an average scattering resistance,\(\langle R_s(E) \rangle\), as:
It was shown in Ref. [MRJB07] that the length- and energy-dependent resistance through a long wire of length \(L\) containing many, randomly placed defects with an average defect–defect distance \(d\), can be well approximated as:
which may be written as
which defines the energy-dependent scattering mean free path as
Note
Note that we have now introduced the total length of the wire \(L\), which in this formalism is an arbitrary parameter.
The corresponding transmission through the long wire is \(T(L,E) = 1/R(L,E)\). This means that the conductance of a wire with length \(L\) can be estimated as
which generalizes the previous relation \(G=G_0 T(E_F)\) to finite temperatures. The function \(f(E,E_F)\) is the Fermi-Dirac distribution function for electrons in the left and right electrodes with average Fermi level \(E_F\).
The conductivity \(\sigma\) of the system is given by
and we can now estimate the mobility \(\mu\) as
where \(e\) is the elementary charge and \(n\) is the charge density. The Fermi level at a given doping density will be estimated from band structure calculations with compensation charge doping.
Summary of required calculations¶
In summary, the calculations we need in order to calculate a mean free path are:
Transmission spectrum, \(T_0(E)\), of the pristine (defect free) nanowire. A transmission spectrum for each defect – in this case for the same element in four different dopant positions. From these calculations we get the average transmission \(\langle T(E) \rangle\).
In order to calculate the mobility at a given doping density, we further need:
A bandstructure calculation (also using the nanowire unit cell) with compensation charge doping, in order to calculate the Fermi level for the desired doping density. Defected silicon nanowires¶
Two different types of nanowires will be considered:
The pristine deviceis constructed from pure, defect-free, relaxed silicon, passivated with hydrogen at the surface; The defected devicesare generated by substituting one Si atom with P (in 4 different positions) and are relaxed as bulk.
More information on how to build the structures can be found at the end of the tutorial, in the Appendix: Building the nanowires.
Computational method: The calculations are performed with DFT and NEGF, using a GGA xc-functional and the FHI-SZP basis sets. We use 1x1x51 k-points for the NEGF calculations. Transmission spectra¶
The calculated transmission spectra are available via download. They have been calculated using 201 points in an energy range between \(E-E_\mathrm{F} = 0.5 eV\) and \(E-E_\mathrm{F} = 2.0 eV\).
Elastic scattering mean free path¶
In this section, we will analyze how the presence of an impurity atom impacts the transmission through the nanowire, and we will also estimate the mean free path for the conduction electrons. The script
T_and_mfp.py calculates and plots the transmission and the mean free path using the theory described in the theoretical background.
The script does the following:
Aligns the conduction band minima, to give the two datasets a common energy reference. Calculates the scattering resistance \(R_{s}(E)\) from the average transmission and the contact resistance \(R_{c}(E)\) from the pristine transmission. Saves the data to a file, for re-use later. Calculates the mean free path from \(R_{c}(E)\) and \(R_{s}(E)\) and the average distance between dopant atoms. Plots of transmissions and mean free path as functions of energy.
Run the script as
atkpython T_and_mfp.py or with the Job Manager, in the directory where you have saved the files with the transmission spectra, and the following figure should appear:
The first plot compares the transmission function for the pristine and the defected nanowires. It can be seen that the transmission of the defected nanowires is similar and always lower than that of the pristine nanowire. The second plot shows the mean free path as a function of energy.
Note how, close to the conduction band edge at energies between 1.0 eV and 1.2 eV, the electrons are strongly scattered by the dopant impurity, and the associated transmission functions are very low. This is reflected in a short mean free path. At higher energies E > 1.2 eV, there is less scattering and both the transmission and mean free path are larger.
Fermi levels in doped nanowires¶
The next task is to estimate the impurity limited mobility as a function of doping density. In order to do this, you need to know the position of the Fermi level relative to the conduction band edge. This can be achieved by calculating the band structure for silicon nanowires which are uniformly doped. This can be done in QuantumATK using atomic compensation charges. You can learn about doping in QuantumATK in the technical note here: Doping methods available in QuantumATK and in the tutorial here: Silicon p-n junction. The bandstructure calculations use 1x1x9 k-points and the \(\mathrm{\Gamma} \to \mathrm{Z}\) route. You can download the data files here:
doped-bandstructures.zip.
After unzipping the files you can compare the bandstructures in QuantumATK using the
Bandstructure Analyzer:
Note that in this case, both bandstructures are referenced to the Fermi level, and it is therefore also easiest to align them here. This means that we need to look at the positions of the bands to study the shift in the Fermi level. We see that the conduction band has shifted about 0.5 eV down when going from a doping level of \(10^{14}\ \mathrm{cm}^{-3}\) to \(10^{21}\ \mathrm{cm}^{-3}\), which is more properly described as the Fermi level shifting up with increasing doping.
Doping dependent mobility¶
You now have the ingredients needed to estimate the mobility as function of doping concentration. The python script below loads the calculated data and calculate the conductivity as well as mobility for a \(1\ \mu \mathrm{m}\) long Silicon nanowire. By calculating the mobility in this way, as also described in the theoretical background, the following assumptions are made:
Each dopant atom is fully ionized and contributes with one electron to the conduction band charge density. The transmission of the long wire can be estimated from the single-dopant transmissions.
Warning
Note that the mobility defined in this way is very much an estimate of the mobility of electrons for the pristine wire, as we do not include any explicit scattering mechanisms. It is not directly comparable with the, for the pristine wire more appropriate, mobility that is calculated by f.ex. the Mobility class.
Notice that although the transmission calculations with the dopant atoms to some extend include the doping effect - i.e. that the P atom becomes partly ionized - this effect is not included in the position of the Fermi level, which is determined by the electrodes, which are un-doped in this calculation. The bandstructure calculations with the compensation charge doping are thus needed to consider this effect. You can download the script for these calculations here:
mobility.py. The script does the following:
Calculates the conductances and mobilities starting from the transmission; The conductances and mobilities are plotted as a function of doping level, both with and without taking scattering processes due to dopant atoms into account.
Run the script as
atkpython mobility.py, and the following figure should appear:
Both the mobility and the conductance are calculated and plotted for two cases:
Including dopant scattering in the transmission calculation of the long wire (red circles). For a pristine, but uniformly doped, wire transmission (black squares).
When the dopant scattering is included, two factors affect the conductance:
As for the pristine wire case, the Fermi level moved closer to the conduction band for increased doping density. This effect increases the conductance. Due to the scattering by the dopant impurities, the energy resolved resistance increases linearly with length according to the equations shown in the theory. This effect will decrease the conductance.
From the conductance plot (bottom), one can observe that the two effects approximately cancel out for doping densities \(n > 10^{18}\ \mathrm{cm}^{-3}\), where the conductance is more or less constant. Since the conductance is constant, the mobility will decrease as \(\mu \propto 1/n\) for \(n > 10^{18}\ \mathrm{cm}^{-3}\).
For doping levels \(n < 10^{18}\ \mathrm{cm}^{-3}\), the average dopant-dopant distance becomes larger than the wire length of \(1\ \mu \mathrm{m}\) and there will on average be no dopants in the wire, which is the reason why the doped wire curves approach the pristine wire curves. If we had chosen a longer wire, this onset would happen at lower doping concentrations, as it is related to the ratio \(L/L_\mathrm{mfp}(E)\).
Summary and discussion¶
As mentioned previously, the focus of this tutorial was not so much on how to calculate very accurate and detailed transmission spectra, but rather on how to extract information from them. In the present case of phosphorus dopants in a silicon nanowire we note the following points:
The actual transmission functions and mobilities will depend on the diameter and orientation of the wire. The transmission might also depend on the basis set, and differ from the one presented here, if e.g.a larger basis set (like DoubleZetaPolarized) is used.
One should also be aware that the analysis presented in this tutorial is not intended to be very quantitative. It rather provides a relatively straightforward approach to estimate the order of magnitude of the elastic scattering by defects. Recall that the actual system considered would be a \(1\ \mu \mathrm{m}\) long nanowire (this would include more than 150,000 atoms) with randomly placed dopants.
Finally, one should notice that the methods presented in this tutorial are applicable to other kinds of structural disorder such as surface defects [MJB09], or surface roughness scattering [Mar12], and can be applied to both electron and phonon transmission spectra [MJB09], [Mar12].
Appendix: Building the nanowires¶
Detailed instructions for building a Si nanowire can be found in this tutorial: Silicon nanowire field-effect transistor. However, you may also use the
Nanowire plugin in the Builder. In this case, we have used a nanowire bounded by (111) and (100) facets and a radius of 7.0 Å. References¶
[MJB09] (1, 2) T. Markussen, A.-P. Jauho, and M. Brandbyge. Electron and phonon transport in silicon nanowires: Atomistic approach to thermoelectric properties. Phys. Rev. B, 79:035415, Jan 2009. doi:10.1103/PhysRevB.79.035415.
[MRJB07] (1, 2) T. Markussen, R. Rurali, A.-P. Jauho, and M. Brandbyge. Scaling theory put into practice: First-principles modeling of transport in doped silicon nanowires. Phys. Rev. Lett., 99:076803, Aug 2007. doi:10.1103/PhysRevLett.99.076803.
[Mar12] (1, 2) Troels Markussen. Surface disordered Ge–Si core–shell nanowires as efficient thermoelectric materials. Nano Letters, 12(9):4698–4704, 2012. doi:10.1021/nl302061f. |
Here I asked the question whether the curvature deterined the metric.
Since I am unfortunately completely new to Riemannian geometry, I wanted to ask, if somebody could give and explain a concrete example to me, as far as the following is concerned:
At the MO page (as cited above) I got the following answer to the question
Given a compact Riemannian manifold M, are there two metrics g1 and g2, which are not everywhere flat, such that they are not isometric to one another, but that there is a diffeomorphism which preserves the curvature?If the answer is yes:Can we chose M to be a compact 2-manifold?
On the positive side, if $M$ is compact of dimension $\ge 3$ and has nowhere constant sectional curvature, then combination of results of Kulkarni and Yau show that a diffeomorphism preserving sectional curvature is necessarily an isometry.
Concerning 2-dimensional counter-examples: First of all, every surface which admits an open subset where curvature is (nonzero) constant would obviously yield a counter-example. Thus, I will assume now that curvature is nowhere constant. Kulkarni refers to Kreyszig's "Introduction to Differential Geometry and Riemannian Geometry", p. 164, for a counter-example attributed to Stackel and Wangerin. You probably can get the book through interlibrary loan if you are in the US.
I looked up the example in Kreyszig's "Introduction to Differential Geometry and Riemannian Geometry", p. 164:
If we rotate the curve $x_3=\log x_1$ about the $x_3$-axis in space, we obtain the surface of revolution $X(u_1,u_2)=(u_2\cos(u_1), u_2\sin(u_1),\log(u_2))$, $u_2>0$. This is diffeomorphic to the helicoid $X(u_1,u_2) =(u_2\cos(u_1),u_2\sin(u_1),u_1)$.
I think, these manifolds are not compact (but I assumed compactness of the manifold in my question on MO).I don't understand, how to manipulate this example in order to get a compact manifold.
Thank you for your help. |
No. There is a counterexample for each $n\geq4$.
I will only show a counterexample for $n=4$; you can get one for any higher$n$ by appending $0$'s.
Let $A_{1}=\left\{ \left( x,y,z,w\right) \in\left\{ 0,1\right\}^{4}\ \mid\ x+y\leq1\text{ and }z+w\leq1\right\} $. Let $A_{2}=\left\{0,1\right\} ^{4}\setminus A_{1}$. Then, $\left( A_{1},A_{2}\right) $ iswhat you call an order-preserved partition (I have seen this being called an"admissible partition"). Your conjecture claims that there exist $p_{1},p_{2},p_{3},p_{4}\in\left( 0,1\right) $ and $\theta\in\mathbb{R}_{+}$ suchthat the function $f:\left\{ 0,1\right\} ^{4}\rightarrow\mathbb{R},\ \left(x,y,z,w\right) \mapsto p_{1}^{1-x}p_{2}^{1-y}p_{3}^{1-z}p_{4}^{1-w}$satisfies $A_{1}\subseteq f^{-1}\left( \left( -\infty,\theta\right)\right) $ and $A_{2}\subseteq f^{-1}\left( \left( \theta,+\infty\right)\right) $.
Assume that this were true. Then, there would exist $q_{1},q_{2},q_{3},q_{4}\in\left( 0,1\right) $ and $\eta\in\mathbb{R}$ such that the function$g:\left\{ 0,1\right\} ^{4}\rightarrow\mathbb{R},\ \left( x,y,z,w\right)\mapsto q_{1}x_{1}+q_{2}x_{2}+q_{3}x_{3}+q_{4}x_{4}$ satisfies $A_{1}\subseteqg^{-1}\left( \left( \eta,\infty\right) \right) $ and $A_{2}\subseteqg^{-1}\left( \left( -\infty,\eta\right) \right) $. (Indeed, take$q_{i}=\ln p_{i}$ and $\eta=\ln p_{1}+\ln p_{2}+\ln p_{3}+\ln p_{4}-\ln\theta$. Then, the inequalities $f\left( x,y,z,w\right) \lesseqgtr\theta$ become$g\left( x,y,z,w\right) \gtreqless\eta$ after taking natural logarithms andslightly rewriting.)
Now, the two points $\left( 1,0,1,0\right) $ and $\left( 0,1,0,1\right) $both belong to $A_{1}$ and therefore to $g^{-1}\left( \left( \eta,\infty\right) \right) $ (since $A_{1}\subseteq g^{-1}\left( \left(\eta,\infty\right) \right) $). Since $g^{-1}\left( \left( \eta,\infty\right) \right) $ is a convex set, the midpoint between these twopoints (that is, the point $\left( \dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2}\right) $) must therefore also belong to $g^{-1}\left(\left( \eta,\infty\right) \right) $.
But the two points $\left( 1,1,0,0\right) $ and $\left( 0,0,1,1\right) $belong to $A_{2}$ and therefore to $g^{-1}\left( \left( -\infty,\eta\right)\right) $ (since $A_{2}\subseteq g^{-1}\left( \left( -\infty,\eta\right)\right) $). Since $g^{-1}\left( \left( -\infty,\eta\right) \right) $ is aconvex set, the midpoint between these two points (that is, the point $\left(\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2}\right) $) must thereforealso belong to $g^{-1}\left( \left( -\infty,\eta\right) \right) $.
So the point $\left( \dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2}\right) $ must belong to $g^{-1}\left( \left( \eta,\infty\right)\right) $ and to $g^{-1}\left( \left( -\infty,\eta\right) \right) $ atthe same time. But this is absurd, since $g^{-1}\left( \left( \eta,\infty\right) \right) \cap g^{-1}\left( \left( -\infty,\eta\right)\right) =\varnothing$. So we have a contradiction.
I am wondering whether it is possible to fix your conjecture by (essentially)assuming that the kind of counterexamples above (viz., two points in $A_{1}$having the same midpoint as two points in $A_{2}$) does not exist.
A stronger requirement that definitely makes your conjecture true is the following:
(1) no convex combination of the points in $A_1$ can be a convex combination of the points in $A_2$ at the same time.
Indeed, if
(1) is true, then the convex hull of $A_1$ is disjoint from the convex hull of $A_2$; but (from basic linear optimization theory) we know that this entails that there exists a hyperplane separating $A_1$ from $A_2$; this immediately translates into the existence of $g$ and $\eta$ as above. |
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