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Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.(b) Find all such matrices with rank 2.Solution.(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.First we look at the rank 1 case. […]
Diagonalizable by an Orthogonal Matrix Implies a Symmetric MatrixLet $A$ be an $n\times n$ matrix with real number entries.Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.Proof.Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$.The orthogonality of the […]
Symmetric Matrices and the Product of Two MatricesLet $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.(a) The product $AB$ is symmetric if and only if $AB=BA$.(b) If the product $AB$ is a diagonal matrix, then $AB=BA$.Hint.A matrix $A$ is called symmetric if $A=A^{\trans}$.In […]
Quiz 13 (Part 1) Diagonalize a MatrixLet\[A=\begin{bmatrix}2 & -1 & -1 \\-1 &2 &-1 \\-1 & -1 & 2\end{bmatrix}.\]Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$.That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]
Questions About the Trace of a MatrixLet $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix.(a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of […]
Normal Nilpotent Matrix is Zero MatrixA complex square ($n\times n$) matrix $A$ is called normal if\[A^* A=A A^*,\]where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$.A matrix $A$ is said to be nilpotent if there exists a positive integer $k$ such that $A^k$ is the zero […]
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Petri net
Definition from Dictionary, a free dictionary
Teach only love for that is what you are.A Course In Miracles
EnglishWikipedia Noun
Singular
Plural
Petri net({{{1}}}) One of several mathematical representations of discrete distributed systems, a 5-tuple <math>(S,T,F,M_0,W)\!</math>, where <math>S</math> is a set of places. <math>T</math> is a set of transitions. <math>S</math> and <math>T</math> are disjoint, i.e. no object can be both a place and a transition <math>F</math> is a set of arcs known as a flow relation. The set <math>F</math> is subject to the constraint that no arc may connect two places or two transitions, or more formally: <math>F \subseteq (S \times T) \cup (T \times S)</math>. <math>M_0 : S \to \mathbb{N}</math> is an initial marking, where for each place <math>s \in S</math>, there are <math>n_s \in \mathbb{N}</math> tokens. <math>W : F \to \mathbb{N^+}</math> is a set of arc weights, which assigns to each arc <math>f \in F</math> some <math>n \in \mathbb{N^+}</math> denoting how many tokens are consumed from a place by a transition, or alternatively, how many tokens are produced by a transition and put into each place. <math>S</math> is a set of Translations
mathematical representations of discrete distributed systems
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Suppose $f=f(x)=f(x_1,\cdots,x_n)$ is a homogeneous polynomial of degree $n$ in the ring $k[x_1,\cdots,x_n]$, with $k$ a field. Denote by $H_f$ the hypersurface given by $f=0$. Suppose that for every smooth point $z$ of $H_f$, $f(x)$ is divisible by the linear form $f_x(x,z) = \sum_{i=1}^n x_i \frac{\partial f(\xi)}{\partial \xi_i}|_{\xi = z}$.
Question 1: Why is it true that $H_f$ coincides with the tangent hyperplane in the neighborhood of $z$? (i can see that $\sum_{i=1}^n x_i \frac{\partial f(\xi)}{\partial \xi_i}|_{\xi = z}$ is the equation of the tangent hyperplane at $z$).
Question 2: Why is $f$ a union of hyperplanes?
PS: I can see the validity of 1 and 2 intuitively, i am looking for a rigorous algebraic geometric argument, requiring as little background as possible.
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I am studying literature on the nonlinear Schroedinger equation. More generally, I am wondering how to tackle an equation of motion such as
$\bigl(i\partial_t-\Delta\bigr)\psi(\vec{x},t)+V(\vec{x},t)=0$
where the potential is of the form
$V(\vec{x},t)=\int d\vec{x}_1...d\vec{x}_n~ \psi(\vec{x}-\vec{x}_1,t)...\psi(\vec{x}-\vec{x}_n,t)$,
that means the potential is nonlocal and nonlinear in the field $\psi$.
Any hints towards literature, also numerical strategies would be really welcome!
Thanks!
Comment: The question refers in particular to the issue of how to deal with the integral in the interaction potential. Is there e.g. a nice way to discretize this object?
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I was wondering if it feasible to think of the Polyakov action as a generalisation of the point-particle action $$ S_0 = \frac{1}{2}\int d\tau(\eta^{-1}\dot{X}^\mu\dot{X}_\mu - \eta m^2) $$ by considering the Lagrangian $$ L_0(\tau) = \frac{1}{2}(\eta^{-1}(\tau)\dot{X}^\mu(\tau)\dot{X}_\mu(\tau) - \eta(\tau) m^2) $$ as the Lagrangian density $$ \mathcal{L}_1(\tau, \sigma) = \frac{1}{2}(\eta^{-1}(\tau, \sigma)\dot{X}^\mu(\tau, \sigma)\dot{X}_\mu(\tau, \sigma) - \eta(\tau, \sigma) m^2) + T(\tau,\sigma) $$ of the Polyakov action, and I'm guessing a potential $T$ would be needed as the 'point' is no longer free.
I have tried working the other direction, differentiaing the Lagrangian of the Polyakov action and performing some integration by parts to obtain $$ \mathcal{L}_1 = \frac{\partial S_p}{\partial \sigma} = \frac{1}{4\pi\alpha'} \sqrt{h}(h^{\gamma\delta}\frac{\partial h_{\gamma\delta}}{\partial\sigma}\frac{\partial h^{\alpha\beta}}{\partial \sigma}\partial_\alpha X^\mu \partial_\beta X_\mu+h^{\alpha\beta}\partial_\alpha X^\mu \partial_\beta X_\mu+h^{\alpha\beta}\partial_\alpha \frac{\partial X^\mu}{\partial \sigma}\partial_\beta X_\mu) $$ where $X^\mu X_\mu \equiv g_{\mu\nu}X^\mu X^\nu $.
However I am not entirely sure how to proceed from here - I guess my problem is that I expect the resulting Lagrangian to be independent of $\sigma$ but I'm unsure how to deal with this.
If anybody could point me to a paper which does something along these lines, or could elucidate as to why this is not possible then I would be very grateful.
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Let's say we have two modes, with the following labeling of occupation number states:
$ \lvert \Psi \rangle = \begin{pmatrix} 0,0 \\ 0,1 \\ 1,0 \\ 1,1 \end{pmatrix} $
An example of (what I assume to be) fermionic creation operators for the two modes is
$\hat a_1^\dagger = \begin{pmatrix} 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \quad \hat a_2^\dagger = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}$
These operators obey full anti-commutation relations.
$\{\hat a_1,\hat a_1^\dagger\} = \{\hat a_2,\hat a_2^\dagger\} = 1$
$a^\dagger_1 a^\dagger_1 = a^\dagger_2 a^\dagger_2 = 0$
$\{\hat a_1,\hat a_2^\dagger\} = \{\hat a_1,\hat a_2\} = 0$
If we don't include the ($-$) sign, then operators corresponding to the same mode still anti-commute, but those corresponding to different modes commute.
$\hat b_1^\dagger = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \quad \hat b_2^\dagger = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}$
$\{\hat b_1,\hat b_1^\dagger\} = \{\hat b_2,\hat b_2^\dagger\} = 1$
$b^\dagger_1 b^\dagger_1 = b^\dagger_2 b^\dagger_2 = 0$
$[\hat b_1^\dagger,\hat b_2^\dagger] = [\hat b_1^\dagger,\hat b_2] = 0$
It looks like we started constructing a boson Fock space, but only included states for which the occupation numbers are 0 or 1. Is there some reason these operators aren't suitable, other than the observation that all elementary particles are either fermions or bosons? Are there any quasi-particles in condensed matter physics that behave like this?
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I suggest to create a tag synonym analysis-and-odes $\to$ ca.classical-analysis-and-odes. The tag analysis-and-odes has only five questions and empty tag info - so it does not seem to have any distinction from ca.classical-analysis-and-odes. Four of the five questions in this tag were asked by t...
I'd like to propose creation of continuum-theory tag. To me (as an outsider, but still a bit interested in this topic) it seems that continuum theory is an area of general topology which enjoys some interest both among topologists and among mathematicians in general. (For example, one part of Op...
Let $P$ be a compact connected set in the plane and $x,y\in P$. Is it always possible to connect $x$ to $y$ by a path $\gamma$ such that the length of $\gamma\backslash P$ is arbitrary small? Comments: Be aware of pseudoarc --- it is a compact connected set which contains no nontrivial p...
Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of diameter $<\varepsilon$ in $X$ such that $x\in C_1$, $y\in C_n$ and $C_i\cap C_{i+1}\ne\emptyset...
This problem was motivated by the MO problems: "Running most of the time in a connected set", "Is every metric continuum almost path connected?" and "Are $\varepsilon$-connected components dense?". Let $p$ be a positive real number. A metric space $(X,d)$ is called $\ell_p$-chain connected if fo...
Continuum $=$ compact connected metric space. Let $X$ be a continuum. $X$ is indecomposable means that every proper subcontinuum of $X$ is nowhere dense in $X$. It is easy to see that if $X$ is indecomposable then every connected open subset of $X$ is dense in $X$. Question. Are these two cond...
Definition. A closed subset $S$ of a topological space $X$ is called a separator between points $x,y\in X\setminus S$ if the points $x$ and $y$ belong to different connected components of $X\setminus S$. A separator $S$ is called an irreducible separator between $x$ and $y$ is $S$ coincides with ...
Let $\square=[0,1]\times[0,1]$ be the unit square and $f\colon\square\to \square$ is a continuous map that fixes the points on the boundary. Assume $f$ is a limit of homeomorphisms $\square\to \square$. (By Moore's theorem it is equivalent to the condition that for any point $p\in \square$ the i...
The question was motivated by this question of Anton Petrunin. By a metric continuum we understand a connected compact metric space. Let $p$ be a positive real number. A metric continuum $X$ is called $\ell_p$-almost path-connected if for any points $x,y\in X$ and any $\varepsilon>0$ here exist...
Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$. Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve? It is easy to find an open disc which boundary lies in a plane, but the boundary might be crazy; for example it might be Polish ...
Let $X$ be a connected separable metrizable topological space. Call it a cut-point space if $X\setminus \{x\}$ is disconnected for every $x\in X$. Then does $X$ embed into the plane? My thoughts: (0) It is not difficult to see that $X$ must have dimension $1$, and therefore embeds into $\mathb...
This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that have trivial shape in the classic Borsuk sense (and thus also in the sense of Cech homology). How...
Problem. Is each Peano continuum a topological fractal? A compact Hausdorff space $X$ is a topological fractal if $X=\bigcup_{i=1}^n f_i(X)$ for some continuous maps $f_1,\dots,f_n:X\to X$ such that for any sequence $(g_i)_{i\in\omega}\subset\{f_1,\dots,f_n\}^{\omega}$ the intersection $\big...
I found a strange attractor which looks a lot like a solenoid. The attractor continuum is the closure of a continuous line which limits onto itself, and it is locally homeomorphic to Cantor set times Reals. It sits in the Möbius strip. Does the Dyadic solenoid embed into the Möbius strip? Wha...
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Is there a standard or common way to write scientific notation in different bases that doesn't require repeating the base in both the coefficient and the exponent base?
For example, this notation is correct, but is quite verbose:
Implicit base 10: $100 \cdot \tau = 6.283 \times 10^2$ Explicit base 10: $100 \cdot \tau = 6.283_{10} \times 10_{10}^2$ Explicit base 16: $100 \cdot \tau = 2.745_{16} \times 10_{16}^2$
Most calculators & programming languages shorten this to an "exponent" notation such as:
6.283e2
Some rare ones even support a base, such as:
16#2.745#e2#
Unfortunately, these are not very nice notations outside of specific domains.
Ideally, I'm looking for an operator notation that concisely does the function:
$f(coefficient_{b},exponent) = coefficient_{b} \times b^{exponent}$
If there is no such standard or common notation, I'd be happy to hear any suggestions.
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Limit Divergence Criteria
Recall from The Sequential Criterion for a Limit of a Function page, that for a function $f : A \to \mathbb{R}$ and for $c$ being a cluster point of $A$, then $\lim_{x \to c} f(x) = L$ if and only if for all sequences $(a_n)$ from $A$ for which $\lim_{n \to \infty} a_n = c$ we also have that $\lim_{n \to \infty} f(a_n) = L$. We will now formulate what is known as the Limit Divergence Criteria, which will establish criteria to establish whether the value $L$ is not the limit of $f$ as $x \to c$.
Theorem 1 (Limit Divergence Criteria): Let $f : A \to \mathbb{R}$ be a function and let $c$ be a cluster point of $A$. Then for $L \in \mathbb{R}$, $\lim_{x \to c} f(x) \neq L$ if either: 1. There exists a sequence $(a_n)$ from $A$ where $a_n \neq c$ $\forall n \in \mathbb{N}$ such that $\lim_{n \to \infty} a_n = c$ but $\lim_{n \to \infty} f(a_n) \neq L$. 2. There exists a sequence $(a_n)$ from $A$ where $a_n \neq c$ $\forall n \in \mathbb{N}$ such that $\lim_{n \to \infty} a_n = c$ but $(f(a_n))$ does not converge in $\mathbb{R}$. Proof:Let $f : A \to \mathbb{R}$ be a function and let $c$ be a cluster point of $A$. Let $L \in \mathbb{R}$. If either (1)or (2)holds, then we note that this contradicts The Sequential Criterion for a Limit of a function. Since in both cases 1 and 2, not all sequences $(a_n)$ where $a_n \neq c$ $\forall n \in \mathbb{N}$ from $A$ such that $\lim_{n \to \infty} a_n = c$ satisfy $\lim_{n \to \infty} f(a_n) = L$, and so we have that $\lim_{x \to c} f(x) \neq L$. $\blacksquare$
The following diagram represents the first scenario of the Limit Divergence Criteria for which there exists a sequence, in this case $(b_n)$ that converges to $c$, however $(f(b_n))$ does not converge to $L$ and so $\lim_{x \to c} f(x) \neq L$:
The succeeding diagram represents the second scenario of the Limit Divergence Criteria for which there exists a sequence, $(b_n)$ that converges to $c$, however $(f(b_n))$ does not converge in $\mathbb{R}$ and so $\lim_{x \to c} f(x) \neq L$:
Example 1 Using the Divergence Criteria, for the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \frac{1}{x - 1}$ show that $\lim_{x \to 1} \frac{1}{x - 1}$ does not exist in $\mathbb{R}$.
Consider the sequence $(a_n) = \frac{1}{n} + 1$. We note that this sequence $(a_n)$ is from $\mathbb{R}$ (the domain). Also notice that this sequence converges to $1$ as $\lim_{n \to \infty} \frac{1}{n} + 1 = 1$. Now notice that $(f(a_n)) = \frac{1}{\frac{1}{n} + 1 - 1} = n$, and $\lim_{n \to \infty} f(a_n) = \lim_{n \to \infty} n = \infty$.
Therefore, we have a sequence $(a_n)$ from the domain that converges to $1$, however the sequence $(f(a_n))$ does not converge in $\mathbb{R}$. By
(2) of the Divergence Criteria, we have that $\lim_{x \to 1} \frac{1}{x - 1}$ does not exist in $\mathbb{R}$.
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Goal First Optimization This is most definitely not my invention. I merely implemented my interpretation of the work of Jelena Vuckovic and her students. I found it incredibly fascinating and a fundamentally different design paradigm from what I had seen in the half-dozen solvers I've used over the years.
In contrast, frequency-domain (FD) techniques, look at the scene from the point of view of a single frequency. Using the frequency domain version of Maxwell's Equations, we can mathematically express how each point in space interacts with the other. This creates a system of constraints which, if formulated correctly, will be equal in size to the number of unknowns in the system. In other words, the system is
determined and square. Solving this problem reduces to inverting this system.
However, this technique opens up an interesting opportunity. What if the formulation of the problem contained more constraints than unknowns? For instance, one could include constraints on various ports in the system that not only expressed how much energy came in, but also how much energy was leaving. This is never done in commercial software because there is no physical way to create a port which "sucks" energy out of the system. It will create systems which (while desired) are impossible, and any solution would be fundamentally flawed. The system will now be
over-determined and rectangular. However, while unphysical, the resulting solution from this minimization problem would be the fields that are nearest to being physical that satisfies these constraints. Since we are prioritizing satisfying the goal over the physics, Vockovic and her team refers to this as "goal-first." Is this field useful?
What is interesting about these fields is that they provide a suggestion for how to change the structure. Given that we know both the electric and magnetic fields, we can solve Maxwell's equations for permittivity to yield
$$\mathbf{\epsilon} = \nabla \times \mathbf{H}/(j \omega \mathbf{E})$$ This formula can be applied to these unphysical fields from multiple desired port constraints to yield several permittivity suggestions. These suggestions can be averaged and used to calculate a "permittivity step direction" which is used to calculate a new permittivity. Various filters can be applied to this process in an attempt to coax out manufacturable structures. From here, the process simply repeats. With each iteration, the residue decreases until unphysical fields are actually not very unphysical at all.
I wrote my own Finite Difference Frequency Domain (FDFD) code in Mathematica, complete with PMLs. Several tweaks need to be made to consider the rows of the system matrix as constraints to be minimized. For instance, they should all share a common unit. Additionally, the source terms are now constraints and represent new rows in the system. Finally, one must find a quick Least-Squares Minimization solver. I found Matlab's to be better for this purpose than Mathematica's. My code sends the matrices to a Matlab server to be solved and then returns the result for further processing and filtering.
As a test case, I considered a multiplexor interfacing ten waveguides. The goal (albeit trivial) is to transfer the energy directly across the system so that the top waveguide on the left transfers its energy to the top waveguide on the right. Fig 1 above shows how the permittivity evolves through the optimization process. You can see that with each iteration the energy becomes more tightly confined to the target waveguide. Fig 2 shows the resulting permittivity. Finally, Fig3 below, shows the final field profiles.
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From Newton’s law, we know that the time rate change of the momentum of a particle is equal to the net force acting on the particle and is in the direction of that force.\(F_{net} = \frac{dp}{dt}\)
This clearly makes us understand that linear momentum can be changed only by a net external force. If there is no net external force,
p cannot change.
Implies that there is no change in the momentum or the
momentum is conserved. i.e
In order to apply conservation of momentum, you have to choose the system in such a way that the net external force is zero.
Example:
In the example given below, the two cars of masses m1 and m2 are moving with velocities v1 and v2 respectively before the collision. And their velocities change to \(v_{1}^{‘}~and~v_{2}^{‘}\) after collision. To apply the law of conservation of linear momentum, you cannot choose any one of the car as the system. If it so, then there is an external force on the car by another car. So we choose both the cars as our system of interest. This is why in all collisions, if both the colliding objects are considered as system, then linear momentum is always conserved (irrespective of type of collision).
Problem
A bullet of mass m leaves a gun of mass M kept on a smooth horizontal surface. If the speed of the bullet relative to the gun is v, the recoil speed of the gun will be_______________________.
No external force acts on the system (gun + bullet) during their impact (till the bullet leaves the gun). Therefore the momentum of the system remains constant. Before the impact (gun + bullet) was at rest. Hence its initial momentum is zero. Therefore just after the impact, its momentum of the system (gun+bullet will be zero)\(\Rightarrow M\vec{V}_b+M\vec{V}_g = 0\) \(\Rightarrow m[\vec{V}_{bg}+\vec{V}_g]+M\vec{V}_g = 0\) \(\vec{V}_g = -\frac{m\vec{V}_{bg}}{M+m}, where \left | \vec{V}_{bg} \right | = velocity\;of\;bullet\;relative\;to\;the\;gun = v\) \(v_g = \frac{mv}{M+m}(opposite\;to\;the\;direction\;of\;v)\)
Types Of Collision
The common normal at the point of contact between the bodies is known as line of impact. If mass centers of the both the colliding bodies are located on the line of impact, the impact is called central impact and if mass centers of both or any one of the colliding bodies are not on the line of impact, the impact is called eccentric impact.
Direct impact/collision
If velocities vectors of the colliding bodies are directed along the line of impact, the impact is called a direct or head-on impact;
Example
Oblique impact/collision
If velocity vectors of both or of any one of the bodies are not along the line of impact, the impact is called an oblique impact.
Example
Consider a ball of mass m1 moving with velocity v1 collides with another ball of mass m2 approaching with velocity v2. After collision, they both move away from each other making an angle with the line of impact.
Coefficient of restitution
When two objects collide head-on, their velocity of separation after impact is in constant ratio to their velocity of approach before impact.\(e = \frac{velocity\;of\;separation}{velocity\;of\;approch}\) \(0\le e\le 1\) \(velocity\;of\;separation = e(velocity\;of\;approach)\)
The constant
e is called as the coefficient of restitution. Elastic Collision
An elastic collision between two objects is one in which total kinetic energy (as well as total momentum) is the same before and after the collision.
Example
On a billiard board, a ball with velocity v collides with another ball at rest. Their velocities are exchanged as it is an elastic collision.
Inelastic Collision
An inelastic collision is one in which total kinetic energy is not the same before and after the collision (even though momentum is constant).
Examples
When the colliding objects stick together after the collision, as happens when a meteorite collides with the Earth, the collision is called perfectly inelastic.
In other inelastic collision, the velocities of the objects are reduced and they move away from each other.
Velocities of colliding bodies after collision in 1- dimension
Let there be two bodies with masses m
1 and m 2 moving with velocity u 1 and u 2 . They collide at an instant and acquire velocities v 1 and v 2 after collision. Let the coefficient of restitution of the colliding bodies be e. Then, applying Newton’s experimental law and the law of conservation of momentum, we can find the value of velocities v 1and v 2.
Conserving momentum of the colliding bodies before and the after the collision\(m_1u_1+m_2u_2 = m_1v_1+m_2v_2\) ……(i)
Applying Newton’s experimental law\(We\;have\;\frac{v_2-v_1}{u_2-u_1}=-e\) \(v_2 = v_1-e(u_2 – u_1)\)…………………….(ii)
Putting (ii) in (i), we obtain\(m_1u_1+m_2u_2 =m_1v_1+ m_2{v_1-e(u_2 – u_1)}\) \(v_1 = u_1\frac{(m_2-em_2)}{m_1+m_2}+u_2\frac{m_2(1+e)}{m_1+m_2}\)……….(iii) \(From\;(ii)\;v_2 = v_1-e(u_2-u_1)\) \(=u_1\frac{m_1(1+e)}{m_1+m_2}+u_2 \frac{(m_2-m_1e))}{m_1+m_2}\)…………(iv)
When the collision is elastic; e = 1\(Finally,\;v_1 = \left ( \frac{m_1+m_2}{m_1+m_2} \right )u_1+\left ( \frac{2m_2}{m_1+m_2} \right )u_2\) \(Finally,\;v_1 =\left ( \frac{2m_1}{m_1+m_2} \right )u_1+ \left ( \frac{m_2-m_1}{m_1+m_2} \right )u_2\)
Having derived the velocities of colliding objects in 1 dimensional collision (elastic and inelastic), let us try to set conditions and draw useful conclusions.
Elastic Collision in Two Dimension
A particle of mass m1 moving with velocity v1 along x-direction makes an elastic collision with another stationary particle of mass m2. After collision the particles move with different directions with different velocities.
Applying
law of conservation of momentum,
\(m_1V_1 = m_1{v_1}’\cos\theta_1+m_2{v_2}’\cos\theta_2\)
Along x-axis:
\(0= m_1{v_1}’\sin\theta_1+m_2{v_2}’\sin\theta_2\)
Along y axis:
Conservation of kinetic energy,\(\frac{1}{2}m_1{v_1}^{2} = \frac{1}{2}m_1{{v_1}’}^{2}+\frac{1}{2}m_2{{v_2}’}^{2}\)
Given the values of \(\theta_1 and \;\theta_2\) , we can calculate the values of \({v_1}’ and \;{v_2}'\) by solving the above examples.
JEE Focus: Understand the conservation principle Make the right choice of system to apply the same Variable Mass System
Problem
After perfectly inelastic collision between two identical particles moving with same speed in different directions, the speed of the particles becomes (2/3)
rd of the initial speed. The angle between the two particles before collision is
Let \(\theta\) be the desired angle. Linear momentum of the system will remain conserved. Hence\(P^{2} = P_{1}^{2}+P_{2}^{2}+2P_1P_2\cos\theta\) \(=\left [ 2m\left ( \frac{2}{3} \right )v \right ]^{2}=(mv)^{2}+2(mv)^{2}+2(mv)(mv)\cos\theta\) \(\Rightarrow \frac{16}{9} = 2+2\cos\theta\) \(\Rightarrow 2\cos\theta= \frac{-2}{9}\) \(\Rightarrow \cos\theta= \frac{-1}{9}\) \(\Rightarrow \theta = 96.37^{0}\)
Hence,(C) is the correct choice.
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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 ...
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Now showing items 1-1 of 1
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 ...
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Probability Seminar Spring 2019 Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 PM.
If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu
January 31, Oanh Nguyen, Princeton
Title:
Survival and extinction of epidemics on random graphs with general degrees
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$. Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.
Wednesday, February 6 at 4:00pm in Van Vleck 911 , Li-Cheng Tsai, Columbia University
Title:
When particle systems meet PDEs
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..
Title:
Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.
February 14, Timo Seppäläinen, UW-Madison
Title:
Geometry of the corner growth model
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).
February 21, Diane Holcomb, KTH
Title:
On the centered maximum of the Sine beta process Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette. Wednesday, February 27 at 1:10pm Jon Peterson, Purdue
Title:
Functional Limit Laws for Recurrent Excited Random Walks
Abstract:
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.
March 7, TBA March 14, TBA March 21, Spring Break, No seminar March 28, Shamgar Gurevitch UW-Madison
Title:
Harmonic Analysis on GLn over finite fields, and Random Walks
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the
character ratio:
$$ \text{trace}(\rho(g))/\text{dim}(\rho), $$
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant
rank. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).
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This question already has an answer here:
Choosing between $z$-test and $t$-test 2 answers
Suppose we have that $X_1, \ldots, X_n \sim N(\mu, \sigma^2)$ where $\sigma^2$ is known. If we wanted to test the hypothesis of:
$$ H_0: \mu = \mu_0 \ \ \text{and} \ \ H_1: \mu \neq \mu_0 $$
a lot of books just jump directly to using a z-score based test statistic of:
$$ Z = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}} \sim N(0,1) $$
However, I do not understand why or how they jump to this statistic aside from the fact it appears the normal distribution is easy to calculate from.
My question is, my understanding is that formally, we can choose any test statistic $T(X)$ such that:
$$ P(T(X) \in R\mid H_0) = \alpha $$
where $R$ is a rejection region to be found and $\alpha$ the significance level.
I am wondering how one can formally derive the $Z$ score approach above? Thanks.
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Consider the transverse-field Ising chain (TFIC) in a transverse-field $B$:
$$H_{TFIC}(B)\equiv -\sum_{j=1}^{N-1} \sigma^x_j\sigma^x_{j+1}+B\sum_{j=1}^N \sigma^z$$
At $B=0$, we have the classical Ising chain, and so there is a set of groundstates for this system with zero entanglement, namely the all-up and all-down states, respectively.
Let's try to understand the ferromagnetic phase of this model: we start by adiabatically turning on $B$, represented by the time-dependent unitary evolution operator $U(B)$ defined below:
$$U(B)\equiv \lim_{T\to \infty}\mathcal T\exp\left(-i\int_0^Tdt\, H(tB/T) \right)$$
From the perspective of the groundstate subspace, because the ferromagnetic phase $B<1$ of the TFIC is gapped, then $U(B)$ must be a local unitary (LU) operator as long as $B<1$:
$$|\psi_0(B)\rangle \,=U(B)|\uparrow\rangle^{\otimes N},\,\,\,\,\,\,\,~~~~~|\psi_1(B)\rangle \,=U(B)|\downarrow\rangle^{\otimes N}$$
Since $U(B)$ is a LU operator, the symmetry-broken groundstates $|\psi_0(B)\rangle, |\psi_1(B)\rangle$ of the transverse-field Ising chain are short-range entangled (SRE) states.
My question is about the
Heisenberg picture for the ferromagnetic phase of the TFIC, which is a much more general viewpoint from the perspective of calculating observables. An observable (e.g. magnetization) at $B\neq 0$ looks like
$$\langle\uparrow|^{\otimes N}U^\dagger (B)\, \sigma^x_j\, U(B)|\uparrow\rangle^{\otimes N}$$
Let's now just examine the operator in the middle:
$$\sigma^x_j(B) \equiv U^\dagger (B)\, \sigma^x_j\, U(B)$$
Clearly, since the TFIC is short-range entangled, $\text{supp}(\sigma^x_j(B))$ decays exponentially with distance from $j$.
At $B=0$, the support vanishes except at $j$. How quickly does this support grow as $B<1$ increases from zero?
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The Annals of Probability Ann. Probab. Volume 8, Number 1 (1980), 115-130. De Finetti's Theorem for Markov Chains Abstract
Let $Z = (Z_0, Z_1, \cdots)$ be a sequence of random variables taking values in a countable state space $I$. We use a generalization of exchangeability called partial exchangeability. $Z$ is partially exchangeable if for two sequences $\sigma, \tau \in I^{n+1}$ which have the same starting state and the same transition counts, $P(Z_0 = \sigma_0, Z_1 = \sigma_1, \cdots, Z_n = \sigma_n) = P(Z_0 = \tau_0, Z_1 = \tau_1, \cdots, Z_n = \tau_n)$. The main result is that for recurrent processes, $Z$ is a mixture of Markov chains if and only if $Z$ is partially exchangeable.
Article information Source Ann. Probab., Volume 8, Number 1 (1980), 115-130. Dates First available in Project Euclid: 19 April 2007 Permanent link to this document https://projecteuclid.org/euclid.aop/1176994828 Digital Object Identifier doi:10.1214/aop/1176994828 Mathematical Reviews number (MathSciNet) MR556418 Zentralblatt MATH identifier 0426.60064 JSTOR links.jstor.org Subjects Primary: 60J05: Discrete-time Markov processes on general state spaces Secondary: 62A15 Citation
Diaconis, P.; Freedman, D. De Finetti's Theorem for Markov Chains. Ann. Probab. 8 (1980), no. 1, 115--130. doi:10.1214/aop/1176994828. https://projecteuclid.org/euclid.aop/1176994828
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Suppose there are $n$ data values $x_1<x_2<\ldots<x_{n-1}<x_n$，and I've found a partition number $k$, such that$$\left|\frac{1}{k}\sum_{i=1}^k(x_i-\hat{\mu_k})^2-\frac{1}{n-k}\sum_{j=k+1}^n(x_j-\hat{\mu}_{n-k})^2\right|$$is minimal. Here $\hat{\mu}_k$ is the mean of the first $k$ values, and $\hat{\mu}_{n-k}$ is the mean of the last $n-k$ values.
But is this optimal partition unique for any set of mutually different data values? How could I prove the uniqueness (or the opposite)? Furthermore, under what conditions would the solution be unique?
-- EDIT 1 --
@Glen_b has given a nice answer which let me notice that the originalproblem description is incomplete in some sense. The method to partition two datasets by minimizing their variance difference is a heuristic way to do
binary classification, and it works in practice. So I'm thinking of the underlying theoretical aspects of the particular problem. In practice the data are affected by noise and will never distributed in some regular symmetric style. Now if I assume the data are generated by a mixture of two Gaussian distribution with equal variance, is it possible to prove that the method mentioned above generates a meaningful result?
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Row Echelon Form of a Matrix
If we have a matrix $A$ that represents a system of linear equations, we can reduce $A$ to what is known as
Row Echelon Form (often times abbreviated as REF) in order to solve the system. We must first look at the definition for a matrix to be in this form.
Definition: Let $A$ be an $m \times n$ matrix. Then $A$ is said to be in Reduced Echelon Form if $A$ satisfies the following three properties. 1. All of the rows that do not consist entirely of zeroes will have their first nonzero entries be $1$ which we defined as leading $1$s. 2. For any two rows that are not entirely comprised of zeroes, the leading $1$ in the row below occurs farther to the right than the leading $1$ in the higher rows. 3. Any rows consisting entirely of zeroes are placed at the bottom of the matrix.
For example, the following matrices are in REF (as you should verify):(1)
Given a matrix $A$, we can use the elementary row operations to convert any matrix to REF.
Example 1 Use elementary row operations to put matrix $A = \begin{bmatrix} 2 & 4 & 6 \\ 2 & 4 & 2\\ 1 & 3 & 1 \end{bmatrix}$ in REF.
Our first step operation is to take row $1$ and multiply it by $\frac{1}{2}$ ($\frac{1}{2}R_1 \to R_1$):(2)
Now let's multiply row $2$ by $\frac{1}{2}$ ($\frac{1}{2}R_2 \to R_2$):(3)
Now let's take row $2$ and subtract row $1$ from it ($R_2 - R_1 \to R_2$):(4)
Now let's interchange rows $2$ and $3$ ($R_2 \leftrightarrow R_3$):(5)
Now let's take row $2$ and subtract row $1$ from it ($R_2 - R_1 \to R_2$):(6)
Lastly let's multiply row $3$ by $-\frac{1}{2}$ ($-\frac{1}{2}R_3 \to R_3$):(7)
We are done. $A$ is now in REF.
Example 2 Use elementary row operations to put matrix $A = \begin{bmatrix} 2 & 1\\ 8 & 4 \end{bmatrix}$ into REF:
First we will multiply row $1$ by $\frac{1}{2}$ ($\frac{1}{2} R_1 \to R_1$):(8)
Now let's take row $2$ and subtract $8$ times row $1$ ($R_2 - 8R_1 \to R_2$):(9)
We are done. $A$ is now in REF.
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Recall that an offset is just a predictor variable whose coefficient is fixed at 1. So, using the standard setup for a Poisson regression with a log link, we have:$$\log \mathrm{E}(Y) = \beta' \mathrm{X} + \log \mathcal{E}$$where $\mathcal{E}$ is the offset/exposure variable. This can be rewritten as$$\log \mathrm{E}(Y) - \log \mathcal{E} = \beta' \...
Offsets can be used in any regression model, but they are much more common when working with count data for your response variable. An offset is just a variable that is forced to have a coefficient of $1$ in the model. (See also this excellent CV thread: When to use an offset in a Poisson regression?)When used correctly with count data, this will let ...
You can use an offset: glm with family="binomial" estimates parameters on the log-odds or logit scale, so $\beta_0=0$ corresponds to log-odds of 0 or a probability of 0.5. If you want to compare against a probability of $p$, you want the baseline value to be $q = \textrm{logit}(p)=\log(p/(1-p))$. The statistical model is now\begin{split}Y & \sim \...
If you are interested in the probability of an incident given N days of patients on ward then you want a model either like:mod1 <- glm(incident ~ 1, offset=patients.on.ward, family=binomial)the offset represents trials, incident is either 0 or 1, and the probability of an incident is constant (no heterogeneity in tendency to generate incidents) and ...
Offsets in Poisson regressionsLets start by looking at why we use an offset in a Poisson regression. Often we want to due this to control for exposure. Let $\lambda$ be the baseline rate per unit of exposure and $t$ be the exposure time in the same units. The expected number of events will be $\lambda \times t$.In a GLM model we are modelling the ...
It is correct you to get cases instead of rates since you are predicting cases. If you want to obtain the rates you should use the predict method on a new data set having all columns equal to data but the population column identically equal to 1, so to have log(populaton)=0. In this case you will get the number of cases of one unit of population, i.e. the ...
There are several issues here:You need to use the observed counts as your response variable. You should not use the densities (g_den).If the observed counts are from differing areas, you need to take the log of those areas as a new variable:larea = log(area)You can control for the differing areas for the observations in two different ways:...
This answer comes in two parts, the first a direct answer to the question and the second a commentary on the model you're proposing.The first part relates to the use of Numbers as an offset along with having it on the r.h.s. of the equation. The effect of doing this will simply be to subtract 1 from the estimated coefficient of Numbers, thereby ...
This also confused me. I thought, "what is the point of explicitly including an offset instead of just pretending that the response divided by the offset / exposure is the $y$ value?".You actually get two different loss functions if you do so.The correct way (use an exposure/offset $s_i$)Model $\log \lambda_i = \log s_i + \theta^T x$ so that $\lambda_i ...
1) What is the best way of determining when to use neg. binom. vs. poisson?A common way (not necessarily the best --- what's 'best' depends on your criteria for bestness) to decide this would be to see if there's overdispersion in a Poisson model (e.g. by looking at the residual deviance.For example, look at summary(glm(count~spray,InsectSprays,family=...
I don't know why glm() doesn't blow up. To figure that out, you'll have to unpack all of the underlying code. (In addition, if your only question is how the R code works, this question is off topic here.)What I can say is that you are not modeling the rates correctly. If you want to model rates instead of counts, you need to include an offset in the ...
I don't know where you heard that a Poisson or negative binomial with an offset is preferable to a binomial model for a number of individuals surviving out of an initial number; I would normally prefer a binomial as it is closer to the actual stochastic process we think is going on. Note that the binomial model would be a binomial GLM,$$n_{\textrm{surv}} \...
You can always include an offset in any GLM: it's just a predictor variable whose coefficient is fixed at 1. Poisson regression just happens to be a very common use case.Note that in a binomial model, the analogue to log-exposure as an offset is just the binomial denominator, so there's usually no need to specify it explicitly. Just as you can model a ...
Time offsets can usually be viewed as your model estimating the rate an event occurs per unit time, with the offset controlling for how long you observed different subjects.In poisson models you are always estimating a rate that something happens, but you never get to observe this rate directly. You do get to observe the number of times that an event ...
According to the answer in:https://stackoverflow.com/questions/34896004/xgboost-offset-exposurexgboost can handle offset term as in glm or gbm using setinfo, but this method is not documented very well.In your example, the code would be:setinfo(xgbMatrix,"base_margin",log(Insurance$Holders))
First, from the help-page of gam (bold font added by me):offset: Can be used to supply a model offset for use in fitting. Notethat this offset will always be completely ignored when predicting,unlike an offset included in formula: this conforms to the behaviourof lm and glm.So for predicting, you should use the formula-specification. Further, ...
One way would be to adopt a formal model-based tree. The glmtree() function in the partykit package implements the general MOB algorithm for model-based recursive partitioning (Zeileis et al. 2008, Journal of Computational and Graphical Statistics, 17(2), 492-514). This supports Poisson responses and also allows for the inclusion of offsets. Furthermore, ...
You can include terms with fixed coefficients using an offset. Technically, an offset is a predictor with coefficient fixed at 1, so you will first need to create a new variable that has the linear combination of the $X$'s with coefficients estimated from the first model.Model for first data set:$$logit(P(Y=1)) = \beta_1X_1 + \dots + \beta_kX_k,$$giving ...
It looks like you divided the fish counts by the volume (or perhaps area) of water surveyed. In that case an offset is indeed appropriate, you should use the log of whatever you divided by. Perhapsmodel1 <- glm(g_den ~ method + site + depth + offset(log(area)), poisson)(edited from earlier incorrect version, missing the log)The reason for the error ...
(with your R code, you could replace "poisson" with "quasipoisson" to avoid all the warnings that get generated. Nothing else of import will change. See (*) below). Your reference use the term "multiplicative glm" which I think just means a glm with log link, since a log link can be thought of as a multiplicative model. Your own example shows that the ...
Look at confidence interval for parameters of your GLM:> set.seed(1)> x = rbinom(100, 1, .7)> model<-glm(x ~ 1, family = "binomial")> confint(model)Waiting for profiling to be done...2.5 % 97.5 %0.3426412 1.1862042This is a confidence interval for log-odds.For $p=0.5$ we have $\log(odds) = \log \frac{p}{1-p} = \log 1 = 0$. ...
The answer by @t.f is the principled one, but can only be used as is for linear models: For glm's (generalized linear models, and other non-linear models needing iterative fitting methods) it cannot be used. So, the concept of an offset was invented: It is a variable in a regression model with a known coefficient (of one.)Building on that R example, the ...
So the response you want to model is "Number of calls per bird" and the troublesome lines are where you didn't observe any birds? Just drop those rows. They add no information to the thing you are trying to model.
Rounding your response variable to an integer is NOT OK. For simplicity, lets assume you're conducting a Poisson regression. What you're modeling is the following:$ \begin{align*}E(Y|x) &= \beta^{T}x + \beta_{0} \\[0.5em]\log \left( \frac{\mbox{No. of Deer}}{\mbox{Area}} \right) &= \beta^{T}x + \beta_{0} \\[0.5em]\log(\mbox{No. of Deer}) - \...
$R^2$ is computed in terms of the sum of squares of fitted values $\text{MSS}=\sum (\hat{y}_i - \bar y_i)^2$ (assuming an intercept term is present) and sum of squares of residuals $\text{RSS} = \sum \left(y_i - \hat{y}_i\right)^2$ as$$R^2 = \frac{\text{MSS}}{\text{MSS} + \text{RSS}}:$$it is the fraction of the total sum of squares "explained" by the fit....
While I don't recommend looking at the source code for glm for those that wish to preserve their mental health, I looked at the source code to glm. The reason R doesn't blow up seems to be that it just doesn't bother to do the kind of defensive checks it probably should.The main iteratively re-weighted least squares loop works by using the methods ...
The point of the offset is that you do not explicitly transform the response. The rate resulting from the standardization would typically not be an integer and a Poisson model would not fit well then.Instead one keeps the count response for which a count distribution like Poisson is appropriate and includes log(exposure) as an offset. Then you get$$ \log(...
If you are going to model using the Poisson you have to have integer values for your response variable. You then have two optionsUse area or some other suitable denominator as an offset. This would usually need to be logged firstInclude area or etc as a predictor variable. Again this would usually be included as a log because you are modelling the log ...
An offset is generally just a coefficient set to a specific value. To get more than one offset, in general you just need to combine the different variables in a way that is consistent to get that fixed value.In a Poisson equation if you set $Z$ as the offset (or exposure its sometimes called):$$\log(\mathbb{E}[Y]) = \beta_0 + \beta_1X + 1\cdot Z$$And ...
The offset does act similarly for both Poisson and NB. The offset has two functions. For Poisson models, the actual number of events defines the variance, so that's needed. It also provides the denominator, so you can compare rates. It's unite-less.Just using a ratio will mess up the standard errors. Having a model,that deals with the offset as most ...
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The values used will often be OK.
A larger than usual motor inductance may cause problems.
The snubber's job is to protect the switch contacts from inductive turnoff transients from the motor. Stopping the transient at source (across the motor) or at destination (across the contacts) both work. Arguably, having it at the switch is better as it deals with the energy that will do damage, as opposed to energy that may do damage, so it is more focused and it also then deals with other spikes that may happen along.
If you look at your circuit you'll note that in both cases the snubber connects from the motor-switch connection point to one leg of the mains. If the mains impedance is low at the spike frequency (-ies) then both are about equivalent.
The circuit current continues instantaneously at switch off. If it all flows through the snubber then it will pass through the 120 ohm resistor, so the voltage spike will
initially will be \$V=IR = 10\mathrm{A} \times 120\mathrm{\Omega} = 1200\mathrm{V}\$. While that is a lot it is usually within the switch break capability (or else), and there are usually other impedances present which will also help to damp it.
The snubbing current will flow only until the capacitor charges to the driving voltage. If the motor inductance is large the capacitor may charge to a higher or much higher voltage.
The capacitor needs to be large enough to not be charged to the point where current decays through charging of the cap before the resistor dissipates the energy. To be sure that the component values present will do the job, you need to know motor inductance.
Energy in inductor is \$E=\frac{1}{2}LI^2\$
Capacitor will "ring" with an energy of \$E=\frac{1}{2}CV^2\$
The resistor needs to dissipate this energy.
Energy = \begin{align}\frac{1}{2}Li^2 &= \frac{1}{2}CV^2 \\ \\\Rightarrow V &= \sqrt{\frac{Li^2}{C}}\end{align}
Then there is some \$L/R\$ time constant as well and ...
You can start to calculate this (if you know L) or simulate it,
but in most cases the values shown are OK for typical equipment.
Place a scope across the contacts. What peak V do you see (use a suitable probe!). Do the contacts spark? They shouldn't.
Note that increasing C improves snubbing action
but also increases losses from the mains in normal operation. Note also that a capacitor across a mains switch may be frowned at in some contexts.
Added:
Dario said: One problem with placing the RS across the switch is that now you have some current in the circuit in the switched off mode. ...
User_long_gone responded: I'm absolutely certain that the 4-5 MILLIAMPS of current flowing through a 0.1 microfarad capacitor at 60 Hz will present no problem to a motor circuit. Wasteful of energy? It's less than 1/2 watt.
It's worth noting that
The snubber across the motor may not bother the motor itself but may well severely bother anyone silly enough to think that the switch being off means that the circuit is "safe" or "dead". If the switch is in the phase/live lead the motor side of the switch may be near ground due to relative impedances. But there is no certainty that this will always be the way the connection is made - even if regulations say that it should be.
2 "Even" 1/2 a Watt of pointlessly wasted energy in an appliance is frowned on in modern scenarios.
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I am curious to find the energies of Dirac delta potential inside the ISW (walls at $x=0,L$) $$ H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-L/2) $$
using Wilson-Sommerfeld Quantization (WSQ),
so I looked at this integral (Here, $p_1=\sqrt{2 m E}, p_2=\sqrt{2 m (E- V_0\delta(x-L/2))}~$) $$\oint p~ dx = 2 \int_0^{L/2-\epsilon} p_1~ dx + 2 \int_{L/2-\epsilon}^{L/2+\epsilon} p_2 ~dx+ 2 \int_{L/2+\epsilon}^{L} p_1~ dx $$ $$= 2* \sqrt{2 m E}*(L-2 \epsilon) + 2 \int_{L/2-\epsilon}^{L/2+\epsilon} p_2 (=\sqrt{2m (E-V_0 \delta(x-L/2))})~dx $$ The Integral term is zero E.g.- this integral (a particular case) , and WSQ gives the energies of ISW Only.
If you recall that momentum is discontinuous for a Dirac delta potential, so integral term should not be zero!!!.
Now the Ques is: can we treat given Hamiltonian $H$ using WSQ ?, If yes, then how to tackle the integral (we have tackled this integral in Schrodinger's formalism)?
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@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?
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Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
In this paper, we study the relations between the long-time dynamical behavior of the perturbed reaction-diffusion equations and the exact reaction-diffusion equations with concave and convex nonlinear terms and prove that bounded sets of solutions of the perturbed reaction-diffusion equations converges to the trajectory attractor $ \mathscr{U}_0 $ of the exact reaction-diffusion equations when $ t\rightarrow\infty $ and $ \varepsilon\rightarrow 0^+. $ In particular, we show that the trajectory attractor $ \mathscr{U}_ \varepsilon $ of the perturbed reaction-diffusion equations converges to the trajectory attractor $ \mathscr{U}_0 $ of the exact reaction-diffusion equations when $ \varepsilon\rightarrow 0^+. $ Moreover, we derive the upper and lower bounds of the fractal dimension for the global attractor of the perturbed reaction-diffusion equations.
Keywords:Reaction-diffusion equations, perturbation, trajectory attractor, global attractor, convergence, fractal dimension. Mathematics Subject Classification:Primary: 35K57, 37L30, 35B40; Secondary: 35B25. Citation:Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5673-5694. doi: 10.3934/dcdsb.2019101
References:
[1]
A. V. Babin and M. I. Vishik,
[2] [3]
V. V. Chepyzhov, E. S. Titi and M. I. Vishik,
On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system,
[4]
V. V. Chepyzhov and M. I. Vishik,
[5] [6] [7] [8]
E. C. Crooks, E. N. Dancer and D. Hilhorst,
On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions,
[9]
E. Feireisl, Ph. Laurencot and F. Simondon,
Global attractors for degenerate parabolic equations on unbounded domain,
[10]
J. K. Hale,
[11]
D. Henry,
[12] [13]
O. A. Ladyzhenskaya,
Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, New York, 1991.
doi: 10.1017/CBO9780511569418.
Google Scholar
[14]
J. L. Lions,
[15] [16] [17]
J. C. Robinson,
Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.
doi: 10.1007/978-94-010-0732-0.
Google Scholar
[18] [19] [20]
M. I. Vishik, E. S. Titi and V. V. Chepyzhov,
On the convergence of trajectory attractors of the three-dimensional Navier-Stokes $\alpha$-model as $\alpha\rightarrow0$.(Russian),
[21] [22]
G. C. Yue,
Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces,
[23] [24] [25]
C. K. Zhong, M. H. Yang and C. Y. Sun,
The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations,
show all references
References:
[1]
A. V. Babin and M. I. Vishik,
[2] [3]
V. V. Chepyzhov, E. S. Titi and M. I. Vishik,
On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system,
[4]
V. V. Chepyzhov and M. I. Vishik,
[5] [6] [7] [8]
E. C. Crooks, E. N. Dancer and D. Hilhorst,
On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions,
[9]
E. Feireisl, Ph. Laurencot and F. Simondon,
Global attractors for degenerate parabolic equations on unbounded domain,
[10]
J. K. Hale,
[11]
D. Henry,
[12] [13]
O. A. Ladyzhenskaya,
Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, New York, 1991.
doi: 10.1017/CBO9780511569418.
Google Scholar
[14]
J. L. Lions,
[15] [16] [17]
J. C. Robinson,
Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.
doi: 10.1007/978-94-010-0732-0.
Google Scholar
[18] [19] [20]
M. I. Vishik, E. S. Titi and V. V. Chepyzhov,
On the convergence of trajectory attractors of the three-dimensional Navier-Stokes $\alpha$-model as $\alpha\rightarrow0$.(Russian),
[21] [22]
G. C. Yue,
Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces,
[23] [24] [25]
C. K. Zhong, M. H. Yang and C. Y. Sun,
The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations,
[1]
Vladimir V. Chepyzhov, Mark I. Vishik.
Trajectory attractor for reaction-diffusion system with diffusion
coefficient vanishing in time.
[2]
Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero.
Structure and regularity of the global attractor of a reaction-diffusion
equation with non-smooth nonlinear term.
[3]
B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan.
Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type.
[4]
Linfang Liu, Xianlong Fu, Yuncheng You.
Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$.
[5]
Yejuan Wang, Peter E. Kloeden.
The uniform attractor of a multi-valued process generated by
reaction-diffusion delay equations on an unbounded domain.
[6]
Boris Andreianov, Halima Labani.
Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems.
[7] [8]
Juliette Bouhours, Grégroie Nadin.
A variational approach to reaction-diffusion equations with forced speed in dimension 1.
[9]
Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik.
On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system.
[10]
Shengfan Zhou, Min Zhao.
Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise.
[11]
José A. Langa, James C. Robinson, Aníbal Rodríguez-Bernal, A. Suárez, A. Vidal-López.
Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations.
[12]
Eduardo Liz, Gergely Röst.
On the global attractor of delay differential equations with unimodal feedback.
[13] [14] [15]
Tomás Caraballo, David Cheban.
On the structure of
the global attractor for non-autonomous dynamical systems with weak convergence.
[16]
Alain Miranville, Xiaoming Wang.
Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations.
[17]
Moncef Aouadi, Alain Miranville.
Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory.
[18]
María Astudillo, Marcelo M. Cavalcanti.
On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion.
[19] [20]
2018 Impact Factor: 1.008
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Though the underlying concepts that are required to build and train a neural network are
difficult, It is very easy to implement it in code. So in this post, let’s Code a neural network by hand Use to build a neural network keras
Firstly let’s see how can we build our own neural network with just raw python code. For this let’s assume our task is to build a model that just
XOR the input. It might seem very easy but believe me, it is the first difficult step in training any neural network as the XOR itself is not linear i.e it is non-linear.
So our model should be able to classify things
non-linearly which is not possible by simple Perceptron. But this is where the Multi Layer Perceptron comes into handy. These are especially suited for such problems indeed. Neural network by hand import numpy as np import matplotlib.pyplot as plt X = np.array([ [0,0,1],[0,1,1],[1,0,1],[1,1,1] ]) y = np.array([[0,1,1,0]]).T
Before starting off, let’s import necessary libraries. $X$ is our input with
no.of.features=3 and
no.of.samples=4 and $y$ is the output. We need to build a model that is capable of predicting the output given the input.
#input features --> 3, hidden_units --> 5, output_units -->1 (D,M,K) = (3,5,1) #initialize weights w1 = np.random.uniform(size=(D,M)) w2 = np.random.uniform(size=(M,K)) learning_rate = 1 cost = []
We then initialize our weights $w1$ and $w2$,
learning_rate=1 and an empty cost list to collect the cost at each iterations which helps us visualize the cost function.
Let’s create an activation function to carry out the operations, here
sigmoid. def sigmoid(z,deriv=False): if deriv: return sigmoid(z)*(1-sigmoid(z)) return 1.0/(1+np.exp(-z))
Note that we don’t use
softmax as our task is just binary classification where sigmoid is just enough to characterize the output.Then we iterate through 5000 times to train our model. And the training is followed as shown below. Feedforward
$z1 = X \cdot w1$
$a1 = g(z1)$
$z2 = a1 \cdot w2$
$a2 = g(z2)$
$cost = \frac{1}{2N} \sum_{i=1}^{N}(a_2-y)^2$
Backpropagation
$\delta_3 = (a_2-y)$
$\delta_2 = \delta_3(g(\bar z_2))$
$\delta_1 = \delta_2 \cdot {w_2}^T g(\bar z_1)$
$\Delta w_2 = a_1 \cdot \delta_2$
$\Delta w_1 = X^T \cdot \delta_1$
$w1 = w1 – \alpha \Delta w_1$
$w2 = w2 – \alpha \Delta w_2$
Though the equations are available in the previous post, I mentioned them here as a matter of completeness.
for i in xrange(5000): #feedforward z1 = X.dot(w1) a1 = sigmoid(z1) z2 = a1.dot(w2) a2 = sigmoid(z2) #append cost cost.append(0.5*np.mean((a2-y)**2)) #backpropagate delta_3 = (a2-y) #shape: (N,K) delta_2 = delta_3*sigmoid(z2,deriv=True) #shape: (N,K) w2_delta = a1.T.dot(delta_2) #shape: (M,K) delta_1 = delta_2.dot(w2.T)*sigmoid(z1,deriv=True) #shape: (N,M) w1_delta = X.T.dot(delta_1) #shape: (D,M) w1 = w1 - learning_rate*w1_delta w2 = w2 - learning_rate*w2_delta
The code follows the equations I’ve mentioned before, so it wouldn’t be a great deal of explaining each of them.
plt.plot(cost) plt.show() print a2
Finally let’s plot the cost function and also print the output predicted output for our inputs to the console.
$ python mlp.py [[ 0.01839197] [ 0.98414062] [ 0.98608678] [ 0.01521415]]
We can see that our model is performing well. Let’s do the same using keras in the next section.
Code with keras Keras is one of the best deep learning library out there which makes building the most complex neural networks in an easy and an intuitive way. Keras primarily relies on either Theano or Tensorflow for its computation grid. We will go into much of it in the upcoming posts. For now let’s see, how we can actually build the same neural network as above. from keras.models import Sequential from keras.layers import Dense from keras.optimizers import SGD
Let’s start off by importing necessary classes. Keras have awesome modularity which makes imports easier.
models.Sequential is some type of a rack we need our neural network to be fitted in.
layers.Dense allows us to construct interconnected network layers. And let’s choose Stochastic Gradient Descent for our training which is then available through
optimizers.SGD
import numpy as np X = np.array([ [0,0,1],[0,1,1],[1,0,1],[1,1,1] ]) y = np.array([[0,1,1,0]]).T
Then import numpy and initialize our inputs. Let’s proceed to build the neural network using
keras.
model = Sequential() model.add(Dense(input_dim=3,output_dim=5,activation="sigmoid")) model.add(Dense(output_dim=1,activation="sigmoid")) sgd = SGD(lr=1.0)
We create a Sequential model object and then add layers as shown above.
input_dim need to be defined only for the input layer which needs
no.of.features and there on we need to supply only
output_dim for each successive layers which helps us building neural network.
Also the activation function is provided as a keyword_argument but can be further extended with keras.activations. SGD with
learning_rate=1.0 is initialized.
model.compile(optimizer=sgd,loss="mean_squared_error") model.fit(X,y,nb_epoch=5000,batch_size=32) print model.predict_classes(X)
And the next step is compiling the model, we choose SGD as optimizer and the
loss="mean_squared_error", since we are dealing with binary classification only,
is not so helpful. And finally let’s print the predicted output to the console. cross-entropy Epoch 4998/5000 4/4 [==============================] - 0s - loss: 5.6662e-04 Epoch 4999/5000 4/4 [==============================] - 0s - loss: 5.6647e-04 Epoch 5000/5000 4/4 [==============================] - 0s - loss: 5.6633e-04 4/4 [==============================] - 0s [[0] [1] [1] [0]]
From the above we can say that our
keras model is super easy to build and it is very handy in building very complex networks which involves deep layers where building a neural network by hand is very difficult.
In the next post, let’s discuss about the various types of optimization techniques that helps us minimizing the cost effectively. And let’s get our hands dirty with
theano and tensorflow. And then let’s head on to recognizing handwritten digits.
|
Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
We provide a more clear technique to deal with general synchronization problems for SDEs, where the multiplicative noise appears nonlinearly. Moreover, convergence rate of synchronization is obtained. A new method employed here is the techniques of moment estimates for general solutions based on the transformation of multi-scales equations. As a by-product, the relationship between general solutions and stationary solutions is constructed.
Keywords:Synchronization, stationary solutions, multi-scale, stochastic differential equations, moment estimates. Mathematics Subject Classification:Primary: 60H10; Secondary: 34F05. Citation:Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103
References:
[1]
V. S. Afraimovich and H. M. Rodrigues, Uniform dissipativeness and synchronization of nonautonomous equations,
[2]
L. Arnold,
[3]
S. A. Azzawi, J. Liu and X. Liu,
Convergence rate of synchronization of systems with additive noise,
[4]
S. A. Azzawi, J. Liu and X. Liu, The synchronization of stochastic differential equations with linear noise,
[5]
T. Caraballo, I. D. Chueshov and P. E. Kloeden,
Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain,
[6]
T. Caraballo and P. E. Kloeden,
The persistence of synchronization under environmental noise,
[7] [8]
T. Caraballo, P. E. Kloeden and B. Schmalfuss,
Exponentially stable stationary solutions for stochastic evolution equations and their pertubation,
[9] [10]
J. Duan and W. Wei,
[11] [12] [13] [14] [15]
X. Liu, J. Duan, J. Liu and P. E. Kloeden,
Synchronization of systems of Marcus canonical equations driven by $\alpha$-stable noises,
[16]
Y. Liu, X. Wan and E. Wu,
Finite time synchronization of Markovian neural networks with proportional delays and discontinuous activations,
[17]
J. Lu, D. W. C. Ho and Z. Wang,
Pinning stabilization of linearly coupled stochastic neural networks via minimum number of controllers,
[18] [19] [20]
A. Pikovsky, M. Rosenblum and J. Kurths,
Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12. Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511755743.
Google Scholar
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B. Schmalfuss and R. Schneider,
Invariant manifolds for random dynamical systems with slow and fast variables,
[24]
S. Strogatz,
Sync: The Emerging Science of Spontaneous Order, Hyperion Press, New York, 2003.
Google Scholar
[25]
T. Su and X. Yang,
Finite-time synchronization of competitive neural networks with mixed delays,
[26]
X. Yang, J. Lu and D. W. C. Ho,
Synchronization of uncertain hybrid switching and impulsive complex networks,
[27] [28]
W. Zhang, C. Li and T. Huang,
Fixed-time synchronization of complex networks with nonidentical nodes and stochastic noise perturbations,
[29]
C. Zhou, W. Zhang and X. Yang,
Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations,
show all references
References:
[1]
V. S. Afraimovich and H. M. Rodrigues, Uniform dissipativeness and synchronization of nonautonomous equations,
[2]
L. Arnold,
[3]
S. A. Azzawi, J. Liu and X. Liu,
Convergence rate of synchronization of systems with additive noise,
[4]
S. A. Azzawi, J. Liu and X. Liu, The synchronization of stochastic differential equations with linear noise,
[5]
T. Caraballo, I. D. Chueshov and P. E. Kloeden,
Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain,
[6]
T. Caraballo and P. E. Kloeden,
The persistence of synchronization under environmental noise,
[7] [8]
T. Caraballo, P. E. Kloeden and B. Schmalfuss,
Exponentially stable stationary solutions for stochastic evolution equations and their pertubation,
[9] [10]
J. Duan and W. Wei,
[11] [12] [13] [14] [15]
X. Liu, J. Duan, J. Liu and P. E. Kloeden,
Synchronization of systems of Marcus canonical equations driven by $\alpha$-stable noises,
[16]
Y. Liu, X. Wan and E. Wu,
Finite time synchronization of Markovian neural networks with proportional delays and discontinuous activations,
[17]
J. Lu, D. W. C. Ho and Z. Wang,
Pinning stabilization of linearly coupled stochastic neural networks via minimum number of controllers,
[18] [19] [20]
A. Pikovsky, M. Rosenblum and J. Kurths,
Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12. Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511755743.
Google Scholar
[21] [22] [23]
B. Schmalfuss and R. Schneider,
Invariant manifolds for random dynamical systems with slow and fast variables,
[24]
S. Strogatz,
Sync: The Emerging Science of Spontaneous Order, Hyperion Press, New York, 2003.
Google Scholar
[25]
T. Su and X. Yang,
Finite-time synchronization of competitive neural networks with mixed delays,
[26]
X. Yang, J. Lu and D. W. C. Ho,
Synchronization of uncertain hybrid switching and impulsive complex networks,
[27] [28]
W. Zhang, C. Li and T. Huang,
Fixed-time synchronization of complex networks with nonidentical nodes and stochastic noise perturbations,
[29]
C. Zhou, W. Zhang and X. Yang,
Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations,
[1]
Thierry Cazenave, Flávio Dickstein, Fred B. Weissler.
Multi-scale multi-profile global solutions of parabolic equations in
$\mathbb{R}^N $.
[2] [3]
Thomas Y. Hou, Pengfei Liu.
Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients.
[4]
Thomas Blanc, Mihai Bostan, Franck Boyer.
Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach.
[5]
Michel Potier-Ferry, Foudil Mohri, Fan Xu, Noureddine Damil, Bouazza Braikat, Khadija Mhada, Heng Hu, Qun Huang, Saeid Nezamabadi.
Cellular instabilities analyzed by multi-scale Fourier series: A review.
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Zhen Wang, Xiong Li, Jinzhi Lei.
Second moment boundedness of linear stochastic delay differential equations.
[8]
Kai Liu.
Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives.
[9]
Emiliano Cristiani, Elisa Iacomini.
An interface-free multi-scale multi-order model for traffic flow.
[10]
Fuke Wu, George Yin, Le Yi Wang.
Razumikhin-type theorems on moment exponential stability of
functional differential equations involving two-time-scale Markovian switching.
[11]
Jean-Philippe Bernard, Emmanuel Frénod, Antoine Rousseau.
Modeling confinement in Étang de Thau: Numerical simulations and multi-scale aspects.
[12]
Wen-ming He, Jun-zhi Cui.
The estimate of the multi-scale homogenization method
for Green's function on Sobolev space $W^{1,q}(\Omega)$.
[13]
Grigor Nika, Bogdan Vernescu.
Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension.
[14]
Weidong Zhao, Jinlei Wang, Shige Peng.
Error estimates of the $\theta$-scheme for backward stochastic differential
equations.
[15]
Markus Gahn.
Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces.
[16]
Miroslava Růžičková, Irada Dzhalladova, Jitka Laitochová, Josef Diblík.
Solution to a stochastic pursuit model using moment equations.
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Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang.
[19]
Tianling Jin, Jingang Xiong.
Schauder estimates for solutions of linear parabolic integro-differential equations.
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Yong Li, Zhenxin Liu, Wenhe Wang.
Almost periodic solutions and stable solutions for stochastic differential equations.
2018 Impact Factor: 1.008
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Intro:
This has been a long coming, endless traps have been setting the math evolutionary progress back throughout time, and you, Physics and Math people have been forced to use simple text equation display techniques to show complex math. Well, be that no longer, as i am about to introduce you to the world that no longer follows the standard text rules, it defies the normal thought, depends only on how far you are willing to take it and has no rivals in the realm of
Actually, it is quite sad that this hasn't happened any earlier, we are a scientific community, so the need for something like this, has bound to be around from very early on in the game (back when Uncle AL, TeleMad, FreeThinker, Gahd and Tormod ran the "Math and Physics" show). But i do have this to say, the software really hasn't caught up with this until early last year, so i guess it's just been ignored or something. Latex itself is a language, similar to HTML, but with major differences; it is a typesetting language made to be able to generate files that will be printable on many different types of printers. Latex is actually a dialect of Tex that was created by D.E.Knuth, it includes many plugins to the original Tex, including the one that i will describe to you today, one created for math.
So what is the major difference between Math v1.0 and Math v2.0? Well its simply this, say you have an expression, and in Math v1.0 you would write it like such:
f(x)=integral from -infinity to x of (e^(-t^2))/(radical(pi^x))dt
uh, yeah, its kinda hard to follow, and some people would actually need to write it out to understand what is going on.... Wouldn't it be easier if you said just:
[math]f(x)=\int_{-\infty}^x \frac{(e^{-t^2})}{\sqrt{\pi^x}}dt[/math]
So, finally arching your attention? I sure hope so, because i am excited as well, and it is not that difficult to write this expression, in latex it looks like this:
f(x)=\int_{-\infty}^x\frac{(e^{-t^2})}{\sqrt{\pi^x}}dt
It may seem a bit much, but trust me, this will be a breeze once you get it.
Basics:
Basics:
BB Tags Hypography uses a [math] tag to signify the beginning and [/math] tag to signify the end of a latex section of the post. Tags are surrounded by square brackets such as [tag] and respectively [/tag]. I strongly urge you to not forget to close the tags, it is not vital, but it is a good practice, even if you only plan to display an expression
Tex Characters:
Most characters in latex are rendered as the regular characters, such as a-z 0-9 () {} [] * $ % and so forth [math]a-z 0-9 () [] * \$ \%[/math]. Some need escaping, such is the case with $ and % in our previous case, escaping means putting a \ in front like \$. But latex introduces some characters that have a meaning and render things differently (for math purposes) such characters would include the underscore (_) and the carat(^). Those characters are used to identify sub and superscript, respectively; for example a_b renders as [math]a_b[/math], a^b renders as [math]a^b[/math]. Latex also has a load of characters that are not defined by regular keyboards, they are special characters and follow the following syntax: \name. Things like \alpha, \beta, \gamma, \delta, \pi and many many others are included ftp://tug.ctan.org/pub/tex-archive/info/symbols/comprehensive/SYMLIST
Those character commands are case sensitive, so \delta [math]\delta[/math]is different from \Delta [math]\Delta[/math].
to be continued...
Continued
Size:
Latex supports many size options, the new software lets us support all of it, but for simplification reasons, any text you type, enclose it in the \text{} tag, you can actually control color, font and size from within those, but to simplify the size for you:
\tiny [math]\text{\tiny{tiny}}[/math]
\small [math]\text{\small{small}}[/math]
\normalsize (default)[math]\text{\normalsize{normal}}[/math]
\Large [math]\text{\Large{large}}[/math]
\LARGE [math]\text{\LARGE{even larger}}[/math]
\huge [math]\text{\huge{huge}}[/math]
I highly recommend using the default size... its just text
You have probably noticed, but latex in math mode also removes spaces thus not making it ideal to write normal text. Another thing to notice is that it outputs an image, however it is partly so, this image is dynamically generated and actually does not get saved on our server; it only exists in your browser, hence linking to it from other websites may be a bit harder then you'd expect, and if the outside traffic picks up too much, i will have to block anyone from the outside domain to be able to use this program...
Spaces and Styling:
Spaces and Styling:
Spaces are not mandatory, however i encourage my readers to use them to avoid confusion, for example you can write \frac{2}{x} and it will render as [math]\frac{2}{x}[/math], however this is fine it still presents a problem, if the fraction is x over 2 then \frac{x}{2} will not work, latex will render the command as fracx and that is not a valid command and hence a problem is due. To avoid confusion i recommend sticking to a good syntax style, such as using popper curly braces and spaces, you can either write that fraction as \frac x2 or using the prototype you should really get used to writing \frac{x}{2} that way you will not have to remember the braces when a complex fraction is due and the fraction or any function for that matter will come out right the first time. On the topic of spaces, i can see that people will ask about newlines and things of that manner. You can actually write text and use newlines and things of that nature. A new line is represented by a \\ and if you needed to write text, you can use the \text{} mode to write it and spaces and such will be used. For example y=\left\lbrace\begin{array}{c c}{2x+5} & \text{if x is less then 1/2} \\ {\pi{x}}^e & \text{if x is more then 1}\end{array}\right. [math]y=\left\lbrace\begin{array}{c c}{2x+5} & \text{if x is less then 1/2} \\ {\pi{x}}^e & \text{if x is more then 1}\end{array}\right.[/math] Latex actually provides spacing commands \, \: \; \quad \qquad, those take no arguments and (a\,b\:c\;d\ e\quad f\qquad g) renders as [math](a\,b\:c\;d\ e\quad f\qquad g)[/math]
Latex has a couple of fonts, here's how to use them:
\mathnormal - default - [math]\mathnormal{A B C d e f 1 2 3 \delta \Delta \infty \lceil \rceil}[/math]
\mathrm - default without italic lower case - [math]\mathrm{A B C d e f 1 2 3 \delta \Delta \infty \lceil \rceil}[/math]
\mathit - italic -[math]\mathit{A B C d e f 1 2 3 \delta \Delta \infty \lceil \rceil}[/math]
\mathbf - bold - [math]\mathbf{A B C d e f 1 2 3 \delta \Delta \infty \lceil \rceil}[/math]
\mathsf - sans serif - [math]\mathsf{A B C d e f 1 2 3 \delta \Delta \infty \lceil \rceil}[/math]
\mathtt - mono space - [math]\mathtt{A B C d e f 1 2 3 \delta \Delta \infty \lceil \rceil}[/math]
\mathcal - caligraphy - [math]\mathcal{A B C d e f 1 2 3 \delta \Delta \infty \lceil \rceil}[/math]
\mathfrak - fraktur - [math]\mathfrak{A B C d e f 1 2 3 \delta \Delta \infty \lceil \rceil}[/math]
\mathbb - blackboard bold - [math]\mathbb{A B C d e f 1 2 3 \delta \Delta \infty \lceil \rceil}[/math]
you can also choose to bold a symbol (such as a greek symbol)
\boldsymbol
[math]\boldsymbol{\Delta} \Delta[/math]
also there are times when latex does not render something quite to your personal spacing specifications, sometimes you need to add or subtract a small space to "nudge" things into place, for this they created a set of spaces for "nudging"
\, 3/18th of a quad
\: 4/18
\; 5/18
\! -3/18
the negative space is handy for places like this:
[math]\left(\begin{array}{c} n \\ r \end{array}\right) = \frac{n!}{r!(n-r)!}[/math]
it looks good, but it could look better
[math]\left(\!\!\!\begin{array}{c} n \\ r \end{array}\!\!\!\right) = \frac{n!}{r!(n-r)!}[/math]
Brackets, Braces and More:
Brackets, Braces and More:
Ofcourse latex supports every imaginable and unimaginable bracet and brace you can ever imagine, from the simple [] () to over and under braces, and more. The curly brackets have their own symbol, as they are used in the syntax, \lbrace and \rbrace will render as [math]\lbrace\rbrace[\math]. Also braces brackets and such render as their default size unless they are specified to do differently, so if you have a complex fraction that you need to be bracketed, say [\frac{(\frac{x+3}{7})+5}{3x+8}] will render as [math][\frac{(\frac{x+3}{7})+5}{3x+8}][/math] however to actually extend those brackets you can use \left and \right flags to do the job, so with the addition of them \left[\frac{\left(\frac{x+3}{7}\right)+5}{3x+8}\right] [math]\left[\frac{\left(\frac{x+3}{7}\right)+5}{3x+8}\right][/math].
Now as promissed the unimaginable stuff. These things are referred to as math accents, and they include vector signs and things of that nature. So tags like \vec{} \hat{} \tilde{} \dot{} \ddot{} will output as [math]\vec{x} \hat{x} \tilde{x} \dot{x} \ddot{x}[/math]. If you need an expression under the sign, the developers have also thought about you, the \widevec \widehat \widetilde do just that [math]\vec{xyz} \hat{xyz} \tilde{xyz}[/math]. Not done yet, also available are \underline{} \overline{} that underline and overline text, as well as \overbrace and \underbrace that make horizontal braces as such [math]\overbrace{x+2}[/math]. There's more to add though, as the sub and superscript come into play here, using it you can explain expressions such is that a1 a2 .. an are just referred to as ai in the matter, you can write that as \overbrace{a_1,a_2...a_n}^{a_i} [math]\overbrace{a_1,a_2...a_n}^{a_i}[/math] with the underbrace remember that you are trying to put stuff under the brace, so use the subscript sign (_) to accomplish the task. Also once again, notice that proper use of brackets is the key, it is easy to make a mistake in the expression, so use the advanced mode preview fearute.
Also available symbols:
\langle and \range [math]\left\langle xyz\right\rangle[/math]
\| [math]\left\| xyz\right\|[/math]
[B]Common Math Needs:[/B]
Matixees, use the \begin{matrix} and \end{matrix} to display one
\left[\begin{matrix} a1,1 & a1,2 & ... & a1,n \\ a2,1 & a2,2 & ... & a2,n \\ ..... & ..... & ..... & ..... \\ am,1 & am,2 & ... & am,n \end{matrix}\right]
[math]\left[\begin{matrix} a1,1 & a1,2 & ... & a1,n \\ a2,1 & a2,2 & ... & a2,n \\ ..... & ..... & ..... & ..... \\ am,1 & am,2 & ... & am,n \end{matrix}\right][/math]
matrix uses no relimiters
pmatrix ()
bmatrix []
Bmatrix {}
vmatrix |
Vmatrix || ||
example:
A_{m,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{pmatrix}
[math]
A_{m,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{pmatrix}
[/math]
like matrices arrays allow you to space your content horizontally as well as vertically, they also allow you to define the lining for it:
\begin{array}{c|c}
1 & 2 \\
\hline
3 & 4
\end{array}
[math]
\begin{array}{c|c}
1 & 2 \\
\hline
3 & 4 \\
\end{array}
[/math]
similarly
[math]
\begin{array}{c|c||}
\hline
1 & 2 \\
\hline
3 & 4 \\
\hline
5 & 6 \\
\hline
\end{array}
[/math]
or if you get really creative with them:
[math]
\begin{array}{|r|l|}
\hline
7C0 & hexadecimal \\
3700 & octal \\ \cline{2-2}
11111000000 & binary \\
\hline \hline
1984 & decimal \\
\hline
\end{array}
[/math]
There is more then a dozen common math symbols recognized in latex:
\arccos [math]\arccos{\left ( \frac{\pi}{2} \right )}[/math]
\arcsin [math]\arcsin{\left ( \frac{\pi}{2} \right )}[/math]
\arctan [math]\arctan{(1)}[/math]
\arg [math]\arg{\left ( \frac{-1-i}{i} \right )}[/math]
\cos [math]\cos{(\pi)}[/math]
\cosh [math]\cosh{(x)}[/math]
\cot [math]\cot{\left( \frac{3\pi}{2} \right )}[/math]
\coth [math]\coth{(x)}=\frac{e^{2x}+1}{e^{2x}-1}[/math]
\csc [math]\csc{(y)}[/math]
\deg [math]f_m([X])=\deg{f[Y]}[/math]
\det_ [math]\Delta_{\overline{a}} \det(A)=\overline{a} \times \overline{c}[/math]
\dim [math]\dim{X}=-1[/math]
\exp [math]\exp{(x)}=e^x[/math]
\gcd_ [math]\gcd{(a,}=2 \sum_{k=1}^{a-1} \lfloor kb/a \rfloor +a+b-ab[/math]
\hom [math]\hom{(X, Y)}[/math]
\inf_ [math]\inf{\{1,2,3\}}=1[/math]
\ker [math]\ker{T}:=\{ v \in V: Tv=0_w\}[/math]
\lg [math]\lg{(2)}=\log_2{(2)}[/math]
\lim_ [math]\lim_{x \to \infty}{(2x+1)}[/math]
\liminf_ [math]\liminf_{n \to \infty}{(x_n)}[/math]
\limsup_ [math]\limsup_{n \to \infty}{(x_n)}[/math]
\ln [math]\ln{(2)}=\log_{e}{(2)}[/math]
\log [math]\log{(2)}=\log_{10}{(2)}[/math]
\max_ [math]\lim_{0 \to 1}\max{(x, 1-x)}dx=\frac{3}{4}[/math]
\min_ [math]\lim_{0 \to 1}\min{(x, 1-x)}dx=\frac{1}{4}[/math]
\Pr_ no clue when this is used, if someone figures it out, let me know, i will post example
\sec [math]\sec{(20)}[/math]
\sin [math]\sin{\theta} = \cos{\left( \frac{\pi}{2} - \theta \right )}[/math]
\sinh [math]\sinh{(x)}=-i \sin{(ix)}[/math]
\sup_ [math]\sup{(X_n)}[/math]
\tan [math]\tan{(x)}[/math]
\tanh [math]\tanh{(x)}=\frac{\sinh(x)}{\cosh(x)}[/math]
\infty [math]\infty[/math]
[B]Stuff that didn't fit anywhere else[/B]
\not can be used with other symbols \not\in [math]\not\in[/math]
\cancel [math]\cancel{ABC}[/math]
\overset{a}{=} [math]\overset{a}{=}[/math]
\underset{a}{=} [math]\underset{a}{=}[/math]
\overrightarrow{abc} [math]\overrightarrow{abc}[/math]
\overleftarrow{abc} [math]\overleftarrow{abc}[/math]
\widetilde{abc} [math]\overwidetilde{abc}[/math]
\widehat{abc} [math]\overwidehat{abc}[/math]
\overline{abc} [math]\overline{abc}[/math]
\underline{abc} [math]\underline{abc}[/math]
\subtrack
[math]
\sum_{\substack{
0<i<m \\
0<j<n
}}
P(i,j)
[/math]
more math stuff:
roots:
\sqrt [math]\sqrt{2}[/math]
root of another power
\sqrt[#] [math]\sqrt[5]{2}[/math]
\exists - [math]\exists[/math]
\forall - [math]\forall[/math]
\neg - [math]\neg[/math]
brackets:
() \, [] \, \{\} \, || \, \|\| \, \langle\rangle \, \lfloor\rfloor \, \lceil\rceil
[math]() \, [] \, \{\} \, || \, \|\| \, \langle\rangle \, \lfloor\rfloor \, \lceil\rceil
[/math]
You are free to practice latex in this thread: http://hypography.co...ice-ground.html
PLEASE ASK ALL YOUR QUESTIONS HERE
I will try to answer them as well I can, and eventually others will be able to answer them as well as they can, but it will create only one thread to go to for answers about latex synthax.
Here is a good reference, and this is my reference for some of the things on this page: LaTeX/Mathematics - Wikibooks, collection of open-content textbooks
Thanks for your time, use math v2.0 wisely, and become free in your expression of math...
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Change of Variables in Double Integrals
Recall from the Evaluating Double Integrals in Polar Coordinates page that if the region $D = \{ (r, \theta) : h_1(\theta) ≤ r ≤ h_2(\theta), \alpha ≤ \theta ≤ \beta \}$ then:(1)
What we have really done is defined a transformation in using these substitution to make evaluating certain integrals simpler. We can apply this technique more generally by changing variables. Suppose that $x$ and $y$ are functions of $u$ and $v$, that is $x = x(u, v)$ and $y = y(u, v)$. Then every $(x, y) \in S \subset \mathbb{R}^2$ is mapped to $(u, v) \in D \subset \mathbb{R}^2$. More importantly, we will want a transformation that is one-to-one from $S$ and onto $D$ so that points in $S$ are mapped to distinct points in $D$ and every point in $D$ is mapped to.
Consider a one-to-one transformation defined by the equations $x = x(u, v)$ and $y = y(u, v)$. Then:(2)
We now need to express the area element, $dA$, from the variables $x$ and $y$ to the variables $u$ and $v$. Suppose that the value of $u$ is fixed such that $u = c$. Then we get that $x = x(c, v)$ and $y = y(c, v)$ from the transformation equations, and these equations together define a parametric curve $\left\{\begin{matrix} x = x(c, v) \\ y = y(c, v) \end{matrix}\right.$ in $\mathbb{R}^3$ that we'll call a $u$-curve. Similarly, if the value of $v$ is fixed such that $v = c$ then $x = x(u, c)$ and $y = y(u, c)$ from the transformation equations, and these equations together define a parametric curve $\left\{\begin{matrix} x = x(u, c) \\ y = y(u, c) \end{matrix}\right.$ in $\mathbb{R}^3$ that we'll call a $v$-curve.
Now consider the area element that is bounded by the $u$-curves for $u$ and $u + du$ and the $v$-curves for $v + dv$:
Let $P$, $Q$, and $R$ be the points shown in the diagram above. As the area elements get smaller, the area of the parallelogram formed by the vectors $\vec{PQ}$ and $\vec{PR}$ approximates the area of the area elements and the error approaches zero since for small values of $du$ and $dv$, the $u$ and $v$ curves are approximately straight. Thus:(3)
We will not go too much into detail, but it can be shown that thus:(4)
The notation $\biggr \rvert \frac{\partial (x, y)}{\partial (u, v)} \biggr \rvert$ represents the absolute value of the Jacobian Determinant. The following theorem summarizes the change of variables for double integrals with respect to the transformation equations $x = x(u, v)$ and $y = y(u, v)$.
Theorem 1: Let $D$ be a region on the $xy$-plane and let $S$ be a region on the $uv$-plane, and let the equations $x = x(u, v)$ and $y = y(u, v)$ define a one-to-one transformation. Suppose that the functions $x$, $y$, and their first partial derivatives with respect to the variables $u$ and $v$ are continuous on the domain $S$ and that $f(x, y)$ is integrable on $D$. Then: $\iint_D f(x, y) \: dx \: dy = \iint_S f(x(u, v), y(u, v)) \biggr \rvert \frac{\partial (x, y)}{\partial (u, v)} \biggr \rvert \: du \: dv$.
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An equity has a value of 100 Euros, and pay a dividend of 5 Euros in 6 months. The interest rate of 6 months is 5% and the interest rate for 1 year is 6%. I would like to compute the value of the price of this equity after 1 year ?
We consider the forward value, which can be employed to estimate the equity value.
Let $T_1=0.5$ be the dividend payment time, and $T=1$. Moreover, let $r_1=5\,\%$ be the annualized interest rate to $T_1$, $r=6\,\%$ be the interest rate to $T$, and $d=5$ be the dividend payment. Then, the forward value, under the risk-neutral measure with the deterministic interest rate assumption, is given by \begin{align*} F &= E(S_T)\\ &=E\big( E(S_T\mid\mathcal{F}_{T_1})\big)\\ &=E\left( e^{rT} E\left(\frac{S_T}{e^{rT}}\mid\mathcal{F}_{T_1}\right)\right)\\ &=E\left(\frac{e^{rT}}{e^{r_1T_1}} S_{T_1} \right)\\ &=E\left(\frac{e^{rT}}{e^{r_1T_1}} (S_{T_1-} -d)\right)\\ &=\frac{e^{rT}}{e^{r_1T_1}} E(S_{T_1-}) - \frac{e^{rT}}{e^{r_1T_1}}\,d\\ &=S_0\,e^{rT} - \frac{e^{rT}}{e^{r_1T_1}}\,d\\ &=101.01. \end{align*}
The relationship between interest rates and equity prices being at best unstable and weak, I'll assume that the level of interest rate is irrelevant here. So the answer to your question (price of the equity in a year) is 95, everything else being equal. Of course it's unlikely that the equity will actually price at 95 in a year due to market movements, but that's a different story.
If you ask for the forward value of the equity, you need to discount that future value with the relevant interest rates.
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Let G = (V, E) be a graph; a set S ⊆ V is a total k-dominating set if every vertex v ∈ V has at least k neighbors in S. The total k-domination number γkt(G) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of a graph. In particular, we investigate the relationship between the total k-domination number of a...
Let G = (V, E) be a simple graph without isolated vertices and minimum degree δ, and let k ∈ {1 − ⌈δ/2⌉, . . . , ⌊δ/2⌋} be an integer. Given a set M ⊂ V, a vertex v of G is said to be k-controlled by M if [...] δM(v)≥δG(v)2+k $\delta _M (v) \ge {{\delta _G (v)} \over 2} + k$ , where δM(v) represents the number of neighbors of v in M and δG(v) the degree of v in G. A set M is called an open k-monopoly...
Ruskey and Savage asked the following question: Does every matching in a hypercube Qn for n ≥ 2 extend to a Hamiltonian cycle of Qn? Fink confirmed that every perfect matching can be extended to a Hamiltonian cycle of Qn, thus solved Kreweras’ conjecture. Also, Fink pointed out that every matching can be extended to a Hamiltonian cycle of Qn for n ∈ {2, 3, 4}. In this paper, we prove that every matching...
The Clar number of a fullerene graph with n vertices is bounded above by ⌊n/6⌋ − 2 and this bound has been improved to ⌊n/6⌋ − 3 when n is congruent to 2 modulo 6. We can construct at least one fullerene graph attaining the upper bounds for every even number of vertices n ≥ 20 except n = 22 and n = 30.
In this note we extend the Mulder-Nebeský characterization of the interval function of a connected graph to the disconnected case. One axiom needs to be adapted, but also a new axiom is needed in addition.
Complete multipartite graphs range from complete graphs (with every partite set a singleton) to edgeless graphs (with a unique partite set). Requiring minimal separators to all induce one or the other of these extremes characterizes, respectively, the classical chordal graphs and the emergent unichord-free graphs. New theorems characterize several subclasses of the graphs whose minimal separators...
The permanental polynomial [...] π(G,x)=∑i=0nbixn−i $\pi (G,x) = \sum\nolimits_{i = 0}^n {b_i x^{n - i} }$ of a graph G is symmetric if bi = bn−i for each i. In this paper, we characterize the graphs with symmetric permanental polynomials. Firstly, we introduce the rooted product H(K) of a graph H by a graph K, and provide a way to compute the permanental polynomial of the rooted product H(K). Then...
In this paper, we study the power domination problem in Knödel graphs WΔ,2ν and Hanoi graphs [...] Hpn . We determine the power domination number of W3,2ν and provide an upper bound for the power domination number of Wr+1,2r+1 for r ≥ 3. We also compute the k-power domination number and the k-propagation radius of [...] Hp2 .
A Roman dominating function (or just RDF) on a graph G = (V, E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF f is the value f(V (G)) = ∑u∈V(G) f(u). An RDF f can be represented as f = (V0, V1, V2), where Vi = {v ∈ V : f(v) = i} for i = 0, 1, 2. An RDF f = (V0, V1, V2) is...
The concept of generalized k-connectivity κk(G), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. The pendant tree-connectivity τk(G) was also introduced by Hager in 1985, which is a specialization of generalized k-connectivity but a generalization of the classical connectivity. Another generalized connectivity of a graph G, named k-connectivity...
A graph is called 2-self-centered if its diameter and radius both equal to 2. In this paper, we begin characterizing these graphs by characterizing edge-maximal 2-self-centered graphs via their complements. Then we split characterizing edge-minimal 2-self-centered graphs into two cases. First, we characterize edge-minimal 2-self-centered graphs without triangles by introducing specialized bi-independent...
In the paper, we show that the incidence chromatic number χi of a complete k-partite graph is at most Δ + 2 (i.e., proving the incidence coloring conjecture for these graphs) and it is equal to Δ + 1 if and only if the smallest part has only one vertex (i.e., Δ = n − 1). Formally, for a complete k-partite graph G = Kr1,r2,...,rk with the size of the smallest part equal to r1 ≥ 1 we have χi(G)={Δ(G)+1if...
Let A = {1, 2, . . . , tm+tn}. We shall say that A has the (m, n, t)-balanced constant-sum-partition property ((m, n, t)-BCSP-property) if there exists a partition of A into 2t pairwise disjoint subsets A1, A2, . . . , At, B1, B2, . . . , Bt such that |Ai| = m and |Bi| = n, and ∑a∈Ai a = ∑b∈Bj b for 1 ≤ i ≤ t and 1 ≤ j ≤ t. In this paper we give sufficient and necessary conditions for a set A to have...
When a graceful labeling of a bipartite graph places the smaller labels in one of the stable sets of the graph, it becomes an α-labeling. This is the most restrictive type of difference-vertex labeling and it is located at the very core of this research area. Here we use an extension of the adjacency matrix to count and classify α-labeled graphs according to their size, order, and boundary value.
We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily...
We continue our systematic search for symmetric Hadamard matrices based on the so called propus construction. In a previous paper this search covered the orders 4v with odd v ≤ 41. In this paper we cover the cases v = 43, 45, 47, 49, 51. The odd integers v < 120 for which no symmetric Hadamard matrices of order 4v are known are the following: 47, 59, 65, 67, 73, 81, 89, 93, 101, 103, 107, 109,...
Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in general.
In this work a comparison is presented between elastic, plastic, and fracture analysis of the monumental arch bridge of Porta Napoli, Taranto (Italy). By means of a FEM model and applying the Mery’s Method, the behavior of the curved structure under service loads is verified, while considering the Safe Theorem approach byHeyman, the ultimate carrying capacity of the structure is investigated. Moreover,...
Financed by the National Centre for Research and Development under grant No. SP/I/1/77065/10 by the strategic scientific research and experimental development program:SYNAT - “Interdisciplinary System for Interactive Scientific and Scientific-Technical Information”.
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The Annals of Statistics Ann. Statist. Volume 24, Number 6 (1996), 2350-2383. Heuristics of instability and stabilization in model selection Abstract
In model selection, usually a "best" predictor is chosen from a collection ${\hat{\mu}(\cdot, s)}$ of predictors where $\hat{\mu}(\cdot, s)$ is the minimum least-squares predictor in a collection $\mathsf{U}_s$ ofpredictors. Here
s is a complexity parameter; that is, the smaller s, the lower dimensional/smoother the models in $\mathsf{U}_s$.
If $\mathsf{L}$ is the data used to derive the sequence ${\hat{\mu}(\cdot, s)}$, the procedure is called unstable if a small change in $\mathsf{L}$ can cause large changes in ${\hat{\mu}(\cdot, s)}$. With a crystal ball, one could pick the predictor in ${\hat{\mu}(\cdot, s)}$ having minimum prediction error. Without prescience, one uses test sets, cross-validation and so forth. The difference in prediction error between the crystal ball selection and the statistician's choice we call predictive loss. For an unstable procedure the predictive loss is large. This is shown by some analytics in a simple case and by simulation results in a more complex comparison of four different linear regression methods. Unstable procedures can be stabilized by perturbing the data, getting a new predictor sequence ${\hat{\mu'}(\cdot, s)}$ and then averaging over many such predictor sequences.
Article information Source Ann. Statist., Volume 24, Number 6 (1996), 2350-2383. Dates First available in Project Euclid: 16 September 2002 Permanent link to this document https://projecteuclid.org/euclid.aos/1032181158 Digital Object Identifier doi:10.1214/aos/1032181158 Mathematical Reviews number (MathSciNet) MR1425957 Zentralblatt MATH identifier 0867.62055 Subjects Primary: 62H99: None of the above, but in this section Citation
Breiman, Leo. Heuristics of instability and stabilization in model selection. Ann. Statist. 24 (1996), no. 6, 2350--2383. doi:10.1214/aos/1032181158. https://projecteuclid.org/euclid.aos/1032181158
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The radio spectrum is a very precious resource like real estate and must be utilized judiciously. Pulse shaping filters control the spectral leakage of the transmitted signal in a wireless channel due to the strict restrictions to comply with a spectral mask. This is even more important for the upcoming 5G wireless systems which are based on a variety of wireless transmission protocols (such as mobile networks, Internet of Things (IoT) and machine to machine communications) combined in one comprehensive standard. Even for wired channels, there is always a natural bandwidth of the medium (copper wire, coaxial cable, optical fiber) that imposes upper limits on its utilization.
The design of a good pulse shaping filter starts with the smallest possible bandwidth exhibited by a rectangular spectrum. However, that abrupt transition in the frequency domain gives rise to long tails in the time domain. To avoid this problem, a smoother rolloff of the spectrum is desired for which we can extend the bandwidth in any shape as long as it has odd symmetry around half the symbol rate $\pm 0.5R_M$ to satisfy Nyquist no-ISI (Inter-Symbol Interference) criterion. This extension can be logically conceived as a convolution between the rectangular mass of width $R_M$ and an even symmetric taper of width $\alpha R_M$ where $0 < \alpha \le 1$. This even symmetry preserves the odd symmetry around $\pm 0.5R_M$ in the resultant filter. The smoothest spectral shape one can imagine is a sine or cosine. A half-cosine of width $\alpha R_M$ -- an even symmetric shape -- is convolved in frequency domain with a rectangular spectrum to generate the most commonly used pulse known as a Raised Cosine (RC) filter. The parameter $\alpha$ is the excess bandwidth or rolloff factor in the resultant desired spectrum. Since the convolution in time domain is multiplication in frequency domain, an RC filter is divided into two parts in frequency domain: one at the Tx and one at the Rx, both of which are square-root of the original RC filter and are known as Square-Root Raised Cosine (SRRC) filters. The Tx SRRC filter implements the shaping filter that determines the spectral mask while the Rx SRRC filter implements the matched filter that maximizes the SNR at the Rx. The Raised Cosine concept is a good starting point for pulse shape design and its closed-form mathematical expression is good for analytical purpose. Nevertheless, there are two major drawbacks in using an SRRC pulse for shaping the spectrum.
Since the transition band of an RC pulse is half cycle of a cosine, the transition band of an SRRC pulse is a quarter cycle of a cosine. Its abrupt termination at the stopband results in a discontinuity causing a limit to the sidelobe (SL) suppression that an SRRC pulse can achieve. As a consequence of truncation in time domain, the pulse is no more absolutely band-limited within $0.5(1+\alpha)R_M$ and assumes infinite support in frequency in the form of sidelobes. This is because the truncation in time domain (i.e., multiplication by a rectangular window) causes subsequent convolution in frequency domain between the SRRC spectrum and a sinc signal. This operation moves the half amplitude values away from the odd symmetry points of $F = \pm 0.5R_M$ violating the Nyquist no-ISI criterion and inducing increased ISI.
This leads us to other pulse shape design procedures that produce a Nyquist filter with improved stopband attenuation preferably without any degradation in peak ISI. We discuss two main design techniques for finding a superior pulse shaping filter: transformation of a lowpass filter based on Parks-McClellan algorithm to a Nyquist filter, and convolution of a frequency domain window with a rectangular spectrum.
Transformed lowpass filter
The standard method is by starting with an initial lowpass filter that is designed according to the Parks-McClellan algorithm whose passband and stopband edges are matched to the rolloff boundaries of the Nyquist spectrum. The Parks-McClellan algorithm is an iterative algorithm for finding the optimal FIR filter based on Remez exchange algorithm and Chebyshev approximation theory such that the maximum error between the desired and the actual frequency response is minimized. Filters designed this way exhibit an equiripple behavior in their frequency responses and thus are also known as equiripple filters, where equiripple implies equal ripple within the passband and the stopband that are not necessarily the same (in fact, mostly they are not).
Naturally, this lowpass filter crosses the band edge $F = 0.5R_M$ with more attenuation than $-3$ dB level required for a Nyquist spectrum. Since the transition band belongs to the filter designer, the passband edge frequency can be pushed forward towards $-3$ dB level. This can be implemented in a software routine through a few iterations of increasing the passband edge frequency based on a gradient descent method, just like an offline adaptive filter.
For a sampling rate $F_S$, passband frequency $F_{\text{pass}}$, stopband frequency $F_{\text{stop}}$ and a positive constant $\mu$ that controls the rate of convergence and the approximation error, the procedure in the $n$-th iteration is listed below:
design a lowpass filter using Parks-McClellan algorithm with frequency set $\big\{0~~ F_{\text{pass}}[n] ~~F_{\text{stop}}~~ F_S/2\big\}$, find the error between $-3$ dB and the filter attenuation in dB at $0.5R_M$ as \begin{equation*} e[n] = -3 – P_{\text{dB}}(0.5R_M), \end{equation*} update the passband frequency as \begin{equation*} F_{\text{pass}}[n+1] = (1+\mu e[n])F_{\text{pass}}[n] \end{equation*}
For most cases, a few iterations are enough for transforming it into a Nyquist filter. There is a weighting option available as well that can place more emphasis on a desired frequency band at the expense of the remaining bands. For example, more stopband attenuation can be achieved by weighting it at a cost of increased in-band ripple.
For a better visual understanding, we create a length $49$ square-root Nyquist filter using a transformed lowpass filter with three different excess bandwidths, namely $\alpha = $ $[0.15,~0.2,~0.25]$ and a group delay equal to $6$. Next, their frequency response is plotted along with the measure of sidelobe attenuation. Finally, two square-root Nyquist filters are convolved and downsampled at $1$ sample/symbol to observe the respect peak ISI levels. The results are drawn in Figure below.
Window based filter
The other procedure, devised by fred harris, is based on the convolution of a smooth taper of width $\alpha R_M$ with a rectangular spectrum of width $R_M$. To affect maximum smoothness, this taper should simply be a good spectral window with a narrow mainlobe width and low sidelobe levels. One such candidate is a Kaiser window which is an approximation to the prolate-spheroidal window for which the ratio of the mainlobe energy to the sidelobe energy is maximized. Given a fixed length, a parameter $\beta$ controls the sidelobe height which decreases with $\beta$ at a cost of increase in the mainlobe width. The coefficients for Kaiser window $w(n)$ are given by
\begin{equation*} w(n) = \begin{cases} \frac{I_0 \left(\pi \beta \sqrt{1-\left(\frac{n}{N/2}\right)^2} \right)}{I_0(\pi\beta)} & -N \le n \le +N \\ 0 & ~~\text{otherwise} \end{cases} \end{equation*} where $I_0(\cdot)$ is the zero-order modified Bessel function of the first kind. Again, we create a length $49$ square-root Nyquist filter using a frequency domain window based filter with similar excess bandwidths $\alpha = $ $[0.15,~0.2,~0.25]$, $\beta=12$ and a group delay of $6$. Next, their frequency response is plotted along with the measure of sidelobe attenuation. Finally, two square-root Nyquist filters are convolved and downsampled at $1$ sample/symbol to observe the respect peak ISI levels. The results are drawn in Figure above and compared with the lowpass based design.
Since Parks-McClellan algorithm minimizes the error in the pass and stop bands, it generates optimal filter coefficients and has consequently become the standard method in FIR filter design. Moreover, the iterative lowpass process is more flexible because any sidelobe level can be exchanged with the in-band ripple by utilizing the penalty weights. On the other hand, the Kaiser window technique is not as flexible. Due to the convolution of the spectra, the stopband ripple and the in-band ripple are always the same amplitude.
Although Figure above demonstrates in each case that the sidelobe attenuations exhibited by the lowpass filter are significantly better than the window based filter, along with its peak ISI being either comparable or even better, the lowpass filter design is nevertheless overall superior with respect to sidelobe levels only and window based technique is superior in terms of peak ISI in most settings. There is room for choosing one over another depending on the system requirements.
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I think you're either misusing the word "renormalizable" or using it in a very old-fashioned way. Perturbative quantum gravity is renormalizable at all loop orders in the sense that divergences can be removed by the addition of local counterterms. Like all EFTs for which the leading interaction is irrelevant (in this case the Einstein-Hilbert term) such counterterms are necessarily higher order in the derivative expansion. The claim about pure gravity is that it is
finite at one-loop, meaning that there are no UV divergent one-loop amplitudes.
One can show that in \(d=4\) for \(\mathcal{N}\geq 1\)supersymmetry, pure supergravity is also finite at two-loops. Let's be clear what this means, it says that any two loop amplitude calculated with vertex factors from the supersymmetrization of the Einstein-Hilbert action is non-divergent if calculated with a supersymmetry preserving regulator. One way to show this is by a combination of counterterm analysis and on-shell methods. You begin by showing that the only possible counterterm would have to be a supersymmetrization of \((R_{\mu\nu\rho\sigma})^3\)and then show that no such supersymmetrization exists. A very efficient way to do the second step is to first see that such a counterterm would produce tree-level on-shell 3-point amplitudes for three gravitons with helicity assignments \((+2,+2,+2)\). However, such an amplitude can never satisfy the on-shell supersymmetry Ward identities and therefore cannot be present in a supersymmetric model.
Similar arguments can be extended to higher loop orders. The most interesting results have been obtained in \(\mathcal{N}\geq 5\) supergravity. At leading order in the derivative expansion all supergravities have an electromagnetic duality symmetry which acts linearly on the graviphotons and chiral fermions and non-linearly on the scalars (if present). At tree-level the non-linear symmetry gives rise to low-energy theorems which say that any amplitude vanishes in the limit that the momentum of a scalar is taken to zero. For \(\mathcal{N}<5\) there is a non-vanishing \((\text{Diff})^2(\text{EM-duality})\) anomaly which leads to a breakdown of the low-energy theorem at loop order. For \(\mathcal{N}\geq 5\) this anomaly vanishes and so we can regularize loop integrals in such a way that the duality symmetry is unbroken. Consequently the on-shell amplitudes arising from any potential counterterms must satisfy the vanishing scalar low-energy theorem.
For example, in https://arxiv.org/abs/1009.1643 it was shown explicitly in \(\mathcal{N}=8\) supergravity that the first on-shell amplitudes which satisfy \(\mathcal{N}=8\) supersymmetry, \(\text{SU}(8)_R\) symmetry and \(E_{7(7)}\) duality symmetry (importantly the low-energy theorems for the scalars) correspond to an interaction term of the form \(D^8R^4\) which in turn corresponds to a 7-loop counterterm. So we expect \(\mathcal{N}=8\) supergravity in \(d=4\) to be UV finite up to six-loops.
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Coverings of a Subset in Euclidean Space
Definition: Let $S \subseteq \mathbb{R}^n$. A Covering of $S$ is a collection of subsets of $\mathbb{R}^n$, $\mathcal F$, such that $\displaystyle{S \subseteq \bigcup_{A \in \mathcal F} A}$. If $\mathcal F$ is a collection of open sets, then $\mathcal F$ is called an Open Covering of $S$ and if $\mathcal F$ is a collection of closed sets, then $\mathcal F$ is called a Closed Covering of $S$.
For example, consider the subset $S = (0, 1) \subseteq \mathbb{R}^n$ and the collection of sets:(1)
It's not hard to see that $\mathcal F$ is a covering of $S = (0, 1)$. Furthermore, the collection $\mathcal F^* = \left \{ \left (0, \frac{1}{2} \right ), \left [ \frac{1}{2}, 1 \right ) \right \}$ is also a covering of $S = (0, 1)$. In particular, $\mathcal F^*$ is a finite covering of $S$.
For another example, consider the subset $S = \{ (x, y) \in \mathbb{R}^2 : x \geq 0 , y \geq 0 \}$. Furthermore consider the following collection of sets:(2)
Once again, it should not be too hard to see that $\mathcal F$ is a covering of $S$:
We see that this is a countably infinite covering of $S$.
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Double Series of Real Numbers and Cauchy Products Review
Table of Contents
Double Series of Real Numbers and Cauchy Products Review
We will now review some of the recent material regarding double series of real numbers and Cauchy products.
On the Double Series of Real Numberswe said that if $(a_{mn})_{m,n=1}^{\infty}$ is a double sequence of real numbers then the corresponding Double Sequence of Partial Sumsis the double sequence $(s_{mn})_{m,n=1}^{\infty}$ defined for all $m, n \in \mathbb{N}$ by:
\begin{align} \quad s_{mn} = \sum_{j=1}^{m} \left ( \sum_{k=1}^{n} a_{jk} \right ) \end{align}
We said that the corresponding Double Seriesdenoted $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ (which denotes the sum of all terms in the double sequence $(a_{mn})_{m,n=1}^{\infty}$ Convergesto $A \in \mathbb{R}$ if $(s_{mn})_{m,n=1}^{\infty}$ converges to $A$, and Divergesif it does not converge to any $A \in \mathbb{R}$. On the Absolute and Conditional Convergence of Double Series of Real Numberspage we said that a double series $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ is Absolutely Convergentif $\displaystyle{\sum_{m,n=1}^{\infty} \mid a_{mn} \mid}$ converges, and Conditionally Convergentif $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ converges and $\displaystyle{\sum_{m,n=1}^{\infty} \mid a_{mn} \mid}$ diverges. As expected, we saw that if the double series $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ converges absolutely, then $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ converges. We then began to look at the product of series on The Product of Two Series of Real Numberspage. We said that if $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ are two series of real numbers then the Productof these series is given by the partial sum sequence $(S_n)_{n=1}^{\infty}$ where for each $n \in \mathbb{N}$ we define:
\begin{align} \quad S_n = \left ( \sum_{i=0}^{n} a_i \right ) \left ( \sum_{j=0}^{n} b_k \right ) \end{align}
To visualize how we determine the product of two series, consider the following array form of the terms in the expansion of the product:
\begin{align} \quad \begin{matrix} a_0b_0 & a_0b_1 & a_0b_2 & \cdots & a_0b_n & \cdots \\ a_1b_0 & a_1b_1 & a_1b_2 & \cdots & a_1b_n & \cdots \\ a_2b_0 & a_2b_1 & a_2b_2 & \cdots & a_2b_n & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \ddots \\ a_nb_0 & a_nb_1 & a_nb_2 & \cdots & a_nb_n & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \ddots \end{matrix} \end{align}
Then each $S_n$ for $n \in \{0, 1, 2, ... \}$ is the sum of the top left $n+1$ by $n+1$ array of terms. We saw that if $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ both converge to $A$ and $B$ then the product of these series will also converge - specifically to the product $AB$. On The Cauchy Product of Two Series of Real Numberspage we looked at a different type of product between two series of real numbers. If $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ are two series of real numbers, then the Cauchy Productof these series is the following sum:
\begin{align} \quad \sum_{n=0}^{\infty} c_n = \sum_{n=0}^{\infty} \underbrace{ \left ( \sum_{k=0}^{n} a_kb_{n-k} \right )}_{c_n} \end{align}
We then looked into the convergence of Cauchy products of series on the Convergence of Cauchy Products of Two Series of Real Numbers. We saw that if $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ both converge absolutely and converge to $A$ and $B$ respectively, then the Cauchy product $\displaystyle{\sum_{n=0}^{\infty} c_n}$ converges to $AB$. We also noted that if $\displaystyle{\sum_{n=0}^{\infty} a_n}$ converges absolutely and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ only converges conditionally, say to $A$ and $B$ respectively, then $\displaystyle{\sum_{n=0}^{\infty} c_n}$ converges (not necessarily absolutely) to $AB$. On the Evaluating Cauchy Products of Two Series of Real Numberswe looked at some examples of evaluating the Cauchy product of two series of real numbers. Lastly, on The Cauchy Product of Power Serieswe looked at a nice result on Cauchy products for power series. We saw that if $\displaystyle{\sum_{n=0}^{\infty} a_nx^n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_nx^n}$ are two power series that are absolutely converge to $A(x)$ and $B(x)$ respectively, then the Cauchy product $\displaystyle{\sum_{n=0}^{\infty} c_nx^n}$ converges to $A(x)B(x)$.
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Moscow-Beijing topology seminar 报告人简介 2019-10-30
Title: Stringc structures
and modular invariants
Speaker: 黄瑞芝 Huang Ruizhi （中科院）
Abstract: Spin structure and
its higher analogies play important roles in index theory and mathematical
physics. In particular, Witten genera for String manifolds have nice geometric
implications. As a generalization of the work of Chen-Han-Zhang (2011), we
introduce the general Stringc structures based on the algebraic topology of
Spinc groups. It turns out that there are infinitely many distinct universal
Stringc structures indexed by the infinite cyclic group. Furthermore, we can
also construct a family of the so-called generalized Witten genera for Spinc
manifolds, the geometric implications of which can be exploited in the presence
of Stringc structures. As in the un-twisted case studied by Witten, Liu, etc,
in our context there are also integrality, modularity, and vanishing theorems
for effective non-abelian group actions. We will also give some applications.
This a joint work with Haibao Duan and Fei Han.
2019-10-23
Title: Classification of
Links with Khovanov Homology of Minimal Rank
Speaker: 谢羿 Xie Yi（北京大学）
Abstract: Khovanov homology is
a link invariant which categorifies the Jones polynomial. It is related to
different Floer theories (Heegaard Floer, monopole Floer and instanton Floer)
by spectral sequences. It is also known that Khovanov homology detects the
unknot, trefoils, unlinks and Hopf links. In this talk, I will give a brief
introduction to Khovanov homology and use the instanton Floer homology to prove
that links with Khovanov homology of minimal rank must be iterated connected
sums and disjoint unions of Hopf links and unknots. This is joint work with
Boyu Zhang.
2019-10-16
Title: Spin generalization
of Dijkraaf-Witten TQFTs
Speaker: Pavel Putrov (ICTP)
Abstract: In my talk I will
consider a family of spin topological quantum field theories (spin-TQFTs) that
can be considered as spin-version of Dijkgraaf-Witten TQFTs. Although
relatively simple, such spin-TQFTs provide non-trivial invariants of
(higher-dimensional) links and manifolds, and provide examples of
categorification of such quantum invariants.
2019-10-9
Title: Virtual Knots and
Links and Perfect Matchings of Trivalent Graphs
Speaker: Louis H. Kauffman (UIC and NSU)
Abstract: In this talk we
discuss a mapping from Graphenes (oriented perfect matching structures on
trivalent graphs with cyclic orders at their vertices) to Virtual Knots and
Links. We show how, with an appropriate set of moves on Graphenes, our mapping
K: Graphenes —> Virtual Knots and Links is an equivalence of categories.
This means that we can define new invariants of graphenes by using invariants
of virtual knots and links, including Khovanov homology for graphenes. The
equivalence K allows us to explore problems about graphs, such as coloring
problems and flow problems, in terms of knot theory. The talk will introduce
many examples of this correspondence and discuss how the classical coloring problems
for graphs are illuminated by the topology of virtual knot theory.
2019-9-25
Title: Simplicial
structures in braid/knot theory and data analytics
Speaker: 吴杰Wu Jie
Abstract: In this talk, we
will explain simplicial technique on braids and links as well as its
applications in data science.
2019-9-18
Title: Computation of Spin
cobordism groups
Speaker: 万喆彦 Wan Zheyan (YMSC)
Abstract: Adams spectral
sequence is a powerful tool for computing homotopy groups of spectra. In
particular, it was used for computing homotopy groups of sphere spectrum, which
are stable homotopy groups of spheres. By the generalized Pontryagin-Thom
isomorphism, the Spin cobordism group \Omega_d^{Spin}(X) is exactly the
homotopy group \pi_d(MSpin\wedge X_+) where MSpin is the Thom spectrum, X_+ is
the disjoint union of X and a point. In my talk, I will introduce spectra,
Adams spectral sequence and compute the Spin cobordism groups for a special
topological space X. These are contained in my joint work with Juven Wang
(arXiv: 1812.11967).
2019-9-11
Title: Word problem in
certain G_n^k groups
Speaker: Denis Fedoseev （莫斯科国立大学）
Abstract: In the present talk
we discuss the word and, to lesser extent, the conjugacy problems in certain
groups G_n^k, which were introduced by V. Manturov.
We prove that the word problem is algorithmically solvable in the groups G_4^3 and G_5^4.
The talk is based on the joint work with V. Manturov and A. Karpov.
2019-9-4
Title: Coxeter arrangements
in three dimensions
Speaker: 王军 Wang Jun （首都师范大学）
Abstract: Let $\mathcal{A}$ be
a finite real linear hyperplane arrangement in three dimensions. Suppose
further that all the regions of $\mathcal{A}$ are isometric. The open quesion
asked by C.Klivans and E.Swartz in 2001 is that does there exist a real central
hyperplane arrangement with all regions isometric that is not a Coxeter
arrangement.
In 2016, R.Ehrenborg, C.Klivans and N.Reading proved that in three dimensions, $\mathcal{A}$ is necessarily a Coxeter arrangement. As it is well known that the regions of a Coxeter arrangement are isometric, this characterizes three-dimensional Coxeter arrangements precisely as those arrangements with isometric regions. It is an open question whether this suffices to characterize Coxeter arrangements in higher dimensions.
In this talk, we will intruduce Coxeter arrangement and the proof of R.Ehrenborg, C.Klivans and N.Reading in three dimensions.
2019-8-28
Title: Pictures instead of
coefficients: the label bracket
Speaker: Vassily Manturov （莫斯科国立技术大学）
Abstract: We shall consider
skein-relations where instead of coefficients we draw small pictures.
This way, starting from the Kauffman bracket formalism, we get to a knot invariant (valued in pictures with modulo relations) which dominates
not only the Kauffman bracket but also the Kuperberg bracket, the HOMFLY polynomial, the arrow polynomial and the Kuperberg picture-valued invariant for virtual knots and knotoids.
This is a joint work with Alyona Akimova and Louis Kauffman.
https://arxiv.org/abs/1907.06502
2019-8-21
Title: Knots with identical
Khovanov homology
Speaker: 柏升 Bai Sheng（北京化工大学）
Abstract: This is Liam
Watson's paper in Algebraic &
Geometric Topology 7 (2007) 1389–1407. He gives a recipe for constructing
families of distinct knots that have identical Khovanov homology and give
examples of pairs of prime knots, as well as infinite families, with this
property.
2019-8-14
Title: A Discrete Morse
Theory for Hypergraphs
Speaker: 任世全Ren
Shiquan（清华大学）
Abstract: A hypergraph can be
obtained from a simplicial complex by deleting some non-maximal simplices. By
[A.D. Parks and S.L. Lipscomb, Homology and hypergraph acyclicity: a combinatorial
invariant for hypergraphs. Naval Surface Warfare Center, 1991], a hypergraph
gives an associated simplicial complex. By [S. Bressan, J. Li, S. Ren and J.
Wu, The embedded homology of hypergraphs and applications. Asian J. Math. 23
(3) (2019), 479-500], the embedded homology of a hypergraph is the homology of
the infimum chain complex, or equivalently, the homology of the supremum chain
complex. In this paper, we generalize the discrete Morse theory for simplicial
complexes by R. Forman [R. Forman, Morse theory for cell complexes. Adv. Math.
134 (1) (1998), 90-145], [R. Forman, A user’s guide to discrete Morse theory.
Séminaire Lotharingien de Combinatoire 48 Article B48c, 2002], [R. Forman,
Discrete Morse theory and the cohomology ring. Trans. Amer. Math. Soc. 354 (12)
(2002), 5063-5085] and give a discrete Morse theory for hypergraphs. We use the
critical simplices of the associated simplicial complex to construct a
sub-chain complex of the infimum chain complex and a sub-chain complex of the
supremum chain complex, then prove that the embedded homology of a hypergraph
is isomorphic to the homology of the constructed chain complexes. Moreover, we
define discrete Morse functions on hypergraphs and compute the embedded
homology in terms of the critical hyperedges. As by-products, we derive some
Morse inequalities and collapse results for hypergraphs.
2019-8-7
Title: Pre-image classes
and an application in knot theory
Speaker: 赵学志Zhao
Xuezhi（首都师范大学）
Abstract: Let $f: X\to Y$ be a
map and $B$ be a non-empty closed subset of $Y$. We consider the pre-image
$f^{-1}(B)$ according to Nielsen fixed theory. Can we give a lower bound for
the number of components $g^{-1}(B)$, where $g$ is an arbitrary map in the
homotopy class of $f$? This is a natural generalization of root theory. We
shall apply this theory to give invariants for knots and links.
2019-7-31
Title: Mysteries of
approximation in $\mathbb{R}^4$
Speaker: N.G.Moshchevitin（莫斯科国立大学）
Abstract: We will discuss some
unsolved problems in the theory of Diophantine Approximation.
It turns out that certain questions related to approximation of subspaces of $\mathbb{R}^4$ by rational subspaces remain unclear. We discuss some of them as well as related topics.
2019-7-24
Title: Quandle and its
applications in knot theory
Speaker: 程志云Cheng
Zhiyun（北京师范大学）
Abstract: In this talk I will
give a brief introduction to quandle theory. Several applications of quandle in
knot theory will be discussed.
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Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain
School of Mathematics, Sun Yat-sen University, No.135 Xingangxi Road, Haizhu District, Guangzhou 510275, China
Based on the $ H^2 $ existence of the solution, we investigate weighted estimates for a mixed boundary elliptic system in a two-dimensional corner domain, when the contact angle $ \omega\in(0,\pi/2) $. This system is closely related to the Dirichlet-Neumann operator in the water-waves problem, and the weight we choose is decided by singularities of the mixed boundary system. Meanwhile, we also prove similar weighted estimates with a different weight for the Dirichlet boundary problem as well as the Neumann boundary problem when $ \omega\in(0,\pi) $.
Keywords:Weighted elliptic estimates, corner domain, mixed boundary problem, Dirichlet-Neumann operator. Mathematics Subject Classification:Primary: 35Q31, 35J25; Secondary: 35Q35. Citation:Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264
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J. Banasiak and G. F. Roach,
On mixed boundary value problems of Dirichlet oblique-derivative type in plane domains with piecewise differentiable boundary,
[2]
M. Sh. Birman and G. E. Skvortsov, On the quadratic integrability of the highest derivatives of the Dirichlet problem in a domain with piecewis smooth boundary,
[3]
M. Borsuk and V. A. Kondrat'ev,
[4]
M. Costabel and M. Dauge,
General edge asymptotics of solutions of second order elliptic boundary value problems, Ⅰ,
[5]
M. Costabel and M. Dauge,
General edge asymptotics of solutions of second order elliptic boundary value problems Ⅱ.,
[6]
M. Dauge,
[7]
M. Dauge, S. Nicaise, M. Bourlard and M. S. Lubuma,
Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques Ⅰ: résultats généraux pour le problème de Dirichlet,
[8]
G. I. Eskin, General boundary values problems for equations of principle type in a plane domain with angular points,
[9]
P. Grisvard,
[10]
P. Grisvard,
[11]
V. A. Kondrat'ev,
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V. A. Kozlov, V. G. Mazya and J. Rossmann,
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Ya. B. Lopatinskiy, On one type of singular integral equations,
[17]
V. G. Maz'ya and J. Rossmann,
[18]
V. G. Maz'ya, The solvability of the Dirichlet problem for a region with a smooth irregular boundary,
[19]
V. G. Maz'ya, The behavior near the boundary of the solution of the Dirichlet problem for an elliptic equation of the second order in divergence form,
[20]
V. G. Maz'ya and B. A. Plamenevskiy,
On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points,
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V. G. Maz'ya and B. A. Plamenevskiy,
Coefficients in the asymptotics of the solutions of an elliptic boundary value problem in a cone,
[23]
V. G. Maz'ya and J. Rossmann,
On a problem of Babu$\breve {\rm{s}}$ka (Stable asymptotics of the solution to the Dirichlet problem for elliptic equations of second order in domains with angular points),
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[1]
Sándor Kelemen, Pavol Quittner.
Boundedness and a priori estimates of solutions
to elliptic systems
with Dirichlet-Neumann boundary conditions.
[2]
Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel.
Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions.
[3]
Claudia Anedda, Giovanni Porru.
Boundary estimates for solutions of weighted semilinear elliptic
equations.
[4] [5]
Dorina Mitrea, Marius Mitrea, Sylvie Monniaux.
The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains.
[6]
Yanni Guo, Genqi Xu, Yansha Guo.
Stabilization of the wave equation with
interior input delay and mixed Neumann-Dirichlet boundary.
[7]
Haitao Yang, Yibin Zhang.
Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data.
[8]
Doyoon Kim, Hongjie Dong, Hong Zhang.
Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces.
[9]
Bastian Gebauer, Nuutti Hyvönen.
Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem.
[10] [11]
Mourad Bellassoued, David Dos Santos Ferreira.
Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map.
[12]
Ping Li, Pablo Raúl Stinga, José L. Torrea.
On weighted mixed-norm Sobolev estimates for some basic parabolic equations.
[13]
Raúl Ferreira, Julio D. Rossi.
Decay estimates for a nonlocal $p-$Laplacian evolution problem
with mixed boundary conditions.
[14]
Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian.
Homogenization of the boundary value for the Dirichlet problem.
[15]
Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez.
Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets.
[16]
Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann.
The Dirichlet to Neumann map - An application to the Stokes problem
in half space.
[17] [18] [19]
Kevin Arfi, Anna Rozanova-Pierrat.
Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by
[20]
E. N. Dancer, Danielle Hilhorst, Shusen Yan.
Peak solutions for the Dirichlet problem of an elliptic system.
2018 Impact Factor: 1.143
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Infinitely many segregated solutions for coupled nonlinear Schrödinger systems
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
$ \left\{\begin{array}{ll} -\Delta u+(1+\delta a(x))u = \mu_1 u^3+\beta uv^2 &\mbox{in }\mathbb{R}^3,\\ -\Delta v+(1+\delta b(x))v = \mu_2 v^3+\beta u^2v &\mbox{in }\mathbb{R}^3,\\ u\to 0,\quad v\to 0, &\mbox{as } |x|\to\infty \end{array}\right. $
$ \mu_1>0 $
$ \mu_2>0 $
$ \beta\in\mathbb{R} $
$ \delta\in\mathbb{R} $
$ a(x) $
$ b(x) $
$ C^\alpha $
$ 0<\alpha<1 $
$ 0<\delta_0<1 $
$ 0<\beta_0<\min\{\mu_1,\mu_2\} $
$ 0<\delta<\delta_0 $
$ 0<\beta<\beta_0 $ Mathematics Subject Classification:Primary: 35J60, 35J75; Secondary: 35J80. Citation:Lushun Wang, Minbo Yang, Yu Zheng. Infinitely many segregated solutions for coupled nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6069-6102. doi: 10.3934/dcds.2019265
References:
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W. Ao, J. Wei and J. Zeng,
An optimal bound on the number of interior peak solutions for the Lin-Ni-Takagi problem,
[3]
A. Bahri and Y. Li,
On a minimax produre for the existence of a positive solution for certain scaler field equation in $ {\Bbb R}^n$,
[4]
T. Bartsch, N. Dancer and Z. Wang,
A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,
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G. Cerami, D. Passaseo and S. Solimini,
Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients,
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E. N. Dancer, J. Wei and T. Weth,
A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,
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N. Hirano,
Multiple existence of nonradial positive solutions for a coupled nonlinear Schrödinger system,
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N. Soave,
On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition,
[18]
Y. Sato and Z. Wang,
On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system,
[19] [20] [21]
show all references
References:
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W. Ao, J. Wei and J. Zeng,
An optimal bound on the number of interior peak solutions for the Lin-Ni-Takagi problem,
[3]
A. Bahri and Y. Li,
On a minimax produre for the existence of a positive solution for certain scaler field equation in $ {\Bbb R}^n$,
[4]
T. Bartsch, N. Dancer and Z. Wang,
A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,
[5] [6]
G. Cerami, D. Passaseo and S. Solimini,
Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients,
[7]
E. N. Dancer, J. Wei and T. Weth,
A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,
[8] [9] [10]
N. Hirano,
Multiple existence of nonradial positive solutions for a coupled nonlinear Schrödinger system,
[11] [12] [13] [14] [15] [16] [17]
N. Soave,
On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition,
[18]
Y. Sato and Z. Wang,
On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system,
[19] [20] [21]
[1] [2]
Zhongwei Tang.
Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions.
[3]
Jing Yang.
Segregated vector Solutions for nonlinear Schrödinger systems with electromagnetic potentials.
[4] [5] [6]
Zhanping Liang, Yuanmin Song, Fuyi Li.
Positive ground state solutions of a quadratically coupled schrödinger system.
[7]
Xu Zhang, Shiwang Ma, Qilin Xie.
Bound state solutions of Schrödinger-Poisson system with critical exponent.
[8]
Silvia Cingolani, Mnica Clapp, Simone Secchi.
Intertwining semiclassical solutions to a Schrödinger-Newton system.
[9]
Haidong Liu, Zhaoli Liu.
Positive solutions of a nonlinear Schrödinger system with nonconstant potentials.
[10] [11] [12] [13] [14]
Mingzheng Sun, Jiabao Su, Leiga Zhao.
Infinitely many solutions for a Schrödinger-Poisson system with
concave and convex nonlinearities.
[15]
Claudianor O. Alves, Minbo Yang.
Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$.
[16]
Weiwei Ao, Liping Wang, Wei Yao.
Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials.
[17] [18]
Sitong Chen, Xianhua Tang.
Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system.
[19]
Sitong Chen, Junping Shi, Xianhua Tang.
Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity.
[20]
2018 Impact Factor: 1.143
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Standing waves for Schrödinger-Poisson system with general nonlinearity
1.
School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China
2.
School of Mathematics and Statistics, Hunan University of Technology and Business, Changsha, 410205 Hunan, China
$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\varepsilon^2\Delta u+V(x)u+\psi u = f(u),\,\, x\in\mathbb{R}^3,\\ -\varepsilon^2\Delta\psi = u^2,\,\,u>0,\,\, u\in H^1(\mathbb{R}^3),\\ \end{array} \right. \end{eqnarray*} $
$ \varepsilon>0 $
$ V $
$ f, $
$ V $
$ u\rightarrow\frac{f(u)}{u^3} $
$ f(u) = |u|^{p-2}u $
$ 3<p<6 $ Mathematics Subject Classification:35J50, 58E05. Citation:Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266
References:
[1] [2] [3] [4] [5]
J. Byeon and L. Jeanjean,
Standing waves for nonlinear Schrödinger equations with a general nonlinearity,
[6]
J. Byeon and L. Jeanjean,
Standing waves with a critical frequency for nonlinear Schrödinger equations Ⅱ,
[7] [8]
S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb{R}^3$,
[9] [10]
S. T. Chen and X. H. Tang,
Geometrically distinct solutions for Klein-Gordon-Maxwell systems with superlinear nonlinearities,
[11]
S. T. Chen and X. H. Tang,
Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity,
[12] [13]
G. M. Figueiredo, N. Ikoma and J. R. Santos Junior,
Existence and concentration result for the Kirchhoff type equations with general nonlinearities,
[14]
D. Gilbarg and N. S. Trudinger,
[15] [16]
X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth,
[17]
Y. He and G. B. Li,
Standing waves for a class of Schrödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents,
[18]
I. Ianni and G. Vaira,
On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,
[19] [20]
I. Ianni and G. Vaira,
Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅰ: necessary condition,
[21] [22] [23] [24]
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case, part 2,
[25] [26]
N. S. Papageorgiou, V. D. Rǎdulescu and D. Repovs,
[27] [28] [29] [30] [31] [32]
D. Ruiz and G. Vaira,
Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of potential,
[33] [34] [35]
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials,
[36]
X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozǎev type for Kirchhoff-type problems with general potentials,
[37]
X. H. Tang, X. Y. Lin and J. S. Yu,
Nontrivial solutions for Schrödinger equation with local super-quadratic conditions,
[38] [39]
J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang,
Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb{R}^{3}$,
[40]
J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang,
Existence of multiple positive solutions for Schrödinger-Poisson systems with critical growth,
[41] [42]
J. Zhang, W. Zhang and X. H. Tang,
Ground state solutions for Hamiltonian elliptic system with inverse square potential,
[43] [44] [45]
show all references
References:
[1] [2] [3] [4] [5]
J. Byeon and L. Jeanjean,
Standing waves for nonlinear Schrödinger equations with a general nonlinearity,
[6]
J. Byeon and L. Jeanjean,
Standing waves with a critical frequency for nonlinear Schrödinger equations Ⅱ,
[7] [8]
S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb{R}^3$,
[9] [10]
S. T. Chen and X. H. Tang,
Geometrically distinct solutions for Klein-Gordon-Maxwell systems with superlinear nonlinearities,
[11]
S. T. Chen and X. H. Tang,
Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity,
[12] [13]
G. M. Figueiredo, N. Ikoma and J. R. Santos Junior,
Existence and concentration result for the Kirchhoff type equations with general nonlinearities,
[14]
D. Gilbarg and N. S. Trudinger,
[15] [16]
X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth,
[17]
Y. He and G. B. Li,
Standing waves for a class of Schrödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents,
[18]
I. Ianni and G. Vaira,
On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,
[19] [20]
I. Ianni and G. Vaira,
Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅰ: necessary condition,
[21] [22] [23] [24]
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case, part 2,
[25] [26]
N. S. Papageorgiou, V. D. Rǎdulescu and D. Repovs,
[27] [28] [29] [30] [31] [32]
D. Ruiz and G. Vaira,
Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of potential,
[33] [34] [35]
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials,
[36]
X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozǎev type for Kirchhoff-type problems with general potentials,
[37]
X. H. Tang, X. Y. Lin and J. S. Yu,
Nontrivial solutions for Schrödinger equation with local super-quadratic conditions,
[38] [39]
J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang,
Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb{R}^{3}$,
[40]
J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang,
Existence of multiple positive solutions for Schrödinger-Poisson systems with critical growth,
[41] [42]
J. Zhang, W. Zhang and X. H. Tang,
Ground state solutions for Hamiltonian elliptic system with inverse square potential,
[43] [44] [45]
[1]
Sitong Chen, Junping Shi, Xianhua Tang.
Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity.
[2] [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] [8]
Mingzheng Sun, Jiabao Su, Leiga Zhao.
Infinitely many solutions for a Schrödinger-Poisson system with
concave and convex nonlinearities.
[9]
Margherita Nolasco.
Breathing modes for the Schrödinger-Poisson system with a multiple--well external potential.
[10]
Claudianor O. Alves, Minbo Yang.
Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$.
[11] [12]
Juntao Sun, Tsung-Fang Wu, Zhaosheng Feng.
Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$.
[13]
Zhengping Wang, Huan-Song Zhou.
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$.
[14]
Sitong Chen, Xianhua Tang.
Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system.
[15] [16] [17]
Miao-Miao Li, Chun-Lei Tang.
Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent.
[18]
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.
[19]
Lun Guo, Wentao Huang, Huifang Jia.
Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $.
[20]
Jaeyoung Byeon, Louis Jeanjean.
Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity.
2018 Impact Factor: 1.143
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Probability Seminar Spring 2019 Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 PM.
If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu
January 31, Oanh Nguyen, Princeton
Title:
Survival and extinction of epidemics on random graphs with general degrees
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$. Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.
Wednesday, February 6 at 4:00pm in Van Vleck 911 , Li-Cheng Tsai, Columbia University
Title:
When particle systems meet PDEs
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..
Title:
Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.
February 14, Timo Seppäläinen, UW-Madison
Title:
Geometry of the corner growth model
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).
February 21, Diane Holcomb, KTH
Title:
On the centered maximum of the Sine beta process Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.
Title: Quantitative homogenization in a balanced random environment
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).
Wednesday, February 27 at 1:10pm Jon Peterson, Purdue
Title:
Functional Limit Laws for Recurrent Excited Random Walks
Abstract:
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.
March 21, Spring Break, No seminar March 28, Shamgar Gurevitch UW-Madison
Title:
Harmonic Analysis on GLn over finite fields, and Random Walks
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the
character ratio:
$$ \text{trace}(\rho(g))/\text{dim}(\rho), $$
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant
rank. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM). April 4, Philip Matchett Wood, UW-Madison
Title:
Outliers in the spectrum for products of independent random matrices
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.
April 11, Eviatar Procaccia, Texas A&M
Stabilization of Diffusion Limited Aggregation in a Wedge.
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.
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It may be helpful to note that the event of interest can be modeled using a discrete random variable $X$: $X = i$ if $i$ of three dices result in the same number, $i = 0, 2, 3$. You are interested in determining $P[X = 2]$ (or perhaps $P[X = 2] + P[X = 3]$).
Taking $P[X = 2]$ for example, it can be computed as follows:\begin{align}P[X = 2] = \frac{\binom{3}{2}\times\binom{6}{1}\times\binom{5}{1}}{6^3} = \frac{15}{36}.\end{align}
The logic goes like this: imagine you are allocating three distinguishable balls into $6$ boxes labeled from $1$ to $6$. Without restriction, there are $6^3$ ways to do so. If it is required that exactly $2$ of three balls are allocated in the same box (hence the remaining ball must be put in another box), you can first determine which two balls will be allocated to the same box ($\binom{3}{2}$ ways), then you are free to choose which one they go ($\binom{6}{1}$ ways), finally, the third ball can go to any one of the remaining $5$ boxes.
In this way, you may enumerate the whole distribution of $X$.
|
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Now showing items 1-10 of 33
The ALICE Transition Radiation Detector: Construction, operation, and performance
(Elsevier, 2018-02)
The Transition Radiation Detector (TRD) was designed and built to enhance the capabilities of the ALICE detector at the Large Hadron Collider (LHC). While aimed at providing electron identification and triggering, the TRD ...
Constraining the magnitude of the Chiral Magnetic Effect with Event Shape Engineering in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(Elsevier, 2018-02)
In ultrarelativistic heavy-ion collisions, the event-by-event variation of the elliptic flow $v_2$ reflects fluctuations in the shape of the initial state of the system. This allows to select events with the same centrality ...
First measurement of jet mass in Pb–Pb and p–Pb collisions at the LHC
(Elsevier, 2018-01)
This letter presents the first measurement of jet mass in Pb-Pb and p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV and 5.02 TeV, respectively. Both the jet energy and the jet mass are expected to be sensitive to jet ...
First measurement of $\Xi_{\rm c}^0$ production in pp collisions at $\mathbf{\sqrt{s}}$ = 7 TeV
(Elsevier, 2018-06)
The production of the charm-strange baryon $\Xi_{\rm c}^0$ is measured for the first time at the LHC via its semileptonic decay into e$^+\Xi^-\nu_{\rm e}$ in pp collisions at $\sqrt{s}=7$ TeV with the ALICE detector. The ...
D-meson azimuthal anisotropy in mid-central Pb-Pb collisions at $\mathbf{\sqrt{s_{\rm NN}}=5.02}$ TeV
(American Physical Society, 2018-03)
The azimuthal anisotropy coefficient $v_2$ of prompt D$^0$, D$^+$, D$^{*+}$ and D$_s^+$ mesons was measured in mid-central (30-50% centrality class) Pb-Pb collisions at a centre-of-mass energy per nucleon pair $\sqrt{s_{\rm ...
Search for collectivity with azimuthal J/$\psi$-hadron correlations in high multiplicity p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 and 8.16 TeV
(Elsevier, 2018-05)
We present a measurement of azimuthal correlations between inclusive J/$\psi$ and charged hadrons in p-Pb collisions recorded with the ALICE detector at the CERN LHC. The J/$\psi$ are reconstructed at forward (p-going, ...
Systematic studies of correlations between different order flow harmonics in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(American Physical Society, 2018-02)
The correlations between event-by-event fluctuations of anisotropic flow harmonic amplitudes have been measured in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV with the ALICE detector at the LHC. The results are ...
$\pi^0$ and $\eta$ meson production in proton-proton collisions at $\sqrt{s}=8$ TeV
(Springer, 2018-03)
An invariant differential cross section measurement of inclusive $\pi^{0}$ and $\eta$ meson production at mid-rapidity in pp collisions at $\sqrt{s}=8$ TeV was carried out by the ALICE experiment at the LHC. The spectra ...
J/$\psi$ production as a function of charged-particle pseudorapidity density in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV
(Elsevier, 2018-01)
We report measurements of the inclusive J/$\psi$ yield and average transverse momentum as a function of charged-particle pseudorapidity density ${\rm d}N_{\rm ch}/{\rm d}\eta$ in p-Pb collisions at $\sqrt{s_{\rm NN}}= 5.02$ ...
Energy dependence and fluctuations of anisotropic flow in Pb-Pb collisions at √sNN=5.02 and 2.76 TeV
(Springer Berlin Heidelberg, 2018-07-16)
Measurements of anisotropic flow coefficients with two- and multi-particle cumulants for inclusive charged particles in Pb-Pb collisions at 𝑠NN‾‾‾‾√=5.02 and 2.76 TeV are reported in the pseudorapidity range |η| < 0.8 ...
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Search
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Search for new resonances in $W\gamma$ and $Z\gamma$ Final States in $pp$ Collisions at $\sqrt{s}=8\,\mathrm{TeV}$ with the ATLAS Detector
(Elsevier, 2014-11-10)
This letter presents a search for new resonances decaying to final states with a vector boson produced in association with a high transverse momentum photon, $V\gamma$, with $V= W(\rightarrow \ell \nu)$ or $Z(\rightarrow ...
Fiducial and differential cross sections of Higgs boson production measured in the four-lepton decay channel in $\boldsymbol{pp}$ collisions at $\boldsymbol{\sqrt{s}}$ = 8 TeV with the ATLAS detector
(Elsevier, 2014-11-10)
Measurements of fiducial and differential cross sections of Higgs boson production in the ${H \rightarrow ZZ ^{*}\rightarrow 4\ell}$ decay channel are presented. The cross sections are determined within a fiducial phase ...
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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 ...
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I post this answer to check my understanding.
Imagine a wavefunction in 1 dimensions with a known energy and momentum it's wavefunction will be:
$$\Psi(x, t) = e^{i(kx-\omega t)} = e^{i(px-E t)/\hbar}$$
With some calculus and algebra you can derive the momentum operator and get this:
$$-i\hbar \partial_x \Psi = p \Psi$$
There $-i\hbar \partial_x$ is the momentum operator (I used $\partial_x$ for sorthand for partial derivation). The $p$ is the momentum we measured: the eigen-value of the operator.
Since we prepared the state with a known momentum, the measurement of the momentum doesn't have any effect on the state.
Now imagine a state that is a superposition of 3 possible momenta, so it's a sum of 3 states for each momentum:
$$\Psi = \Psi_1 + \Psi_2 + \Psi_3$$
The superposition principle allows this. Applying the momentum operator on them, you'll get this:
$$-i\hbar \partial_x \left( \Psi_1 + \Psi_2 + \Psi_3 \right) = p_1\Psi_1 + p_2\Psi_2 + p_3\Psi_3 $$
That means our state have 3 different momenta at the same time, but the measurement must give one of the 3 possible eigenvalues. You can get the probability of collapse to a particular state by calculating the
$$ \langle \Psi_i| \Psi\rangle = \int_{-\infty}^\infty\Psi_i^*(x,t) \Psi(x,t) dx$$
Where the asterisk means the complex conjugate. And on the bra-side there must be one of the eigenstates of the operator (that is a pure plane wave with known momentum).
So to answer your question (partially):
After the measurement the Copenhagen interpretation says that state immediately changes to one of the eigenstates. The many worlds interpretation says there is no such collapse instead all the eigenstates can coexist simultaneously in parallel worlds. If the nature have chosen $p_1$ as the measurement result, you'll know that the state is now $\Psi_1$ which is then renormalized to ensure $\langle \Psi_1 | \Psi_1 \rangle = 1$. This renormalization just a technical step for convenience since the Schrodinger-equation doesn't care if you multiply the wave function with an arbitrary constant number. You can see states as infinite dimensional vectors (you can use dimensional analogy of the finite dimensional vectors). And only the directions of these vectors matter. Not the length.
An operator doesn't change the direction of an eigenstate.
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what important elements can relate to electrical resistance when considering about the resistance effect and force? i find some equations may be concerned,like the Ohm's law,R=V/I.V is voltage and I is currents.Also R=gL/A g is resistivity and L is length, A is cross section also how to measure the resistance in experiments?
closed as unclear what you're asking by Neil_UK, Bruce Abbott, Daniel Grillo, Dmitry Grigoryev, JIm Dearden Aug 30 '16 at 16:00
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
The main four things are the mobility of the charges, the charge density, the average cross section area of the conductor, and the length of the conductor.
The essential idea is that free electrons can exist as a kind of cloud within a material. (The so-called 'electron-gas' theory.) When there is an applied electric field between the two ends, this field
accelerates the cloud of charges in a particular direction. As they accelerate under that force, they eventually collide with an atom (electrons are very much lighter than heavy metallic atoms), where it is presumed that they simply come to an abrupt stop. They then start back up again under the electric field accelerating potentials, accelerating until the next impact happens. On average, there is a certain average velocity called the "drift velocity" and given the time between collisions you can then get the "mean free path" that they achieve before hitting another atom, again, and stopping. The average speed reached is \$v = \mu \mathscr{E}\$, where \$\mathscr{E}\$ is the electric field usually in \$\frac{volts}{meter}\$ and \$\mu\$ is the mobility in \$\frac{meter^2}{volt\cdot second}\$. You can experimentally approach getting the mobility of a material using one of several methods, which include time of flight (using thicker materials) and using an impulse voltage and observing the resulting current impulse response function (thinner materials.) Similar concepts apply to positive charges.
Energy is lost into increasing the vibrational energy of the impacted atoms. This becomes heat.
The reason why length would affect the total resistance (not resistivity) should be pretty obvious, as more length means more impacts and more lost energy. A larger charge density (number of charges within a volume times their charge and then divided by the volume they occupy, or \$\frac{N\cdot q}{L\cdot A}\$) clearly leads to more charges per unit time crossing some chosen cross-section and therefore increases the net current given some electric field. So clearly a larger charge density should reduce the resistance. Finally, the mobility itself is directly related to drift velocity (scaled by the applied electric field) and so better mobility (higher average velocity given a specific electric field) means lower resistance. The resulting equation is something like \$R = \frac{L}{p\mu A}\$, where \$L\$ is the length, \$A\$ is the cross section area, \$p\$ is the charge density, and \$\mu\$ is the mobility of the material.
It's actually a pretty simple idea to apply, when you think about it. The mobility is the tricky measurement to get right. But if you have it, you can figure out a lot.
For example, say you have a bit of silicon that is \$2cm\$ long and has a square cross-section of \$2mm \times 2mm\$. The mobility of silicon is already tested in the lab and is \$\mu = 1300 \frac{cm^2}{V\cdot s}\$. You now impress \$V= 10V\$ between the ends of the bar and measure \$I=80mA\$. You can now get the conductivity as:
\$ \sigma = \frac{L\cdot I}{A\cdot V} = \frac{2cm\cdot 80mA}{4mm^2\cdot 10V} = 40 (\Omega\cdot m)^{-1}\$
And can now compute the drift velocity under a \$10V\$ accelerating field between the ends as,
\$v = \frac{I\cdot \mu}{A\cdot \sigma} = 65 \frac{m}{s} \$
Just think of it as a cloud of electrons with equal spacing between them like a gas, sitting in a volume of material. There will be an effective charge density as a result of that idea. And if you apply an accelerating force (voltage), then there will be an electric field potential set up, measured in \$\frac{volts}{meter}\$, and this will now accelerate the cloud. These charges will periodically smack into atoms along the way, coming to a stop and depositing some equivalent thermal energy, then will accelerate back up to speed again before the next impact. It's a very simple concept that works pretty well.
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Let
$$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$
be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral sequence in homology is of the form
$$E^2_{pq} = H_p(Q;H_q(K;M)) \Longrightarrow H_{p+q}(G;M).$$
All the sources I've consulted construct this via a Grothendieck spectral sequence. For something I am doing, what would be great is if there is a free resolution $R_{\bullet} \rightarrow \mathbb{Z}$ of the trivial $\mathbb{Z}[G]$-module $\mathbb{Z}$ and a filtration $\mathcal{F}_{\bullet} R_{\bullet}$ of $R_{\bullet}$ such that for all $\mathbb{Z}[G]$-modules $M$, the above spectral sequence is the one associated to the filtration
$$\mathcal{G}_k(R_{\bullet} \otimes M) = (\mathcal{F}_k R_{\bullet}) \otimes M$$ of the chain complex $R_{\bullet} \otimes M$ computing $H_p(G;M)$. It would be even better if this filtration resolution was reasonably explicit, though I could get by with something non-explicit if I had to. Does anyone know where I could find this?
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This harmonic oscillator suffers damping. Why is that? I don't
understand the mechanism of this damping.
The electron pair is an oscillating electric dipole, so it radiates energy according to the Larmor formula ${dE \over dt}\sim-|\ddot P|^2$ ($P$ is the magnitude of the polarisation). The total energy of the oscillator (kinetic + potential) transforms into the energy of the produced EM waves, that's why the damping arises: the oscillator's energy decreases.
In order to find the absorption coefficient of the material, we must
introduce the damping. I know how to find the index of refraction, but
without introducing the dumping. So why analysing the dumping is
necessary?
This can be shown by a rigorous mathematical analysis. In short, if you take the dumping into account, then the electric susceptibility $\chi_e$ is a complex quantity, its real part gives you the refractive index $n$, and its imaginary part gives you the absorption coefficient $\alpha$.
Edit. You start by developing the differential equation relating the polarisation $\vec P$ and the macroscopic electric field $\vec E$, this task is not quite obvious. This equation is similar to the forced harmonic oscillations's equation of motion: $${d^2\vec P \over dt^2}+a{d\vec P \over dt}+\omega_0^2 \vec P=b\vec E$$
The damping is represented by the term ${d\vec P \over dt}$. You can interpret this equation as follows: the electric field forces (or induces) the polarisation.
The solution of this equation is trivial, is given by:
$$\vec P= \vec P_0 e^{i\omega t}$$ By plunging this solution into the equation we find the following relation:
$$\vec P=k(\omega)\vec E \tag 1\\$$ Where $k(\omega)$ is some complex parameter depends on the the electric field frequency $\omega$. $k(\omega)$ is complex due to the term ${d\vec P \over dt}$ in the equation. On the other hand, we have this relation:
$$\vec P=\chi_e \epsilon_0 \vec E \tag 2\\$$$\chi_e$ is the electric susceptibility and $\epsilon_0$ is the permittivity of free space. By comparison, between $(1)$ and $(2)$: $$\chi_e={k(\omega) \over \epsilon_0}$$Since $k(\omega)$ is complex, so is $\chi_e$! The final step is writing $\chi_e$ in the algebraic form i.e $\chi_e=\chi_{e,r}+i\chi_{e,i}$, where the subscripts $r$ and $i$ refers to real and imaginary, respectively.
The math is straightforward. As I said before, the real part is related to the refraction index $n$ and the imaginary part is related the absorption coefficient $\alpha$.
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panel-qtet.Rmd
The goal is to estimate the Quantile Treatment Effect on the Treated (QTET) which is given by
\[ F^{-1}_{Y_{1t}|D=1}(\tau) - F^{-1}_{Y_{0t}|D=1}(\tau) \]
for \(\tau \in (0,1)\) and where \(Y_{1t}\) are treated potential outcomes in period \(t\), \(Y_{0t}\) are untreated potential outcomes in period \(t\) and \(D\) indicates whether an individual is a member of the treated group or not.
Thus, the key identification challenge is for the counterfactual distribution of untreated potential outcomes for the treated group \(F_{Y_{0t}|D=1}(y)\) – once we identify this, we can invert it to get the quantiles.
We consider the case with three periods of panel data and where individuals are first treated in the last period – period \(t\). In period \(t-1\), \(Y_{it-1} = Y_{i0t-1}\) and in period \(t-2\), Y_{it-2} = Y_{i0t-2}$ for all individuals; that is, in the first two periods no one is treated so that we observe everyone’s untreated potential outcomes.
Assumption 1 (Distributional Difference in Differences)
\[ \Delta Y_{0t} \perp D \]
This is an extension of the most common mean DID assumption (\(E[\Delta Y_{0t}|D=1] = E[\Delta Y_{0t}|D=0]\) to full independence. But, unlike in the mean DID case, this assumption is not strong enough to point identify the QTET. We also invoke the following additional assumption.
Assumption 2 (Copula Stability Assumption)
\[ C_{\Delta Y_{0t}, Y_{0t-1} | D=1}(u,v) = C_{\Delta Y_{0t-1}, Y_{0t-2} | D=1}(u,v) \]
The CSA says that if, in the periods before treatment, we observe the largest gains in outcomes going to, say, those at the top of the distribution, then in the current period, we would also observe the same thing. Together, the Distributional DID Assumption and the Copula Stability Assumption imply that the counterfactual distribution of untreated potential outcomes for the treated group is identified. It is given by
\[ F_{Y_{0t}|D=1}(y) = E[1\{F^{-1}_{\Delta Y_{t}|D=0}(F_{\Delta Y_{t-1}|D=1}(\Delta Y_{t-1})) + F^{-1}_{Y_{t-1}|D=1}(F^{-1}_{Y_{t-2}|D=1}(Y_{t-2})) \leq y\} | D=1] \]
And then we can invert this to obtain the QTET.
We also can allow the Distributional DID Assumption to hold conditional on covariates. This may be important when the path of outcomes depends on covariates – for example, the path of earnings even in the absence of treatment is likely to depend on education, experience, etc. In this case, first step estimation depends on the propensity score, but is still straightforward. The
panel.qtet contains all the tools needed to do the estimation.
##load the package library(qte)
## Registered S3 methods overwritten by 'ggplot2':## method from ## [.quosures rlang## c.quosures rlang## print.quosures rlang
##load the data data(lalonde) ## Run the panel.qtet method on the observational data with no covariates pq1 <- panel.qtet(re ~ treat, t=1978, tmin1=1975, tmin2=1974, tname="year", x=NULL, data=lalonde.psid.panel, idname="id", se=FALSE, probs=seq(0.1, 0.9, 0.1)) summary(pq1)
## ## Quantile Treatment Effect:## ## tau QTE## 0.1 1987.35## 0.2 -7366.04## 0.3 -7992.15## 0.4 -6597.37## 0.5 -4702.88## 0.6 -2741.80## 0.7 -771.12## 0.8 580.00## 0.9 -250.77## ## Average Treatment Effect: 2326.51
## Run the panel.qtet method on the observational data conditioning on ## age, education, black, hispanic, married, and nodegree. ## The propensity score will be estimated using the default logit method. pq2 <- panel.qtet(re ~ treat, t=1978, tmin1=1975, tmin2=1974, tname="year", xformla=~age + I(age^2) + education + black + hispanic + married + nodegree, data=lalonde.psid.panel, idname="id", se=FALSE, probs=seq(0.1, 0.9, 0.1)) summary(pq2)
## ## Quantile Treatment Effect:## ## tau QTE## 0.1 0.00## 0.2 -2159.20## 0.3 -3378.66## 0.4 -2922.88## 0.5 -1935.37## 0.6 -88.03## 0.7 1514.30## 0.8 2633.73## 0.9 3174.61## ## Average Treatment Effect: 401.65
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From Jacod and Shiryaev's Limit Theorems for Stochastic Processes, we get the following definitions.
Definitions: A process with independent increments (abbreviated PII) $X = (X_t)_{t \geq 0}$ on a stochastic basis $(\Omega, \mathcal{F}, \mathbb{F} = (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$ is a càdlàg adapted real-valued process with $X_0 = 0$ and for all $0 \leq s \leq t < +\infty$, $X_t - X_s$ is independent of $\mathcal{F}_s$. A Lévy process (also called process with independent and stationary increments) on a stochastic basis $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ is a PII $X$ such that the distribution of the increment $X_t - X_s$ depends only on $t-s$, for all $0 \leq s \leq t$.
Lévy processes are ubiquitous in mathematical finance. For example, most models for the return of financial assets (Brownian motion, Kou, Merton, CGMY, etc...) are Lévy processes.
I would like to know if processes with independent increments which are not Lévy (i.e. not stationary) are used in finance. One possible application could be models with seasonal changes in parameters of the distribution. Thanks a lot !
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To directly answer the question: Simplex noise is patented, whereas Perlin noise is not. Other than that, Simplex noise has many advantages that are already mentioned in your question, and apart from the slightly increased implementation difficulty, it is the better algorithm of the two.I believe the reason why many people still pick Perlin noise is simply ...
The interpolation looks fine. The main problem here is that the hash function you're using isn't very good. If I look at just one octave, and visualize the hash result by outputting hash(PT).x, I get something like this:This is supposed to be completely random per grid square, but you can see that it has a lot of diagonal line patterns in it (it almost ...
It's unfortunate that people commonly recommend this. Blending between two (or four, etc.) translated copies of a noise function in that way is a pretty bad idea. Not only is it expensive, it doesn't even produce correct-looking results!On the left is some Perlin noise. On the right is two instances of Perlin noise, stacked and blended left-to-right....
I'd consider just going with 3D noise and evaluating it on the surface of the sphere.For gradient noise which is naturally in the domain of the surface of the sphere, you need a regular pattern of sample points on the surface that have natural connectivity information, with roughly equal area in each cell, so you can interpolate or sum adjacent values. I ...
As usual with numerical methods and samplings, it also depends of your quality threshold of what you consider "isotropic". And of what you would consider as a being or not a "grid-based noise algorithm".For instance Gabor Noise reproduces a target spectrum, for instance blue noise, which in Fourier domain is a simple isotropic ring.Now if you consider ...
The benefit of perlin noise is the overall distribution of frequencies. Since value noise uses simple values that are interpolated, there is a higher chance, that a row of several values only differs a little. The consequence is, that some regions of your picture may contain little changes and some regions a lot of changes.By using gradients you are ...
I would say yes with a small asterisk.When generating a perlin noise texture, using multiple of octaves of noise like you are talking about, the point of adding higher octaves (higher frequency lower amplitude) is to add high frequency details to the noise.When making mipmaps of a texture, the point is to remove high frequency content that would cause ...
TL;DR: 2*1LSB triangular-pdf dithering breaks in edgecases at 0 and 1 due to clamping. A solution is to lerp to a 1bit uniform dither in those edgecases.I am adding a second answer, seeing as this turned out a bit more complicated than I originally thought. It appears this issue has been a "TODO: needs clamping?" in my code since I switched from normalized ...
Perlin noise is just a base block, not very interesting by itself. You don't need to modify it, but to combine and filter it in interesting ways. Look at how to make fractal Brownian motion (fBm) with it for example, which combines octaves based on few parameters to get a richer texture.The question of terrain rendering is a difficult one and a topic of ...
A 2D Fourier transform is performed by first doing a 1D Fourier transform on each row of the image, then taking the result and doing a 1D Fourier transform on each column. Or vice versa; it doesn't matter.Just as a 1D Fourier transform allows you to decompose a function into a sum of (1D) sine waves at various frequencies, a 2D Fourier transform ...
As you've surmised, the transform() function transforms points from one co-ordinate space to another. (There are also vtransform() and ntransform() for transforming direction vectors and normal vectors, respectively.) The string argument names the co-ordinate space to transform into.The Renderman Shading Guidelines have this to say about it:At the ...
Is denoising ALWAYS about doing a low pass filter / blur?No, but this is the most obvious technique. A good denoiser isn't just a filter that runs on the image, but actually performs the reconstruction; i.e. it's a function from random samples to an image, not a function from an image to an image.Or are there other ideas and techniques for removing ...
First of all - a number must not occur twice, that is implied since we're talking about permutations. So filling the table with a simple random(255) function won't work.Secondly, you need to ensure that there are no premature recurrence patterns:Consider the values 1,2,3,4 - the permutation table 4,3,2,1 is not a very good one because of its short cyclic ...
A sine wave remapped to [0, 1] and raised to a power will give you periodic ridges: (Desmos graph)That could be a good place to start. It will make perfectly straight, even ridges; but you could then perturb the X position where the sine is evaluated using low-frequency Perlin noise, which will make the ridges bend and waver while still going mostly along ...
Yep, you've got that right. In Perlin's reference implementation of "improved noise", the noise will be periodic, repeating after 256 units along each axis. It's usually not very noticeable even if you have a large extent of noise visible, since there's no large-scale features for the eye to track.But there's no particular reason it needs to tile after 256 ...
I am not sure I can fully answer your question, but I will add some thoughts and maybe we can arrive at an answer together :)First, the foundation of the question is a bit unclear to me: Why do you consider it desirable to have clean black/white when every other color has noise? The ideal result after dithering is your original signal with entirely uniform ...
Animated noise can be created by using time as an extra dimension. So instead of 2D noise, you'd use 3D noise with time as the z-axis position, like ofNoise(x, y, time).To control the level of detail, you'd use octaves of noise: multiple noise layers with different scales and amplitudes, mixed together. The basic Perlin routines just generate a single ...
Why don't you just use Perlin noise twice on the same grid, or volume? Each with slightly different parameters (a phase shift, or different pseudo-random vectors). In this case both component of your float2 are smoothly defined by a Perlin Noise field. A float2 $v$ could be defined as $v = \{P(u,v), P(u+0.5, v+0.5)\},$ where $P(u,v)$ is the Perlin noise ...
r a similar problem (a tree of combined noise functions, evaluated on the GPU), I found a good method is to generate a shader from the expression tree. Each predefined node corresponds to a single shader function, e.g.float simplexNode(vec3 pos) {// ... implementation of simplex noise}orfloat sumNode(float val1, float val2) {return val1 + val2; ...
It certainly isn't always about low pass filters (see for example here on WP on "Noise Reduction") but you have to keep in mind that in your case the noise will always have a high frequency because you can basically consider each pixel with a independent noise realization. So any way of removing noise in this situation will have a low pass effect.
Since the question was somewhat clarified I will formalize both the question and the answer for future readers.Having a differentiable scalar field $f : \mathbb{R}^4 \rightarrow \mathbb{R}$ we want to find the gradient of the field with respect to $\theta, \phi$ on the 2-manifold defined parametrically by:$$(x(\theta,\phi), y(\theta,\phi) z(\theta,\phi), ...
Worley noise, also known as cellular noise, has the same property. It just as easily implemented as Perlin noise and easily extends to higher dimensions. Thus the slicing of 4D Worley noise will produce a 3D Worley noise. However, it is not necessarily a noise function but rather a texture function, producing cellular-like characteristics. With FBM applied ...
If you can't change the textures, I see 2 possibilities:Use mirrored repeat on the noise textures so they tile seamlessly. (taken from here)Do like Gimp does (or many other alternatives) and create tilable versions of the noise textures. The idea is to blend the left and right, and bottom and top edges together in pairs. Here's how to do it in an image ...
You can use the gradient of the noise/hash which for a function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ would be $n$ dimensional (depending on the application this may not work for you). Another possibility, as Reynolds mentioned is to generate the noise by calling the function multiple times.
First, (1/2) I didn't understand how the hash table is actually used. The workflow I think I understood is the following :For an input (coordinate), we find the both nearest pre-defined coordinates (i.e. : those present as indices in A). It's a multiple of255.The lower one is used to lookup in BWe get a number between [0;255] that we use to ...
I've simplified Mikkel Gjoel's idea of dithering with triangular noise to a simple function that only needs a single RNG call. I've stripped away all unecessary bits so it should be pretty readable and understandable what's going on:// Dithers and quantizes color value c in [0, 1] to the given color depth.// It's expected that rng contains a uniform ...
Perlin noise not good for real planet surface because planet surface is not random. Planet structure is create by geology/physics and interaction between different parts.This video show geology simulator have name PlaTec (have link in text below video):https://www.youtube.com/watch?v=bi4b45tMEPELink have source code at SourceForge web site too.
Here’s a video that talks about a couple different techniques.One idea mentioned in the video is to take the values from 0.0 to 1.0 you get from the noise and only care about ones within a certain range, (mapping the rest to 0.0). Further, you can map the values in that range to different values, following a curve. For instance if you take the values ...
"Downscaling by skipping rows and columns – any example images?"You asked for example images: The following is a sequence of 50 frames from a well known (and hopefully freely available) sequence, subsampled as you suggested.I think you might find simply subsampling/decimating to be a bit too noisy.Are you sure you can't at least do some box filtering?
A basic nearest neighbour scale function would work like so. Here I also use 24:8 fixed point arithmetic to make it more micro friendly.const int originalWidth = 320, originalHeight = 240;int targetWidth = 50, targetHeight = 50;char *pSource = sourceBuffer;char *pDest = destBuffer; // you can use RGB instead of charint yStep = (originalWidth <...
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I have found a definition of phase margin of amplifier system from Texas Instruments application report. This definition looks like this:$$\phi = tan^{-1} (A \beta)$$where \$ A\$ is amplifiers open-loop gain (aka direct gain) and \$ \beta \$ is feedback return signal ratio - or \$ A \beta \$ known as
loop gain. Now, \$ A\beta \$ would typically be a value ranging \$ 1000000 \$ to \$ 10000 \$ (in opamp amplifier systems, where open-loop gain is usually around \$ 120 dB \$).
Such values of \$ A\beta \$ inserted into upper definition of phase margin always equals (approximately) \$ \phi = 90° \$. So, using that equation for definition of phase margin must be definitely wrong, because it is not possible, for amplifier's phase margin to be \$ 90° \$ in all scenarios possible. Unless we would be discussing an example with \$ A\beta < 100 \$, which is very unlikely to happen.
Also, it would seem more logical if phase margin definition equation would be described as a function, dependent on poles of amplifier or \$s\$, damping factor or \$\zeta\$, frequency or \$\omega\$, etc.
I know how to find phase margin (and gain margin) from already drawn Bode plot, but I cannot solve it, using mathematical ways, not graphical.
Can anyone tell me, if this is the actual formula for calculation of phase margin? Or are there more data needed to solve such case? Would "fully defined" transfer function provide enough data for proper calculation of phase margin?
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Let $E$ be a set of finite outer measure and $F$ a collection of closed, bounded intervals that cover $E$ in the sense of Vitali. Show that there is a countable disjoint collection {${I_k}$}$_{k=1}^\infty$ of intervals in $F$ for which
$m$*$[E$~$\bigcup_{k=1}^{\infty}I_k] = 0$.
This is the same as the Vitali Covering Lemma except for each $\epsilon > 0$, $m$*$[E$~$\bigcup_{k=1}^{\infty}I_k] < \epsilon$. In that proof, my book (Royden, 4th) uses
Theorem 11: Let $E$ be any set of real numbers. Then, each of the following four assertions is equivalent to the measurability of E.
i) For each $\epsilon > 0$, there is an open set $O$ containing $E$ for which $m$*$(O$~$E)<\epsilon$.
ii) There is a $G_\delta$ set $G$ containing $E$ for which $m$*$(G$~$E)=0$.
iii), iv) etc.
Would this proof boil down to i) -> ii)?
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The following is quoted from the Mathematical Reviews.
MR0544896 (80j:12002) Reviewed
Bhaskaran, M. Construction of genus field and some applications. J. Number Theory 11 (1979), no. 4, 488–497. 12A35 (12A65)
Let $k$ be a finite algebraic number field and $K$ its Hilbert class field, i.e., $K/k$ is maximally abelian such that every finite prime divisor of $k$ is unramified in $K$. Let ${\bf Q}$ be the rational number field and $A$ its maximal abelian extension. The author calls the intersection $\tilde{K} = K \cap Ak$ the narrow genus field of $k$, and proves that $\tilde{K}$ is obtained as a compositum field of $k$ and $\Omega^{(p)}$'s, i.e., $\tilde{K}=k\prod_{p} \Omega^{(p)}$, where $p$ runs over every positive rational integer and each $\Omega^{(p)}$ is a cyclic extension of ${\bf Q}$ that has a power of the ideal $(p)$ as its conductor.
{Reviewer's remarks: This seems rather odd, for the following reason. Let $g(x)$ be a polynomial of degree 4 with rational integer coefficients such that $g(x) \equiv x^4 +1 \mod 2^m$, where $m$ is a sufficiently large integer; let $\alpha$ be a root of the equation $g(x)=0$ and set $k={\bf Q}(\alpha)$. If we can take $\alpha$ such that $g(x)$ is irreducible over ${\bf Q}$ and the Galois closure $L$ of $k/{\bf Q}$ has Galois group isomorphic with the symmetric group $S_4$, then clearly $k$ cannot have any quadratic subfield and so $k \cap A = {\bf Q}$, and $[k(\sqrt{i})\colon k] = 4$. Clearly, as $m$ is large, the prime ideal $(2)$ of ${\bf Q}$ is completely ramified in $k$ and $k(\sqrt{i})\subset\tilde{K}$. Since $\mathrm{Gal}(k(\sqrt{i})/k) \cong \mathrm{Gal}({\bf Q}(\sqrt{i})/{\bf Q})$ is a Klein four-group, we can easily obtain a contradiction from the author's theorem stated above. Thus, if it is valid, we must have the conclusion that $\mathrm{Gal}(L/{\bf Q})$ is not isomorphic with $S_4$ for every irreducible $g(x)$ as stated above. This is odd.}
Reviewed by K. Masuda
Question: Could anyone explain why $k(\sqrt{i})\subset\tilde{K}$, i.e. why the dyadic prime ideal of $k$ is unramified at the extension $k(\sqrt{i})/k$? Here $i=\sqrt{-1}$.
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The term divergence means a function $D$, which, given two probability distributions $P,Q$, assigns a non-negative real number $D(P,Q)$ such that $D(P,Q) = 0$ iff $P(x)=Q(x) \forall x$.
The relative entropy (or the Kullback-Leibler divergence) $$D_f(P,Q) = \sum_x P(x)\log\left(\frac{P(x)}{Q(x)}\right)$$ is a classical example of a divergence function.
For any convex real function $f$ on $(0,\infty)$ with $f(1)=0$, Csiszar (independently Ali and Silvey) proposed a divergence given by $$D_f(P,Q) = \sum_x Q(x)f\left(\frac{P(x)}{Q(x)}\right).$$
When $f(x)=x\log x$, we get the relative entropy.
When $f(x) = \frac{x^\alpha-1}{\alpha-1}, \alpha >0, \alpha\neq 1$, $$D_f(P,Q) = \frac{1}{\alpha-1}\left(\sum_x P(x)^{\alpha}Q(x)^{1-\alpha}-1\right).$$ Can someone give a reference for the actual name of this divergence?
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This is not a complete answer so far, I will come back completing it when time. Using results and notation from What is the moment generating function of the generalized (multivariate) chi-square distribution?, we can write$$ \| Y \|^2 = Y^T Y$$which we can write as a sum of $n$ independent scaled chisquare random variables (here it is central chisquare since the expectation of $Y$ is zero) $Y^T Y = \sum_1^n \lambda_j U_j^2$ with the following definitions taken from the link above. Use the spectral theorem to write $\Sigma ^{1/2} A \Sigma^{1/2} = P^T \Lambda P$ with $P$ orthogonal and $\Lambda$ diagonal with diagonal elements $\lambda_j$, $U=P Y$ then have independent standard normal components. Then we find the mgf of $Y^T Y$ as $$ M_{Y^T Y}(t) = \exp\left(-\frac12 \sum_1^n \log(1-2\lambda_j t) \right)$$Then one can continue finding the expectation using the method from the link in comments. The expectation of $\sqrt{Y^T Y}$ is given by $$ \DeclareMathOperator{\E}{\mathbb{E}} D^{1/2} M (0) = \Gamma(1/2)^{-1}\int_{-\infty}^0 |z|^{-1/2} M'(z) \; dz$$where $M(t)$ is the mgf above and $'$ denotes differentiation. Using maple we can calculate
M := (t,lambda1,lambda2,lambda3) -> exp( -(1/2)*( log(1-2*t*lambda1)+log(1-2*t*lambda2)+log(1-2*t*lambda3) ) )
which is the case with $n=3$. As an example, if the covariance matrix $\Sigma$ is the equicorrelation matrix with offdiagonal elements $1/2$, that is,$$ \Sigma=\begin{pmatrix} 1 & 1/2 & 1/2 \\ 1/2 & 1 & 1/2 \\ 1/2 & 1/2 & 1 \end{pmatrix}$$then the eigenvalues $\lambda_j$ can be calculated as $2, 1/2, 1/2$. Then we can calculate
int( diff(M(z,2,1/2,1/2),z)/sqrt(abs(z)),z=-infinity..0 )/GAMMA(1/2)
obtaining the result. $$1/3\,{\frac {\sqrt {3}{\rm arctanh} \left(\sqrt {3}/2\right)+6}{\sqrt {\pi}}}$$It is probably better to just use the option
numeric=true, we will not find some simple general symbolic answer. An example with other eigenvalues gives another symbolic form of the result:
int( diff(M(z,5,1,1/2),z)/sqrt(abs(z)),z=-infinity..0 )/GAMMA(1/2)
$$-1/15\,{\frac {\sqrt {5} \left( 10\,i\sqrt {5}{\it EllipticF} \left( 3\,i,1/3 \right) -9\,i\sqrt {5}{\it EllipticE} \left( 3\,i,1/3 \right) -30 \right) }{\sqrt {\pi}}}$$ (I now also suspect that at least one of this results returned by maple is in error, see https://www.mapleprimes.com/questions/223567-Error-In-A-Complicated-Integralcomplex?sq=223567)
Now I will turn to numerical methods, programmed in R. First, one idea is to use the saddlepoint approximation or one of the other methods from the answer here: Generic sum of Gamma random variables combined with the relation $f_{\text{chi}}(x)=2xf_{\text{chi}^2}(x^2)$, but first I will look into use of the R package
CompQuadForm, which implements several methods for the cumulative distribution function, and then numerical differentiation with the package
numDeriv, code follows:
library(CompQuadForm)
library(numDeriv)
make_pchi2 <- function(lambda, h=rep(1, length(lambda)), delta=rep(0, length(lambda)), sigma=0, ...) {
function(q) {
vals <- rep(0, length(q))
warning_indicator <- 0
for (j in seq(along=q)) {
res <- davies(q[j], lambda=lambda, h=h, delta=delta, sigma=sigma, ...)
if (res$ifault > 0) warning_indicator <- warning_indicator+1
vals[j] <- res$Qq
}
if (warning_indicator > 0) warning("warnings reported", warning_indicator)
vals <- 1-vals # converting to cumulative probabilities
return(vals)
}
}
pchi2 <- make_pchi2(c(2, 0.5), c(1, 2))
# Construction a density function via numerical differentiation:
make_dchi <- function(pchi) {
function(x, method="simple") {
side <- ifelse(x==0, +1, NA)
vals <- rep(0, length(x))
for (j in seq(along=x)) {
vals[j] <- grad(pchi, x[j]^2, method=method, side=side[j])
}
vals <- 2*x*vals
return(vals)
}
}
dchi <- make_dchi(pchi2)
plot( function(x) dchi(x, method="Richardson"), from=0, to=7, main="density of generalized chi distribution", xlab="x", ylab="density", col="blue")
which gives this result:
and we can test with:
integrate(function(x) dchi(x, method="Richardson"), lower=0, upper=Inf)
0.9999565 with absolute error < 0.00011
Back to (approximation) for the expected value. I will try to get some approximation for the moment generating function of $Y^T Y $ for large $n$. Start with the expression for the mgf$$ M_{Y^T Y}(t) = \exp\left(-\frac12 \sum_1^n \log(1-2\lambda_j t) \right)$$found above. We concentrate on its logarithm, the cumulant generating function (cgf). Assume that when $n$ increases, then the eigenvalues will approximatly come from some distribution with density $g(\lambda)$on $(0,\infty)$ (in the numerical examples we will just use the uniform distribution on $(0,1)$). Then the sum above divided into $n$ will be a Riemann sum for the integral$$I(t) = \int_0^\infty g(\lambda) \log(1-2\lambda t)\; d\lambda$$ for the uniform distribution example we will find$$ I(t) = \frac{2t\ln(1-2t)-\ln(1-2t)-2t}{2t}$$Then the mgf will be (approximately) $M_n(t)=e^{-\frac{n}2 I(t)}$ and the expected value of $\| Y \|$ is $$ \Gamma(1/2)^{-1} \int_{-\infty}^0 |z|^{-1/2} M_n'(z) \; dz$$This seems to be a difficult integral, some work seems to show that the integrand has a vertical asymptote at zero, for instance maple refuses to do a numerical integration (but uses abnormally long time). So it might be that a better strategy is to use numerical integration using the density approximations above. Nevertheless, this approach seems to indicate that the dependence on $n$ should not be very dependent on the eigenvalue distribution.
Going for numerical integration in R, that also shows to be difficult. First some code:
E <- function(lambda, h=rep(1, length(lambda)), lower, upper, ...) {
pchi2 <- make_pchi2(lambda, h, acc=0.001, lim=50000)
dchi <- make_dchi(pchi2)
val <- integrate(function(x) x*dchi(x), lower=lower, upper=upper, subdivisions=10000L, stop.on.error=FALSE)$value
return(val)
}
Some experimentation with this code, using the integration limits
lower=-Inf, upper=0, shows that for a large number of eigenvalues, the result is 0! The reason is that the integrand is zero "almost everywhere", and the integration routine misses the dump. But plotting the integrand first, and choosing a sensible interval, we can get reasonable results. For the example I am using uniformly distributed eigenvalues between 1 and 5, with varying $n$. So for $n=5000$ the following code:
E(seq(from=1, to=5, length.out=5000), lower=-130, upper=-110)
works. This way we can compute the following results:
n Expected value Approximation
[1,] 5 3.656979 3.890758
[2,] 15 6.581486 6.738991
[3,] 25 8.559964 8.700000
[4,] 35 10.161615 10.293979
[5,] 45 11.544113 11.672275
[6,] 55 12.777101 12.904185
[7,] 65 13.901465 14.028328
[8,] 75 14.941275 15.068842
[9,] 85 15.915722 16.042007
[10,] 95 16.830085 16.959422
[11,] 105 17.697715 17.829694
[12,] 500 38.702620 38.907583
[13,] 1000 54.749000 55.023631
[14,] 5000 122.449200 123.036580
Where the last column is calculated as $1.74 \sqrt{n}$, and is found by regression analysis. We show this as a plot:
The fit is quite good, indicating that a reasonable guess is that the expectation grows with the square root of $n$, with a prefactor which probably depends on the eigenvalue distribution.
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Partial Derivatives Examples 1
We will now look at some more examples of computing partial derivatives for functions of several variables. Be sure to review the Partial Derivatives page before hand! More examples can be found on the Partial Derivatives Examples 2 page.
Before we look at the following examples, it will be important to recall the following differentiation rules for single variable real-valued functions:
Rule for Power Functions:$\frac{d}{dx} x^n = nx^{n-1}$. Rules for Trigonometric Functions:$\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \cos x = - \sin x$, and $\frac{d}{Dx} \tan x = \sec ^2 x$. Rules for Exponential Functions:$\frac{d}{dx} a^x = a^x \ln (a)$ and $\frac{d}{dx} e^x = e^x$. Rules for Logarithmic Functions:$\frac{d}{dx} \log_a (x) = \frac{1}{x \ln a}$ and $\frac{d}{dx} \ln x = \frac{1}{x}$. The Chain Rule:$\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$. The Product Rule:$\frac{d}{dx} (f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$. The Quotient Rule:$\frac{d}{dx} \left ( \frac{f(x)}{g(x)} \right ) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$ provided that $g(x) \neq 0$. Example 1 Let $f(x,y) = x^3 \cos (2xy) - xy^2 e^y$. Find $\frac{\partial}{\partial x} f(x,y)$ and $\frac{\partial}{\partial y} f(x,y)$.
To calculate $\frac{\partial}{\partial x} f(x,y)$ we will need to apply the product rule to the first term, and so:(1)
To calculate $\frac{\partial}{\partial y} f(x,y)$ we will also need to apply the product rule, this time to the second term of the function, and thus:(2)
Example 2 Let $f(x, y) = (1 + xy)^{6}$. Find $\frac{\partial}{\partial x} f(x,y)$ and $\frac{\partial}{\partial y} f(x,y)$.
To calculate $\frac{\partial}{\partial x} f(x,y)$ we will need to apply the chain rule, and so:(3)
To calculate $\frac{\partial}{\partial y} f(x, y)$ we will need to apply the chain rule once again, and so:(4)
Example 3 Let $f(x, y) = \ln (x^2 + y^4 + 1)$. Find $\frac{\partial}{\partial x} f(x,y)$ and $\frac{\partial}{\partial y} f(x,y)$.
To calculate $\frac{\partial}{\partial x} f(x,y)$ we will need to apply the rule for differentiating logarithmic functions:(5)
To calculate $\frac{\partial}{\partial y} f(x,y)$ we will need to apply the rule for differentiating logarithmic functions once again:(6)
Example 4 Let $f(x, y) = \frac{x^2 - y^2}{x^3 + y^3}$. Find $\frac{\partial}{\partial x} f(x,y)$ and $\frac{\partial}{\partial y} f(x,y)$.
To calculate $\frac{\partial}{\partial x} f(x,y)$ we will need to apply the quotient rule:(7)
To calculate $\frac{\partial}{\partial y} f(x,y)$ we will also need to apply the quotient rule:(8)
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In 2-space, a line can algebraically be expressed by simply knowing a point that the line goes through and its slope. This can be expressed in the form $y - y_0 = m(x - x_0)$. In 3-space, a plane can be represented differently. We will still need some point that lies on the plane in 3-space, however, we will now use a value called the
normal that is analogous to that of the slope. Point-Normal Form of a Plane
We must first define what a normal is before we look at the point-normal form of a plane:
Definition: A Normal Vector usually denoted $\vec{n} = (a, b, c)$, is a vector that is perpendicular to a plane $\Pi$. That is $\vec{n} \perp \Pi$.
Suppose that the points $P_0 = (x_0, y_0, z_0)$ and $P = (x, y, z)$ lay on the plane $\Pi$. The vector $\vec{P_0P}$ runs along this plane. Now let $\vec{n} = (a, b, c)$ be a normal of this plane. It thus follows that $\vec{P_0P} \perp \vec{n}$. From the dot product, we know that $\vec{P_0P} \cdot \vec{n} = 0$, which when expanded gives us the point-normal form of a plane:(1)
Definition: Let $\Pi$ be a plane in $\mathbb{R}^3$. Then the Point-Normal Form Equation of the plane $\Pi$ is $0 = \vec{n} \cdot \vec{P_0 P} = (a, b, c) \cdot (x - x_0, y - y_0, z - z_0)$ where $\vec{n}$ is any normal vector to $\Pi$ and $\vec{P_0}$ is any point on $\Pi$.
Often times we expand the last formula out to get $ax - ax_0 + by - by_0 + cz - cz_0 =$, and we let $d = -ax_0 - by_0 - cz_0$ to get $ax + by + cz + d = 0$.
Example 1 Find the equation of a plane that passes through the point $P(-2, 3, 4)$ and is perpendicular to the vector $\vec{n} = (1, 3, -7)$.
We can solve this question by inputting what we know into the formula, that is:(2)
We are done as we have an equation of a plane that satisfies our conditions. Alternatively we could make this equation neater and write it is $x + 3y - 7z + 21 = 0$.
Example 2 Determine the normal vector of the following equation of a plane $2x + 3y -6z + 3 = 0$.
The general equation of a plane is in the form $ax + by + cz + d = 0$. From this we can easily pull the normal to be $\vec{n} = (a, b, c) = (2, 3, -6)$.
Example 3 Determine the point-normal form of a plane that goes through the points $P(1, 4, 2)$, $Q(-10, 4, 3)$, and $R(2, 2, 4)$.
It is important to recognize that we will need both a single point and the normal vector to determine the point-normal form of this line. We already have a point given to us, in fact, we have three! We can use either as it does not matter as long as both lie on the plane (and both do according to the question). Hence, we will need to find a normal vector. We can do this with the use of the
cross product, however, we will need to find two parallel vectors first. We can easily do this with utilization of the points given. Let's use the vector formed from the cross product of $\vec{PQ} = (-11, 0, 1)$ and $\vec{PR} = (1, -2, 2)$
Thus we can now input our normal and one of the three points into our point-normal form a line to obtain $2(x - 1) + 23(x - 4) + 22(x - 2) = 0$.
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Recently I have seen that the Einstein-Hilbert action (eventually with the Gibbon-Hawkings boundary term) is not all in the description of spacetime. There can exist also various additional terms in the action that give information about the topology of the spacetime manifold. These terms are:
- A Pontryagin term ($S_{Pont} = \int_\Omega \epsilon_{abcd} R^{ab} \wedge R^{cd}$ with curvature tensor $R^{ab}$ and the spacetime manifold $\Omega$)
- An Euler term that gives the Euler characteristic
- A Nieh-Yan term related to the spacetime torsion
A complete description of gravity is given by
$S = S_{EinstHilb} + S_{Pont} + S_{Euler} + S_{Nieh-Yan} + S_{gaugefixing} + S_{matter}$
(last term is dependent on the choice of gauge) and the partition function has the form
$Z = \int D[\phi] \int D[e_I^a] \int D[\omega_{IJ}^a] e^{iS}$
with matter fields $\phi$, tetrad $e_I^a$ and spin connection $\omega_{IJ}^a$. I know that the partition function will be UV divergent in the case when Einstein-Hilbert term (for classical General Relativity) is present. To resolve this trouble I assume that the Einstein-Hilbert action will not be quantized such that I will have a semiclassical theory; gravitational fields can be assumed to be classical that obey the equation
$R_{IJ}-\frac{1}{2}R g_{IJ} = 8 \pi G T_{IJ}$.
Gauge fixing I neglect also for simplification.
But all other topological terms I can quantize, since these are simply topological invariants; only numbers. I have seen e.g. in String theory that an expansion over all possible topological stuctures can be performed. Now I decompose gravitational fields in form of the following:
$e_I^a = e_I^a|_0 + e'_I^a$
$\omega_{IJ}^a = \omega_{IJ}^a|_0 + \omega'_{IJ}^a$
Here, the subscript 0 denotes the field that has no contribution to gravity, but induces nonzero topological invariants, depending on the manifold structure that I have. Primed fields are fields obtained by classical gravity, these do not change topological terms. Finally, I have the following path integral:
$Z = \sum_{Topologies}g_1^{n_{Pont}}g_2^{n_{Euler}}g_3^{n_{Nieh-Yan}} \int D[\phi] \int D[e'_I^a] \int D[\omega'_{IJ}^a] \delta(\delta_{e^a,\omega^a}S_{EinstHilb})e^{iS_{matter}}|_{n_{Pont},n_{Euler},n_{Nieh-Yan}}$.
The couplings $g_1,g_2,g_3$ are couplings corresponding to the topological structure and $n_{Pont},n_{Euler},n_{Nieh-Yan}$ characterizes the manifold topology. Is this path integral correct? Due to coordinate and Lorentz invariance I can fix gauge such that the location of topological features does not affect path integral; is a aum over topologies sufficient or I must integrate over fields to represent different topologies?
Now if I transform spacetime manifold integrals into momentum space I will get e.g.
$\int_{\Omega} d^4x \psi^*(x) \psi(x) = \int_{\mathbb{R}}d^4x 1_\Omega(x) \psi^*(x) \psi(x) = \frac{1}{(2 \pi)^12} \int d^4x \int d^4K 1_\Omega^{fouriertransformed}(K) e^{iKx} \int d^4k e^{-ikx} \phi^*(k) \int d^4k' e^{ik'x} \phi(k') = \frac{1}{(2 \pi)^8} \int d^4k \int d^4k' \int d^4 K \delta(k-k'+K) 1_\Omega^{fouriertransformed}(K) \phi^*(k) \phi(k') $
meaning that there will be excess of energy and momentum due to eventual boundaries of the spacetime manifold $\Omega$; this excess occurs if the distribution $1_\Omega$ is somewhere zero because of internal boundaries in spacetime.
Next Question: This energy-momentum excesses represent Heisenberg's uncertainty, where some spacetime regions that are "cut out" of spacetime define a given length and time interval. A minimum energy and momentum uncertainty arises. What would we observe if spacetime would have microscopic holes? Would we observe that some particles seem to pop out from nothingless (physically, energy is borrowed to create particle-antiparticle pairs) that will vanish a shorter time later near the holes of spacetime?
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In asset allocation, you usually send reports to your clients where you will report the volatility of its portfolio. Assuming you only have monthly returns, you will compute volatility over a considered period of $n$ months with the classic sample volatility estimate:
$$\sigma_s=\sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i-\bar{x})^2}, \quad \bar{x}=\frac{1}{n}\sum_{i=1}^N x_i$$
The result you will get for $\sigma_s$ depends very much on the number of months $n$ you decide to take into account. Hence, I think that this measure can become pretty abstract to unsophisticated investors and they might find it pretty different from the "feeling" the have of the volatility of their portfolio, what I call the "experienced" volatility.
My question is, has there been any research aiming to find out which $n$ is the best to make sure that the measure is closest to the experienced volatility?
I believe this is very much linked to behavioral finance and it might very well depend on the risk-aversion or the sophistication of the investor.
I tried to answer my question by proposing the following:
I assume that investors will be biased by the most recent events in the market; if they have been with you 10 years, they will remember 2008-2012 and would have forgot the quiet and lucrative early years. Hence, I took $n=36$, 3 years, as I thought is was taking a recent enough sample, yet had enough data ($n>30$, intuitively... it might be arguable) to measure a properly the estimate $\sigma_s$.
Does this make sense?
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Define a group homomorphism $\psi: G\to \Aut(N)$ by $\psi(g)(n)=gng^{-1}$ for all $g\in G$ and $n\in N$.We need to check:
The map $\psi(g)$ is an automorphism of $N$ for each $g\in G$.
The map $\psi$ is in fact a group homomorphism from $G$ to $\Aut(N)$.
The assumption that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime implies that $G=\ker(\psi)$.
This implies that $N$ is in the center of $G$.
Proof.
We define a group homomorphism $\psi: G \to \Aut(N)$ as follows.For each $g\in G$, we first define an automorphism $\psi(g)$ of $N$.Define $\psi(g): N \to N$ by\[\psi(g)(n)=gng^{-1}.\]
Note that since $N$ is a normal subgroup of $G$, the output $\psi(g)(n)=gng^{-1}$ actually lies in $N$.
We prove that so defined $\psi(g)$ is a group homomorphism from $N$ to $N$ for each fixed $g\in G$.For $n_1, n_2 \in N$, we have\begin{align*}\psi(g)(n_1n_2)&=g(n_1n_2)g^{-1} && \text{by definition of $\psi(g)$}\\&=gn_1g^{-1}gn_2g^{-1} && \text{by inserting $e=g^{-1}g$}\\&=\psi(g)(n_1) \psi(g)(n_2) && \text{by definition of $\psi(g)$}.\end{align*}It follows that $\psi(g)$ is a group homomorphism, and hence $\psi(g)\in \Aut(N)$.
We have defined a map $\psi:G\to \Aut(N)$. We now prove that $\psi$ is a group homomorphism.For any $g_1, g_2$, and $n\in N$, we have\begin{align*}\psi(g_1 g_2)(n)&=(g_1g_2)n(g_1 g_2)^{-1}\\&=g_1 g_2 n g_2^{-1} g_1^{-1}\\&=g_1 \psi(g_2)(n) g_1^{-1}\\&=\psi(g_1)\psi(g_2)(n).\end{align*}
Thus, $\psi: G\to \Aut(N)$ is a group homomorphism.By the first isomorphism theorem, we have\[G/\ker(\psi)\cong \im(\psi)< \Aut(N). \tag{*}\]Note that if $g\in N$, then $\psi(g)(n)=gng^{-1}=n$ since $N$ is abelian.It yields that the subgroup $N$ is in the kernel $\ker(\psi)$.
Then by the third isomorphism theorem, we have\begin{align*}G/\ker(\psi) \cong (G/N)/(\ker(\psi)/N). \tag{**}\end{align*}
It follows from (*) and (**) that the order of $G/\ker(\psi)$ divides both the order of $\Aut(N)$ and the order of $G/N$. Since the orders of the latter two groups are relatively prime by assumption, the order of $G/\ker(\psi)$ must be $1$. Thus the quotient group is trivial and we have\[G=\ker(\psi).\]
This means that for any $g\in G$, the automorphism $\psi(g)$ is the identity automorphism of $N$.Thus, for any $g\in G$ and $n\in N$, we have $\psi(g)(n)=n$, and thus $gng^{-1}=n$.As a result, the subgroup $N$ is contained in the center of $G$.
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Group Homomorphisms From Group of Order 21 to Group of Order 49Let $G$ be a finite group of order $21$ and let $K$ be a finite group of order $49$.Suppose that $G$ does not have a normal subgroup of order $3$.Then determine all group homomorphisms from $G$ to $K$.Proof.Let $e$ be the identity element of the group […]
Basic Properties of Characteristic GroupsDefinition (automorphism).An isomorphism from a group $G$ to itself is called an automorphism of $G$.The set of all automorphism is denoted by $\Aut(G)$.Definition (characteristic subgroup).A subgroup $H$ of a group $G$ is called characteristic in $G$ if for any $\phi […]
Isomorphism Criterion of Semidirect Product of GroupsLet $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism.The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation\[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),\]where $a_i […]
Group Homomorphism, Preimage, and Product of GroupsLet $G, G'$ be groups and let $f:G \to G'$ be a group homomorphism.Put $N=\ker(f)$. Then show that we have\[f^{-1}(f(H))=HN.\]Proof.$(\subset)$ Take an arbitrary element $g\in f^{-1}(f(H))$. Then we have $f(g)\in f(H)$.It follows that there exists $h\in H$ […]
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Metric Spaces Review
Metric Spaces Review
We will now review some of the definitions of a metric and a metric space and review some examples of metric spaces that we saw recently.
On the Metric Spacespage we first defined a special type of function on a set $M$ known as a Metricwhich is a function $d : M \times M \to [0, \infty)$ which takes each pair $(x, y)$ and maps it to some nonnegative real number called the Distancebetween $x$ and $y$. We say such a function $d$ is a metric of $M$ if it satisfies the following three properties: The first property is that for all $x, y \in M$ we must have that $d$ is symmetric:
\begin{align} \quad d(x, y) = d(y, x) \end{align}
The second property that $d$ must have is that the distance from $x$ to $y$ equals $0$ if and only if $x$ and $y$ are the same point, that is:
\begin{align} \quad d(x, y) = 0 \quad \Leftrightarrow x = y \end{align}
The third property is known as the triangle inequality. It says that if $z$ is any intermediary point, then the distance from $x$ to $y$ must be less than or equal to the distance from $x$ to $z$ plus the distance from $z$ to $y$. In other words, the distance of any non-direct "path" from $x$ to $y$ is always greater than or equal to the distance of the direct path from $x$ to $y$. So, for all $x, y, z \in M$ we must have that:
\begin{align} \quad d(x, y) \leq d(x, z) + d(z, y) \end{align}
If $d$ is a metric as summarized above, then the set $M$ with a metric $d$ defined on $M$ is called a Metric Spaceand is denoted as the pair $(M, d)$, or sometimes simply as $M$ for brevity, and if $S \subseteq M$ then $(S, d)$ is said to be a Metric Subspaceof $(M, d)$ (with the metric $d$ restricted to elements in $S$). On the Some Metrics Defined on Euclidean Spacepage we looked at an important metric defined for all $\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n$ by:
\begin{align} \quad d(\mathbf{x}, \mathbf{y}) = \sum_{k=1}^{n} \mid x_k - y_k \mid \end{align}
On The Chebyshev Metricpage we looked at another important metric defined for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$ known as the Chebyshev Metricgiven by:
\begin{align} \quad d(\mathbf{x}, \mathbf{y}) = d(\mathbf{x}, \mathbf{y}) = \max_{1 \leq k \leq n} \{ \mid x_k - y_k \mid \} \end{align}
On The Discrete Metricpage we looked at a more abstract metric known as the Discrete Metricdefined for all $x, y \in M$ ($M$ an arbitrary set) by:
\begin{align} \quad d(x, y) = \left\{\begin{matrix} 0 & \mathrm{if} \: x = y\\ 1 & \mathrm{if} \: x \neq y \end{matrix}\right. \end{align}
We looked at another abstract metric on the The Standard Bounded Metricknown as the Standard Bounded Metricdefined for all $x, y \in M$ and with respect to any other metric $d$ by:
\begin{align} \quad \bar{d}(x, y) = \mathrm{min} \{ 1, d(x, y) \} \end{align}
After looked at all of those examples we then looked at a generalization of the triangle inequality property of a metric on The Polygonal Inequality for Metric Spacespage known as the Polygonal Propertywhich says that if $(M, d)$ is a metric space and $x_1, x_2, ..., x_m \in M$ then:
\begin{align} \quad d(x_1, x_m) \leq \sum_{k=1}^{m-1} d(x_k, x_{k+1}) \end{align}
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Partial Derivatives Examples 2
We will now look at even more examples of computing partial derivatives for functions of several variables. Be sure to review the Partial Derivatives page beforehand! More examples can be found on the Partial Derivatives Examples 1 page.
Before we look at the following examples, it will be important to recall the following differentiation rules for single variable real-valued functions:
Rule for Power Functions:$\frac{d}{dx} x^n = nx^{n-1}$. Rules for Trigonometric Functions:$\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \cos x = - \sin x$, and $\frac{d}{Dx} \tan x = \sec ^2 x$. Rules for Exponential Functions:$\frac{d}{dx} a^x = a^x \ln (a)$ and $\frac{d}{dx} e^x = e^x$. Rules for Logarithmic Functions:$\frac{d}{dx} \log_a (x) = \frac{1}{x \ln a}$ and $\frac{d}{dx} \ln x = \frac{1}{x}$. The Chain Rule:$\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$. The Product Rule:$\frac{d}{dx} (f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$. The Quotient Rule:$\frac{d}{dx} \left ( \frac{f(x)}{g(x)} \right ) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$ provided that $g(x) \neq 0$. Example 1 Let $z = \ln (x^5e^x 2^y \cos y )$. Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.
To compute $\frac{\partial z}{\partial x}$ we will use the rule for logarithmic differentiation and the product rule:(1)
To compute $\frac{\partial z}{\partial y}$ we will also use the rule for logarithmic differentiation and the product rule:(2)
Example 2 Let $w = 2 \sin (xy) \cos (yz) \tan (xz)$. Find $\frac{\partial w}{\partial x}$, $\frac{\partial w}{\partial y}$, and $\frac{\partial w}{\partial z}$.
To compute $\frac{\partial x}{\partial w}$ we will need to use the product rule:(3)
To compute $\frac{\partial y}{\partial w}$ we will also need to use the product rule:(4)
And lastly, to compute $\frac{\partial w}{\partial z}$ we will once again need to use the product rule:(5)
Example 3 Let $w = \cos ( \sin (x^2y^4z^5)) + \ln (xyz)$. Find $\frac{\partial w}{\partial x}$, $\frac{\partial w}{\partial y}$, and $\frac{\partial w}{\partial z}$.
Applying the chain rule we obtain that:(6)
Example 4 Let $w = \frac{x^2y + y\cos z}{e^x + y^2e^z}$. Find $\frac{\partial w}{\partial x}$, $\frac{\partial w}{\partial y}$, and $\frac{\partial w}{\partial z}$.
Applying the quotient rule and we get that:(9)
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Quotient Normed Linear Spaces
Let $(X, \| \cdot \|_X)$ be a normed linear space and let $M$ be a linear subspace of $X$. We ultimately want to turn $X / M$ into a normed linear space where:(1)
Where $x + M = \{ x + m : m \in M \}$.
We define addition of $x + M, y + M \in X/M$ by:(2)
And for all $\alpha \in \mathbb{R}$ we define scalar multiplication by:(3)
The following proposition tells us that we can always define a particular seminorm on $X / M$.
Proposition 1: Let $(X, \| \cdot \|_X)$ be a normed linear space and let $M$ be a linear subspace of $X$. Define $\| \cdot \| : X / M \to [0, \infty)$ for all $x + M \in X/M$ by $\| x + M \| = \inf \{ \| x + m \|_X : m \in M \}$. Then $\| \cdot \|$ is a seminorm on $X / M$. Proof:There are two things to show. First let $\alpha \in \mathbb{R}$ and let $x + M \in X/M$. If $\alpha = 0$ then $\| \alpha (x + M) \| = \| 0 + M \| = \inf \{ \| m \|_X : m \in M \} = 0$ since $M$ is a subspace of $X$ and so $0 \in M$. If $\alpha \neq 0$ then since $\frac{1}{\alpha}m \in M$ we have that: Now let $x + M, y + M \in X / M$. Then: So $\| \cdot \|$ is a seminorm on $X / M$. $\blacksquare$
Note that in general, $\| \cdot \|$ need not be a norm on $X/M$, that is, it may be that $\| x + M \| = 0$ does not imply that $x + M = 0 + M$. If $M$ is a closed subspace of $X$ then it is.
Proposition 2: If $(X, \| \cdot \|_X)$ is a normed linear space and let $M$ be a closed linear subspace of $X$. Then $\| \cdot \| : X/M \to [0, \infty)$ defined for all $x + M \in X/M$ by $\| x + M \| = \inf \{ \| x + m \|_X : m \in M \}$ is a norm on $X/M$. Proof:Proposition 1 establishes that $\| \cdot \|$ is a seminorm on $X/M$. So all that remains to show is that $\| x + M \| = 0$ if and only if $x + M = 0 + M$. First suppose that $\| x + M \| = 0$. Then $\inf \{ \| x + m \|_X : m \in M \} = 0$. Observe that $\| x + m \|_X = 0$ if and only if $x + m = 0$, that is, $m = -x$. So there must exist a sequence $(m_n) \subset M$ such that $(m_n)$ converges to $-x$. Since $M$ is closed, $-x \in M$. Since $M$ is a subspace of $X$ we have that $x \in M$. But then $x + M = 0 + M$. Conversely, suppose that $x + M = 0 + M$. Then $\| x + M \| = \| 0 + M \| = \inf \{ \| m \|_X : m \in M \}$. Since $M$ is a subspace of $X$ we have that $0 \in M$ and so $\| 0 + M \| = 0$. $\blacksquare$
If $M$ is not closed, then $M$ does not contain all of its accumulation points. Let $t$ be an accumulation point of $M$. Note that $t \neq 0$ since $0 \in M$ by the definition of $M$ being a subspace. Let $(m_n)$ be a sequence in $M$ which converges to $t$. Then $\inf \{ \| t - m \|_X : m \in M \} = 0$. So $\| t + M \| = \| 0 + M \|$. But $t + M \neq 0 + M = M$ since $t \not \in M$.
Definition: Let $(X, \| \cdot \|_X)$ be a normed linear space and let $M$ be a linear subspace of $X$. The Quotient Map from $X$ to $X/M$ is defined to be the map $Q : X \to X/M$ defined for all $x \in X$ by $Q(x) = x + M$. Sometimes the map $Q$ defined above is called the Natural Map from $X$ to $X/M$.
Proposition 3: Let $(X, \| \cdot \|_X)$ be a normed linear space and let $M$ be a closed linear subspace of $X$. Then $\| Q(x) \| \leq \| x \|$ for all $x \in X$, and $Q$ is continuous on $X$. Proof:Let $x \in X$. Then: Since $0 \in M$. For continuity, let $x_0 \in X$ and let $\epsilon > 0$ be given. Suppose that $\delta = \epsilon$. If $\| x_0 - x \|_X < \delta = \epsilon$ then from above we have that: So $Q$ is continuous at $x_0$ and hence $Q$ is continuous on all of $X$. $\blacksquare$
Theorem 4: Let $(X, \| \cdot \|_X)$ be a normed linear space and let $M$ be a closed linear subspace of $X$. If $X$ is a Banach space then $X/M$ is a Banach space. Proof:Let $(x_n + M)$ be a Cauchy sequence in $X/M$. Then there must exist a subsequence $(x_{n_k}+ M)$ of $(x_n + M)$ such that: Set $y_1 = 0$. Consider the value $\| (x_{n_1} - x_{n_2}) + M \| + \frac{1}{2}$. Note that $\| (x_{n_1} - x_{n_2}) + M \| \leq \| (x_{n_1} - x_{n_2}) + M \| + \frac{1}{2}$ clearly, and so there exists a $y_2 \in M$ for which: Now consider the value $\| (x_{n_2} - x_{n_2}) + M \| + \frac{1}{4}$. We can choose a $y_3 \in M$ for which: We continue on in this manner to obtain a sequence $(y_k)$ in $M$ for which: Therefore $(x_{n_k} + y_k)$ is a Cauchy sequence in $X$, and since $X$ is a Banach space there exists an $x \in X$ such that $(x_{n_k} + y_k)$ converges to $x$. Since $Q$ is continuous by the previous proposition, we have that $Q(x_{n_k} + y_k)$ converges to $Q(x)$, that is, $(x_{n_k} + y_k) + M$ converges to $x + M$. Since $y_k \in M$ for each $k$ we have that $(x_{n_k} + M)$ converges to $x_0 + M \in X/M$. So every Cauchy sequence in $X/M$ converges in $X/M$. $\blacksquare$
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This reduction is for average constrained CSMC to importance weighted binary classification. It is nearly identical to the filter tree reduction of unconstrained CSMC, with the addition that infeasible choices automatically lose any tournament they enter (if both entrants are infeasible, one is chosen arbitrarily). Because of this rule the underlying binary classifier is only invoked for distinctions between feasible choices and the regret analysis is essentially the same.
Algorithm:Forfeit Filter Tree Train
Data:Constrained CSMC training data set $S$. Input:Importance-weighted binary classification routine $\mbox{Learn}$. Input:A binary tree $T$ over the labels with internal nodes $\Lambda (T)$. Result:Trained classifiers $\{\Psi_n | n \in \Lambda (T) \}$. For each $n \in \Lambda (T)$ from leaves to roots: $S_n = \emptyset$. For each example $(x, \omega, c) \in S$: Let $a$ and $b$ be the two classes input to $n$ (the predictions of the left and right subtrees on input $(x, \omega)$ respectively). If $a \in \omega$, predict $b$ for the purposes of constructing training input for parent node (``$a$ forfeits''); else if $b \in \omega$, predict $a$ for the purposes of constructing training input for parent node (``$b$ forfeits''); else (when $a \not \in \omega$ and $b \not \in \omega$), $S_n \leftarrow S_n \cup \{ (x, 1_{c_a < c_b}, | c_a - c_b | ) \}$; Let $\Psi_n = \mbox{Learn} (S_n)$. Return $\{\Psi_n | n \in \Lambda (T) \}$.
Algorithm:Forfeit Filter Tree Test
Input:A binary tree $T$ over the labels with internal nodes $\Lambda (T)$. Input:Trained classifiers $\{\Psi_n | n \in \Lambda (T) \}$. Input:Instance realization $(x, \omega)$. Result:Predicted label $k$. Let $n$ be the root node. Repeat until $n$ is a leaf node: If all the labels of the leaves in the left-subtree of $n$ are in $\omega$, traverse to the right child; else if all the labels of the leaves in the right-subtree of $n$ are in $\omega$, traverse to the left child; else if $\Psi_n (x) = 1$, traverse to the left child; else (when $\Psi_n (x) = 0$ and at least one label in each subtree is not in $\omega$), traverse to the right child. Return leaf label $k$. Regret AnalysisThe regret analysis for the forfeit filter tree is very similar to the regret analysis for the filter tree, with additional arguments for forfeiture cases.
The average constrained CSMC problem is characterized by a distribution $D = D_x \times D_{\omega|x} \times D_{c|\omega,x}$, where $c: K \to \mathbf{R}$ takes values in the extended reals $\mathbf{R} = \mathbb{R} \cup \{ \infty \}$, and the components of $c$ which are $\infty$-valued for a particular instance are revealed as part of the problem instance via $\omega \in \mathcal{P} (K)$ (i.e., $\omega$ is a subset of $K$). The regret of a particular classifier $h: X \times \mathcal{P} (K) \to K$ is given by \[ r_{av} (h) = E_{(x, \omega) \sim D_x \times D_{\omega|x}} \left[ E_{c \sim D_{c|\omega,x}} \left[ c (h (x, \omega)) - \min_{k \in K}\, E_{c \sim D_{c|\omega,x}} \left[ c (k) \right] \right] \right]. \] Let $\Psi = (T, \{\Psi_n | n \in \Lambda (T) \})$ denote a particular forfeit filter tree (i.e., a choice of a binary tree and a particular set of node classifiers), and let $h^\Psi$ denote the classifier that results from the forfeit filter tree.
The regret analysis leverages an induced importance-weighted binary distribution $D^\prime (\Psi)$ over triples $(x^\prime, y, w)$ defined as follows:
Draw $(x, \omega, c)$ from $D$. Draw $n$ uniform over the internal nodes of the binary tree. Let $x^\prime = (x, n)$. Let $a$ and $b$ be the two classes input to $n$ (the predictions of the left and right subtrees on input $x$ respectively). If $a \in \omega$, create importance-weighted binary example $(x^\prime, 0, 0)$; else if $b \in \omega$, create importance-weighted binary example $(x^\prime, 1, 0)$; else (when $a \not \in \omega$ and $b \not \in \omega$), create importance-weighted binary example $(x^\prime, 1_{c_a < c_b}, | c_a - c_b |)$.
In other words, there is no importance-weighted regret at node $n$ if either the left or the right subtree at $n$ entirely consists of labels that are infeasible for this instance, or if the classifier makes the correct decision.
Theorem:Regret Bound
This bound is essentially the same as for the filter tree reduction of unconstrained CSMC to importance-weighted classification, so ultimately average constrained CSMC is no more difficult than unconstrained CSMC (in that both reduce to binary classification with the same regret bound). Ultimately this is not surprising, since average constrained CSMC is essentially unconstrained CSMC augmented with a supreme penalty value ($\infty$) and additional instance information about where the penalties are. This is fortunate, however, because average constrained CSMC is a useful primitive for encoding constraints into reductions.
For all average constrained CSMC distributions $D$ and all forfeit filter trees $\Psi$, \[ r_{av} (h^\Psi) \leq (|K| - 1) q (\Psi), \] where $q (\Psi)$ is the importance-weighted binary regret on the induced subproblem.
Proof:See Appendix. Minimax CounterexampleTo show the forfeit trick won't work on minimax constrained CSMC, consider a null feature space problem with 3 classes and deterministic cost vector $c = (x, y, z)$ with $x < z < y$. Further suppose our binary tree first plays 1 vs. 2, then plays (winner of 1 vs. 2) vs. 3. Training with $\omega = \emptyset$ the forfeit filter tree will learn to take both left branches from the root and choose 1, leading to zero CSMC regret. If an adversary then comes along and sets $\omega = \{ 1 \}$ at test time, the forfeit filter tree will choose 2 creating regret since 3 is the best choice once 1 is infeasible.
If this sounds totally unfair, keep in mind that a regression reduction with zero regret can handle arbitrarily imposed constraints at test time without incurring additional regret. This is because regression not only determines the best class, but actually the order of all the classes by cost. (This doesn't mean reduction to regression is better; actually the converse, it shows regression is solving a harder problem than is typically necessary).
This also suggests that minimax constrained CSMC, unlike average constrained CSMC,
isharder than unconstrained CSMC.
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We first claim that there is a unique Sylow $11$-subgroup of $G$.Let $n_{11}$ be the number of Sylow $11$-subgroups in $G$.
By Sylow’s theorem, we know that\begin{align*}&n_{11}\equiv 1 \pmod{11}\\&n_{11}|21.\end{align*}By the first condition, $n_{11}=1, 12, 23 \cdots$ and only $n_{11}=1$ divides $21$.Thus, we have $n_{11}=1$ and there is only one Sylow $11$-subgroup $P_{11}$ in $G$, and hence it is normal in $G$.
Now we consider the action of $G$ on the normal subgroup $P_{11}$ given by conjugation.The action induces the permutation representation homomorphism\[\psi:G\to \Aut(P_{11}),\]where $\Aut(P_{11})$ is the automorphism group of $P_{11}$.
Note that $P_{11}$ is a group of order $11$, hence it is isomorphic to the cyclic group $\Zmod{11}$.Recall that\[\Aut(\Zmod{11})\cong (\Zmod{11})^{\times}\cong \Zmod{10}.\]
The first isomorphism theorem gives\begin{align*}G/\ker(\psi) \cong \im(\psi) < \Aut(P_{11})\cong \Zmod{10}.\end{align*}
Hence the order of $G/\ker(\psi)$ must be a divisor of $10$.Since $|G|=231=3\cdot 7 \cdot 11$, the only possible way for this is $|G/\ker(\psi)|=1$ and thus $\ker(\psi)=G$.
This implies that for any $g\in G$, the automorphism $\psi(g): P_{11}\to P_{11}$ given by $h\mapsto ghg^{-1}$ is the identity map.Thus, we have $ghg^{-1}=h$ for all $g\in G$ and $h\in H$.It yields that $P_{11}$ is in the center $Z(G)$ of $G$.
Every Group of Order 72 is Not a Simple GroupProve that every finite group of order $72$ is not a simple group.Definition.A group $G$ is said to be simple if the only normal subgroups of $G$ are the trivial group $\{e\}$ or $G$ itself.Hint.Let $G$ be a group of order $72$.Use the Sylow's theorem and determine […]
Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.Hint.Use Sylow's theorem.(See Sylow’s Theorem (Summary) for a review of Sylow's theorem.)Recall that if there is a unique Sylow $p$-subgroup in a group $GH$, then it is […]
Non-Abelian Group of Order $pq$ and its Sylow SubgroupsLet $G$ be a non-abelian group of order $pq$, where $p, q$ are prime numbers satisfying $q \equiv 1 \pmod p$.Prove that a $q$-Sylow subgroup of $G$ is normal and the number of $p$-Sylow subgroups are $q$.Hint.Use Sylow's theorem. To review Sylow's theorem, check […]
Group of Order $pq$ Has a Normal Sylow Subgroup and SolvableLet $p, q$ be prime numbers such that $p>q$.If a group $G$ has order $pq$, then show the followings.(a) The group $G$ has a normal Sylow $p$-subgroup.(b) The group $G$ is solvable.Definition/HintFor (a), apply Sylow's theorem. To review Sylow's theorem, […]
Sylow Subgroups of a Group of Order 33 is Normal SubgroupsProve that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.Hint.We use Sylow's theorem. Review the basic terminologies and Sylow's theorem.Recall that if there is only one $p$-Sylow subgroup $P$ of $G$ for a fixed prime $p$, then $P$ […]
A Subgroup of Index a Prime $p$ of a Group of Order $p^n$ is NormalLet $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer.Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$.Then prove that $H$ is a normal subgroup of $G$.(Michigan State University, Abstract Algebra Qualifying […]
Are Groups of Order 100, 200 Simple?Determine whether a group $G$ of the following order is simple or not.(a) $|G|=100$.(b) $|G|=200$.Hint.Use Sylow's theorem and determine the number of $5$-Sylow subgroup of the group $G$.Check out the post Sylow’s Theorem (summary) for a review of Sylow's […]
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Now showing items 1-9 of 9
Production of $K*(892)^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$ =7 TeV
(Springer, 2012-10)
The production of K*(892)$^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$=7 TeV was measured by the ALICE experiment at the LHC. The yields and the transverse momentum spectra $d^2 N/dydp_T$ at midrapidity |y|<0.5 in ...
Transverse sphericity of primary charged particles in minimum bias proton-proton collisions at $\sqrt{s}$=0.9, 2.76 and 7 TeV
(Springer, 2012-09)
Measurements of the sphericity of primary charged particles in minimum bias proton--proton collisions at $\sqrt{s}$=0.9, 2.76 and 7 TeV with the ALICE detector at the LHC are presented. The observable is linearized to be ...
Pion, Kaon, and Proton Production in Central Pb--Pb Collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(American Physical Society, 2012-12)
In this Letter we report the first results on $\pi^\pm$, K$^\pm$, p and pbar production at mid-rapidity (|y|<0.5) in central Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV, measured by the ALICE experiment at the LHC. The ...
Measurement of prompt J/psi and beauty hadron production cross sections at mid-rapidity in pp collisions at root s=7 TeV
(Springer-verlag, 2012-11)
The ALICE experiment at the LHC has studied J/ψ production at mid-rapidity in pp collisions at s√=7 TeV through its electron pair decay on a data sample corresponding to an integrated luminosity Lint = 5.6 nb−1. The fraction ...
Suppression of high transverse momentum D mesons in central Pb--Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV
(Springer, 2012-09)
The production of the prompt charm mesons $D^0$, $D^+$, $D^{*+}$, and their antiparticles, was measured with the ALICE detector in Pb-Pb collisions at the LHC, at a centre-of-mass energy $\sqrt{s_{NN}}=2.76$ TeV per ...
J/$\psi$ suppression at forward rapidity in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(American Physical Society, 2012)
The ALICE experiment has measured the inclusive J/ψ production in Pb-Pb collisions at √sNN = 2.76 TeV down to pt = 0 in the rapidity range 2.5 < y < 4. A suppression of the inclusive J/ψ yield in Pb-Pb is observed with ...
Production of muons from heavy flavour decays at forward rapidity in pp and Pb-Pb collisions at $\sqrt {s_{NN}}$ = 2.76 TeV
(American Physical Society, 2012)
The ALICE Collaboration has measured the inclusive production of muons from heavy flavour decays at forward rapidity, 2.5 < y < 4, in pp and Pb-Pb collisions at $\sqrt {s_{NN}}$ = 2.76 TeV. The pt-differential inclusive ...
Particle-yield modification in jet-like azimuthal dihadron correlations in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(American Physical Society, 2012-03)
The yield of charged particles associated with high-pT trigger particles (8 < pT < 15 GeV/c) is measured with the ALICE detector in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV relative to proton-proton collisions at the ...
Measurement of the Cross Section for Electromagnetic Dissociation with Neutron Emission in Pb-Pb Collisions at √sNN = 2.76 TeV
(American Physical Society, 2012-12)
The first measurement of neutron emission in electromagnetic dissociation of 208Pb nuclei at the LHC is presented. The measurement is performed using the neutron Zero Degree Calorimeters of the ALICE experiment, which ...
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We all know it does mean revert. The question is why. What's making volatility mean-revert? Is it some sort of cyclical behaviour of option traders? The way it's calculated? Why?
Volatility is mean reverting if the underlying security doesn't drop to zero.
If the security has some underlying "value" then its price is co-integrated with that "value". The volatility is the uncertainty of that price as it tracks the security's "value".
Edit 12/03/2011 =================================================
@pteetor,
I may have missed something, but the question was "
Why is volatility mean-reverting?". I realize that the standard answer is that the VIX (I'm assuming he's asking about the VIX) is related to the historical volatility of the S&P. A simple version of that relationship provides a reasonable R^2 (see Fig. 1). It relates the VIX to the S&P "wiggliness" (30-day standard deviation of the daily log differences of the S&P), but it doesn't explain why more or less "wiggliness" takes place. To explain that, I have to look at the underlying fundamentals.
Figure 2 below, shows the S&P Price (in gray) and what I think is the underlying S&P
VALUE (in red). Both lines refer to the left hand scale. This VALUE is calculated from my estimate of sustainable earnings and the appropriate P/E ratio. It is what I think an Investor would set for the "value" (money generating value) of the S&P.
The blue line is the VIX, and is read on the right hand scale. On the right hand side of the graph, I have divided the VIX range into three regions. From past experience, a VIX of 20 or less seems to be a time of "Don't Worry, Be Happy". Personally, that's when I worry the most, but the market seems to be in a care-free state, so I tagged it as such. Next, for a VIX from about 20 to 40, the market seems to be in an "I'm Nervous" state (not care-free, but also not panicky). For a VIX above 40, a "Panic" state seems to show up.
Our current VIX of 28 puts us in the "I'm Nervous" state.
Now to the issue of
WHY.
Don't the Happy/Nervous/Panicky states of the VIX have to be consistent with the level of the S&P Price (not just its "wiggliness")? If I'm "Happy" then I'm happy with the level of the Price. If I'm "Nervous" aren't I nervous about the level of the Price? If I'm "Panicky" then isn't the level of the Price nose-diving?
As an
Investor, the only way I can be "Happy" with the level of the Price is for the Price to be somewhere near or below my estimate of the VALUE of the S&P. That happened from 1991 to 1997, 2003 to 2007, and part of 2010 and 2011.
As an
Investor, I will be "Nervous" with the level of the Price when the Price is too high compared to the VALUE or when some external "thing" is going on (for example, the Euro-mess). That happened in 1990, 1997 to 2003, part of 2007, 2008, 2009, 2010, and 2011.
And, as stated above, when "Panic" takes place, the market sells everything and the Price level nose-dives (1998 LTCM/Russian thing, 9/11/2001, July and Sept 2002, Sept 2008, the Debt-mess in 2010 and 2011).
So, if all of the above fits the hypothesis, then "
Why is volatility mean-reverting?" can be answered as.....the market has, and probably will continue to spend most of its time in the "Don't Worry, Be Happy" state or the "I'm Nervous" state (i.e. reverting to a state that is not extreme).
I agree that my three-sentence answer at the top of this post left out a lot of details, but it is the short version of the same answer.
Edit 12/09/2011 =======================================================
@pteetor,
I must be getting old and forgetful. In your comment below, you asked for references (which I forgot to include above). Here are a few:
With a little Googling, you'll find more.
In my answer above, I purposely didn't get into my technique for setting the "value" of the S&P. It always starts an argument about which is best, Modified Gordon Models, Modified Miller-Modigliani Models, Modified XXXX Models, or whatever. The bottom line is, no matter what valuation technique you use, you'll always find some form of "equity risk premium" that is related to volatility. It's just common sense......there has to be a "price" for being "Nervous" or "Happy".
Another issue that usually comes up has to do with using Historical Volatility versus Implied Volatility. All I can say is, the volatility part of the equity risk premium existed long before options were traded.
Volatility is mean reverting because you can prove by contradiction that it cannot be otherwise. You have an intuitive understanding of why, but you need something closer to a proof.
Assume volatility is not mean reverting. At time t, the effect of the random component of the volatility on its level will be $\sigma \cdot \sqrt{t}$
For an arbitrarily large
t (far enough into the future), the probability that the final volatility is in the same order of magnitude as where it started approaches 0. This means if you look at it evolve for long enough, you are certain it will eventually become arbitrarily large or arbitrarily small, which would be nonsense.
Therefore volatility cannot NOT be mean reverting.
This is a hard question because you have to explain i) why does volatility mean revert, AND the opposite effect ii) why does volatility exhibit clustering/persistence? GARCH models can describe this behavior but they do not explain.
We need a model of investor behavior that can explain both phenomenon. I think Rama Cont has a strong answer in his 2005 paper Volatility Clustering in Financial Markets: Empirical Facts and Agent–Based Models:
Many market microstructure models – especially those with learning or evolution– converge over large time intervals to an equilibrium where prices and other aggregate quantities cease to fluctuate randomly. By contrast, in the present model, prices fluctuate endlessly and the volatility exhibits mean-reverting behavior. Suppose we are in a period of “low volatility”; the amplitude |rt| of returns is small.
Agents who update their thresholds will therefore update them to small values, become more sensitive to news arrivals, thus generating higher excess demand and thus increasing the amplitude of returns. Conversely, in a period of high volatility, agents will update their threshold values to high values and become less reactive to the incoming signal: this increase in investor inertia will thus decrease the amplitude of returns. The mean reversion time in the volatility corresponds here to the time it takes for agents to adjust their thresholds to current market conditions, which is of order τc = 1/q. When the amplitude of the noise is small it can be shown that volatility decays exponentially in time and increases through upward “jumps”. This behavior is actually similar to that of a class of stochastic volatility models, introduced by Barndorff-Nielsen and Shephard and successfully used to describe various econometric properties of returns.
It is mean reverting due to the addition of new information into the market.
Imagine a very very simple case: the world only has two traders trading chicken feet. Trader A and B both think it's worth around \$10-\$11 per foot because historically, they have been trading that range all else constant. So vol is low.
Suddenly, GSK announces chicken feet cause cancer, trader A thinks they're worthless and trader B thinks some people dont care so he thinks they're worth \$5. It'll trade in the \$0 - \$5 range (higher vol). As the truth slowly unfolds, their opinions will converge (lower vol).
Becareful though, this doesn't mean you can capitalize on shorting vol when vol is above the mean (vice versa), realized vol may kill you. And market usually reflects mean reversion if you look at the term structure during crisis, front few months gets bid way more than the back months so you'll have a very descending term structure.
I would give a kind of commonsense answer: Volatility is basically measuring the range of fluctuation of the underlying. There is always some fluctuation (noise), some average range, but there are times when people are panicking and esp. risk-averse (falling stocks, crashes, high vola) and there are times when people are esp. calm (rising stocks, low vola).
So its a little bit how our world is and what drives markets in general (fear, greed -> lots of psychology) ...and no, there is nothing special about options traders here...
Do you believe it would be otherwise? Certainly we wouldn't expect low volatility to revert upwards, so you must be asking about why volatility spikes are short-lived.
Volatility spikes are often news related, either micro (merger announcement, bankruptcy notice, patent rejected, etc) or macro (rate changes, commodity bubble bursting, etc). That news isn't perpetual; the dust settles at some point and the world moves on.
Related question: Why do media frenzies die down? Same answer.
It is not the volatility that mean reverts, but the price! The volatility (attempts to) measure the deviations from the "equilibrium" or the "stable price" (even though we might never observe it).
It is not really a cyclical behaviour of traders, but one can look at it as a
continuous game of tug of war between the bulls and bears where any player can change sides at any time depending on her perception of future movements of price.
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Now showing items 1-10 of 24
Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ALICE detector at the LHC. Transverse ...
Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement is performed in the central pseudorapidity ...
Inclusive photon production at forward rapidities in proton-proton collisions at $\sqrt{s}$ = 0.9, 2.76 and 7 TeV
(Springer Berlin Heidelberg, 2015-04-09)
The multiplicity and pseudorapidity distributions of inclusive photons have been measured at forward rapidities ($2.3 < \eta < 3.9$) in proton-proton collisions at three center-of-mass energies, $\sqrt{s}=0.9$, 2.76 and 7 ...
Rapidity and transverse-momentum dependence of the inclusive J/$\mathbf{\psi}$ nuclear modification factor in p-Pb collisions at $\mathbf{\sqrt{\textit{s}_{NN}}}=5.02$ TeV
(Springer, 2015-06)
We have studied the transverse-momentum ($p_{\rm T}$) dependence of the inclusive J/$\psi$ production in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV, in three center-of-mass rapidity ($y_{\rm cms}$) regions, down to ...
Measurement of pion, kaon and proton production in proton–proton collisions at √s = 7 TeV
(Springer, 2015-05-27)
The measurement of primary π±, K±, p and p¯¯¯ production at mid-rapidity (|y|< 0.5) in proton–proton collisions at s√ = 7 TeV performed with a large ion collider experiment at the large hadron collider (LHC) is reported. ...
Two-pion femtoscopy in p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV
(American Physical Society, 2015-03)
We report the results of the femtoscopic analysis of pairs of identical pions measured in p-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. Femtoscopic radii are determined as a function of event multiplicity and pair ...
Measurement of charm and beauty production at central rapidity versus charged-particle multiplicity in proton-proton collisions at $\sqrt{s}$ = 7 TeV
(Springer, 2015-09)
Prompt D meson and non-prompt J/$\psi$ yields are studied as a function of the multiplicity of charged particles produced in inelastic proton-proton collisions at a centre-of-mass energy of $\sqrt{s}=7$ TeV. The results ...
Charged jet cross sections and properties in proton-proton collisions at $\sqrt{s}=7$ TeV
(American Physical Society, 2015-06)
The differential charged jet cross sections, jet fragmentation distributions, and jet shapes are measured in minimum bias proton-proton collisions at centre-of-mass energy $\sqrt{s}=7$ TeV using the ALICE detector at the ...
Centrality dependence of high-$p_{\rm T}$ D meson suppression in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(Springer, 2015-11)
The nuclear modification factor, $R_{\rm AA}$, of the prompt charmed mesons ${\rm D^0}$, ${\rm D^+}$ and ${\rm D^{*+}}$, and their antiparticles, was measured with the ALICE detector in Pb-Pb collisions at a centre-of-mass ...
K*(892)$^0$ and $\Phi$(1020) production in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(American Physical Society, 2015-02)
The yields of the K*(892)$^0$ and $\Phi$(1020) resonances are measured in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV through their hadronic decays using the ALICE detector. The measurements are performed in multiple ...
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Now showing items 1-5 of 5
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 ...
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 ...
Elliptic flow of muons from heavy-flavour hadron decays at forward rapidity in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV
(Elsevier, 2016-02)
The elliptic flow, $v_{2}$, of muons from heavy-flavour hadron decays at forward rapidity ($2.5 < y < 4$) is measured in Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV with the ALICE detector at the LHC. The scalar ...
Centrality dependence of the pseudorapidity density distribution for charged particles in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2013-11)
We present the first wide-range measurement of the charged-particle pseudorapidity density distribution, for different centralities (the 0-5%, 5-10%, 10-20%, and 20-30% most central events) in Pb-Pb collisions at $\sqrt{s_{NN}}$ ...
Beauty production in pp collisions at √s=2.76 TeV measured via semi-electronic decays
(Elsevier, 2014-11)
The ALICE Collaboration at the LHC reports measurement of the inclusive production cross section of electrons from semi-leptonic decays of beauty hadrons with rapidity |y|<0.8 and transverse momentum 1<pT<10 GeV/c, in pp ...
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Unit Tangent Vectors to a Space Curve
Recall from the Derivatives of Vector-Valued Functions page that if $\vec{r}(t) = (x(t), y(t), z(t))$ is a vector-valued function defined for $t$ an the interval $I = [a, b]$ and is differentiable, then the derivative $\vec{r'}(t)$ gives us the tangent vector corresponding to each $t \in [a, b]$. We also noted that $\vec{r'}(t)$ could represent a velocity vector-valued function for a point travelling along the curve traced by $\vec{r}(t)$ at time $t$. The length/magnitude of these vectors represent the speed of such a particle at time $t$. Of course, the speed need not be constant, and so if we were to make these velocity vectors have unit length/magnitude $1$, then the result is what is known as a unit tangent vector.
Definition: Let $\vec{r}(t) = (x(t), y(t), z(t))$ be a vector-valued function for $t \in [a, b]$ that is differentiable. Then the Unit Tangent Vector at $t$ denoted $\hat{T}(t)$ is the tangent vector at the point $\vec{r}(t)$ that has magnitude/length $1$, that is $\hat{T} = \frac{\vec{r'}(t)}{\| \vec{r'}(t) \|} = \frac{\vec{v}(t)}{\| \vec{v}(t) \|}$.
The following graph represents some unit vectors for an arbitrary curve $C$. Notice that the length of each vector is equal to the unit length, $1$.
Let's now look at an example of computing a unit tangent vector.
Example 1 Find the unit tangent vector to the curve defined by the vector-valued function $\vec{r}(t) = (t^2, 2t, 3)$ at $t = 2$.
We must first differentiate $\vec{r}(t)$ to get $\vec{r'}(t)$. Differentiating component by component we have that $\vec{r'}(t) = (2t, 2, 0)$. Now to compute $\hat{T}$, we will divide this vector by its magnitude. We must first compute the magnitude of $\vec{r'}(t)$ though, which isn't too hard as $\| \vec{r'}(t) \| = \sqrt{(2t)^2 + (2)^2 + (0)^2} = \sqrt{4t^2 + 4}$, and so:(1)
Now to figure out the unit tangent vector for $t = 2$, we just need to plug in $2$ into the above equation and so $\hat{T}(2) = \frac{1}{\sqrt{20}} (4, 2, 0) = \frac{1}{2 \sqrt{5}} (4, 2, 0) = \left ( \frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}}, 0 \right )$ is the unit tangent vector at $t = 2$.
Example 2 Find the unit tangent vector to the curve defined by the vector-valued function $\vec{r}(t) = (3\sin t, 2 \cos t, 3t^2 - 1)$ at $t = \pi$.
Once again, we must first differentiate $\vec{r}(t)$ to get $\vec{r'}(t)$. Differentiating component by component we have that $\vec{r'}(t) = (3 \cos t, -2 \sin t, 6t)$. Now to compute $\hat{T}$ we must divide this vector by its magnitude. We note that $\| \vec{r'}(t) \| = \sqrt{(3 \cos t)^2 + (-2 \sin t)^2 + (6t)^2} = \sqrt{9 \cos ^2 t + 4 \sin ^2 t + 36t^2}$, and so:(2)
Plugging in $t = \pi$ we have that $\hat{T}(\pi) = \frac{1}{\sqrt{9 + 36\pi^2}} (-3, 0, 6\pi) = \left ( \frac{-3}{\sqrt{9 + 36\pi^2}}, 0, \frac{6 \pi}{\sqrt{9 + 36\pi^2}} \right )$.
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Consider the following Lagrangian for a massive vector field $A_{\mu}$ in Euclidean space time: $$\mathcal L = \frac{1}{4} F^{\alpha \beta}F_{\alpha \beta} + \frac{1}{2}m^2 A^{\alpha}A_{\alpha}$$ where $F_{\alpha \beta} = \partial_{\alpha}A_{\beta} - \partial_{\beta}A_{\alpha}$ which means $$\mathcal L = \frac{1}{4} (\partial^{\alpha}A^{\beta} - \partial^{\beta}A^{\alpha})(\partial_{\alpha}A_{\beta} - \partial_{\beta}A_{\alpha}) + \frac{1}{2}m^2A^{\alpha}_{\alpha} \tag{1}$$ The canonical energy-momentum tensor is supposed to be, using the relation
$$T^{\mu \nu}_c = -\eta^{\mu \nu} \mathcal L + \frac{\partial \mathcal L}{\partial (\partial_{\mu}\Phi)}\partial^{\nu}\Phi,\,\,\tag{2}$$
$$T^{\mu \nu}_c = F^{\mu \alpha}\partial^{\nu}A_{\alpha} - \eta^{\mu \nu}\mathcal L$$
Then from $T^{\mu \nu}_B = T^{\mu \nu}_c + \partial_{\rho}B^{\rho \mu \nu}$, it is found that $$B^{\alpha \mu \nu} = F^{\alpha \mu}A^{\nu}\tag{3}$$ using the formula $$B^{\mu \rho \nu} = \frac{1}{2}i \left\{\frac{\partial \mathcal L}{\partial (\partial_{\mu}A_{\gamma})} S^{\nu \rho}A_{\gamma} + \frac{\partial \mathcal L}{\partial (\partial_{\rho}A_{\gamma})} S^{\mu \nu}A_{\gamma} + \frac{\partial \mathcal L}{\partial (\partial_{\nu}A_{\gamma})} S^{\mu \rho}A_{\gamma}\right\}$$ My question is how is this equation obtained and how did they obtain $(3)$? Did they make use of the explicit form of the spin matrix for a vector field? (My question is from Di Francesco et al 'Conformal Field Theory' P.46-47).
Here is my attempt: $$B^{\alpha \mu \nu} = \frac{i}{2}\left\{F^{\alpha \gamma}S^{\nu \mu}A_{\gamma} + F^{\mu \gamma}S^{\alpha \nu}A_{\gamma} + F^{\nu \gamma}S^{\alpha \mu}A_{\gamma}\right\}$$ from simplifying the above. Concentrate on the first term. Then $$F^{\alpha \gamma}S^{\nu \mu}A_{\gamma} = F^{\alpha \gamma}\eta_{\gamma c}\eta^{\nu a}\eta^{\mu b} (S_{ab})^c_dA^d$$ Inputting the form of $S$ for a vector field, I get $$F^{\alpha \gamma}\eta_{\gamma c}\eta^{\nu a}\eta^{\mu b}(\delta^c_a \eta_{bd} - \delta^c_b \eta_{da})A^d$$ But simplifying this and writing the other terms does not yield the result. Did I make a mistake upon insertion of the spin matrix?
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The expected boson occupation for a state with energy $\epsilon_j$ is given by
$$\langle n_j \rangle = \frac{1}{e^{\beta (\epsilon_j - \mu)}-1} = \frac{1}{z^{-1}e^{\beta\epsilon_j}-1} = \frac{z}{e^{\beta \epsilon_j}-z}$$
Here $\beta = \frac{1}{kT}$ and I have defined the fugacity $z=e^{\beta\mu}$ where $\mu$ is the chemical potential. Without changing the physics we can add an arbitrary offset to the energies so that the ground state energy, $\epsilon_0 = 0$. We see that for $\langle n_j \rangle$ to be positive it is necessary that $0<z<1$.
To understand BEC we must understand the behaviour of $z$ as a function of temperature. For concreteness I assume a fixed atom number $N$ and a 3D isotropic harmonic oscillator with frequency $\omega_0$. The energy levels are then space by $\hbar \omega$. It is then reasonable to define a dimensionless temperature $\tilde{T} = \frac{kT}{\hbar \omega_0}$. Below I plot $z$ as a function of $\tilde{T}$ for a non-interacting bosonic gas in a 3D harmonic oscillator potential for various atom numbers $N$.
This plot shows fugacity $z$ versus temperature $\tilde{T}$ for atom number $N = \{10^1, 10^2, 10^3, 10^4\}$ from left to right. We see that as $T$ decreases $z$ increases linearly towards $1$ until $T_c$ at which point $z$ saturates to 1. As $T$ is lowered below $T_c$ $z$ still increases but now more slowly since it has saturated. The transition from linear growth to saturate becomes sharper and more "phase transition-y" as atom number $N$ is increased.
Let us now consider the ground state population.
$$\langle n_0 \rangle = \frac{z}{1-z}$$
We see that as $z \rightarrow 1$ that $\langle n_0 \rangle$ will become very large.
Now consider the first excited state population.
$$\langle n_1 \rangle = \frac{z}{e^{\frac{\epsilon_1}{kT}}-z}$$
I note that for experimental BECs the quantity $\frac{\epsilon}{kT} \ll 1$*. I will now consider two limits of this function, the $T>T_c$ and $T<T_c$ limits.
For $T>T_c$ we have that $z<1$ so we can approximate $e^{\beta \epsilon_1} \approx 1$ and write
\begin{align}\langle n_1 \rangle \approx \frac{z}{1-z}\end{align}
As temperature decreases towards $T_c$ this function increases since $z$ increases towards $1$.
For $T<T_c$ we have $z\approx 1$ and we can no longer approximate $e^{\beta \epsilon_1} \approx 1$ so we have
$$\langle n_1 \rangle \approx \frac{1}{e^{\beta \epsilon_1}-1} \approx \frac{1}{1+\beta \epsilon_1-1} \approx \frac{kT}{\epsilon_1}$$
We see that this function
decreases as $T$ is decreased.
Thus we see that the excited state population $\langle n_1 \rangle$ decreases as $T$ is either increased or decreased away from $T_c$. Thus, the excited state population has a maximum at $T=T_c$. The question then of whether the excited state can be macroscopically occupied is a question of how large the excited state population is at the critical temperature. For a 3D Harmonic oscillator we have
$$kT_c \approx \hbar \omega N^{\frac{1}{3}}$$
So the fraction of atoms in the first excited states at the transition is
$$\frac{n_1}{N} \approx \frac{N^{\frac{1}{3}}}{N} = N^{-\frac{2}{3}}$$
So we see that the excited state fraction decreases as the total atom number $N$ increases and we move deeper and deeper into the thermodynamic limit. Below I plot the excited state fraction as a function of $\frac{T}{T_c}$ for $N = \{10^2, 10^3, 10^4, 10^5\}$ atoms.
See W. Ketterle and N.J. van Druten, Phys. Rev. A 54, 656 atW. Ketterle and N.J. van Druten, Phys. Rev. A 54, 656 for a more thorough discussion of finite atom number effects in Bose-Einstein condensation.
*This is a critically important point about Bose-Einstein condensation. The energy corresponding to the critical temperature is MUCH larger than the energy corresponding to the first excited state. There is a trivial single particle effect which is that if you decrease the temperature so much that $kT \ll \epsilon_i$ then of course you expect to find most particles in the ground state. This would be true even of a classical gas of distinguishable particles. I emphatically point out that this is
not the physics of BEC.
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We give two proofs. The first one uses Bays’ theorem and the second one simply uses the definition of the conditional probability.
Solution 1.
Let $E$ be the event that the first coin lands heads. Let $F$ be the event that at least one of two coins lands heads.By Bayes’ rule, the required probability can be calculated by the formula:\[P(E \mid F) = \frac{P(E) \cdot P(F \mid E)}{P(F)}.\]We know $P(E)=1/2$. When the first coin lands heads, then of course at least one of two coins lands heads. So, we have $P(F \mid E) = 1$.
Since $F = \{\text{hh}, \text{ht}, \text{th}\}$, we see that $P(F) = 3/4$.Plugging these values into the formula, we obtain\[P(E \mid F) = \frac{\frac{1}{2}\cdot 1}{\frac{3}{4}} = \frac{2}{3}.\]
Solution 2.
Let $E$ be the event that the first coin lands heads. Let $F$ be the event that at least one of two coins lands heads.Then we have $F = \{\text{hh}, \text{ht}, \text{th}\}$.Also, we have\[E \cap F = \{\text{hh}, \text{ht}\}.\]Thus, the required probability is given by\begin{align*}P(E \mid F) &= \frac{|E \cap F|}{|F|}\\&= \frac{2}{3}.\end{align*}
Independent and Dependent Events of Three Coins TossingSuppose that three fair coins are tossed. Let $H_1$ be the event that the first coin lands heads and let $H_2$ be the event that the second coin lands heads. Also, let $E$ be the event that exactly two coins lands heads in a row.For each pair of these events, determine whether […]
Probability of Having Lung Cancer For SmokersLet $C$ be the event that a randomly chosen person has lung cancer. Let $S$ be the event of a person being a smoker.Suppose that 10% of the population has lung cancer and 20% of the population are smokers. Also, suppose that we know that 70% of all people who have lung cancer […]
Independent Events of Playing CardsA card is chosen randomly from a deck of the standard 52 playing cards.Let $E$ be the event that the selected card is a king and let $F$ be the event that it is a heart.Prove or disprove that the events $E$ and $F$ are independent.Definition of IndependenceEvents […]
Jewelry Company Quality Test Failure ProbabilityA jewelry company requires for its products to pass three tests before they are sold at stores. For gold rings, 90 % passes the first test, 85 % passes the second test, and 80 % passes the third test. If a product fails any test, the product is thrown away and it will not take the […]
Pick Two Balls from a Box, What is the Probability Both are Red?There are three blue balls and two red balls in a box.When we randomly pick two balls out of the box without replacement, what is the probability that both of the balls are red?Solution.Let $R_1$ be the event that the first ball is red and $R_2$ be the event that the […]
Complement of Independent Events are IndependentLet $E$ and $F$ be independent events. Let $F^c$ be the complement of $F$.Prove that $E$ and $F^c$ are independent as well.Solution.Note that $E\cap F$ and $E \cap F^c$ are disjoint and $E = (E \cap F) \cup (E \cap F^c)$. It follows that\[P(E) = P(E \cap F) + P(E […]
Conditional Probability Problems about Die RollingA fair six-sided die is rolled.(1) What is the conditional probability that the die lands on a prime number given the die lands on an odd number?(2) What is the conditional probability that the die lands on 1 given the die lands on a prime number?Solution.Let $E$ […]
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The Kernel of a Group Homomorphism
The Kernel of a Group Homomorphism
Definition: Let $(G, \cdot)$ and $(H, *)$ be two groups and let $f : G \to H$ be a group homomorphism. Let $e_2$ be the identity of $H$. Then the Kernel of $f$ is defined as the subgroup (of $G$) denoted $\mathrm{ker} (f) = f^{-1} ( \{ e_2 \} ) = \{ x \in G : f(x) = e_2 \}$. Note that $\mathrm{ker} (f) \neq \emptyset$ since we know that if $e_1$ is the identity of $G$ then $f(e_1) = e_2$, i.e., $e_1 \in \mathrm{ker} (f)$.
The kernel of a group homomorphism has many nice attributes - some of which we acknowledge below.
Proposition 1: Let $(G, \cdot)$ and $(H, *)$ be groups and let $f : G \to H$ be a group homomorphism. Then $f$ is injective if and only if $\ker(f) = \{ e_1 \}$. Proof:$\Rightarrow$ Suppose that $f$ is injective. Let $x \in \ker (f)$. Then $f(x) = e_2$. But since $f$ is a homomorphism we have that $f(e_1) = e_2$. Since $f$ is injective we must have that $x = e_1$ and so $\ker (f) = \{ e_1 \}$. $\Leftarrow$ Suppose that $\ker (f) = \{ e_1 \}$. Let $x, y \in G$ and suppose that $f(x) = f(y)$. Then $f(x) * [f(y)]^{-1} = e_2$. Since $f$ is a homomorphism we have that $f(x \cdot y^{-1}) = e_2$. So $x \cdot y^{-1} \in \ker (f)$, so $x \cdot y^{-1} = e_1$, i.e., $x = y$. So $f$ is injective. $\blacksquare$
Corollary 2: Let $(G, \cdot)$ and $(H, *)$ be groups and let $f : G \to H$ be a group homomorphism. Let $e_1$ be the identity of $G$ and let $e_2$ be the identity of $H$. If $G$ is a simple group then either $f(G) = \{ e_2 \}$ or $f$ is injective. Proof:Since $G$ is a simple group and the kernel of $f$ is a subgroup of $G$ that must be normal, then $\mathrm{ker} (f) = \{ e_1 \}$ or $\mathrm{ker} (f) = G$ (since these are the only two normal subgroups of $G$). If $\mathrm{ker} (f) = \{ e_1 \}$ then by Proposition 1, $f$ is injective. Otherwise $\mathrm{ker} (f) = G$ which means that for all $x \in G$, $f(x) = e_2$. Thus $f(G) = \{ e_2 \}$. $\blacksquare$
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Sec 3.1 Introduction Equivalence Principle:
\(\quad\)
At every space-time point with arbitrary gravitational fields, there exists local(Inertial) frame with respect to which the laws of physics in a sufficiently small region takes the same form as in Mincowski space-time with no gravitational field. Sec 3.2 Motion in a gravitational field
\(\quad\)Equivalence of motion gives that (here I inset code
<script type=”text/x-mathjax-config”>// <![CDATA[
MathJax.Hub.Config({ TeX: { equationNumbers: { autoNumber: “AMS” } } }); // ]]></script>
\(\quad\)to number the equation)\begin{equation}\dfrac{d^{2}\tilde{x}^{\mu}}{d\tau^{2}}=0\label{1}\end{equation}
where \(\tilde{x}^{\mu}\) is the
and \(\tau\) denotes the coordinate of geodesic (Because every neighborhood \(\mathcal{U}\) of \(0_{p}\in T_{p}M\) the exponential map \(\exp\) is a diffeomorphism, there must exists geodesic coordinates \(u=(u^{1},\cdots,u^{m})\) such that \(\displaystyle x=\exp_{p}\sum_{i}u^{i}\dfrac{\partial}{\partial x^{i}}\).(A theorem in differential manifolds)). proper time
\(\quad\)Transform the geodesic coordinate to the general one \(x^{\mu}\):
$$\dfrac{d\tilde{x}^{\mu}}{d\tau}=\dfrac{\partial\tilde{x}^{\mu}}{\partial x^{\sigma}}\dfrac{d x^{\sigma}(\tau)}{d\tau},$$
then \eqref{1} becomes
$$\dfrac{d}{d\tau}\left(\dfrac{d\tilde{x}^{\mu}}{d\tau}\right)=\dfrac{\partial^{2}\tilde{x}^{\mu}}{\partial x^{\sigma}\partial x^{\rho}}\dfrac{d x^{\sigma}}{d\tau}\dfrac{d x^{\rho}}{d\tau}+\dfrac{\partial\tilde{x}^{\mu}}{\partial x^{\sigma}}\dfrac{d^{2}x^{\sigma}}{d\tau^{2}}=0.$$
\(\quad\)Multiply \(\dfrac{\partial x^{\nu}}{\partial\tilde{x}^{\mu}}\) on the left, we get
Geodesic Equation:
\begin{equation}\label{2}\dfrac{d^{2}x^{\nu}}{d\tau^{2}}+\varGamma^{\nu}_{\rho\sigma}\dfrac{d x^{\rho}}{d\tau}\dfrac{d x^{\sigma}}{d\tau}=0,\quad\text{where }\varGamma^{\nu}_{\rho\sigma}\equiv\dfrac{\partial x^{\nu}}{\partial\tilde{x}^{\mu}}\dfrac{\partial^{2}\tilde{x}^{\mu}}{\partial x^{\sigma}\partial x^{\rho}}\end{equation}
\(\quad\)For massive particles, \(ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}=g_{\mu\nu}d\tilde{x}^{\mu}d\tilde{x}^{\nu}\Rightarrow g_{\mu\nu}=\eta_{\rho\sigma}\dfrac{\partial\tilde{x}^{\rho}}{\partial x^{\mu}}\dfrac{\partial\tilde{x}^{\sigma}}{\partial x^{\nu}}\), so we are going to find connection between \(g_{\mu\nu}\) and \(\varGamma^{\nu}_{\rho\sigma}\).
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Functions that are Not Binary Operations
Recall from the Unary and Binary Operations on Sets page that if $S$ is a set then a function $f : S \times S \to S$ is said to be a binary operation and we must be specific in having that:
Each pair $(x, y) \in S \times S$ is mapped into $S$. $f(x, y)$ is defined. Each pair $(x, y) \in S \times S$ is mapped to only one element in $S$.
We will now look at some examples of sets $S$ and functions $f : S \times S \to S$ that are not binary functions.
First consider the set of nonnegative real numbers $S = \{ x \in \mathbb{R} : x > 0 \}$ and consider the function $* : S \times S \to S$ defined for $a, b \in S$ by:(1)
We can clearly see that since $a, b > 0$ that $a * b$ is well defined and each pair $(a, b) \in S \times S$ is mapped to only one element in $S$. That said, $*$ is not a binary operation on $S$ because the first condition is not satisfied. Take $a, b \in S$ where $0 < b \leq 1$. Then $\ln b \leq 0$ and hence $a \ln b \leq 0$. Therefore $a * b = a \ln b \leq 0$ so $(a * b) \not \in S$. So some pairs $(a, b) \in S \times S$ are mapped out of $S$, so $*$ is not a binary operation on $S$.
Of course, if we let $S^* = \{ x \in \mathbb{R} : x > 1 \}$ then $\ln b > 0$ for all $b \in S^*$ so $a * b > 0$ for all $(a, b) \in S^* \times S^*$ so indeed, $*$ is a binary operation on the set $S^*$.
For another example, consider the set of rational numbers $\mathbb{Q}$ and the function $* : \mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$ defined for $x, y \in \mathbb{Q}$ by:(2)
In this instance, we have that $*$ is merely division on the set of rational numbers. Unfortunately, $*$ is not a binary operation since the pair $(a, 0) \in \mathbb{Q} \times \mathbb{Q}$ is such that $a * 0 = \frac{a}{0}$ which is undefined. Instead, we can define the binary operation above if we instead consider the set $\mathbb{Q} \setminus \{ 0 \}$.
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when you have a chiral superfield in the adjoint representation in your theory (i.e the Higgs $\Sigma_{24}$ in $SU(5)$), you can write it as a $5\times5$ Matrix: $(\Sigma_{24})_i^j=\Sigma^a (t^a)_i^j$ (sum over a), where the $t^a$ are the generatorns of the $SU(5)$.
My question is: is it possible that the $\Sigma_a$ are real, or do they have to be complex for $SU(5)$ SUSY GUT?
I'm asking because some steps in the calculation of the minimum (vev) seem to need real $\Sigma_a$'s, since that would make $\Sigma_{24}$ Hermitian. But on the other side I'm not sure if real scalars would make problems with supersymmetry (degrees of freedom).
thanks
Update:
The problem with real $\Sigma_a$ is the following:
Suppose you want to write down the superfield $S$ in components. For one complex scalar $\Phi$ it would look as follows: $S=\Phi+\sqrt{2} \theta \Psi+\theta\theta F$. Looking at the degrees of freedome on-shell $\rightarrow dof(\Phi)=2, dof(\Psi)=2,dof(F)=0$
Now looking at the adjoint representation of $\Sigma$, wich is a 24-dimesional multiplett ($\Sigma_1,...,\Sigma_{24}$)(this are the $\Sigma_a$ from above, (btw. $\Sigma_a=\Sigma^a$, sorry for mixing notation)). Now each field in this supermultiplett has to be a superfield, like above $\Sigma_a=\Phi_a+\sqrt{2} \theta \Psi_a+\theta\theta F_a$. Counting now the degress of freedome reveals the problem: $\rightarrow dof(\Phi_a)=1, dof(\Psi_a)=2,dof(F_a)=0$. That shows that the fermionic dof. are not equal to the bosonic dof. wich contradicts SUSY.
Edit:
$i,j$ in $\Sigma_i^j$ are matrix indices and run from 1 to 5.
$a$ in $t^a$ and $\Sigma^a$ runs from 1 to 24 (24 = dimension of adjoint representation of $SU(5)$)
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The goal is to remove the white line after the
1. in the second
enumerate in the picture, so it will look like the first
enumerate.
I don't like the first
enumerate because it uses inline math, needs the white line in the code to make a new paragraph, and uses
\displaystyle which I think is not the way to go: it feels like a 'hack'. Maybe I am mistaken, in that case say so, but I think there must be a good solution.
MWE:
\documentclass[english]{exam}\usepackage[fleqn]{amsmath}\begin{document} \begin{enumerate} %result looks fine, but code ugly \item $\displaystyle \pi$ $\displaystyle 2\pi$ \end{enumerate} \begin{enumerate} %code looks fine, but result ugly \item \[\pi\] \[2\pi\] \end{enumerate}\end{document}
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Search
Now showing items 1-7 of 7
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 ...
Azimuthally differential pion femtoscopy relative to the second and thrid harmonic in Pb-Pb 2.76 TeV collisions from ALICE
(Elsevier, 2017-11)
Azimuthally differential femtoscopic measurements, being sensitive to spatio-temporal characteristics of the source as well as to the collective velocity fields at freeze-out, provide very important information on the ...
Investigations of anisotropic collectivity using multi-particle correlations in pp, p-Pb and Pb-Pb collisions
(Elsevier, 2017-11)
Two- and multi-particle azimuthal correlations have proven to be an excellent tool to probe the properties of the strongly interacting matter created in heavy-ion collisions. Recently, the results obtained for multi-particle ...
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 ...
Elliptic flow of muons from heavy-flavour hadron decays at forward rapidity in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV
(Elsevier, 2016-02)
The elliptic flow, $v_{2}$, of muons from heavy-flavour hadron decays at forward rapidity ($2.5 < y < 4$) is measured in Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV with the ALICE detector at the LHC. The scalar ...
Centrality dependence of the pseudorapidity density distribution for charged particles in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2013-11)
We present the first wide-range measurement of the charged-particle pseudorapidity density distribution, for different centralities (the 0-5%, 5-10%, 10-20%, and 20-30% most central events) in Pb-Pb collisions at $\sqrt{s_{NN}}$ ...
Beauty production in pp collisions at √s=2.76 TeV measured via semi-electronic decays
(Elsevier, 2014-11)
The ALICE Collaboration at the LHC reports measurement of the inclusive production cross section of electrons from semi-leptonic decays of beauty hadrons with rapidity |y|<0.8 and transverse momentum 1<pT<10 GeV/c, in pp ...
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Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations
1.
Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka 560-0043, Japan
2.
Department of Mathematics, College of Science, Yanbian University, No. 977 Gongyuan Road, Yanji City, Jilin Province, 133002, China, and, Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico
$\begin{matrix} i{{\partial }_{t}}u+\frac{1}{2}\Delta u=\lambda {{\left| u \right|}^{p-1}}u,\left( t,x \right)\in {{\mathbb{R}}^{+}}\text{ }\!\!\times\!\!\text{ }{{\mathbb{R}}^{n}}, \\ u\left( 0,x \right)={{u}_{0}}\left( x \right),x\in {{\mathbb{R}}^{n}}, \\ \end{matrix}$
$n=1,2$
$3$
$\lambda =\lambda _{1}+i\lambda _{2},$
$\lambda _{j}∈ \mathbb{R},$
$j=1,2,$
$\lambda _{2}<0$
$p=1+\frac{2}{n}-μ ,$
$μ >0$ Mathematics Subject Classification:Primary: 35Q55; Secondary: 35B40. Citation:Nakao Hayashi, Chunhua Li, Pavel I. Naumkin. Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2089-2104. doi: 10.3934/cpaa.2017103
References:
[1] [2] [3] [4] [5] [6]
N. Hayashi, C. Li and P. I. Naumkin, Time decay for nonlinear dissipative Schrödinger equations in optical fields
[7]
N. Hayashi, Li, Chunhua and P. I. Naumkin, Dissipative nonlinear Schrödinger equations with singular data,
[8]
G. Jin, Y. Jin and C. Li,
The initial value problem for nonlinear Schödinger equations with a dissipative nonlinearity in one space dimension,
[9]
N. Kita and A. Shimomura,
Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity,
[10]
N. Kita and A. Shimomura,
Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data,
[11]
C. Li and N. Hayashi,
Critical nonlinear Schrödinger equations with data in homogeneous weighted $\mathbf{L}^{2}$ spaces,
[12]
J. -L. Lions,
show all references
References:
[1] [2] [3] [4] [5] [6]
N. Hayashi, C. Li and P. I. Naumkin, Time decay for nonlinear dissipative Schrödinger equations in optical fields
[7]
N. Hayashi, Li, Chunhua and P. I. Naumkin, Dissipative nonlinear Schrödinger equations with singular data,
[8]
G. Jin, Y. Jin and C. Li,
The initial value problem for nonlinear Schödinger equations with a dissipative nonlinearity in one space dimension,
[9]
N. Kita and A. Shimomura,
Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity,
[10]
N. Kita and A. Shimomura,
Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data,
[11]
C. Li and N. Hayashi,
Critical nonlinear Schrödinger equations with data in homogeneous weighted $\mathbf{L}^{2}$ spaces,
[12]
J. -L. Lions,
[1]
Jean-Michel Roquejoffre, Luca Rossi, Violaine Roussier-Michon.
Sharp large time behaviour in $ N $-dimensional Fisher-KPP equations.
[2] [3] [4] [5]
Marcello D'Abbicco, Ruy Coimbra Charão, Cleverson Roberto da Luz.
Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient.
[6] [7] [8] [9] [10] [11]
Dongfen Bian, Boling Guo.
Global existence and large time
behavior of solutions to the electric-magnetohydrodynamic equations.
[12] [13] [14]
Marco Cappiello, Fabio Nicola.
Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations.
[15]
Yongsheng Jiang, Huan-Song Zhou.
A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials.
[16]
Linghai Zhang.
Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations.
[17]
Yingshan Chen, Shijin Ding, Wenjun Wang.
Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations.
[18]
Jun-Ren Luo, Ti-Jun Xiao.
Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping.
[19]
Zhuan Ye.
Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations.
[20]
2018 Impact Factor: 0.925
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It seems in the literature that there is a certain notion of a “macroscopic wavefunction” associated with a Bose-Einstein system (see this PSE answer) which exhibits a global $U(1)$-phase symmetry. After the BEC phase transition, this symmetry is supposedly broken corresponding to the sudden vanishing of the chemical potential (which is possibly an order parameter for that symmetry). There is another PSE answer which suggested I should probably read the second chapter of
Quantum Liquids by A. J. Leggett to understand this gauge-symmetric aspect of Bose Einstein condensation, which I tried but find very different from the intuition I have been given.
For a background on how I have understood Bose Einstein condensation, please read the section “Background” below. I do not find any place in the language I am familiar with to describe anything like gauge symmetry and that the chemical potential tracks that symmetry.
Can somebody explain to me how gauge-theoretic perspectives about the BEC can be developed from the background I have?
Thank you!
Background:
The Bose distribution is defined by average occupation numbers, $n_p \equiv (e^{\beta(\varepsilon_p - \mu)}-1)^{-1}$, of single-particle states $|p\rangle$ with energies $\varepsilon_p = p^2/2m$, such that $\sum_p n_p = N$ and $\mu< \min_p \varepsilon_p = 0$ (in order to ensure that $n_p \ge 0$).
The partition function is given by
$$ \mathcal Z = \sum_{N=0}^\infty \ \sum_{\{n_p\ |\sum n_p=N\}} \exp\left(-\beta\sum_p(\varepsilon_p-\mu)n_p \right) \ = \prod_p\sum_{n_p=0}^\infty e^{-\beta(\varepsilon_p-\mu)n_p} = \prod_p \frac{1}{1-e^{-\beta(\varepsilon_p-\mu)}}\,. $$
The grand potential is $\Phi = -\beta^{-1}\log\mathcal Z$, and the average particle number is
$$ N \equiv -\left( \frac{\partial \Phi}{\partial \mu}\right) = \sum_pn_p \approx \frac{V}{(2\pi\hbar)^3}\int d^3p\ n_p = \frac{V}{\lambda^3}Li_{3/2}\left(e^{\beta\mu}\right) \equiv N_{\text{excited}}\,, $$
where the sum is approximated by an integral assuming the large momentum states are occupied more significantly. $V$ is the volume occupied by $N$ particles, thus defining a specific volume $v = V/N$. Here, $\lambda = \hbar\sqrt{2\pi\beta/m}$ is the so-called thermal wavelength. In terms of the specific volume $v$, we can find the dependence of the chemical potential on the temperature with constant $v$.
$$ \frac{\mu(T)}{kT_c} = \begin{cases} \sim 0 &\text{for } T<T_c\,,\\ ({T}/{T_c}) \log \left({Li_{3/2}}^{-1}(\zeta(3/2)(T/T_c)^{-3/2})\right) &\text{for } T>T_c\,. \end{cases} $$
As you see below, at $T=T_c(v)=2\pi\hbar^2m^{-1}(\zeta(3/2)v)^{-2/3}$, a phase transition occurs.
If $N_0$ is the number of bosons in the ground state, and $N = N_0 + N_{\text{excited}}$, then in the thermodynamic limit,
$$ \lim_{\substack{N\to\infty \\ v \text{ fixed} }} \frac{N_0}{N} = \begin{cases} 1-(T/T_c)^{3/2} &\text{for } T < T_c\,,\\ 0 &\text{for } T > T_c\,. \end{cases}$$
This is Bose Einstein Condensation.
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Okay, so, in the traditional Bernoulli Urn problem, we have an urn with a number N, possibly infinite, of coloured balls, and there are k possible colours. That one I grok.
However, what if I don't actually
know what k is? That is, what if I have an urn with N balls and an unknown but finite and strictly positive number of possible colours?
The main question is, in fact, what my priors should be. What's the prior that there is exactly one colour? Exactly two? At least two? How do I update on the relative frequencies of each colour? Is this problem even solvable?
My first lines of thinking are to have a vector of parameters $\vec \theta \in \mathbb R^\infty$ such that the first parameter is the number of colours in the urn (let's call it $\alpha$) and the remaining are the relative frequencies of each colour. If $P(A=n|\vec\theta)$ is the probability that the colour of the next draw will be n given the knowledge contained by $\vec\theta$, we'd have:
$\vec\theta = (\alpha, p_1, p_2, p_3, ...)$ $\alpha \in \mathbb N^*$ $\left(\sum\limits_{n=1}^\infty P(\alpha = n) \right)= 1$ $\left(\sum\limits_{n=1}^\infty p_n\right) = 1$ $\forall n > \alpha : p_n = 0$ $\forall n \in \mathbb N^* : P(A=n|\vec\theta) = p_n$
However, this is just wild speculation on my part. I'm mostly curious about whether this is even in principle solvable. What I'd want to know is a way to compute both the prior (objective/uninformative) and posterior distributions of $P(\vec\theta)$ or, in other words, the pdfs $P(\alpha)$, $P(p_1)$, $P(p_2)$, etc. How to start with them and how to update on them.
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Please reference this paper for notation in this question.
I'm trying to understand two claims made in the above paper (they may be related). First, in the construction of $\mathcal{H}_\lambda$ on page 4, we start with the usual irreducible representation $V_\lambda$ of the semisimple Lie algebra $\mathfrak{g}$, declare the positive powers of $z$ to act trivially and the central element $c$ to act by multiplication by $l$, and then induce an action of the rest of the affine Lie algebra $\hat{\mathfrak{g}}$ on $V_\lambda$. The result is called $\mathcal{V}_\lambda$ and it is isomorphic to $U(\hat{\mathfrak{g}}_-) \otimes_{\mathbb{C}} V_\lambda$ as a $\hat{\mathfrak{g}}_-$-module. As a consequence, it is a highest weight module, and by theory analogous to that of semisimple Lie algebras, it has a maximal submodule.
I understand all of that. What I do not understand is the claim that this maximal submodule $\mathcal{Z}_\lambda$ is generated by $(X_\theta \otimes z^{-1})^{l - \lambda(H_\theta) + 1} v_\lambda$. Following Remark 3.6 on page 8, I managed to prove that this element is annihilated by $\hat{\mathfrak{g}}_+$, and indeed that no lower nonzero power of $X_\theta \otimes z^{-1}$ acting on $v_\lambda$ is annihilated by all of $\hat{\mathfrak{g}}_+$. Thus I'm lead to believe that this should somehow imply that $\mathcal{Z}_\lambda$ is a highest weight module with $(X_\theta \otimes z^{-1})^{l - \lambda(H_\theta) + 1} v_\lambda$ its highest weight vector, and hence the claim follows, but I can't see why this is the case.
The second claim is labeled (*) on page 7: The endomorphism $X_{-\theta} \otimes f$ of $\mathcal{H}$ is locally nilpotent for all $f \in \mathcal{O}(U)$. From what I can tell, this is equivalent to the claim that $X_{-\theta} \otimes f$ is a locally nilpotent endomorphism of $\mathcal{H}_\lambda$ for any $f \in \mathcal{C}((z))$. Following the given hint, I can see why this is true for $f \in \mathcal{C}[[z]]$, but I don't know about negative powers of $z$. I think this might be related to the first claim: Perhaps once we have pushed any element of $\mathcal{V}_\lambda$ into a suitably low weight space by acting on it by $X_{-\theta} \otimes z^{-1}$ repeatedly, it actually falls into some proper submodule of $\mathcal{V}_\lambda$, and is therefore quotiented out when we pass to $\mathcal{H}_\lambda$. Again, I can't see why this is actually true though.
Any ideas on either claim are greatly appreciated.
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In This MO question, Werner said that Hecke operator "changes" characters. I'm looking for any explicit theory of this kind, about Hecke operator with characters.
Actually, there are somewhat explicit example I found, for $\Gamma_{0}(4)$. Define $\chi:\Gamma_{0}(4)\to \mathbb{C}^{\times}$ as $\chi(T)=\chi(R)=e^{2\pi i/8}, \chi(-I)=1$, where $$ T=\begin{pmatrix}1&1\\0&1\end{pmatrix}, R=\begin{pmatrix}1&0\\4&1\end{pmatrix}, -I=\begin{pmatrix}-1&0\\0&-1\end{pmatrix} $$ which are generators of $\Gamma_{0}(4)$. Now let $f$ be a weight 0 modular form on $\Gamma_{0}(4)$ with character $\chi$, i.e. it satisfies $$ f(z+1)=e^{2\pi i /8}f(z), \,\,\,\,f\left(\frac{z}{4z+1}\right)=e^{2\pi i/8}f(z). $$ Now define $g(z)$ as $$ g(z)=f(3z)+f\left(\frac{z}{3}\right)+e^{10\pi i/8}f\left(\frac{z+1}{3}\right)+e^{4\pi i /8}f\left(\frac{z+2}{3}\right) $$ which is something looks like $T_{3}f$. Then with some tedious computations, I found that $$ g(z+1)=e^{6\pi i /8}g(z),\,\,\,\,g\left(\frac{z}{4z+1}\right)=e^{6\pi i /8}g(z) $$ holds. I just want to ask for any known results similar to this example.
Miyake's book about modular forms mentioned about Hecke operator with character and his Hecke operator does not changes character. But it assumes that the character can be extended to some bigger set multiplicatively, but the book doesn't give any necessary or sufficient conditions for this. For example, it is possible to define Hecke operator $T_{p}$ if we can extend the character $\chi:\Gamma_{0}(N)\to \mathbb{C}^{\times}$ to the bigger set $$\Gamma_{0}(N)\cup\Gamma_{0}(N)\begin{pmatrix}1&0\\0&p\end{pmatrix}\Gamma_{0}(N)$$ multiplicatively.
EDIT : To show the functional equation of $g$, I just computed a lot. The first one is not much complicated : we have\begin{align}g(z+1)&=f(3z+3)+f\left(\frac{z+1}{3}\right)+\zeta_{8}^{5}f\left(\frac{z+2}{3}\right)+\zeta_{8}^{2}f\left(\frac{z+3}{3}\right) \\&=\zeta_{8}^{3}\left[f(3z)+f\left(\frac{z}{3}\right)+\zeta_{8}^{5}f\left(\frac{z+1}{3}\right)+\zeta_{8}^{2}f\left(\frac{z+2}{3}\right)\right] \\&=\zeta_{8}^{3}g(z)\end{align}where $\zeta_{8}=e^{2\pi i /8}$. Second one is too long to write here..
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Thompson, 1987, doesn't address this directly, but did consider an algorithm for specific configurations of the probabilities. He was looking for the worst case scenario when the bounds of error were not constant, and suggested a search over many configurations.
Here is my modification of Thompson's algorithm. This version reduces the search to your a priori configuration and, perhaps, some near neighbors.
Select a set of vectors $P =(P_1,P_2,P_3,P_4,P_5), \sum_j P_j=1)$, which are plausible according to your a priori beliefs. You state that you have one configuration in mind, and it is likely that small departures from that configuration will have little effect on the calculation. Still, I recommend that, for safety, you consider plausible departures from this configuration.
Choose an overall $\alpha$ level and bound on error $d_j$ for estimating $P_j$. Your question has $d_j = 0.05$ and $\alpha = 0.05$.
If $p_j$ is the sample proportion for category $j$, a simultaneous $1-\alpha$ confidence for the $P_J$ will be of the form:
$$\left[\,p_j -d_j,\, p_j +d_j\,\right]$$
with the desired property that
$$\text{Pr}(|\,P_j - p_j\,| > d_j\text{, for any j}) \le \alpha.$$
Compute the worst case $n$ for your $\alpha$ from the table in Sample size for categorical data
Starting from your initial configuration of probabilities , compute $\sum \alpha_j$, where $\alpha_j = 2(1-\Phi(z_j))$ and
$$z_j = \frac{ d_j\sqrt{n}}{\sqrt{P_j(1-P_j)}}$$
($ 1- \Phi(z)$ is the probability that a standard Normal variable is $\ge z$.)
If $\sum \alpha_j< \alpha$, repeat with a smaller value of $n$. If $\sum \alpha_j>\alpha$, repeat with a larger value, until the smallest $n$ is found such that $\sum \alpha_j \leq \alpha$
If you believe in your initial configuration, you can stop there. Otherwise repeat for every vector of probabilities in your plausible set and choose the largest $n$.
Reference
Thompson, Steven K. 1987. Sample size for estimating multinomial proportions. The American Statistician 41, no. 1: 42-46.
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I wish to plot the function $f(x)=\sin(\omega x).$ One property of this function is that it is periodic in $x$ with period $\frac{2 \pi}{\omega}$.
I wish to plot $f(x)$ in the region $x\in (-\frac{2\pi}{\omega},\frac{2\pi}{\omega})$, with ticks on $$-\frac{2\pi}{\omega},-\frac{3}{2}\frac{\pi}{\omega},-\frac{\pi}{\omega},-\frac{1}{2}\frac{\pi}{\omega}, 0,\frac{1}{2}\frac{\pi}{\omega},\frac{\pi}{\omega},\frac{3}{2}\frac{\pi}{\omega},\frac{2\pi}{\omega}$$
which means quarter-steps of the periodicity in $x$.
When I define this function in Mathematica I do the following:
f[x_] := Sin[ω x]
Since I want to plot it, the only command line(s) that really plots something is
ω = 5Plot[f[x], {x, -((2 π)/ω), (2 π)/ω}]
This advances me a bit but is not exactly what I want to get.
I want to keep $\omega$ unset, and to see the axis labels with ticks on multiplies of $\frac{2\pi}{\omega}$ and not just numbers appearing there.
I know I can manage all this using
Ticks, but I wonder whether Mathematica can do it automatically.
For now it is all simple but it becomes much more complicated when plotting, for example, 2 variables scalar-function where each variable has its corresponding period, as in this solution to Laplace equation, with certain boundary conditions:
$$V(x,y)=\frac{4V_0}{\pi}\sum_{n=1,3,5} \frac{1}{n} \frac{\cosh(n\,\pi\,x/a)}{\cosh(n\,\pi\,b/a)}\sin(n\,\pi\,y/a)$$
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Difference between revisions of "SageMath"
m (→Starting Sage Notebook Server throws an ImportError: flag for deletion)
(→Installation: add warning from discussion)
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== Installation ==
== Installation ==
− + +
|the []
* {{Pkg|sagemath}} contains the command-line version;
* {{Pkg|sagemath}} contains the command-line version;
Revision as of 15:52, 24 January 2016
SageMath (formerly
Sage) is a program for numerical and symbolic mathematical computation that uses Python as its main language. It is meant to provide an alternative for commercial programs such as Maple, Matlab, and Mathematica.
SageMath provides support for the following:
Calculus: using Maxima and SymPy. Linear Algebra: using the GSL, SciPy and NumPy. Statistics: using R (through RPy) and SciPy. Graphs: using matplotlib. An interactive shellusing IPython. Access to Python modulessuch as PIL, SQLAlchemy, etc. Contents 1 Installation 2 Usage 3 Optional additions 4 Install Sage package 5 Troubleshooting 6 See also Installation Warning:Most if not all of the standard sage packages are available as Arch packages and exposed as (optional) dependencies of , so there is no need to install them with
sage -i.
contains the command-line version; for HTML documentation and inline help from the command line. includes the browser-based notebook interface.
The optional dependencies for various features that will be disabled if the needed packages are missing.package has number of
Usage
SageMath mainly uses Python as a scripting language with a few modifications to make it better suited for mathematical computations.
SageMath command-line
SageMath can be started from the command-line:
$ sage
For information on the SageMath command-line see this page.
Note, however, that it is not very comfortable for some uses such as plotting. When you try to plot something, for example:
sage: plot(sin,(x,0,10))
SageMath opens a browser window with the Sage Notebook.
Sage Notebook
A better suited interface for advanced usage in SageMath is the Notebook. To start the Notebook server from the command-line, execute:
$ sage -n
The notebook will be accessible in the browser from http://localhost:8080 and will require you to login.
However, if you only run the server for personal use, and not across the internet, the login will be an annoyance. You can instead start the Notebook without requiring login, and have it automatically pop up in a browser, with the following command:
$ sage -c "notebook(automatic_login=True)" Jupyter Notebook
SageMath also provides a kernel for the Jupyter notebook. To use it, install and , launch the notebook with the command
$ jupyter notebook
and choose "SageMath" in the drop-down "New..." menu. The SageMath Jupyter notebook supports LaTeX output via the
%display latex command and 3D plots if is installed.
Cantor
Cantor is an application included in the KDE Edu Project. It acts as a front-end for various mathematical applications such as Maxima, SageMath, Octave, Scilab, etc. See the Cantor page on the Sage wiki for more information on how to use it with SageMath.
Cantor can be installed with the official repositories.package or as part of the or groups, available in the
Documentation
For local documentation, one can compile it into multiple formats such as HTML or PDF. To build the whole SageMath reference, execute the following command (as root):
# sage --docbuild reference html
This builds the HTML documentation for the whole
reference tree (may take longer than an hour). An option is to build a smaller part of the documentation tree, but you would need to know what it is you want. Until then, you might consider just browsing the online reference.
For a list of documents see
sage --docbuild --documents and for a list of supported formats see
sage --docbuild --formats.
Optional additions SageTeX
If you have installed TeX Live on your system, you may be interested in using SageTeX, a package that makes the inclusion of SageMath code in LaTeX files possible. TeX Live is made aware of SageTeX automatically so you can start using it straight away.
As a simple example, here is how you include a Sage 2D plot in your TEX document (assuming you use
pdflatex):
include the
sagetexpackage in the preamble of your document with the usual
\usepackage{sagetex} create a
sagesilentenvironment in which you insert your code:
\begin{sagesilent} dob(x) = sqrt(x^2 - 1) / (x * arctan(sqrt(x^2 - 1))) dpr(x) = sqrt(x^2 - 1) / (x * log( x + sqrt(x^2 - 1))) p1 = plot(dob,(x, 1, 10), color='blue') p2 = plot(dpr,(x, 1, 10), color='red') ptot = p1 + p2 ptot.axes_labels(['$\\xi$','$\\frac{R_h}{\\max(a,b)}$']) \end{sagesilent} create the plot, e.g. inside a
floatenvironment:
\begin{figure} \begin{center} \sageplot[width=\linewidth]{ptot} \end{center} \end{figure} compile your document with the following procedure: $ pdflatex <doc.tex> $ sage <doc.sage> $ pdflatex <doc.tex> you can have a look at your output document.
The full documentation of SageTeX is available on CTAN.
Install Sage package
If you installed sagemath from the official repositories, it is not possible to install sage packages using the sage option
sage -i packagename.
Instead, you should install the required packages system-wide. For example, if you need
jmol (for 3D plots):
$ sudo pacman -S jmol
An alternative would be to have a local installation of sagemath and to manage optional packages manually.
Troubleshooting TeX Live does not recognize SageTex
If your TeX Live installation does not find the SageTex package, you can try the following procedure (as root or use a local folder):
Copy the files to the texmf directory: # cp /opt/sage/local/share/texmf/tex/* /usr/share/texmf/tex/ Refresh TeX Live: # texhash /usr/share/texmf/ texhash: Updating /usr/share/texmf/.//ls-R... texhash: Done. Starting Sage Notebook Server throws an ImportError
The Sage Notebook Server is in an extra package. So, if you get an ImportError when launching
% sage --notebook ┌────────────────────────────────────────────────────────────────────┐ │ Sage Version 6.4.1, Release Date: 2014-11-23 │ │ Type "notebook()" for the browser-based notebook interface. │ │ Type "help()" for help. │ └────────────────────────────────────────────────────────────────────┘ Please wait while the Sage Notebook server starts... Traceback (most recent call last): File "/usr/bin/sage-notebook", line 180, in <module> launcher(unknown) File "/usr/bin/sage-notebook", line 58, in __init__ from sagenb.notebook.notebook_object import notebook ImportError: No module named sagenb.notebook.notebook_object
you most likely do not have the packageinstalled.
sage -i doesn't work
If you have installed Sage from the official repositories, then you have to install your additional packages system-wide. See Install Sage package
3D plot fails in notebook
If you get the following error while trying to plot a 3D object:
/usr/lib/python2.7/site-packages/sage/repl/rich_output/display_manager.py:570: RichReprWarning: Exception in _rich_repr_ while displaying object: Jmol failed to create file '/home/nicolas/.sage/temp/archimede/3188/dir_cCpcph/preview.png', see '/home/nicolas/.sage/temp/archimede/3188/tmp_JVpSqF.txt' for details RichReprWarning, Graphics3d Object
then you probably miss the jmol package. See Install Sage package to install it.
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SM + W' and Z' at NLO in QCD Contact Authors
Benjamin Fuks
LPTHE - CNRS - UPMC (Paris VI) fuks AT lpthe.jussieu.fr
Richard Ruiz
Universite Catholique de Louvain (CP3) richard.ruiz AT uclouvain.be Model Description
This effective model extends the Standard Model (SM) field content by introducing the massive vector fields W'
+/- and Z' bosons, which are electrically charged and neutral, respectively. To remain model independent, couplings to SM gauge bosons and scalars are omitted.
Here,
i and
j denote flavor indices, P
L/Rare the usual left/right-handed chirality projectors, V CKM is the CKM matrix, and
g and
\theta_W are the weak coupling constant and mixing angle respectively. We choose coupling normalizations facilitating the mapping to the reference Sequential Standard Model (SSM) Lagrangian
{\cal L}_{\rm SSM} [ 3 ].The real-valued quantities
\kappa_{L,R}^q and
\zeta_{L,R}^qserve as overall normalization of the new interactions relative to the strength of the weak coupling constant.
The interactions involving charged lepton \ell and massless neutrino \nu_\ell fields are parametrized by
The quantities
\kappa_L^\ell are real-valued and serve as normalizations for leptonic coupling strengths. As no right-handed neutrinos are present in the SM, the correspondingright-handed leptonic new physics couplings are omitted (
\zeta_R^\nu = \kappa_R^\ell = 0).
From our general Lagrangians, the SSM limit is obtained by imposing the coupling strengths to be equal to the SM weak couplings up to an overall normalization factor:
In the canonical SSM, the overall normalizations are further trivially fixed as
NLO in QCD Corrections
The above Lagrangian in the Feynman Gauge was implemented into FeynRules 2.3.10. QCD renormalization and R2 rational counter terms were determined using NLOCT 1.02 and FeynArts 3.8. Feynman rules were collected into a single UFO, available below. In the UFO file, five massless quarks are assumed as well as vanishing off-diagonal CKM matrix entries. For additional details, see [ 4 ] and references therein. These additions permit tree-level calculations at LO and NLO in QCD, loop-induced calculations at LO in QCD, and NLO calculations matched to NNLL(Veto) resummation using MadGraph_aMC@NLO.
Model Files VPrime_NLO_UFO.tgz: Standalone UFO folder. vPrimeNLO.fr: Main model file. Relies on sm.fr (default FR model file) being declared elsewhere. vPrimeNLO_Notebook.nb: Mathematica notebook file that generates UFO file from FeynRules model files. Allows user to also run quick sanity checks (optional) on model. WZPrimeAtNLOfiles.tgz: Standalone package containing vPrimeNLO.fr, vPrimeNLO_Notebook.nb, massless.rst (default FR file), diagonalCKM.rst (default FR file), and sm.fr (default FR file). WZPrimeAtNLOfiles_withUFO.tgz: Combination of WZPrimeAtNLOfiles.tgz and VPrime_NLO_UFO.tgz. Notes To download any of the packages and unpack via the terminal, use the commands:
~/Path $ tar -zxvf VPrime_NLO_UFO.tgz
~/Path $ tar -zxvf WZPrimeAtNLOfiles_withUFO.tgz
This model contains six free parameters by default: Two masses: mzp, mwp, with default values of 3 TeV. Two widths: wzp, wwp, with default values of 89.59 and 101.27 GeV, respectively. Two coupling normalizations: kL, kR, with default values 1.0 and 0.0, respectively. Validation The model file was validated at NLO in [ 4 ] by comparing against Standard Model predictions and W 'predictions from [ 5 ], as well as private LO and NLO calculations. See Table III of [ 4 ] for representative total widths that have been checked against analytic computation. List of studies that have used the model file Please email to update this space. References Please cite [3,4] for the model file, [4,5] for NLO corrections, and [6] for automated veto resummation. Attachments (5) vPrimeNLO.fr (7.3 KB) - added by richardphysics3 years ago. W' Z' FeynRules model file in Feynman gauge vPrimeNLO_Notebook.nb (16.1 KB) - added by richardphysics3 years ago. W' Z' NLO FR mathematica notebook that loads model file, performs basic sanity checks, and outputs to desired format, e.g., UFO. VPrime_NLO_UFO.tgz (21.3 KB) - added by richardphysics3 years ago. Stand-alone NLO UFO model file: assumes top quark is the only massive fermion + diagonal CKM WZPrimeAtNLOfiles.tgz (11.9 KB) - added by richardphysics3 years ago. Standalone package containing vPrimeNLO.fr, vPrimeNLO_Notebook.nb, massless.rst (default FR file), diagonalCKM.rst (default FR file), and sm.fr (default FR file). WZPrimeAtNLOfiles_withUFO.tgz (46.7 KB) - added by richardphysics3 years ago. Combination of WZPrimeAtNLOfiles.tgz and VPrime_NLO_UFO.tgz
Download all attachments as: .zip
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The offset tree is a reduction from cost-sensitive multiclass classification (CSMC) with partial feedback to binary classification. It is designed for when there is a single decision amongst a fixed set of classes (like vanilla CSMC) and historical data where only the value of the decision chosen has been revealed (unlike vanilla CSMC). As a side note, this difference is fundamental, since there are reductions from CSMC to binary classification which have regret independent of the number of classes; whereas the offset tree regret bound (proportional to $|A| - 1$) is a provable lower bound.
So a natural question to ask is: if I can reduce a full information version of a problem (e.g., SSP without recourse) to set of CSMC subproblems, can I reduce a partial information version of the problem to a set of CSMC with partial feedback subproblems (and leverage the offset tree)? I don't know for sure, but I suspect so.
In order to build intuition, however, I'm going to start with something easier. Since I actually want to reduce problems to constrained CSMC, I need something that can handle constrained CSMC with partial feedback. That suggests defining the forfeit offset tree analogous to the forfeit filter tree.
The setup is as follows. There is a distribution $D = D_x \times D_{\omega|x} \times D_{r|\omega,x}$ where $r: A \to [0, 1] \cup \{ -\infty \}$ takes values on the unit interval augmented with $-\infty$, and the components of $r$ that are $-\infty$ valued for a particular instance are revealed as part of the problem instance via $\omega \in \mathcal{P} (A)$ (i.e., $\omega$ is a subset of $A$). The regret of a particular deterministic policy $h: X \times \mathcal{P} (A) \to A$ is \[ v (h) = E_{(x, \omega) \sim D_x \times D_{\omega|x}} \left[ \max_{k \in A}\; E_{r \sim D_{r|\omega,x}} \left[ r (k) \right] - E_{r \sim D_{r|\omega,x}} \left[ r (h (x, \omega)) \right] \right]. \] So far this is identical to CSMC with cosmetic changes to align with the reinforcement learning literature: costs (now called ``rewards'') have been negated and compressed into the unit interval; and classes are now called ``actions.'' This is because the only real difference is in the partial nature of the training data, which manifests itself only via the induced distribution for subproblems. To define the induced distribution I'll assume that the historical policy is using a known conditional distribution over actions given an instance $p (a | x, \omega)$; in practice there are ways to relax this assumption.
Algorithm:Forfeit Offset Tree Train
Data:Partially labelled constrained CSMC training data set $S$. Input:Importance-weighted binary classification routine $\mbox{Learn}$. Input:A binary tree $T$ over the labels with internal nodes $\Lambda (T)$. Result:Trained classifiers $\{\Psi_n | n \in \Lambda (T) \}$. For each $n \in \Lambda (T)$ from leaves to roots: $S_n = \emptyset$. For each example $(x, \omega, a, r (a), p (\cdot | x, \omega)) \in S$: Let $\lambda$ and $\phi$ be the two classes input to $n$ (the predictions of the left and right subtrees on input $(x, \omega)$ respectively). If $\lambda \in \omega$, predict $\phi$ for the purposes of constructing training input for parent node (``$\lambda$ forfeits''); else if $\phi \in \omega$, predict $\lambda$ for the purposes of constructing training input for parent node (``$\phi$ forfeits''); else (when $\lambda \not \in \omega$ and $\phi \not \in \omega$) if $a = \lambda$ or $a = \phi$: If $a = \lambda$ then $a^\prime = \phi$; else $a^\prime = \lambda$. Let $y = 1_{a^\prime = \lambda}$, i.e., $a^\prime$ is from the left subtree of $n$. If $r (a) < \frac{1}{2}$, $S_n \leftarrow S_n \cup \left\{ \left( x, y, \frac{p (a | x, \omega) + p (a^\prime | x, \omega)}{p (a | x, \omega)} \left(\frac{1}{2} - r (a)\right) \right) \right\}$; else $S_n \leftarrow S_n \cup \left\{ \left( x, 1 - y, \frac{p (a | x, \omega) + p (a^\prime | x, \omega)}{p (a | x, \omega)} \left(r (a) - \frac{1}{2}\right) \right) \right\}$. Let $\Psi_n = \mbox{Learn} (S_n)$. Return $\{\Psi_n | n \in \Lambda (T) \}$.
Algorithm:Forfeit Offset Tree Test
The Forfeit Offset Tree is nearly identical to the offset tree reduction of unconstrained partially labelled CSMC, with the addition that infeasible choices automatically lose any tournament they enter (if both entrants are infeasible, one is chosen arbitrarily). Because of this rule the underlying binary classifier is only invoked for distinctions between feasible choices and the regret analysis is essentially the same. Note that even if the historical exploration policy $p (a | x, \omega)$ never chooses an infeasible action, forfeiture is still important for determining the ``alternate action'' when training an internal node. If the historical exploration policy actually chooses infeasible actions, those training examples are skipped.
Input:A binary tree $T$ over the labels with internal nodes $\Lambda (T)$. Input:Trained classifiers $\{\Psi_n | n \in \Lambda (T) \}$. Input:Instance realization $(x, \omega)$. Result:Predicted label $k$. Let $n$ be the root node. Repeat until $n$ is a leaf node: If all the labels of the leaves in the left-subtree of $n$ are in $\omega$, traverse to the right child; else if all the labels of the leaves in the right-subtree of $n$ are in $\omega$, traverse to the left child; else if $\Psi_n (x) = 1$, traverse to the left child; else (when $\Psi_n (x) = 0$ and at least one label in each subtree is not in $\omega$), traverse to the right child. Return leaf label $k$. Regret AnalysisThe regret analysis for the forfeit offset tree is very similar to the regret analysis for the offset tree, with additional arguments for forfeiture cases.
Let $\Psi = (T, \{\Psi_n | n \in \Lambda (T) \})$ denote a particular forfeit offset tree (i.e., a choice of a binary tree and a particular set of node classifiers), and let $h^\Psi$ denote the policy that results from the forfeit offset tree. The regret analysis leverages an induced importance-weighted binary distribution $D^\prime (\Psi)$ over triples $(x^\prime, y, w)$ defined as follows:
Draw $(x, \omega, r)$ from $D$. Draw $n$ uniform over the internal nodes $\Lambda (T)$ of the binary tree. Let $x^\prime = (x, n)$. Let $\lambda$ and $\phi$ be the two classes input to $n$ (the predictions of the left and right subtrees on input $x$ respectively). If $\lambda \in \omega$, create importance-weighted binary example $(x^\prime, 0, 0)$; else if $\phi \in \omega$, create importance-weighted binary example $(x^\prime, 1, 0)$; else (when $\lambda \not \in \omega$ and $\phi \not \in \omega$): Draw $a$ from $p (a | x, \omega)$. If $a \neq \lambda$ and $a \neq \phi$, reject sample; else (when $a = \lambda$ or $a = \phi$): If $a = \lambda$, $a^\prime = \phi$; else $a = \phi$, $a^\prime = \lambda$. Let $y = 1_{a^\prime = \lambda}$ If $r (a) < \frac{1}{2}$, create importance-weighted binary example \[\left( x^\prime, y, \frac{p (a | x, \omega) + p (a^\prime | x, \omega)}{p (a | x, \omega)} \left(\frac{1}{2} - r (a)\right) \right);\] else, create importance-weighted binary example \[ \left( x^\prime, 1 - y, \frac{p (a | x, \omega) + p (a^\prime | x, \omega)}{p (a | x, \omega)} \left(r (a) - \frac{1}{2}\right) \right). \]
Theorem:Forfeit Offset Tree Regret Bound
This is the same bound as for the offset tree, showing that the introduction of constraints does not alter the fundamental properties of the problem setting. Parenthetically, in order to understand the origin of the $\frac{1}{2}$ factor, one has to reduce all the way to binary classification, in which case the worst-case bound on the expected importance becomes a factor, and this is minimized by the choice of $\frac{1}{2}$ (or the median reward at each internal node if this can be known apriori).
For all partially labelled CSMC distributions $D$; all historical policies $p$ such that $p (a | x, \omega) > 0$ whenever $a \not \in \omega$; and all forfeit offset trees $\Psi$, \[ v (h^\Psi) \leq (|A| - 1) q (\Psi) \] where $q (\Psi)$ is the importance-weighted binary regret on the induced subproblem.
Proof:See Appendix.
A natural next step is the look at cost-sensitive best m with partial feedback; perhaps the forfeit offset tree will be useful in that context.
AppendixThis is the proof of the regret bound theorem.
Consider a fixed $(x, \omega)$. It is useful to talk about the conditional policy regret experienced at an internal node $n$, \[ v (h^\Psi | x, \omega, n) = \max_{k \in \Gamma (n)} E_{r \sim D_{r|\omega,x}} \left[ r (k) \right] - E_{r \sim D_{r|\omega,x}} \left[ r (h^\Psi_n (x, \omega)) \right]. \] where $h_n^\Psi$ is the prediction at internal node $n$. When $n$ is the root of the tree, $v (h^\Psi | x, \omega, n)$ is the forfeit offset tree policy regret conditional on $(x, \omega)$.
The proof strategy is to bound $v (h^\Psi | x, \omega, n) \leq \sum_{m \in \Lambda (n)} q_m (\Psi | x, \omega)$ via induction. The base case is trivially satisfied for trees with only one leaf (no internal nodes) since it evaluates to $0 \leq 0$. To show the recursion at a particular internal node $n$, let $\lambda$ and $\phi$ be the predictions of the left subtree ($n_\lambda$) and right subtree ($n_\phi$).
Case 1: $\Gamma (n_\lambda) \setminus \omega = \emptyset$. In this case $\lambda \in \omega$ and forfeits, so $\phi$ is chosen. There must be a maximizer in the right subtree, since all values in the left subtree are $-\infty$. Furthermore $q_m (\Psi | x, \omega) = 0$ for both $m = n$ and for $m \in \Lambda (n_\lambda)$ by definition. Therefore \[ \begin{aligned} v (h^\Psi | x, \omega, n) &=
\max_{k \in \Gamma (n)} E_{r \sim D_{r|\omega,x}} \left[ r (k) \right] - E_{r \sim D_{r|\omega,x}} \left[ r (\phi) \right] \\ &= \max_{k \in \Gamma (n_\phi)} E_{r \sim D_{r|\omega,x}} \left[ r (k) \right] - E_{r \sim D_{r|\omega,x}} \left[ r (\phi) \right] \\ &= v (h^\Psi | x, \omega, n_\phi) \\ &\leq \sum_{m \in \Lambda (n_\phi)} q_m (\Psi | x, \omega) \\ &= \sum_{m \in \Lambda (n)} q_m (\Psi | x, \omega). \end{aligned} \]
Case 2: $\Gamma (n_\lambda) \setminus \omega \neq \emptyset$ and $\Gamma (n_\phi) \setminus \omega = \emptyset$. In this case $\phi \in \omega$ and $\lambda \not \in \omega$, so $\phi$ forfeits and $\lambda$ is chosen. There must be a maximizer in the left subtree, since all values in the right subtree are $-\infty$. Furthermore $q_m (\Psi | x, \omega) = 0$ for both $m = n$ and for $m \in \Lambda (n_\phi)$ by definition. Therefore \[ \begin{aligned} v (h^\Psi | x, \omega, n) &=
\max_{k \in \Gamma (n)} E_{r \sim D_{r|\omega,x}} \left[ r (k) \right] - E_{r \sim D_{r|\omega,x}} \left[ r (\lambda) \right] \\ &= \max_{k \in \Gamma (n_\lambda)} E_{r \sim D_{r|\omega,x}} \left[ r (k) \right] - E_{r \sim D_{r|\omega,x}} \left[ r (\lambda) \right] \\ &= v (h^\Psi | x, \omega, n_\lambda) \\ &\leq \sum_{m \in \Lambda (n_\lambda)} q_m (\Psi | x, \omega) \\ &= \sum_{m \in \Lambda (n)} q_m (\Psi | x, \omega). \end{aligned} \]
Case 3: $\Gamma (n_\lambda) \setminus \omega \neq \emptyset$ and $\Gamma (n_\phi) \setminus \omega \neq \emptyset$. This is the ``normal'' offset tree case, where both $\lambda \not \in \omega$ and $\phi \not \in \omega$ so no forfeiture happens. The expected importance weights conditioned on $(x, \omega, r)$ and $\lambda \not \in \omega$ and $\phi \not \in \omega$ are \[ \begin{aligned} w_{\lambda|r} &= E_{a \sim p} \biggl[ 1_{a = \lambda} 1_{r (a) \geq \frac{1}{2}}\frac{p (\lambda | x, \omega) + p (\phi | x, \omega)}{p (\lambda | x, \omega)} \left(r (\lambda) - \frac{1}{2}\right) \\ &\quad \quad \quad \quad + 1_{a = \phi} 1_{r (a) < \frac{1}{2}} \frac{p (\lambda | x, \omega) + p (\phi | x, \omega)}{p (\phi | x, \omega)} \left(\frac{1}{2} - r (\phi)\right) \biggr] \biggl/ \\ &\quad \quad E_{a \sim p} \left[ 1_{a = \lambda} + 1_{a = \phi} \right] \\ &= \left( r (\lambda) - \frac{1}{2} \right)_+ + \left( \frac{1}{2} - r (\phi) \right)_+, \\ w_{\phi|r} &= \left( r (\phi) - \frac{1}{2} \right)_+ + \left( \frac{1}{2} - r (\lambda) \right)_+, \\ \end{aligned} \] where $\left( x \right)_+ = \max (x, 0)$. Thus \[ | w_\lambda - w_\phi | = \left| E_{r \sim D_{r|\omega,x}} \left[ w_{\lambda|r} - w_{\phi|r} \right] \right| = \left| E_{r \sim D_{r|\omega,x}} [r (\lambda) - r (\phi)] \right|, \] i.e., the importance-weighted regret at an internal node is equal to the policy regret with respect to the two actions input to that node.
Assume without loss of generality that the classifier chooses $\phi$. If the maximizer comes from the right subtree, then \[ \begin{aligned} v (h^\Psi | x, \omega, n) &= \max_{k \in \Gamma (n_\phi)} E_{r \sim D_{r|\omega,x}} \left[ r (k) \right] - E_{r \sim D_{r|\omega,x}} \left[ r (\phi) \right] \\ &= v (h^\Psi | x, \omega, n_\phi) \\ &\leq \sum_{m \in \Lambda (n_\phi)} q_m (\Psi | x, \omega) \\ &\leq \sum_{m \in \Lambda (n)} q_m (\Psi | x, \omega). \end{aligned} \] If the maximizer comes from the left subtree, then \[ \begin{aligned} v (h^\Psi | x, \omega, n) &= \max_{k \in \Gamma (n_\lambda)} E_{r \sim D_{r|\omega,x}} \left[ r (k) \right] - E_{r \sim D_{r|\omega,x}} \left[ r (\phi) \right] \\ &= E_{r \sim D_{r|\omega,x}} \left[ r (\lambda) - r (\phi) \right] + v (h^\Psi | x, \omega, n_\lambda) \\ &= q_n (\Psi | x, \omega) + v (h^\Psi | x, \omega, n_\lambda) \\ &\leq q_n (\Psi | x, \omega) + \sum_{m \in \Lambda (n_\lambda)} q_m (\Psi | x, \omega) \\ &\leq \sum_{m \in \Lambda (n)} q_m (\Psi | x, \omega). \end{aligned} \] Terminating the induction at the root yields \[ v (h^\Psi | x, \omega) \leq \sum_{n \in \Lambda (T)} q_n (\Psi | x, \omega) = |\Lambda (T)| q (\Psi | x, \omega). \] Taking the expectation of both sides with respect to $D_x \times D_{\omega|x}$ and noting $|\Lambda (T)| = (|A| - 1)$ completes the proof.
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Motivation: As is stated in the former post, a left ideal $I$ of a maximal order $\mathcal{O}$ in a quaternion algebra ramified at $p$ ($p$ is a prime) and $\infty$ can be used to construct modular form $\theta_I$ of weight 2 with Fricke eigenvalue $-1:$
$$\theta_I(\tau)=\sum_{x\in I}e^{2\pi i\tau\frac{N(x)}{N(I)}}$$ One can use the quotient of different linear combinations of theta functions $\theta_I$ to construct explicit equations of modular curves $X_0^{+}(p)$. It is of great interest to investigate the exact positions of the poles of certain quotients (or the zeros of certain linear combination of theta functions from quaternion algebra).
Experiment: Let $p=37$ (a case which is known to E. Hecke). Suppose the maximal order $\mathcal{O}$ in $A(37)$ is the order given by the Proposition 5.2 in A. Pizer's paper, then one can find the class number of left-$\mathcal{O}$ ideals in $A(37)$ is $3$, and there are two different theta functions associated to these ideals:
$$\theta_{I_i}(\tau)=\sum_{x\in\mathbb{Z}^4}q^{x^{T}M_ix},q=e^{\pi i\tau},i=1,2$$
while the $M_1$ and $M_2$ are two Gram matrices attached to two even lattices of dimension $4$:
$$M_1=\left(\begin{matrix}2 & 0 & 1 & 1\\ 0 & 4 & 1 &2\\ 1 & 1 & 10 & 1\\ 1 & 2 & 1 & 20 \end{matrix}\right),M_2=\left(\begin{matrix}4 & 1 & 0 & 1\\ 1 & 6 & 3 &1\\ 0 & 3 & 8 & 2\\ 1 & 1 & 2 & 10 \end{matrix}\right).$$
Then $\phi=(\theta_{I_1}-\theta_{I_2})/2$ is a cusp form on $\Gamma_0(p)$ with Fricke eigenvalue $-1$($\phi$ is such cusp form with zeros at cusps of the possible highest order).
We note that $$\Phi(\tau)=\phi(\tau)\prod_{k=0}^{p-1}\phi\left(\frac{\tau+k}{p}\right)$$ is a modular form of weight $2(p+1)$ on $\Gamma(1)$. As $\Gamma_0(37)$ has two elliptic point of order $2$ and two elliptic points of order $3$, then $$\varphi=\frac{\Phi}{E_4^4E_6^2\Delta^4}$$ is a modular function on $\Gamma(1)$ (everywhere holomorphic except a double pole at $\infty$), $E_4,E_6,\Delta$ are Eisenstein series and modular discriminant, respectively. It is possible to determine numerically the first few coefficients of the $q$-expansion of $\varphi$, and one can determine that $$\varphi(\tau)=(j(\tau)-66^3)^2,$$ which leads to the conclusion that $\phi(\tau)$ vanishes at $(\pm12+2i)/37$.
Experiment Results: One can extend these calculations to some other primes. I tried to construct modular forms of weight $2$ on $\Gamma_0(p)$ with Fricke eigenvalue $-1$ and zeros of highest possible order at the cusp of $\Gamma_0(p)$ for a few primes, and all the zeros of such cusp forms are quadratic irrationals. The results are listed in the table below. Notations: We denote $\#0$ to be the highest possible order of zeros at the cusp of $\Gamma_0(p)$ for the linear combinations of theta functions. Let $-D,D>0$, $(a_i,b_i,c_i)$ be all the possible reduced binary quadratic forms with discriminant $-D$ and $\tau_{(i,-D)}=(-b_i+\sqrt{-D})/(2a_i)$. We denote a zero of those modular forms by $(\tau_{(i,-D)},k),k\in\mathbb{Z}$, since those modular forms vanish at $(\tau_{(i,-D)}+k)/p$.
\begin{array}{|c|r|c|c|c|c|c|c|} \hline p & \#0 & 1 & 2 & 3 & 4 & 5 & 6\\ \hline 37&1& (\tau_{(1,-16)},12) & & & & & \\ 43&2& (\tau_{(1,-12)},13)& & & & & \\ 67& 3& (\tau_{(1,-12)},8) & (\tau_{(1,-7)},12)& & & & \\ 73& 4& (\tau_{(1,-27)},26)& & & & & \\ 97& 6& (\tau_{(1,-12)},26)& & & & & \\ 163& 7& (\tau_{(1,-7)},26)& (\tau_{(1,-11)},29)& (\tau_{(1,-27)},13)& (\tau_{(1,-44)},57)& (\tau_{(2,-44)},35)& (\tau_{(3,-44)},90)\\ 193& 9& (\tau_{(1,-8)},34)& (\tau_{(1,-16)},31) & (\tau_{(1,-24)},90)& (\tau_{(2,-24)},45)& (\tau_{(1,-75)},37)& (\tau_{(2,-75)},77)\\ \hline \end{array}
Question: What is the reason that all the zeros of these modular forms are quadratic irrationals?
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In the standard GARCH(1,1) model with normal innovations
$\sigma^2_t=\omega+\alpha\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $
the likelihood of $m$ observations occurring in the order in which they are observed is
$\sum_{t=1}^m\left[-\ln(\sigma^2_{t})-{\left(\frac{\epsilon^2_{t}}{\sigma^2_{t}}\right)}\right] $
This expression, with the usual caveats of optimization, allows us to obtain the MLE estimates of the GARCH(1,1) parameters.
However, in the GJR-GARCH(1,1) model by Glosten et al. (1993), the conditional variance is
$ \sigma^2_t=\omega+(\alpha+\gamma I_{t-1})\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $
where $ I_{t-1} $ is the indicator function:
$I_{t-1}(\epsilon_{t-1})=1 $ when $\epsilon_{t-1}<0$ and $I_{t-1}(\epsilon_{t-1})=0$ otherwise. Question: Is there a closed-form expression for the likelihood function in the GJR-GARCH(1,1) with normal innovations? EDIT: Per comments, the likelihood function in the GJR-GARCH(1,1) model is the same than in the standard GARCH(1,1): Can someone provide a reference/explanation to justify this? If we use empirical innovations instead of normal ones (e.g: a Filtered Historical Simulation/FHS approach), would this change the functional form of the likelihood function? (my guess is that empirical innovations does not affect the likelihood function, but any reference or explanation will be highly welcome)
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Say we've previously used a neural network or some other classifier C with $N$ training samples $I:=\{I_1,...I_N\}$ (that has a sequence or context, but is ignored by C) the, belonging to $K$ classes. Assume, for some reason (probably some training problem or declaring classes), C is confused and doesn't perform well. The way we assign a class using C to each test data $I$ is: $class(I):= arg max _{ {1 \leq j \leq K} } p_j(I)$, where $p_j(I)$ is the probability estimate of $I$ corresponding to the $j$-th class, given by C.
Now, on top of this previous classifier C, I'd like to use a Hidden Markov Model (HMM) to "correct" the mistakes made by the previous context-free classifier C, by taking into account the contextual/sequential information not used by C.
Hence let in my HMM, the hidden state $Z_i$ denote the true class of the $i$-th sample $I_i$, and $X_i$ be the predicted class by C. My question is: how could we use the probabilistic information $cl(I):= arg max _{ {1 \leq j \leq K} } p_j(I)$ to train this HMM? I understand that the
confusion matrix of C can be used to define the emission prob. of the HMM, but how do we define the transition and start/prior prob.? I'm tempted to define the start/prior prob. vector as $\pi:=(p_1(x_1), ..., p_K(x_1))$. But I may be wrong. This is my main question. A follow up question: One can define an HMM in the above way (using confusion matrix and the prob. information from C); call the resulting parameter set $\Theta_0$. But after doing so, is it advisable to estimate the parameters to best fit the data $I$ used for C, while initializing a parameter set with the values mentioned in the previous paragraph?
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Let $(T_t)$ be a strongly continuous semigroup of positive operators on $C(K)$, where $K$ is a compact space. Assume also that $T_t1 =1 $ for every $t\geq 0$. (This is also called a Feller Semigroup.)
Since $K$ is compact we know that there exists a probability measure $\mu$ on $K$ satisfying $\mu T^*_t = \mu $ for every $t\geq 0$ (i.e. $\mu$ is invariant).
My question is: to show that $\mu$ is the unique invariant probability distribution, is it sufficient to show that $(T_t)$ is irreducible?
Recall that a semigroup is by defintion irreducible if the resolvent $R_\lambda=(\lambda-L)^{-1}$ ($L$ is the generator of $(T_t)$) maps for sufficiently large $\lambda$ nonnegative nonzero functions into strictly positive functions.
I thought this should be true by applying some version of the Krein-Rutman theorem, but did not find a suitable reference.
The closest I found is Proposition 3.5. on p. 185 of this book http://www.springer.com/mathematics/algebra/book/978-3-540-16454-8 , from which, if I understand well, I can just conclude that $\text{dim (ker } L) = 1$, but not $\text{dim (ker } L^*) = 1$.
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Well there are problems in your question and analysis. First off, there have been a few SE questions recently about this "Keplerian" treatment of dark matter. The shell theorem, that the gravitational field is the equivalent of that due to the mass inside radius $r$, and that exterior masses can be ignored is
only true for spherically symmetric mass distributions or cases where most of the mass is centrally concentrated inside radius $r$. Any book that fails to point this out is making a serious omission, probably in the interests of simplifying the argument. Real work in this area does not make that assumption (e.g. Sofue 2011).
Even adopting the shell theorem, it does not imply $ v \propto r^{-1/2}$, it implies $v \propto (M(R)/r)^{1/2}$.
The correct argument for dark matter is that if we assume that the visible matter and gas traces the mass and has a certain mass-to-light ratio, then we find that (i) the rotation speeds of stars and gas are too high and that (ii) one would expect the velocities to drop as $r^{-1/2}$
at large radii, whereas they appear to actually be flat or even increasing.
The latter point works because the visible matter implies that there is hardly any mass at large radii, and so the Keplerian approximation is valid there.
Now, as for your model. If $\rho \propto r^{-1}$ in shells, this would imply that annuli with a given thickness $\Delta r$ contain similar quantities of mass! $\Delta M = 2\pi r z\rho\,\Delta r$ (where $z$ is a thickness for the disk). So, rather than having a galaxy whose
azimuthally integrated luminosity decreased with distance from the centre, your Galaxy, without dark matter, would have to have constant integrated luminosity, as a function of radius as you move away from the centre (or to satisfy my critics below, if you are observing the Galaxy face on, the surface brightness of light would fall as just $r^{-1}$). And, of course without defining some sort of cut-off radius, the total mass of your Galaxy would soon become large! However, if you assume that most of this matter is dark then indeed you might be able to explain the rotation curve of the Galaxy using such a density law!
Below I show an example (using surface brightness) for M31 (taken from Corteau et al. (2012)), using various luminosity indicators, on which I have marked a $\rho \propto r^{-1}$ dependence. A $r^{-1}$ does work reasonably in the inner part of the disk (it is actually a bit shallower than that because of the bulge), but at some point $r> 10$ kpc, the luminous matter just runs out and the observed intensity distribution becomes steeper than $r^{-1}$.
In fact the most commonly used prescription for dark matter is the Navarro, Frenk & White dark matter profile, $$ \rho(r) = \frac{\rho_0 R_s}{r(1 + r/R_s)^2}, $$where $R_s$ is a length scale (of order 15 kpc for the Milky Way and M31). When $r< R_s$, this
does scale as $1/r$ and does explain the flat rotation curve!
i.e. Your analysis that a $\rho \propto 1/r$ relationship leads to a flat rotation curve is roughly correct. However, the fact that intensity in our Galaxy (and others) falls off more steeply than $r^{-1}$ in the outer parts of the Galaxy leads to the conclusion that the matter is ... dark! There is an additional problem with the normalisation as well. Even in the inner parts of the disk, the mass implied by the luminous matter is insufficient (by factors of a few) to explain the rotation speeds.
So now to answer the last part of your question - how do we know that $\rho$ (of luminous matter) does not fall as $1/r$ in the disk. This is just a question of counting up stars and estimating the contribution of gas from HI surveys (and dust, though this is negligible). There is no single source of this information (though here is an example I pick at random that uses SDSS number counts), it is agglomerated from many different surveys at different wavelengths and built up to give a coherent picture. The underlying assumptions are that we understand the types and mixture of stars that make up the overall stellar populations. Our understanding could be incorrect, but the way in which it would need to be incorrect to explain rotation curves is to have lots (and I mean orders of magnitude) more dim stars that contribute mass but no light at large radii (i.e. dark matter, though baryonic, which doesn't help you with other pieces of evidence for dark matter).
For example, the luminosity of the M31 data shown above drops off steeper than $r^{-1}$. If you were to extrapolate a $r^{-1}$ relationship, then to ensure that the mass
did go as $r^{-1}$ you would need the mass to luminosity ratio to increase by some factors of ten. To do this would require orders of magnitude more faint stars to bright stars than are observed in the local disk.
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The following is another solution that uses "analytic geometry," at a lower level of technical sophistication. The downside is that the equations look uglier and less structured than they could be.
We will use the not very hard to prove fact that if a line has equation $ax+by+c=0$,then the perpendicular distance from $(u,v)$ to the line is$$\frac{|au+bv+c|}{\sqrt{a^2+b^2}}.$$Note that in the numerator we have an absolute value. That will cause some headaches later.
(The formulas that follow would look simpler if we adjusted the equation of any line $ax+by+c=0$ by dividing each coefficient by $\sqrt{a^2+b^2}$, but we will not do that.)
Given two points on each of our lines, we can find equations for the lines.Suppose that these equations turn out to be$$a_1x+b_1y+c_1=0\qquad\text{and}\qquad a_2x+b_2y+c_2=0.$$
Let $(u,v)$ be the center of the circle, and let $r$ be its radius.Then from the "distance to a line" formula, we have$$\frac{a_1u+b_1v+c_1}{\sqrt{a_1^2+b_1^2}}=\pm r \qquad\text{and}\qquad
\frac{a_2u+b_2v+c_2}{\sqrt{a_2^2+b_2^2}}=\pm r.$$
Unfortunately that gives in general $4$ possible systems of two linear equations, which correspond to the generally $4$ (in the parallel case, $3$) pieces into which the lines divide the plane. At the end of this post are some comments about how to identify which signs to choose.
But suppose for now that we have identified the two relevant equations. We can use them to "eliminate" $r$, and obtain a linear equation $ku+lv+m=0$ that links $u$ and $v$.
Since $(u,v)$ is the center of the circle, and $r$ is the radius, we have$$(u-x_p)^2+(u-y_p)^2=r^2.$$
Use one of our linear expressions for $r$ to substitute for the $r^2$ term.We obtain a quadratic equation in $u$ and $v$. Use the equation $ku+lv+m=0$ to eliminate one of the variables. We are left with a quadratic equation in the other variable. Solve.
Note that in general we will get two solutions, since, almost always, there are two circles that work, a small circle with $(x_p,y_p)$ on the other side of the circle from where the two given lines meet, and a big circle with $(x_p,y_p)$ on the near side of the circle from where the two given lines meet.
Sign issues: It remains to deal with how to choose the $\pm$ signs in the distance equations. One approach that works reasonably well is, in our line equations $a_ix+b_iy+c_i=0$, always to choose the coefficient of $y$ to be positive. (This can be done unless the line is vertical.) Then if $a_1x_p+b_1y_p+c_1$ is positive, use $(a_1x_p+b_1y_p+c_1)/\sqrt{a_1^2+b_1^2}=r$, and if it is negative use $-r$. Do the same with the other line. The reason this works is that that if the coefficient of $y$ is positive, then $a_ix+b_iy+c_i$ is positive for fixed $x$ and large $y$. So if $a_ix_p+b_iy_p+c_i$ is positive, then $(x_p,y_p)$ lies "above" the line, and if $a_ix_p+b_iy_p+c_i$ is negative, then $(x_p,y_p)$ lies below the line.
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Table of Contents
Basic Theorems on the Composition of Two Functions
Recall from The Composition of Two Functions page that if $f : A \to B$ and $g : B \to C$ are functions then the composition of $f$ and $g$ is the function $g \circ f : A \to C$ defined for all $x \in A$ by $(g \circ f)(x) = g(f(x))$.
On that page, we proved that:
If $f$ and $g$ are injective then $g \circ f$ is injective. If $f$ and $g$ are surjective then $g \circ f$ is surjective. If $f$ and $g$ are bijective then $g \circ f$ is bijective.
We will now look at some slightly stronger theorems regarding the composition of two functions.
Theorem 1: Let $f : A \to B$, $g : B \to C$, and let $g \circ f$ be injective. Then $f$ is injective. Proof:Let $g \circ f$ be injective. Suppose that $f(x) = f(y)$. Then by applying $g$ to both sides of the equation above we have that: Since $g \circ f$ is injective, we have that from the equation above that then $x = y$. Therefore $f$ is injective. $\blacksquare$
Theorem 2: Let $f : A \to B$, $g : B \to C$, and let $g \circ f$ be surjective. Then $g$ is surjective. Proof:Let $g \circ f$ be surjective. Then for each $y \in C$ there exists $x \in A$ such that $g(f(x)) = y$. Let $b = f(x)$. Then for every $y \in C$ there exists a $f(x) = b \in B$ such that: Therefore $g$ is surjective. $\blacksquare$
Corollary 1: Let $f : A \to B$, $g : B \to C$, and let $g \circ f$ be bijective. Then $f$ is injective and $g$ is surjective. Proof:Let $g \circ f$ be bijective. Then $g \circ f$ is both injective and surjective. By Theorem 1 we have that then $f$ is injective, and by Theorem 2 we have that then $g$ is surjective. $\blacksquare$
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A bijective ring homomorphism is called an isomorphism.
If there is an isomorphism from $R$ to $S$, then we say that rings $R$ and $S$ are isomorphic (as rings).
Proof.
Suppose that the rings are isomorphic. Then we have a ring isomorphism\[f:2\Z \to 3\Z.\]Let us put $f(2)=3a$ for some integer $a$. Then we compute $f(4)$ in two ways.First we have\[f(4)=f(2+2)=f(2)+f(2)=3a+3a=6a.\]
Next we have\[f(4)=f(2\cdot 2)=f(2)\cdot f(2)=3a\cdot 3a=9a^2.\]These are equal and hence we have\[6a=9a^2.\]The only integer solution is $a=0$.
But then we have $f(0)=0=f(2)$, which contradicts that $f$ is an isomorphism (hence in particular injective).Therefore, there is no such isomorphism $f$, thus the rings $2\Z$ and $3\Z$ are not isomorphic.
Characteristic of an Integral Domain is 0 or a Prime NumberLet $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.Definition of the characteristic of a ring.The characteristic of a commutative ring $R$ with $1$ is defined as […]
Generators of the Augmentation Ideal in a Group RingLet $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by\[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\]where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring […]
The Preimage of Prime ideals are Prime IdealsLet $f: R\to R'$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R'$.Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.Proof.The preimage of an ideal by a ring homomorphism is an ideal.(See the post "The inverse image of an ideal by […]
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Stochastic partial differential equation models for spatially dependent predator-prey equations DCDS-BHome This Issue
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Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment Detailed analytic study of the compact pairwise model for SIS epidemic propagation on networks
Institute of Mathematics, Eötvös Loránd University Budapest, Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary
The global behaviour of the compact pairwise approximation of SIS epidemic propagation on networks is studied. It is shown that the system can be reduced to two equations enabling us to carry out a detailed study of the dynamic properties of the solutions. It is proved that transcritical bifurcation occurs in the system at $ \tau = \tau _c = \frac{\gamma n}{\langle n^{2}\rangle-n} $, where $ \tau $ and $ \gamma $ are infection and recovery rates, respectively, $ n $ is the average degree of the network and $ \langle n^{2}\rangle $ is the second moment of the degree distribution. For subcritical values of $ \tau $ the disease-free steady state is stable, while for supercritical values a unique stable endemic equilibrium appears. We also prove that for subcritical values of $ \tau $ the disease-free steady state is globally stable under certain assumptions on the graph that cover a wide class of networks.
Keywords:SIS epidemic, pairwise approximation, transcritical bifurcation, global stability, network process. Mathematics Subject Classification:Primary: 34C23, 34D23, 92C42. Citation:Noémi Nagy, Péter L. Simon. Detailed analytic study of the compact pairwise model for SIS epidemic propagation on networks. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 99-115. doi: 10.3934/dcdsb.2019174
References:
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K. T. D. Eames and M. J. Keeling,
Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases,
[3] [4] [5]
J. K. Hale,
[6]
T. House and M. J. Keeling,
Insights from unifying modern approximations to infections on networks,
[7]
M. J. Keeling, D. A. Rand and A. J. Morris,
Correlation models for childhood epidemics,
[8] [9]
H. Matsuda, N. Ogita, A. Sasaki and K. Sato,
Statistical mechanics of population: The lattice Lotka-Volterra model,
[10] [11]
M. A. Porter and J. P. Gleeson,
[12]
P. L. Simon, M. Taylor and I. Z. Kiss,
Exact epidemic models on graphs using graph automorphism driven lumping,
[13]
M. Taylor, P. L. Simon, D. M. Green, T. House and I. Z. Kiss,
From Markovian to pairwise epidemic models and the performance of moment closure approximations,
show all references
References:
[1] [2]
K. T. D. Eames and M. J. Keeling,
Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases,
[3] [4] [5]
J. K. Hale,
[6]
T. House and M. J. Keeling,
Insights from unifying modern approximations to infections on networks,
[7]
M. J. Keeling, D. A. Rand and A. J. Morris,
Correlation models for childhood epidemics,
[8] [9]
H. Matsuda, N. Ogita, A. Sasaki and K. Sato,
Statistical mechanics of population: The lattice Lotka-Volterra model,
[10] [11]
M. A. Porter and J. P. Gleeson,
[12]
P. L. Simon, M. Taylor and I. Z. Kiss,
Exact epidemic models on graphs using graph automorphism driven lumping,
[13]
M. Taylor, P. L. Simon, D. M. Green, T. House and I. Z. Kiss,
From Markovian to pairwise epidemic models and the performance of moment closure approximations,
[1]
Shouying Huang, Jifa Jiang.
Global stability of a network-based SIS epidemic model with a general nonlinear
incidence rate.
[2]
Toshikazu Kuniya, Yoshiaki Muroya.
Global stability of a multi-group SIS epidemic model for population migration.
[3]
Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang.
Backward bifurcation and global stability in
an epidemic model with treatment and vaccination.
[4] [5]
Jianquan Li, Zhien Ma.
Stability analysis for SIS epidemic models with vaccination and constant population size.
[6] [7] [8] [9]
Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya.
On the global stability of an SIRS epidemic model with distributed delays.
[10]
Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya.
Global stability for a class of discrete SIR epidemic models.
[11]
Toshikazu Kuniya, Mimmo Iannelli.
$R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission.
[12] [13]
Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel.
The saddle-node-transcritical bifurcation in a population model with constant rate harvesting.
[14]
Russell Johnson, Francesca Mantellini.
A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles.
[15]
C. Connell McCluskey.
Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence.
[16]
Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi.
Global stability for epidemic
model with constant latency and infectious periods.
[17] [18]
Deqiong Ding, Wendi Qin, Xiaohua Ding.
Lyapunov functions and global stability for a discretized multigroup SIR epidemic model.
[19]
Yongli Cai, Yun Kang, Weiming Wang.
Global stability of the steady states of an epidemic model incorporating intervention strategies.
[20]
2018 Impact Factor: 1.008
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What is the slope of the curve 8x2+5y2=r2 8x^2 + 5y^2 = r^28x2+5y2=r2 (where r≠0)r \neq 0)r=0) at a point (x1,y1) (x_{1}, y_{1}) (x1,y1) that lies on the curve?
What is the slope of the tangent line of the curve x2=y2 x^2 = y^2 x2=y2 at the point (15,−15)? (15, -15)?(15,−15)?
If y4=17x,y^{4}=17x,y4=17x, what is dydx?\displaystyle \frac{dy}{dx}?dxdy?
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If y3−4y2=x5+3x4,y^3-4y^2=x^5+3x^4,y3−4y2=x5+3x4, what is dydx?\displaystyle \frac{dy}{dx}?dxdy?
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