text:
## On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid.(English)Zbl 0982.76014

Summary: We study certain classes of flows of ideal incompressible fluid which with time gradually lose their smoothness. The loss of smoothness is expressed as infinite growth of vorticity as $$t\to\infty$$ for three-dimensional flows, and as an increase in the gradient of vorticity for plane and axisymmetric flows. Examples of such flows in the plane and axisymmetric cases are flows with a rectilinear streamline; this can be established using a special local Lyapunov function. Incompressible flows of a dusty medium are another example (it turns out that collapse is impossible for such flows, but the vorticity and the rate of deformation, as a rule, grow unboundedly). Other examples can be constructed by composition of shear flows. Here we show that in the vorticity metric almost all stationary plane flows are unstable with respect to three-dimensional disturbances, and in the vorticity gradient metric plane and axisymmetric flows with a rectilinear streamline are unstable.

### MSC:

 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 76E99 Hydrodynamic stability 35Q35 PDEs in connection with fluid mechanics
Full Text:

### References:

 [1] DOI: 10.1007/BF01474610 · Zbl 0008.06901 [2] Yudovich V. I., Dokl. Akad. Nauk SSSR 136 pp 564– (1961) [3] Yudovich V. I., Zh. Vychisl. Mat. Mat. Fiz. 3 pp 1032– (1963) [4] Yudovich V. I., Dokl. Akad. Nauk SSSR 146 pp 561– (1962) [5] Yudovich V. I., Mat. Sb. 106 pp 662– (1964) [6] Kato T., Arch. Ration. Mech. Anal. 27 pp 188– (1968) [7] DOI: 10.4310/MRL.1995.v2.n1.a4 · Zbl 0841.35092 [8] DOI: 10.1007/s002050050128 · Zbl 0926.35123 [9] Kra{ı\u{}}ko A. N., PMM USSR 46 pp 972– (1982) [10] Friedlander S. J., Ann. IHP, Analyse Nonlineare 14 pp 187– (1997) [11] Arnold V. I., PMM USSR 36 pp 255– (1972) [12] DOI: 10.1063/1.165888 · Zbl 1055.76518 [13] DOI: 10.1017/S0022112080000699 · Zbl 0504.76080 [14] DOI: 10.1016/0022-0396(81)90012-7 · Zbl 0427.35012 [15] DOI: 10.1063/1.867002 · Zbl 0651.76018 [16] DOI: 10.5802/aif.233 · Zbl 0148.45301 [17] Abrashkin A. A., Izv. Vyssh. Uchebn. Zaved., Radiofiz. 6 pp 783– (1996) [18] Abrashkin A. A., Dokl. Akad. Nauk SSSR 276 pp 76– (1974) [19] DOI: 10.1063/1.858153 · Zbl 0746.76050 [20] Arnold V. I., Izv. Vyssh. Uchebn. Zaved., Matem. 5 pp 3– (1966) [21] Ukhovski{ı\u{}} M. R., Mekh. Zhidk. Gaza 3 pp 3– (1967) [22] Serfati P., J. Math. Pures Appl. 74 pp 95– (1995) [23] Serre D., Comptes Rendus Acad. Sc. Paris 328 pp Serie– (1999) [24] DOI: 10.2307/1971423 · Zbl 0698.45010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Maple 14 Questions and PostsMaple 14 Questions and Posts Feed

These are Posts and Questions associated with the product, Maple 14

### Trace in Physics package at Maple14...

January 22 2016
1 0

I try to repeat lines (25)-(28) at

http://www.maplesoft.com/support/help/maple/view.aspx?path=Physics%2fTrace#commentform

I use Maple 14. However, instead of (28) I get the following result:

It means that Maple 14 does not perceive p_\mu, k_\nu and m as scalar quantities. I would like to ask how to define these variables correctly.

### Eigenvalues and Characteristic Polynomial...

January 17 2016
1 5

Hi!

I am comptuing the eigenvalues and the characteristic polynomial of a 8 by 8 symmetric matrix, say M. Thus, we define the matrix M, and compute its charast. plynm. by

and its eigenvalues with the command

Well, Maple returns the charast. polynm. an dthe eigenvalues. But, if we compute p(E[k]), for k=1,...,8, thats is, the values of the polynomial p(x) in the eingenvalues, Maple not turns cero!!! I'm really confused ... anyone know what could be happening?

Maple attached file with this example. Thank very much for your help!!

### Error when fsolving ...

November 05 2015
1 8

Error, (in fsolve/polynom) Digits cannot exceed 38654705646

I am using fsolve to find numerical approximations to the roots of many fairly large polynomials (degrees up to ~80).  I often get this error message and I'm not sure why.  Is there any workaround?  Any help is much appreciated.

### Apollonius (geometry) problem...

September 25 2015
3 6

Hello,

I am trying to use Apollonius procedure from geometry package. Here is an example:

restart;with(geometry): circle( c1, [ point( c1c, 0 , 0 ), 5 ] );circle( c2, [ point( c2c, 5 , 4 ), 2 ] );circle( c3, [ point( c3c, 13 , 0 ), 3 ] );A := Apollonius( c1, c2, c3 );

Unfortunatelly the Apollonius method does not give any result. The only message is:

intersection: there is no point of intersection

Anyone know what is wrong in my code?

Best regards

Rafał Nowak

### Surface/volume of revolution in polar coords...

September 09 2015
0 4

Hi there,

How can we pl0t the volume or surface of revolution of cardioid r=1-cos(\theta) about  polar axis or the vertical axis in maple14.

Any help will be appreciated.

M.R.Yegan

### I would be getting my factors delivered to my home...

August 31 2015
0 0

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August 31 2015
0 0

Packers and Movers Bangalore

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Relocation is a procedure that is risky as well as uncomfortable, as you would be in constant fear of harm of your valuable goods. To avoid this, we usually do the hiring of the movers and packers Bhopal. To help you in looking the Best Packers and Moving organizations Bhopal that would satisfy all your needs, we have established a website called moving service packers.com. To look for for a moving organizations more quickly, we have installed special features like getting completely free several estimates by just providing initial details like contact details and types of alternatives you would be requiring.  You can also create use of our fast and easy look for feature where you have to provide the name of location and types of alternatives you would be requiring and within a moment allin formation regarding every moving organizations would be offered to you.

In the procedure of moving, product packaging holdsan important position as improper product packaging and use of faulty product packaging materials will lead to harm of products. Hence in this article we provide you with some of the useful guidelines which would be beneficial to follow and create the job of the moving organizations as well as for you, easy.

As products like furnishings and breakable products should be packaged by the group of professionals, you can manage the product packaging of non- breakable products like garments, books, beddings etc. In this way you can also cut the expenses as well plenty of duration of product packaging.

Leave the product packaging of large products like furnishings and electronics to the packers and movers Bhopal, as these are challenging to package and should require professional side.

Precious products like gold jewelry, documentation, family heirlooms etc. should be loaded by your own. It is better to package them in the storage containers that would be quickly identifiable by you and do not get them loaded in the automobile.

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### Top Hydrogen peroxide teeth whitener...

June 12 2015
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### Error, (in soru) illegal use of a formal parameter...

May 19 2015
1 2

Hello. I have a question. If you can help me, i am pleasure.

Have nice day. :)

restart;
soru := proc(n,x)
local top::0;
for x from 2 to n do
top =top+(((x^(2+i))*top)^(1/(n+2-i)));
print(top);
od;
end proc;

soru(5,1);

Error, (in soru) illegal use of a formal parameter

### Error, reserved word do unexpected...

May 19 2015
1 2

Hello. I have a question. If you can help me, i am pleasure.

Have nice day. :)

Question: Enter a number after maple find(understand) to prime number or normal number.

restart;
> asal := proc(n)
> local m;
> if n<2 then
> m := s=n/2+1;
> print(m);
> else
> for i from 2 to n do
> if irem(n,i,0) then
> print("asal değil");
> if i = m then
> print("asaldir");
> end do;
> end if;
> end if;
> end proc;

### do not draw diagrams...

May 08 2015
0 0

Hi

I want to run the following code repeatedly displays but do not draw diagrams.

>restart;read(orbit.sav)
> dsnumsort := proc (numpr::list, Coor::list)
local i, j, n; global C1, C2, C3, V1, V2, V3;
n := nops(Coor); print("Order of the variables:");
for i from 2 to 2*n+1 do
for j to n do
if [numpr[i]] = select(has, numpr, diff(Coor[j](t), t)) then C[j] := i-1; V[j] := i;
print(Coor[j], C[j], " ", diff(Coor[j](t), t), V[j])
fi;
od;
od;
end:
>dx := diff(x(t), t, t) = -G*Mz*x(t)/(x(t)^2+y(t)^2)^(3/2);
>dy := diff(y(t), t, t) = -G*Mz*y(t)/(x(t)^2+y(t)^2)^(3/2);
>IniC := x(0) = 7*10^6, (D(x))(0) = 1, y(0) = 0, (D(y))(0) = 9*10^3;
>Ns := dsolve({dx, dy, IniC}, {x(t), y(t)}, numeric);
>dsnumsort(Ns(0), [x, y]):
>for i from 0 to 328 do T := 40*i; NsT := Ns(T);
>X[i], Y[i], Vx[i], Vy[i] := (eval([x(t), y(t), diff(x(t), t), diff(y(t), t)], NsT))[]
>od:
>i := 'i'; j := i+1;
>with(plots);
> plot([seq([X[i]-XS[j], Y[i]-YS[j]], i = 0 .. 328)], labels = ['dx', 'dy']);
>plot([seq([Vx[i]-VxS[j], Vy[i]-VyS[j]], i = 0 .. 328)], labels = ['dx', 'dy'])
> R := sqrt(XS[j]^2+YS[j]^2);
>sin := YS[j]/R; cos := X[j]/R;
> GCX := (X[i]-XS[j])*cos-(Y[i]-YS[i])*sin;
> GCY := (X[i]-XS[j])*sin+(Y[i]-YS[i])*cos;
> GCVx := (Vx[i]-VxS[j])*cos-(Vy[i]-VyS[i])*sin;
> GCVy := (Vx[i]-VxS[j])*sin+(Vy[i]-VyS[i])*cos;
> plot([seq([GCX, GCY], i = 0 .. 328)], labels = ['dx', 'dy']);
>plot([seq([GCVx, GCVy], i = 0 .. 327)], labels = ['dVx', 'dVy'])

### do not draw diagrams...

May 08 2015
0 0

Hi,

I want to run this code, but I do not draw diagrams. You can help me?

>restart;read(orbit.sav)
> dsnumsort := proc (numpr::list, Coor::list)
local i, j, n; global C1, C2, C3, V1, V2, V3;
n := nops(Coor); print("Order of the variables:");
for i from 2 to 2*n+1 do
for j to n do
if [numpr[i]] = select(has, numpr, diff(Coor[j](t), t)) then C[j] := i-1; V[j] := i;
print(Coor[j], C[j], " ", diff(Coor[j](t), t), V[j])
fi;
od;
od;
end:
>dx := diff(x(t), t, t) = -G*Mz*x(t)/(x(t)^2+y(t)^2)^(3/2);
>dy := diff(y(t), t, t) = -G*Mz*y(t)/(x(t)^2+y(t)^2)^(3/2);
>IniC := x(0) = 7*10^6, (D(x))(0) = 1, y(0) = 0, (D(y))(0) = 9*10^3;
>Ns := dsolve({dx, dy, IniC}, {x(t), y(t)}, numeric);
>dsnumsort(Ns(0), [x, y]):
>for i from 0 to 328 do T := 40*i; NsT := Ns(T);
>X[i], Y[i], Vx[i], Vy[i] := (eval([x(t), y(t), diff(x(t), t), diff(y(t), t)], NsT))[]
>od:
>i := 'i'; j := i+1;
>with(plots);
> plot([seq([X[i]-XS[j], Y[i]-YS[j]], i = 0 .. 328)], labels = ['dx', 'dy']);
>plot([seq([Vx[i]-VxS[j], Vy[i]-VyS[j]], i = 0 .. 328)], labels = ['dx', 'dy'])
> R := sqrt(XS[j]^2+YS[j]^2);
>sin := YS[j]/R; cos := X[j]/R;
> GCX := (X[i]-XS[j])*cos-(Y[i]-YS[i])*sin;
> GCY := (X[i]-XS[j])*sin+(Y[i]-YS[i])*cos;
> GCVx := (Vx[i]-VxS[j])*cos-(Vy[i]-VyS[i])*sin;
> GCVy := (Vx[i]-VxS[j])*sin+(Vy[i]-VyS[i])*cos;
> plot([seq([GCX, GCY], i = 0 .. 328)], labels = ['dx', 'dy']);
>plot([seq([GCVx, GCVy], i = 0 .. 327)], labels = ['dVx', 'dVy'])

### do not draw diagrams...

May 08 2015
0 0

Hi,

I want to run this code, but I do not draw diagrams. You can help me?

>restart;read(orbit.sav)
> dsnumsort := proc (numpr::list, Coor::list)
local i, j, n; global C1, C2, C3, V1, V2, V3;
n := nops(Coor); print("Order of the variables:");
for i from 2 to 2*n+1 do
for j to n do
if [numpr[i]] = select(has, numpr, diff(Coor[j](t), t)) then C[j] := i-1; V[j] := i;
print(Coor[j], C[j], " ", diff(Coor[j](t), t), V[j])
fi;
od;
od;
end:
>dx := diff(x(t), t, t) = -G*Mz*x(t)/(x(t)^2+y(t)^2)^(3/2);
>dy := diff(y(t), t, t) = -G*Mz*y(t)/(x(t)^2+y(t)^2)^(3/2);
>IniC := x(0) = 7*10^6, (D(x))(0) = 1, y(0) = 0, (D(y))(0) = 9*10^3;
>Ns := dsolve({dx, dy, IniC}, {x(t), y(t)}, numeric);
>dsnumsort(Ns(0), [x, y]):
>for i from 0 to 328 do T := 40*i; NsT := Ns(T);
>X[i], Y[i], Vx[i], Vy[i] := (eval([x(t), y(t), diff(x(t), t), diff(y(t), t)], NsT))[]
>od:
>i := 'i'; j := i+1;
>with(plots);
> plot([seq([X[i]-XS[j], Y[i]-YS[j]], i = 0 .. 328)], labels = ['dx', 'dy']);
>plot([seq([Vx[i]-VxS[j], Vy[i]-VyS[j]], i = 0 .. 328)], labels = ['dx', 'dy'])
> R := sqrt(XS[j]^2+YS[j]^2);
>sin := YS[j]/R; cos := X[j]/R;
> GCX := (X[i]-XS[j])*cos-(Y[i]-YS[i])*sin;
> GCY := (X[i]-XS[j])*sin+(Y[i]-YS[i])*cos;
> GCVx := (Vx[i]-VxS[j])*cos-(Vy[i]-VyS[i])*sin;
> GCVy := (Vx[i]-VxS[j])*sin+(Vy[i]-VyS[i])*cos;
> plot([seq([GCX, GCY], i = 0 .. 328)], labels = ['dx', 'dy']);
>plot([seq([GCVx, GCVy], i = 0 .. 327)], labels = ['dVx', 'dVy'])

May 05 2015
0 8

Hi,

> whit*evalf; tx := evalf((t1+t2)*(1/2));
> t1 := 326*40; t2 := 327*40;
> y1, y2 := eval([y1, y2], Ns(t1), Ns(t2));
>while (t1 < tx) and (tx < t2) do:
yx := rhs(Ns(tx)[c2]):
if yx > 0 then
y2 := yx: t2 := tx:
else
y1 := yx: t1 := tx:
fi;
od;
>Tx = evalf(tx);
> Hx := floor((1/3600)*tx):
> Mx := floor((tx-3600*Hx)*(1/60)):
> Sx := tx-3600*Hx-60*Mx:
> Hx, Mx, Sx;
> XS := [seq(X[i], i = 0 .. 328)]: YS := [seq(Y[i], i = 0 .. 328)]:
> VxS := [seq(Vx[i], i = 0 .. 328)]: VyS := [seq(Vy[i], i = 0 .. 328)]:
> save(G,Mz,Xs,VxS,VyS, 'orbit.sav'):

can you help me?

April 11 2015
0 5

hi friends

I encountered a problem and I can not draw the plot of this code

> sol := fsolve({diff(S, x) = 0, diff(S, y) = 0}, {x, y});

> with(VectorCalculus);
> with(linalg);
> s1 := evalf(subs(sol, linalg[grad](S, [x, y])));

> with(VectorCalculus);
> with(LinearAlgebra);
> s2 := evalf(subs(sol, linalg[hessian](S, [x, y]))); pmp0 := [x-subs(sol, x), y-subs(sol, y)]; sapprox := s0+evalm(&*(&*(transpose(pmp0), s2), pmp0));
> with(Statistics);
>
> with(stats); statevalf[icdf, chisquare[4]](.95);

> with(VectorCalculus);
> with(plottools);
> with(plots);
> with(linalg); ellips := {seq(stats*([statevalf[icdf, chisquare[4]]])(c) = sapprox, c = [.5, .95, .999])};
> plots(ellips(x, y), x = 950 .. 1000, y = 700 .. 750, grid = [50, 50], view = [950 .. 1000, 700 .. 750]);

can you helpe me?Thank you

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format: plain


text:
## Find differential equation

We are to find a differential equation which can be used to solvefor vo(t) as a function of vin(t). We do notneed to solve the equation, just find it. Helpful ideas: Idealmodel (v+ = v- and i+ = i- = 0), KCL (sum of currents is zero), KVL(sum of voltage is zero), and v = L(di/dt) for inductors. Pleasehelp me find this differential equation. I have no idea where tobegin?


text:
# Structured Estimation in High-Dimensions

### http://www2.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-110.pdf

High-dimensional statistical inference deals with models in which the the number of parameters $p$ is comparable to or larger than the sample size $n$. Since it is usually impossible to obtain consistent procedures unless $p/n \to 0$, a line of recent work has studied models with various types of low-dimensional structure, including sparse vectors, sparse and structured matrices, low-rank matrices, and combinations thereof. Such structure arises in problems found in compressed sensing, sparse graphical model estimation, and matrix completion. In such settings, a general approach to estimation is to solve a regularized optimization problem, which combines a loss function measuring how well the model fits the data with some regularization function that encourages the assumed structure. We will present a unified framework for establishing consistency and convergence rates for such regularized $M$-estimators under high-dimensional scaling. We will then show how this framework can be utilized to re-derive a few existing results and also to obtain a number of new results on consistency and convergence rates, in both $\ell_2$-error and related norms.

An equally important consideration is the computational efficiency in performing inference in the high-dimensional setting. This high-dimensional structure precludes the usual global assumptions---namely, strong convexity and smoothness conditions---that underlie much of classical optimization analysis. We will discuss ties between the statistical inference problem itself and efficient computational methods for performing the estimation. In particular, we will show that the same underlying statistical structure can be exploited to prove global geometric convergence of the gradient descent procedure up to \emph{statistical accuracy}. This analysis reveals interesting connections between statistical precision and computational efficiency in high-dimensional estimation.

BibTeX citation:

@phdthesis{Negahban:EECS-2012-110,
Author = {Negahban, Sahand N},
Title = {Structured Estimation in High-Dimensions},
School = {EECS Department, University of California, Berkeley},
Year = {2012},
Month = {May},
URL = {http://www2.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-110.html},
Number = {UCB/EECS-2012-110},
Abstract = {High-dimensional statistical inference deals with models in which the the number of parameters $p$ is comparable to or larger than the sample size $n$. Since it is usually impossible to obtain consistent procedures unless $p/n \to 0$, a line of recent work has studied models with various types of low-dimensional structure, including sparse vectors, sparse and structured matrices, low-rank matrices, and combinations thereof. Such structure arises in problems found in compressed sensing, sparse graphical model estimation, and matrix completion. In such settings, a general approach to estimation is to solve a regularized optimization problem, which combines a loss function measuring how well the model fits the data with some regularization function that encourages the assumed structure. We will present a unified framework for establishing consistency and convergence rates for such regularized $M$-estimators under high-dimensional scaling. We will then show how this framework can be utilized to re-derive a few existing results and also to obtain a number of new results on consistency and convergence rates, in both $\ell_2$-error and related norms.

An equally important consideration is the computational efficiency in performing inference in the high-dimensional setting. This high-dimensional structure precludes the usual global assumptions---namely, strong convexity and smoothness conditions---that underlie much of classical optimization analysis. We will discuss ties between the statistical inference problem itself and efficient computational methods for performing the estimation. In particular, we will show that the same underlying statistical structure can be exploited to prove global geometric convergence of the gradient descent procedure up to \emph{statistical accuracy}. This analysis reveals interesting connections between statistical precision and computational efficiency in high-dimensional estimation.}
}

EndNote citation:

%0 Thesis
%A Negahban, Sahand N
%T Structured Estimation in High-Dimensions
%I EECS Department, University of California, Berkeley
%D 2012
%8 May 11
%@ UCB/EECS-2012-110
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-110.html
%F Negahban:EECS-2012-110


text:
# How inflation creates a universe from nothing?

I have a basic, mostly purely conceptual understanding of Quantum Field Theory, and after lots of Youtube (thanks PBS Spacetime!) I have an idea of how inflation works to turn the vacuum into a universe. Please correct me if I'm wrong.

There are quantum fields present everywhere in the universe at once. Excitations on those fields, caused by energy, are the vibrations that we perceive as particles

A field at its lowest possible energy state still has quantum fluctuations since, due to the laws of Quantum Mechanics, it is impossible for the field to have precisely zero energy. These quantum fluctuations are small vibrations in the field that usually quickly form and cancel each other out, and they can be thoight of as (although this is mostly to aid in visualization, since they don't have the exact same properties) 'virtual particles' popping in and out of existence.

At the event horizon of a black hole, some of the vibrations are 'cut off' because of the one-way boundary that is the evnt horizon, so the vibrations that would be cancelled out by these don't, so they are perceived as 'real particles', what we call Hawking Radiation.

So my understanding is this. An early universe is in the vacuum state, with only quantum fluctuations permeating the cosmos. The process of inflation causes points that were previously very close to suddenly become extremely far apart, going from a distance in the quantum scale of things to lightyears apart in a tiny fraction of a second. Since points of the quantum field which were very close together are suddenly way too far away to communicate, this effectively 'cuts off' the vibrations in one spot from the rest of the universe, turning those quantum fluctuations into 'real particles'

Is this correct?

• it is different than hawking radiation, which happens at the event horizon of a black hole. The black hole has a lot of energy/mass . Please see this en.wikipedia.org/wiki/Inflation_(cosmology) – anna v Jun 16 at 6:44
• Apparently science is becoming religion. Everyone blindly believes in what no one has ever seen and stories of "creation" are told that have nothing to do with reality. – safesphere Jun 16 at 7:51
• @safesphere you must know then , if you know "that they have nothing to do with reality", you must know reality.. In physics one is always talking with mathematical models, that is all. If the models do not fit data and observations they will be falsified. – anna v Jun 16 at 9:30
• Which law of quantum mechanics forbids zero as an eigenvalue of the Hamiltonian? – Feynmans Out for Grumpy Cat Jun 16 at 19:02
• On a more general note, what is known as "vacuum fluctuation" is not something that quantum mechanics inherently requires (such as the superposition principle or non-commuting operators, etc.). Rather, it is a feature (albeit quantum mechanical) of specific types of QFTs, namely, interacting QFTs. A free QFT doesn't have vacuum fluctuations and is perfectly quantum mechanical. – Feynmans Out for Grumpy Cat Jun 16 at 19:05

I think it is not possible to give a clear "yes or no" answer to your question, because it is a question about a research area where there remain many models which do not agree with one another, and we simply don't know which if any are right. The research area being inflation theory.

Inflation or something like it may have happened, or it may not have happened. The biggest unknowns here are to do with entropy. Attempts to model the early universe in detail typically evoke (without always realising that they have done so) extremely special states of affairs. This makes it hard to assess whether or not a given theory has not so much explained something as shown that it would be the outcome of something even more inexplicable. Inflation does not escape this problem.

I think the main message here is that something rather odd is happening in our day in the interaction between research science and the wider public. The distinction between carefully constructed and tested ideas and mere speculation is blurred in many popular books, and You Tube channels are even worse. In elementary particle physics, progress over the last 80 years has required a partnership between experiment and theory. There are occasional examples where theoretical understanding put in place something well out of the range of experiment but which proved to be correct (Higgs mechanism being a good example). But there are also many examples of cases where experiments yielded surprises. Inflation is an attempt to grapple with physics at the energy scale $$\ge 10^{15}$$ GeV. Experiments have accessed up to $$10^4$$ GeV.

I think the best way to respond to your question is to encourage continuing interest in these areas, but also to encourage a greater role for the attitude "well we really don't know yet".

But one thing we do know is that every scientific model ever put forward for anything has invoked a continuity between one thing and another, between a prior situation and a consequent situation. The idea that physics suggests that something could come from nothing is simply a misdirection, a deliberate miss-use of words, presumably in an effort to gain readers or something like that. I mention this simply because the title of your question suggests that you may have been miss-directed into this sort of juggling with the meanings of words.

Among the authors well-placed to comment here, and who does a reasonably balanced job I think, is Sean Carroll.

Added edit to answer specific point at the end of the question.

Either with or without inflation, space is reckoned to have started from an early state presumably described by quantum gravity, and it grew extremely fast at early times. This resulted in energy density fluctuations being present on pretty much all distance scales. This is modeled theoretically by using quantum theory to provide a value for the standard deviation of the distribution, and then subsequently treating that distribution as a classical field having fluctuations over space and time with the given standard deviation. The move here from quantum to classical is rather glossed-over in the research literature; it is connected to the subtleties involved in the process called symmetry-breaking. (What is spontaneous symmetry breaking in QUANTUM systems?)

Anyway the main point for your question is that this is not like Hawking radiation. The fluctuations are already reckoned to be classical, or are treated as classical, whether or not there was a subsequent inflation to stretch them out. (I don't work directly in this research area; I got the above information from a book by Hobson, Efstathiou and Lasenby, and from various review and other papers).

• Right, but does the inflation model work as I described it? – user140323 Jun 17 at 10:53
• @user140323 I added an extra section to reply to this. – Andrew Steane Jun 17 at 15:37

Theories of inflation are not the same as universe-from-nothing theories.

There was already “something” when inflation started. In inflation, the universe (both spacetime and quanta in that spacetime) already existed, but it then became exponentially larger due to the energy and pressure of a theorized scalar “inflaton” field that would be somewhat similar to a Higgs field.

In universe-from-nothing theories, the universe emerges from a no-spacetime-and-no-quanta state, perhaps via quantum tunneling.

If what you meant was “How is inflation able to create so much matter and radiation?”, then there are two possible answers. The first is that energy conservation may not apply to the universe as a whole. The second is that, if it does apply, it holds because the gravitational potential energy of all that new matter and radiation is negative (just like the gravitational potential energy of the earth and the Sun is negative) and it balances all the positive energy of the created matter and radiation.

• "In universe-from-nothing theories, the universe emerges from a no-spacetime-and-no-quanta state, perhaps via quantum tunneling." I think this comes down to an issue of nomenclature, but even in theories where we tunnel to the initial state there is already "something" around to tunnel. True "something-from-nothing" theories are metaphysical in nature. See arxiv.org/abs/1802.02231, for example. – gabe Jun 16 at 17:26
• I don’t agree. See sciencedirect.com/science/article/abs/pii/0370269382908668 – G. Smith Jun 16 at 17:32
• My point is that even such scenarios presuppose the existence of things like quantum fields, which is decidedly not "nothing". Again, this is primarily a metaphysical point on what "nothing" really means. As pointed out in Carroll's review: "This has nothing at all to do with the origin of the universe itself, and certainly nothing to do with why there is a quantum wave function in the first place." Your answer is certainly correct if we are free to define "nothing" as a "no-spacetime-and-no-quanta state" (which may seem reasonable), but leaves the question of why there are "states" at all. – gabe Jun 16 at 17:59
• OK, we are in agreement that math-y stuff pre-exists in these theories. – G. Smith Jun 16 at 18:02

It was the Big Bang which created the universe, and the "nothing" from which it was created was most likely the fireball of a previous universe whitch collapsed into a Big Crunch followed by a Big Bounce. This is not the prevailing theory at the moment, though it hasn't been completely discarded. One-off Big Bangs which have no apparent cause and literally create a universe out of nothing and eventually dissipate into the dark nothingness of space raise all sorts of problems from the point of view of logic, which are avoided by a Big Crunch. Theories come and go and so does the evidence on which they are founded, so perhaps the Big Crunch will come back into favour some day.

I am very sceptical about the inflation scenario, and think we should await more evidence on this. Likewise with virtual particles, as there is good evidence to cast doubt on them. The Cassimir effect, for example, can be explained by more mundane mechanisms. The 40 orders of magnitude discrepancy between the vacuum energy required by QM and the cosmological observations with which it conflicts does not inspire confidence in the virtual particle theory.

• I don't see any problems with the Big Bang as a white singularity or conceptually even with the Big Crunch as the black singularity. Both are geodesic incomplete meaning spacetime starts at the Big Bang and ends at the Big Crunch. However, I cannot see how you can mathematically transition from the Big Crunch to Big Bang. There is no connection of geodesics. These concepts are not fireball objects that exist in space and could follow each other in time. They are singularities creating and destroying space and time along with matter. Can you show any connection from the Big Crunch to Big Bang? – safesphere Jun 16 at 15:24
• The blue-shifted universe will collapse into a fireball similar to the one it arose from before it disappears into a black hole singularity. There has been much speculation about black holes emerging as white holes in some other time or location. I'll leave that particular matter in abeyance for the time being,but on the grander scale of universes, this is perhaps what constitutes the Big Bounce, where the recycled universe will emerge again as a white hole. – Michael Walsby Jun 16 at 19:04
• Understood (and +1, as I don't think the downvotes are justified just for bringing up the well known concept of the Big Bounce). – safesphere Jun 16 at 19:12
• ... the concept is valid; however, the answer could be improved with more hard facts and references and fewer phrases such as "most likely", "theories come and go" and "all sorts of problems". – rghome Jun 17 at 8:41
• The CMBR was originally thought to be evidence for the Big Bounce. Robert Dicke believed in a cyclical cosmology. – John Duffield Jun 20 at 21:16


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You are here: MIMS > EPrints
MIMS EPrints

## 2007.199: Some local definability theory for holomorphic functions

2007.199: A.J. Wilkie (2007) Some local definability theory for holomorphic functions. Modnet.

Full text available as:

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## Abstract

Let \mathcal{F} be a collection of holomorphic functions and let \mathbb{R}(PR(\mathcal{F})) denote the reduct of the structure \mathbb{R}_{an} to the ordered field operations together with the set of proper restrictions (see below) of the real and imaginary parts of all functions in \mathcal{F}. We ask the question: Which holomorphic functions are locally definable (ie have their real and imaginary parts locally definable) in the structure \mathbb{R}(PR(\mathcal{F}))? It is easy to see that the collection of all such functions is closed under composition, partial differentiation, implicit definability (via the Implicit Function Theorem in one dependent variable) and Schwarz Reflection. We conjecture that this exhausts the possibilities and we prove as much in the neighbourhood of generic points. More precisely, we show that these four operations determine the natural pregeometry associated with \mathbb{R}(PR(\mathcal{F}))-definable, holomorphic functions.

Item Type: Other MSC 2000 > 03 Mathematical logic and foundationsMSC 2000 > 11 Number theory 2007.199 Mrs Louise Healey 21 November 2007


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# Multicolumn list environment

I want to generate an environment that allows to establish the number of columns.

I tried the following environment but I do not know if the \AtBeginEnvironment commands are compatible with \newenvironment

...

\usepackage{etoolbox}
\usepackage{multicol}

\newenvironment{ncomunas}[1][2]{%
\AtBeginEnvironment{itemize}{\begin{multicols}{#1}}%
}
{%
\AtEndEnvironment{itemize}{\end{multicols}}%
}

• Welcome to TeX.SX! Please make your code compilable (if possible), or at least complete it with \documentclass{...}, the required \usepackage's, \begin{document}, and \end{document}. That may seem tedious to you, but think of the extra work it represents for TeX.SX users willing to give you a hand. Help them help you: remove that one hurdle between you and a solution to your problem. – Stefan Pinnow Sep 24 '18 at 4:28
• If I understand well, you're probably after the functionalities of the tasks package. – Bernard Sep 24 '18 at 8:38
• Have a look at the paralist package, which offers some extra lists (less space, in-line, etc.). – Keks Dose Sep 24 '18 at 10:46

Just wrap multicols around itemize:

\documentclass{article}
\usepackage{multicol}
\usepackage{lipsum} % for context

\newenvironment{ncomunas}[1][2]
{\begin{multicols}{#1}\begin{itemize}}
{\end{itemize}\end{multicols}}

\begin{document}

\lipsum[2]

\begin{ncomunas}[3]
\item abc
\item def
\item ghi
\item jkl
\item mno
\end{ncomunas}

\lipsum[3]

\end{document}

You should also consider the tasks package.


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## News Forum

### So Many Sequences (Weekly Brain Potion)

So Many Sequences (Weekly Brain Potion)

For this week's brain potion we will explore some various problems involving sequences. Caution, you'll probably want a calculator for some of these!

For each of the sequences below: (i) Describe the pattern and calculate the first few numbers. (ii) What number do you think the sequence is approaching?

1. Consider the following pattern of numbers $$\displaystyle \frac{4}{1}, \frac{4}{1} - \frac{4}{3}, \frac{4}{1} - \frac{4}{3} + \frac{4}{5}, \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7}, \ldots$$
2. Consider now this pattern of numbers: $$\displaystyle 2\times \frac{2}{1}, 2\times \frac{2}{1}\frac{2}{3}, 2\times \frac{2}{1}\frac{2}{3}\frac{4}{3}, 2\times \frac{2}{1}\frac{2}{3}\frac{4}{3}\frac{4}{5}, 2\times \frac{2}{1}\frac{2}{3}\frac{4}{3}\frac{4}{5}\frac{6}{5},\ldots$$
3. Here's another pattern: $$3, 3 + \frac{4}{2\cdot3\cdot4}, 3 + \frac{4}{2\cdot3\cdot4} - \frac{4}{4\cdot5\cdot6}, 3 + \frac{4}{2\cdot3\cdot4} - \frac{4}{4\cdot5\cdot6} + \frac{4}{6\cdot7\cdot8}, \ldots$$
4. One last pattern: $$3, 3 + 1^2, 3 + \frac{1^2}{6 + 3^2}, 3 + \frac{1^2}{6 + \dfrac{3^2}{6+5^2}}, 3 + \frac{1^2}{6 + \dfrac{3^2}{6+\dfrac{5^2}{6+7^2}}}, \ldots$$

As a challenge, use Excel/Google sheets or write a program to help calculate the values of these sequences!


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# Hoofin predates Huffington Post by two years

Archive.org record of Huffington Post, May 10, 2005

To the friendly e-mailer:

Au contraire! Hoofin had been blogging at least two years before Arianna Huffington and the Huffington Post ever showed up on the internet. (May 9, 2005) Hoofin was posting in early 2003.

If I were posting as “Hoofington Post” or something, then yes, I could see a copy cat accusation. But Huffington showed up sometime after Hoofin. (And, yes, infinitely much more successfully so–she’s trying and has the dough from hubby.)

So . . .


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# A standard piece of paper is 8.5 inches by 11 inches. A piece of legal-size paper is 8.5 inches by 14 inches. By what scale factor k would you need to dilate the standard paper so that you could fit two pages on a single piece of legal paper?

Question
A standard piece of paper is 8.5 inches by 11 inches. A piece of legal-size paper is 8.5 inches by 14 inches. By what scale factor k would you need to dilate the standard paper so that you could fit two pages on a single piece of legal paper?

2020-12-26
You would need that the length of the standard paper be equal to the width of the length of the legal paper. Hence, the scale factor must be the ratio of the width of the legal paper to the length of length of the standard paper: $$\displaystyle{k}=\frac{{8.5}}{{11}}=\frac{{17}}{{22}}$$
To check if the widths of the two dilated standard papers fit along the length of the legal paper, we multiply the scale factor to 2 times the width of the standard paper:
$$\displaystyle{2}\cdot{\left(\frac{{17}}{{22}}\right)}{\left({8.5}\right)}\sim{13.1}\in.{<}{14}\in$$</span>
So, the two pages will fit for this value of k.

### Relevant Questions

Saw a board 8 ft 4 in. long into nine equal pieces. If the loss per cut is 1/8 in., how long will each piece be?
$$\displaystyle{\left(\frac{{5}}{{8}}\right)}+{\left(\frac{{11}}{{12}}\right)}$$
An empty tank for storing water from a borehole has a volume of 480 m³. If it is filled by a pump that pumps water at a rate of 16 l/s, how many hours will it take the pump to fill this tank?
A family is canoeing downstream (with the current). Their speed relative to the banks of the river averages 2.75 mi/h. During the return trip, they paddle upstream (against the current), averaging 1.5 mi/h relative to the riverbank.
a. Write an equation for the rate of the canoe downstream.
b. Write an equation for the rate of the canoe upstream.
c. Solve the system to find the family’s paddling speed in still water.
d. Find the speed of the current of the river.
$$\displaystyle{\left({2.66}×{10}^{{13}}\right)}÷{\left({9.5}×{10}^{{5}}\right)}$$
Simplify by using order of operations: $$\displaystyle\frac{{{10}-{3}\cdot{2}}}{{{1}+{1}}}$$


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# How much degrees does a pentagon have?

The sum of interior angles is $180 \times 3 = {540}^{o}$ and each angle of a regular pentagon is $\frac{540}{5} = {108}^{o}$

#### Explanation:

There is a cool way to view this (the link below has a complete description):

We know that the sum of the interior angles of a triangle is ${180}^{o}$.

A rectangle can be divided into 2 triangles, each of which have interior angles equaling ${180}^{o}$, and so $2 \times 180 = {360}^{o}$ and that is the sum of the interior angles of a rectangle.

Each angle is $\frac{360}{4} = {90}^{o}$

And we can do the same type of thing with any polygon. Take for example, the pentagon:

The sum of interior angles is $180 \times 3 = {540}^{o}$ and each angle of a regular pentagon is $\frac{540}{5} = {108}^{o}$

https://www.mathsisfun.com/geometry/interior-angles-polygons.html


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statics problem, help needed

dbag

Hello

Problem description: determine the lenght of h with respect to b so that the water flows.

ive gotten as far as come up with an equation for the sum of moments and it goes as follows: $$\displaystyle bh*1/2b-1/2hb*1/3b=0$$ the answer is wrong and i dont know why. moment arm on the b is 1/2b and the moment arm on h is 1/3h because of the way water pressure works , meaning its a uniform load in the shape od triangle. the answer to this problem is supposed to be $$\displaystyle sqrt3 *b$$. Any help appreciated.

Attachments

• 262.1 KB Views: 0

oz93666

force acting on face length b .....= dencity x b x width x gravity x h , this acts at center of face so the turning torque is... dencity x 1/2 x b x b x width x gravity x h

Force acting on face h is ....= 1/2 dencity x h x width x gravity x h
This force acts 1/3 of the way up (www.youtube.com/watch?v=WkM1tCZDM2c)
so torque is ..... = 1/3 h x 1/2 dencity x h x width x gravity x h

tipping occurs when two torques are equal

so b x b = 1/3 x h x h so 3 b squared = h squared .... h = root 3 b

2 people

dbag

thank you very much


text:
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00:15
Problem 10

# In Exercises $6-14, x=12, y=10,$ and $z=24 .$ Write each ratio in simplest form. $\frac{x+y}{z+y}$

## Discussion

You must be signed in to discuss.

## Video Transcript

in this question. It says that X is equal to 12. Why is equal to 10 and Z is equal to 24 Notice? I just put a line through my disease. You don't have to do that. It just so it doesn't look like a two. Okay, now lost the ratio of X plus y over Z plus. Why so just substitute. Here we go. X is 12. Why is 10 z is 24? And why over here again is 10. So you get 22 over 34 now if you reduce that to lowest terms by dividing by two on the top and bottom. So I went to you get 11 Bobby to over 17. So the answer is 11 17 and that isn't simplest form. Be really, really careful. I just want to mention this. It's very tempting to try to cross out those tens right there, but you can't do that over. In addition, sign correct. Remember, add add and then


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# pandas.Series.shift¶

Series.shift(periods=1, freq=None, axis=0)[source]

Shift index by desired number of periods with an optional time freq

Parameters: periods : int Number of periods to move, can be positive or negative freq : DateOffset, timedelta, or time rule string, optional Increment to use from the tseries module or time rule (e.g. ‘EOM’). See Notes. axis : {0, ‘index’} shifted : Series

Notes

If freq is specified then the index values are shifted but the data is not realigned. That is, use freq if you would like to extend the index when shifting and preserve the original data.

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# Centripetal acceleration in Kepler orbit in cartesian coordinate

I have some difficult in deriving the formula for centripetal acceleration of a planet in solar system moving in elliptical orbit with constant areal velocity $\dot S$.

The book report the formula in cartesian coordinate:

$$a_c=\frac{4\dot S^2(1+(\frac{cy}{b^2})^2)}{((x+c)^2+y^2)}\frac{a^4b}{(a^4+c^2x^2)^\frac 3 2}$$

With:
a = semi major axis
b = semi minor axis
$c = \sqrt{a^2-b^2}$ semi focal distance

I try using polar coordinate find $a_c$ and then return to cartesian but i get stuck.

I guess that the last part is the curvature and the first is $r^2\dot\theta$ but have no idea for the others.

In that figure, the Sun is at the origin and the origin is chosen to coincide with the right focal point of the ellipse. Now there is enough information to give a Cartesian description of the ellipse, $$\frac{(x+c)^2}{a^2} + \frac{y^2}{b^2} =1$$ where $(x,y)$ is the position of the planet at some instant in time. Use the formula (Stewart Calculus 7th edition, p891), $$\vec{a} = a_T \vec{T} + a_{N}\vec{N} = \frac{|\vec{v}(t)\times \vec{a}(t)|}{|\vec{v}(t)|}\vec{N} + \frac{|\vec{v}(t)\cdot \vec{a}(t)|}{|\vec{v}(t)|}\vec{T}$$ where $\vec{N}$ is a unit vector directed towards the center of curvature and hence normal to the velocity and $\vec{T}$ is a unit vector in the direction of velocity. The goal is to compute $a_N$. To avoid polar coordinates and what not we use $a_N = \kappa v^2$ (Stewart Calculus p891 7th ed). From page 881 of Stewart Calculus 7th ed. the curvature, $$\kappa(x) = \frac{|f''(x)|}{(1+f'(x)^2)^{\frac{3}{2}}}$$ which explains the power of $\frac{3}{2}$ you have. Theoretically speaking this approach is sketchy since $y$ is clearly not a function of $x$ whereas this formula works for functions only; however, there comes the beauty of symmetry, if we consider only the upper half of the ellipse, which is a graph of a function, the radius of curvature there, at any given abscissa $x$, will be the same for the lower half of the ellipse at the same abscissa.
Using implicit differentiation one gets, $$\frac{(x+c)}{a^2} + \frac{yy'}{b^2} = 0 \Rightarrow y' = -\frac{-b^2(x+c)}{ya^2}$$ Using implicit differentiation twice one gets, $$\frac{1}{a^2} + \frac{1}{b^2}(y'^2+yy'')=0 \Rightarrow |y''| = \frac{\frac{b^2}{a^2}+y'^2}{y}= \frac{\frac{b^2}{a^2}+\frac{b^4(x+c)^2}{y^2a^4}}{y} =\frac{b^4}{a^2 y^3}$$ Thus, $$y'^2+1= y|y''|-\frac{b^2}{a^2} +1= \frac{b^4}{a^2y^2}-\frac{b^2}{a^2}+1=\frac{b^4-b^2y^2+a^2y^2}{a^2y^2}=\frac{c^2y^2+b^4}{a^2y^2}$$ Therefore, $$\displaystyle \kappa(x) = \frac{b^4a}{(b^4+c^2y^2)^{\frac{3}{2}}}$$ Now, in polar coordinates $\vec{v} = \dot{\rho}\vec{e}_{\rho} + \rho\dot{\phi}\vec{e}_{\phi} \Rightarrow v^2 = \dot{\rho}^2 +(\rho\dot{\phi})^2$.
The area swept between the angles $\phi_0$ and $\phi$ is $S(\phi) = \frac{1}{2}\int_{\phi_0}^{\phi} (\rho(\phi))^2 d\phi$ hence $\dot{S} =\frac{1}{2} \rho^2 \dot{\phi}$ which is a constant due to conservation of angular momentum. From the figure I attached one sees that $\rho + \rho' = 2a =\rho + \sqrt{(\rho \sin{\phi})^2+(2c+\rho \cos{\phi})^2}$ hence $$\displaystyle \rho = \frac{a(1-e^2)}{1+e\cos{\phi}}$$ where $e=\frac{c}{a}$ is the eccentricity of the ellipse. Differentiating and using the formula obtained for $\dot{S}$ , $$\dot{\rho} = \frac{2e\dot{S}}{a(1-e^2)}\sin{\phi}=\frac{2e\dot{S}}{\rho a(1-e^2)}\rho\sin{\phi}=\frac{2e\dot{S}} {a(1-e^2)}\frac{y}{\sqrt{x^2+y^2}}$$ Now, to calculate $\rho \dot{\phi} = 2\frac{\dot{S}}{\rho}=2\frac{\dot{S}}{\sqrt{x^2+y^2}}$. Therefore, $$v^2 = \dot{\rho}^2+(\rho\dot{\phi})^2 = \frac{4\dot{S}^2}{x^2+y^2}+\frac{4e^2\dot{S}^2}{a^2(1-e^2)^2}\frac{y^2}{x^2+y^2}$$ after some careful algebraic changes, $$v^2 = \frac{4\dot{S}^2\left( 1+\left(\frac{cy}{b^2} \right)^2\right)}{x^2+y^2}$$ and finally $$a_c =\frac{4\dot{S}^2\left( 1+\left(\frac{cy}{b^2} \right)^2\right)}{x^2+y^2}\frac{b^4 a}{(b^4+c^2y^2)^{\frac{3}{2}}}$$


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# Criterion for a group to be abelian

Let $G$ be a group of order $pq$ , where $p,q$ are primes and $p<q$ . Show that if $p \nmid (q-1)$ (i.e $q \not \equiv 1\mod p )$ then $G$ is abelian.

I was able to use Sylow's theorems to prove that the group is cyclic and hence abelian (Sketch of what I did: the number of $p$-Sylow subgroups is $1$ or $q$ but since $q \not \equiv 1\mod p$ there is one $p$-Sylow subgroup and it is normal , similarly I showed that the number of $q$-Sylow subgroups is $1$ and also normal I then took their direct product, of the sylow subgroups that is, both are cyclic since they are of prime orders I then showed that the multiplication of generators generates the direct product which is isomorphic to $G$ it follows that G is abelian )

Besides the fact that the above proof looks very messy, I'm looking for a proof that does not use any Sylow theorem.

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What do you know about semidirect products? –  mixedmath Mar 25 '13 at 14:00
@mixedmath I've never heard of them, I just looked them up on Wikipedia. They seem similar to internal direct products but only requiring one of the subgroups to be normal. –  user10444 Mar 25 '13 at 14:04

Yes, this can be done without Sylow as follows:

By Cauchy's theorem, the group has a subgroup of order $p$ and one of order $q$. If $p < q$ then the subgroup of order $q$ has index $p$ which is the smallest prime dividing the order of the group, so it is normal (see various proofs of this at Normal subgroup of prime index).

Now we know that the group is a semidirect product of $C_p$ with $C_q$ and since there is no non-trivial homomorphism from $C_p$ to $\rm{Aut}(C_q)$, since the latter has order $q-1$ which $p$ does not divide, this semidirect product is in fact direct, and we are done.

Just saw that you were not familiar with semidirect products, so here is a way without them:

We want to show that not only is the subgroup of order $q$ normal, it is in fact central. To see this, let $H$ be the $q$-Sylow subgroup (that normality implies uniqueness for Sylow subgroups is a lot easier to prove than the general theorems. For a generalization with an elementary proof, see Rotman introduction to theory of groups exercise).

We know from the N/C theorem that $G/C_G(H)$ is isomorphic to a subgroup of $\rm{Aut}(H)$ But we also know that since $H\leq C_G(H)$ we have that $|G/C_G(H)|$ divides $p$, and since $p$ does not divide the order of $\rm{Aut}(H)$ the only possibility is that $C_G(H) = G$ so $H$ is central.

Now let $K$ be any subgroup of order $p$. Since $|HK| = |G|$ we see that any element of $G$ can be written uniquely as $hk$ with $h\in H$ and $k\in K$ and since $H$ is central, elements of this form commute, so the group is abelian.

Added: The N/C theorem is the following: Let $G$ be a group and $H$ a subgroup of $G$. Then $C_G(H)$ is a normal subgroup of $N_G(H)$ and $N_G(H)/C_G(H)$ is isomorphic to a subgroup of $\rm{Aut}(H)$. The proof is by defining a homomorphism from $N_G(H)$ to $\rm{Aut}(H)$ by letting any $g\in N_G(H)$ go to the action on $H$ by conjugation and noting that this homomorphism has exactly $C_G(H)$ as its kernel.

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thank you for the proof, I am actually unfamiliar with most of the theorems besides Cauchy's theorem but you posted sources which is great, I'll understand them and get back to this. –  user10444 Mar 25 '13 at 14:28
@user10444 I added a short remark on the N/C theorem. –  Tobias Kildetoft Mar 25 '13 at 14:42

By Cauchy's Theorem, $G$ contains subgroups of order $p$ and $q$. Let $x,y\in G$ such that $\langle x\rangle\cong\mathbb{Z}_p,\langle y\rangle\cong\mathbb{Z}_q$. Since $\mathbb{Z}_p\not\leq Aut(\mathbb{Z}_q)$, so $x$ centralises $y$, i.e., $x^{-1}yx=y$, and hence $x,y$ commute. Then $|\langle x,y\rangle|=pq$, and so $G=\langle x,y\rangle$ is abelian.

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Is the derived group the commutator subgroup (the elements of the form $xyx^{-1}y^{-1}$ )? Also isn't there a missing order in the first case? –  user10444 Mar 25 '13 at 14:37
Yes, the derived group is an alternative name for the commutator group. And the missing case you meant $|G'|=pq$? –  Easy Mar 25 '13 at 14:40
@Easy Indeed the case $|G'| = pq$ is the tricky one (showing that it cannot happen, that is), since once you show that the group is not perfect, the rest follows. –  Tobias Kildetoft Mar 25 '13 at 14:43
@TobiasKildetoft, you are right, I have just amended my method. –  Easy Mar 25 '13 at 14:58


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# Is a volumetric rate frame-invariant in general relativity?

Imagine that I have a radioactive material with a long half life. The atoms in this material decay at a certain rate $R$. The rate is the decay constant times the number density $R = \lambda N$. It has dimensionality:

$$\left( \frac{ \text{decays} }{m^3 s} \right)$$

Imagine that the material is on board a spaceship traveling at some significant fraction of the speed of light. Length is contracted and time is dilated.

$$\Delta t' = \Delta t \gamma = \frac{\Delta t}{\sqrt{1-v^2/c^2}}$$

$$L'=\frac{L}{\gamma}=L\sqrt{1-v^{2}/c^{2}}$$

The volumetric decay rate according to the lab reference frame is found by correcting for both the increased density (due to length contraction) and the decreased decay constant (due to time dilation).

$$N' = L' A = \frac{N}{\gamma}$$

$$R' = \frac{N'}{N} \frac{\Delta t'}{\Delta t} R = R$$

It's the same volumetric decay rate! Amusingly, the $Q$ value of the decay would be greater, but that's aside the point.

Question:

What if the material was put in a large gravity well? If you use the coordinates from outside the gravity well, would you obtain this same result?

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I don't know whether it applies to all physically possible metrics, but the volumetric decay rate you define does stay constant in a Schwarzschild metric. Well, it does if the box is small compared to the curvature i.e. the time dilation etc is constant thoughout the box. I would need to think more about what happens if the box is very large.

Anyhow the Scharzschild metric is:

$$ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1}dr^2 + r^2 d\Omega^2$$

The time dilation is easy, as we see time moving more slowly for the box by a factor of $(1 - 2M/r)^{1/2}$. I had to think a bit about length contraction, but I think this is a sensible way to define it:

The Schwarzschild radial co-ordinate $r$ is defined as the radius of a circle with circumference $2\pi r$. So we can take a shell with circumference $2\pi r$ and another with circumference $2\pi (r + dr)$ and that defines our ruler of length $dr$. But the observer standing alongside the box would measure a different radial distance between the shells. Specifically they would measure the distance to be $dr/(1 - 2M/r)^{1/2}$. This distance is bigger than the observer at infinity measures, and therefore this means the shell observer's ruler is shorter than ours by a factor of $(1 - 2M/r)^{1/2}$. This factor is exactly the same as the time dilation factor, which means the time dilation and length contraction balance out, and the volumetric decay rate stays the same.

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Actually, a previous answer of yours inspired this question. I thought the equation $d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2$ would be used to answer, not for a specific geometry, but possibly completely generally. My thinking was that the universality of distance between two spacetime points would lead to agreement on a "4-D volume", which led me to volumetric decay rates. One last point: it shouldn't matter the size or shape of the box, no? Everything here could be a differential volume. Anyway, your answer is very clear and easy to understand. –  Alan Rominger Oct 4 '12 at 13:20
I'm fairly certain you could make up a metric where the time dilation and length contraction didn't balance, then work backwards to find the match stress-energy tensor. What I'm not sure about is whether that stress-energy tensor would be physical i.e. not require exotic matter. In any case, questions like yours need careful thought. In SR we can always choose frames with the origins at the point of interest (the box) so the box is at (0, 0) in both frames and can easily be compared. In GR this is not the case and you need to think carefully how you do the comparison. –  John Rennie Oct 4 '12 at 14:58


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# Question #c1a8e

Oct 4, 2014

a. Difference Quotient

Identify f(x) and f(x+h)

$f \left(x\right) = \sqrt{3 x}$

$f \left(x + h\right) = \sqrt{3 \left(x + h\right)}$

Make substitutions and rationalize the numerator

$f ' \left(x\right) = {\lim}_{h \to 0} \frac{\sqrt{3 \left(x + h\right)} - \sqrt{3 x}}{h} \cdot \frac{\sqrt{3 \left(x + h\right)} + \sqrt{3 x}}{\sqrt{3 \left(x + h\right)} + \sqrt{3 x}}$

Remember difference of perfect squares

$f ' \left(x\right) = {\lim}_{h \to 0} \frac{3 \left(x + h\right) - \left(3 x\right)}{h \cdot \left(\sqrt{3 \left(x + h\right)} + \sqrt{3 x}\right)}$

Distribute the $3$

$f ' \left(x\right) = {\lim}_{h \to 0} \frac{3 x + 3 h - 3 x}{h \cdot \left(\sqrt{3 \left(x + h\right)} + \sqrt{3 x}\right)}$

Simplify

$f ' \left(x\right) = {\lim}_{h \to 0} \frac{3 h}{h \cdot \left(\sqrt{3 \left(x + h\right)} + \sqrt{3 x}\right)}$

Cancel the factor $h$

$f ' \left(x\right) = {\lim}_{h \to 0} \frac{3}{\sqrt{3 \left(x + h\right)} + \sqrt{3 x}}$

Now you can substitute in 0 for $h$

$\frac{3}{\sqrt{3 \left(x + 0\right)} + \sqrt{3 x}} =$

Simplify and manipulate the square roots

$\frac{3}{\sqrt{3 \left(x\right)} + \sqrt{3 x}} = \frac{3}{\sqrt{3 x} + \sqrt{3 x}} = \frac{3}{2 \sqrt{3 x}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3 \sqrt{3}}{6 \sqrt{x}} = \frac{\sqrt{3}}{2 \sqrt{x}} = \frac{\sqrt{3}}{2 \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} = \frac{\sqrt{3 x}}{2 x}$

b. Product Rule

$f \left(x\right) = u \cdot v = \sqrt{3} \cdot \sqrt{x} = \sqrt{3} \cdot {x}^{\frac{1}{2}}$

$u ' = 0$
$v ' = \left(\frac{1}{2}\right) {x}^{\frac{1}{2} - 1} = \left(\frac{1}{2}\right) {x}^{\frac{1}{2} - \frac{2}{2}} = \left(\frac{1}{2}\right) {x}^{- \frac{1}{2}} = \frac{1}{2 \sqrt{x}}$

Apply the product rule by making the appropriate substitutions

$f ' \left(x\right) = u \cdot v ' + v \cdot u ' = \sqrt{3} \cdot \frac{1}{2 \sqrt{x}} + \sqrt{x} \cdot 0 = \frac{\sqrt{3}}{2 \sqrt{x}} = \frac{\sqrt{3}}{2 \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} = \frac{\sqrt{3 x}}{2 x}$

c. Factor out the constant of $\sqrt{3}$

$f \left(x\right) = \sqrt{3} \cdot {x}^{\frac{1}{2}}$

$f ' \left(x\right) = \sqrt{3} \cdot \frac{d}{\mathrm{dx}} \left({x}^{\frac{1}{2}}\right) = \sqrt{3} \cdot \left(\frac{1}{2}\right) {x}^{\frac{1}{2} - 1} = \sqrt{3} \cdot \left(\frac{1}{2}\right) {x}^{\frac{1}{2} - \frac{2}{2}} = \sqrt{3} \cdot \left(\frac{1}{2}\right) {x}^{- \frac{1}{2}} = \sqrt{3} \cdot \left(\frac{1}{2 {x}^{\frac{1}{2}}}\right) = \frac{\sqrt{3}}{2 {x}^{\frac{1}{2}}} = \frac{\sqrt{3}}{2 \sqrt{x}} = \frac{\sqrt{3}}{2 \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} = \frac{\sqrt{3 x}}{2 x}$


text:
Indico has been upgraded to v2.3. Please see our blog for a list of improvements.

22-27 March 2015
Hotel do Bosque
Brazil/East timezone

Decay constants of the pion and its excitations in holographic QCD

24 Mar 2015, 15:30
30m
Hotel do Bosque

Hotel do Bosque

Rodovia Mário Covas (Rio-Santos) BR - 101 Sul, Km 533, Angra dos Reis, RJ, Brazil
Oral presentation Field theoretical approaches to QCD

Speaker

Prof. Gastão Krein (UNESP)

Description

There is a remarkable prediction of QCD that the leptonic decay constants of the excited states of the pion are dramatically suppressed relative to that of the ground-state pion - in the chiral limit, the decay constants of the excited pions are exactly zero. Although within a quark model perspective a suppression of a leptonic decay constant for excited states is expected, as it is proportional to the wave-function at the origin, there is, however, no a priori reason, within such a perspective, for the dramatic suppression predicted by QCD. Lattice computations give conflicting results. In the present talk we present results of a recent study of the structure of excited pions within a chiral holographic QCD model; we present results on the leptonic decay constants of pion's excited states and present new predictions concerning their quark mass dependence. Comparisons are made with corresponding results obtained in light-front holography.

Primary author

Prof. Gastão Krein (UNESP)

 Slides


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# Use of MathJax (or something similar) in .Rd files

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## Use of MathJax (or something similar) in .Rd files

 Hello All, I am wondering if there has ever been any discussion/consideration given to incorporating MathJax (or something similar) into R for rendering equations in .Rd files. I know that equations are rendered beautifully in the pdf manuals, but I suspect the majority of users primarily look at the html help files when using R. While I am comfortable reading something like "B_x(a,b) = integral_0^x t^(a-1) (1-t)^(b-1) dt", it would definitely be nicer if this was shown as a 'proper' equation. Best, Wolfgang ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Use of MathJax (or something similar) in .Rd files

 You can probably already do this using the \if{html} conditional, and either including the matchjax js files in the package, or getting them from a CDN. Gabor On Thu, Apr 30, 2020 at 9:13 AM Viechtbauer, Wolfgang (SP) <[hidden email]> wrote: > > Hello All, > > I am wondering if there has ever been any discussion/consideration given to incorporating MathJax (or something similar) into R for rendering equations in .Rd files. I know that equations are rendered beautifully in the pdf manuals, but I suspect the majority of users primarily look at the html help files when using R. While I am comfortable reading something like "B_x(a,b) = integral_0^x t^(a-1) (1-t)^(b-1) dt", it would definitely be nicer if this was shown as a 'proper' equation. > > Best, > Wolfgang > > ______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-devel______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Use of MathJax (or something similar) in .Rd files

 Interesting. I gave this a try, but couldn't make this work. One would have to infuse something like or into the .html file, but just adding this to the .Rd file (directly or wrapped with \if{html}{}) doesn't do that (it just gets printed). If somebody has already successfully done this, I would be interested in hearing how. Best, Wolfgang >-----Original Message----- >From: Gábor Csárdi [mailto:[hidden email]] >Sent: Thursday, 30 April, 2020 11:51 >To: Viechtbauer, Wolfgang (SP) >Cc: r-devel >Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files > >You can probably already do this using the \if{html} conditional, and >either including the matchjax js files in the package, or getting them >from a CDN. > >Gabor > >On Thu, Apr 30, 2020 at 9:13 AM Viechtbauer, Wolfgang (SP) ><[hidden email]> wrote: >> >> Hello All, >> >> I am wondering if there has ever been any discussion/consideration given >to incorporating MathJax (or something similar) into R for rendering >equations in .Rd files. I know that equations are rendered beautifully in >the pdf manuals, but I suspect the majority of users primarily look at the >html help files when using R. While I am comfortable reading something like >"B_x(a,b) = integral_0^x t^(a-1) (1-t)^(b-1) dt", it would definitely be >nicer if this was shown as a 'proper' equation. >> >> Best, >> Wolfgang >> >> ______________________________________________ >> [hidden email] mailing list >> https://stat.ethz.ch/mailman/listinfo/r-devel______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Use of MathJax (or something similar) in .Rd files

 In reply to this post by Gábor Csárdi This sounds promising. Also remember that user-defined macros are possible, so even if you need a lot of contortions to enter the code, you can simplify it at the Rd level. Duncan Murdoch On 30/04/2020 5:50 a.m., Gábor Csárdi wrote: > You can probably already do this using the \if{html} conditional, and > either including the matchjax js files in the package, or getting them > from a CDN. > > Gabor > > On Thu, Apr 30, 2020 at 9:13 AM Viechtbauer, Wolfgang (SP) > <[hidden email]> wrote: >> >> Hello All, >> >> I am wondering if there has ever been any discussion/consideration given to incorporating MathJax (or something similar) into R for rendering equations in .Rd files. I know that equations are rendered beautifully in the pdf manuals, but I suspect the majority of users primarily look at the html help files when using R. While I am comfortable reading something like "B_x(a,b) = integral_0^x t^(a-1) (1-t)^(b-1) dt", it would definitely be nicer if this was shown as a 'proper' equation. >> >> Best, >> Wolfgang >> >> ______________________________________________ >> [hidden email] mailing list >> https://stat.ethz.ch/mailman/listinfo/r-devel> > ______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-devel> ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Use of MathJax (or something similar) in .Rd files

 In reply to this post by Wolfgang Viechtbauer-2 On 30/04/2020 6:15 a.m., Viechtbauer, Wolfgang (SP) wrote: > Interesting. I gave this a try, but couldn't make this work. One would have to infuse something like > > > > or > > > > into the .html file, but just adding this to the .Rd file (directly or wrapped with \if{html}{}) doesn't do that (it just gets printed). If somebody has already successfully done this, I would be interested in hearing how. You'd need to wrap that code in \out (inside the conditional, i.e. something like \if{html}{\out{}} Duncan Murdoch > > Best, > Wolfgang > >> -----Original Message----- >> From: Gábor Csárdi [mailto:[hidden email]] >> Sent: Thursday, 30 April, 2020 11:51 >> To: Viechtbauer, Wolfgang (SP) >> Cc: r-devel >> Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files >> >> You can probably already do this using the \if{html} conditional, and >> either including the matchjax js files in the package, or getting them >>from a CDN. >> >> Gabor >> >> On Thu, Apr 30, 2020 at 9:13 AM Viechtbauer, Wolfgang (SP) >> <[hidden email]> wrote: >>> >>> Hello All, >>> >>> I am wondering if there has ever been any discussion/consideration given >> to incorporating MathJax (or something similar) into R for rendering >> equations in .Rd files. I know that equations are rendered beautifully in >> the pdf manuals, but I suspect the majority of users primarily look at the >> html help files when using R. While I am comfortable reading something like >> "B_x(a,b) = integral_0^x t^(a-1) (1-t)^(b-1) dt", it would definitely be >> nicer if this was shown as a 'proper' equation. >>> >>> Best, >>> Wolfgang >>> >>> ______________________________________________ >>> [hidden email] mailing list >>> https://stat.ethz.ch/mailman/listinfo/r-devel> ______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-devel> ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Use of MathJax (or something similar) in .Rd files

 Thanks Gabor and Duncan! It works. For those interested, I added this to the beginning of the \details{} section: \if{html}{\out{ }} And then I can use: \if{html}{\out{$$B_{x(a,b)} = \int_0^x t^{a-1} (1-t)^{b-1} dt$$}} or \if{html}{\out{$B_{x(a,b)} = \int_0^x t^{a-1} (1-t)^{b-1} dt$}} to render the equation (inline or as 'displaymath'). I'll try to add MathJax locally next. Best, Wolfgang >-----Original Message----- >From: Duncan Murdoch [mailto:[hidden email]] >Sent: Thursday, 30 April, 2020 12:22 >To: Viechtbauer, Wolfgang (SP); r-devel >Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files > >On 30/04/2020 6:15 a.m., Viechtbauer, Wolfgang (SP) wrote: >> Interesting. I gave this a try, but couldn't make this work. One would >have to infuse something like >> >> >> >> or >> >> >> >> into the .html file, but just adding this to the .Rd file (directly or >wrapped with \if{html}{}) doesn't do that (it just gets printed). If >somebody has already successfully done this, I would be interested in >hearing how. > >You'd need to wrap that code in \out (inside the conditional, i.e. >something like > >\if{html}{\out{}} > >Duncan Murdoch > >> >> Best, >> Wolfgang >> >>> -----Original Message----- >>> From: Gábor Csárdi [mailto:[hidden email]] >>> Sent: Thursday, 30 April, 2020 11:51 >>> To: Viechtbauer, Wolfgang (SP) >>> Cc: r-devel >>> Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files >>> >>> You can probably already do this using the \if{html} conditional, and >>> either including the matchjax js files in the package, or getting them >>>from a CDN. >>> >>> Gabor >>> >>> On Thu, Apr 30, 2020 at 9:13 AM Viechtbauer, Wolfgang (SP) >>> <[hidden email]> wrote: >>>> >>>> Hello All, >>>> >>>> I am wondering if there has ever been any discussion/consideration given >>> to incorporating MathJax (or something similar) into R for rendering >>> equations in .Rd files. I know that equations are rendered beautifully in >>> the pdf manuals, but I suspect the majority of users primarily look at >the >>> html help files when using R. While I am comfortable reading something >like >>> "B_x(a,b) = integral_0^x t^(a-1) (1-t)^(b-1) dt", it would definitely be >>> nicer if this was shown as a 'proper' equation. >>>> >>>> Best, >>>> Wolfgang ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Use of MathJax (or something similar) in .Rd files

 Thank you.  This is a very useful idea! Best, Ravi ________________________________ From: R-devel <[hidden email]> on behalf of Viechtbauer, Wolfgang (SP) <[hidden email]> Sent: Thursday, April 30, 2020 6:59:02 AM To: r-devel Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files Thanks Gabor and Duncan! It works. For those interested, I added this to the beginning of the \details{} section: \if{html}{\out{ }} And then I can use: \if{html}{\out{$$B_{x(a,b)} = \int_0^x t^{a-1} (1-t)^{b-1} dt$$}} or \if{html}{\out{$B_{x(a,b)} = \int_0^x t^{a-1} (1-t)^{b-1} dt$}} to render the equation (inline or as 'displaymath'). I'll try to add MathJax locally next. Best, Wolfgang >-----Original Message----- >From: Duncan Murdoch [mailto:[hidden email]] >Sent: Thursday, 30 April, 2020 12:22 >To: Viechtbauer, Wolfgang (SP); r-devel >Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files > >On 30/04/2020 6:15 a.m., Viechtbauer, Wolfgang (SP) wrote: >> Interesting. I gave this a try, but couldn't make this work. One would >have to infuse something like >> >> >> >> or >> >> >> >> into the .html file, but just adding this to the .Rd file (directly or >wrapped with \if{html}{}) doesn't do that (it just gets printed). If >somebody has already successfully done this, I would be interested in >hearing how. > >You'd need to wrap that code in \out (inside the conditional, i.e. >something like > >\if{html}{\out{}} > >Duncan Murdoch > >> >> Best, >> Wolfgang >> >>> -----Original Message----- >>> From: G�bor Cs�rdi [mailto:[hidden email]] >>> Sent: Thursday, 30 April, 2020 11:51 >>> To: Viechtbauer, Wolfgang (SP) >>> Cc: r-devel >>> Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files >>> >>> You can probably already do this using the \if{html} conditional, and >>> either including the matchjax js files in the package, or getting them >>>from a CDN. >>> >>> Gabor >>> >>> On Thu, Apr 30, 2020 at 9:13 AM Viechtbauer, Wolfgang (SP) >>> <[hidden email]> wrote: >>>> >>>> Hello All, >>>> >>>> I am wondering if there has ever been any discussion/consideration given >>> to incorporating MathJax (or something similar) into R for rendering >>> equations in .Rd files. I know that equations are rendered beautifully in >>> the pdf manuals, but I suspect the majority of users primarily look at >the >>> html help files when using R. While I am comfortable reading something >like >>> "B_x(a,b) = integral_0^x t^(a-1) (1-t)^(b-1) dt", it would definitely be >>> nicer if this was shown as a 'proper' equation. >>>> >>>> Best, >>>> Wolfgang ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel        [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
 Hi All, After some tinkering, and with support from Duncan, I put together a package that allows for easy inclusion of MathJax equations in Rd files. The package has been submitted to CRAN, but those who want to try this out already can get it here: https://github.com/wviechtb/mathjaxror in other words: install.packages("remotes") remotes::install_github("wviechtb/mathjaxr") Package authors can then use the \loadmathjax macro to enable its use and the \mjeqn{latex}{ascii} and \mjdeqn{latex}{ascii} macros for including inline and displayed equations. Feedback and suggestions more than welcome! Best, Wolfgang >-----Original Message----- >From: Ravi Varadhan [mailto:[hidden email]] >Sent: Thursday, 30 April, 2020 15:58 >To: Viechtbauer, Wolfgang (SP); r-devel >Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files > >Thank you.  This is a very useful idea! > >Best, >Ravi >________________________________________ >From: R-devel <[hidden email]> on behalf of Viechtbauer, >Wolfgang (SP) <[hidden email]> >Sent: Thursday, April 30, 2020 6:59:02 AM >To: r-devel >Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files > >Thanks Gabor and Duncan! It works. For those interested, I added this to the >beginning of the \details{} section: > >\if{html}{\out{ > >}} > >And then I can use: > >\if{html}{\out{$$B_{x(a,b)} = \int_0^x t^{a-1} (1-t)^{b-1} dt$$}} > >or > >\if{html}{\out{$B_{x(a,b)} = \int_0^x t^{a-1} (1-t)^{b-1} dt$}} > >to render the equation (inline or as 'displaymath'). I'll try to add MathJax >locally next. > >Best, >Wolfgang > >>-----Original Message----- >>From: Duncan Murdoch [mailto:[hidden email]] >>Sent: Thursday, 30 April, 2020 12:22 >>To: Viechtbauer, Wolfgang (SP); r-devel >>Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files >> >>On 30/04/2020 6:15 a.m., Viechtbauer, Wolfgang (SP) wrote: >>> Interesting. I gave this a try, but couldn't make this work. One would >>have to infuse something like >>> >>> >>> >>> or >>> >>> >>> >>> into the .html file, but just adding this to the .Rd file (directly or >>wrapped with \if{html}{}) doesn't do that (it just gets printed). If >>somebody has already successfully done this, I would be interested in >>hearing how. >> >>You'd need to wrap that code in \out (inside the conditional, i.e. >>something like >> >>\if{html}{\out{}} >> >>Duncan Murdoch >> >>> >>> Best, >>> Wolfgang >>> >>>> -----Original Message----- >>>> From: Gábor Csárdi [mailto:[hidden email]] >>>> Sent: Thursday, 30 April, 2020 11:51 >>>> To: Viechtbauer, Wolfgang (SP) >>>> Cc: r-devel >>>> Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files >>>> >>>> You can probably already do this using the \if{html} conditional, and >>>> either including the matchjax js files in the package, or getting them >>>>from a CDN. >>>> >>>> Gabor >>>> >>>> On Thu, Apr 30, 2020 at 9:13 AM Viechtbauer, Wolfgang (SP) >>>> <[hidden email]> wrote: >>>>> >>>>> Hello All, >>>>> >>>>> I am wondering if there has ever been any discussion/consideration >given >>>> to incorporating MathJax (or something similar) into R for rendering >>>> equations in .Rd files. I know that equations are rendered beautifully >in >>>> the pdf manuals, but I suspect the majority of users primarily look at >>the >>>> html help files when using R. While I am comfortable reading something >>like >>>> "B_x(a,b) = integral_0^x t^(a-1) (1-t)^(b-1) dt", it would definitely be >>>> nicer if this was shown as a 'proper' equation. >>>>> >>>>> Best, >>>>> Wolfgang ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
 The package is really nice, I recommend it.  Package writers should install it, but users of their help pages don't need it unless they're working offline.  Even then, because of the way MathJax works, they get something that's fairly readable, e.g. "The probability density function of a normal distribution is given by $f(x) = \frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2},$ where $$\mu$$ denotes the mean of the distribution and $$\sigma$$ its standard deviation." For a user who is online, or who has the mathjaxr package installed, this would be displayed with the equation as a display equation, and mu and sigma displayed properly inline.  To obtain that output I used this input in an Rd file: "The probability density function of a normal distribution is given by \mjdeqn{f(x) = \frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2},}{f(x) = 1/(sqrt(2 pi) sigma) e^-((x - mu)^2/(2 sigma^2)),} where \mjeqn{\mu}{mu} denotes the mean of the distribution and \mjeqn{\sigma}{sigma} its standard deviation." A frill that might be possible to add would be a test for a remote connection, and a fallback to the ascii display if it's not there.  It already has a display-time test for the presence of the MathJax files locally. Maybe once all the kinks are worked out this could be incorporated into the R distribution.  However, it's not really needed there, since it works so well as a contributed package already. Duncan Murdoch On 05/05/2020 9:11 a.m., Viechtbauer, Wolfgang (SP) wrote: > Hi All, > > After some tinkering, and with support from Duncan, I put together a package that allows for easy inclusion of MathJax equations in Rd files. The package has been submitted to CRAN, but those who want to try this out already can get it here: > > https://github.com/wviechtb/mathjaxr> > or in other words: > > install.packages("remotes") > remotes::install_github("wviechtb/mathjaxr") > > Package authors can then use the \loadmathjax macro to enable its use and the \mjeqn{latex}{ascii} and \mjdeqn{latex}{ascii} macros for including inline and displayed equations. > > Feedback and suggestions more than welcome! > > Best, > Wolfgang > >> -----Original Message----- >> From: Ravi Varadhan [mailto:[hidden email]] >> Sent: Thursday, 30 April, 2020 15:58 >> To: Viechtbauer, Wolfgang (SP); r-devel >> Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files >> >> Thank you.  This is a very useful idea! >> >> Best, >> Ravi >> ________________________________________ >> From: R-devel <[hidden email]> on behalf of Viechtbauer, >> Wolfgang (SP) <[hidden email]> >> Sent: Thursday, April 30, 2020 6:59:02 AM >> To: r-devel >> Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files >> >> Thanks Gabor and Duncan! It works. For those interested, I added this to the >> beginning of the \details{} section: >> >> \if{html}{\out{ >> >> }} >> >> And then I can use: >> >> \if{html}{\out{$$B_{x(a,b)} = \int_0^x t^{a-1} (1-t)^{b-1} dt$$}} >> >> or >> >> \if{html}{\out{$B_{x(a,b)} = \int_0^x t^{a-1} (1-t)^{b-1} dt$}} >> >> to render the equation (inline or as 'displaymath'). I'll try to add MathJax >> locally next. >> >> Best, >> Wolfgang >> >>> -----Original Message----- >>> From: Duncan Murdoch [mailto:[hidden email]] >>> Sent: Thursday, 30 April, 2020 12:22 >>> To: Viechtbauer, Wolfgang (SP); r-devel >>> Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files >>> >>> On 30/04/2020 6:15 a.m., Viechtbauer, Wolfgang (SP) wrote: >>>> Interesting. I gave this a try, but couldn't make this work. One would >>> have to infuse something like >>>> >>>> >>>> >>>> or >>>> >>>> >>>> >>>> into the .html file, but just adding this to the .Rd file (directly or >>> wrapped with \if{html}{}) doesn't do that (it just gets printed). If >>> somebody has already successfully done this, I would be interested in >>> hearing how. >>> >>> You'd need to wrap that code in \out (inside the conditional, i.e. >>> something like >>> >>> \if{html}{\out{}} >>> >>> Duncan Murdoch >>> >>>> >>>> Best, >>>> Wolfgang >>>> >>>>> -----Original Message----- >>>>> From: Gábor Csárdi [mailto:[hidden email]] >>>>> Sent: Thursday, 30 April, 2020 11:51 >>>>> To: Viechtbauer, Wolfgang (SP) >>>>> Cc: r-devel >>>>> Subject: Re: [Rd] Use of MathJax (or something similar) in .Rd files >>>>> >>>>> You can probably already do this using the \if{html} conditional, and >>>>> either including the matchjax js files in the package, or getting them >>>> >from a CDN. >>>>> >>>>> Gabor >>>>> >>>>> On Thu, Apr 30, 2020 at 9:13 AM Viechtbauer, Wolfgang (SP) >>>>> <[hidden email]> wrote: >>>>>> >>>>>> Hello All, >>>>>> >>>>>> I am wondering if there has ever been any discussion/consideration >> given >>>>> to incorporating MathJax (or something similar) into R for rendering >>>>> equations in .Rd files. I know that equations are rendered beautifully >> in >>>>> the pdf manuals, but I suspect the majority of users primarily look at >>> the >>>>> html help files when using R. While I am comfortable reading something >>> like >>>>> "B_x(a,b) = integral_0^x t^(a-1) (1-t)^(b-1) dt", it would definitely be >>>>> nicer if this was shown as a 'proper' equation. >>>>>> >>>>>> Best, >>>>>> Wolfgang > > ______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-devel> ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel


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# Séminaires : Séminaire Théorie des Nombres

 Equipe(s) : fa, tn, tga, Responsables : Marc Hindry, Bruno Kahn, Wieslawa Niziol, Cathy Swaenepoel Email des responsables : cathy.swaenepoel@imj-prg.fr Salle : Adresse : Description http://www.imj-prg.fr/tn/STN/stnj.html

 Orateur(s) G. Stevens - , Titre p-Adic L-Functions and the Evil Twin Date 28/04/2003 Horaire 14:00 à 16:00 Diffusion Résume In the classical theory of modular forms, the Eichler-Shimura theoremassociates parabolic cohomology classes to classical cusp forms.In attempting to extend the Eichler-Shimura theory to $p$-adic analyticfamilies of modular forms one encounters subtle problems of localfreenessand semi-simplicity of Hecke algebras acting on certain overconvergentcohomology spaces. To solve these problems, more concrete knowledgeisneeded about these cohomology spaces.This talk will give an analytic construction of a $p$-adic $L$-functionassociated to the "evil twin" of a classical ordinary eigenform.Adichotomy arises out of this construction: the criticalvalues of the $p$-adic $L$-function either (1) all vanish, or (2) agreewith the complex $L$-values up to standard multipliers. I willexplainhow this is related to semi-simplicity of the Hecke algebra actingontheoverconvergent cohomolgy. Moreover, I will describe joint workwithRobertPollack, in which we establish a sufficient condition for the latterconclusion in terms of modular forms modulo $p$. I willalso presentsome intriquing numerical data on the disribution of the zeroes ofthese$p$-adic $L$-functions that arose out of our work. Salle Adresse


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• ### Experimental search for the origin of low-energy modes in topological materials(1709.04000)

July 27, 2019 cond-mat.supr-con
Point-contact spectroscopy of several non-superconducting topological materials reveals a low temperature phase transition that is characterized by a Bardeen-Cooper-Schrieffer-type of criticality. We find such a behavior of differential conductance for topological surfaces of non-magnetic and magnetic Pb$_{1-y-x}$Sn$_y$Mn$_x$Te. We examine a possible contribution from superconducting nanoparticles, and show to what extent our data are consistent with Brzezicki's et al. theory [arXiv:1812.02168], assigning the observations to a collective state adjacent to atomic steps at topological surfaces.
• ### Disorder-generated non-Abelions(1707.02929)

Aug. 8, 2017 cond-mat.mes-hall
Two classes of topological superconductors and Majorana modes in condensed matter systems are known to date: one, in which impurity disorder strongly suppresses topological superconducting gap and is detrimental to Majorana modes, and the other, where Majorana fermions are protected by disorder-robust superconductor gap. In this work we predict a third class of topological superconductivity and Majorana modes, in which they appear exclusively in the presence of impurity disorder. Observation and control of Majorana fermions and other non-Abelions often requires a symmetry leading to a gap in a single-particle spectra. Disorder introduces states into the gap and enables conductance and proximity-induced superconductivity via the in-gap states. We show that disorder-enabled topological superconductivity can be realized in a quantum Hall ferromagnet, when helical domain walls are coupled to an s-wave superconductor. Solving a general quantum mechanical problem of impurity bound states in a system of spin-orbit coupled Landau levels, we show that disorder-induced Majorana modes emerge in a setting of the quantum Hall ferromagnetic transition in a CdMnTe quantum wells at a filling factor $\nu=2$. Recent experiments on transport through electrostatically controlled single domain wall in this system indicated the vital role of disorder in conductance, but left an unresolved question whether this could intrinsically preclude generation of Majorana fermions. The proposed resolution of the problem, demonstrating emergence of Majorana fermions exclusively due to impurity disorder, opens a path forward. We show that electrostatic control of domain walls in an integer quantum Hall ferromagnet allows manipulation of Majorana modes. Similar physics can emerge for ferromagnetic transitions in the fractional quantum Hall regime leading to the formation and control of higher order non-Abelian excitations.
• ### Superconducting detector for visible and near-infrared quantum emitters(1612.05838)

Dec. 17, 2016 quant-ph
Further development of quantum emitter based communication and sensing applications intrinsically depends on the availability of robust single-photon detectors. Here, we demonstrate a new generation of superconducting single-photon detectors specifically optimized for the 500-1100 nm wavelength range, which overlaps with the emission spectrum of many interesting solid-state atom-like systems, such as nitrogen-vacancy and silicon-vacancy centers in diamond. The fabricated detectors have a wide dynamic range (up to 350 million counts per second), low dark count rate (down to 0.1 counts per second), excellent jitter (62 ps), and the possibility of on-chip integration with a quantum emitter. In addition to performance characterization, we tested the detectors in real experimental conditions involving nanodiamond nitrogen-vacancy emitters enhanced by a hyperbolic metamaterial.
• ### Soft photon registration at Nuclotron(1510.00517)

Oct. 2, 2015 nucl-ex
First results of a soft photon yield in nucleus-nuclear interactions at 3.5 GeV per nucleon are presented. These photons have been registered at Nuclotron (LHEP, JINR) by an electromagnetic calorimeter built in the SVD Collaboration. The obtained spectra confirm the excess yield in the energy region less than 50 MeV in comparison with theoretical predictions and agree with previous experiments at high-energy interactions.
• ### Quantum optical device accelerating dynamic programming(quant-ph/0506081)

Nov. 23, 2010 quant-ph
In this paper we discuss analogue computers based on quantum optical systems accelerating dynamic programming for some computational problems. These computers, at least in principle, can be realized by actually existing devices. We estimate an acceleration in resolving of some NP-hard problems that can be obtained in such a way versus deterministic computers


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# Posts from January 2015

• ### Optimisation of a composite function involving a differentiable monotonic component

While doing optimisation problems with calculus I often find myself needing to optimise expressions such as $$\sqrt{\sin x}$$, though often more complicated. Using the chain rule on the square root is time-consuming, and it seems logical that as a monotonically increasing function, the… »


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# Acid-base titration: Unknown base in burette or flask?

While finding the concentration of $$\ce{NaOH}$$ via titration with $$\pu{0.05M}$$ Oxalic acid solution (in our lab) we put NaOH in the burette whereas added phenolphthalein to oxalic acid in titration flask. But in this video the demonstrator does just opposite ( i.e., $$\ce{NaOH}$$ in flak with phenolphthalein). I am confused in deciding which one might be correct. My view is that either would do the job but I need some clarification.

• Both are perfectly correct methods as long as the volume measurements are accurate. – M. Farooq Nov 1 '19 at 16:12
• @M.Farooq - I disagree. It is easier to detect the change from colorless to colored. So, if possible, the titrant should be a base when phenolphthalein is the indicator. – MaxW Nov 1 '19 at 19:41
• No matter what you do students will always come to you holding the flask asking if this is the right end-point :-) For a skilled analytical chemist, both techniques should be fine and of course colorless to color is easier to detect. One is less painful than the other. – M. Farooq Nov 1 '19 at 20:42