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# Do magnets redshift light?

Do magnets redshift light?

Suppose we have an extremely powerful magnet (say the size of the Sun) and we have a smaller paramagnetic material above it (say. Titanium Brick which is indestructible) . Due to magnetic attraction it will fall toward the big magnet.

Now suppose that when it hits the surface of the bigger magnet, the smaller paramagnetic material (Titanium Brick) is somehow converted into photons (light) equivalent to its mass using $E=mc^2$, plus its kinetic energy, $KE_1$. These photons are shot upwards via an perfect mirror perpendicular to the paramagnetic material above. When they reach the magnet's original position they are again converted into mass in the form of the titanium brick.

If this is repeated many times it seems we will gain free energy, since the iron brick gains kinetic energy when it falls, but light is not effected by magnetic fields. (Or am I wrong?) But this will breach the energy conservation laws in specific "Energy cannot be created nor destroyed".

Therefore I must conclude light is (or should be) redshifted (loses energy) as it travels through magnetic field. The problem arises when you take into consideration that light is not charged so it should not be effected by any magnetic fields but my intuition tells me light should still be affected so is my conclusion valid or not? Is light red shifted in presence of an magnetic field?

If light is not red shifted then what is lost to ensure the system is not gaining energy but in this experiment it does not make sense since we are using photons which are not effected by magnetic fields and then recreating the para magnetic material (like Iron brick we used) so we should not loose energy as we are simply recreating the same object again.

• I have voted to close this question because the title of it and the content do not correspond. The content is science fiction scenario, the title is a legitimate main stream physics question. Dec 13 '21 at 14:21

I love the question! I wanted to share my thoughts and a possible explanation of this phenomena.

Lets look at the process(we'll start from a non-magnetized object):

1. Two bodies, one of which is a huge magnet and the other a very small ferromagnetic object are in free space.
2. No forces act upon them in the starting moment (gravity is unneded). The ferromagnet is not yet magnetised(the magnetic domains are still random)
3. We assume a swift magnetisation of the small object, they begin to attract each other magnetically. Energy used up can be calculated from the magnetic energy/coenergy in the hysteresis loop of the material. $W_m^*=\int_V\int_H B\mathrm{d}H\mathrm{d}V$
4. The actual force that attracts them magnetically is the Relucance force, this force wants to reduce the total magnetic resistance of a path and moves two objects into a position of minimal resistance for the magnetic flux.
5. The attraction process happens because of the same energy/coenergy mechanism. A force results from the change of coenergy $F=-\frac{\mathrm{d}W_m^*}{\mathrm{d}x}$.
6. As the small object gets closer the value of $W_m^*$ is decreased (you must consider the volume). Intuitive explanation: everything wants to get to the lowest energy state.
7. The increase in the kinetic Energy happens in parallel $E_k=\int_xF\mathrm{d}x$, therefore we can look on the initial value of $W_m^*$ as the potential energy.
8. In the moment the small object is converted to photons with $E=mc^2$ the $W_m^*$ ist mostly (depending on the moment) converted to $E_k$.

If I didn't make any mistakes, there should be no energy loss. The big magnet supplies the energy

• I changed the question to be para-magnetic rather than ferromagnetic as the energy is balanced but I cannot see balanced energy in an system if I put para-magnetic so I changed the question.
– user43495
May 30 '14 at 15:19
• For this observation, the difference would be that the paramagnetic materials have a lower amplitude of the force, and that they have a linear $B(H)$ characteristic. On the other hand if you argue that the hysteresis can somehow create an energy imbalance, well that would be worth thinking about. May 30 '14 at 15:27
• The hysteresis create an energy imbalance if paramagnetic material is used instead? That in mind, wouldn't energy be gained?
– user43495
Jul 22 '14 at 18:17

Remember that in addition to the pure mass of your smaller "test" magnet, you need to create it's energetically non-trivial magnetic structure. This should account for the energy gained through the electromagnetic attraction.

Barring non-linear quantum optics effects light is not influenced by magnetic fields. This is because the equations of motion are linear, therefore one electromagnetic field can not influence another without the help of charged matter.

• I just re-edited this question to rather than have smaller magnet, we just have 1 big magnet and 1 smaller ferromagnetic material but since this does not need me to create it's energetically non-trivial magnetic structure. This should NOT account for the energy gained through the electromagnetic attraction. That in mind, now we should be gain free-energy but since this would break laws of energy conservation as we are creating free energy.
– user43495
May 30 '14 at 12:48
• @user43495 It should. Ferromagnetic materials act as they do because they do have a non-trivial magnetic structure, see the wikipedia May 30 '14 at 12:49
• Ok thanks but my only question is surely energy from the photons can be converted into anything as all we are doing is changing the energy into the same object as we started with so we don't really need to put any energy to make it energetically non-trivial magnetic structure.
– user43495
May 30 '14 at 12:54

There are a lot of things wrong with this thought experiment, imo:

1. a piece of ferromagnetic material will immediately display a north and a south pole to the magnet lines. The north pole will be attracted the south will be repelled, and it will end up like iron filings when it hits the surface.

2. in the real world if it hits a surface it will break up into its component parts, depending on the energy. To get annihilation the energy should be huge and there will be many fragments atoms and photons

3. with the break up the ferromagnetism disappears and the individual particles will be bouncing off according to the rules of particles scattering ( look at crossections in the particle data book)

4. These will have every which angular distribution ( photons too) and will fly off according to their momentum and energy given by the big bounce. They will never cohere enough to fall back to any structure called solid matter to envisage it ending up at its previous spot whole . Energy is conserved in elementary particle interactions which means if you gather all the bits you will find the rest mass and kinetic energies all there.

5. Supposing the big magnet was made of antimatter, again the same problem of not reconstructability: annihilation of protons and neutrons gives mostly pions , the charged decay into muons and neutrinos and pi0 into two photons, that will never join up at the top, both because of angular distributions and because the crossection of gamma gamma to pi0 is tiny ( higher order diagrams and the electromagnetic coupling is 1/137 ). Nothing annihilates into just photons, and even if it did, the photons would disperse worse than a bagful of cats.

• This is similar to this experiment: mth.uct.ac.za/omei/gr/chap5/node2.html
– user43495
May 30 '14 at 14:32
• As far as I am concerned that one is wrong too. The "magical" intervention nullifies all conservation laws. Physics has no magic. Thought experiments are fine as long as they obey the laws of physics. May 30 '14 at 14:45
• The experiment quoted has no magic. It has photon distributions and the effect of gravity on them, and it shows an effect. It does not validate magical thinking. May 30 '14 at 14:52
• Nor does mine, fine let me edit it and suppose the observer has some magical method of converting all this energy into a photon of the same energy [ this is a thought experiment after all! ] just as the quoted thought experiment said
– user43495
May 30 '14 at 15:09
• I meant the Mossbauer effect experiment, they measured the effect of gravitation on a distribution of photons. There is no magic there, and they found an effect. The thought experiment is wrong, because there is no physics way of doing it and "magic" introduces the possibility of balancing conservation laws "illegally" May 30 '14 at 18:45


text:
7×7 MIMO System Using Extended 13-Element ESPAR Antenna

Title & Authors
7×7 MIMO System Using Extended 13-Element ESPAR Antenna
Bok, Junyeong; Lee, Seung Hwan; Ryu, Heung-Gyoon;

Abstract
Multiple-input and multiple-output (MIMO) technique is used in many communication fields in order to increase the channel capacity. However, this MIMO system has difficulty of miniaturization of antenna size due to the multiple RF chains Also, multiple RF chain raises some problems which increase power consumption at RF circuit and degrade the system performance due to the interference between RF chains. Because of these reasons, beamspace MIMO (BS-MIMO) technique with only single RF chain was proposed for MIMO transmission. This BS-MIMO system basically uses electronically steerable parasitic array radiator (ESPAR) antenna. Existing ESPAR antenna has a 5-element structure. So, it is possible to do only $\small{3{\times}3}$ MIMO transmission. Therefore, in order to extend BS-MIMO dimension, extension of ESPAR antenna structure is essential. In this paper, we show that BS-MIMO dimension can be increased according to the extension of structure of the ESPAR antenna, as in the conventional MIMO techniques. For example, we show that it is possible to design the $\small{7{\times}7}$ BS-MIMO transmissions with the 13-element ESPAR antenna. Also, when the number of parasitic elements of ESPAR antenna increases by two elements, MIMO dimension is expanded by 1.
Keywords
BS-MIMO;ESPAR antenna;single RF chain;pattern modulation;MIMO;
Language
Korean
Cited by
References
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M. A. Jensen and J. W. Wallace. "Termination-dependent diversity performance of coupled antennas: Network theory analysis," IEEE Trans. Antennas and Propagation, vol. 52, no. 1. pp. 98-105, Jan. 2004.

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A. F. Molisch, G. Kristensson, J. B. Andersen, and B. K. Lau. "Impact of matching network on bandwidth of compact antenna arrays," IEEE Trans. Antennas and Propagation, vol. 54, no. 11, pp. 3225-3238, Nov. 2006.

6.
R.Y. Mesleh, H. Haas, S. Sinanovic, A. Chang Wook, and Y. Sangboh, "Spatial modulation," IEEE Trans. Veh. Technol., vol.57, no.4, pp. 2228-2241, Jul. 2008.

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R. Rajashekar, K.V.S. Hari, and L. Hanzo, "Antenna selection in spatial modulation systems," IEEE Commun. Lett., vol. 17, no. 3, pp. 521-524, Mar. 2013.

8.
J. Yoon and S. Lee, "Compact meander-type antenna with a two layer structure for bluetooth operation," J. KICS, vol. 35, no. 4, pp. 89-93, Apr. 2010.

9.
A. Kalis, A. G. Kanatas, and C. Papadias, "A novel approach to MIMO transmission using a single rf front end" IEEE J. Selected Areas in Commun., vol. 26, no. 6, pp. 972-980, Aug. 2008.

10.
C. Y. An, S. H. Lee, and H. G. Ryu, "Beamspace MIMO system using ESPAR antenna with single RF chain," J. KICS, vol. 38, no. 10, pp. 885-892, Oct. 2013.

11.
O. N. Alrabadi, J. Perruisseau-Carrier, and A. Kalis, "MIMO transmission using a single RF source: Theory and antenna design," IEEE Trans. Antennas and Propagation, vol. 60, no. 2, pp. 654-664, Feb. 2012.

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V. Barousis, A. G. Kanatas, and A. Kalis, "Single RF MIMO systems: Exploiting the capabilities of parasitic antennas," IEEE Veh. Technol. Conf. (VTC Fall), pp. 1-5, Sept. 2011.

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text:
American Institute of Mathematical Sciences

June  2012, 32(6): 2207-2221. doi: 10.3934/dcds.2012.32.2207

On dynamical behavior of viscous Cahn-Hilliard equation

 1 Department of Mathematics, School of Science, Tianjin University, Tianjin, 300072, China, China

Received  February 2011 Revised  October 2011 Published  February 2012

In this paper, we consider the initial and Dirichlet boundary value problem of the viscous Cahn-Hilliard equation with a general nonlinearity $f$, that is $$d((1-\alpha)u-\alpha\Delta u)+(\Delta^2u-\Delta f(u))dt= 0,$$where $\alpha\in[0,1]$. Firstly, we establish the existence and continuity results on weak solutions and attractors to this problem. Secondly, we show the $\alpha$-uniform attractiveness of the attractors $A_\alpha$.
Citation: Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
References:
 [1] N. D. Alikakos, P. W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differential Equations, 90 (1991), 81-135. doi: 10.1016/0022-0396(91)90163-4.  Google Scholar [2] P. Bates and P. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time-scales for coarsening, Phys. D, 43 (1990), 335-348. doi: 10.1016/0167-2789(90)90141-B.  Google Scholar [3] F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. I. Computations, Nonliearity, 8 (1995), 131-160. doi: 10.1088/0951-7715/8/2/002.  Google Scholar [4] A. N. Carvalho and T. Dlotko, Dynamics of the viscous Cahn-Hilliard equation, J. Math. Anal. Appl., 344 (2008), 703-725. doi: 10.1016/j.jmaa.2008.03.020.  Google Scholar [5] A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514. doi: 10.1016/0362-546X(94)00205-V.  Google Scholar [6] T. Dlotko, On the Cahn-Hilliard equation with a logarithmic free energy $H^2$ and $H^3$, J. Differential Equations, 113 (1994), 381-393. doi: 10.1006/jdeq.1994.1129.  Google Scholar [7] C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. II. Analysis, J. Differeential Equations, 128 (1996), 387-414. doi: 10.1006/jdeq.1996.0101.  Google Scholar [8] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. doi: 10.1002/mana.200310186.  Google Scholar [9] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar [10] J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. (4), 154 (1989), 281-326. doi: 10.1007/BF01790353.  Google Scholar [11] J. K. Hale, X.-B. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp., 50 (1988), 89-123. doi: 10.1090/S0025-5718-1988-0917820-X.  Google Scholar [12] J. K. Hale, Dynamics of numerical approximations, Appl. Math. Comput., 89 (1998), 5-15. doi: 10.1016/S0096-3003(97)81644-X.  Google Scholar [13] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar [14] D. S. Li and P. E. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasg. Math. J., 46 (2004), 131-141. doi: 10.1017/S0017089503001605.  Google Scholar [15] D. S. Li and C. K. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Differential Equations, 149 (1998), 191-210. doi: 10.1006/jdeq.1998.3429.  Google Scholar [16] D. S. Li and X. X. Zhang, Strongly positively-invariant attractor for periodic processes, J. Math. Anal. Appl., 241 (2000), 10-29. doi: 10.1006/jmaa.1999.6499.  Google Scholar [17] A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590.  Google Scholar [18] B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Differential Equations, 14 (1989), 245-297.  Google Scholar [19] A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in "Material Instabilities in Continuum Mechanics" (Edinburgh, 1985–1986), Oxford Sci. Publ., Oxford Univ. Press, New York, (1988), 329-342.  Google Scholar [20] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar [21] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

show all references

References:
 [1] N. D. Alikakos, P. W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differential Equations, 90 (1991), 81-135. doi: 10.1016/0022-0396(91)90163-4.  Google Scholar [2] P. Bates and P. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time-scales for coarsening, Phys. D, 43 (1990), 335-348. doi: 10.1016/0167-2789(90)90141-B.  Google Scholar [3] F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. I. Computations, Nonliearity, 8 (1995), 131-160. doi: 10.1088/0951-7715/8/2/002.  Google Scholar [4] A. N. Carvalho and T. Dlotko, Dynamics of the viscous Cahn-Hilliard equation, J. Math. Anal. Appl., 344 (2008), 703-725. doi: 10.1016/j.jmaa.2008.03.020.  Google Scholar [5] A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514. doi: 10.1016/0362-546X(94)00205-V.  Google Scholar [6] T. Dlotko, On the Cahn-Hilliard equation with a logarithmic free energy $H^2$ and $H^3$, J. Differential Equations, 113 (1994), 381-393. doi: 10.1006/jdeq.1994.1129.  Google Scholar [7] C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. II. Analysis, J. Differeential Equations, 128 (1996), 387-414. doi: 10.1006/jdeq.1996.0101.  Google Scholar [8] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. doi: 10.1002/mana.200310186.  Google Scholar [9] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar [10] J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. (4), 154 (1989), 281-326. doi: 10.1007/BF01790353.  Google Scholar [11] J. K. Hale, X.-B. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp., 50 (1988), 89-123. doi: 10.1090/S0025-5718-1988-0917820-X.  Google Scholar [12] J. K. Hale, Dynamics of numerical approximations, Appl. Math. Comput., 89 (1998), 5-15. doi: 10.1016/S0096-3003(97)81644-X.  Google Scholar [13] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar [14] D. S. Li and P. E. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasg. Math. J., 46 (2004), 131-141. doi: 10.1017/S0017089503001605.  Google Scholar [15] D. S. Li and C. K. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Differential Equations, 149 (1998), 191-210. doi: 10.1006/jdeq.1998.3429.  Google Scholar [16] D. S. Li and X. X. Zhang, Strongly positively-invariant attractor for periodic processes, J. Math. Anal. Appl., 241 (2000), 10-29. doi: 10.1006/jmaa.1999.6499.  Google Scholar [17] A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590.  Google Scholar [18] B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Differential Equations, 14 (1989), 245-297.  Google Scholar [19] A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in "Material Instabilities in Continuum Mechanics" (Edinburgh, 1985–1986), Oxford Sci. Publ., Oxford Univ. Press, New York, (1988), 329-342.  Google Scholar [20] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar [21] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar
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Maximum value of Von Neumann Entropy

Homework Statement

Prove that the maximum value of the Von Neumann entropy for a completely random ensemble is $ln(N)$ for some population $N$

Homework Equations

$S = -Tr(ρ~lnρ)$
$<A> = Tr(ρA)$

The Attempt at a Solution

Using Lagrange multipliers and extremizing S

Let $~S = -Tr(ρ~lnρ) + γ (<1> - 1) = -Tr(ρ~lnρ) + γ (Tr(ρ) - 1)$

$δS = Tr( -δρ~ lnρ - ρ \frac{δρ}{ρ} + γ δρ) = 0$

$∑_k [-lnρ_{kk} - 1 + γ] δρ = 0$

⇒ $ρ_{kk} = e^{γ-1}$

Let $γ = \frac{1}{N}$

$ρ_{kk} = e^{\frac{1}{N}-1} → \frac{1}{N}$ for N very large.

Thus, $S = -Tr(ρ~lnρ) = -∑_k \frac{1}{N} ln(\frac{1}{N}) = lnN$

Is my solution valid?

Related Advanced Physics Homework Help News on Phys.org
Actually, I think after the line $ρ_{kk} = e^{γ-1}$ everything is wrong. I should use the result $Tr(ρ)=1$ for a completely random ensemble.

So we take the sum from 1 to N, $∑_k ρ_{kk} = ∑_k e^{γ-1} =Ne^{γ-1} = 1~$ ⇒ $~e^{γ-1} = \frac{1}{N}~$

Thus, $S = -Tr(ρ~lnρ) = -∑_k \frac{1}{N} ln(\frac{1}{N}) = lnN$

I think this is kinda correct. Any comment?


text:
# Kalman decomposition using Hautus test

In linear control theory, the Kalman decomposition is used as a similarity transformation to decompose a given linear time-invariant system

$$\dot{x}(t)=Ax(t)+Bu(t)$$ $$y(t) = Cx(t) + Du(t)$$

into its controllable & observable, controllable & not observable, not controllable & observable and not controllable & not observable subsystems.

A system is controllable if $\mathcal{C}=[B\quad AB \quad A^2B \ldots A^{n-1}B]^T$ has rank $n$ (number of rows/columns of the system matrix $A$).

A system is observable if $\mathcal{O}=[C \quad CA \quad CA^2 \ldots CA^{n-1}]^T$ has rank $n$ (number of rows/columns of the system matrix $A$).

It seems possible to construct the similarity transformation by using the eigenvectors of the Hautus test for controllability and observability.

It states a system is controllable if for all eigenvalues $\lambda$ of $A$ the matrix $[A-\lambda I\quad B]$ has full rank $n$ for a $n \times n$ system matrix $A$. A similar statement reads: A system is observable if for all eigenvalues $\lambda$ of $A$ the matrix $[A^T - \lambda I \quad C^T]$ has full rank $n$ for a $n \times n$ system matrix $A$.

The user fibonatic from Engineering Stack Exchange suggested that one could construct the similarity transformation for the Hautus test. I am wondering, how this is possible.

This is the same example as in the post on Engineering Stack Exchange.

$$\dot{x} = \begin{bmatrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}x + \begin{bmatrix}0 & 1 \\ 1 & 0 \\ 0 & 1 \end{bmatrix} u$$ $$y = \begin{bmatrix}1 & 1 & 1 \end{bmatrix} x$$

The similarity transfromation is given by $$M = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & -1 \end{bmatrix}.$$ This is what I have tried. The three eigenvalues of $A$ are all $\lambda =1$ the eigenvectors are $v_1 = [1, 0,0]^T, v_2=[0,0,1]^T$ and $v_3=[1,0,1]^T$. The rank of the $[A-\lambda B]$ is 2. That means there is a deficit in rank for every eigenvector. If I include the eigenvectors to a "similarity" transformation

$$\tilde{M} = \begin{bmatrix} 1 & 0 & 1\\ 0 & 0 & 0\\ 0 & 1 & 1\\ \end{bmatrix}$$

it turns out that it is singular and cannot be the similarity transformation that I need.

So how can I use the Hautus test to construct the similarity transformation for the Kalman decomposition?

• It seems that the user fibonatic assumes that the matrix $A$ has $n$ linearly independent eigenvectors when he says "...for the columns of $M$ you can use the corresponding eigenvectors...". In your example, the matrix is not diagonalizable, so I guess you may need generalized eigenvectors too. – A.Γ. Oct 28 '17 at 14:13
• Thank you/спасибо. I will have to look that up as soon as I have the time. So when A has $n$ linearly independent eigenvectors then I can directly use this method and if it does not have $n$ linearly independent eigenvectors then I need to use generalized eigenvectors. – MrYouMath Oct 28 '17 at 14:18

A.Γ. is right that you need to use the generalised eigenvectors, so do a Jordan decomposition instead of an eigendecomposition. So in case of the example in your question the $A$ matrix has the following generalised eigenvectors

$$V = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$$

where the Jordan canonical form of $A$ consists out of two blocks of size two and one, both with the eigenvalue one.

However $\text{rank}\left[A-\lambda\,I \quad B\right]$ and $\text{rank}\left[A^\top-\lambda\,I \quad C^\top\right]$ tells you nothing about which block and corresponding generalized eigenvectors are causing those matrices to lose rank. So this would be the limitation of using the Hautus test in order to obtain the Kalman decomposition. The structure of a Jordan block actually always ensures that the largest rank loss caused by that block is at most one. So all this method would tell you is how many blocks have rank loss, but not which and by how much. For example when

$$A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}, \quad B_1 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad B_2 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$$

where $A$ is one Jordan block of size three with eigenvalue zero, but for both $B_1$ and $B_2$ does $\left[A-\lambda\,I \quad B_i\right]$ lose rank. But for $B_1$ only one state is uncontrollable, while for $B_2$ two states are uncontrollable.

So this method fails when the matrix has an eigenvalue with algebraic multiplicity larger than one, but the rank loss (either for controllability, observability or both) is less then that. Because in that case it is not clear which subset of (generalised) eigenvectors needs to be taken.

• +1: Thank you for your answer. I will have to read it more carefully then I will accept the answer. – MrYouMath Oct 31 '17 at 19:39

Eigenvector $$v_3=v_1+v_2$$ which implies that your eigenvectors aren't independent.


text:
An infinite subset in $3^\omega$ with “large” projections?

Problem. Is there an infinite set $I\subset 3^\omega$ such that for any infinite subset $J\subset I$ there exists $n\in\omega$ such that $\{x(n):x\in J\}=3$?

Here $3^\omega$ is the set of functions from $\omega$ to $3=\{0,1,2\}$.

Claim: For every integer $k>1$ there exists a subset $I\subseteq k^\omega$ of cardinality $\frak{c}$ such that for every subset $J\subseteq I$ such that $|J|=k$ there exists $n\in\omega$ such that $|\{x(n)\mid x\in J\}|=k$.
Proof: By Theorem 7.7 of Jech "Set Theory" there exists an independent family $\mathcal{F}$ of subsets of $\omega$ of cardinality $\frak{c}$. Independent means that for every distinct members of $\mathcal{F}$ $F_1,F_2,\ldots,F_s,F_1',F_2',\ldots,F'_t$ the intersection $$F_1\cap F_2\cap \ldots\cap F_s\cap (\omega\setminus F_1')\cap (\omega\setminus F_2')\cap \ldots\cap (\omega\setminus F'_t)$$ is nonempty. Partition $\mathcal{F}$ into subsets of size $k-1$: $S_i=\{F_{i1},F_{i2},\ldots,F_{ik-1}\}$. We have $\frak{c}$ such subsets.
Define the subset $I=\{x_i\}$ as: $$x_i(n) = |\{F\in S_i\mid n\in F\}|$$
Let $J\subseteq I$ be a subset of size $k$. Let $S_0,S_1,\ldots,S_{k-1}$ be the subsets of $\mathcal{F}$ that correspond to the elements of $J$. Since the family $S_0\cup S_1\cup \ldots\cup S_{k-1}$ is independent we can find an $n\in\omega$ which belongs to no members of $S_0$, to one member of $S_1$, and so on.


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# InterSpeech 2019 – Real-Time One-Pass Decoder for Speech Recognition Using LSTM Language Models

This year I was presenting our work on ASR in Graz at the InterSpeech conference, one of the most relevant conferences about speech technologies. I had the opportunity to talk and interact with people from the top-companies regarding ASR, such as people from Amazon Alexa’s or Apple Siri’s team. It was very enlightening to discuss the problems that you should face in the industry in contrast to academia, where we have more freedom to explore and risk.

Regarding our work, we presented our one-pass ASR system that benefits from the neural network-based state-of-the-art language models. To provide some insights on this, I will elaborate a bit to provide some background.

Generally speaking, the ASR system used to perform the recognition is usually called the decoder, as they decode the audio signal/utterance (sequence of vectors representing the audio signal) into words (sequence of strings). During the decoding, you work at acoustic level (the input is the audio information), and this part is managed by the acoustic model, while the structure of the search and the sequences of words that you consider are managed by the language model.

In general, the decoding process involves using these models during one, two or more iterative steps to obtain the final sequence of words at the output. The point is to reduce the number of different hypotheses considered on each step to focus the search of the best ones. In most of the ASR frameworks (i.e: Kaldi as the most popular one) the first decoding (first-pass / one-pass) is critical, as you start from scratch and you should consider several hypotheses to perform the decoding. After performing this first step of decoding, you can produce some kind of compact representation of a reduced set of hypotheses that you can refine in posterior steps (second-pass), usually in a graph form called “word lattice” that represents the potential combinations of word sequences, along with different scores.

The key idea is that the kind of models that you can use on each step is limited: during the first pass, as several hypotheses are competing and the search space is huge, you should limit the complexity of the model that you use, well due to the size of the model or the computational complexity of getting the scores. These limitations affect the language model (LM) directly, as the acoustic model (AM) is not as integrated during the search as the LM. AM could be considered as a feature extractor that provides vectors of features to the decoder to process. As the decoding is usually performed time-synchronous, we can say that for each query to the AM at each time step, the LM has been queried thousands of times.

Due to this limitation, the use of the complex and big LMs or state-of-the-art neural-based LMs was mostly limited to the second step of recognition, where the set of hypotheses is reduced in the word lattice and the decoding is not so demanding. This involves that you should perform two steps to leverage the best LMs, and you don’t benefit from them during the first step, potentially reducing the performance of the system.

There are several approaches that propose methods to solve this issue, but they usually involve limiting the potential of the neural model using simpler models (i.e: Feedforward neural networks) or increasing the time required to perform the first-pass decoding substantially if they use the complex ones (i.e: Long Short-Term Memory (LSTM) recurrent neural networks). In addition to the complexity of these models regarding their structure, this kind of neural networks poses an additional challenge. Unlike Feedforward neural networks that are stateless (they process the input and provide an output based only on this input) their internal state should be kept (LSTMs process not only the input but their internal state to provide the output) during the whole search process.

In our work, we managed to integrate the current best neural-based LM, an LM based on LSTM recurrent neural networks, during the first-pass of decoding. Additionally, we perform this integration keeping the speed of the decoder under the real-time regime. You can check all the details of this work in our technical paper. I’m going to try to summarize informally the main contributions very briefly with some results and conclusions.

Regarding the integration, our decoder follows a structure that allows us to use the LMs efficiently and with fewer limitations, unlike most typical Kaldi decoders. The structure that we follow is very similar to the one proposed in this thesis (advance topic!). In brief, this decoder has an internal organization that eases the use of advance LMs and LMs techniques such as LM look-ahead, a technique that helps to guide the search. There are other aspects such us advance pruning methods that also help to reduce the decoding effort without reducing the accuracy. Concerning LSTM LM itself, the main improvements were related to the use of GPU to alleviate the computational demand and the use of the variance regularization (VR) that reduces the complexity of getting the scores of neural models. If you want to know more, I refer to the paper where you can find a detailed description of our system and the techniques that we have included.

Some results, in ASR the main metric is the Word Error Rate (WER), that gives an idea of how far is your output from the correct sentence. It is an error rate so, the lower the better. To provide some figures, based on my experience, about 15-20% WER involves that the output has enough quality to be used to assist professional transcribers, and 5-10% is considered to have enough quality to provide useful transcriptions without supervision, i.e: for MOOC courses. On the other hand, as we are concerned about the speed of the decoder, we need another metric that can help us with that. In this case, we used Real-Time Factor (RTF) a very straightforward metric that is computed with the following formula:

$RTF = \frac{\text{time to recognize the utterance}}{\text{utterance duration}}$

For example, 1 hour of audio that it's transcribed in 1 hour has an RTF of 1, if that is transcribed in 30 minutes, it has an RTF of 0.5. On beneath of all this, we wanted a system that can work at <1 RTF, because this will allow us to think on streaming applications for the future with this decoder.

Now that we have the metrics, let's talk about the tasks to evaluate our approach. We selected the academic well-known datasets LibriSpeech (LS) and TED-LIUM release 3 (TL). LS is a big corpus with ~1K hours for training with people reading books. On the other hand, TL contains ~400 hours of TED talks.

To summarize the results, I will include just one figure from the paper that illustrates the comparison among the decoders that we have in our framework: HCS one-pass, HCS two-pass and WFST. WFST is the common decoder structure that is implemented in Kaldi, and HCS is the structure we are following for this work, with one or two steps of decoding. Results show that the one-pass decoder produces significant improvements in WER compared to WFST, especially when RTF is greater than 0.4. Considering a very similar RTF performance, the one-pass decoding approach achieves relative improvements in WER ∼12% and ∼6% in LibriSpeech and TED-LIUM, respectively. Comparing HCS decoders, one-pass shows a consistent improvement in RTF, reducing WER (∼6%) in the case of LibriSpeech while obtaining a similar accuracy in TED-LIUM but just performing one decoding step.

To conclude, in this work we have developed and evaluated our novel one-pass decoder that integrates a state-of-the-art LM based on LSTM RNN efficiently, obtaining very competitive results with an RTF (<1) that paves the way to consider the use of this system under streaming conditions. Indeed, our future efforts will go in that direction in order to obtain a streaming ASR that integrates the best neural-based models.


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# Direct standardization of mortality rates

## 2021-09-20

Mortality rates in different regions of the United States between different countries are dependent on the demographic distributions. Differences in the age distribution can cause mortality rates that are actually equivalent to appear as if they differ. These differences could just as easily hide a difference in mortality rates between two regions or even make the difference go in the opposite direction. Direct standardization is a simple approach to adjust mortality rates

suppressMessages(suppressWarnings(library(broom)))
suppressMessages(suppressWarnings(library(dplyr)))
suppressMessages(suppressWarnings(library(ggplot2)))
suppressMessages(suppressWarnings(library(magrittr)))

### References

Omar B. Ahmad, Cynthia Boschi-Pinto, Alan D. Lopez, Christopher JL Murray, Rafael Lozano, Mie Inoue. Age standardization of rates: A new WHO standard. GPE Discussion Paper Series: No.31, World Health Organization 2001. Available in pdf format.


text:
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Remarks on complex difference equations. (English) Zbl 1093.39018
Authors’ abstract: R. Halburd and R. Korhonen [Existence of finite order meromorphic solutions as a detector of integrability in difference equations. (to appear)] have shown that the existence of sufficiently many meromorphic solutions of finite order is enough to single out a discrete form of the second Painlevé equation from a more general class $f\left(z+1\right)+f\left(z-1\right)=R\left(z,f\right)$ of complex difference equations. A key lemma in their reasoning is to show that $f\left(z\right)$ has to be of infinite order, provided that ${\text{deg}}_{f}R\left(z,f\right)\le 2$ and that a certain growth condition for the counting function of distinct poles of $f\left(z\right)$ holds. In this paper, we prove a generalization of this lemma to higher order difference equations of more general type. We also consider related complex functional equations.
MSC:
 39A12 Discrete version of topics in analysis 30D05 Functional equations in the complex domain, iteration and composition of analytic functions 30D35 Distribution of values (one complex variable); Nevanlinna theory 39B12 Iterative and composite functional equations 39B32 Functional equations for complex functions 34M55 Painlevé and other special equations; classification, hierarchies 39A10 Additive difference equations
References:
 [1] M. Ablowitz, R. Halburd and B. Herbst, On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), 889–905. · Zbl 0956.39003 · doi:10.1088/0951-7715/13/3/321 [2] P. M. Cohn, Algebra, Vol. 1. John Wiley & Sons, London - New York - Sydney, (1974). [3] R. Goldstein, Some results on factorisation of meromorphic functions, J. London Math. Soc. (2) 4 (1971), 357–364. · Zbl 0223.30036 · doi:10.1112/jlms/s2-4.2.357 [4] R. Goldstein, On meromorphic solutions of certain functional equations, Aequationes Math. 18 (1978), 112–157. · Zbl 0384.30009 · doi:10.1007/BF01844071 [5] B. Grammaticos, T. Tamizhmani, A. Ramani and K. Tamizhmani, Growth and integrability in discrete systems, J. Phys. A 34 (2001), 3811–3821. · Zbl 1006.39018 · doi:10.1088/0305-4470/34/18/309 [6] V. Gromak, I. Laine and S. Shimomura, Painlevé Differential Equations in the Complex Plane, Studies in Mathematics 28 (2002). [7] R. Halburd and R. Korhonen, Existence of finite order meromorphic solutions as a detector of integrability in difference equations, submitted. [8] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and K. Tohge, Complex difference equations of Malmquist type, Comp. Methods Func. Theory 1 (2001), 27–39. · doi:10.1007/BF03320974 [9] J. Heittokangas, I. Laine, J. Rieppo and D. Yang, Meromorphic solutions of some linear functional equations, Aequationes Math. 60 (2000), 148–166. · Zbl 0963.39028 · doi:10.1007/s000100050143 [10] G. Hiromi and M. Ozawa, On the existence of analytic mappings between two ultrahyperelliptic surfaces, Kodai Math. Sem. Report 17 (1965), 281–306. · Zbl 0154.07903 · doi:10.2996/kmj/1138845125 [11] I. Laine, Nevanlinna Theory and Complex Differential Equations, Studies in Mathematics 15, W. de Gruyter, Berlin, 1993. [12] A. Shidlovskii, Transcendental Numbers, Studies in Mathematics 12, W. de Gruyter, Berlin, 1989. [13] H. Silvennoinen, Meromorphic solutions of some composite functional equations, Ann. Acad. Sci. Fenn. Math. Diss. 133 (2003), 1–39. [14] G. Weissenborn, On the theorem of Tumura and Clunie, Bull. London Math. Soc. 18 (1986), 371–373. · Zbl 0598.30044 · doi:10.1112/blms/18.4.371


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# Integer linear programming, energy constrained max-flow problem, column generation

We have graph $(V,A)$, $V$ is teh set of nodes, $A$ is the set of arcs. There is a source node $s \in V$ and a sink $t \in V$. Each node $i$ has a battery with capacity $E_i$. Sending flow on arc $(i,j)$ requires energy from the battery at node $i$, which amounts $e_{ij}$ per unit flow. The objective is to find the maximal flow $s$ to $t$, where the flow is required to be integral.

My formulation for this problem (is it ok?): \begin{align*} max & \sum_{(s,i) \in A} f_{si} \\ s.t. & \sum_{(i,j) \in A} f_{ij} - \sum_{(j,i) \in A} f_{ji} = 0, \quad \forall i \in N\setminus \{ s,t \} \\ & \sum_{(i,j) \in A} f_{ij} \cdot e_{ij} \leq E_i, \quad \forall i \in N\setminus \{ t \} \\ & \sum_{(s,i) \in A} f_{si} \in \{ 0,1,\dots,\lfloor E_s \rfloor \} \end{align*}

I would like to show also the following

(a) If we omit energy constraints, the constraint matrix is totally unimodular, that is every square non-singular submatrix is unimodular (its determinant is $-1,0$ or $1$)

(b) Network flow can be decomposed into a number of $s-t$ paths. Give an integer linear programming formulation for this problem based on paths.

(c) Describe how the relaxation of linear program can be solved by column generation. Include a formulation of the pricing problem, describe how to solve the problem.

I have problems with all three points.

In (a), based on my formulation (which could be wrong), i can't see how the constraint matrix is totally unimodular. It suffices to show that the matrix is composed only by the entries $-1,0$ and $1$ and that in each column there is at most one non-zero element but i can't see it.

In (b) i would denote $S$ as the set of all $s-t$ paths and write objective function as \begin{align*} max & \sum_{s-t \in S} f_{s-t} \end{align*} but then i have problems with constraints.

In (c) i don't know how to start. I searched on the web for some introductory literature on column generation but could't find any.

Can someone help me?

a) In my understanding, without the energy constraint, the direction of energy flow is not constrained, e.g. every energy that flows in one direction can flow into the opposite direction. Which would make sense to result in an invertible (unimodular) matrix.

b) A s−t path is a $\lambda$ variable (see literature). The variable is put into every capayity constraint, in which it uses capacity. e.g.

With $\lambda_{1478}$ as a flow from battery 1 to 4 to 7 to 8.

\begin{align} \lambda_{1478}\ +\ \cdots \ <&= E_1\\ \lambda_{1478}\ +\ \cdots \ <&= E_4\\ \lambda_{1478}\ +\ \cdots \ <&= E_7 \end{align}

c)


text:
Theorem

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.

For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.

The operation $+$ is commutative on $\map \MM {m, n}$.

That is:

$\mathbf A + \mathbf B = \mathbf B + \mathbf A$

for all $\mathbf A$ and $\mathbf B$ in $\map \MM {m, n}$.

Proof 1

From:

Integers form Ring
Rational Numbers form Ring
Real Numbers form Ring
Complex Numbers form Ring

the standard number systems $\Z$, $\Q$, $\R$ and $\C$ are rings.

Hence we can apply Matrix Entrywise Addition over Ring is Commutative.

$\Box$

The above cannot be applied to the natural numbers $\N$, as they do not form a ring.

However, from Natural Numbers under Addition form Commutative Monoid, the algebraic structure $\struct {\N, +}$ is a commutative monoid.

The result follows from Commutativity of Hadamard Product.

$\blacksquare$

Proof 2

Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be matrices whose order is $m \times n$.

Then:

 $\ds \mathbf A + \mathbf B$ $=$ $\ds \sqbrk a_{m n} + \sqbrk b_{m n}$ Definition of $\mathbf A$ and $\mathbf B$ $\ds$ $=$ $\ds \sqbrk {a + b}_{m n}$ Definition of Matrix Entrywise Addition $\ds$ $=$ $\ds \sqbrk {b + a}_{m n}$ Commutative Law of Addition $\ds$ $=$ $\ds \sqbrk b_{m n} + \sqbrk a_{m n}$ Definition of Matrix Entrywise Addition $\ds$ $=$ $\ds \mathbf B + \mathbf A$ Definition of $\mathbf A$ and $\mathbf B$

$\blacksquare$


text:
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Only one translucent cloud, G192-67, isclearly embedded within the LB and well isolated.These maps of the distribution of local neutral interstellar NaI gas arealso briefly compared with the distribution of both interstellar dustand neutral HI gas within 300 pc.Tables 1 and 2 are only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp:cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/411/447 Formation scenarios for the young stellar associations between galactic longitudes l = 280degr - 360degrWe investigate the spatial distribution, the space velocities and agedistribution of the pre-main sequence (PMS) stars belonging toOphiuchus, Lupus and Chamaeleon star-forming regions (SFRs), and of theyoung early-type star members of the Scorpius-Centaurus OB association.These young stellar associations extend over the galactic longituderange from 280degr to 360degr , and are at a distance interval ofaround 100 and 200 pc. This study is based on a compilation ofdistances, proper motions and radial velocities from the literature forthe kinematic properties, and of basic stellar data for the constructionof Hertzsprung-Russel diagrams. Although there was no well-known OBassociation in Chamaeleon, the distances and the proper motions of agroup of 21 B- and A-type stars, taken from the Hipparcos Catalogue,lead us to propose that they form a young association. We show that theyoung early-type stars of the OB associations and the PMS stars of theSFRs follow a similar spatial distribution, i.e., there is no separationbetween the low and the high-mass young stars. We find no difference inthe kinematics nor in the ages of these two populations studied.Considering not only the stars selected by kinematic criteria but thewhole sample of young early-type stars, the scattering of their propermotions is similar to that of the PMS stars and all the young starsexhibit a common direction of motion. The space velocities of theHipparcos PMS stars of each SFR are compatible with the mean values ofthe OB associations. The PMS stars in each SFR span a wide range of ages(from 1 to 20 Myr). The ages of the OB subgroups are 8-10 Myr for UpperScorpius (US), and 16-20 Myr for Upper Centaurus Lupus (UCL) and forLower Centaurus Crux (LCC). Thus, our results do not confirm that UCL isolder than the LCC association. Based on these results and theuncertainties associated with the age determination, we cannot say thatthere is indeed a difference in the age of the two populations. Weanalyze the different scenarios for the triggering of large-scalestar-formation that have been proposed up to now, and argue that mostprobably we are observing a spiral arm that passes close to the Sun. Thealignment of young stars and molecular clouds and the average velocityof the stars in the opposite direction to the Galactic rotation agreewith the expected behavior of star formation in nearby spiral arms.Tables 1 to 4 are only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/404/913 Rotational velocities of A-type stars in the northern hemisphere. II. Measurement of v sin iThis work is the second part of the set of measurements of v sin i forA-type stars, begun by Royer et al. (\cite{Ror_02a}). Spectra of 249 B8to F2-type stars brighter than V=7 have been collected at Observatoirede Haute-Provence (OHP). Fourier transforms of several line profiles inthe range 4200-4600 Å are used to derive v sin i from thefrequency of the first zero. Statistical analysis of the sampleindicates that measurement error mainly depends on v sin i and thisrelative error of the rotational velocity is found to be about 5% onaverage. The systematic shift with respect to standard values fromSlettebak et al. (\cite{Slk_75}), previously found in the first paper,is here confirmed. Comparisons with data from the literature agree withour findings: v sin i values from Slettebak et al. are underestimatedand the relation between both scales follows a linear law ensuremath vsin inew = 1.03 v sin iold+7.7. Finally, thesedata are combined with those from the previous paper (Royer et al.\cite{Ror_02a}), together with the catalogue of Abt & Morrell(\cite{AbtMol95}). The resulting sample includes some 2150 stars withhomogenized rotational velocities. Based on observations made atObservatoire de Haute Provence (CNRS), France. Tables \ref{results} and\ref{merging} are only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.125.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/393/897 Catalogue of Apparent Diameters and Absolute Radii of Stars (CADARS) - Third edition - Comments and statisticsThe Catalogue, available at the Centre de Données Stellaires deStrasbourg, consists of 13 573 records concerning the results obtainedfrom different methods for 7778 stars, reported in the literature. Thefollowing data are listed for each star: identifications, apparentmagnitude, spectral type, apparent diameter in arcsec, absolute radiusin solar units, method of determination, reference, remarks. Commentsand statistics obtained from CADARS are given. The Catalogue isavailable in electronic form at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcar?J/A+A/367/521 High S/N Echelle spectroscopy in young stellar groups. II. Rotational velocities of early-type stars in SCO OB2.We investigate the rotational velocities of early-type stars in the ScoOB2 association. We measure v.sin(i) for 156 established and probablemembers of the association. The measurements are performed with threedifferent techniques, which are in increasing order of expectedv.sin(i): 1) converting the widths of spectral lines directly tov.sin(i), 2) comparing artificially broadened spectra of low v.sin(i)stars to the target spectrum, 3) comparing the HeI λ4026 lineprofile to theoretical models. The sample is extended with literaturedata for 47 established members of Sco OB2. Analysis of the v.sin(i)distributions shows that there are no significant differences betweenthe subgroups of Sco OB2. We find that members of the binary populationof Sco OB2 on the whole rotate more slowly than the single stars. Inaddition, we find that the B7-B9 single star members rotatesignificantly faster than their B0-B6 counterparts. We test varioushypotheses for the distribution of v.sin(i) in the association. Theresults show that we cannot clearly exclude any form of randomdistribution of the direction and/or magnitude of the intrinsicrotational velocity vector. We also investigate the effects of rotationon colours in the Walraven photometric system. We show that positions ofB7-B9 single dwarfs above the main sequence are a consequence ofrotation. This establishes the influence of rotation on the Walravencolours, due primarily to surface gravity effects. The Relation between Rotational Velocities and Spectral Peculiarities among A-Type StarsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1995ApJS...99..135A&db_key=AST Ca II H and K Filter Photometry on the UVBY System. II. The Catalog of ObservationsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1995AJ....109.2828T&db_key=AST Vitesses radiales. Catalogue WEB: Wilson Evans Batten. Subtittle: Radial velocities: The Wilson-Evans-Batten catalogue.We give a common version of the two catalogues of Mean Radial Velocitiesby Wilson (1963) and Evans (1978) to which we have added the catalogueof spectroscopic binary systems (Batten et al. 1989). For each star,when possible, we give: 1) an acronym to enter SIMBAD (Set ofIdentifications Measurements and Bibliography for Astronomical Data) ofthe CDS (Centre de Donnees Astronomiques de Strasbourg). 2) the numberHIC of the HIPPARCOS catalogue (Turon 1992). 3) the CCDM number(Catalogue des Composantes des etoiles Doubles et Multiples) byDommanget & Nys (1994). For the cluster stars, a precise study hasbeen done, on the identificator numbers. Numerous remarks point out theproblems we have had to deal with. SANTIAGO 91, a right ascension catalogue of 3387 stars (equinox J2000).The positions in right ascension of 3387 stars belonging to the Santiago67 Catalogue, observed with the Repsold Meridian Circle at Cerro Calan,National Astronomical Observatory, during the period 1989 to 1994, aregiven. The average mean square error of a position, for the wholeCatalogue, is +/-0.009 s. The mean epoch of the catalogue is 1991.84. The position corrections of 1400 stars observed with PA II in San Juan.Not Available Fifth fundamental catalogue. Part 2: The FK5 extension - new fundamental starsThe mean positions and proper motions for 3117 new fundamental starsessentially in the magnitude range about 4.5 to 9.5 are given in thisFK5 extension. Mean apparent visual magnitude is 7.2 and is on average2.5 magnitudes fainter then the basic FK5 which has a mean magnitude of4.7. (The basic FK5 gives the mean positions and proper motions for theclassical 1535 fundamental stars). The following are discussed: theobservational material, reduction of observations, star selection, andthe system for the FK5 extension. An explanation and description of thecatalog are given. The catalog of 3117 fundamental stars for the equinoxand epoch J2000.0 and B1950.0 is presented. The parallaxes and radialvelocities for 22 extension stars with large forecasting effects aregiven. Catalogs used in the compilation of the FK5 fundamental catalogare listed. Early type high-velocity stars in the solar neighborhood. IV - Four-color and H-beta photometryResults are presented from photometric obaservations in the Stromgrenuvby four-color and H-beta systems of early-type high-velocity stars inthe solar neighborhood. Several types of photometrically peculiar starsare selected on the basis of their Stromgren indices and areprovisionally identified as peculiar A stars, field horizontal-branchstars, metal-poor stars near the Population II and old-disk turnoffs,metal-poor blue stragglers, or metallic-line A stars. Numerousphotometrically normal stars were also found. Walraven photometry of nearby southern OB associationsHomogeneous Walraven (VBLUW) photometry is presented for 5260 stars inthe regions of five nearby southern OB associations: Scorpio Centaurus(Sco OB2), Orion OB1, Canis Major OB1, Monoceros OB1, and Scutum OB2.Derived V and (B - V) in the Johnson system are included. Physical parameters of stars in the Scorpio-Centaurus OB associationWalraven photometry is presented of established and probable members ofthe Scorpio-Centaurus OB association. For each star, effectivetemperature and surface gravity are derived using Kurucz (1979)atmosphere models. From the Straizys and Kuriliene (1981) tables,absolute magnitudes are calculated. Distance moduli and visualextinctions are determined for all stars. From a comparison of theHR-diagrams of the stars in each subgroup with theoretical isochrones,the ages of the three subgroups are derived. The distances to the threesubgroups are shown to be different; there is a general trend (alsowithin each subgroup) for the distances to be larger at higher galacticlongitudes. The visual extinction in the youngest subgroupUpper-Scorpius, is well correlated with the IRAS 100-micron map. Thedistance toward the Ophiuchus dark clouds is found to be 125 pc, basedon the photometric distances to the stars. Most of the early-type starsin Upper-Scorpius are located at the far side of the dark clouds. Properties of Am, Delta Del and Delta SCT stars in the VBLUW systemThe spread with respect to the main sequence in the two-color diagramsfor 115 Am, Delta Del and Delta Sct, for which VBLUW photometricobservations have been obtained, is primarily attributed to gravityeffects. Exceptions to this rule can be recognized from a comparison oftheir positions in these diagrams and in the reddening-independentdiagram L-U/B-L, where some Delta Sct stars have experienced relativelyhigh reddenings, while others are presumably metal-poor. It is notedthat gravities and temperatures compare satisfactorily with those ofDavis Philip et al. (1976), which were based on uvby-beta photometry. Prediction of spectral classification from photometric observations - Application of the UVBY beta photometry and the MK spectra classification. II - General caseAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1980A&A....85...93M&db_key=AST Prediction of spectral classification from photometric observations-application to the UVBY beta photometry and the MK spectral classification. I - Prediction assuming a luminosity classAn algorithm based on multiple stepwise and isotonic regressions isdeveloped for the prediction of spectral classification from photometricdata. The prediction assumes a luminosity class with reference touvbybeta photometry and the MK spectral classification. The precisionattained is about 90 percent and 80 percent probability of being withinone spectral subtype respectively for luminosity groups I and V and forluminosity groups III and IV. A list of stars for which discrepanciesappear between photometry and spectral classification is given. UBVRI photometry of 225 AM starsUBVRI photometry of 225 Am stars taken from Mendoza's (1974) catalog ispresented. The results are compared with those obtained by Feinstein(1974) for 21 of the stars and with the values of Johnson et al. (1966).It is assumed that in the first approximation the (V-I) color index ofan unreddened Am star is equal to that of a normal main-sequence star; astandard main sequence is defined for A and early F stars, and thefive-color photometry is analyzed by means of plots of U-V vs. V-I, B-Vvs. V-I, and V-R vs. V-I. Mean color deficiencies of Am stars areexamined, and it is suggested that an unreddened star located below themain-sequence A0-F2 line in the (V-I, U-V) plane is a photometric Amstar. It is concluded that: (1) photometric Am stars have colordeficiencies (as a function of V-I) which, on the average, are 0.07 magin (U-V) color index and 0.025 mag in (B-V) color index; (2) Am starswith V-R less than 0.25 mag may also have a color deficiency of about0.01 mag; (3) Am stars with V-R greater than 0.3 mag may have a colorexcess of approximately 0.01 mag; and (4) Am stars with V-R between 0.25and 0.3 mag may have normal colors. Detection of errors in spectral classification by cluster analysisCluster analysis methods are applied to the photometric catalogue ofuvby-beta measurements by Hauck and Lindemann (1973) and point out 249stars the spectral type of which should be reconsidered or thephotometric indices of which should be redetermined. Bright metallic-line and pulsating A stars.Abstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1976PASP...88..402E&db_key=AST Mitteilungen uber Veraenderliche des Bamberger Liste.Not Available Multicolor photometry of metallic-line stars. III. A photometric catalogueAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1974RMxAA...1..175M&db_key=AST Possible new members of a subgroup of the Scorpio-Centaurus association.Abstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1972AJ.....77..474G&db_key=AST K-Line Photometry of Southern a StarsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1971ApJS...23..421H&db_key=AST Line strengths for southern OB stars-II. Observations with moderate dispersionAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1969MNRAS.144...31B&db_key=AST
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# Synopsis: The nuclear option

Optical clocks based on nuclear energy levels may offer unprecedented precision and a chance to find out whether fundamental constants change over time.

Atomic clock makers look for two important characteristics when they choose atoms to build their time pieces. First, the energy levels that generate the clock signal must be narrow and well defined. And second, the energy levels should be immune from external electromagnetic fields so that the clock rate doesn’t vary with, for example, position near Earth.

Several atomic species with these qualities exist, but researchers would like to take optical clocks to an even higher level of precision and stability. One approach would be to find nuclear energy levels with the right features for clocks. Nuclei have extremely sharp energy levels, but typically the excitation energies range from kilo-electron-volts to mega-electron-volts—far above the several-$\text{eV}$ photon energies that are available to make optical clocks. Now, Corey Campbell, Alexander Radnaev, and Alexander Kuzmich of the Georgia Institute of Technology report in Physical Review Letters their studies of thorium nuclei that may be just right for clock building.

One likely candidate, thorium-$229$, has a pair of low-lying nuclear states separated only by $7.6\phantom{\rule{0.333em}{0ex}}\text{eV}$. As a first step in harnessing these levels, the authors trap and cool the triply charged form of $\text{Th}$-$229$ into a regular array called a Wigner crystal, which allows spatial localization of each nucleus. Future work will include addressing single ions in the crystal and spectroscopic studies of the nuclear clock transition. This could not only lead to clocks of unusual precision, but it might also help answer questions about whether fundamental physical constants are changing over time. – David Voss

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Some terms in $H_0$ can be absorbed into the definition of $\epsilon$. To understand the validity of this substitution, you must go back to chapter 7 and calculate according to eqn. (7.2)-(7.6). You will find "we can absorb the positive coefficient into $\epsilon$ to get..."

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Imagine a QFT with some particle content. Some of these fields will be massless and some massive. For simplicity, consider a massles scalar field and a massive scalar field with mass $M$. If we are working at some energy $E\ll M$, we won't see the massive field (as happened with the Higgs before LHC, for example). This is the IR CFT. Why IR? Because we ...

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I think that this nomenclature has nothing to do with the regulators. "UV CFT" and "IR CFT" actually refers to the end point of the RG flow which is triggered by a relevant operator that perturbs an UV fixed point, and it ends (barring certain non-unitary QFT) into a fixed point at lower energy scales, hence the name IR.

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# On a Mean Reverting Dividend Strategy with Brownian Motion

11 Pages Posted: 13 Nov 2009 Last revised: 20 Nov 2009

See all articles by Benjamin Avanzi

## Benjamin Avanzi

UNSW Australia Business School, School of Risk and Actuarial Studies

## Bernard Wong

UNSW Australia Business School, School of Risk & Actuarial Studies

Date Written: November 19, 2009

### Abstract

In actuarial risk theory, the introduction of dividend pay-outs in surplus models goes back to Bruno de Finetti (1957). Dividend strategies that can be found in the literature often yield pay-out patterns that are inconsistent with actual practice. One issue is the high variability of the dividend payment rates over time. We aim at addressing that problem by specifying a dividend strategy that yields stable dividend pay-outs over time.

In this paper, we model the surplus of a company with a Brownian risk model. Dividends are paid at a constant rate $g$ of the company's modified surplus (after distribution of dividends), which operates as a buffer reservoir to yield a regular flow of shareholders' income. The dividend payment rate reverts around the drift of the original process $\mu$, whereas the modified surplus itself reverts around the level $l=\mu/g$.

We determine the distribution of the present value of dividends when the surplus process is never absorbed. After introducing an absorbing barrier $a$ (inferior to the initial surplus) and stating the Laplace transform of the time of absorption, we derive the expected present value of dividends until absorption. The latter is then also determined if dividends are not paid whenever the surplus is too close to the absorbing barrier. The calculation of the optimal value of the parameter $l$ (and equivalently $g$) is discussed. We conclude by comparing both barrier and mean reverting dividend strategies.

Keywords: Dividends, Brownian motion, Ornstein-Uhlenbeck process, Mean reverting

JEL Classification: G35, G32, G22, C44

Suggested Citation

Avanzi, Benjamin and Wong, Bernard, On a Mean Reverting Dividend Strategy with Brownian Motion (November 19, 2009). UNSW Australian School of Business Research Paper No. 2009ACTL12. Available at SSRN: https://ssrn.com/abstract=1504401 or http://dx.doi.org/10.2139/ssrn.1504401

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