text:
# Is the trade dying a slow death??

D

#### David Stuart

I am 23 year experenced controls guy (Engineer- Designer-Programmer) I have been self employed for 3 +years . I cant find any work or projects
. Been looking since this last may ... Is it time to change my course in life .... Is this trade dying a slow death ?? Whats your opinion?? How can you survive out there today ?? What would you do ??

E

#### Eric Allen

Dave. I know the feeling. It's primarily a regional based profession. Here in the Detroit area, there are tons of jobs in controls. In other areas of the country, there are people who dont even know what a PLC is. I work primarily in the water and water treatment industry. I moved over from automotive knowing that sooner or later the big three would take a hit. My hunch played out. Like everything else in this business, its highly cyclical. As long as automation needs are needed so will the need for the people who program the machinery. Todays improvements on programming and PLC design are making it easy for alsmost anyone to do it now if you know what you are doing.

Good Luck

A

#### Al Bailey

I manage the controls group for a small OEM in Oregon. We do the controls for our own machinery and also contract SI projects.

Things have been slow over the last year, but things are picking up. I honestly believe that in order for US Companies to compete, they need to continue to automate. This improves quality and decreases production costs. I cannot think of a better field to be in for the future.

Good luck and I hope things work out.

W

#### Walt Boyes

There is an article on the Control Magazine website from a recent issue, "Who Will Do the Work?" by David Spitzer and Walt Boyes that addresses this issue.

There is an article on the Inside ISA website at http://www.isa.org that talks about former ISA Executive Director Glen Harvey's speech at the 2001 Honors and Awards Banquet in Houston...it is exactly about this issue.

In addition, there are the articles I've written for automationtechies.com on the world of work in automation and control. You can find them at http://www.automationtechies.com/sitepages/pid577.php. They are a detailed look at what to do in your position.

I hope this helps.

Walt Boyes

---------SPITZER AND BOYES, LLC-------------
"Consulting from the engineer
to the distribution channel"
www.spitzerandboyes.com
[email protected]
21118 SE 278th Place
Maple Valley, WA 98038
253-709-5046 cell 425-432-8262 home office
fax:1-253-981-0285
--------------------------------------------

W

#### William Sturm

Man, that is tough... You may want to change your niche or your target market. Diversify a little. I think that the controls industry is a good field to be in, but not always. You need a strong niche where you can add value, and then start knocking on doors.

Bill

T

#### Thomas Ng

Here in the Western world, we've been going through a very severe and long-lasting economic downturn.
People are getting laid-off, fired or downsized
- not because they were in anyway "not good enough", because the senior executives made some
overly-optimistic guesses in the past. A lot of
good people at HP-Compaq are suffering unemployment because Carly wanted to roll the dice.
This is very depressing to hear, and
job-hunting has pretty discouraging for these
past few years.
Maybe the best thing is to temporarily do
something (ANYTHING!) else, because things won't
pick up for at least a couple of years. Once things get back on track, you won't have time
to do anything but work, work, work.

R

#### Ranjan Acharya

Be prepared to move around, even emigrate. English is a widely spoken (and abused) language, so that should not really be a bother. Think Canada, Australia, New Zealand, United Kingdom, Continental Europe / Scandinavia, India, Hong Kong, Singapore, ... even a short term ex-pat contract could see you over the hump until your domestic economy picks up. I guess you should stay away from certain countries ....

Be prepared to re-train and diversify.

Don't get put off by the cyclical nature of a market-based economy.

I don't think controls are in trouble.

RA

J

#### Jim Pinto

Automation List :

The automation business is declining -
and has been for a few years.

You might like to read my article -
"Why is Industrial Automation Declining?" at:

In a poor economy, don't simply sit back and wait
for the axe to fall. Be pro-active - use your personal
strengths, plus new ideas and tools, to find new
opportunities for continued growth and success.
There are always good opportunities for good people.
And good people find those opportunities!

My article "In a declining economy, what's a good
automation-techie to do?" has just been published on
the Automation Techies website, Dec. 2002 at :
http://www.automationtechies.com/sitepages/art633.php

You'll find it too on the JimPinto.com website at:
http://www.jimpinto.com/writings/whatodo.html

Cheers:
jim
----------/
Jim Pinto
email : [email protected]
web: www.JimPinto.com
San Diego, CA., USA
----------/

B

#### Bob Peterson

One of the problems with a one-man shop is that you spend your time working and not beating the bushes for new work. When your contract is up, its hard sometimes to find a new one quickly. Some one-man shops I have spoken with claim they spend as much as 20% of their time looking for new contracts to take up as old ones are finished. If you are heavily tied to a single client, you can get into trouble because you don't realize whats going on in the rest of the world. When times are tough, it can be very difficult for a one-man operation to find work, as you are finding out.

People are cutting back, and they are looking to outsource chunks or whole projects rather than contracting a guy for a while. there is usually no economic advantage to a client to hire you just as an engineer (unless you work cheap, in which case you probably are not much of a bargain), its probably more cost effective over time to hire someone, even with a training cycle included. but if you have the capability to do a bigger chunk of a project, you may be more attractive to more clients. This has the advantage of giving you more of a base to sell yourself to, but requires you to work within a larger organization.

Don't get me wrong here. SIs are slow as well, but the ones that are more than rent-an-engineer companies have something to offer that you don't and it can make all the difference. Plus they have salesmen out beating the bush who are probably better at cold calling than you are, and will starve if they don't sell something to bring in commissions, thus they have a serious incentive to sell stuff.

I am not a big fan of temp services, but there are some temp services around that rent out engineering help. From what I have heard, usually they do not pay all that well, but when things are slow it may be worth a look.

Another area you might consider is doing field service work. There is a serious lack of competent field service help available (the key word being competent). You might be able to work out a deal with an OEM who's equipment you are familiar with (from programming them) or an equipment supplier, or even a local electrical distributor. Life on the road is not all that much fun, but it might pay the bills for a while.

You might also want to reduce your rate temporarily. A lot of companies seriously understate what it costs them to have an engineer on staff. I have seen people that charge themselves as little as $35-40 for internal engineering services, and have a huge sticker shock when they go outside to get help. I have run the numbers a lot of times, and the real cost for me is over$50 an hour. That is before any overhead or profit, just what it costs for me to show up to work everyday, based on one years cost to have me on staff divided by the number of typical billable hours in one year. Since wages and benefits are reasonably close, I see no way anyone could hire someone internally to do the same work for such a low number. And usually, the people working on them internally are bogged down by meetings, and politics, and other inefficiencies you are not afflicted with. But even so the numbers are starkly different than what they see as the cost of an hour of engineering time.

Another helpful idea might be to take on projects on a fixed price basis. Customers are often willing to pay more than if they did it on a T&M basis, just so they have a known quantity to deal with. this requires you to have a good idea what it is you are bidding on, and a good idea of how good you will be at what they need done.

Bob Peterson

C

#### Curt Wuollet

Hi all
With automation slow for the longest time and the tech sector in general about dead, I'm looking at HVACR as an alternative. Low tech, but at least somewhat related and short of people. Not having much luck finding someone who'll help me make the jump.

They seem to be looking more for brawn than brains and seem to be sceptical that I could actually fix furnaces. Might be worth looking into, especially in metro areas where there are energy management systems, etc.

Regards
cww

Z

#### Zelenka

Dear Sir,

If this is a case in USA, can you imagine what is in the east Europe countries where exactly the western economy and trade closed all industry (for exception of some food industry that has to be local.

Complete engineering companies broke, and a lot of engineers are having their small engineering offices, each covering his speciality, but when a bigger project is planned it is very difficult to get it. So the home business and small companies will brake.

That's a bit of economy. The world is one. The market is one. if you want to take some more, you have to take it from somebody else.

Regards

R

#### Ralph Mackiewicz

> That's a bit of economy. The world is one. The market is one. if you
> want to take some more, you have to take it from somebody else.

I hate to burst your bubble but, fortunately for the human race, the economy (global or local) doesn't work like that. This zero sum model of economic activity does not follow the real world. In the real world the economy grows. Wealth, a measure of economic activity, is not something static that you can only take from someone else. Wealth is created by the exchange of goods and services. If this weren't true then we would all still have to grow our own food just to survive to the ripe old age of 35 like our ancestors from centuries ago. The lifestyles of people all over the world (with the few exceptions of those still living as they did in the stone age) is a direct repudiation of the view of a static (zero sum) economic model.

Regards,
Ralph Mackiewicz

S

#### Steve Myres, PE

A big "Hear, Hear!" for that statement.

One thing, though. I disagree that wealth is created by the EXCHANGE of goods and services. I think wealth is created when we PRODUCE goods and services.

If I take an empty tract of land and some milled lumber which I use to build a house, I am left with an asset worth more than the total cost I paid for the various parts. The difference is the value of my planning, building, etc.

This difference exists whether or not I ever exchange the house with another person.

We create wealth when we add value or utility. This can be by moving an object from a geographical location where it has a certain value to a location where its value is greater. It could be by purchasing a PLC and then reselling it with code included and tested, etc.

But enough nitpicking. I posted mostly to add my voice to your rebuttal of the zero sum attitude of the other poster.

J

#### Jiri Baum

Admittedly, we still haven't quite got the hang of distributing this wealth in anything remotely like a fair manner, so while we now have enough food for everyone and to spare, people still die of hunger.

Jiri
--
Jiri Baum <[email protected]> http://www.csse.monash.edu.au/~jirib
MAT LinuxPLC project --- http://mat.sf.net --- Machine Automation Tools

B

#### Bob Peterson

Just what is "fair"?

Kepe in mind the vast majority of people who are hungry are hungry because of their own government's policies, often resulting in deliberate starvation. Its not as if some loco wealth redistribution scheme will solve the problem of mass murder by governments.

Bob Peterson

R

#### Ralph Mackiewicz

> One thing, though. I disagree that wealth is created by the EXCHANGE
> of goods and services. I think wealth is created when we PRODUCE
> goods and services.

You are absolutely correct. Exchange was not the correct word to use. The ability to exchange the result of production facilitates new production and wealth. Restricting exchange inhibits production and wealth. Freedom is the grease that lubricates the wheels of production and thereby the creation of wealth.

Regards,
Ralph Mackiewicz


text:
# Probability that the sum of $k$ distinct integers selected from $1, 2, \dots, n$ is divisible by $n$

Would you please help me solve Problem 7 of Section 4.5, "Combinatorial Number Theory," in An Introduction to the Theory of Numbers, Niven, Zuckerman, Montgomery, 5th ed., Wiley (New York), 1991:

Let $n$ and $k$ be positive, relatively-prime integers with $n > k$. Prove that if $k$ distinct integers are selected at random from $1, 2, \dots, n$, the probability that their sum is divisible by $n$ is $1/n$.

The authors describe three methods for solving combinatorial problems: (1) the pigeonhole principle, (2) the one-to-one correspondence procedure, in which elements in a finite set or between two sets are paired off to determine the number of elements, and (3) the inclusion-exclusion principle. I know that the denominator of the probability fraction is $n \choose k$, so I tried (without success) to use the second method to count the number of desired outcomes to be $(n - 1)!/((n - k)! k!)$.

Because $n$ and $k$ are relatively prime, I know that $n \choose k$ is divisible by $n$ (see Example 10, notwithstanding the mistake/typo in the proof). In that case, numerical evidence suggests that the sum in question is uniformly distributed modulo $n$. If I can prove that conjecture, I am done.

I found similar problems with particular values for $n$ and $k$, but the solutions do not help me solve this general case.

Think of the numbers as integers modulo~$n$. Consider the map $A=\{a_1,a_2,\ldots,a_k\}\mapsto\{a_1+1,a_2+1,\ldots,a_k+1\}$. This adds $k$ modulo $n$ to the sum of the elements of $A$.

• Can you explain more? How does this solve the problem? – crskhr Jun 2 '18 at 15:32

The exponential formula tells us that the cycle index $Z(P_k)$ of the unlabeled set operator

$$\def\textsc#1{\dosc#1\csod} \def\dosc#1#2\csod{{\rm #1{\small #2}}}\textsc{SET}$$

on $k$ slots has OGF

$$Z(P_k) = [w^k] \exp\left(\sum_{l\ge 1} (-1)^{l-1} a_l \frac{w^l}{l}\right).$$

The desired probability is given by

$${n\choose k}^{-1} \frac{1}{n} \sum_{p=0}^{n-1} \left. Z(P_k) \left(\sum_{q=1}^n z^q\right)\right|_{z=\exp(2\pi ip/n)}.$$

This is

$${n\choose k}^{-1} \frac{1}{n} \sum_{p=0}^{n-1} \left. [w^k] \exp\left(\sum_{l\ge 1} (-1)^{l-1} \left(\sum_{q=1}^n z^{ql}\right) \frac{w^l}{l}\right)\right|_{z=\exp(2\pi ip/n)}.$$

Evaluating the contribution for $p=0$ first we get

$${n\choose k}^{-1} \frac{1}{n} [w^k] \exp\left(\sum_{l\ge 1} (-1)^{l-1} n \frac{w^l}{l}\right) \\ = {n\choose k}^{-1} \frac{1}{n} [w^k] \exp\left(n \log(1+w)\right) \\ = {n\choose k}^{-1} \frac{1}{n} [w^k] (1+w)^n = {n\choose k}^{-1} \frac{1}{n} {n\choose k} = \frac{1}{n}.$$

It remains to evaluate the contribution from $1\le p\le n-1.$ Now for these $p$ if $l$ is a multiple of $m = n/\gcd(p, n)$ we have

$$\sum_{q=1}^n \exp(2\pi ip/n)^{ql} = n.$$

We get zero otherwise. This yields for the remaining terms without the scalar in front

$$\sum_{p=1}^{n-1} [w^k] \exp\left(\sum_{l\ge 1} (-1)^{ml-1} n \frac{w^{ml}}{ml} \right) = \sum_{p=1}^{n-1} [w^k] \exp\left(-\frac{n}{m} \sum_{l\ge 1} \frac{(-w)^{ml}}{l} \right) \\ = \sum_{p=1}^{n-1} [w^k] \exp\left(-\frac{n}{m}\log\frac{1}{1-(-w)^m}\right) = \sum_{p=1}^{n-1} [w^k] (1-(-w)^{n/\gcd(p, n)})^{\gcd(p, n)} \\ = \sum_{p=1}^{n-1} [w^k] (1+(-1)^{1+n/\gcd(p, n)} w^{n/\gcd(p, n)})^{\gcd(p, n)}.$$

Using an Iverson bracket this becomes

$$\sum_{p=1}^{n-1} [[n/\gcd(p,n)|k]] \times {\gcd(p, n)\choose k\gcd(p,n)/n} (-1)^{k\gcd(p,n)/n+k}.$$

Putting it all together we thus obtain

$$\bbox[5px,border:2px solid #00A000]{ \frac{1}{n} + (-1)^k {n\choose k}^{-1} \frac{1}{n} \sum_{p=1}^{n-1} [[n/\gcd(p,n)|k]] \times {\gcd(p, n)\choose k\gcd(p,n)/n} (-1)^{k\gcd(p,n)/n}.}$$

Note that $n/\gcd(p,n)$ is a divisor of $n$ that is at least two (we would need $p=n$ to get $n/\gcd(p,n) = 1$ but $p\lt n$). This means when $\gcd(k, n) = 1$ the Iverson bracket fails for all $p$ and only the first term remains, which is what we wanted to prove.

As a sanity check we evaluate the formula when $k=n.$ The sum of the one subset is $n(n+1)/2$ so we should get for the probability one when $n$ is odd and zero when it is even. We find

$$\frac{1}{n} + (-1)^n {n\choose n}^{-1} \frac{1}{n} \sum_{p=1}^{n-1} {\gcd(p, n)\choose \gcd(p,n)} (-1)^{\gcd(p,n)} \\ = \frac{1}{n} + (-1)^n \frac{1}{n} \sum_{p=1}^{n-1} (-1)^{\gcd(p,n)} = (-1)^n \frac{1}{n} \sum_{p=1}^{n} (-1)^{\gcd(p,n)} \\ = (-1)^n \frac{1}{n} \sum_{d|n} \sum_{\gcd(p,n) = d} (-1)^d = (-1)^n \frac{1}{n} \sum_{d|n} (-1)^d \sum_{\gcd(q,n/d) = 1} 1 \\ = (-1)^n \frac{1}{n} \sum_{d|n} (-1)^d \varphi(n/d).$$

We evaluate this using formal Dirichlet series, starting from $\sum_{d|n} \varphi(d) = n$ which yields

$$\sum_{n\ge 1} \frac{\varphi(n)}{n^s} = \frac{\zeta(s-1)}{\zeta(s)}.$$

Furthermore we have

$$\sum_{n\ge 1} \frac{(-1)^n}{n^s} = -\left(1-\frac{2}{2^s}\right)\zeta(s).$$

This means that

$$\sum_{n\ge 1} \frac{1}{n^s} \sum_{d|n} (-1)^d \varphi(n/d) = -\left(1-\frac{2}{2^s}\right) \zeta(s-1).$$

Now we have for $n$ even

$$- (-1)^n \frac{1}{n} [n^{-s}] \left(1-\frac{2}{2^s}\right) \zeta(s-1) = - (-1)^n \frac{1}{n} (n - 2 [(n/2)^{-s}] \zeta(s-1)) \\ = - (-1)^n \frac{1}{n} (n - 2 \times n/2) = 0.$$

We obtain zero as required. On the other hand with $n$ odd we find

$$- (-1)^n \frac{1}{n} [n^{-s}] \left(1-\frac{2}{2^s}\right) \zeta(s-1) = - (-1)^n \frac{1}{n} [n^{-s}] \zeta(s-1) \\ = - (-1)^n \frac{1}{n} \times n = 1,$$

again as required. This concludes the sanity check. and indeed the entire argument.

format: plain


text:
# zbMATH — the first resource for mathematics

Geometry of enstrophy and dissipation, grid resolution effects and proximity issues in turbulence. (English) Zbl 1156.76406
Summary: We perform a multi-scale non-local geometrical analysis of the structures extracted from the enstrophy and kinetic energy dissipation-rate, instantaneous fields of a numerical database of incompressible homogeneous isotropic turbulence decaying in time obtained by DNS in a periodic box. Three different resolutions are considered: 256$$^3$$, 512$$^3$$ and 1024$$^3$$ grid points, with $$k_{\max}\overline \eta$$ approximately 1, 2 and 4, respectively, the same initial conditions and $$Re_\lambda\approx 77$$. This allows a comparison of the geometry of the structures obtained for different resolutions. For the highest resolution, structures of enstrophy and dissipation evolve in a continuous distribution from blob-like and moderately stretched tube-like shapes at the large scales to highly stretched sheet-like structures at the small scales. The intermediate scales show a predominance of tube-like structures for both fields, much more pronounced for the enstrophy field. The dissipation field shows a tendency towards structures with lower curvedness than those of the enstrophy, for intermediate and small scales. The 256$$^3$$ grid resolution case $$k_{\max}\overline \eta\approx 1$$ was unable to detect the predominance of highly stretched sheet-like structures at the smaller scales in both fields. The same non-local methodology for the study of the geometry of structures, but without the multi-scale decomposition, is applied to two scalar fields used by existing local criteria for the eduction of tube- and sheet-like structures in turbulence, $$Q$$ and $$[A_{ij}]_+$$, respectively, obtained from invariants of the velocity-gradient tensor and alike in the 1024$$^3$$ case. This adds the non-local geometrical characterization and classification to those local criteria, assessing their validity in educing particular geometries. Finally, we introduce a new methodology for the study of proximity issues among structures of different fields, based on geometrical considerations and non-local analysis, by taking into account the spatial extent of the structures. We apply it to the four fields previously studied. Tube-like structures of $$Q$$ are predominantly surrounded by sheet-like structures of $$[A_{ij}]_+$$, which appear at closer distances. For the enstrophy, tube-like structures at an intermediate scale are primarily surrounded by sheets of smaller scales of the enstrophy and structures of dissipation at the same and smaller scales. A secondary contribution results from tubes of enstrophy at smaller scales appearing at farther distances. Different configurations of composite structures are presented.

##### MSC:
 76F65 Direct numerical and large eddy simulation of turbulence
Full Text:
##### References:
 [1] DOI: 10.1017/S0022112005004040 · Zbl 1070.76033 · doi:10.1017/S0022112005004040 [2] DOI: 10.1023/B:APPL.0000044408.46141.26 · Zbl 1081.76564 · doi:10.1023/B:APPL.0000044408.46141.26 [3] DOI: 10.1063/1.858333 · doi:10.1063/1.858333 [4] DOI: 10.1017/S0022112008001511 · Zbl 1145.76024 · doi:10.1017/S0022112008001511 [5] DOI: 10.1103/PhysRevLett.79.1253 · doi:10.1103/PhysRevLett.79.1253 [6] DOI: 10.1098/rspa.1926.0043 · doi:10.1098/rspa.1926.0043 [7] DOI: 10.1137/05064182X · Zbl 1122.65134 · doi:10.1137/05064182X [8] Richardson, Weather Prediction by Numerical Process. (1922) · JFM 48.0629.07 [9] DOI: 10.1063/1.858441 · Zbl 0775.76078 · doi:10.1063/1.858441 [10] DOI: 10.1146/annurev.fluid.30.1.31 · Zbl 1398.76084 · doi:10.1146/annurev.fluid.30.1.31 [11] DOI: 10.1063/1.863442 · doi:10.1063/1.863442 [12] Onsager, Phys. Rev. 68 pp 286– (1945) [13] DOI: 10.1017/S002211200800092X · Zbl 1151.76522 · doi:10.1017/S002211200800092X [14] DOI: 10.1063/1.2771661 · Zbl 1182.76566 · doi:10.1063/1.2771661 [15] Batchelor, Proc. R. Soc. Lond. A 199 pp 238– (1949) [16] DOI: 10.1017/S0022112098003024 · Zbl 0933.76035 · doi:10.1017/S0022112098003024 [17] DOI: 10.1063/1.866513 · doi:10.1063/1.866513 [18] DOI: 10.1017/S0022112004009802 · Zbl 1107.76328 · doi:10.1017/S0022112004009802 [19] DOI: 10.1063/1.869361 · Zbl 1185.76770 · doi:10.1063/1.869361 [20] DOI: 10.1017/S0022112091003786 · Zbl 0749.76033 · doi:10.1017/S0022112091003786 [21] DOI: 10.1063/1.863957 · Zbl 0536.76034 · doi:10.1063/1.863957 [22] Landau, Fluid Mechanics. (1959) [23] DOI: 10.1017/S002211207400070X · Zbl 0273.76037 · doi:10.1017/S002211207400070X [24] DOI: 10.1017/S0022112062000518 · Zbl 0112.42003 · doi:10.1017/S0022112062000518 [25] Kolmogorov, Dokl. Nauk. SSSR. 30 pp 301– (1941) [26] Kolmogorov, Dokl. Nauk. SSSR. 32 pp 16– (1941) [27] DOI: 10.1016/0262-8856(92)90076-F · doi:10.1016/0262-8856(92)90076-F [28] DOI: 10.1017/S0022112085001136 · Zbl 0587.76080 · doi:10.1017/S0022112085001136 [29] DOI: 10.1017/S0022112061000974 · Zbl 0103.19903 · doi:10.1017/S0022112061000974 [30] DOI: 10.1017/S0022112093002393 · Zbl 0800.76156 · doi:10.1017/S0022112093002393 [31] DOI: 10.1063/1.858282 · doi:10.1063/1.858282 [32] DOI: 10.1038/nature01334 · doi:10.1038/nature01334 [33] DOI: 10.1017/S0022112095000462 · Zbl 0847.76007 · doi:10.1017/S0022112095000462 [34] DOI: 10.1063/1.869866 · Zbl 1147.76537 · doi:10.1063/1.869866 [35] DOI: 10.1143/JPSJ.72.983 · doi:10.1143/JPSJ.72.983 [36] DOI: 10.1017/S0022112006009128 · Zbl 1156.76405 · doi:10.1017/S0022112006009128 [37] DOI: 10.1017/S0022112094003319 · Zbl 0800.76157 · doi:10.1017/S0022112094003319 [38] DOI: 10.1063/1.2147610 · Zbl 1188.76063 · doi:10.1063/1.2147610 [39] DOI: 10.1017/S002211208100181X · Zbl 0476.76051 · doi:10.1017/S002211208100181X [40] DOI: 10.1017/S0022112092002325 · doi:10.1017/S0022112092002325 [41] DOI: 10.1017/S0022112007009251 · Zbl 1159.76334 · doi:10.1017/S0022112007009251 [42] DOI: 10.1063/1.870348 · Zbl 1149.76561 · doi:10.1063/1.870348 [43] DOI: 10.1063/1.1410981 · Zbl 1184.76230 · doi:10.1063/1.1410981 [44] DOI: 10.1063/1.858546 · doi:10.1063/1.858546 [45] He, Phys. Rev. Lett. 81 pp 4639– (1998) [46] DOI: 10.1103/PhysRevA.38.6287 · doi:10.1103/PhysRevA.38.6287 [47] DOI: 10.1103/PhysRevE.77.026303 · doi:10.1103/PhysRevE.77.026303
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:
# Re: st: Quantile Regression With Groupwise Fixed Effects

 From "Austin Nichols" To statalist@hsphsun2.harvard.edu Subject Re: st: Quantile Regression With Groupwise Fixed Effects Date Thu, 5 Jun 2008 11:27:36 -0400

Thomas Mayock <tjm04e@fsu.edu>:
There are a number of relevant threads in the Statalist archives.  You
http://www.econ.uiuc.edu/~roger/research/panel/long.pdf
or at least p.4, which states "In least squares applications the usual
strategy would be to transform y and X to deviations from individual
means, and then compute \hat{\beta} from the transformed data. For
quantile regression this decomposition of projections isn't available
and we are required to deal directly with the full problem."

You can't specify clustering for -qreg but -bootstrap- supports the
-cluster- option, if you are mainly worried about getting standard
errors slightly more robust to clustering.  But I don't know how
-bootstrap- with the -cluster- option will perform in your data--you
may want to run simulations to see its small-sample properties in data
like yours.

On Thu, Jun 5, 2008 at 10:40 AM, Thomas Mayock <tjm04e@fsu.edu> wrote:
> Dear Statalisters
> I am attempting to estimate a quantile regression model with many (over 100) geographic fixed effects. My initial attempt at estimation was to just include group indicator dummies. The drawback to this approach is computational cost: it takes several hours for the model to run.
>    As the group fixed effects are of no interest in the study, I was thinking of performing the group-wise within transformation on the data, then implementing qreg. As I am new to quantile regression, however, I was unsure as to whether or not this is valid. Is anyone aware of other studies that have faced this issue or methodological papers that address a clustering structure in cross-sectional quantile regression?
>     Any information is greatly appreciated.
>
> Tom
> *
> *   For searches and help try:
> *   http://www.stata.com/support/faqs/res/findit.html
> *   http://www.stata.com/support/statalist/faq
> *   http://www.ats.ucla.edu/stat/stata/
>
*
*   For searches and help try:
*   http://www.stata.com/support/faqs/res/findit.html
*   http://www.stata.com/support/statalist/faq
*   http://www.ats.ucla.edu/stat/stata/


text:
# Confusion about Weinberg's discussion of equivalence principle. Help understanding formula

I am reading Steven Weinberg's book Gravitation and Cosmology. On pages 67-68, he begins his discussion of the Principle of Equivalence of Gravitation and Indertia by saying the following:

The Principle of Equivalence rests on the equality of gravitational and inertial mass as demonstrated by Galileo, Huygens, Newton, Bessel, and Eötvös. Einstein reflected that, as a consequence, no external static homogeneous gravitational field could be detected in a freely falling elevator, for observers, their test bodies, and the elevator itself would respond to the field with the same acceleration. This can be easily proved for a system of particles $N$, moving with nonrelativistic velocities under the influences of forces $\mathbf{F}(\mathbf{x}_N-\mathbf{x}_M)$ (e.g. electrostatic or gravitational forces) and an external gravitational field $\mathbf{g}$. The equations of moation are $$m_N \frac{d^2\mathbf{x}_N}{dt^2} = m_N\mathbf{g} + \sum_M \mathbf{F}(\mathbf{x}_N-\mathbf{x}_M)$$ Suppose we perform a non-Galilean space-time coordinate transformation $$\mathbf{x}'=\mathbf{x}-\tfrac{1}2{}\mathbf{g}t^2 \phantom{XXXX} \mathbf{t}'=\mathbf{t}$$ Then $\mathbf{g}$ will be canceled by an inertial "force," and the equation of motion will become $$m_N \frac{d^2\mathbf{x}'_N}{dt'^2} = \sum_M \mathbf{F}(\mathbf{x}'_N-\mathbf{x}'_M)$$ Hence the original observer O who uses coordinates $\mathbf{x}t$, and his freely falling friend O' who uses $\mathbf{x}'t'$, will detect no difference in the laws of mechanics, except that O will say that he feels a gravitational field and O' will say that he does not.

Remark on My Confusion

To me, the first equation looks like it has both a gravitational force $m_N\mathbf{g}$ and some other force $\sum_M \mathbf{F}(\mathbf{x}_N-\mathbf{x}_M)$. The third equation then looks like it only has $\sum_M \mathbf{F}(\mathbf{x}_N-\mathbf{x}_M)$. If I am understanding him correctly, what he is saying is that if we pick a coordinate transformation such as $\mathbf{x}'=\mathbf{x}-\tfrac{1}2{}\mathbf{g}t^2$, then the force $m_N\mathbf{g}$ will seemingly disappear in the $\mathbf{x}'$ reference frame.

But I don't understand where it goes. Does this mean the term $\frac{d^2\mathbf{x}_N}{dt^2}$ changes from the $\mathbf{x}'$ frame to the $\mathbf{x}$ frame? In other words, if $m_N \frac{d^2\mathbf{x}_N}{dt^2}=m_N \frac{d^2\mathbf{x}'_N}{dt'^2}$ and $\sum_M \mathbf{F}(\mathbf{x}_N-\mathbf{x}_M)=\sum_M \mathbf{F}(\mathbf{x}'_N-\mathbf{x}'_M)$, then the equations don't make sense. That would be like saying $a = b + c$ and $a = b$ but $c \neq 0$.

My Question:

(1) Does $m_N \frac{d^2\mathbf{x}_N}{dt^2}=m_N \frac{d^2\mathbf{x}'_N}{dt'^2}$?

(2) Does $\sum_M \mathbf{F}(\mathbf{x}_N-\mathbf{x}_M)=\sum_M \mathbf{F}(\mathbf{x}'_N-\mathbf{x}'_M)$?

1. No.

2. Yes, because the forces only depend on the separations. (Translation symmetry.)

Elaboration of 1.: We have $\mathbf{x}'=\mathbf{x}-\tfrac{1}{2}\mathbf{g}t^2$. Let us calculate the acceleration: $$\frac{d^2\mathbf{x}'}{dt^2}=\frac{d^2}{dt^2}(\mathbf{x}-\tfrac{1}{2}\mathbf{g}t^2)=\frac{d^2\mathbf{x}}{dt^2}-\mathbf{g}$$ Thus the force is $$m\frac{d^2\mathbf{x}'}{dt^2}=m\frac{d^2\mathbf{x}}{dt^2}-m\mathbf{g}$$ Since $\mathbf{F}=\mathbf{F}'$ we have $$m\frac{d^2\mathbf{x}'}{dt^2}=\mathbf{F}'=m\frac{d^2\mathbf{x}}{dt^2}-m\mathbf{g}=\mathbf{F}$$ Thus $$m\frac{d^2\mathbf{x}'}{dt^2}=\mathbf{F}'$$ $$m\frac{d^2\mathbf{x}}{dt^2}=m\mathbf{g}+\mathbf{F}$$ the unprimed frame experiences a gravitational force $m\mathbf{g}$ whereas the primed frame does not.

Elaboration of 2.: We have $\mathbf{x}'=\mathbf{x}-\tfrac{1}{2}\mathbf{g}t^2$. Thus $$\mathbf{x}_N'-\mathbf{x}_M'=(\mathbf{x}_N-\tfrac{1}{2}\mathbf{g}t^2)-(\mathbf{x}_M-\tfrac{1}{2}\mathbf{g}t^2)=\mathbf{x}_N-\mathbf{x}_M$$ implies $$\mathbf{F}(\mathbf{x}_N'-\mathbf{x}_M')=\mathbf{F}(\mathbf{x}_N-\mathbf{x}_M)$$

• That was much simpler than I expected. What do $\mathbf{x}_N$ and $\mathbf{x}_M$ stand for? – Stan Shunpike Feb 10 '15 at 16:39
• I take it $M$ is an index set denoting the various bodies in the gravitational field. $\mathbf{x}_N$ is the $N$th body. – Ryan Unger Feb 10 '15 at 16:48
• By translation symmetry, you mean 2 is invariant under transformation of $\mathbf{x}$ to $\mathbf{x}'$ and you demonstrate that in your second elaboration, correct? Is this spatial translation? – Stan Shunpike Feb 10 '15 at 18:31
• Translation symmetry means that if the positions of both particle $N$ and $M$ changes by the same amount, then the force between them does not change. The coordinate transformation to the accelerated frame is a time-dependent translation. Thus the force is invariant. – Ryan Unger Feb 10 '15 at 19:03
• I get it. In the first case, there is a force but not an accelerating frame. In the second, there is an accelerating frame but no experienced gravitational force. Yet the laws or mechanics remain the same because the new equation still holds. – Stan Shunpike Feb 10 '15 at 20:00


text:
Leveraging Reddit for Suicidal Ideation Detection
Contributor: , ,

Suicide is a major public-health problem that exists in virtually every part of the world. Hundreds of thousands of people commit suicide every year. The early detection of suicidal ideation is critical for suicide prevention. However, there are challenges associated with conventional suicide-risk screening methods. At the same time, individuals contemplating suicide are increasingly turning to social media and online forums, such as Reddit, to express their feelings and share their struggles with suicidal thoughts.

• suicidal ideation detection
• machine learning
• natural language processing
• text mining

## 1. Introduction

Suicide is a global public-health problem. According to the World Health Organization, approximately 703,000 people commit suicide every year [1]. It is the world’s fourth leading cause of death among young people aged 15 to 29 years old. Moreover, it is estimated that there are more than 20 attempts for every completed suicide [2].
The causes of suicide are largely complicated and result from the interaction of multiple factors that can be grouped into three categories: health factors, environmental factors, and factors related to personal history, such as childhood abuse or previous suicide attempts [3,4]. Other examples of suicide risk factors include mental disorder, physical illness, substance abuse, domestic violence, bullying, relationship problems, and other stressful life events. Due to the complexity of the problem, no single risk factor can be reliably used to predict suicide [5]. For instance, despite the strong association between suicide and depression, a depression diagnosis alone has a limited ability to predict suicide. More recently, the issue of suicide has been further exacerbated by the impact of the COVID-19 pandemic [6]. In particular, social isolation—which resulted from measures imposed to curb the spread of the virus—was linked to increased suicide risk.
People with suicide risk fall into two classes: ideators and attempters [7]. Suicidal ideation is a broad term that describes thoughts and behaviors ranging from being preoccupied with death to planning a suicide attempt [8]. The suicidal ideations can be passive and active. Passive suicidal ideation involves thinking about suicide and wishing to be dead, whereas active suicidal ideation implies intending and planning an attempt to take one’s own life [8]. While it is believed that passive suicidal ideation poses a lower risk, both types need to be carefully assessed by mental health professionals, since passive suicidal ideation can rapidly transform into the active form [9]. This can happen when a person’s circumstances or health condition worsen.
The early detection of suicidal ideation expressed by an at-risk individual is key to effective prevention, as it facilitates timely intervention by mental health professionals [10]. However, there are several challenges associated with suicide prevention. They include (1) social stigma, (2) limited access to professional help, and (3) inadequate training of clinicians in dealing with suicidal patients [11]. The combination of these factors creates a new challenge—(4) fragmented professional care, which entails having large time gaps between mental health assessments [11].
At the same time, an increasing number of at-risk individuals are turning to online communication channels to express their feelings and discuss their suicidal thoughts [12,13,14]. This tendency prompted research that focuses on detecting suicide risk and other mental health issues on social networks and online forums by applying machine learning (ML) and natural language processing (NLP) techniques [10,13,15]. The quantifiable signals in user-generated online data aid researchers in gaining insight into an individual’s emotional state and detecting cues indicative of suicidality [16,17]. The feasibility of such an approach has been demonstrated by numerous studies on different mental health disorders. For examples, the authors of [18] used the textual data from Facebook posts of consenting study participants to predict depression diagnoses recorded in their electronic medical records with high accuracy, using a logistic regression model. In the study of [19], using pre-trained machine learning models, the researchers detected negative changes in Twitter users’ sentiment, stress, anxiety, and loneliness measures after the declaration of emergency in the US due to the COVID-19 pandemic.

## 2. Leveraging Reddit for Suicidal Ideation Detection

#### 2.1. Detection of Suicidal Ideation on Social Media

The social stigma related to having suicidal ideations has a particularly significant effect. The fear of social stigma has been shown to discourage individuals at risk of suicide from discussing their experiences in person and seeking support [22,24,25,26]. Further, it obstructs the extant suicide-risk screening methods, such as questionnaires and interviews, since they require patients to explicitly disclose their intentions to commit suicide [27]. According to a meta-analysis of 71 studies, on average, nearly 80% of people in non-psychiatric settings—primary healthcare patients, general population, military personnel, and incarcerated individuals—who died by suicide did not reveal their suicidal intentions when they were surveyed before their suicide attempt [28]. Thus, there is a need for novel suicidality detection methods that do not require face-to-face interactions [24]. In this case, detecting suicidal ideations on online platforms can be more effective since the anonymity of social media and forums enables people to openly share their struggles with suicidal thoughts without fear of judgment [11,16,29,30].
Although the Columbia-Suicide Severity Rating Scale (C-SSRS) has been widely used as a screening instrument, the administration of C-SSRS may place a burden on health-care providers [31]. Therefore, another motivation for detecting suicidal ideations on online platforms is to reduce the load on the health-care system. The goal is to create a tool that would automatically and instantaneously detect if a user is exhibiting any signs of suicidality based on their online activity before engagement with providers. Ideally, these screening tools should be highly scalable and adaptable so that they can be used with a variety of data sources and be readily integrated into existing health-care IT systems [10,31]. The adoption of such suicidal ideation detection tools can assist mental health professionals and even those without specialized training (e.g., primary-care physicians and social workers) in quickly identifying individuals at risk and making informed decisions [26].
Studying the online activity for suicidal ideation detection can also help address the challenges of fragmented care for existing patients [32]. Given that about 70% of psychiatric patients are active on social media, mental health professionals can monitor their online activity to obtain information relevant to patients’ mental state during gaps in patient–clinician interactions [11]. In this scenario, suicidality detection tools can be employed to automatically detect signals of deteriorating mental condition and alert health-care providers, prompting them to attend to a patient under their care [31].

#### 2.2. Reddit as a Source for Suicidal Ideation Detection

Reddit has generated particular interest among researchers due to its distinctive characteristics. Reddit is a popular online forum, covering a wide range of topics, with subcommunities called subreddits [21]. Currently, there are over 13 billion posts and comments distributed across more than 100,000 active communities [33]. More than 50 million active unique users interact with the platform in a single day. Researchers choose Reddit over other platforms as the source of data for several reasons.

Reddit posts have a higher character limit of 40,000 characters compared to Twitter, which only allows 280 characters [25]. It gives users more space to express their suicidal thoughts and describe their emotional state in more detail. The large posts provide a better insight into the author’s mental state [26]. By analyzing long passages of text, the researchers capture and extract textual features that sufficiently indicate suicidal ideations [10,27].
Reddit has numerous specialized support forums dedicated to various mental health topics [26]. For example, the r/SuicideWatch subreddit is a subcommunity of 366,000 members where people share their suicidal thoughts, seek help, and provide support to others dealing with suicidal ideations [36,37]. This subreddit is extensively used by researchers as a source of suicidal posts to serve as positive samples in their datasets [10]. What further supports the validity of r/SuicideWatch as a source of genuine suicide-related posts is that this subreddit is monitored by moderators [25]. The moderators remove any irrelevant posts and posts that violate the community rules, e.g., abuse, criticism, and spam [37].

#### 2.3.1. Data Collection

The first step in the process of building a classifier is obtaining a dataset containing sufficient posts for each class label. Having an accurate dataset with labeled examples is critical for the success of the ML model. The dataset is used to train and then test the model. The model’s predictive performance and its generalizability strongly depend on the quality and amount of training data. There are two broad data collection approaches adopted by the studies: collecting data directly from Reddit and using datasets created by other researchers.

#### 2.3.2. Data Annotation

Supervised ML algorithms require annotated datasets. During the training stage, the algorithm generates a function that maps the relationship between the features and the target variables. To train the model to detect posts with suicidal ideations, the researchers need examples of posts annotated as suicidal and not suicidal. For the multiclass classification problem, posts with annotations for different suicide risk levels are required.

#### 2.3.3. Data Preprocessing

The data collected from Reddit consist of raw, unstructured text and contain noise that can negatively impact the predictive performance of the model. The noise includes punctuation, special characters, URLs, emails, etc. The raw text needs to be converted into a numerical representation before it can be fed into a classifier. During the preprocessing stage, the input data are cleaned and standardized. Therefore, it is an important step that lays the foundation for feature extraction and classification.

#### 2.3.4. Feature Engineering

To use ML algorithms, researchers need to extract features from the data. These features then serve as an input to a classifier algorithm. Therefore, the quality of extracted features is one of the factors that significantly affects the predictive performance of the model. Most studies combined techniques to extract different types of features. The researchers primarily focused on extracting features from the textual content of posts. However, several studies also considered statistical metadata, such as the number of posts per user, the frequency of posting, and the number of votes [13,26].

#### 2.3.5. Model Development

All the studies in the corpus frame their contributions as building a predictive model that detects suicidal ideations from Reddit data. They tested multiple algorithms with different sets of features and proposed best-performing models. In total, 21 supervised ML algorithms were explored by the researchers. Most studies (18 out of 26 studies) included deep learning techniques. The researchers chose deep learning because, when used in conjunction with word embeddings, the deep-learning-based models can effectively detect suicidal ideations without the need for feature engineering.

#### 2.3.6. Model Validation

Once the predictive model is trained, the performance of the model is evaluated. The most common evaluation metrics include accuracy, precision, recall, and F1-score. However, two studies also calculated the area under the curve (AUC) metric [11,26].
For suicidality detection task, true positive (TP) represents the number of posts that were correctly classified as suicidal. True negative (TN) represents the number of posts that were correctly classified as non-suicidal. False positive (FP), also known as Type I error, represents the number of non-suicidal posts that were misclassified as suicidal. False negative (FN), also known as Type II error, represents the number of suicidal posts that were misclassified as non-suicidal.
Accuracy measures the overall portion of correct predictions [51]. It is a ratio of all correctly classified posts to the total number of posts:
$Accuracy=\frac{TP+TN}{TP+FP+TN+FN}$
Precision is a ratio of correctly classified suicidal posts to the total number of posts classified as suicidal (both correctly and incorrectly) [51]:
$Precision=\frac{TP}{TP+FP}$
Recall, also called sensitivity or true-positive rate, is the ratio of correctly classified suicidal posts to the total number of suicidal posts, including both correctly classified posts and posts that should have been classified as suicidal [51]:
$Recall=\frac{TP}{TP+FN}$
This metric is especially useful for selecting the best model where there is a high cost of false-negative predictions [61]. In the suicidal ideation detection model, false positives are more tolerable than false negatives [62]. In other words, it is better to raise a false alarm by incorrectly predicting someone as suicidal than to miss someone who is indeed at risk of suicide.
F1-score is the harmonic mean of precision and recall:
$F1=2×\frac{Precision×Recall}{Precision+Recall}$
For multiclass classification problems, the macro-averaged F1-score can be determined by calculating individual F1-scores for each class and finding their unweighted mean.
The receiver operating characteristic curve is a graph that plots the true-positive rate (Equation (3)) against the false-positive rate (Equation (5)) at different classification thresholds [55]. It provides a graphical representation of the classifier’s performance and a larger area under the curve indicates better performance.

This entry is adapted from 10.3390/ijerph191610347

This entry is offline, you can click here to edit this entry!
Feedback


text:
# Boxes stacked one above the other

Eight boxes, A, B, C, D, E, F, G, and H are kept one above another not necessarily in the order above.

• Three boxes are between H and G and neither of them is at the top or bottom position.
• Two boxes are between F and C and box F is above box C.
• Two boxes are between G and C.
• The number of boxes between A and H is same as the number between A and G.
• Box D is above box E and more than one box is between D and E.
• Box E is not at the bottom position.

Source

Is the answer to the above question:

F
D
H
C
A
E
G
B

• Yeah! that's the answer. – mocarsha2019 Feb 6 at 13:53

## 2 Answers

Yes, I believe that this is the only solution. My reasoning, which I'm sure is what you had, but to reaffirm:

Reasoning (note: Top->Bottom, (#) is the clue sentence):

Using clues (2) and (3), the only options are _F__C__G or F__C__G_. Since G is not the bottom (1), it's the latter.

Then it follows:

H must be 3 above G (1): F_HC__G_. A must be (4) F_HCA_G_. Based on (5 & 6): FDHCAEG_

And finally:

FDHCAEGB (the only possible answer)

D 4.
G 1. 3.
B
A 3.
F 2.
H 1. 3.
E 4.
C 2.
There may be more possible solutions. The numbers point to the hints given!
B was not mentioned, so I left B at an empty place.

• Hi Anders, welcome to Puzzling! Please hide your answer using the spoiler tag (">!"). Also, can you clarify your answer and expand on your logic, please? I'm having a hard time reading it in the current format and I'm unsure how you reached all of your conclusions (for example, G and C should have 2 boxes between them, but your answer seems to have 5 between them). Thanks and happy puzzling! – Greg Feb 6 at 14:18


text:
# Time dependence of the Lagrangian of a free particle?

I am working through Landau's book on Classical Mechanics. I understand the logic and physics of isotropy and homogeneity of space-time behind the derivation of the Lagrangian for a free particle, but I am confused regarding its time dependence. When we calculate the action as the integral of the Lagrangian for a wiggly trajectory, the velocity is obviously dependent on time and so should be the Lagrangian. Of course, we extremize the action to find the true trajectory of motion. But this time dependence of Lagrangian for wiggly trajectories is very confusing to me because this indicates to me that Lagrangian is dependent on time for a free particle. Where am I making the mistake here?

Also, when we have a particle in a position dependent potential, the velocity (and kinetic energy) is again dependent on time for any trajectory we choose and even for the true trajectory. But then again we write the velocity as independent of time in the Lagrangian. Why is that so?

• Thanks for the response. But I would also like some insight into my second question. Why do we treat velocity as independent of time even in the case where we have an interaction ? Time is homogeneous but we know that for a true trajectory (or motion) velocity is changing with time in this case. – singularity Mar 3 '15 at 20:33

I) In the Lagrangian $L(q(t),\dot{q}(t),t)$, one must distinguish between implicit time dependence via the variables $q(t)$ and $\dot{q}(t)$, and explicit time dependence.$^1$

However, the implicit time dependence in the Lagrangian $L$ only makes sense in the context of a fixed (but arbitrary, possibly virtual) path $$\tag{1} [t_i,t_f]~\stackrel{q}{\longrightarrow}~\mathbb{R}^n.$$ The implicit time dependence would typically be different for another path.

II) In fact a (possibly virtual) path (1) is technically speaking not the input for a Lagrangian $L$. Rather the Lagrangian $$\tag{2} \mathbb{R}^n\times\mathbb{R}^n\times [t_i,t_f]~\stackrel{L}{\longrightarrow}~\mathbb{R}$$ is a function $$\tag{3} (q,v,t)~\mapsto~ L(q,v,t)$$ (as opposed to a functional) that only depends on

1. an instant $t\in[t_i,t_f]$,

2. an instantaneous position$^2$ $q\in\mathbb{R}^n$, and

3. an instantaneous velocity $v\in\mathbb{R}^n$;

not the past, nor the future.

Notice that we here use the symbol $v$ rather than the notation $\dot{q}\equiv \frac{dq}{dt}$. This is because the ability to differentiate $\frac{dq}{dt}$ would imply that we know (at least a segment of) a path (1) rather than just information about an instantaneous state $(q,v,t)$ of the system.

III) In contrast, the action $$\tag{4} S[q] ~:=~ \int_{t_i}^{t_f}dt \ L(q(t),\dot{q}(t),t)$$ is a functional (as opposed to a function) that depends on a (possibly virtual) path (1).

For more details, such as, e.g., an explanation how calculus of variations works, why $q$ and $v$ are independent variables in the Lagrangian (3) but dependent variables in the action (4), etc.; see e.g. this related Phys.SE post and links therein.

--

$^1$ By the way, if the Lagrangian $L(q,v)$ has no explicit time dependence, then the energy

$$\tag{5} h(q,v)~:=~v^i \frac{\partial L(q,v)}{\partial v^i}-L(q,v)$$ is conserved, cf. Noether's theorem.

$^2$ Here $q\in\mathbb{R}^n$ denotes an $n$-tuple, as opposed to eq. (1) where $q$ denotes a path/curve.

• Very interesting ! I am reading the link you have provided here as well. But one question that arises here is how would you differentiate between implicit time dependence and explicit dependence in the case where Lagrangian actually is explicitly time dependent ? How would we make the difference precise ? I think I will try to work out the math here. Thank you. – singularity Mar 3 '15 at 21:00
• @stringcosmologyapplicant To satisfy the minimum action we must find a relation between the variables of the Lagrangian. So, we admit that the position and velocity depend on time. But before we find this relation, we know nothing of a particular dependence of the velocity on time. – Sofia Mar 3 '15 at 21:21

Think about an action principle as an abstract mapping of trajectories $\mathbf r = (\vec r(t), t_0, t_1)$ to some number $S(\mathbf r)$ which no longer is explicitly time-dependent.

Now, sometimes, the action can be computed from another function. The other function just has a bunch of parameters, which we can call $\{\alpha_i\}, ~ \{\beta_i\}, ~ \tau$. From the perspective of this function, those are just numbers.

We say that this function is a Lagrangian if we compute the action from it by doing a time integral:

$S(\mathbf r) = \int_{t_0}^{t_1} dt~ L(\{\alpha_i = r_i(t)\}, ~ \{\beta_i = \dot r_i(t)\},~ \tau = t)$

In other words, the function doesn't "know about" physics just yet; the physics is imbued by the least-action principle. The Lagrangian is just a function which takes a bunch of variables and produces a number. Sure, if you "know" that this time integral will be done you can apply that function to a real particle's trajectory and then you'd find that it was time-dependent, but by itself, it's just a function, used by the action principle for all trajectories, to compute their action.

So when we skip writing the parameters as $\{\alpha_i\}, ~ \{\beta_i\}, ~ \tau$ and instead write them as $\{r_i\}, ~ \{v_i\}, ~ t$ that is just us reusing these symbols as names for the numbers that we're eventually going to put into the function. This "punning" (as the programming language Haskell calls it) is a little confusing at first but it helps us remember what was what when we later go to substitute the values in.

You say " When we calculate the action as the integral of the Lagrangian for a wiggly trajectory, the velocity is obviously dependent on time and so is the Lagrangian".

How exactly is the velocity dependent on time? Before applying the least action principle and find the trajectory of the object, we have a Lagrangian dependent on velocity (through the kinetic energy term), and position and (eventually) time (through the potential energy). We have no idea how the velocity depends on time. There is a continuum of forms of dependences, because there is a continuum of forms of trajectories that the object may follow in principle. This is why, before minimizing the action, we take in the Lagrangian the velocity as a variable in itself.

We don't know the trajectory before minimizing the action, s.t. we have no relationship between velocity and time.

• Thanks for the response. I am sorry but your last sentence is a bit confusing to me. Are you saying that arbitrary trajectories may have time dependence of velocity but since we have chosen a Lagrangian which has time-independent velocity, the final equation of motion would have no relationship between time and velocity. – singularity Mar 3 '15 at 20:45
• @stringcosmologyapplicant No, no. I am saying that before applying the least action principle and find the trajectory of the object, we have a Lagrangian dependent on velocity I the kinetic energy term, and position and (eventually) time in the potential energy. We have no idea how the velocity depends on time. There is a continuum of forms of dependence, because there is a continuum of forms of trajectories available. This is why, in the Lagrangian we take the velocity as a variable in itself. (I continue) – Sofia Mar 3 '15 at 21:06
• @stringcosmologyapplicant Only when we learn what is the trajectory, by choosing the path that minimizes the action, we learn that the velocity depends on time in a certain way. But, as long as we don't define a trajectory, i.e. $\vec r(t), \vec {\dot r(t)}$ what we know about the velocity? Do we know on what it depends? – Sofia Mar 3 '15 at 21:08
• So it seems it doesn't make sense to talk about time dependence before applying the least action principle, since we do not really know the actual time dependence and the actual trajectory. – singularity Mar 3 '15 at 21:36
• @stringcosmologyapplicant yes. Before imposing the condition of minimal action, the velocity is treated as an independent variable because at a given time $\tau$ between $t_1$ and $t_2$ and at an whatsoever place in space, $\vec r$, the velocity $\vec v$ can be 5m/s, 0.32m/s, -200m/s, God knows what and in which direction. It behaves as an independent variable. Only by imposing the condition of minimum action, will we find that it has to depend on time in a certain way. And the condition of minimum action leads to the Euler-Lagrange eqs., which lead us explicitly to the form of dependence. – Sofia Mar 3 '15 at 21:55


text:
### 统计代写|生物统计代写biostatistics代考| RANDOM SAMPLING

statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写生物统计biostatistics代写方面经验极为丰富，各种生物统计biostatistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|生物统计代写biostatistics代考|OBTAINING REPRESENTATIVE DATA

The purpose of sampling is to get a sufficient amount of data that is representative of target population so that statistical inferences can be made about the distribution and the parameters of the target population. Because a sample is only a subset of the units in the target population, it is generally impossible to guarantee that the sample data are representative of the target population; however, with a well-designed sampling plan, it will be unlikely to select a sample that is not representative of the target population. To ensure the likelihood that the sample data will be representative of the target population, the following components of the sampling process must be considered:

• Target Population The target population must be well defined, accessible, and the researcher should have a good understanding of the structure of the population. In particular, the researcher should be able to identify the units of the population, the approximate number of units in the population, subpopulations, the approximate shape of the distributions of the variables being studied, and the relevant parameters that need to be estimated.
• Sampling Units The Sampling units are the units of the population that will be sampled. A sampling unit may or may not be a unit of the population. In fact, in some sampling plans, the sampling unit is a collection of population units. The sampling unit is also the smallest unit in the target population that can be selected.
• Sampling Element A sampling element is an object on which measurements will be made. A sampling element may or may not be a sampling unit. When the sampling unit consists of several population units, it is called a cluster of units. If each

population unit in a cluster will be measured, then the sampling elements are the population units within the sampled clusters. In this case, the sampling element is a subunit of the sampling unit.

• Sampling Frame The sampling frame is the list of sampling units that are available for sampling. The sampling frame should be nearly equal to the target population. When the sampling frame is significantly different from the target population, it makes it less unlikely that a sample representative of the target population will be obtained, even with a well-designed sampling plan. Sampling frames that fail to include all of the units of the target population are said to undercover the target population and may lead to biased samples.
• Sample Size The sample size is the number of sampling units that will be selected. The sample size will be denoted by $n$ and must be sufficiently large to ensure the reliability of the statistical analysis. The variability in the target population plays a key role in determining the sample size necessary for the desired level of reliability associated with a statistical analysis.

## 统计代写|生物统计代写biostatistics代考|Probability Samples

The statistical theory that provides the foundation for the estimation or testing of research hypotheses about the parameters of a population is based on the sampling structure known as probability sampling. A probability sample is a sample that is selected in a random fashion according to some probability model. In particular, a probability sample is a sample chosen so that each of the possible samples is known in advance and the probability of drawing each sampling unit is known. Random samples are samples that arise through a sampling plan based on probability sampling.

Probability sampling allows flexibility in the sampling plan and can be designed specifically for the target population being studied. That is, a probability sampling plan allows a sample to be designed so that it will be unlikely to produce a sample that is not representative of the target population. Furthermore, probability samples allow for confidence statements and hypothesis tests to be made from the observed sample with a high degree of reliability.

Samples of convenience are samples that are not based on probability samples and are also referred to as nonprobability samples. The statistical theory that justifies the use of confidence statements and tests of hypotheses does not apply to nonprobability samples; therefore, confidence statements and test of the research hypotheses based on nonprobability samples are erroneous applications of statistics and should not be trusted.
In a random sample, the chance that a particular unit of the population will be selected is known prior to sampling, and the units available for sampling are selected at random according to these probabilities. The procedure for drawing a random sample is outlined below.

## 统计代写|生物统计代写biostatistics代考|Simple Random Sampling

The first sampling plan that will be discussed is the simple random sample. A simple random sample of size $n$ is a sample consisting of $n$ sampling units selected in a fashion that every possible sample of $n$ units has the same chance of being selected. In a simple random sample, every possible sample has the same chance of being selected, and moreover, each sampling unit has the same chance of being drawn in a sample. Simple random sampling is a reasonable sampling plan for sampling homogeneous or heterogeneous populations that do not have distinct subpopulations that are of interest to the researcher.
Example 3.3
Simple random sampling might be a reasonable sampling plan in the following scenarios:
a. A pharmaceutical company is checking the quality control issues of the tablet form of a new drug. Here, the company might take a random sample of tablets from a large pool of available drug tablets it has recently manufactured.
b. The Federal Food and Drug Administration (FDA) may take a simple random sample of a particular food product to check the validity of the information on the nutrition label.
c. A state might wish to take a simple random sample of medical doctors to review whether or not the state’s continuing education requirements are being satisfied.
d. A federal or state environment agency may wish to take a simple random sample of homes in a mining town to investigate the general health of the town’s inhabitants and contamination problems in the homes resulting from the mining operation.

The number of possible simple random samples of size $n$ selected from a sampling frame listing of $N$ sampling units is
$$\left(\begin{array}{l} N \ n \end{array}\right)=\frac{N !}{n !(N-n) !}$$
The probability that any one of the possible simple random samples of $n$ units selected from a sampling frame of $N$ units is
$$\frac{1}{\frac{N !}{n !(N-n) !}}=\frac{n !(N-n) !}{N !}$$

## 统计代写|生物统计代写biostatistics代考|OBTAINING REPRESENTATIVE DATA

• 目标人群 目标人群必须明确定义、易于访问，并且研究人员应对人群结构有很好的了解。特别是，研究人员应该能够识别总体单位、总体中单位的大致数量、亚总体、所研究变量分布的大致形状以及需要估计的相关参数。
• 抽样单位 抽样单位是要抽样的总体单位。抽样单位可能是也可能不是人口的单位。事实上，在一些抽样计划中，抽样单位是人口单位的集合。抽样单位也是目标人群中可以选择的最小单位。
• 采样元件 采样元件是将对其进行测量的对象。采样元件可以是也可以不是采样单元。当抽样单位由若干人口单位组成时，称为单位群。如果每个

• 抽样框架 抽样框架是可用于抽样的抽样单位列表。抽样框架应该几乎等于目标人群。当抽样框架与目标人群显着不同时，即使采用精心设计的抽样计划，也不太可能获得代表目标人群的样本。未能包括目标总体的所有单位的抽样框架被称为隐藏目标总体，并可能导致样本有偏差。
• 样本大小 样本大小是要选择的抽样单位的数量。样本大小将表示为n并且必须足够大以确保统计分析的可靠性。目标人群的可变性在确定与统计分析相关的所需可靠性水平所需的样本量方面起着关键作用。

## 统计代写|生物统计代写biostatistics代考|Simple Random Sampling

：一家制药公司正在检查一种新药片剂的质量控制问题。在这里，该公司可能会从其最近生产的大量可用药片中随机抽取片剂样本。

C。一个州可能希望对医生进行简单的随机抽样，以审查该州的继续教育要求是否得到满足。
d。联邦或州环境机构可能希望对采矿城镇中的房屋进行简单的随机抽样，以调查该镇居民的总体健康状况以及采矿作业导致的房屋污染问题。

(ñ n)=ñ!n!(ñ−n)!

1ñ!n!(ñ−n)!=n!(ñ−n)!ñ!

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。


text:
## Clustering Using SVD on Omics Data

For Michigan, I’ve since applied the SVD based sorting algorithm, which can be used for clustering, to real data. The goal is to in some way model the relationship of proteins and metabolites, come up with modules, or groups of proteins and metabolites that were related, and then use these later in a regression model to see which groups are associated with outcomes such as cardiovascular disease (CVD) or cholesterol.

The data here is something like this. Suppose, for example, we have $p$ proteins, $n$ patients and $m$ metabolites. The protein and metabolite, matrix are then $P_{p \times n}$ and $M_{m \times n}$ respectively. For this next step, recall the definition of covariance. Suppose we have matrices $X$ and $Y$. The definition is as follows, where $E$ is the mathematical expectation.

$cov(X, Y) = E[(X - \mu_x)(Y - \mu_y)]$

We do something similar to render a protein-metabolite cross covariance matrix. That is, we scale (or standardize) the matrix by each row, for $P$, along the protein axis, and each column, for $M$ along the metabolite axis. After, scaling, we take the inner product, we get a matrix I’ll call $L$, which is as follows.

$L_{m \times p} = M^{T}_{m \times n} P_{n \times p}$

The cross-covariance matrix looks like this.

What a mess!

Next, we’d like to determine if there’s any modules, as we’ve done in the previous post on simulated data, which can be found here: https://likelyllc.com/2022/06/13/using-svd-for-clustering/. In this case, we sort by using both the first, left, singular vector and first, right singular vector. The code snippets are below, with full code to follow. Remember to sort the names as well, so the plot renders with the correct names!

# order by first left singular vector
K_ab_svd_tmp = svd(K_ab);
U = K_ab_svd_tmp$u; Uj = U[,j]; Uj_order = order(Uj, decreasing = T); Uj_sorted = Uj[Uj_order]; K_ab_reordered = matrix(NA, nrow = nrow(K_ab), ncol = ncol(K_ab)); for (j in 1:ncol(K_ab)) { K_ab_reordered[, j] = K_ab[,j][Uj_order]; } prot_names = prot_names[Uj_order]; # order by first right singular vector K_ab_svd_re_ordered_tmp = svd(K_ab_reordered); V = K_ab_svd_re_ordered_tmp$v;
Vti = Vt[i,];
Vti_order = order(Vti, decreasing = T);
Vti_sorted = Vti[Vti_order];
K_ab_reordered_2 = matrix(NA, nrow = nrow(K_ab), ncol = ncol(K_ab));
for (i in 1:nrow(K_ab)) {
K_ab_reordered_2[i, ] = K_ab_reordered[i,][Vti_order];
}
meta_names = meta_names[Vti_order];

Recall this diagnostic plot from the previous post, where we can see there are three clear clusters, on simulated data:

Let’s take a look at these plots but for metabolites and proteins. We can compute the differences, explicitly, too.

Looks like there are no clear clusters, as determined by singular gaps, for the proteins, however, looks like we can see about 3 clusters for the proteins, depending on how you’d like to define them. For the former, this happens sometimes with real data! Although we like to categorize, sometimes in real life this isn’t possible. Furthermore, we can compute these explicitly.

After sorting using this strategy, we get the following plot. This, in my opinion, is a lot more easy to interpret than clustering using hierarchical clustering.

Full code below.

## now we use the spectral sorting on the real data with K_ab
library(doBy);
library(ggplot2);
set.seed(1);

setwd("/home/andre/venk/svd_clustering/");

df = expand.grid(x = 1:N, y = 1:M);
df = cbind(df, as.vector(K_ab));
colnames(df)[3] = "inner_prod";
p = ggplot(df, aes(x, y)) + geom_tile(aes(colour = inner_prod), size = 12) +
high = "red", mid = "white")  +
theme(axis.title=element_blank(),
axis.text.x=element_text(angle=90, hjust=1, vjust=0.5)) +
coord_equal() + coord_flip() + theme(legend.position = "none");
p

K_ab_svd_tmp = svd(K_ab);
U = K_ab_svd_tmp$u; V = K_ab_svd_tmp$v;
svds = K_ab_svd_tmp$d; prot_names = rownames(K_ab); meta_names = colnames(K_ab); j = 1; k = 3; Uj = U[,j]; Uj_order = order(Uj, decreasing = T); ## this is to find biggest gaps Uj_sorted = Uj[Uj_order]; diffi = abs(diff(Uj_sorted)); largest_diffs = sort(which.maxn(diffi, n = k - 1)); # clusters = c(rep(1, largest_diffs[1]), # rep(2, largest_diffs[2] - largest_diffs[1]), # rep(3, N - largest_diffs[2])); K_ab_reordered = matrix(NA, nrow = nrow(K_ab), ncol = ncol(K_ab)); for (j in 1:ncol(K_ab)) { K_ab_reordered[, j] = K_ab[,j][Uj_order]; } prot_names = prot_names[Uj_order]; N = nrow(K_ab); M = ncol(K_ab); df = expand.grid(x = 1:N, y = 1:M); df = cbind(df, as.vector(K_ab_reordered)); colnames(df)[3] = "inner_prod"; p = ggplot(df, aes(x, y)) + geom_tile(aes(colour = inner_prod), size = 12) + scale_color_gradient2(low = "blue", high = "red", mid = "white") + theme(axis.title=element_blank(), axis.text.x=element_text(angle=90, hjust=1, vjust=0.5)) + coord_equal() + coord_flip(); p ## now we try the row sorting as well ## we'll take a leap and do it for K_ab_svd_re_ordered_tmp = svd(K_ab_reordered); U = K_ab_svd_re_ordered_tmp$u;
V = K_ab_svd_re_ordered_tmp$v; svds = K_ab_svd_re_ordered_tmp$d;

i = 1;
Vt = t(V);
Vti = Vt[i,];
Vti_order = order(Vti, decreasing = T);
Vti_sorted = Vti[Vti_order];
plot(Vti_sorted);

K_ab_reordered_2 = matrix(NA, nrow = nrow(K_ab), ncol = ncol(K_ab));
for (i in 1:nrow(K_ab)) {
K_ab_reordered_2[i, ] = K_ab_reordered[i,][Vti_order];
}
meta_names = meta_names[Vti_order];

df = expand.grid(x = prot_names, y = meta_names);
df = cbind(df, as.vector(K_ab_reordered_2));
colnames(df)[3] = "inner_prod";
p = ggplot(df, aes(x, y)) + geom_tile(aes(colour = inner_prod), size = 12) +


text:
## anonymous 5 years ago what is the taylor series of cos(x/2) about a=pi

This is your first term $f(\ \cos \pi/2)+f'(\ \cos \pi/2)(x-[\ \cos \pi/2])$


text:
# Homology Whitehead theorem for non simply connected spaces

(One version of) the Whitehead theorem states that a homology equivalence between simply connected CW complexes is a homotopy equivalence. Does the following generalisation hold true?

Suppose $X,Y$ are two connected CW complexes and $f:X\to Y$ is a continuous map that induces isomorphisms of the fundamental groups and on homology. Then $f$ is a weak equivalence.

It would be enough to show that the map $\tilde{f}$ obtained by lifting $f$ to the universal covers (for some arbitrary choice of base points) is a homology equivalence, but I've never studied the homology spectral sequence, so I don't know the relationship between the homology of the universal cover $\widetilde{X}$ and that of $X$, and I don't know how to prove that $\tilde{f}$ is a homology equivalence.

Also, are there any good, reasonably self contained and short accounts of the homology of covering spaces, preferably online?

EDIT 1 I've seen some lecture notes (second paragraph on page 5) where this theorem is used (without further comment). They used this proposition to establish that a certain type of homotopy colimit is well defined up to weak equivalence.

EDIT 2 Theorem 6.71 from Kirk and Davis (page 179 of the document or 164 internally) is probably what I need, but it involves isomorphisms with local coefficient systems for homology for the base spaces.

EDIT 3 Thank you @studiosus for the reference. I don't quite understand yet, could you help out? If I get it (using the theorem of the source), there are finite connected two dimensional complexes $X,Y$ and a homotopy equivalence $X\vee S^2\simeq Y\vee S^2$, yet $X$ and $Y$ aren't homotopy equivalent. From the assumption it follows immediately that $$X\hookrightarrow X\vee S^2\to Y\vee S^2\twoheadrightarrow Y$$ gives an isomorphism of the fundamental groups. From $$\tilde{H}(X)\oplus\tilde{H}(S^2)\simeq\tilde{H}(X\vee S^2)\simeq\tilde{H}(Y\vee S^2)\simeq\tilde{H}(X)\oplus\tilde{H}(S^2)\;,$$ $\tilde{H}(S^2)\simeq \Bbb Z[2]$ and the fundamental theorem of finitely generated abelian groups, it follows at once that $\tilde{H}(X)\simeq\tilde{H}(Y)$ abstractly. The only problem I have is that it doesn't seem to me that we get a map $X\to Y$ that induces isomorphisms on $\pi_1$ and is a homology equivalence, actually, I don't see why there should be a homology equivalence $X\to Y$ at all.

• This generalisation seems unlikely (mainly because the homotopy group version of Whitehead's theorem seems close to a 'sharp' result), but I can't think of a counterexample at the moment. – Dan Rust Apr 13 '14 at 12:26
• Could you outline why it's enough to show the lift $\tilde{f}$ is a homology equivalence? I understand that this would show $\tilde{f}$ is a homotopy equivalence, but why would there be a homotopy inverse $\tilde{g}$ to $\tilde{f}$ which respects the projection mappings? Is there an equivariant Whitehead's theorem? – Jason DeVito Apr 14 '14 at 23:45
• @JasonDeVito If $\tilde{f}$ is a homology equivalence, it follows from simple connectedness and Whitehead's homology theorem that it is a homotopy equivalence $\widetilde{X}\to\widetilde{Y}$. Since a covering projection induces an isomorphisms $p_*:\pi_n(\widetilde{X})\simeq\pi_n( X)$ for all $n\geq 2$, the commutative diagram $$\begin{array}{ccc}\pi_n(\widetilde{X})&\xrightarrow{\tilde{f}_*}&\pi_n( \widetilde{Y} )\\\downarrow&&\downarrow\\\pi_n(X)&\xrightarrow{f_*}&\pi_n(Y)\end{array}$$ that $f_*$ is an isomorphism for all $\pi_n$, $n\geq 2$. – Olivier Bégassat Apr 15 '14 at 0:23
• @JasonDeVito Since by hypothesis $f_*$ is an isomorphism on $\pi_1$ and $\pi_0$, $f$ is a weak equivalence between CW complexes and Whitehead's theorem tells us that it is a homotopy equivalence. – Olivier Bégassat Apr 15 '14 at 0:26
• I think that a counterexample is given at the top of p. 281 in McDuff and Segal's article on group completion (available here: maths.ed.ac.uk/~aar/papers/mcdsegal.pdf) but maybe there is an issue of basepoints. They consider the shift map on the infinite symmetric group (finitely supported permutations of $\mathbb{N}$). – Dan Ramras Apr 15 '14 at 3:04

The answer is no. An easy example is provided in Allen Hatcher's Algebraic Topology, Example 4.35.

http://www.math.cornell.edu/~hatcher/AT/ATch4.pdf

There, a CW-complex $X$ is formed by attaching an appropriate $(n+1)$-cell to the wedge $S^1 \vee S^n$ of a circle and an $n$-sphere. The inclusion of the $1$-skeleton $S^1 \to X$ induces an isomorphism on integral homology and on homotopy groups $\pi_i$ for $i < n$ but not on $\pi_n$.

That example is also worked out as Problem 1 here:

http://www.home.uni-osnabrueck.de/mfrankland/Math527/Math527_HW10_sol.pdf

• Thanks for your answer, I somehow hadn't noticed it until today ... – Olivier Bégassat Oct 22 '16 at 15:17

There are two finite 2-dimensional complexes $A, B$ which are not homotopy-equivalent but are homology-equivalent, i.e., there exists a continuous map $$f: A\to B$$ inducing isomorphism of fundamental groups and homology groups. See the last paragraph (page 522) of this paper, the actual example is due to Dunwoody.

Edit 1: The homology equivalence part follows from Dyer's paper, see the link above (read the very last paragraph of his paper). Dyer's results are by no means obvious: He proves that equality of Euler characteristics plus one extra condition ($m=1$, whatever $m$ is) imply homology equivalence of 2-d complexes with isomorphic fundamental groups. Then Dyer verifies his conditions in the case of Dunwoody's example (the only nontrivial condition is $m=1$, since equality of Euler characteristics is clear).

Edit 2: The correct version of the "homological Whitehead's theorem" is indeed requires a cohomology isomorphisms with sheaf coefficients, see the discussion and reference here.

• Thanks for the reference, I have a question though, too long for a comment. I added it in the main body of the question, I'd appreciate it if you could take a look at it! – Olivier Bégassat Apr 15 '14 at 16:54
• Just in case you already read my question, there was a mistake in my EDIT to the original question, the source I linked to wasn't the article by Dunwoody. – Olivier Bégassat Apr 15 '14 at 19:16
• @OlivierBégassat: See the edit. – Moishe Kohan Apr 15 '14 at 20:11
• Ok, thank you for the explanation. I'll try to understand Dyer's result tomorow. – Olivier Bégassat Apr 15 '14 at 20:44
• I left an answer to tell you what I understood from your examples, could you take a look at it and correct my understanding of it? – Olivier Bégassat Apr 20 '14 at 16:20

...it turns out there are non-trivial high dimensional smooth knots $S^n\subset S^{n+2}$ such that $\pi_1$ of the knot complement $X$ is infinite cyclic. The inclusion of the meridian $S^1 \subset X$ is a homology isomorphism, a $\pi_1$ isomorphism but is never a weak equivalence since by a result of Levine if it were then the knot would be trivial (we should assume $n \ge 3$ here).

• That's very interesting. I asked John Klein for a reference in the comment section of his answer; do you know of a reference for this? – Olivier Bégassat Apr 18 '14 at 12:38
• @Olivier The first part (the inclusion of the meridian is a homology iso) follows from Alexander duality. The reference for the second part (codimension 2 sphere is unknotted iff its complement is homotopy equivalent to $S^1$) is, I believe, J. Levine, Unknotting spheres in codimension 2, Topology 4, 9–16 (1965) – Grigory M Apr 18 '14 at 12:52
• Actually, the second part is due to John Stallings "On topologically unknotted spheres", Annals of Math., 1963. – Moishe Kohan Apr 18 '14 at 16:39
• @studiosus Oh, looks like you're right – Grigory M Apr 18 '14 at 16:45

This is meant as a question directed at @studiosus and anybody who's familiar with his example.

M. Dyer's theorem seems to state, as a special case, that

Theorem Suppose $X,Y$ are finite, at most 2 dimensional, connected CW complexes with isomorphic fundamental groups (so called $[G,2]$-complexes, where $\pi_1(X)\simeq G\simeq\pi_1(Y)$). If $X$ and $Y$ have the same Euler characteristic $\chi$, and $\chi>\chi_{\min}(G,2)$, then $X$ and $Y$ are homology equivalent.

(From what I understood, when $\chi>\chi_{\min}(G,2)$ then automatically $m=1$, see the first sentence under his theorem)

To the trefoil knot group presentation $G=\langle a,b\mid a^3b^{-2}=1\rangle$ corresponds a $[G,2]$-complex with one $0$ cell, two $1$ cells and one $2$ cell ($Y$ in Dyer's paper) so that $\chi(Y)=0\geq\chi_{\min}(G,2)$, and thus with $X=Y\vee S^2$, which is manifestly still a $[G,2]$-complex, $1=\chi(X)>0\geq\chi_{\min}(G,2)$. This $X$ corresponds to the presentation $G=\langle a,b\mid a^3b^{-2}=1, 1=1\rangle$.

There is another presentation for $G$ defined in Dunwoody's paper with a complicated presentation (still on two generators and two relations, it's on the last page of his paper, so with $\chi=1$), and I take it, that the realization $Z$ of this presentation has $X\not\simeq Z$ (while, I take it, $X\vee S^2\simeq Z\vee S^2$).

So since $X$ and $Z$ are $[G,2]$-spaces with identical Euler characteristic equal to $1>0\geq\chi_{\min}(G,2)$, they should automatically have $m=1$ and thus be homology equivalent by M.Dyer's therem (for him this means: there is a map that is an isomorphism on $\pi_1$ and $H_*$), yet they aren't homotopy equivalent by Dunwoody's theorem.

Is this correct?

$\chi_{\min}(G,2)$ is the minimum over all Euler characteristics of $[G,2]$-complexes.The fact that $\chi_{\min}(G,2)$ is well defined strikes me as believable, at least not implausible, and I'm willing to accept it. Indeed, it seems that every $[G,2]$-complex $X$ should define a presentation $P$ of $G$, and a subcomplex $X'$ with the same $1$-skeleton, but possibly less $2$-cells, thus with smaller Euler characteristic, homotopy equivalent to the $[G,2]$-complex obtained from the presentation $P$. So it seems that, in order to show $\chi_{\min}(G,2)$ is well defined, one can forget about general $[G,2]$-complex and only consider those associated to presentations of $G$. Adding a new relation increases the Euler characteristic by one. Adding a new generator should introduce at least one new relation, thus at least one new $2$-cell, so it seems that ($\#$ of $2$-cells)$-$($\#$ of $1$-cells)$+1$ can only increase by adding new generators and relations to a given presentation of $G$. So it seems one can do the following:

1. From any $[G,2]$-complex $X$ extract a subcomplex $X'$ with fewer $2$ cells and identical $1$ skeleton (thus reducing $\chi$) that remains a $[G,2]$-complex and is homotopy equivalent to a $[G,2]$-complex associated to a presentation of $G$.
2. From a given presentation of $G$ extract a "minimal" presentation of $G$. Intuitively one that can't be reduced by either considering less generators or relations. From the intuitive argument above it follows that adding new generators or relations to a presentation can only increase teh Euler characteristic, so passing to "minimal" presentations should reduc the Euler characteristic of the associated $[G,2]$-complex.
3. Only study the Euler characteristics of $[G,2]$-complexes associated to "minimal" presentations of $G$.
4. Somehow show that they all have the same Euler characteristic, which then must be $\chi_{\min}(G,2)$.

EDIT 1 Proposition 1 on page 14 of these notes gives a lower bound for the Euler characteristic of any $[G,2]$-complex associated to any presentation of (finitely presentable) $G$ in terms of the rank of its first and second homology groups. So steps 3 and 4 aren't necessary, 4 might even be wrong.

• I did not look at the staff before Edit 1, but what you say after Edit 1 is correct. The trick is to show that $0=\chi_{min}(G,2)$ in Dunwoody's example. You can see this from the fact that the presentation complex for $G$ is $K(G,1)$. – Moishe Kohan Apr 20 '14 at 18:13
• @studiosus But as I understand it, it is only required that the Euler characteristics be equal and $m=1$, and it seems to me he is saying that $m=1$ whenever $\chi>\chi_{\min}(G,2)$. Since we know by example (the complex associated to the presentation $\langle a,b\mid a^3=b^2\rangle$) that there is a $[G,2]$-complex with $\chi=0$ we have $0\geq\chi_{\min}(G,2)$, and so, for any $[G,2]$-complex with $\chi\geq 1$, $m=1$. My point is that it apparantly doesn't matter that $\chi_{\min}(G,2)=0$ or not, it only matters that it be less than our $\chi$. Am I correctly interpreting the article? – Olivier Bégassat Apr 20 '14 at 18:59
• Also, are the spaces $X$ and $Z$ the actual spaces he uses to find homology equivalent (in $\pi_1$ and $H_*$), non homotopy equivalent spaces? – Olivier Bégassat Apr 20 '14 at 19:02
• True on both counts. – Moishe Kohan Apr 20 '14 at 19:40
• Ok, if you say so ^^ I awarded you the bounty, I guess I need to read the articles more in depth to believe their claims and my own explanation ^^ – Olivier Bégassat Apr 20 '14 at 19:47

You know already that the answer for your question is no, nevertheless, there is a generalisation which you might like:

Suppose that $$X$$, $$Y$$ are path-connected, simple CW-complexes. If $$f:X\rightarrow Y$$ induces isomorphism in $$H_n$$ for all $$n\geq 0$$, then $$f$$ is a homotopy equivalence.

• Hi Mihail, I only noticed your answer today, sorry about that. What is the definition of a simple CW complex ? – Olivier Bégassat Jul 31 '19 at 12:39
• @OlivierBégassat A topological space is called simple if the action of a fundamental group on higher homotopy groups is trivial. For example, this is the case when the fundamental group is trivial. – Mihail Aug 1 '19 at 22:54
• Thanks! ${}{}{}$ – Olivier Bégassat Aug 1 '19 at 22:55
• I think that one also requires that the fundamental group is abelian usally. At least, that is the generalization that I know – Andres Mejia Nov 17 '20 at 23:47
• It acts on itself my inner automorphisms – Andres Mejia Mar 15 at 3:43


text:
# Eutectic system

(Redirected from Eutectic point)
A phase diagram for a fictitious binary chemical mixture (with the two components denoted by A and B) used to depict the eutectic composition, temperature, and point. (L denotes the liquid state.)

A eutectic system is a mixture of chemical compounds or elements that have a single chemical composition that solidifies at a lower temperature than any other composition made up of the same ingredients. This composition is known as the eutectic composition and the temperature at which it solidifies is known as the eutectic temperature.

On a phase diagram the intersection of the eutectic temperature and the eutectic composition gives the eutectic point.[1] Non-eutectic mixtures will display solidification of one component of the mixture before the other. Not all binary alloys have a eutectic point; for example, in the silver-gold system the melt temperature (liquidus) and freeze temperature (solidus) both increase monotonically as the mix changes from pure silver to pure gold.[2]

## Eutectic reaction

Four eutectic structures: A) lamellar B) rod-like C) globular D) acicular.

The eutectic reaction is defined as follows:[3]

$\text{Liquid} \xrightarrow[\text{cooling}]{\text{eutectic temperature}} \alpha \,\, \text{solid solution} + \beta \,\, \text{solid solution}$

This type of reaction is an invariant reaction, because it is in thermal equilibrium; another way to define this is the Gibbs free energy equals zero. Tangibly, this means the liquid and two solid solutions all coexist at the same time and are in chemical equilibrium. There is also a thermal arrest for the duration of the change of phase during which the temperature of the system does not change.[3]

The resulting solid macrostructure from a eutectic reaction depends on a few factors. The most important factor is how the two solid solutions nucleate and grow. The most common structure is a lamellar structure, but other possible structures include rodlike, globular, and acicular.[4]

## Non-eutectic compositions

Compositions of eutectic systems that are not at the eutectic composition are called either hypoeutectic or hypereutectic. Hypoeutectic compositions are those to the left of the eutectic composition while hypereutectic compositions are those to the right. As the temperature of a non-eutectic composition is lowered the liquid mixture will precipitate one component of the mixture before the other.[3]

## Types

### Alloys

Eutectic alloys have two or more materials and have a eutectic composition. When a non-eutectic alloy solidifies, its components solidify at different temperatures, exhibiting a plastic melting range. Conversely, when a well-mixed, eutectic alloy melts, it does so at a single, sharp temperature. The various phase transformations that occur during the solidification of a particular alloy composition can be understood by drawing a vertical line from the liquid phase to the solid phase on the phase diagram for that alloy.

Some uses include:

## Other critical points

### Eutectoid

Iron-carbon phase diagram, showing the eutectoid transformation between austenite (γ) and pearlite.

When the solution above the transformation point is solid, rather than liquid, an analogous eutectoid transformation can occur. For instance, in the iron-carbon system, the austenite phase can undergo a eutectoid transformation to produce ferrite and cementite, often in lamellar structures such as pearlite and bainite. This eutectoid point occurs at 727 °C (1,341 °F) and about 0.76% carbon.[10]

### Peritectoid

A peritectoid transformation is a type of isothermal reversible reaction that have two solid phases reacting with each other upon cooling of a binary, ternary, ..., $n\!$ alloy to create a completely different and single solid phase.[11] The reaction plays a key role in the order and decomposition of quasicrystalline phases in several alloy types.[12]

### Peritectic

Peritectic transformations are also similar to eutectic reactions. Here, a liquid and solid phase of fixed proportions react at a fixed temperature to yield a single solid phase. Since the solid product forms at the interface between the two reactants, it can form a diffusion barrier and generally causes such reactions to proceed much more slowly than eutectic or eutectoid transformations. Because of this, when a peritectic composition solidifies it does not show the lamellar structure that is found with eutectic solidification.

Such a transformation exists in the iron-carbon system, as seen near the upper-left corner of the figure. It resembles an inverted eutectic, with the δ phase combining with the liquid to produce pure austenite at 1,495 °C (2,723 °F) and 0.17% carbon.

Gold-aluminium phase diagram (German). Top axis title reads "Weight-percent Gold", lower axis title reads "Atomic-percent Gold"

Peritectic decomposition. Up to this point in the discussion transformations have been addressed from the point of view of cooling. They also can be discussed noting the changes that occur to some solid chemical compounds as they are heated. Rather than melting, at the peritectic decomposition temperature, the compound decomposes into another solid compound and a liquid. The proportion of each is determined by the lever rule. The vocabulary changes slightly. Just as the cooling of water, which leads to ice, is termed freezing, the warming of ice leads to melting. In the Al-Au phase diagram, for example, it can be seen that only two of the phases melt congruently, AuAl2 and Au2Al. The rest peritectically decompose.

## Eutectic calculation

The composition and temperature of a eutectic can be calculated from enthalpy and entropy of fusion of each components.[13]

The Gibbs free energy, G, depends on its own differential

$G = H - TS \Rightarrow {\left\{ \begin{array}{l} H = G + TS \\ \\ {\left( {\frac{\partial G}{\partial T}} \right)_P = - S} \end{array} \right.} \Rightarrow H = G - T\left( {\frac{\partial G}{\partial T}} \right)_P .$

Thus, the G/T derivative at constant pressure is calculated by the following equation

$\left( {\frac{\partial G / T}{\partial T}} \right)_P = \frac{1}{T}\left( {\frac{\partial G}{\partial T}} \right)_P - \frac{1}{T^{2}}G = - \frac{1}{T^{2}}\left( {G - T\left({\frac{\partial G}{\partial T}} \right)_P } \right) = - \frac{H}{T^{2}}$

The chemical potential $\mu _{i}$ is calculated if we assume the activity is equal to the concentration.

$\mu _i = \mu _i^\circ + RT\ln \frac{a_i}{a} \approx \mu _i^\circ + RT\ln x_i$

At the equilibrium, $\mu_i =0$, thus $\mu_i^\circ$ is obtained by:

$\mu _i = \mu _i^\circ + RT\ln x_i = 0 \Rightarrow \mu _i^\circ = - RT\ln x_i.$

Using and integrating gives

$\begin{array}{l} \left( {\frac{\partial \mu _i / T}{\partial T}} \right)_P = \frac{\partial }{\partial T}\left( {R\ln x_i } \right) \Rightarrow R\ln x_i = - \frac{H_i ^\circ }{T} + K \\ \\ \end{array}$

The integration constant K may be determined for a pure component with a melting temperature $T^\circ$ and an enthalpy of fusion $H^\circ$ Eq.

$x_i = 1 \Rightarrow T = T_i^\circ \Rightarrow K = \frac{H_i^\circ }{T_i^\circ }$

We obtain a relation that determines the molar fraction as a function of the temperature for each component.

$R\ln x_i = - \frac{H_i ^\circ }{T} + \frac{H_i^\circ }{T_i^\circ }$

The mixture of n components is described by the system

$\begin{array}{l} \left\{ {{\begin{array}{*{20}c} {\ln x_i + \frac{H_i ^\circ }{RT} - \frac{H_i^\circ }{RT_i^\circ } = 0} \\ {\sum\limits_{i = 1}^n {x_i = 1} } \\ \end{array} }} \right. \\ \\ \end{array}$ $\begin{array}{l} \left\{ {{\begin{array}{*{20}c} {\forall i < n \Rightarrow \ln x_i + \frac{H_i ^\circ }{RT} - \frac{H_i^\circ }{RT_i^\circ } = 0} \\ {\ln \left( {1 - \sum\limits_{i = 1}^{n - 1} {x_i } } \right) + \frac{H_n ^\circ }{RT} - \frac{H_n^\circ }{RT_n^\circ } = 0} \\ \end{array} }} \right. \\ \\ \end{array}$

that can be solved by

$\begin{array}{c} \left[ {{\begin{array}{*{20}c} {\Delta x_1 } \\ {\Delta x_2 } \\ {\Delta x_3 } \\ \vdots \\ {\Delta x_{n - 1} } \\ {\Delta T} \\ \end{array} }} \right] = \left[ {{\begin{array}{*{20}c} {1 / x_1 } & 0 & 0 & 0 & 0 & { - \frac{H_1^\circ }{RT^{2}}} \\ 0 & {1 / x_2 } & 0 & 0 & 0 & { - \frac{H_2^\circ }{RT^{2}}} \\ 0 & 0 & {1 / x_3 } & 0 & 0 & { - \frac{H_3^\circ }{RT^{2}}} \\ 0 & 0 & 0 & \ddots & 0 & { - \frac{H_4^\circ }{RT^{2}}} \\ 0 & 0 & 0 & 0 & {1 / x_{n - 1} } & { - \frac{H_{n - 1}^\circ }{RT^{2}}} \\ {\frac{ - 1}{1 - \sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 - \sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 - \sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 - \sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 - \sum\limits_{1 = 1}^{n - 1} {x_i } }} & { - \frac{H_n^\circ }{RT^{2}}} \\ \end{array} }} \right]^{ - 1} .\left[ {{\begin{array}{*{20}c} {\ln x_1 + \frac{H_1 ^\circ }{RT} - \frac{H_1^\circ }{RT_1^\circ }} \\ {\ln x_2 + \frac{H_2 ^\circ }{RT} - \frac{H_2^\circ }{RT_2^\circ }} \\ {\ln x_3 + \frac{H_3 ^\circ }{RT} - \frac{H_3^\circ }{RT_3^\circ }} \\ \vdots \\ {\ln x_{n - 1} + \frac{H_{n - 1} ^\circ }{RT} - \frac{H_{n - 1}^\circ }{RT_{n - 1i}^\circ }} \\ {\ln \left( {1 - \sum\limits_{i = 1}^{n - 1} {x_i } } \right) + \frac{H_n ^\circ }{RT} - \frac{H_n^\circ }{RT_n^\circ }} \\ \end{array} }} \right] \end{array}$

## References

1. ^ Smith & Hashemi 2006, pp. 326–327.
2. ^ http://www.crct.polymtl.ca/fact/phase_diagram.php?file=Ag-Au.jpg&dir=SGTE
3. ^ a b c Smith & Hashemi 2006, p. 327.
4. ^ Smith & Hashemi 2006, pp. 332–333.
5. ^ Muldrew, Ken; Locksley E. McGann (1997). "Phase Diagrams". Cryobiology—A Short Course. University of Calgary. Retrieved 2006-04-29.
6. ^ Senese, Fred (1999). "Does salt water expand as much as fresh water does when it freezes?". Solutions: Frequently asked questions. Department of Chemistry, Frostburg State University. Retrieved 2006-04-29.
7. ^ "Molten salts properties". Archimede Solar Plant Specs.
8. ^ Fichter, Lynn S. (2000). "Igneous Phase Diagrams". Igneous Rocks. James Madison University. Retrieved 2006-04-29.
9. ^ Davies, Nicholas A.; Beatrice M. Nicholas (1992). "Eutectic compositions for hot melt jet inks". US Patent & Trademark Office, Patent Full Text and Image Database. United States Patent and Trademark Office. Retrieved 2006-04-29.
10. ^ Iron-Iron Carbide Phase Diagram Example
11. ^ IUPAC Compendium of Chemical Terminology, Electronic version. "Peritectoid Reaction" Retrieved May 22, 2007.
12. ^ Numerical Model of Peritectoid Transformation. Peritectoid Transformation Retrieved May 22, 2007.
13. ^ International Journal of Modern Physics C, Vol. 15, No. 5. (2004), pp. 675-687

### Bibliography

• Smith, William F.; Hashemi, Javad (2006), Foundations of Materials Science and Engineering (4th ed.), McGraw-Hill, ISBN 0-07-295358-6.


text:
Volume 358 - 36th International Cosmic Ray Conference (ICRC2019) - DM - Dark Matter
The search for dark matter with metastable mediators with the IceCube observatory
C. Tönnis*  on behalf of the IceCube Collaboration
Full text: pdf
Pre-published on: August 20, 2019
Published on: July 02, 2021
Abstract
The IceCube neutrino observatory is a 3D array of photodetectors installed in the Antarctic ice. It consists of 5,160 photomultiplier-tubes spread among 86 vertical strings making a total detector volume of more than a cubic kilometer. It detects neutrinos via Cherenkov light of charged relativistic particles from neutrino interactions with the detector volume. IceCube is, due to its size and photosensor spacing, particularly sensitive to high-energy neutrinos. In this analysis we search for dark matter that annihilates into a metastable mediator that subsequently decays into Standard Model particles. These models yield an enhanced high-energy neutrino flux from dark matter annihilation inside the Sun compared to models without a mediator. Neutrino signals that are produced directly inside the Sun are strongly attenuated at higher energies due to interactions with the solar plasma. In the models considered here, the mediator can escape the Sun before producing any neutrinos, thereby avoiding attenuation. IceCube is ideal to search for this enhanced high-energy neutrino signal. We present the sensitivities of an analysis of six years of IceCube data looking for dark matter in the Sun considering mediator lifetimes between 1 ms to 10 s and dark matter masses between 200 GeV and 10 TeV. We show that IceCube is sensitive to spin--dependent cross--sections of $3.45 \times 10^{-34}~\rm cm^2$ for dark matter masses of 1 TeV.
DOI: https://doi.org/10.22323/1.358.0548
How to cite

Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete.

Open Access