url
stringlengths 14
2.42k
| text
stringlengths 100
1.02M
| date
stringlengths 19
19
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|---|
http://www.amt.edu.au/wfnmc/mc20011.html
|
### An article from "Mathematics Competitions", Vol 14, No 1, 2001
The Influence of Mathematics Competitions on Teaching: Benefits and Dangers
János Surányi
Professor János Surányi has been active in the organisation of mathematics competitions, in mathematics education and the popularisation of mathematics for many years. He restarted the world famous Kozepiskolai Matematikai Lapok (Kömal), and has been actively involved with the Kürschák (formerly the Eotvos Competition. He has been a member of ICMI and other international commissions. His research interests include mathematical logic, number theory and combinatorics.
Professor Surányi receiving his award from Professor Ron Dunkley, President of WFNMC, in Tokyo on Friday 4 August, 2000.
People often find it chic not to like mathematics, even to understand nearly nothing of it. Mathematicians are sometimes characterised as impractical, awkward persons. Quiz-games organised repeatedly by Hungarian television during the 60s on different subjects, including mathematics, proved surprisingly useful in reducing this general repugnance to mathematics. People who hardly understood the questions, let alone the solutions, watched the contesting pairs of students with great excitement. Seeing that normal, lively children, far from the dry-as-dust' type have found the solution in 2 or 3 minutes, these people have lost their proudness for being ignorant of mathematics [1]. Unfortunately, these quiz-games have been discontinued from that period.
Mathematical competitions, organised yearly partly for raising interest in mathematics and partly for the identification of mathematically gifted young people, date back somewhat more than 100 years [2] [3]. According to Freudenthal's report, the first one amongst such contests was the Hungarian Eotvos, later the Kürschák, Competition, first held in 1894 and becoming widely known outside Hungary as well. The collection of problems of these competitions appeared from time to time in foreign languages as well, with solutions and comments, which partly shed some light on the background of some of the problems and partly expounded some fields of mathematics in connection with particular problems [4].
Contests improve the capacity to solve problems, and whoever recognises the pleasure of it and discovers its beauty will enjoy it forever.
As for the benefits of problem solving. Pólya writes in the introduction of his excellent book How to solve it? [5]:
A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest: but if it challenges your curiosity and brings into play your inventive faculties, and you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.
Problem solving as described on Pólya's book creates an attitude which is not easily lost and is helpful in many situations other than mathematical ones.
However, not all gifted children are good at competitions and in particular, not all of them can be winners. Good contest results also depend on having good nerves, quick reaction time and other psychological components as well as talent, and failure can discourage participants, not only from competitions, but from mathematics itself.
Problem solving is an essential component of mathematicization but certainly not the only one. Moreover, emphasis is given to some topics appropriate for good contest problems and for some types of problems, whereas some other fields do not fit into competitions. These are, however, only warnings which must be heeded and taken into consideration in order for competitions to play their beneficial role.
In the case of two-round competitions, the problem of discouragement can be lessened essentially by giving more, say 8, problems in the first round, amongst them sufficiently easy ones but also others which are hard enough to select the participants for the second round. It is not necessary to solve all the problems, the participants should be able to choose to some extent according to their interests.
An overgrowth of competitions - and we see these symptoms in Hungary - and generally on an overdosing of talented youngsters with mathematics can turn them against their first interest and the danger of this occurring can not be neglected.
The emerging different difficulties require skills and effort from the teacher. Without going into details, I will illustrate this by an example and a counterexample in that traditional teaching methods must be changed as well [6].
By traditional teaching, the teacher gives the definition and supports it with some examples, require the pupils to repeat the process and thus believe the students have grasped the concept.
Let us take the natural numbers as a result of counting. Children use these seemingly well, add and subtract in the range of natural numbers. However, an experiment shows that the situation is not completely encouraging. I must admit that I was skeptical initially whether the experiment would provide any difficulty at all to the students. Five match boxes were laid out on a table and before each one was placed a coin. This was shown to 7-8 year old children, one at a time, and the question was asked: Were there more boxes or coins?'. Naturally the answer was: the same.' Afterwards, the boxes were drawn further apart in front of the children while the coins were left unmoved. A large proportion of the children then said that there were more boxes than coins. Some even tried to guess how many more.
Similarly, children often believe that the space objects occupy influences the number of them.
An example in the other direction shows how children develop real notions. A group of first form pupils have discovered negative numbers while practising addition and subtraction climbing and descending stairs. They numbered each step as the stairs led to the cellar, they called the new numbers cellar numbers and denoted them by c_1, c_2, and so on. They have had this notation for some time. They operated with addition and subtraction in this domain quite some time before it was prescribed in the syllabus. When the conventional notation was introduced, this reminded them of the subtraction sign where the cellar numbers' had come from.
Test type competitions have become a world wide movement nowadays, which to my mind, is a completely abortive process. The short time available for the solution of a problem does not allow for what the major aim and benefit of these competitions should be, namely creative thinking. Instead, it reduces the competition to a game of trial and error.
Maybe this type of competition is inevitable where there are hundreds of thousands of competitors, but this is not the case in a small country such as Hungary with only ten million inhabitants.
Nowadays, more and more disciplines demand different domains and aspects of mathematics and this has the danger that the use of mathematics can be reduced to the formal introduction of specific procedures. It would be far preferable instead to have students trained to intelligently read mathematical texts which would enable them to later acquire knowledge not yet taught. For more details see [6].
Mathematical competition problems can also supply the demands of teachers for the provision of various problems and types of problems, generally supplementing the teachers' own collections.
Competitions must be and are generally independent of the teaching practice in the sense that their results should not influence students' marks obtained at school. This is, however, not the case in the assessment of the efficiency of the work of the teacher by some headmasters and sometimes even by the teacher himself. Moreover, having the students sit for mathematics competitions cannot replace the work of the teacher and the demands on him in the classroom: a low level of mathematics teaching seldom produces good mathematicians and good competitors.
In the case of the sufficiently well prepared teachers there is the additional danger that it is not difficult at all and can even be more comfortable for them to train only their best students and not pay much attention to the rest.
The opposite can occur as well: the teacher does not inform his pupils about competitions and student journals in mathematics (described below) as students can then raise difficult problems which lead to much effort and trouble for the teacher in answering these.
I mention several problems which teachers have to solve and to overcome in their everyday work: preparation for the lessons taking into account considerations for the difficulties the students may have understanding; corrections of their papers and so on. All this assumes a thorough grounding and much attention from the teacher. This demands much appreciation of the social as well as educational aspects, a characteristic often lacking.
Significant tools for out of school mathematical education are journals of mathematics for students - if and where they exist. They arouse interest in mathematics quite considerably. In Hungary, the first journal of this type has been published regularly (10 issues a year) since 1895 with some breaks occurring in the two world wars. This journal played an essential part in my becoming a mathematician. This partly explains also why after the Second World War I was for quite a long time the editor of the journal.
I have referred repeatedly to the different tasks and difficulties encountered by teachers. To supplement their work, periodicals for mathematics teachers are also published. These contain, in addition to articles from experts and teachers on different mathematical and didactic questions, problems to solve. Solutions by students are not accepted in this context.
Mathematical competitions are also organised in Hungary for university students. They have 10 days to work at home on usually 10 problems from very different branches of mathematics [7] [8]. Not even the suspicion of consultation with others, either colleagues or experts, has been in evidence since 1950. Here, however, secondary school students are allowed to participate and from time to time some of them receive prizes.
#### References
1. Fried F., Gyarmati E., Surányi J, (1968) Ki Miben Tudós? - (Who is the Expert and in What?), Budapest, p 163 (in Hungarian)
2. Freudenthal H. (Ed), (1969), ICMI Report on Mathematics Contests in Secondary Education (Olympiades) I. {\it Educational Studies in Mathematics}. 2, D. Reidel, Dordrecht, pp 425-440
3. Roman T., (1974), Les Olympiades Mathéematiques en Roumanie, Educational Studies in Mathematics, 5, D. Reidel, Dordrecht, pp 425-440
4. Hajós G, Neukomn G., Surányi J., (1955), Matematikai Versenyttelek. Vol I, Tankonyvkiad\'o, Budapest, Vol II (1929-1963), Vol III Surányi J., (1964-87), Vol IV, Surányi J., (1988-97), TypoTEX, Budapest (1998), Translated into English as Hungarian Problem Books I and II, also in Japanese, Romanian and Russian.
5. Pólya G., (1945), How To Solve It?, Princeton University Press, Princeton
6. Surányi J., (1968), Remarques sur les Taches de l'Enseignment des Math\'ematiques et des Obstacles, Modernisation de l'Enseignment Mathématique dans les Pays Européans, Colloque International UNESCO, Bucharest, pp 104-110
7. Szäsz G., Gehér I., Kovács I., Pintér L., (1996), Contests in Higher Mathematics, Hungary 1949-1961, Akadémiai Kiadó, Budapest
8. Székely G. (Ed), (1996), Contests in Higher Mathematics - Miklós Schweitzer Competition 1962-1991, Springer Verlag, New York.
Professor János Surányi
Zichy Jenóu, 39
Budapest H-1066
HUNGARY
|
2013-06-20 00:01:09
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5040289163589478, "perplexity": 1826.9485282488065}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368709906749/warc/CC-MAIN-20130516131146-00044-ip-10-60-113-184.ec2.internal.warc.gz"}
|
https://www.physicsforums.com/threads/pdes-examples-i-can-do-real-equations-no.84184/
|
# PDE's: examples I can do, real equations No.
1. Aug 4, 2005
### crepincdotcom
Hey all,
I've been working my way through various Calc books I have to reach this point: the infamous Partial Differential. In school I was in Algebra 2 last year due to some track issues shall we say... but I was so set on learning how to use some of the great physics equations in my code such as the Navier-Stokes and Wave equation that I managed to plod my way through differentiation and now have arrived at partials.
I've got the theory well enough, and if you give me an equation of the form f(x,y) = something I can find $$\frac{\partial f(x,y)}{\partial x}$$ or $$\frac{\partial f(x,y)}{\partial y}$$. I was so excited when I checked my answers to some problems in the book and they were right!
I ran to The Oracle (Google) to find a physics equation I could code and see results. I found the Wave equation: the simplest one, so I was told.
Turns out the simplest real equation is beyond me: $$T \frac{\partial^2 y}{\partial x^2} = U \frac{\partial^2 y}{\partial t^2} =$$. (I think I have that correct)
Now: second order equations I could probably handle, but really, what is that thing! I mean, there are TWO sets of $$\frac{\partial^2 }{\partial V^2}$$ (where V is some variable). I mean, the idea is that knowing x and t, you can find the height y of a tiny section of string. I simply have no idea how to go about putting in an X or T, getting an equation in terms or Y, etc. All I know is that through some magic, there are solutions to that equation of the form y(x,t).
Thanks for any help you can give, sorry I have to bother all of you "learned scholars",
-Jack Carrozzo
http://www.crepinc.com/
Jack {[at]} crepinc.com
2. Aug 4, 2005
### Woxor
It would be a mistake to approach this problem by "plugging in" values for x and t; that's a luxury you have when you have solved the problem and discovered what y(x,t) is. However, you DO have to have additional specifications to the problem in order to find out exactly what y(x,t) is. These are called "boundary conditions," or "initial conditions," depending on the application. They consist of equations like "y(0,0) = 3," or "y(-4.5,t) = cos(t)." These equations are necessary to specify the solution completely.
However, it is possible to solve this equation generally, and then plug the boundary conditions in at the end, usually in order to determine values for various constants or other. As a warning: unless you've taught yourself some integration as well, solving differential equations will be pretty difficult.
Here's how you can solve the wave equation:
First assume that y(x,t) can be expressed as the product of a function of x with a function of t; that is, let y(x,t) = f(x)*g(t). Now if we find f and g, we will know y.
Using the product rule, plug f and g into the wave equation:
$$T \frac{\partial^2}{\partial x^2}(f(x)g(t)) = U \frac{\partial^2}{\partial t^2}(f(x)g(t))$$
This simplifies to:
$$Tf''(x)g(t) = Uf(x)g''(t)$$
Rearranging, we get:
$$\frac{g(t)}{g''(t)} = \frac{Uf(x)}{Tf''(x)}$$
Clearly, the left-hand side is a function of t alone (that is, not of x and t), and the right-hand side is a function of x (and not t). So the fact that they equal each other must mean that they are both, in fact, equal to a constant. Call this N. Thus, you have:
$$\frac{g(t)}{g''(t)} = \frac{Uf(x)}{Tf''(x)} = N$$
Rearrange this into two equations:
$$g(t) = Ng''(t)$$
$$f(x) = \frac{NT}{U}f''(x)$$
So our partial differential equation of two variables has reduced to two ordinary differential equations. Given the simple form of these ODEs, you can plug in trig functions of the form $$g(t) := Acos(\sqrt{N}t)+Bsin(\sqrt{N}t)$$ to find the answer to the first one, and use $$f(x) := Ccos(\sqrt{\frac{NT}{U}}x)+sin(\sqrt{\frac{NT}{U}}x)$$ on the second. (You don't have to have another constant in front of the sin function because it's just going to be multiplied by other arbitrary constants anyway -- you'll see what I mean if you work it out).
You have six unknown constants: A, B, C, N, T, and U. You can use the above two equations to find two of them (in terms of the others). This gives you the "general solution" to the PDE.
Then, you'll need about two boundary conditions to further specify the constants, and you'll be left with the fact that T/U can be treated as a single constant, and that any solution y(x,t) will also give you E*y(x,t) as a solution, so that's another degree of freedom right there. Effectively five constants, and four restrictions (two equations, two boundary conditions), with one degree of freedom left over. If you have another boundary/initial condition, you can find y(x,t) completely.
Of course, this is, unfortunately, a simplistic treatment of it. You'd do better to spend a LOT of time researching this type of problem; PDEs are things you have to spend a lot of time on, as opposed to something like an arithmetic problem.
Last edited: Aug 4, 2005
3. Aug 4, 2005
### crepincdotcom
Wow.
I'm going to have to spend some time ripping that apart tonight, thanks a lot for the great response. You suggest that I spend a lot of time researching this, and I agree that that would probably be a good idea. Do you know of a good place to find some info? I've looked around quite a bit on google, and haven't found anything that I either didn't already know or expected that you knew something I hadn't yet learned.
Thanks again,
-Jack Carrozzo
http://www.crepinc.com/
Jack {[at]} crepinc.com
4. Aug 4, 2005
### Woxor
Off-hand, I don't know any specific resources besides MathWorld and http://planetmath.org/encyclopedia [Broken], but the key ideas involved are PDEs, the wave equation, separation of variables (which is what you call the method I used), boundary conditions, and ODEs. That might help a little on your google searches.
By the way, if you haven't studied integration, do that first if at all possible. After that, if you haven't studied any ordinary differential equations, you almost have to do that before starting partial differential equations.
Last edited by a moderator: May 2, 2017
5. Dec 1, 2005
### crepincdotcom
I've spent quite some time since this post's inception working on this.... here's where I'm stuck at the moment.
By what therom, name, or process does this part take place?
I haven't read anywhere about doing this. Is it problam-specific that trig functions happen to work, or a general idea?
Thanks,
-Jack Carrozzo
http://www.crepinc.com/
6. Dec 2, 2005
### Spectre5
Those come from solving the ordinary differential equations of f and g separately. Do you have any experience with solving ODE's? If not, then you should really study them first...such as separation of variables, integrating factor method, etc, etc.
The way you actually get cos and sin in the answer though, come by trying exponential solutions of the form exp(rt) [t is the independent variable]...then you will get equation for r when you put that solution into the ODE. exp(rt) is never zero, so you will end up with an equation for "r". This is known as the auxilary equation. Solve it (quadratic formula, assuming a second order ODE) to get your values of r that satisfy the equation. Sin and Cos come into play becuase you will get imaginery numbers for r, and you then use euler's identity to get the oh-so-familar sin/cos forms.
If you still don't understand this, post back and I will work it out slower.
Last edited: Dec 3, 2005
7. Dec 4, 2005
### Pseudo Statistic
It's somewhat a general idea that trig functions will, after a certain number of derivatives are taken, be pretty much the same as the original function multiplied or divided by constants. (Thanks to our good old friend the chain rule)
Alternatively you could have said that the original function was e^mx, similar to what Spectre5 said, and end up with what we call the "auxillary equation", try to solve it using the quadratic formula, and find that the roots are imaginary...
That's where you're going to have to use sine and cosine-- unless you like keeping things in exponential-imaginary form. :}
Here's an example... let's say you had this, a second order differential equation with constant co-efficients:
y'' + 3y' + 2y = 0
We can guess that the original function will be something along the lines of y = e^mx, then:
y' = me^mx, y'' = m^2 e^mx
Nice, alright, let's plug it into the differential equation:
m^2 e^mx + 3m e^mx + 2 e^mx = 0
Factorizing...
e^mx (m^2 + 3m + 2) = 0
Whaddaya know, a quadratic equation in m...
Because e^mx will never, ever be zero (so far as we're concerned) just pull it out:
m^2 + 3m + 2 = 0
We call this the auxillary equation, solve it using the quadratic formula, you'll find:
m = -2, m = -1
Nice, so we have two ms, but don't we only have one term for m in our proposed solution to the differential equation? Not quite; there are some rules for what the original solution is after solving the auxillary equation:
If the two roots, we'll call them m1 and m2, are unequal and real, then the solution to the differential equation is:
y = Ae^(m1 x) + Be^(m2 x)
Where A and B are some constants you can find thanks to initial-conditions.
If the two roots m1 and m2 are equal, the solution to the differential equation is:
y = (A + Bx)e^mx
If the two roots m1 and m2 are complex conjugates (in the form a +/- bi), you can write the solution as:
y = [e^(ax)][A cos Bx + B sin Bx]
You can indeed test these out and find that they do, indeed, work..
Back to our original question, we found:
m1 = -2, m2 = -1, so the solution is:
y = ae^(-2x) + Be^(-x)
Hope that made a little bit of sense. :rofl:
8. Dec 5, 2005
### crepincdotcom
Thanks a lot, both of you. I'm going to spend a few hours pouring over this tonight to attempt to wrap my mind around it, and see what I come up with.
For clarification, I'm wondering how we can simply "assume" that if we have an ODE of some form that f(x) takes a particular form. Is it that f(x) can take any form we want, given our contraints?
And Yes spectre, I have a (somewhat small) experience with BASIC ODE's, but I can't say I'm exactly comfertable with them. I understand the concept, but generally they way I learn best is to integrate an idea into a program, and I haven't see any good ODE examples I can use for this yet.
Thanks again,
-Jack Carrozzo
http://www.crepinc.com/
9. Dec 5, 2005
### Spectre5
Yes, you can use ANY form of f(x), IF it works as a solution...try some other random forms of f, and you shall find that they do not work (their constants will be forced to be zero, thus creating the already-known, trivial solution, zero).
For example, using Psuedo's example, try a solution of the form
A*x^2 + B*x + C. Plug that in, and you will find that the only constants A, B, and C that make it a solution are A = B = C = 0. Thus y=0. We already knew that, it doesn't help any.
We know to use e^rx by experience (it works for many of these problems, but definitely NOT always!!
(i.e. cauchy-euler equation, bernoulli equation, ricatti equation, etc, etc, etc)
|
2017-11-21 08:56:32
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8279123902320862, "perplexity": 554.738674322947}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806327.92/warc/CC-MAIN-20171121074123-20171121094123-00430.warc.gz"}
|
http://aas.org/archives/BAAS/v25n4/aas183/abs/S4005.html
|
Rotation-Induced Radio Variability from the T Tauri Star HDE283572?
Session 40 -- Young Stars and T Tauri Stars
Display presentation, Thursday, January 13, 9:30-6:45, Salons I/II Room (Crystal Gateway)
## [40.05] Rotation-Induced Radio Variability from the T Tauri Star HDE283572?
J.L.Hand (Kansas University), C.J.Lonsdale, R.B.Phillips (MIT Haystack Observatory)
We report the results of a month-long monitoring observation of the Pre-Main Sequence star HDE283572 (G5 IV). The VLA was used to measure total and polarized intensity at $\lambda$20, 6, and 3.6 cm in 34 snapshots in May and June 1993, to investigate possible correlations of nonthermal radio activity with rotational phase.
Previous claims for stellar rotation - radio correlations were in some cases unconvincing because of sparse phase coverage, or because widely separated radio measurements sampled disparate rotation cycles. We chose the target source for our study HDE283572 for attributes favorable to observing putative modulation, such as an inclination nearly equator-on and a rotation period (1.55 days) that is a non-integral number of terrestrial days, so that observations repeated at the same LST would slew in stellar rotation phase. Observation dates were requested to sample several consecutive rotations periods, in the event radio activity arose in persistent regions anchored in stellar longitude.
Our results suggest that elevated levels of radio emission are absent from one hemisphere of the star, but rigorous statistical tests are in progress. Independently of the search for rotational effects in the radio emission, the observations provide a rich data base of nonthermal radio spectra vs. time and activity level for this rapidly-rotating, newly convective PMS star.
|
2015-08-04 23:02:12
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6151893138885498, "perplexity": 8592.883998018657}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042992201.62/warc/CC-MAIN-20150728002312-00236-ip-10-236-191-2.ec2.internal.warc.gz"}
|
http://mathhelpforum.com/calculus/194827-area-integral-circle.html
|
# Math Help - area integral on a circle
1. ## area integral on a circle
any help for this ...
Given the function f : $f(x,y)=(4-x^2)^{-y^2/2}$
which is on a circle area $D={x^2+y^2\leq 4}
$
with center (0,0) and radius of 2.
compute $\int \int_{D}^{.}f(x,y)$
2. ## Re: area integral on a circle
What kind of help do you want? Do you know how to integrate at all? Integrating over a disc is a fairly standard problem in Calculus. Do you have any idea how to find the limits of integration in xy coordinates? What about polar coordinates?
3. ## Re: area integral on a circle
yes i would like the solution please.
4. ## Re: area integral on a circle
any help ?????????
5. ## Re: area integral on a circle
Originally Posted by kotsos
yes i would like the solution please.
You won't get the solution here. You can get hints, but you are expected to do your OWN work!
As was suggested, try converting everything to polars.
|
2015-02-27 15:41:44
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.911096453666687, "perplexity": 1120.0806844930598}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936461332.16/warc/CC-MAIN-20150226074101-00174-ip-10-28-5-156.ec2.internal.warc.gz"}
|
https://people.maths.bris.ac.uk/~matyd/GroupNames/480i1/C2xD4xD15.html
|
Copied to
clipboard
## G = C2×D4×D15order 480 = 25·3·5
### Direct product of C2, D4 and D15
Series: Derived Chief Lower central Upper central
Derived series C1 — C30 — C2×D4×D15
Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C23×D15 — C2×D4×D15
Lower central C15 — C30 — C2×D4×D15
Upper central C1 — C22 — C2×D4
Generators and relations for C2×D4×D15
G = < a,b,c,d,e | a2=b4=c2=d15=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 3092 in 472 conjugacy classes, 135 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], C5, S3 [×8], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], D5 [×8], C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], D6 [×30], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C22×C4, C2×D4, C2×D4 [×11], C24 [×2], Dic5 [×2], C20 [×2], D10 [×30], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3 [×19], C22×C6 [×2], D15 [×4], D15 [×4], C30, C30 [×2], C30 [×4], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5 [×19], C22×C10 [×2], S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], Dic15 [×2], C60 [×2], D30 [×10], D30 [×20], C2×C30, C2×C30 [×4], C2×C30 [×4], C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4 [×2], D4×C10, C23×D5 [×2], C2×S3×D4, C4×D15 [×4], D60 [×4], C2×Dic15, C157D4 [×8], C2×C60, D4×C15 [×4], C22×D15, C22×D15 [×10], C22×D15 [×8], C22×C30 [×2], C2×D4×D5, C2×C4×D15, C2×D60, D4×D15 [×8], C2×C157D4 [×2], D4×C30, C23×D15 [×2], C2×D4×D15
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C22×S3 [×7], D15, C22×D4, C22×D5 [×7], S3×D4 [×2], S3×C23, D30 [×7], D4×D5 [×2], C23×D5, C2×S3×D4, C22×D15 [×7], C2×D4×D5, D4×D15 [×2], C23×D15, C2×D4×D15
Smallest permutation representation of C2×D4×D15
On 120 points
Generators in S120
(1 73)(2 74)(3 75)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(52 120)(53 106)(54 107)(55 108)(56 109)(57 110)(58 111)(59 112)(60 113)
(1 92 28 107)(2 93 29 108)(3 94 30 109)(4 95 16 110)(5 96 17 111)(6 97 18 112)(7 98 19 113)(8 99 20 114)(9 100 21 115)(10 101 22 116)(11 102 23 117)(12 103 24 118)(13 104 25 119)(14 105 26 120)(15 91 27 106)(31 80 46 65)(32 81 47 66)(33 82 48 67)(34 83 49 68)(35 84 50 69)(36 85 51 70)(37 86 52 71)(38 87 53 72)(39 88 54 73)(40 89 55 74)(41 90 56 75)(42 76 57 61)(43 77 58 62)(44 78 59 63)(45 79 60 64)
(1 73)(2 74)(3 75)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(37 120)(38 106)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)(16 17 18 19 20 21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75)(76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 30)(14 29)(15 28)(31 46)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(91 107)(92 106)(93 120)(94 119)(95 118)(96 117)(97 116)(98 115)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 108)
G:=sub<Sym(120)| (1,73)(2,74)(3,75)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113), (1,92,28,107)(2,93,29,108)(3,94,30,109)(4,95,16,110)(5,96,17,111)(6,97,18,112)(7,98,19,113)(8,99,20,114)(9,100,21,115)(10,101,22,116)(11,102,23,117)(12,103,24,118)(13,104,25,119)(14,105,26,120)(15,91,27,106)(31,80,46,65)(32,81,47,66)(33,82,48,67)(34,83,49,68)(35,84,50,69)(36,85,51,70)(37,86,52,71)(38,87,53,72)(39,88,54,73)(40,89,55,74)(41,90,56,75)(42,76,57,61)(43,77,58,62)(44,78,59,63)(45,79,60,64), (1,73)(2,74)(3,75)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,120)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,30)(14,29)(15,28)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(91,107)(92,106)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)>;
G:=Group( (1,73)(2,74)(3,75)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113), (1,92,28,107)(2,93,29,108)(3,94,30,109)(4,95,16,110)(5,96,17,111)(6,97,18,112)(7,98,19,113)(8,99,20,114)(9,100,21,115)(10,101,22,116)(11,102,23,117)(12,103,24,118)(13,104,25,119)(14,105,26,120)(15,91,27,106)(31,80,46,65)(32,81,47,66)(33,82,48,67)(34,83,49,68)(35,84,50,69)(36,85,51,70)(37,86,52,71)(38,87,53,72)(39,88,54,73)(40,89,55,74)(41,90,56,75)(42,76,57,61)(43,77,58,62)(44,78,59,63)(45,79,60,64), (1,73)(2,74)(3,75)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,120)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,30)(14,29)(15,28)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(91,107)(92,106)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108) );
G=PermutationGroup([(1,73),(2,74),(3,75),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,99),(32,100),(33,101),(34,102),(35,103),(36,104),(37,105),(38,91),(39,92),(40,93),(41,94),(42,95),(43,96),(44,97),(45,98),(46,114),(47,115),(48,116),(49,117),(50,118),(51,119),(52,120),(53,106),(54,107),(55,108),(56,109),(57,110),(58,111),(59,112),(60,113)], [(1,92,28,107),(2,93,29,108),(3,94,30,109),(4,95,16,110),(5,96,17,111),(6,97,18,112),(7,98,19,113),(8,99,20,114),(9,100,21,115),(10,101,22,116),(11,102,23,117),(12,103,24,118),(13,104,25,119),(14,105,26,120),(15,91,27,106),(31,80,46,65),(32,81,47,66),(33,82,48,67),(34,83,49,68),(35,84,50,69),(36,85,51,70),(37,86,52,71),(38,87,53,72),(39,88,54,73),(40,89,55,74),(41,90,56,75),(42,76,57,61),(43,77,58,62),(44,78,59,63),(45,79,60,64)], [(1,73),(2,74),(3,75),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,114),(32,115),(33,116),(34,117),(35,118),(36,119),(37,120),(38,106),(39,107),(40,108),(41,109),(42,110),(43,111),(44,112),(45,113),(46,99),(47,100),(48,101),(49,102),(50,103),(51,104),(52,105),(53,91),(54,92),(55,93),(56,94),(57,95),(58,96),(59,97),(60,98)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,30),(14,29),(15,28),(31,46),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(61,84),(62,83),(63,82),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,90),(71,89),(72,88),(73,87),(74,86),(75,85),(91,107),(92,106),(93,120),(94,119),(95,118),(96,117),(97,116),(98,115),(99,114),(100,113),(101,112),(102,111),(103,110),(104,109),(105,108)])
90 conjugacy classes
class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G ··· 10N 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 15 15 15 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 2 15 15 15 15 30 30 30 30 2 2 2 30 30 2 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4
90 irreducible representations
dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D6 D10 D10 D10 D15 D30 D30 D30 S3×D4 D4×D5 D4×D15 kernel C2×D4×D15 C2×C4×D15 C2×D60 D4×D15 C2×C15⋊7D4 D4×C30 C23×D15 D4×C10 D30 C6×D4 C2×C20 C5×D4 C22×C10 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C10 C6 C2 # reps 1 1 1 8 2 1 2 1 4 2 1 4 2 2 8 4 4 4 16 8 2 4 8
Matrix representation of C2×D4×D15 in GL4(𝔽61) generated by
60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
1 0 0 0 0 1 0 0 0 0 0 1 0 0 60 0
,
60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 1
,
53 23 0 0 38 5 0 0 0 0 1 0 0 0 0 1
,
53 23 0 0 45 8 0 0 0 0 60 0 0 0 0 60
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,1],[53,38,0,0,23,5,0,0,0,0,1,0,0,0,0,1],[53,45,0,0,23,8,0,0,0,0,60,0,0,0,0,60] >;
C2×D4×D15 in GAP, Magma, Sage, TeX
C_2\times D_4\times D_{15}
% in TeX
G:=Group("C2xD4xD15");
// GroupNames label
G:=SmallGroup(480,1169);
// by ID
G=gap.SmallGroup(480,1169);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,185,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^15=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
×
𝔽
|
2021-03-02 11:40:01
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999834299087524, "perplexity": 5711.553660272992}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178363809.24/warc/CC-MAIN-20210302095427-20210302125427-00094.warc.gz"}
|
https://webassets.readthedocs.io/en/latest/custom_filters.html
|
# Creating custom filters¶
Creating custom filters can be easy, or very easy.
Before we get to that though, it is first necessary to understand that there are two types of filters: input filters and output filters. Output filters are applied after the complete content after all a bundle’s contents have been merged together. Input filters, on the other hand, are applied to each source file after it is read from the disk. In the case of nested bundles, input filters will be passed down, with the input filters of a parent bundle are applied before the output filter of a child bundle:
child_bundle = Bundle('file.css', filters='yui_css')
Bundle(child_bundle, filters='cssrewrite')
In this example, because cssrewrite acts as an input filter, what will essentially happen is:
yui_css(cssrewrite(file.css))
To be even more specific, since a single filter can act as both an input and an output filter, the call chain will actually look something like this:
cssrewrite.output(yui_css.output((cssrewrite.input((yui_css.input(file.css)))))
The usual reason to use an input filter is that the filter’s transformation depends on the source file’s filename. For example, the cssrewrite filter needs to know the location of the source file relative to the final output file, so it can properly update relative references. Another example are CSS converters like less, which work relative to the input filename.
With that in mind…
## The very easy way¶
In the simplest case, a filter is simply a function that takes two arguments, an input stream and an output stream.
def noop(_in, out, **kw):
That’s it! You can use this filter when defining your bundles:
bundle = Bundle('input.js', filters=(noop,))
If you are using Jinja2, you can also specify the callable inline, provided that it is available in the context:
{% assets filters=(noop, 'jsmin') ... %}
It even works when using Django templates, although here, you are of course more limited in terms of syntax; if you want to use multiple filters, you need to combine them:
{% assets filters=my_filters ... %}
Just make sure that the context variable my_filters is set to your function.
Note that you currently cannot write input filters in this way. Callables always act as output filters.
## The easy way¶
This works by subclassing webassets.filter.Filter. In doing so, you need to write a bit more code, but you’ll be able to enjoy a few perks.
The noop filter from the previous example, written as a class, would look something like this:
from webassets.filter import Filter
class NoopFilter(Filter):
name = 'noop'
def output(self, _in, out, **kwargs):
def input(self, _in, out, **kwargs):
The output and input methods should look familiar. They’re basically like the callable you are already familiar with, simply pulled inside a class.
Class-based filters have a name attribute, which you need to set if you want to register your filter globally.
The input method will be called for every source file, the output method will be applied once after a bundle’s contents have been concatenated.
Among the kwargs you currently receive are:
• source_path (only for input()): The filename behind the in stream, though note that other input filters may already have transformed it.
• output_path: The final output path that your filters work will ultimatily end up in.
Note
Always make your filters accept arbitrary **kwargs. The API does allow for additional values to be passed along in the future.
### Registering¶
The name wouldn’t make much sense, if it couldn’t be used to reference the filter. First, you need to register the class with the system though:
from webassets.filter import register_filter
register_filter(NoopFilter)
Or if you are using yaml then use the filters key for the environment:
directory: .
url: /
debug: True
updater: timestamp
filters:
- my_custom_package.my_filter
After that, you can use the filter like you would any of the built-in ones:
{% assets filters='jsmin,noop' ... %}
### Options¶
Class-based filters are used as instances, and as such, you can easily define a __init__ method that takes arguments. However, you should make all parameters optional, if possible, or your filter will not be usable through a name reference.
There might be another thing to consider. If a filter is specified multiple times, which sometimes can happen unsuspectingly when bundles are nested within each other, it will only be applied a single time. By default, all filters of the same class are considered the same. In almost all cases, this will be just fine.
However, in case you want your filter to be applicable multiple times with different options, you can implement the unique method and return a hashable object that represents data unique to this instance:
class FooFilter(Filter):
def __init__(self, *args, **kwargs):
self.args, self.kwargs = args, kwargs
def unique(self):
return self.args, self.kwargs
This will cause two instances of this filter to be both applied, as long as the arguments given differ. Two instances with the exact same arguments will still be considered equal.
If you want each of your filter’s instances to be unique, you can simply do:
def unique(self):
return id(self)
### Useful helpers¶
The Filter base class provides some useful features.
#### setup()¶
It’s quite common that filters have dependencies - on other Python libraries, external tools, etc. If you want to provide your filter regardless of whether such dependencies are matched, and fail only if the filter is actually used, implement a setup() method on your filter class:
class FooFilter(Filter):
def setup(self):
import foolib
self.foolib = foolib
def apply(self, _in, out):
self.foolib.convert(...)
#### options¶
Some filters will need to be configured. This can of course be done by passing arguments into __init__ as explained above, but it restricts you to configuring your filters in code, and can be tedious if necessary every single time the filter is used.
In some cases, it makes more sense to have an option configured globally, like the path to an external binary. A number of the built-in filters do this, allowing you to both specify a config variable in the webassets Environment instance, or as an OS environment variable.
class FooFilter(Filter):
options = {
'binary': 'FOO_BIN'
}
If you define a an options attribute on your filter class, these options will automatically be supported both by your filter’s __init__, as well as via a configuration or environment variable. In the example above, you may pass binary when creating a filter instance manually, or define FOO_BIN in Environment.config, or as an OS environment variable.
#### get_config()¶
In cases where the declarative approach of the options attribute is not enough, you can implement custom options yourself using the Filter.get_config() helper:
class FooFilter(Filter):
def setup(self):
self.bin = self.get_config('BINARY_PATH')
This will check first the configuration, then the environment for BINARY_PATH, and raise an exception if nothing is found.
get_config() allows you to specify different names for the setting and the environment variable:
self.get_config(setting='ASSETS_BINARY_PATH', env='BINARY_PATH')
It also supports disabling either of the two, causing only the other to be checked for the given name:
self.get_config(setting='ASSETS_BINARY_PATH', env=False)
Finally, you can easily make a value optional using the require parameter. Instead of raising an exception, get_config() then returns None. For example:
self.java = self.get_config('JAVA_BIN', require=False) or 'java'
### Abstract base classes¶
In some cases, you might want to have a common base class for multiple filters. You can make the base class abstract by setting name to None explicitly. However, this is currently only relevant for the built-in filters, since your own filters will not be registered automatically in any case.
## More?¶
You can have a look inside the webassets.filter module source code to see a large number of example filters.
Assets can be filtered through one or multiple filters, modifying their contents (think minification, compression).
|
2018-11-21 08:36:24
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30477410554885864, "perplexity": 2252.800559163224}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039747369.90/warc/CC-MAIN-20181121072501-20181121094501-00464.warc.gz"}
|
https://www.codecademy.com/courses/probability-mssp/lessons/introduction-to-probability-distributions/exercises/probability-mass-functions
|
Learn
A probability mass function (PMF) is a type of probability distribution that defines the probability of observing a particular value of a discrete random variable. For example, a PMF can be used to calculate the probability of rolling a three on a fair six-sided die.
There are certain kinds of random variables (and associated probability distributions) that are relevant for many different kinds of problems. These commonly used probability distributions have names and parameters that make them adaptable for different situations.
For example, suppose that we flip a fair coin some number of times and count the number of heads. The probability mass function that describes the likelihood of each possible outcome (eg., 0 heads, 1 head, 2 heads, etc.) is called the binomial distribution. The parameters for the binomial distribution are:
• n for the number of trials (eg., n=10 if we flip a coin 10 times)
• p for the probability of success in each trial (probability of observing a particular outcome in each trial. In this example, p= 0.5 because the probability of observing heads on a fair coin flip is 0.5)
If we flip a fair coin 10 times, we say that the number of observed heads follows a Binomial(n=10, p=0.5) distribution. The graph below shows the probability mass function for this experiment. The heights of the bars represent the probability of observing each possible outcome as calculated by the PMF.
### Instructions
Let’s see how the shape of the binomial distribution changes as the sample size changes.
Use the slider to change the value of x fair coin flips, between one and ten. The heights of the resulting bars represent the probability of observing different values of heads from x number of fair coin flips. You can roll your cursor over each bar and see the actual numeric value of the bar height. Taller bars represent more likely outcomes.
Notice that as x increases, the bars get smaller. This is because the sum of the heights of all the bars will always equal 1. So when x is larger, the number of heads we can observe increases, and the probability needs to be divided between more values.
|
2022-07-05 05:44:52
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8807427883148193, "perplexity": 193.13639442023495}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104514861.81/warc/CC-MAIN-20220705053147-20220705083147-00185.warc.gz"}
|
https://proofwiki.org/wiki/Preimage_of_Relation_is_Subset_of_Domain
|
Preimage of Relation is Subset of Domain
Theorem
Let $\mathcal R \subseteq S \times T$ be a relation.
Then the preimage of $\mathcal R$ is a subset of its domain:
$\Preimg {\mathcal R} \subseteq S$
Proof
The preimage of $\mathcal R$ is defined as:
$\Preimg {\mathcal R} = \set {s \in \Dom {\mathcal R}: \exists t \in \Rng {\mathcal R}: \tuple {s, t} \in \mathcal R}$
Hence:
$\displaystyle s$ $\in$ $\displaystyle \Preimg {\mathcal R}$ $\displaystyle \leadsto \ \$ $\displaystyle s$ $\in$ $\displaystyle \Dom {\mathcal R}$ Definition of Preimage of Relation $\displaystyle \leadsto \ \$ $\displaystyle \Preimg {\mathcal R}$ $\subseteq$ $\displaystyle \Dom {\mathcal R}$ Subset of Set with Propositional Function
$\blacksquare$
|
2019-10-18 09:54:28
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9662222266197205, "perplexity": 277.1977250008945}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986679439.48/warc/CC-MAIN-20191018081630-20191018105130-00540.warc.gz"}
|
https://math.stackexchange.com/questions/3554290/finding-lim-frac3n4n
|
# Finding $\lim \frac{3^n}{4^n}$
$$\lim_{n \to \infty} \frac{3^n}{4^n}$$
I know the limit is zero because the denominator grows faster than the numerator in this case... although I still get infinity over infinity.
How do I "show" that the limit is zero? L'Hopital's rule is redundant in this case and doing $$\lim e^{n\ln(3/4)}$$ doesn't help.
• A standard $\epsilon$-$N$ argument should suffice. – JMoravitz Feb 20 '20 at 21:14
• Hint: Write the expression as $\left( \frac 34\right)^n$. Note: you should make it clear that you mean $\lim_{n\to \infty}$. – lulu Feb 20 '20 at 21:15
• @JMoravitz What's that? – Segmentation fault Feb 20 '20 at 21:15
• Well, $\frac{3^n}{4^n} = (\frac{3}{4})^n$ and $\frac{3}{4} < 1$... – PrudiiArca Feb 20 '20 at 21:15
• @InterstellarProbe Why though? – Segmentation fault Feb 20 '20 at 21:16
I don't think Lhopital's rule is redundant.
Call the limit $$L$$, which exists because the sequence is bounded and decreasing.
The derivative of $$b^x$$ is $$\ln bb^x$$.
We get $$L=\ln3/\ln4L\implies L=0$$.
Also, $$e^{n\ln(3/4)}\to0$$, since $$\ln(3/4)$$ is negative.
One textbook proceeds like this.
(1) Bernoulli's inequality: $$(1+x)^n \ge 1+nx\qquad\text{when } x > 0, n \in \mathbb N$$ Hint: induction.
(2) Use (1) to show $$t^n \to \infty\quad\text{as } n \to \infty, \text{when } t > 1$$ Hint: use $$1+x=t$$.
(3) Use (2) to show $$s^n \to 0\quad\text{as } n \to \infty, \text{when } 0 Hint: use $$t=1/s$$.
• +1 too same reason – Satyendra Feb 20 '20 at 22:01
The sequence $$x_{n}=\left(\frac{3}{4}\right)^n$$ has recursive definition $$x_0=1, x_{n+1}=\frac{3}{4}x_n.$$
$$\{x_n\}$$ is decreasing and bounded below by zero. So it has a limit, $$L.$$
But $$L=\lim_{n\to\infty} x_{n+1} = \frac{3}{4}\lim_{n\to\infty} x_{n}=\frac{3}{4}L.$$
So $$L=0.$$
So you really only need that a decreasing sequence bound below has a limit, and simple properties of limits.
Perhaps like this:
$$\dfrac{1}{(4/3)^n}=\dfrac{1}{(1+1/3)^n}<$$
$$\dfrac{1}{1+n/3}<\dfrac{1}{n/3} =3/n.$$
Used: $$(1+x)^n \gt 1+xn$$, for $$x>0$$, $$n$$ positive integer
• +1 Nice answer with Bernouilli's inequality. – Satyendra Feb 20 '20 at 21:59
• LostinSpace.Thanks. – Peter Szilas Feb 20 '20 at 22:04
HINT
Note $$3^n/4^n = (3/4)^n$$ so let $$\epsilon > 0$$. Can you find $$N$$ such that $$(3/4)^n < \epsilon$$ for all $$n > N$$?
This would show that the positive sequence $$(3/4)^n$$ gets arbitrarily small, which is the definition of convergence to $$0$$.
• No, how do I find it? – Segmentation fault Feb 20 '20 at 21:16
• @SilenceOnTheWire $(3/4)^n < \epsilon \iff n > \frac{\ln \epsilon}{\ln (3/4)}$... – gt6989b Feb 20 '20 at 21:17
• @gt6989b $\ln(3/4)<0$, so it flips the direction of the inequality: $$\left(\dfrac{3}{4}\right)^n < \epsilon \Longrightarrow n > \dfrac{\ln \epsilon}{\ln 3 - \ln 4}$$ – InterstellarProbe Feb 20 '20 at 21:20
• @InterstellarProbe +1, thanks, fixed mine, but yours is also formatted better, don't delete it... – gt6989b Feb 20 '20 at 21:22
|
2021-01-18 20:06:43
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 31, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9495042562484741, "perplexity": 2193.8849887390543}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703515235.25/warc/CC-MAIN-20210118185230-20210118215230-00308.warc.gz"}
|
https://motls.blogspot.com/2008/03/wmap-five-year-data-released.html?m=1
|
## Thursday, March 06, 2008
### WMAP: five-year data released
It's been two years since the publication of the WMAP three-year data. As some of the readers know, 3+2=5 so the only sensible thing that the WMAP team can publish right now are the five-year data.
WMAP 5: main page
WMAP 5: papers
WMAP 5: Phil Plait
WMAP 5: Sean Carroll
Theoretical physicists and non-experimenters will primarily care about the cosmological interpretation.
Don't expect a revolution here. Some numbers got a little bit more accurate, following Lord Kelvin's prescription for the completion of science. For example,
• the Hubble constant is 70.1 +- 1.3 km/s per Mpc
• the relative density of dark energy is 72.1 +- 1.5 percent
• the relative density of dark matter is 23.3 +- 1.3 percent
• the relative density of baryonic matter is 4.6 +- 0.2 percent
• the age of the Universe is 13.73 +- 0.12 billion years
• the global mean temperature dropped to 2.725 K by now (outside the urban heat islands such as stars, plus minus the fractions of the degree from the image above), confirming worries about global cooling ;-)
• the recombination occurred on 375,900th (+- 3,100) birthday
Interestingly enough, the team can exclude the existence of very massive neutrino species: the total sum of the neutrino masses is below 0.61 eV, at the 95% confidence level. Entertainingly enough, they have a non-particle-physics measurement of the number of neutrino species: it is 4.4 +- 1.5 which makes the value 3 tolerable. ;-)
Tensor-to-scalar ratio is below 20% at 95% confidence level, making theories with gravity waves and with the spectral index above one disfavored.
#### 1 comment:
1. see for better understanding: the big bang symmetrical splitting black hole in fractal shape and the small copy of splitting black holes inside Carina and Eagle nebulae.
http://bigbang-entanglement.blogspot.com/
|
2018-07-23 07:51:49
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43751779198646545, "perplexity": 2601.5281274115086}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676595531.70/warc/CC-MAIN-20180723071245-20180723091245-00322.warc.gz"}
|
http://mathhelpforum.com/calculus/173615-proving-inequality.html
|
# Math Help - proving an inequality
1. ## proving an inequality
i tried differentiatin both sides, i did that 4 times and i still couldnt find the pattern that gets me to sin any idea how to solve this ??
2. Did you compute $\dfrac{d^n}{dx^n}\left(\dfrac 1{x-i}\right)$ and $\dfrac{d^n}{dx^n}\left(\dfrac 1{x+i}\right)$ ?
3. I think we are using the same book! The last exercise of Chapter IX, right?
Originally Posted by PeaceSoul
i tried differentiatin both sides, i did that 4 times and i still couldnt find the pattern that gets me to sin any idea how to solve this ?
Show that $\displaystyle \frac{d^n}{dx^n} \left(\frac{1}{x^2+1} \right) = \frac{i(-1)^nn!}{2(x+i)^{n+1}}+\frac{i(-1)^{n+1}n!}{2(x-i)^{n+1}}$.
I avoid the differentiation mess by using the following formula:
$\displaystyle \frac{d^n}{dx^n} (ax+b)^m = a^n\left(ax+b\right)^{m-n}\prod_{1 \le k \le n}\left(m-k+1\right)$.
4. yes i did that, but i got no where,,please guide me im really lost :S
5. $\displaystyle \frac{d^n}{dx^n} \left(\frac{1}{x^2+1} \right) = \frac{i(-1)^nn!}{2(x+i)^{n+1}}+\frac{i(-1)^{n+1}n!}{2(x-i)^{n+1}} = (-1)^nn!\bigg(\frac{i}{2(x+i)^{n+1}}-\frac{i}{2(x-i)^{n+1}}\bigg)$.
Denote the stuff under the brackets by $S$. Now, let $x = \cot{\theta}$, then we have:
\begin{aligned} \displaystyle S & = \frac{i}{2(\cot{\theta}+i)^{n+1}}-\frac{i}{2(\cot{\theta}-i)^{n+1}} = \frac{i}{2\left(\frac{\cos{\theta}}{\sin{\theta}}+ i\right)^{n+1}}-\frac{i}{2\left(\frac{\cos{\theta}}{ \sin{\theta}}-i\right)^{n+1}} \\ & = \frac{i}{\frac{2}{ \sin^{n+1}{\theta}}\left(\cos{\theta}+i\sin{\theta }\right)^{n+1}}-\frac{i}{\frac{2}{ \sin^{n+1}{\theta}}\left(\cos{\theta}-i\sin{\theta}\right)^{n+1}}\\ & = \frac{i\sin^{n+1}\left(\cos{\theta}+i\sin{\theta}\ right)^{-n-1}}{2}-\frac{i\sin^{n+1}\left(\cos{\theta}-i\sin{\theta}\right)^{-n-1}}{2} \end{aligned}
Recall that $\left(\cos{\varphi}+i\sin{\varphi}\right)^{t} = \cos{t\varphi} +i\sin{t\varphi}$ (De Moivre's theorem), then:
\displaystyle \begin{aligned} S & = \frac{i\sin^{n+1}\left(\cos[(-n-1)\theta]+i\sin[(-n-1)\theta]\right)}{2}-\frac{i\sin^{n+1}\left(\cos[(-n-1)\theta]-i\sin[(-n-1)\theta]\right)}{2} \\ & = \frac{i\sin^{n+1}\left(\cos[(n+1)\theta]-i\sin[(n+1)\theta]\right)}{2}-\frac{i\sin^{n+1}\left(\cos[(n+1)\theta]+i\sin[(n+1)\theta]\right)}{2} \\& = \frac{i\sin^{n+1}{\theta}\cos[(n+1)\theta)-i^2sin[(n+1)\theta]-i\sin^{n+1}\cos[(n+1)\theta]-i^2\sin^{n+1}{\theta}\sin[(n+1)\theta]}{2} \\ & = \frac{-2i^2\sin^{n+1}\sin[(n+1)\theta]}{2}= -(-1)\sin^{n+1}{\theta}\sin[(n+1)\theta] = \sin^{n+1}{\theta}\sin[(n+1){\theta}]. \end{aligned}
Thus $S = \sin^{n+1}{\theta}\sin[(n+1)\theta]$, therefore $\displaystyle \frac{d^n}{dx^n} \left(\frac{1}{x^2+1} \right) = (-1)^nn!\sin^{n+1}{\theta}\sin[(n+1)\theta]$.
6. Thank you for an amazinggggggggggggg answer !!!!!! You rock
|
2015-04-18 08:49:30
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9994427561759949, "perplexity": 2652.799682602858}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246634257.45/warc/CC-MAIN-20150417045714-00175-ip-10-235-10-82.ec2.internal.warc.gz"}
|
https://calculator.academy/cost-per-unit-calculator/
|
Enter the total number of units and the total cost into the calculator to determine the cost per unit.
## Cost Per Unit Formula
The following equation is used to calculate the Cost Per Unit.
CPU = TC / TU
• Where CPU is the cost per unit ($/unit) • TC is the total cost ($)
• TU is the total number of units
To calculate the cost per unit, simply divide the total cost by the total number of units.
## What is a Cost Per Unit?
Definition:
A cost per unit is a metric used to describe the cost to produce, purchase, buy, etc one unit of anything.
This could be the cost per unit of a manufacturing plant to make a product or a store’s cost to purchase one unit of product for their shelves. It can be used in any manner of way.
## How to Calculate Cost Per Unit?
Example Problem:
The following example outlines the steps and information needed to calculate the cost per unit.
In this example, we are looking at a manufacturing plant that made a certain number of product of a 1 month period.
First, determine the number of units produced. For this problem, the plant was able to produce 30,000 units in this 1 month period.
Next, determine the total cost of those units. This would include material costs, overhead, labor, and other cost. In this case, the total cost is $40,000.00. Finally, calculate the cost per unit using the equation above: CPU = TC / TU CPU = 40,000 / 30,000 CPU =$1.33 /unit
|
2023-02-09 02:51:34
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4933337867259979, "perplexity": 702.1299744116575}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764501066.53/warc/CC-MAIN-20230209014102-20230209044102-00869.warc.gz"}
|
https://samacheerguru.com/samacheer-kalvi-12th-maths-solutions-chapter-1-ex-1-1/
|
# Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.1
## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.1
Exercise 1.1 Class 12 Maths State Board Question 1.
Find the adjoint of the following:
Solution:
12th Maths Exercise 1.1 Answers Question 2.
Find the inverse (if it exists) of the following:
Solution:
For a matrix A, $$\mathrm{A}^{-1}=\frac{1}{|\mathrm{A}|}(\mathrm{adj} \mathrm{A})$$. Where |A| ≠ 0. If |A| = 0 then A is called a singular matrix and so $$\mathrm{A}^{-1}$$ does not exist.
12th Maths Exercise 1.1 Question 3.
If F(α) = $$\left[\begin{array}{ccc}{\cos \alpha} & {0} & {\sin \alpha} \\ {0} & {1} & {0} \\ {-\sin \alpha} & {0} & {\cos \alpha}\end{array}\right]$$ show that $$[\mathrm{F}(\alpha)]^{-1}=\mathrm{F}(-\alpha)$$
Solution:
Let A = F (α)
So $$[\mathrm{F}(\alpha)]^{-1}=\mathrm{A}^{-1}$$
Now
12th Maths Chapter 1 Exercise 1.1 Question 4.
If A = $$\left[\begin{array}{cc}{5} & {3} \\ {-1} & {-2}\end{array}\right]$$ show that A2 – 3A – 7I2 = O2. Hence find A-1.
Solution:
A = $$\left[\begin{array}{cc}{5} & {3} \\ {-1} & {-2}\end{array}\right]$$
To Find A-1
Now we have proved that A2 – 3A – 7I2 = O2
Post multiply by A-1 we get
A – 3I – 7A-1 = O2
If $$\mathbf{A}=\frac{1}{9}\left[\begin{array}{ccc}{-8} & {1} & {4} \\ {4} & {4} & {7} \\ {1} & {-8} & {4}\end{array}\right]$$ prove that A-1 = AT
Solution:
If $$\mathbf{A}=\left[\begin{array}{rr}{8} & {-4} \\ {-5} & {3}\end{array}\right]$$, verify that A(adj A) = (adj A)A = |A| I2
Solution:
Samacheer Kalvi 12 Maths Solutions Question 7.
If $$\mathbf{A}=\left[\begin{array}{ll}{3} & {2} \\ {7} & {5}\end{array}\right]$$, and $$\mathbf{B}=\left[\begin{array}{cc}{-1} & {-3} \\ {5} & {2}\end{array}\right]$$ verify that (AB)-1 = B-1 A-1.
Solution:
Samacheer Kalvi Guru 12th Maths Question 8.
If adj (A) = $$\left[\begin{array}{ccc}{2} & {-4} & {2} \\ {-3} & {12} & {-7} \\ {-2} & {0} & {2}\end{array}\right]$$ find A
Solution:
If adj(A) = $$\left[\begin{array}{ccc}{0} & {-2} & {0} \\ {6} & {2} & {-6} \\ {-3} & {0} & {6}\end{array}\right]$$ find A-1
Solution:
Samacheer Kalvi 12th Maths Guide Question 10.
Find adj(adj(A)) if adj A = $$\left[\begin{array}{ccc}{1} & {0} & {1} \\ {0} & {2} & {0} \\ {-1} & {0} & {1}\end{array}\right]$$
Solution:
12th Samacheer Maths Solutions Question 11.
Solution:
12th Maths 1st Chapter Exercise 1.1 Question 12.
Find the matrix A for which A $$\left[\begin{array}{cc}{5} & {3} \\ {-1} & {-2}\end{array}\right]=\left[\begin{array}{cc}{14} & {7} \\ {7} & {7}\end{array}\right]$$
Solution:
Given A $$\left[\begin{array}{cc}{5} & {3} \\ {-1} & {-2}\end{array}\right]=\left[\begin{array}{cc}{14} & {7} \\ {7} & {7}\end{array}\right]$$
Let $$\mathrm{B}=\left(\begin{array}{cc}{5} & {3} \\ {-1} & {-2}\end{array}\right) \text { and } \mathrm{C}=\left(\begin{array}{cc}{14} & {7} \\ {7} & {7}\end{array}\right)$$
Given AB = C, To find A
Now AB = C
Post multiply by B-1 on both sides
ABB-1 = CB-1 (i.e) A (BB-1) = CB-1
⇒ A(I) = CB-1 (i.e) A = CB-1
To find B-1:
12th Maths Solution Book Question 13.
Given $$\mathbf{A}=\left[\begin{array}{cc}{1} & {-1} \\ {2} & {0}\end{array}\right], \mathbf{B}=\left[\begin{array}{cc}{3} & {-2} \\ {1} & {1}\end{array}\right] \text { and } \mathbf{C}\left[\begin{array}{ll}{1} & {1} \\ {2} & {2}\end{array}\right]$$, find a matrix X such that AXB = C.
Solution:
A × B = C
Pre multiply by A-1 and post multiply by B-1 we get
A-1 A × BB-1 = A-1CB-1 (i.e) X = A-1CB-1
Maths Guide For Class 12 Question 14.
Solution:
Maths 12th Guide Question 15.
Decrypt the received encoded message $$\left[\begin{array}{cc}{2} & {-3}\end{array}\right]\left[\begin{array}{ll}{20} & {4}\end{array}\right]$$ with the encryption matrix $$\left[\begin{array}{cc}{-1} & {-1} \\ {2} & {1}\end{array}\right]$$ and the decryption matrix as its inverse, where the system of codes are described by the numbers 1-26 to the letters A- Z respectively, and the number 0 to a blank space.
Solution:
Let the encoding matrix be $$\left[\begin{array}{cc}{-1} & {-1} \\ {2} & {1}\end{array}\right]$$
So the sequence of decoded matrices is [8 5], [12 16].
### Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.1 Additional Problems
Using elementary transformations find the inverse of the following matrix
Solution:
Using elementary transformations find the inverse of the matrix
Solution:
12th Maths Guide Samacheer Kalvi Question 3.
Using elementary transformation find the inverse of the matrix
Solution:
12th Maths Book Back Answers Question 4.
Using elementary transformations find the inverse of the matrix
Solution:
12th Maths Samacheer Solutions Question 5.
Using elementary transformation, find the inverse of the following matrix
Solution:
12th Samacheer Maths Solution Question 6.
Solution:
Solution:
12th Maths Book Back Questions With Answers Question 8.
Solution:
12th Samacheer Maths Guide Question 9.
Solution:
|
2021-05-08 21:01:37
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8577356934547424, "perplexity": 6461.611577747514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988923.22/warc/CC-MAIN-20210508181551-20210508211551-00193.warc.gz"}
|
https://zbmath.org/?q=an:0288.12005
|
# zbMATH — the first resource for mathematics
Solution of the class number two problem for cyclotomic fields. (English) Zbl 0288.12005
It is shown that the only cyclotomic fields of the form $$\mathbb Q(e^{2\pi i/m})$$ which have class number two are $$\mathbb Q(e^{2\pi i/39})$$ and $$\mathbb Q(e^{2\pi i/56})$$. Methods are the same as used in solving the class number one problem [the author and H. L. Montgomery, J. Reine Angew. Math. 286/287, 248–256 (1976; Zbl 0335.12013)].
Reviewer: John Myron Masley
##### MSC:
11R29 Class numbers, class groups, discriminants 11R18 Cyclotomic extensions 11R42 Zeta functions and $$L$$-functions of number fields
Full Text:
##### References:
[1] Baker, A.: Linear forms in the logarithms of algebraic numbers. Mathematika13, 204?216 (1966) · Zbl 0161.05201 [2] Baker, A.: Imaginary quadratic fields with class number 2. Annals of Math.94, 139?152 (1971) · Zbl 0219.12008 [3] Baker, A., Stark, H.M.: On a fundamental inequality in number theory. Annals of Math.94, 190?199 (1971) · Zbl 0219.12009 [4] Bauer, H.: Numerische Bestimmung von Klassenzahlen reeller zyklischer Zahlkörper. J. of Number Theory1, 161?162 (1969) · Zbl 0167.32301 [5] Carlitz, L.: A characterization of algebraic number fields with class number two. Proc. AMS11, 391?392 (1960) · Zbl 0202.33101 [6] Hasse, H.: Über die Klassenzahl aberscher Zahlkörper. Berlin: Academic Verlag 1952 · Zbl 0046.26003 [7] Iwasawa, K.: A note on class numbers of algebraic number fields. Abh. Math. Sem. Univ. Hamburg20, 257?258 (1956) · Zbl 0074.03002 [8] Iwasawa, K.: A note on ideal class groups. Nagoya Math. J.27, 239?247 (1966) · Zbl 0139.28104 [9] Masley, J.: On the class number of cyclotomic fields. Dissertation, Princeton Univ. 1972 [10] Masley, J., Montgomery, H.L.: Unique factorization in cyclotomic fields. To appear · Zbl 0335.12013 [11] Metsankyla, T.: On prime factors of the relative class numbers of cyclotomic fields. Ann. Univ. Turku. Ser A I, 149 (1971) [12] Metsankyla, T.: On the growth of the first factor of the cyclotomic class number. Ann. Univ. Turku. Ser A I, 155 (1972) [13] Montgomery, H.L., Weinberger, P.: Notes on small class numbers, to appear · Zbl 0285.12004 [14] Schrutka v. Rechtenstamm, G.: Tabelle der (relativ.) Klassenzahlen von Kreiskörpern, Abh. Deutsche Akad. Wiss. Berlin, 1964, Math. Nat. K1. Nr. 2 · Zbl 0199.09803 [15] Stark, H. M.: A complete determination of the complex quadratic fields of class-number one. Mich. Math. J.14, 1?27 (1967) · Zbl 0148.27802 [16] Stark, H. M.: A transcendence theorem for class-number problems. Annals of Math.94, 153?173 (1971) · Zbl 0229.12010
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.
|
2021-10-21 21:29:33
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.683415949344635, "perplexity": 6256.157880044394}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585441.99/warc/CC-MAIN-20211021195527-20211021225527-00307.warc.gz"}
|
https://en.wikibooks.org/wiki/Econometric_Theory/Asymptotic_Convergence
|
# Econometric Theory/Asymptotic Convergence
## Asymptotic Convergence
### Modes of Convergence
#### Convergence in Probability
Convergence in probability is going to be a very useful tool for deriving asymptotic distributions later on in this book. Alongside convergence in distribution it will be the most commonly seen mode of convergence.
##### Definition
A sequence of random variables $\{ X_n ; n=1,2, \cdots \}$ converges in probability to $X_{ }$ if:
$\forall \epsilon, \delta >0,$ $\exists N \; \operatorname{s.t.} \; \forall n \geq N,$ $\Pr \{ |X_n - X| > \delta \}< \epsilon$
an equivalent statement is:
$\forall \delta >0,$ $\lim_{n \to \infty} \Pr \{ |X_n - X| > \delta \}=0$
This will be written as either $X_n \begin{matrix} \begin{matrix} { }_p \\ \longrightarrow \\{ } \end{matrix} \end{matrix} X$ or $\operatorname{plim} X_n = X$.
##### Example
$X_n = \begin{cases} \eta & 1- \begin{matrix} \frac{1}{n} \end{matrix} \\ \theta & \begin{matrix} \frac{1}{n} \end{matrix} \end{cases}$
We'll make an intelligent guess that this series converges in probability to the degenerate random variable $\eta$. So we have that:
$\forall \delta >0,\; \Pr \{ |X_n - \eta| > \delta \} \leq \Pr \{ |X_n - \eta| > 0 \}= \Pr \{ X_n= \theta \}= \begin{matrix} \frac{1}{n} \end{matrix}$
Therefore our definition for convergence in probability in this case is:
$\forall \epsilon , \delta >0,$ $\exists N \quad \operatorname{s.t.} \forall n \geq N,$ $\Pr \{ |X_n - \eta | > \delta \} \leq \Pr \{ |X_n - \eta | > 0 \}=\Pr \{ X_n= \theta \}= \begin{matrix} \frac{1}{n} \end{matrix} < \epsilon$
So for any positive values of $\epsilon \in \mathbb{R}$ we can always find an $N \in \mathbb{N}$ large enough so that our definition is satisfied. Therefore we have proved that $X_n \begin{matrix} { }_p \\ \longrightarrow \\{ } \end{matrix} \eta$.
#### Convergence Almost Sure
Almost-sure convergence has a marked similarity to convergence in probability, however the conditions for this mode of convergence are stronger; as we will see later, convergence almost surely actually implies that the sequence also converges in probability.
##### Definition
A sequence of random variables $\{ X_n ; n=1,2, \cdots \}$ converges almost surely to the random variable $X$ if:
$\forall \delta >0,$ $\lim_{n \to \infty} \Pr \{ \bigcup_{m \geq n} |X_m - X| > \delta, \}=0$
equivalently
$\Pr \{ \lim_{n \to \infty} X_n = X \}=1$
Under these conditions we use the notation $X_n \begin{matrix} \begin{matrix} { }_{a.s.} \\ \longrightarrow \\{ } \end{matrix} \end{matrix} X$ or $\lim_{n \to \infty} X_n = X \operatorname{a.s.}$.
##### Example
Let's see if our example from the convergence in probability section also converges almost surely. Defining:
$X_n = \begin{cases} \eta & 1- \begin{matrix} \frac{1}{n} \end{matrix} \\ \theta & \begin{matrix} \frac{1}{n} \end{matrix} \end{cases}$
we again guess that the convergence is to $\eta$. Inspecting the resulting expression we see that:
$\Pr \{ \lim_{n \to \infty} X_n = \eta \}=1- \Pr \{ \lim_{n \to \infty} X_n \ne \eta \}=1- \Pr \{ \lim_{n \to \infty} X_n= \theta \} \geq 1-\lim_{n \to \infty}\begin{matrix} \frac{1}{n} \end{matrix}=1$
Thereby satisfying our definition of almost-sure convergence.
#### Convergence in Distribution
Convergence in distribution will appear very frequently in our econometric models through the use of the Central Limit Theorem. So let's define this type of convergence.
##### Definition
A sequence of random variables $\{ X_n ; n=1,2, \cdots \}$ asymptotically converges in distribution to the random variable $X$ if $F_{X_n}(\zeta ) \rightarrow F_{X}(\zeta )$ for all continuity points. $F_{X_n}(\zeta )$ and $F_{X_{}}(\zeta )$ are the cumulative density functions of $X_n$ and $X$ respectively.
It is the distribution of the random variable that we are concerned with here. Think of a students-T distribution: as the degrees of freedom, $n$, increases our distribution becomes closer and closer to that of a gaussian distribution. Therefore the random variable $Y_n \sim t(n)$ converges in distribution to the random variable $Y \sim N(0,1)$ (n.b. we say that the random variable $Y_n \begin{matrix} { }_{d} \\ \longrightarrow \\{ } \end{matrix} Y$ as a notational crutch, what we really should use is $f_{Y_n} (\zeta )\begin{matrix} { }_{d} \\ \longrightarrow \\{ } \end{matrix} f_Y(\zeta )$/
##### Example
Let's consider the distribution Xn whose sample space consists of two points, 1/n and 1, with equal probability (1/2). Let X be the binomial distribution with p = 1/2. Then Xn converges in distribution to X.
The proof is simple: we ignore 0 and 1 (where the distribution of X is discontinuous) and prove that, for all other points a, $\lim F_{X_n}(a) = F_X(a)\,$. Since for a < 0 all Fs are 0, and for a > 1 all Fs are 1, it remains to prove the convergence for 0 < a < 1. But $F_{X_n}(a) = \frac{1}{2} ([a \ge \frac{1}{n}] + [a \ge 1])$ (using Iverson brackets), so for any a chose N > 1/a, and for n > N we have:
$n > 1/a \rightarrow a > 1/n \rightarrow [a \ge \frac{1}{n}] = 1 \land [a \ge 1] = 0 \rightarrow F_{X_n}(a) = \frac{1}{2}\,$
So the sequence $F_{X_n}(a)\,$ converges to $F_X(a)\,$ for all points where FX is continuous.
#### Convergence in R-mean Square
Convergence in R-mean square is not going to be used in this book, however for completeness the definition is provided below.
##### Definition
A sequence of random variables $\{ X_n ; n=1,2, \cdots \}$ asymptotically converges in r-th mean (or in the $L^r$ norm) to the random variable $X$ if, for any real number $r>0$ and provided that $E(|X_n|^r) < \infty$ for all n and $r\geq 1$,
$\lim_{n\to \infty }E\left( \left\vert X_n-X\right\vert ^r\right) =0.$
#### Cramer-Wold Device
The Cramer-Wold device will allow us to extend our convergence techniques for random variables from scalars to vectors.
##### Definition
A random vector $\mathbf{X}_n \begin{matrix} { }_{d} \\ \longrightarrow \\{ } \end{matrix} \mathbf{X} \; \iff \; {\mathbf{\lambda}}^{\operatorname{T}}\mathbf{X}_n \begin{matrix} { }_{d} \\ \longrightarrow \\{ } \end{matrix} {\mathbf{\lambda}}^{\operatorname{T}}\mathbf{X} \quad \forall \lVert \mathbf{\lambda} \rVert \ne 0$.
### Central Limit Theorem
Let $\ X_1, X_2, X_3, ...$ be a sequence of random variables which are defined on the same probability space, share the same probability distribution D and are independent. Assume that both the expected value μ and the standard deviation σ of D exist and are finite.
Consider the sum $\ S_n = X_1 + ... + X_n$. Then the expected value of $\ S_n$ is nμ and its standard error is σ n1/2. Furthermore, informally speaking, the distribution of Sn approaches the normal distribution N(nμ,σ2n) as n approaches ∞.
|
2015-08-02 02:29:11
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 56, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9670424461364746, "perplexity": 275.90622674995734}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988930.94/warc/CC-MAIN-20150728002308-00194-ip-10-236-191-2.ec2.internal.warc.gz"}
|
https://tex.stackexchange.com/questions/491449/how-to-create-cram-bonds-with-anchors
|
# How to create cram bonds with anchors
I want to recreate the chemical structure of 1,8-cineol in chemfig. On Wikipedia it's shown like this:
The best I can come up with is this code, with an anchor for the oxygen atom:
\documentclass[a4paper]{article}
\usepackage{chemfig}
\begin{document}
\chemfig{*6(-(-[:-90](-[:-30])(-[:-150])(<[:110,1.8]O?))---?(-)--)}
\end{document}
This results in the following picture:
I have read chemfig's documentation about anchors, but I can't figure out how to make that second bond to the oxygen atom also a cram bond.
From the chemfig manual page 15:
Thusadding a name and bondtype to the hook (?) can be done:
\documentclass{standalone}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{chemfig}
\begin{document}
\chemfig{*6(-(-[:-90](-[:-30])(-[:-150])(<[:110,1.8]O?[a,4]))---?[a,4](-)--)}
\end{document}
Result :
• It is also enough to add [,{>}] after the anchor. – Kess Vargavind May 18 '19 at 10:58
• @KessVargavind that works too, thanks for your useful addition. – Koen Vervloesem May 18 '19 at 11:14
|
2020-05-25 04:50:47
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3285883963108063, "perplexity": 2178.2927948583265}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347387219.0/warc/CC-MAIN-20200525032636-20200525062636-00506.warc.gz"}
|
https://mathoverflow.net/questions/60888/can-we-relate-cech-cohomology-and-derived-functor-cohomology-even-when-the-cover
|
# Can we relate Cech cohomology and derived functor cohomology even when the cover we choose isn't nice?
In my algebraic geometry class this semester, we've learned about Leray's Theorem, which states that for a sheaf $\mathcal{F}$ on a topological space $X$, and $\mathcal{U}$ a countable cover of $X$, if $\mathcal{F}$ is acyclic on every finite intersection of elements of $\mathcal{U}$ then the Cech cohomology $\check{H}^p(\mathcal{U},\mathcal{F})$ and derived functor cohomology $H^p(X,\mathcal{F})$ agree.
The potential for disagreement between them is covered well in these two MO questions. However, what neither of them seem to address is whether we can salvage any information about $H^p(X,\mathcal{F})$ from $\check{H}^p(\mathcal{U},\mathcal{F})$ even when $\mathcal{U}$ does not have the property that $\mathcal{F}$ is acyclic on all finite intersections, which is what I'd like to find out about here. I'm aware of Hartshorne Lemma 3.4.4, which says that there is a natural map $\check{H}^p(\mathcal{U},\mathcal{F})\rightarrow H^p(X,\mathcal{F})$ which is functorial in $\mathcal{F}$, but this is gotten by abstract nonsense - my feeling is that the existence of this map is not conveying much useful information. For all we know (?), all these maps could be the trivial homomorphism.
What I'm imagining is that perhaps the higher cohomology of $\mathcal{F}$ on the finite intersections of $\mathcal{U}$ can be related to the "difference" between $\check{H}^p(\mathcal{U},\mathcal{F})$ and $H^p(X,\mathcal{F})$, and that when the higher cohomology vanishes (i.e. $\mathcal{F}$ is acyclic), we get back the original theorem (that Cech and derived functor agree).
So, is there a useful relationship betwen Cech and derived functor cohomology even when $\mathcal{U}$ is not a nice open cover with respect to $\mathcal{F}$? Am I mistaken in assuming that the map $\check{H}^p(\mathcal{U},\mathcal{F})\rightarrow H^p(X,\mathcal{F})$ is not (particularly) useful?
Also I would like to avoid if possible the operation of taking the limit over all covers of $X$. I want to relate the specific Cech cohomology with respect to the cover $\mathcal{U}$, whatever its failings may be, with the derived functor cohomology.
-
Sounds like you're looking for the spectral sequence relating Cech cohomology and derived functor cohomology: en.wikipedia.org/wiki/… – Andrew Niles Apr 7 '11 at 6:14
Neat! From the description at the bottom, that looks like exactly what I was hoping would exist, though I'm not too familiar with spectral sequences. Why does taking the Cech cohomology of the presheaf that takes $U$ to $H^q(U,\mathcal{F})$ actually related to the Cech cohomology of $\mathcal{F}$? – Zev Chonoles Apr 7 '11 at 6:19
Zev: does the following example clarify things? Imagine the cover $(U)$ is just $X$ again! You can easily compute the Cech cohomology of this cover and see that it gives only the global sections of $F$ and no further information about the cohomology of $F$. On $H^0$ you then get an isomorphism (which you always do anyway) and for all the other $H^i$ you get no information at all. Hence the inf of the information we can get from Cech cohomology is "nothing more than global sections". – Kevin Buzzard Apr 7 '11 at 6:58
If a cover is fixed I would not call it Čech cohomology but the cohomology of the cover; the Čech is what one gets by taking a colimit over all (Čech) covers. – Zoran Skoda Apr 8 '11 at 14:57
There is a Mayer-Vietoris spectral sequence relating the two. This is a "direct" generalization of the Mayer-Vietoris long exact sequence, which is the special case in which your covering has just two open sets.
This is explained, if I recall correctly, in Bott-Tu.
-
So it sounds like the key word here is "spectral sequence", which I've seen once or twice but have not really computed with. At least in theory, would they provide an effective way of computing derived functor cohomology using Cech cohomology on an arbitrary open cover? I guess I was naively hoping just finding the Cech cohomology of $\mathcal{F}$ and the (derived functor) cohomology of $\mathcal{F}$ on the finite intersections would suffice to immediately write down the derived cohomology of $\mathcal{F}$, but we have to go through this gigantic array? – Zev Chonoles Apr 7 '11 at 6:30
That naive idea does not work even in the case of two open sets: that is precisely the reason for the M-V long exact sequence. In a way, what you are trying to do is deceptively simple. – Mariano Suárez-Alvarez Apr 7 '11 at 6:45
Zev, unfortunately, you'll have to wrestle with this or similar spectral sequence in general. The point of the Leray condition, is that it makes the spectral sequence collapse into something more manageable. – Donu Arapura Apr 7 '11 at 6:53
To elaborate a bit further, note that the $E_2$ term of the MV spectral sequence is precisely what you suggested: $$E_{2}^{p,q} = \bigoplus_{i_0,\dots,i_p} H^q(U_{i_0,\dots,i_p},\mathcal{F})$$ i.e. the (derived functor) cohomology of the intersections. The higher differentials give the corrections to $H^{n}(\mathcal{F})$ being the direct sum $\oplus_{p+q = n} E_{2}^{p,q}$. This is an excellent starting point to learn about spectral sequences. And I also recommend Bott and Tu. – Heinrich Hartmann Apr 7 '11 at 16:01
|
2014-04-17 07:33:41
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9441468715667725, "perplexity": 184.43592121080292}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00272-ip-10-147-4-33.ec2.internal.warc.gz"}
|
http://forums.codeguru.com/showthread.php?536121-Problem-with-the-display-of-picture&mode=hybrid
|
Problem with the display of picture
Hi to all! (and sorry for my english..!)
I have a problem that don't know how to solve.
I have a form consisting of:
1 groupBox that contains, inside, 4 pictureBox (which we call to understand pbPrimary).
Both the GroupBox that the PictureBox have the background transparent, so that you can see the background of the form.
"above" each pictureBox, I put another pictureBox (pbSecondary) which also has a transparent background, but are smaller than pbPrimarie.
The pbSecondary simply contain images .png of a cockade. The image has the "transparency" on the color white, so that to give the appearance that it is "glued" on the background image (in this case, the background image that i want is the image that i show in pbPrimary)
During the execution, in pbPrimary appear the image of a guy and later a cockade on the picture of that guy.
What happens is that the cockade that is in pbSecondary, which has a transparent background, DO NOT take the background of pbPrimary BUT 'to the form!
I wish the background of pbSecondary is the image of pbPrimary and NOT the background of the principal form.
Certainly, i don't have imposed something, such as the "owner", or some "priority", but i don't know what needs to be set.
|
2015-03-27 06:09:37
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8609977960586548, "perplexity": 1958.583130503609}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131295084.53/warc/CC-MAIN-20150323172135-00242-ip-10-168-14-71.ec2.internal.warc.gz"}
|
https://electronics.stackexchange.com/questions/203058/fpga-frequency-divider-with-linear-regulation
|
# FPGA Frequency divider with linear regulation
I'm currently developing an FPGA gateware requiring a regulated frequency divider and I wonder if there is any tricky way to get a linear output frequency regulation?
What I mean is that if I use a simple preloaded counter I get a f~1/x relevance, where x is a counter initial value. As I need to regulate frequency in a relatively wide range (say 0.5Hz-10kHz) I'd like it to be a linear function of some argument (of course, not necessarily a counter value-it can be something more complex ;-) ).
• It is linear function of 1/x, as you said. So what is the question? Nov 27 '15 at 17:10
• I know it is. But notice that change of one value of x in case of small values gives large changes in frequency. Whereas change in large values gives little change. I wounder if there is a method that gives more uniform changes in function of parameter. Nov 27 '15 at 18:15
• This is a nature of mathematics, you can't fight it. .Frequencies are generated by digital devices using timers, which by their nature can only count time, which is in the denominator. You can use some analog circuitry to overcome this limitation. Nov 27 '15 at 18:25
• Most FPGAs have at least one PLL that will give you more linear control. Nov 27 '15 at 18:33
• Are you wanting PLL or DDS? Nov 27 '15 at 19:34
At the moment, you are considering using a preloaded counter. This means that each clock cycle you get a fixed increment, in a variable accumulator length that you control, hence the frequency is reciprocal to your control value.
If instead you use a controllable variable increment in a fixed accumulator length, your frequency will be linearly dependent on the control value.
This is how a DDS works. A fixed length accumulator, typically a power of 2, for instance an 18 bit accumulator that counts up to 2^18 (250k-ish long) is incremented each clock cycle odf the system clock. In pigHDL, you would write
int count [17 downto 0];
process(on sys_clock rising):
_ count <= count+freq;
_ output := count[17];
This may, or may not, give you what you need. The MSB of the counter will give you the output, but unless freq is an exact power of 2, the output cycles will not be exactly the same length, they will vary by one count.
The average output frequency = freq * fs / accumulator length.
You can get increased resolution for the frequency by increasing the length of accumulator and frequency word, to the right.
There is no way to remove this one cycle jitter, if you are going to use the clock as a digital source. Bear in mind that for an FPGA, it is bad form to take the MSB output and use it as the clock line to other elements. Much better to turn it into a one cycle ClockEnable, and use that to condition the clock into downstream elements.
If it's for an external source, you can take the top several bits of the accumulator into a DAC, filter the waveform, and use a comparator, this will reduce the jitter considerably.
I will suggest using a binary rate multiplier. I have attached a reference sheet that shows what I mean. You can accurately use the binary multiplication to get a good range of output frequencies. Not exactly linear, but easily controlled.
|
2022-01-28 09:31:36
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5160892605781555, "perplexity": 931.3635695712505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305423.58/warc/CC-MAIN-20220128074016-20220128104016-00450.warc.gz"}
|
https://math.stackexchange.com/questions/901577/proving-finiteness-of-group-from-presentation
|
# Proving Finiteness of Group from Presentation
Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group.
I tried to find a canonical form for elements of $H$. That didn't work. Then I tried to find a finite normal subgroup of $H$ of finite index, but this didn't work either. I'd be grateful if someone could show me a correct method to solve this problem.
• Does $\langle abc\rangle$ mean the smallest normal subgroup of $G$ containing $abc$, or are we expected to also prove the normality? Otherwise $H$ is not a group!? – Jyrki Lahtonen Aug 18 '14 at 6:24
• – user1729 Aug 20 '14 at 12:36
Yes, your group $$H$$ is finite. You should consider Jyrki's comment (hint: the element $$abc$$ is central) but it a certain sense it does not matter - I will explain why the group with presentation $$\langle a, b, c; a^2, b^3, c^5, abc\rangle$$ is finite, which corresponds to the group $$G/\langle\langle abc\rangle\rangle$$, where $$\langle\langle abc\rangle\rangle$$ is the normal closure of the element $$abc$$. If you are wanting $$H$$ to be a group, then this is the group you are meaning.
To see that $$H$$ is infinite, note that it is a triangle group. The triple $$(2, 3, 5)$$ then means that $$H$$ is finite. So, a triangle group is a group with a presentation of the following form. $$T_{p, q, r}=\langle a, b, c; a^p, b^q, c^r, abc\rangle$$ Your group has this form. The infinite-ness of the group $$T_{p, q, r}$$ is determined by the following rules.
1. The group $$T_{p, q, r}$$ is finite if and only if $$\frac1p+\frac1q+\frac1r>1$$. Such groups are called spherical triangle groups.
2. The group $$T_{p, q, r}$$ is infinite if $$\frac1p+\frac1q+\frac1r=1$$. Such groups are called Euclidean triangle groups.
3. The group $$T_{p, q, r}$$ is infinite if $$\frac1p+\frac1q+\frac1r<1$$. Such groups are called Hyperbolic triangle groups.
The names spherical, Euclidean and hyperbolic triangle groups are because these groups act faithfully on surfaces tiles by triangles (with angles $$(\pi/p, \pi/q, \pi/r)$$), and for spherical triangles groups the surfaces is a sphere, for Euclidean triangle group the tiled surface is the Euclidean plane, while for hyperbolic groups the tiled is the hyperbholic plane. The Euclidean and hyperbolic planes are both infinite, so the corresponding groups are infinite, while spheres have finite area so spherical triangle groups must be finite.
Indeed, your group can be viewed as the group of orientation-preserving symmetries of the following tiling of the sphere (image from wikipedia).
|
2019-08-18 07:23:52
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 19, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8226066827774048, "perplexity": 112.33555947507878}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313715.51/warc/CC-MAIN-20190818062817-20190818084817-00465.warc.gz"}
|
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=17&t=47544
|
## Positive and Negative signs?
H-Atom ($E_{n}=-\frac{hR}{n^{2}}$)
MKearney_4G
Posts: 50
Joined: Fri Aug 30, 2019 12:18 am
### Positive and Negative signs?
I am unclear as to why sometimes we say that the energy is negative and sometimes it is positive. Could somebody explain what the signs mean in the context of Energy?
Christineg1G
Posts: 115
Joined: Fri Aug 09, 2019 12:15 am
Been upvoted: 1 time
### Re: Positive and Negative signs?
I believe that if energy is leaving a system or if the force and displacement are moving in the opposite direction, then we use a negative sign to represent that. And vice versa if energy is entering a system, or the force and displacement are moving in the same direction, we would use a positive sign.
Hope this helps!
KDang_1D
Posts: 127
Joined: Fri Aug 30, 2019 12:15 am
Been upvoted: 1 time
### Re: Positive and Negative signs?
Positive energy is an increase in the system (coming in), and negative energy is a decrease in the system (going out)
E=hv is positive because you're simply analyzing energy that is already there. E=(-hR)/n^2 is negative because the electron's energy is being emitted as a photon.
Khushboo_3D
Posts: 60
Joined: Wed Sep 18, 2019 12:19 am
### Re: Positive and Negative signs?
You would have a negative sign when energy is emitted- electron moves from. higher to lower energy level- and a positive sign when energy is absorbed- electron moves from lower to higher energy level.
xenamclean_1G
Posts: 51
Joined: Sat Jul 20, 2019 12:15 am
### Re: Positive and Negative signs?
A negative sign in front of energy would indicate an emission of energy (the electron moves from a higher to a lower energy level. Positive is just the vice-versa of this (a lower to a higher energy level).
Osvaldo SanchezF -1H
Posts: 122
Joined: Wed Sep 18, 2019 12:21 am
### Re: Positive and Negative signs?
The positive and negative signs are just indicative of direction in which the energy is coming from, whether it is emitting into or out the system. It is like a vector which shows the direction in which the energy is going too.
|
2020-07-05 20:15:40
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4815681278705597, "perplexity": 942.1745298136666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655888561.21/warc/CC-MAIN-20200705184325-20200705214325-00209.warc.gz"}
|
https://gamedev.stackexchange.com/questions/186194/how-to-randomly-generate-biome-with-perlin-noise?noredirect=1
|
# How to randomly generate biome with perlin noise?
I want to generate a procedural world using perlin noise. As expected, my terrain is generating with a repetitive patern. I'd like to add biomes, zones separated with higher or lower amplitude to simulate mountains or plain. I first thought I could randomly select rectangles of verticesof random sizes and then give them bigger amplitude to simulate mountains. The problem with that is that It wont seem natural, since I will only have rectangle biomes and also I don't know how to make it so the mountains would connect with the plain to avoid things like this
Here is my script:
public int size = 10;
public float amplitude;
public float frequence;
void GenerateTerrain(float amp, float freq)
{
Mesh meshy;
MeshRenderer meshRenderer = GetComponent<MeshRenderer>();
meshRenderer.sharedMaterial = new Material(Shader.Find("Standard"));
MeshFilter meshFilter = gameObject.GetComponent<MeshFilter>();
List<Vector2> uv = new List<Vector2>();
Vector3[] vert = new Vector3[size * size];
List<int> tri = new List<int>();
List<Vector3> norm = new List<Vector3>();
meshy = new Mesh();
for (int i = 0, x = 0; x < size; x++){
for(int z = 0; z < size; z++){
float y = Mathf.PerlinNoise(x * amp, z * amp) * freq;
vert[i] = new Vector3(x, y, z);
i++;}}
for (int i = 0; i < vert.Length; i++){
norm.Add(-Vector3.forward);}
for (int x = 0; x < size - 1; x++){
for (int z = 0; z < size - 1; z++){
int i = size * x + z;
tri.Add(i);
tri.Add(i + 1);
tri.Add(i + size);
tri.Add(i + size + 1);
tri.Add(i + size);
tri.Add(i + 1);}}
for (int i = 0; i < vert.Length; i++){
uv.Add(new Vector2(vert[i].x, vert[i].z));}
meshy.vertices = vert;
meshy.triangles = tri.ToArray();
meshy.uv = uv.ToArray();
meshy.normals = norm.ToArray();
meshFilter.mesh = meshy;
}
void Update()
{
if (Input.GetKeyDown("u"))
GenerateTerrain(frequence, amplitude);
}
• Also have a look at libnoise (libnoise.sourceforge.net) that solves your problems with tiling and repeatedness, even for planetary size terrain (if there's enough processing power and storage capacity).
– user144188
Nov 1 '20 at 16:35
## 1 Answer
There are three main steps here:
1. Use some method to assign biomes to regions (this is the hard part, with multiple strategies I'll break down shortly)
2. For each point in your mesh or tile/node in your world, determine which biome it's in, as well as which neighbouring biomes it's close to. Compute an interpolation weight representing the influence of each nearby biome.
3. Evaluate your terrain generation logic for each of the nearby biomes separately. (In practice, this means you've computed 3-4 possible heights or other characteristics for this point/node) The final height for this location is a weighted average of the results from the nearby biomes, according to their influence.
It might seem wasteful to compute multiple full biome results for a single point, but this is a necessary evil if you want to get sensible blending between them. If you try to interpolate the generation input parameters alone (particularly the noise frequency or number of octaves) and generate just one height result based on the blended parameters, you'll get non-sensical looking blends with nasty artifacts, especially when far from the origin. You also limit how complex and varied your terrain generators can be, since they all have to use the same rules.
There are lots of different strategies for step 1. I'll classify the main families as "zoned" and "emergent".
# Strategy 1: Zoned
For this strategy, we start by dividing our world into a collection of zones.
A popular method for this is using a Voronoi diagram (also sometimes referred to as a Worley noise basis in procedural generation contexts), where we pseudorandomly scatter points across our map. The set of all locations closer to this point than any other point forms one zone, which is guaranteed to be a convex polygon. This gives us a nice structured region to work with that's still a little more organic than strict rectangles:
(See also this Red Blob Games article for some discussion of how we can tweak the standard Voronoi approach to something we might prefer for world generation)
You can get arbitrarily more complex with your region selection, of course. Maybe applying domain warping to make the borders between regions more organic, etc. Though I recommend starting simple - most of the hard geometric borders will be hidden by the time we do our blending/interpolation, so they might not be a problem.
Next, we assign a biome to each zone.
We could do this randomly, using the zone location/ID to seed a weighted random lookup into a table of biomes. The risk with this is that adjacent biomes make their determinations separately, and so you can get combinations of adjacent zones that don't follow any coherent geographical logic - like a lush jungle completely surrounded by desert, with no fresh water nearby or flowing through it.
We could also use rule-based zone assignments. Maybe we randomly assign a certain fraction of the zones to be water - lakes, seas or oceans. Then we designate any non-water zone adjacent to a water zone must be a beach, cliff, or marsh. Forests and jungles can appear 2 zones away from the nearest water, and mountains only 3+ zones away. You could accomplish this with cellular automata or other adjacency-based rules.
Then for step 2, we can compute where our point-to-be-generated sits relative to its nearest neighbouring zones, and use that to compute the interpolation weights to use for each biome's result. One way we can do that is to compute a Delaunay triangulation of the seed points we used to create our zones. This is the dual of the Voronoi diagram, where each triangle represents a junction between three adjacent zones. We can then use the barycentric coordinates of our point-to-be-generated within this triangle to compute the weight to give to each of the three nearest neighbouring zones.
# Strategy 2: Emergent
Here we try to model/approximate some of the real-world processes that give rise to different biomes, as a way to get more natural / logical relationships between the different structures. The biomes then "emerge" as a consequence of the simulated natural processes that shape your terrain.
Th first step is to generate the underlying drivers of biome formation in our model, like temperature and moisture, to pick a popular pair of inputs. We could generate a moisture map and a temperature map as two separate, low-frequency Perlin noise maps, for example, though going this route these won't have any particular relationship to landforms or location on your map/planet.
Another route is to generate your broad-scale elevation up-front - say the first couple octaves of your terrain height function. This isn't biome-level detail yet, more like the underlying bedrock under the biomes. Some folks will even generate tectonic plates and simulate their motion to find where there should be mountain ranges formed by two plates colliding, etc.
Once you have this coarse-grained elevation detail (and correspondingly, the locations of oceans/seas based which areas are below sea level), you can generate other metrics like temperature and moisture from that. Say, the temperature gets colder as you move higher in elevation or away from the equator in latitude. You can model the moisture based on proximity to an ocean or sea, or the prevailing winds: if winds at a particular latitude run mostly west-to-east, then areas on the east side of a mountain range will tend to be wetter, and areas on the west side dryer, as the mountain wrings the rain out of the air as it passes.
Now you have, for each site in your map, a collection of input values like latitude, elevation, moisture, and temperature. You can use these as coordinates to look up into a biome assignment map.
Here's an example diagram from Navarras on Wikipedia, showing how real-world biomes relate to precipitation (moisture) and temperature:
A game might use a more abstracted version, like this example from an older version of Minecraft, via the Minecraft wiki:
You could even add more inputs like the elevation or latitude into the mix, like this plot of the Holdridge Life Zone Classification Scheme from Wikipedia:
With our computed biome driver values for a given point, we can find which region of this plot we fall into, to determine the predominant biome for this location.
For step 2, we can also determine how close we are to other biomes in this map, but this time instead of computing our distance in barycentric coordinates within our triangle of zones, or world-space distances across our terrain, we're computing our distance in moisture-temperature space (or whatever driving inputs you've chosen for biome selection).
This route ostensibly gives a stronger connection between the landforms of your continents and the details of your biomes, but it comes at the cost of less direct control over the placement of those biomes. With the wrong parameters, you could easily end up with a whole world of deserts, or a world with only scarce islands of a particular biome. And maybe that's desirable for the variety of worlds from your procedural generator. Or maybe it's not, and you'd prefer to sacrifice some geographical logic (that maybe only the hardcore Earth sciences geeks will notice or appreciate anyway) in order to have more hands-on control over the gameplay combinations of biomes the players will encounter, as you can get with zoning-style approaches.
Whichever route we've taken, we now have a list of nearby biomes influencing this point, and a relative weight to apply to each one. Now we can run our terrain generation logic for each biome at this point, and blend the results using those interpolation weights, to arrive at one consensus elevation (and other properties) for this location.
For the emergent scheme, note that you might already have a coarse-grained elevation selected, so your individual biome generators can be more flat, capturing only local terrain shapes, and delegating the broad-scale landforms to the bedrock pass you did earlier. This makes it easier to get a clean blend between adjacent biomes that won't have a sudden artificial-looking ramp between them because they disagree about the average elevation in their domain.
You can even use the coarse elevation value as a bias to your blending function, allowing eg. forest biomes to creep further into the valleys while letting rocky biomes dominate the higher elevations, adding a little non-linearity so the blend feels more organic.
• This is an amazing answer. I am learning so much with this and I will document myself on everything you talked about in order to succeed following these methods. Thanks a lot Oct 6 '20 at 14:26
• Awesome answer! Just got the chance to read through it all after seeing you link it. Couple of things I would add or change if it were me: Instead of Barycentric to interpolate on the Voronoi/Delaunay mesh, I would use normalized sparse convolution. Pick a radius big enough to always contain points, use a falloff function that goes to zero smoothly at that radius, and add up weighted contributions of all points in range. Then divide by the total weight. This is both smoother than barycentric interpolation, and doesn't require the Delaunay or Voronoi mesh at all -- just the points. Apr 10 at 11:20
• In the emergent section, an important issue I run into is that noise values are not uniformly distributed. Therefore, if you want each biome to represent a roughly equal share of the world, you need to either pass the noise output into a function that makes it more uniform, or bias your lookup map to compensate. If your lookup map is a Voronoi diagram itself, then this amounts to placing points more tightly packed in the center, than the outer edges. Apr 10 at 11:20
• Regarding noise, I think it's best to take care what to recommend. Perlin is an older noise with a lot of square bias, which isn't a necessary compromise for most uses. Readily-available Simplex-type noise, and drop-in 3D+ domain rotation, can easily address its problems. However, a huge number of sources teach the old noise in a vacuum rather than in context. If we teach the right thing in newer sources, then slowly this can improve. On one hand, you're only responding to the question. But OTOH, we know the asker is considering the old noise, so it's an impactful opportunity for education. Apr 10 at 11:21
• Sounds like you have plenty of content to share in an answer of your own, @KdotJPG Apr 10 at 11:37
|
2021-10-16 19:09:55
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4007965326309204, "perplexity": 1590.0580525018117}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323584913.24/warc/CC-MAIN-20211016170013-20211016200013-00689.warc.gz"}
|
https://crypto.stackexchange.com/questions/20004/cryptosystems-used-to-generate-public-key-certificate
|
# Cryptosystems used to generate public key certificate
Where can we find the cryptosystems used to generate public key certificate? Are the cryptosystems under the signature algorithm and signature hash algorithm? Do I need to analyze the packets trasmitted between my computer and the server? For example, ciphersuite.
Suppose the following ciphersuites are transmitted from ClientHello to the server.
- TLS_RSA_WITH_AES_128_CBC_SHA256 (0x003c)
- TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA256 (0xc027)
- TLS_ECDHE_ECDSA _WITH_AES_128_GCM_SHA256 (0xc02b)
- TLS_DHE_DSS_WITH_AES_128_CBC_SHA256 (0xc040)
- TLS_RSA_ WITH_RC4_128_MD5 (0xc004)
Can I say that the cryptosystems used to protect the communication between my computer and the server are RSA, ECDHE_RSA, ECDHE_ECDSA and DHE_DSS?
• The question might not ask what you really mean, but in it's current form the answer is no. RSA, ECDHE_RSA, ECDHE_ECDSA and DHE_DSS are what's been used for the key exchange phase. For any authentication to happen a certificate needs to already have been generated. For example during RSA key exchange, secrets generated are encrypted with an RSA public key and the public key is checked to belong to the other party by verifying a certificate provided by a CA. – Edvard Fagerholm Nov 4 '14 at 7:25
• Then may i know where can I obtain the cryptosystems used to generate certificate? – Idonknow Nov 5 '14 at 2:45
• Either you have generated your public/secret key in your computer with your software and a CA sign your public key or a CA generates your public/secret key and further signs your public key. I think you are talking for the latter. – 111 Feb 6 '15 at 9:13
The ciphersuites only define the properties of the SSL channel that is to be created. The certificate - including the private key usage indicated in the certificate - should be consistent with the ciphersuite.
For instance if you have a ciphersuite starting with TLS_RSA then the certificate should allow encryption, and the public/private key should be RSA of course. If you have a TLS_DHE or TLS_ECDHE ciphersuite then the certificate should allow authentication with any supported authentication algorithm. Authentication is generally supported by the certificate of course, as that is the main use case for TLS certificates in the first place.
How the certificate was generated is outside the realm of TLS. Basically it doesn't care, as long as you can create a trust chain to a trusted certificate in the certificate store. For this it does need to be able to verify signatures though, so the algorithms used to sign certificates need to be supported.
The certificate is simply a file containing your public key (RSA or DSA) together with some information on e.g. how the public key is allowed to be used (for example only to authenticate a TLS connection to foo.yourdomain.com). Additionally, the file has been signed using either an RSA, DSA or ECDSA signature. Therefore, you should look up RSA/DSA/ECDSA signatures.
|
2019-05-20 19:11:00
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4230502247810364, "perplexity": 2003.6268967685098}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256100.64/warc/CC-MAIN-20190520182057-20190520204057-00120.warc.gz"}
|
https://byjus.com/maths/surface-areas-volumes/
|
# Surface Areas and Volume
## Area
The space occupied by a two-dimensional flat surface. It is measure in square units.
Generally, Area can be of two types
(i) Total Surface Area
(ii) Curved Surface Area
### Total surface area
Total surface area refers to area including the base(s) and the curved part.
### Curved surface area (lateral surface area)
Refers to area of only the curved part excluding it’s base(s).
### Volume
The amount of space, measured in cubic units, that an object or substance occupies. Some shapes are two-dimensional, so it doesn’t have volumes. Example, Volume of Circle cannot be found, though Volume of the sphere can be. It is so because a sphere is a three-dimensional shape.
Below given is the table for calculating Surface area and Volume for the basic geometrical figures:
Name Perimeter Total Surface Area Curved Surface Area Volume Figure Square 4a a2 —- —- Rectangle 2(w+h) w.h —- —- Parallelogram 2(a+b) b.h —- —- Trapezoid a+b+c+d 1/2(a+b).h —- —- Circle $2 \pi r$ $\pi r^{2}$ —- —- Ellipse $2\pi\sqrt{\left ( \frac{a^{2}+b^{2}}{2} \right )}$ $\pi a.b$ —- —- Triangle a+b+c $\frac{1}{2}\times b \times h$ —- —- Cuboid 4(l+b+h) 2(lb+bh+hl) 2h(l+b) $l \times b \times h$ Cube 6a 6a2 4a2 $a^{3}$ Cylinder —- $\pi r(r+h)$ $2 \pi r h$ $\pi r^{2} h$ Cone —- $\pi r(r+l)$ $\pi r l$ $\frac{1}{3}\pi r^{2} h$ Sphere —- $4 \pi r^{2}$ $4 \pi r^{2}$ $\frac{4}{3}\pi r^{3}$ Hemisphere —- $3 \pi r^{2}$ $2 \pi r^{2}$ $\frac{2}{3}\pi r^{3}$
|
2019-09-22 15:05:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6018961071968079, "perplexity": 2241.4706632343155}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575515.93/warc/CC-MAIN-20190922135356-20190922161356-00524.warc.gz"}
|
https://www.physicsforums.com/threads/unit-conversion-mm-sqrt-hz-degree-to-m-2-hz-rad.628530/
|
# Unit conversion mm/sqrt(Hz)/degree to m^2/Hz/rad
#### robbie.
Hello,
I have two datasets containing power spectral density data to be compared. One of these datasets is presented in units mm/sqrt (Hz)/degree, and I would like to do some transformation so that data is comparable with the other set, which has units m^2/Hz/rad.
Any help on how to do this would be massively appreciated, as this kind of thing is not my strong point!
Many thanks,
Robbie
#### Simon Bridge
Homework Helper
They look like they've each measured something different - they have different dimensions.
The first units look like
$$\left ( \frac{mm^2}{Hz} \right )^{1/2}\text{deg}^{-1}$$
So this is the square-root of the other one scaled for degrees.
I would reverse the per-degree part of the calculation and square it to get the same thing, then bother with converting the units.
#### robbie.
Thank you for your reply, however I am struggling to understand what you mean when you say 'reverse' the per degree part?
#### Simon Bridge
Homework Helper
The numbers had to be calculated somehow that ended up with the units being "per degree" ... whatever they did, do the opposite. eg. if they divided by 360, then multiply by 360.
Put another way:
If there are X (mm2/Hz)1/2 in one degree ... then how many mm2/Hz are there in 1 degree?
### Physics Forums Values
We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
|
2019-05-24 00:45:12
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6332486271858215, "perplexity": 1999.276823574737}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257481.39/warc/CC-MAIN-20190524004222-20190524030222-00526.warc.gz"}
|
http://ramp.studio/problems/sea_ice
|
Arctic sea ice forecast
Description
Balázs Kégl (CNRS), Camille Marini (CNRS), Andy Rhines (UW), Jennifer Dy (NEU), Arindam Banerjee (UMN)
## Introduction
Arctic sea ice cover is one of the most variable features of Earth's climate. Its annual cycle peaks at around 15 million square kilometers in early spring, melting back to a minimum of about 6 million square kilometers in September. These seasonal swings are important for Earth's energy balance, as ice reflects the majority of sunlight while open water absorbs it. Changes in ice cover are also important for marine life and navigation for shipping.
In recent years, Arctic sea ice cover has declined rapidly, particularly during the September minimum. These changes have outpaced the predictions of climate models, and forecasting extent remains a formidable challenge. Typically, skillful predictions are limited to ~2-5 months in advance (Stroeve, et al. "Improving Predictions of Arctic Sea Ice Extent"), while idealized experiments suggest that predictions up to two years in advance should be possible (Guemas et al, 2014).
Better tools to predict ice cover are critical for seasonal and regional climate prediction, and would thus address grand challenges in the study of climate change (World Climate Research Programme: Grand Challenges, 2013)
### The CCSM4 simulator
As a surrogate for observational data, we will use output from a 1300 year simulation using the NCAR CCSM4.0 climate model. The model was run in fully-coupled mode with interactive ocean, atmosphere, and sea ice. The simulation was also performed in an idealized "Pre-Industrial" mode, where greenhouse gas concentrations and other external forcings are held fixed to 1850 levels. This allows us to access a stationary climate over a 1000+ year period, which makes the evaluation of the predictor more robust than if we used real measurements that are both non-stationary and limited to several decades.
### The data
The data is a time series of "images" $z_t$, consisting of different physical variables on a regular grid on the Earth, indexed by lon(gitude) and lat(itude) coordinates. The variables we have made available are:
• ice_area --- the Northern Hemisphere sea ice area, in millions of squared kilometers.
• ts --- surface temperature, most important over the oceans which have a very high heat capacity.
• taux --- zonal (x-direction) surface wind stress. This is the frictional effect of winds on the sea surface and sea ice.
• tauy --- meridional (y-direction) surface wind stress.
• ps --- surface pressure.
• psl --- equivalent sea-level surface pressure. This corrects ps for the effects of topography, though the two should be very similar.
• shflx --- Surface sensible heat flux, the amount of heat transferred from the surface to the atmosphere.
• cldtot --- Total cloud cover (fractional), which has strong effects on radiative energy balance at the surface.
The fields are recorded every month for 1300 years, giving 15,600 time points. The goal is to predict the Northern Hemisphere sea ice area 4 months ahead. Since the most important prediction is the minimum area in September, we will also display the RMSE over predictions in May, predecting that years (minimum) ice area in September.
The pipeline will consists of a time series feature extractor and a predictor. Since the task is regression, the predictor will be a regressor, and the score (to minimize) will be the root mean square error. The feature extractor will have access to the whole data. It will construct a "classical" feature matrix where each row corresponds to a time point. You should collect all information into these features that you find relevant to the regressor. The feature extractor can take anything from the past, that is, it will implement a function $x_t = f(z_1, \ldots, z_t)$. Since you will have access to the full data, in theory you can cheat (even inadvertantly) by using information from the future. We have implemented a randomized test to find such "bugs", but please do your best to avoid this since it would make the results irrelevant.
|
2017-05-29 02:02:54
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5480360388755798, "perplexity": 2167.2142929210695}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463612008.48/warc/CC-MAIN-20170529014619-20170529034619-00494.warc.gz"}
|
https://www.physicsforums.com/threads/derivation-for-this-equation-n-n0-1-2-t-t1-2.234601/
|
# Derivation for this equation: N = N0(1/2)^t/t1/2?
1. May 11, 2008
### nothing123
Anyone know the derivation for this equation: N = N0(1/2)^t/t1/2? I can understand it plugging numbers in but I don't really know how it was derived in the first place.
Thanks!
2. May 11, 2008
### malty
Well, we know that the rate of radioactive decay is proportional to the Number of particles at a time t.
So:
$$-\frac{dN_{(t)}}{dt}=\lambda N_{(t)}$$
Now can you derive it?
3. May 11, 2008
### nothing123
Using what you gave, I am able to derive N = N0e^-kt but the equation I provided is without the decay constant...
4. May 11, 2008
### malty
Well the half life will be when $$N_{(t)}=\frac{N_0}{2}$$. and when $$t=T_{\frac{1}{2}}}$$.
/Not sure how the decay constant disappears tbh.
/No wait I see now.
Last edited: May 11, 2008
5. May 11, 2008
### nothing123
Can you explain it a bit more clearly? Thanks.
6. May 11, 2008
### malty
Sure no problem.
Half life is the when the number of particles is reduced by half.
Hence this occurs when $$N_{(t)}=\frac{N_o}{2})$$ No is the original number of particles.
The time which this occurs will be the half life and we call it $$T_{\frac{1}{2}}$$
So we have:
$$\frac{N_o}{2}=N_o e^{-\lambda T_{\frac{1}{2}}}$$
$$\frac{1}{2}=e^{-\lambda T_{\frac{1}{2}} }$$
$$N_{(t)}=\frac{N_o}{2})$$
but
$$e^{-\lambda T_{\frac{1}{2}}}=\frac{1}{2}$$
so $$-\lambda T_{\frac{1}{2}}= L_n(\frac{1}{2}})$$
$$e^{-\lambda}=\frac{e^{L_n(\frac{1}{2})}}{e^{T_{\frac{1}{2}}}}= \frac{1}{2}e^{-T_{\frac{1}{2}}}$$
Sub that into the equation $$N = N_0e^{-\lambda t}$$
and you got it :D
*Phew that took some time to type*
:)
Last edited: May 11, 2008
7. May 11, 2008
### nothing123
Thanks for the reply but I'm not entirely sure how you got the above lines. My exponent rules might be a big hazy but I don't think e^a/b is the same as e^a/e^b if you know what I'm saying.
8. May 11, 2008
### malty
Not 100% sure what you are refering to.
Here's how I got that line:
$$-\lambda T_{\frac{1}{2}}= L_n(\frac{1}{2}})$$
and
$${-\lambda}=\frac{L_n(\frac{1}{2})}{{T_{\frac{1}{2}}}}$$
9. May 11, 2008
### nothing123
Right, so does e^(ln(1/2)/T1/2) = e^ln(1/2)/e^T1/2?
10. May 11, 2008
### malty
Well yes, if
$$e^{ab}=e^a*e^b$$
then $$e^{-ab}=e^a*e^{-b}=\frac{e^a}{e^b}$$
11. May 11, 2008
### nothing123
Wait, doesn't e^a*e^b = e^(a+b) not e^ab? e^ab would be (e^a)^b, right?
12. May 11, 2008
### malty
Lol yeah your right is does, I have absolutely no idea what made me think it was well that...
Last edited: May 11, 2008
13. May 11, 2008
### malty
Ahh now I remember the equation I used was
$${-\lambda}=\frac{L_n(\frac{1}{2})}{{T_{\frac{1}{2}}} }$$
I took the exponent of this:
$$e^{-\lambda}=\frac{e^{L_n(\frac{1}{2})}}{e^{T_{\frac{1 }{2}}}}= \frac{1}{2}e^{-T_{\frac{1}{2}}}$$
And that is still correct, I believe, my whole escapade with the awful exponent rules was well becasue...I'm tired (and dreaming) :p
14. May 11, 2008
### nothing123
Hmm I don't know if I'm just tired too or what but we seem to be back at square one. Didn't we conclude that e^(ln(1/2)/T1/2) does not equal e^ln(1/2)/e^T1/2?
15. May 11, 2008
### malty
No this is actually true because
$$=e^{-\lambda}=e^{- \frac{ ln\frac{1}{2}}{T_{\frac{1}{2}}}}=e^{(ln\frac{1}{2})^{-T_{\frac{1}{2}}}}=\frac{1}{2}e^{-T_{\frac{1}{2}}}}$$
Least I think it, *is brain dead*
16. May 11, 2008
### nothing123
Ok, I think I figured it out but I don't think the equation you just wrote is correct. e^(ln1/2)/T1/2 = 1/2^(T1/2^-1). Now, subbing this into our original equation, we get N = N0(1/2)^t/t1/2. Thanks so much for your help anyways, it got my brain going.
|
2017-08-18 20:51:34
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7152842283248901, "perplexity": 2259.4674588918506}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105108.31/warc/CC-MAIN-20170818194744-20170818214744-00495.warc.gz"}
|
https://www.r-bloggers.com/problems-in-estimating-garch-parameters-in-r-part-2-rugarch/
|
# Problems in Estimating GARCH Parameters in R (Part 2; rugarch)
January 28, 2019
By
[This article was first published on R – Curtis Miller's Personal Website, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
# Introduction
Now here is a blog post that has been sitting on the shelf far longer than it should have. Over a year ago I wrote an article about problems I was having when estimating the parameters of a GARCH(1,1) model in R. I documented the behavior of parameter estimates (with a focus on $\beta$) and perceived pathological behavior when those estimates are computed using fGarch. I called for help from the R community, including sending out the blog post over the R Finance mailing list.
I was not disappointed in the feedback. You can see some mailing list feedback, and there were some comments on Reddit that were helpful, but I think the best feedback I got was through my own e-mail.
Dr. Brian G. Peterson, a member of the R finance community, sent some thought provoking e-mails. The first informed me that fGarch is no longer the go-to package for working with GARCH models. The RMetrics suite of packages (which include fGarch) was maintained by Prof. Diethelm Würtz at ETH Zürich. He was killed in a car accident in 2016.
Dr. Peterson recommended I look into two more modern packages for GARCH modelling, rugarch (for univariate GARCH models) and rmgarch (for multivariate GARCH models). I had not heard of these packages before (the reason I was aware of fGarch was because it was referred to in the time series textbook Time Series Analysis and Its Applications with R Examples by Shumway and Stoffer), so I’m very thankful for the suggestion. Since I’m interested in univariate time series for now, I only looked at rugarch. The package appears to have more features and power than fGarch, which may explain why it seems more difficult to use. However the package’s vignette is helpful and worth printing out.
Dr. Peterson also had interesting comments about my proposed applications. He argued that intraday data should be preferred to daily data and that simulated data (including simulated GARCH processes) has idiosyncracies not seen in real data. The ease of getting daily data (particularly for USD/JPY around the time of Asian financial crises, which was an intended application of a test statistic I’m studying) motivated my interest in daily data. His comments, though, may lead me to reconsider this application.1 (I might try to detect the 2010 eurozone financial crises via EUR/USD instead. I can get free intraday data from HistData.com for this.) However, if standard error estimates cannot be trusted for small sample sizes, our test statistic would still be in trouble since it involves estimating parameters even for small sample sizes.
He also warned that simulated data exhibits behaviors not seen in real data. That may be true, but simulated data is important since it can be considered a statistician’s best-case scenario. Additionally, the properties of the process that generated simulated data are known a priori, including the values of the generating parameters and whether certain hypotheses (such as whether there is a structural change in the series) are true. This allows for sanity checks of estimators and tests. This is impossible for real-world since we don’t have the a priori knowledge needed.
Prof. André Portela Santos asked that I repeat the simulations but with $\alpha 0.6$ since these values are supposedly more common than my choice of $\alpha = \beta = 0.2$. It’s a good suggestion and I will consider parameters in this range in addition to $\alpha = \beta = 0.2$ in this post. However, my simulations seemed to suggest that when $\alpha = \beta = 0.2$, the estimation procedures nevertheless seem to want to be near the range of large $\beta$. I’m also surprised since my advisor gave me the impression that GARCH processes with either $\alpha$ or $\beta$ large are more difficult to work with. Finally, if the estimators are strongly biased, we might expect to see most estimated parameters to lie in that range, though that does not mean the “correct” values lie in that range. My simulations suggest fGarch struggles to discover $\alpha = \beta = 0.2$ even when those parameters are “true.”” Prof. Santos’ comment leads me to desire a metastudy about what common estimates of GARCH parameters are on real world. (There may or may not be one; I haven’t checked. If anyone knows of one, please share.)
My advisor contacted another expert on GARCH models and got some feedback. Supposedly the standard error for $\beta$ is large, so there should be great variation in parameter estimates. Some of my simulations agreed with this behavior even for small sample sizes, but at the same time showed an uncomfortable bias towards $\beta = 0$ and $\beta = 1$. This might be a consequence of the optimization procedures, as I hypothesized.
So given this feedback, I will be conducting more simulation experiments. I won’t be looking at fGarch or tseries anymore; I will be working exclusively with rugarch. I will explore different optimization procedures supported by the package. I won’t be creating plots like I did in my first post; those plots were meant only to show the existence of a problem and its severity. Instead I will be looking at properties of the resulting estimators produced by different optimization procedures.
## Introduction to rugarch
As mentioned above, rugarch is a package for working with GARCH models; a major use case is estimating their parameters, obviously. Here I will demonstrate how to specify a GARCH model, simulate data from the model, and estimate parameters. After this we can dive into simulation studies.
library(rugarch)
## Loading required package: parallel
##
## Attaching package: 'rugarch'
## The following object is masked from 'package:stats':
##
## sigma
### Specifying a $\text{GARCH}(1, 1)$$\text{GARCH}(1, 1)$ model
To work with a GARCH model we need to specify it. The function for doing this is ugarchspec(). I think the parameters variance.model and mean.model are the most important parameters.
variance.model is a list with named entries, perhaps the two most interesting being model and garchOrder. model is a string specify which type of GARCH model is being fitted. Many major classes of GARCH models (such as EGARCH, IGARCH, etc.) are supported; for the “vanilla” GARCH model, set this to "sGARCH" (or just omit it; the standard model is the default). garchOrder is a vector for the order of the ARCH and GARCH components of the model.
mean.model allows for fitting ARMA-GARCH models, and functions like variance.model in that it accepts a list of named entries, the most interesting being armaOrder and include.mean. armaOrder is like garchOrder; it’s a vector specifying the order of the ARMA model. include.mean is a boolean that, if true, allows for the ARMA part of the model to have non-zero mean.
When simulating a process, we need to set the values of our parameters. This is done via the fixed.pars parameter, which accepts a list of named elements, the elements of the list being numeric. They need to fit the conventions the function uses for parameters; for example, if we want to set the parameters of a $\text{GARCH}(1,1)$ model, the names of our list elements should be "alpha1" and "beta1". If the plan is to simulate a model, every parameter in the model should be set this way.
There are other parameters interesting in their own right but I focus on these since the default specification is an ARMA-GARCH model with ARMA order of $(1,1)$ with non-zero mean and a GARCH model of order $(1,1)$. This is not a vanilla $\text{GARCH}(1,1)$ model as I desire, so I almost always change this.
spec1 <- ugarchspec(mean.model = list(armaOrder = c(0,0), include.mean = FALSE),
fixed.pars = list("omega" = 0.2, "alpha1" = 0.2,
"beta1" = 0.2))
spec2 <- ugarchspec(mean.model = list(armaOrder = c(0,0), include.mean = FALSE),
fixed.pars = list("omega" = 0.2, "alpha1" = 0.1,
"beta1" = 0.7))
show(spec1)
##
## *---------------------------------*
## * GARCH Model Spec *
## *---------------------------------*
##
## Conditional Variance Dynamics
## ------------------------------------
## GARCH Model : sGARCH(1,1)
## Variance Targeting : FALSE
##
## Conditional Mean Dynamics
## ------------------------------------
## Mean Model : ARFIMA(0,0,0)
## Include Mean : FALSE
## GARCH-in-Mean : FALSE
##
## Conditional Distribution
## ------------------------------------
## Distribution : norm
## Includes Skew : FALSE
## Includes Shape : FALSE
## Includes Lambda : FALSE
show(spec2)
##
## *---------------------------------*
## * GARCH Model Spec *
## *---------------------------------*
##
## Conditional Variance Dynamics
## ------------------------------------
## GARCH Model : sGARCH(1,1)
## Variance Targeting : FALSE
##
## Conditional Mean Dynamics
## ------------------------------------
## Mean Model : ARFIMA(0,0,0)
## Include Mean : FALSE
## GARCH-in-Mean : FALSE
##
## Conditional Distribution
## ------------------------------------
## Distribution : norm
## Includes Skew : FALSE
## Includes Shape : FALSE
## Includes Lambda : FALSE
### Simulating a GARCH process
The function ugarchpath() simulates GARCH models specified via ugarchspec(). The function needs a specification objectect created by ugarchspec() first. The parameters n.sim and n.start specify the size of the process and the length of the burn-in period, respectively (with defaults 1000 and 0, respectively; I strongly recommend setting the burn-in period to at least 500, but I go for 1000). The function creates an object that contains not only the simulated series but also residuals and $\sigma_t$.
The rseed parameter controls the random seed the function uses for generating data. Be warned that set.seed() is effectively ignored by this function, so if you want consistent results, you will need to set this parameter.
The plot() method accompanying these objects is not completely transparent; there are a few plots it could create and when calling plot() on a uGARCHpath object in the command line users are prompted to input a number corresponding to the desired plot. This is a pain sometimes so don’t forget to pass the desired plot’s number to the which parameter to avoid the prompt; setting which = 2 will give the plot of the series proper.
old_par <- par()
par(mfrow = c(2, 2))
x_obj <- ugarchpath(spec1, n.sim = 1000, n.start = 1000, rseed = 111217)
show(x_obj)
##
## *------------------------------------*
## * GARCH Model Path Simulation *
## *------------------------------------*
## Model: sGARCH
## Horizon: 1000
## Simulations: 1
## Seed Sigma2.Mean Sigma2.Min Sigma2.Max Series.Mean
## sim1 111217 0.332 0.251 0.915 0.000165
## Mean(All) 0 0.332 0.251 0.915 0.000165
## Unconditional NA 0.333 NA NA 0.000000
## Series.Min Series.Max
## sim1 -1.76 1.62
## Mean(All) -1.76 1.62
## Unconditional NA NA
for (i in 1:4) {
plot(x_obj, which = i)
}
par(old_par)
## Warning in par(old_par): graphical parameter "cin" cannot be set
## Warning in par(old_par): graphical parameter "cra" cannot be set
## Warning in par(old_par): graphical parameter "csi" cannot be set
## Warning in par(old_par): graphical parameter "cxy" cannot be set
## Warning in par(old_par): graphical parameter "din" cannot be set
## Warning in par(old_par): graphical parameter "page" cannot be set
# The actual series
x1 <- [email protected]$seriesSim plot.ts(x1) ### Fitting a $\text{GARCH}(1,1)$$\text{GARCH}(1,1)$ model The ugarchfit() function fits GARCH models. The function needs a specification and a dataset. The solver parameter accepts a string stating which numerical optimizer to use to find the parameter estimates. Most of the parameters of the function manage interfacing with the numerical optimizer. In particular, solver.control can be given a list of arguments to pass to the optimizer. We will be looking at this in more detail later. The specification used for generating the simulated data won’t be appropriate for ugarchfit(), since it contains fixed values for its parameters. In my case I will need to create a second specification object. spec <- ugarchspec(mean.model = list(armaOrder = c(0, 0), include.mean = FALSE)) fit <- ugarchfit(spec, data = x1) show(fit) ## ## *---------------------------------* ## * GARCH Model Fit * ## *---------------------------------* ## ## Conditional Variance Dynamics ## ----------------------------------- ## GARCH Model : sGARCH(1,1) ## Mean Model : ARFIMA(0,0,0) ## Distribution : norm ## ## Optimal Parameters ## ------------------------------------ ## Estimate Std. Error t value Pr(>|t|) ## omega 0.000713 0.001258 0.56696 0.57074 ## alpha1 0.002905 0.003714 0.78206 0.43418 ## beta1 0.994744 0.000357 2786.08631 0.00000 ## ## Robust Standard Errors: ## Estimate Std. Error t value Pr(>|t|) ## omega 0.000713 0.001217 0.58597 0.55789 ## alpha1 0.002905 0.003661 0.79330 0.42760 ## beta1 0.994744 0.000137 7250.45186 0.00000 ## ## LogLikelihood : -860.486 ## ## Information Criteria ## ------------------------------------ ## ## Akaike 1.7270 ## Bayes 1.7417 ## Shibata 1.7270 ## Hannan-Quinn 1.7326 ## ## Weighted Ljung-Box Test on Standardized Residuals ## ------------------------------------ ## statistic p-value ## Lag[1] 3.998 0.04555 ## Lag[2*(p+q)+(p+q)-1][2] 4.507 0.05511 ## Lag[4*(p+q)+(p+q)-1][5] 9.108 0.01555 ## d.o.f=0 ## H0 : No serial correlation ## ## Weighted Ljung-Box Test on Standardized Squared Residuals ## ------------------------------------ ## statistic p-value ## Lag[1] 29.12 6.786e-08 ## Lag[2*(p+q)+(p+q)-1][5] 31.03 1.621e-08 ## Lag[4*(p+q)+(p+q)-1][9] 32.26 1.044e-07 ## d.o.f=2 ## ## Weighted ARCH LM Tests ## ------------------------------------ ## Statistic Shape Scale P-Value ## ARCH Lag[3] 1.422 0.500 2.000 0.2331 ## ARCH Lag[5] 2.407 1.440 1.667 0.3882 ## ARCH Lag[7] 2.627 2.315 1.543 0.5865 ## ## Nyblom stability test ## ------------------------------------ ## Joint Statistic: 0.9518 ## Individual Statistics: ## omega 0.3296 ## alpha1 0.2880 ## beta1 0.3195 ## ## Asymptotic Critical Values (10% 5% 1%) ## Joint Statistic: 0.846 1.01 1.35 ## Individual Statistic: 0.35 0.47 0.75 ## ## Sign Bias Test ## ------------------------------------ ## t-value prob sig ## Sign Bias 0.3946 6.933e-01 ## Negative Sign Bias 3.2332 1.264e-03 *** ## Positive Sign Bias 4.2142 2.734e-05 *** ## Joint Effect 28.2986 3.144e-06 *** ## ## ## Adjusted Pearson Goodness-of-Fit Test: ## ------------------------------------ ## group statistic p-value(g-1) ## 1 20 20.28 0.3779 ## 2 30 26.54 0.5965 ## 3 40 36.56 0.5817 ## 4 50 47.10 0.5505 ## ## ## Elapsed time : 2.60606 par(mfrow = c(3, 4)) for (i in 1:12) { plot(fit, which = i) } ## ## please wait...calculating quantiles... par(old_par) ## Warning in par(old_par): graphical parameter "cin" cannot be set ## Warning in par(old_par): graphical parameter "cra" cannot be set ## Warning in par(old_par): graphical parameter "csi" cannot be set ## Warning in par(old_par): graphical parameter "cxy" cannot be set ## Warning in par(old_par): graphical parameter "din" cannot be set ## Warning in par(old_par): graphical parameter "page" cannot be set Notice the estimated parameters and standard errors? The estimates are nowhere near the “correct” numbers even for a sample size of 1000, and there is no way a reasonable confidence interval based on the estimated standard errors would contain the correct values. It looks like the problems I documented in my last post have not gone away. Out of curiosity, what would happen with the other specification, one in the range Prof. Santos suggested? x_obj <- ugarchpath(spec2, n.start = 1000, rseed = 111317) x2 <- [email protected]$seriesSim
fit <- ugarchfit(spec, x2)
show(fit)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.001076 0.002501 0.43025 0.66701
## alpha1 0.001992 0.002948 0.67573 0.49921
## beta1 0.997008 0.000472 2112.23364 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.001076 0.002957 0.36389 0.71594
## alpha1 0.001992 0.003510 0.56767 0.57026
## beta1 0.997008 0.000359 2777.24390 0.00000
##
## LogLikelihood : -1375.951
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.7579
## Bayes 2.7726
## Shibata 2.7579
## Hannan-Quinn 2.7635
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.9901 0.3197
## Lag[2*(p+q)+(p+q)-1][2] 1.0274 0.4894
## Lag[4*(p+q)+(p+q)-1][5] 3.4159 0.3363
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.768 0.05226
## Lag[2*(p+q)+(p+q)-1][5] 4.986 0.15424
## Lag[4*(p+q)+(p+q)-1][9] 7.473 0.16272
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.2232 0.500 2.000 0.6366
## ARCH Lag[5] 0.4793 1.440 1.667 0.8897
## ARCH Lag[7] 2.2303 2.315 1.543 0.6686
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.3868
## Individual Statistics:
## omega 0.2682
## alpha1 0.2683
## beta1 0.2669
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.846 1.01 1.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.5793 0.5625
## Negative Sign Bias 1.3358 0.1819
## Positive Sign Bias 1.5552 0.1202
## Joint Effect 5.3837 0.1458
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 24.24 0.1871
## 2 30 30.50 0.3894
## 3 40 38.88 0.4753
## 4 50 48.40 0.4974
##
##
## Elapsed time : 2.841597
That’s no better Now let’s see what happens when we use different optimization routines.
## Optimization and Parameter Estimation in rugarch
### Choice of optimizer
ugarchfit()‘s default parameters did a good job of finding appropriate parameters for what I will refer to as model 2 (where $\alpha = 0.1$ and $\beta = 0.7$) but not for model 1 ($\alpha = \beta = 0.2$). What I want to know is when one solver seems to beat another.
As pointed out by Vivek Rao2 on the R-SIG-Finance mailing list, the “best” estimate is the estimate that maximizes the likelihood function (or, equivalently, the log-likelihood function), and I omitted inspecting the log likelihood function’s values in my last post. Here I will see which optimization procedures lead to the maximum log-likelihood.
Below is a helper function that simplifies the process of fitting a GARCH model’s parameters and extracting the log-likelihood, parameter values, and standard errors while allowing for different values to be passed to solver and solver.control.
evalSolverFit <- function(spec, data, solver = "solnp",
solver.control = list()) {
# Calls ugarchfit(spec, data, solver, solver.control), and returns a vector
# containing the log likelihood, parameters, and parameter standard errors.
# Parameters are equivalent to those seen in ugarchfit(). If the solver fails
# to converge, NA will be returned
vec <- NA
tryCatch({
fit <- ugarchfit(spec = spec, data = data, solver = solver,
solver.control = solver.control)
coef_se_names <- paste("se", names([email protected]$coef), sep = ".") se <- [email protected]$se.coef
names(se) <- coef_se_names
robust_coef_se_names <- paste("robust.se", names([email protected]$coef), sep = ".") robust.se <- [email protected]$robust.se.coef
names(robust.se) <- robust_coef_se_names
vec <- c([email protected]$coef, se, robust.se) vec["LLH"] <- [email protected]$LLH
}, error = function(w) {
NA
})
return(vec)
}
Below I list out all optimization schemes I will consider. I only fiddle with solver.control, but there may be other parameters that could help the numerical optimization routines, namely numderiv.control, which are control arguments passed to the numerical routines responsible for standard error computation. This utilizes the package numDeriv which performs numerical differentiation.
solvers <- list( # A list of lists where each sublist contains parameters to
# pass to a solver
list("solver" = "nlminb", "solver.control" = list()),
list("solver" = "solnp", "solver.control" = list()),
list("solver" = "lbfgs", "solver.control" = list()),
list("solver" = "gosolnp", "solver.control" = list(
"n.restarts" = 100, "n.sim" = 100
)),
list("solver" = "hybrid", "solver.control" = list()),
list("solver" = "nloptr", "solver.control" = list("solver" = 1)), # COBYLA
list("solver" = "nloptr", "solver.control" = list("solver" = 2)), # BOBYQA
list("solver" = "nloptr", "solver.control" = list("solver" = 3)), # PRAXIS
list("solver" = "nloptr",
"solver.control" = list("solver" = 4)), # NELDERMEAD
list("solver" = "nloptr", "solver.control" = list("solver" = 5)), # SBPLX
list("solver" = "nloptr",
"solver.control" = list("solver" = 6)), # AUGLAG+COBYLA
list("solver" = "nloptr",
"solver.control" = list("solver" = 7)), # AUGLAG+BOBYQA
list("solver" = "nloptr",
"solver.control" = list("solver" = 8)), # AUGLAG+PRAXIS
list("solver" = "nloptr",
"solver.control" = list("solver" = 9)), # AUGLAG+NELDERMEAD
list("solver" = "nloptr",
"solver.control" = list("solver" = 10)) # AUGLAG+SBPLX
)
tags <- c( # Names for the above list
"nlminb",
"solnp",
"lbfgs",
"gosolnp",
"hybrid",
"nloptr+COBYLA",
"nloptr+BOBYQA",
"nloptr+PRAXIS",
"nloptr+SBPLX",
"nloptr+AUGLAG+COBYLA",
"nloptr+AUGLAG+BOBYQA",
"nloptr+AUGLAG+PRAXIS",
"nloptr+AUGLAG+SBPLX"
)
names(solvers) <- tags
Now let’s run the gauntlet of optimization choices and see which produces the estimates with the largest log likelihood for data generated by model 1. The lbfgs method (low-storage version of the Broyden-Fletcher-Goldfarb-Shanno method, provided in nloptr) unfortunately does not converge for this series, so I omit it.
optMethodCompare <- function(data, spec, solvers) {
# Runs all solvers in a list for a dataset
#
# Args:
# data: An object to pass to ugarchfit's data parameter containing the data
# to fit
# spec: A specification created by ugarchspec to pass to ugarchfit
# solvers: A list of lists containing strings of solvers and a list for
# solver.control
#
# Return:
# A matrix containing the result of the solvers (including parameters, se's,
# and LLH)
model_solutions <- lapply(solvers, function(s) {
args <- s
args[["spec"]] <- spec
args[["data"]] <- data
res <- do.call(evalSolverFit, args = args)
return(res)
})
model_solutions <- do.call(rbind, model_solutions)
return(model_solutions)
}
round(optMethodCompare(x1, spec, solvers[c(1:2, 4:15)]), digits = 4)
## omega alpha1 beta1 se.omega se.alpha1 se.beta1 robust.se.omega robust.se.alpha1 robust.se.beta1 LLH
## ------------------------- ------- ------- ------- --------- ---------- --------- ---------------- ----------------- ---------------- ----------
## nlminb 0.2689 0.1774 0.0000 0.0787 0.0472 0.2447 0.0890 0.0352 0.2830 -849.6927
## solnp 0.0007 0.0029 0.9947 0.0013 0.0037 0.0004 0.0012 0.0037 0.0001 -860.4860
## gosolnp 0.2689 0.1774 0.0000 0.0787 0.0472 0.2446 0.0890 0.0352 0.2828 -849.6927
## hybrid 0.0007 0.0029 0.9947 0.0013 0.0037 0.0004 0.0012 0.0037 0.0001 -860.4860
## nloptr+COBYLA 0.0006 0.0899 0.9101 0.0039 0.0306 0.0370 0.0052 0.0527 0.0677 -871.5006
## nloptr+BOBYQA 0.0003 0.0907 0.9093 0.0040 0.0298 0.0375 0.0057 0.0532 0.0718 -872.3436
## nloptr+PRAXIS 0.2689 0.1774 0.0000 0.0786 0.0472 0.2444 0.0888 0.0352 0.2823 -849.6927
## nloptr+NELDERMEAD 0.0010 0.0033 0.9935 0.0013 0.0039 0.0004 0.0013 0.0038 0.0001 -860.4845
## nloptr+SBPLX 0.0010 0.1000 0.9000 0.0042 0.0324 0.0386 0.0055 0.0536 0.0680 -872.2736
## nloptr+AUGLAG+COBYLA 0.0006 0.0899 0.9101 0.0039 0.0306 0.0370 0.0052 0.0527 0.0677 -871.5006
## nloptr+AUGLAG+BOBYQA 0.0003 0.0907 0.9093 0.0040 0.0298 0.0375 0.0057 0.0532 0.0718 -872.3412
## nloptr+AUGLAG+PRAXIS 0.1246 0.1232 0.4948 0.0620 0.0475 0.2225 0.0701 0.0439 0.2508 -851.0547
## nloptr+AUGLAG+NELDERMEAD 0.2689 0.1774 0.0000 0.0786 0.0472 0.2445 0.0889 0.0352 0.2826 -849.6927
## nloptr+AUGLAG+SBPLX 0.0010 0.1000 0.9000 0.0042 0.0324 0.0386 0.0055 0.0536 0.0680 -872.2736
According the the maximum likelihood criterion, the “best” result is achieved by gosolnp. The result has the unfortunate property that $\beta \approx 0$, which is certainly not true, but at least the standard error for $\beta$ would create a confidence interval that contains $\beta$‘s true value. Of these, my preferred estimates are produced by AUGLAG+PRAXIS, as $\beta$ seems reasonable and in fact the estimates are all close to the truth, (at least in the sense that the confidence intervals contain the true values), but unfortunately the estimates do not maximize the log likelihood, even though they are the most reasonable.
If we looked at model 2, what do we see? Again, lbfgs does not converge so I omit it. Unfortunately, nlminb does not converge either, so it too must be omitted.
round(optMethodCompare(x2, spec, solvers[c(2, 4:15)]), digits = 4)
## omega alpha1 beta1 se.omega se.alpha1 se.beta1 robust.se.omega robust.se.alpha1 robust.se.beta1 LLH
## ------------------------- ------- ------- ------- --------- ---------- --------- ---------------- ----------------- ---------------- ----------
## solnp 0.0011 0.0020 0.9970 0.0025 0.0029 0.0005 0.0030 0.0035 0.0004 -1375.951
## gosolnp 0.0011 0.0020 0.9970 0.0025 0.0029 0.0005 0.0030 0.0035 0.0004 -1375.951
## hybrid 0.0011 0.0020 0.9970 0.0025 0.0029 0.0005 0.0030 0.0035 0.0004 -1375.951
## nloptr+COBYLA 0.0016 0.0888 0.9112 0.0175 0.0619 0.0790 0.0540 0.2167 0.2834 -1394.529
## nloptr+BOBYQA 0.0010 0.0892 0.9108 0.0194 0.0659 0.0874 0.0710 0.2631 0.3572 -1395.310
## nloptr+PRAXIS 0.5018 0.0739 0.3803 0.3178 0.0401 0.3637 0.2777 0.0341 0.3225 -1373.632
## nloptr+NELDERMEAD 0.0028 0.0026 0.9944 0.0028 0.0031 0.0004 0.0031 0.0035 0.0001 -1375.976
## nloptr+SBPLX 0.0029 0.1000 0.9000 0.0146 0.0475 0.0577 0.0275 0.1108 0.1408 -1395.807
## nloptr+AUGLAG+COBYLA 0.0016 0.0888 0.9112 0.0175 0.0619 0.0790 0.0540 0.2167 0.2834 -1394.529
## nloptr+AUGLAG+BOBYQA 0.0010 0.0892 0.9108 0.0194 0.0659 0.0874 0.0710 0.2631 0.3572 -1395.310
## nloptr+AUGLAG+PRAXIS 0.5018 0.0739 0.3803 0.3178 0.0401 0.3637 0.2777 0.0341 0.3225 -1373.632
## nloptr+AUGLAG+NELDERMEAD 0.0001 0.0000 1.0000 0.0003 0.0003 0.0000 0.0004 0.0004 0.0000 -1375.885
## nloptr+AUGLAG+SBPLX 0.0029 0.1000 0.9000 0.0146 0.0475 0.0577 0.0275 0.1108 0.1408 -1395.807
Here it was PRAXIS and AUGLAG+PRAXIS that gave the “optimal” result, and it was only those two methods that did. Other optimizers gave visibly bad results. That said, the “optimal” solution is the preferred on with the parameters being nonzero and their confidence intervals containing the correct values.
What happens if we restrict the sample to size 100? (lbfgs still does not work.)
round(optMethodCompare(x1[1:100], spec, solvers[c(1:2, 4:15)]), digits = 4)
## omega alpha1 beta1 se.omega se.alpha1 se.beta1 robust.se.omega robust.se.alpha1 robust.se.beta1 LLH
## ------------------------- ------- ------- ------- --------- ---------- --------- ---------------- ----------------- ---------------- ---------
## nlminb 0.0451 0.2742 0.5921 0.0280 0.1229 0.1296 0.0191 0.0905 0.0667 -80.6587
## solnp 0.0451 0.2742 0.5921 0.0280 0.1229 0.1296 0.0191 0.0905 0.0667 -80.6587
## gosolnp 0.0451 0.2742 0.5921 0.0280 0.1229 0.1296 0.0191 0.0905 0.0667 -80.6587
## hybrid 0.0451 0.2742 0.5921 0.0280 0.1229 0.1296 0.0191 0.0905 0.0667 -80.6587
## nloptr+COBYLA 0.0007 0.1202 0.8798 0.0085 0.0999 0.0983 0.0081 0.1875 0.1778 -85.3121
## nloptr+BOBYQA 0.0005 0.1190 0.8810 0.0085 0.0994 0.0992 0.0084 0.1892 0.1831 -85.3717
## nloptr+PRAXIS 0.0451 0.2742 0.5921 0.0280 0.1229 0.1296 0.0191 0.0905 0.0667 -80.6587
## nloptr+NELDERMEAD 0.0451 0.2742 0.5920 0.0281 0.1230 0.1297 0.0191 0.0906 0.0667 -80.6587
## nloptr+SBPLX 0.0433 0.2740 0.5998 0.0269 0.1237 0.1268 0.0182 0.0916 0.0648 -80.6616
## nloptr+AUGLAG+COBYLA 0.0007 0.1202 0.8798 0.0085 0.0999 0.0983 0.0081 0.1875 0.1778 -85.3121
## nloptr+AUGLAG+BOBYQA 0.0005 0.1190 0.8810 0.0085 0.0994 0.0992 0.0084 0.1892 0.1831 -85.3717
## nloptr+AUGLAG+PRAXIS 0.0451 0.2742 0.5921 0.0280 0.1229 0.1296 0.0191 0.0905 0.0667 -80.6587
## nloptr+AUGLAG+NELDERMEAD 0.0451 0.2742 0.5921 0.0280 0.1229 0.1296 0.0191 0.0905 0.0667 -80.6587
## nloptr+AUGLAG+SBPLX 0.0450 0.2742 0.5924 0.0280 0.1230 0.1295 0.0191 0.0906 0.0666 -80.6587
round(optMethodCompare(x2[1:100], spec, solvers[c(1:2, 4:15)]), digits = 4)
## omega alpha1 beta1 se.omega se.alpha1 se.beta1 robust.se.omega robust.se.alpha1 robust.se.beta1 LLH
## ------------------------- ------- ------- ------- --------- ---------- --------- ---------------- ----------------- ---------------- ----------
## nlminb 0.7592 0.0850 0.0000 2.1366 0.4813 3.0945 7.5439 1.7763 11.0570 -132.4614
## solnp 0.0008 0.0000 0.9990 0.0291 0.0417 0.0066 0.0232 0.0328 0.0034 -132.9182
## gosolnp 0.0537 0.0000 0.9369 0.0521 0.0087 0.0713 0.0430 0.0012 0.0529 -132.9124
## hybrid 0.0008 0.0000 0.9990 0.0291 0.0417 0.0066 0.0232 0.0328 0.0034 -132.9182
## nloptr+COBYLA 0.0014 0.0899 0.9101 0.0259 0.0330 0.1192 0.0709 0.0943 0.1344 -135.7495
## nloptr+BOBYQA 0.0008 0.0905 0.9095 0.0220 0.0051 0.1145 0.0687 0.0907 0.1261 -135.8228
## nloptr+PRAXIS 0.0602 0.0000 0.9293 0.0522 0.0088 0.0773 0.0462 0.0015 0.0565 -132.9125
## nloptr+NELDERMEAD 0.0024 0.0000 0.9971 0.0473 0.0629 0.0116 0.0499 0.0680 0.0066 -132.9186
## nloptr+SBPLX 0.0027 0.1000 0.9000 0.0238 0.0493 0.1308 0.0769 0.1049 0.1535 -135.9175
## nloptr+AUGLAG+COBYLA 0.0014 0.0899 0.9101 0.0259 0.0330 0.1192 0.0709 0.0943 0.1344 -135.7495
## nloptr+AUGLAG+BOBYQA 0.0008 0.0905 0.9095 0.0221 0.0053 0.1145 0.0687 0.0907 0.1262 -135.8226
## nloptr+AUGLAG+PRAXIS 0.0602 0.0000 0.9294 0.0523 0.0090 0.0771 0.0462 0.0014 0.0565 -132.9125
## nloptr+AUGLAG+NELDERMEAD 0.0000 0.0000 0.9999 0.0027 0.0006 0.0005 0.0013 0.0004 0.0003 -132.9180
## nloptr+AUGLAG+SBPLX 0.0027 0.1000 0.9000 0.0238 0.0493 0.1308 0.0769 0.1049 0.1535 -135.9175
The results are not thrilling. The “best” result for the series generated by model 1 was attained by multiple solvers, and the 95% confidence interval (CI) for $\omega$ would not contain $\omega$‘s true value, though the CIs for the other parameters would contain their true values. For the series generated by model 2 the best result was attained by the nlminb solver; the parameter values are not plausible and the standard errors are huge. At least the CI would contain the correct value.
From here we should no longer stick to two series but see the performance of these methods on many simulated series generated by both models. Simulations in this post will be too computationally intensive for my laptop so I will use my department’s supercomputer to perform them, taking advantage of its many cores for parallelization.
library(foreach)
library(doParallel)
logfile <- ""
# logfile <- "outfile.log"
# if (!file.exists(logfile)) {
# file.create(logfile)
# }
cl <- makeCluster(detectCores() - 1, outfile = logfile)
registerDoParallel(cl)
optMethodSims <- function(gen_spec, n.sim = 1000, m.sim = 1000,
fit_spec = ugarchspec(mean.model = list(
armaOrder = c(0,0), include.mean = FALSE)),
solvers = list("solnp" = list(
"solver" = "solnp", "solver.control" = list())),
rseed = NA, verbose = FALSE) {
# Performs simulations in parallel of GARCH processes via rugarch and returns
# a list with the results of different optimization routines
#
# Args:
# gen_spec: The specification for generating a GARCH sequence, produced by
# ugarchspec
# n.sim: An integer denoting the length of the simulated series
# m.sim: An integer for the number of simulated sequences to generate
# fit_spec: A ugarchspec specification for the model to fit
# solvers: A list of lists containing strings of solvers and a list for
# solver.control
# rseed: Optional seeding value(s) for the random number generator. For
# m.sim>1, it is possible to provide either a single seed to
# initialize all values, or one seed per separate simulation (i.e.
# m.sim seeds). However, in the latter case this may result in some
# slight overhead depending on how large m.sim is
# verbose: Boolean for whether to write data tracking the progress of the
# loop into an output file
# outfile: A string for the file to store verbose output to (relevant only
# if verbose is TRUE)
#
# Return:
# A list containing the result of calling optMethodCompare on each generated
# sequence
fits <- foreach(i = 1:m.sim,
.packages = c("rugarch"),
.export = c("optMethodCompare", "evalSolverFit")) %dopar% {
if (is.na(rseed)) {
newseed <- NA
} else if (is.vector(rseed)) {
newseed <- rseed[i]
} else {
newseed <- rseed + i - 1
}
if (verbose) {
cat(as.character(Sys.time()), ": Now on simulation ", i, "\n")
}
sim <- ugarchpath(gen_spec, n.sim = n.sim, n.start = 1000,
m.sim = 1, rseed = newseed)
x <- [email protected]$seriesSim optMethodCompare(x, spec = fit_spec, solvers = solvers) } return(fits) } # Specification 1 first spec1_n100 <- optMethodSims(spec1, n.sim = 100, m.sim = 1000, solvers = solvers, verbose = TRUE) spec1_n500 <- optMethodSims(spec1, n.sim = 500, m.sim = 1000, solvers = solvers, verbose = TRUE) spec1_n1000 <- optMethodSims(spec1, n.sim = 1000, m.sim = 1000, solvers = solvers, verbose = TRUE) # Specification 2 next spec2_n100 <- optMethodSims(spec2, n.sim = 100, m.sim = 1000, solvers = solvers, verbose = TRUE) spec2_n500 <- optMethodSims(spec2, n.sim = 500, m.sim = 1000, solvers = solvers, verbose = TRUE) spec2_n1000 <- optMethodSims(spec2, n.sim = 1000, m.sim = 1000, solvers = solvers, verbose = TRUE) Below is a set of helper functions I will use for the analytics I want. optMethodSims_getAllVals <- function(param, solver, reslist) { # Get all values for a parameter obtained by a certain solver after getting a # list of results via optMethodSims # # Args: # param: A string for the parameter to get (such as "beta1") # solver: A string for the solver for which to get the parameter (such as # "nlminb") # reslist: A list created by optMethodSims # # Return: # A vector of values of the parameter for each simulation res <- sapply(reslist, function(l) { return(l[solver, param]) }) return(res) } optMethodSims_getBestVals <- function(reslist, opt_vec = TRUE, reslike = FALSE) { # A function that gets the optimizer that maximized the likelihood function # for each entry in reslist # # Args: # reslist: A list created by optMethodSims # opt_vec: A boolean indicating whether to return a vector with the name of # the optimizers that maximized the log likelihood # reslike: A bookean indicating whether the resulting list should consist of # matrices of only one row labeled "best" with a structure like # reslist # # Return: # If opt_vec is TRUE, a list of lists, where each sublist contains a vector # of strings naming the opimizers that maximized the likelihood function and # a matrix of the parameters found. Otherwise, just the matrix (resembles # the list generated by optMethodSims) res <- lapply(reslist, function(l) { max_llh <- max(l[, "LLH"], na.rm = TRUE) best_idx <- (l[, "LLH"] == max_llh) & (!is.na(l[, "LLH"])) best_mat <- l[best_idx, , drop = FALSE] if (opt_vec) { return(list("solvers" = rownames(best_mat), "params" = best_mat)) } else { return(best_mat) } }) if (reslike) { res <- lapply(res, function(l) { mat <- l$params[1, , drop = FALSE]
rownames(mat) <- "best"
return(mat)
})
}
return(res)
}
optMethodSims_getCaptureRate <- function(param, solver, reslist, multiplier = 2,
spec, use_robust = TRUE) {
# Gets the rate a confidence interval for a parameter captures the true value
#
# Args:
# param: A string for the parameter being worked with
# solver: A string for the solver used to estimate the parameter
# reslist: A list created by optMethodSims
# multiplier: A floating-point number for the multiplier to the standard
# error, appropriate for the desired confidence level
# spec: A ugarchspec specification with the fixed parameters containing the
# true parameter value
# use_robust: Use robust standard errors for computing CIs
#
# Return:
# A float for the proportion of times the confidence interval captured the
# true parameter value
se_string <- ifelse(use_robust, "robust.se.", "se.")
est <- optMethodSims_getAllVals(param, solver, reslist)
moe_est <- multiplier * optMethodSims_getAllVals(
paste0(se_string, param), solver, reslist)
param_val <- [email protected]$fixed.pars[[param]] contained <- (param_val <= est + moe_est) & (param_val >= est - moe_est) return(mean(contained, na.rm = TRUE)) } optMethodSims_getMaxRate <- function(solver, maxlist) { # Gets how frequently a solver found a maximal log likelihood # # Args: # solver: A string for the solver # maxlist A list created by optMethodSims_getBestVals with entries # containing vectors naming the solvers that maximized the log # likelihood # # Return: # The proportion of times the solver maximized the log likelihood maxed <- sapply(maxlist, function(l) { solver %in% l$solvers
})
return(mean(maxed))
}
optMethodSims_getFailureRate <- function(solver, reslist) {
# Computes the proportion of times a solver failed to converge.
#
# Args:
# solver: A string for the solver
# reslist: A list created by optMethodSims
#
# Return:
# Numeric proportion of times a solver failed to converge
failed <- sapply(reslist, function(l) {
is.na(l[solver, "LLH"])
})
return(mean(failed))
}
# Vectorization
optMethodSims_getCaptureRate <- Vectorize(optMethodSims_getCaptureRate,
vectorize.args = "solver")
optMethodSims_getMaxRate <- Vectorize(optMethodSims_getMaxRate,
vectorize.args = "solver")
optMethodSims_getFailureRate <- Vectorize(optMethodSims_getFailureRate,
vectorize.args = "solver")
I first create tables containing, for a fixed sample size and model:
• The rate at which a solver attains the highest log likelihood among all solvers for a series
• The rate at which a solver failed to converge
• The rate at which a roughly 95% confidence interval based on the solver’s solution managed to contain the true parameter value for each parameter (referred to as the “capture rate”, and using the robust standard errors)
solver_table <- function(reslist, tags, spec) {
# Creates a table describing important solver statistics
#
# Args:
# reslist: A list created by optMethodSims
# tags: A vector with strings naming all solvers to include in the table
# spec: A ugarchspec specification with the fixed parameters containing the
# true parameter value
#
# Return:
# A matrix containing metrics describing the performance of the solvers
params <- names([email protected]$fixed.pars) max_rate <- optMethodSims_getMaxRate(tags, optMethodSims_getBestVals(reslist)) failure_rate <- optMethodSims_getFailureRate(tags, reslist) capture_rate <- lapply(params, function(p) { optMethodSims_getCaptureRate(p, tags, reslist, spec = spec) }) return_mat <- cbind("Maximization Rate" = max_rate, "Failure Rate" = failure_rate) capture_mat <- do.call(cbind, capture_rate) colnames(capture_mat) <- paste(params, "95% CI Capture Rate") return_mat <- cbind(return_mat, capture_mat) return(return_mat) } ### Model 1, $n = 100$$n = 100$ as.data.frame(round(solver_table(spec1_n100, tags, spec1) * 100, digits = 1)) ## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate ## ------------------------- ------------------ ------------- -------------------------- --------------------------- -------------------------- ## nlminb 16.2 20.0 21.8 29.2 24.0 ## solnp 0.1 0.0 13.7 24.0 15.4 ## lbfgs 15.1 35.2 56.6 67.9 58.0 ## gosolnp 20.3 0.0 20.3 32.6 21.9 ## hybrid 0.1 0.0 13.7 24.0 15.4 ## nloptr+COBYLA 0.0 0.0 6.3 82.6 19.8 ## nloptr+BOBYQA 0.0 0.0 5.4 82.1 18.5 ## nloptr+PRAXIS 15.8 0.0 42.1 54.5 44.1 ## nloptr+NELDERMEAD 0.4 0.0 5.7 19.3 8.1 ## nloptr+SBPLX 0.1 0.0 7.7 85.7 24.1 ## nloptr+AUGLAG+COBYLA 0.0 0.0 6.1 84.5 19.9 ## nloptr+AUGLAG+BOBYQA 0.1 0.0 6.5 83.2 19.4 ## nloptr+AUGLAG+PRAXIS 22.6 0.0 41.2 54.6 44.1 ## nloptr+AUGLAG+NELDERMEAD 11.1 0.0 7.5 18.8 9.7 ## nloptr+AUGLAG+SBPLX 0.6 0.0 7.9 86.5 23.0 ### Model 1, $n = 500$$n = 500$ as.data.frame(round(solver_table(spec1_n500, tags, spec1) * 100, digits = 1)) ## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate ## ------------------------- ------------------ ------------- -------------------------- --------------------------- -------------------------- ## nlminb 21.2 0.4 63.3 67.2 63.8 ## solnp 0.1 0.2 32.2 35.6 32.7 ## lbfgs 4.5 41.3 85.0 87.6 85.7 ## gosolnp 35.1 0.0 69.0 73.2 69.5 ## hybrid 0.1 0.0 32.3 35.7 32.8 ## nloptr+COBYLA 0.0 0.0 3.2 83.3 17.8 ## nloptr+BOBYQA 0.0 0.0 3.5 81.5 18.1 ## nloptr+PRAXIS 18.0 0.0 83.9 87.0 84.2 ## nloptr+NELDERMEAD 0.0 0.0 16.4 20.7 16.7 ## nloptr+SBPLX 0.1 0.0 3.7 91.4 15.7 ## nloptr+AUGLAG+COBYLA 0.0 0.0 3.2 83.3 17.8 ## nloptr+AUGLAG+BOBYQA 0.0 0.0 3.5 81.5 18.1 ## nloptr+AUGLAG+PRAXIS 21.9 0.0 80.2 87.4 83.4 ## nloptr+AUGLAG+NELDERMEAD 0.6 0.0 20.0 24.0 20.5 ## nloptr+AUGLAG+SBPLX 0.0 0.0 3.7 91.4 15.7 ### Model 1, $n = 1000$$n = 1000$ as.data.frame(round(solver_table(spec1_n1000, tags, spec1) * 100, digits = 1)) ## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate ## ------------------------- ------------------ ------------- -------------------------- --------------------------- -------------------------- ## nlminb 21.5 0.1 88.2 86.1 87.8 ## solnp 0.4 0.2 54.9 53.6 54.6 ## lbfgs 1.1 44.8 91.5 88.0 91.8 ## gosolnp 46.8 0.0 87.2 85.1 87.0 ## hybrid 0.5 0.0 55.0 53.6 54.7 ## nloptr+COBYLA 0.0 0.0 4.1 74.5 15.0 ## nloptr+BOBYQA 0.0 0.0 3.6 74.3 15.9 ## nloptr+PRAXIS 17.7 0.0 92.6 90.2 92.2 ## nloptr+NELDERMEAD 0.0 0.0 30.5 29.6 30.9 ## nloptr+SBPLX 0.0 0.0 3.0 82.3 11.6 ## nloptr+AUGLAG+COBYLA 0.0 0.0 4.1 74.5 15.0 ## nloptr+AUGLAG+BOBYQA 0.0 0.0 3.6 74.3 15.9 ## nloptr+AUGLAG+PRAXIS 13.0 0.0 83.4 93.9 86.7 ## nloptr+AUGLAG+NELDERMEAD 0.0 0.0 34.6 33.8 35.0 ## nloptr+AUGLAG+SBPLX 0.0 0.0 3.0 82.3 11.6 ### Model 2, $n = 100$$n = 100$ as.data.frame(round(solver_table(spec2_n100, tags, spec2) * 100, digits = 1)) ## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate ## ------------------------- ------------------ ------------- -------------------------- --------------------------- -------------------------- ## nlminb 8.2 24.2 22.3 34.7 23.9 ## solnp 0.3 0.0 21.1 32.6 21.3 ## lbfgs 11.6 29.5 74.9 73.2 70.4 ## gosolnp 19.0 0.0 31.9 41.2 30.8 ## hybrid 0.3 0.0 21.1 32.6 21.3 ## nloptr+COBYLA 0.0 0.0 20.5 94.7 61.7 ## nloptr+BOBYQA 0.2 0.0 19.3 95.8 62.2 ## nloptr+PRAXIS 16.0 0.0 70.2 57.2 52.8 ## nloptr+NELDERMEAD 0.2 0.0 7.8 27.8 14.1 ## nloptr+SBPLX 0.1 0.0 24.9 91.0 65.0 ## nloptr+AUGLAG+COBYLA 0.0 0.0 21.2 95.1 62.5 ## nloptr+AUGLAG+BOBYQA 0.9 0.0 20.1 96.2 62.5 ## nloptr+AUGLAG+PRAXIS 38.8 0.0 70.4 57.2 52.7 ## nloptr+AUGLAG+NELDERMEAD 14.4 0.0 10.7 26.0 16.1 ## nloptr+AUGLAG+SBPLX 0.1 0.0 25.8 91.9 65.5 ### Model 2, $n = 500$$n = 500$ as.data.frame(round(solver_table(spec2_n500, tags, spec2) * 100, digits = 1)) ## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate ## ------------------------- ------------------ ------------- -------------------------- --------------------------- -------------------------- ## nlminb 1.7 1.6 35.0 37.2 34.2 ## solnp 0.1 0.2 46.2 48.6 45.3 ## lbfgs 2.2 38.4 85.2 88.1 82.3 ## gosolnp 5.2 0.0 74.9 77.8 72.7 ## hybrid 0.1 0.0 46.1 48.5 45.2 ## nloptr+COBYLA 0.0 0.0 8.2 100.0 40.5 ## nloptr+BOBYQA 0.0 0.0 9.5 100.0 41.0 ## nloptr+PRAXIS 17.0 0.0 83.8 85.1 81.0 ## nloptr+NELDERMEAD 0.0 0.0 26.9 38.2 27.0 ## nloptr+SBPLX 0.0 0.0 8.2 100.0 40.2 ## nloptr+AUGLAG+COBYLA 0.0 0.0 8.2 100.0 40.5 ## nloptr+AUGLAG+BOBYQA 0.0 0.0 9.5 100.0 41.0 ## nloptr+AUGLAG+PRAXIS 77.8 0.0 84.4 85.4 81.3 ## nloptr+AUGLAG+NELDERMEAD 1.1 0.0 32.5 40.3 32.3 ## nloptr+AUGLAG+SBPLX 0.0 0.0 8.2 100.0 40.2 ### Model 2, $n = 1000$$n = 1000$ as.data.frame(round(solver_table(spec2_n1000, tags, spec2) * 100, digits = 1)) ## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate ## ------------------------- ------------------ ------------- -------------------------- --------------------------- -------------------------- ## nlminb 2.7 0.7 64.1 68.0 63.8 ## solnp 0.0 0.0 70.1 73.8 69.8 ## lbfgs 0.0 43.4 90.6 91.5 89.9 ## gosolnp 3.2 0.0 87.5 90.3 86.9 ## hybrid 0.0 0.0 70.1 73.8 69.8 ## nloptr+COBYLA 0.0 0.0 2.3 100.0 20.6 ## nloptr+BOBYQA 0.0 0.0 2.5 100.0 22.6 ## nloptr+PRAXIS 14.1 0.0 89.1 91.3 88.5 ## nloptr+NELDERMEAD 0.0 0.0 46.3 55.6 45.4 ## nloptr+SBPLX 0.0 0.0 2.2 100.0 19.5 ## nloptr+AUGLAG+COBYLA 0.0 0.0 2.3 100.0 20.6 ## nloptr+AUGLAG+BOBYQA 0.0 0.0 2.5 100.0 22.6 ## nloptr+AUGLAG+PRAXIS 85.5 0.0 89.1 91.3 88.5 ## nloptr+AUGLAG+NELDERMEAD 0.3 0.0 51.9 58.2 51.3 ## nloptr+AUGLAG+SBPLX 0.0 0.0 2.2 100.0 19.5 These tables already reveal a lot of information. In general it seems that the AUGLAG-PRAXIS method (the augmented Lagrangian method using the principal axis solver) provided in NLOpt does best for model 2 especially for large sample sizes, while for model 1 the gosolnp method, which uses the solnp solver by Yinyu Ye but with random initializations and restarts, seems to win out for larger sample sizes. The bigger story, though, is the failure of any method to be the “best”, especially in the case of smaller sample sizes. While there are some optimizers that consistently fail to attain the maximum log-likelihood, no optimizer can claim to consistently obtain the best result. Additionally, different optimizers seem to perform better with different models. The implication for real-world data–where the true model parameters are never known–is to try every optimizer (or at least those that have a chance of maximizing the log-likelihood) and pick the results that yield the largest log-likelihood. No algorithm is trustworthy enough to be the go-to algorithm. Let’s now look at plots of the estimated distribution of the parameters. First comes a helper function. library(ggplot2) solver_density_plot <- function(param, tags, list_reslist, sample_sizes, spec) { # Given a parameter, creates a density plot for each solver's distribution # at different sample sizes # # Args: # param: A string for the parameter to plot # tags: A character vector containing the solver names # list_reslist: A list of lists created by optMethodSimsf, one for each # sample size # sample_sizes: A numeric vector identifying the sample size corresponding # to each object in the above list # spec: A ugarchspec object containing the specification that generated the # datasets # # Returns: # A ggplot object containing the plot generated p <- [email protected]$fixed.pars[[param]]
nlist <- lapply(list_reslist, function(l) {
optlist <- lapply(tags, function(t) {
return(na.omit(optMethodSims_getAllVals(param, t, l)))
})
names(optlist) <- tags
df <- stack(optlist)
names(df) <- c("param", "optimizer")
return(df)
})
ndf <- do.call(rbind, nlist)
ndf\$n <- rep(sample_sizes, times = sapply(nlist, nrow))
ggplot(ndf, aes(x = param)) + geom_density(fill = "black", alpha = 0.5) +
geom_vline(xintercept = p, color = "blue") +
facet_grid(optimizer ~ n, scales = "free_y")
}
Now for plots.
### Estimated $\omega$$\omega$, model 1
solver_density_plot("omega", tags, list(spec1_n100, spec1_n500, spec1_n1000),
c(100, 500, 1000), spec1)
### Estimated $\alpha$$\alpha$, model 1
solver_density_plot("alpha1", tags, list(spec1_n100, spec1_n500, spec1_n1000),
c(100, 500, 1000), spec1)
### Estimated $\beta$$\beta$, model 1
solver_density_plot("beta1", tags, list(spec1_n100, spec1_n500, spec1_n1000),
c(100, 500, 1000), spec1)
Bear in mind that there are only 1,000 simulated series and the optimizers produce solutions for each series, so in principle optimizer results should not be independent, yet the only time these density plots look the same is when the optimizer performs terribly. But even when an optimizer isn’t performing terribly (as is the case for the gosolnp, PRAXIS, and AUGLAG-PRAXIS methods) there’s evidence of artifacts around 0 for the estimates of $\omega$ and $\alpha$ and 1 for $\beta$. These artifacts are more pronounced for smaller sample sizes. That said, for the better optimizers the estimators look almost unbiased, especially for $\omega$ and $\alpha$, but their spread is large even for large sample sizes, especially for $\beta$‘s estimator. That’s not the case for the AUGLAG-PRAXIS optimizer, though; it appears to produce biased estimates.
Let’s look at plots for model 2.
### Estimated $\omega$$\omega$, model 2
solver_density_plot("omega", tags, list(spec2_n100, spec2_n500, spec2_n1000),
c(100, 500, 1000), spec2)
### Estimated $\alpha$$\alpha$, model 2
solver_density_plot("alpha1", tags, list(spec2_n100, spec2_n500, spec2_n1000),
c(100, 500, 1000), spec2)
### Estimated $\beta$$\beta$, model 2
solver_density_plot("beta1", tags, list(spec2_n100, spec2_n500, spec2_n1000),
c(100, 500, 1000), spec2)
The estimators don’t struggle as much for model 2, but the picture is still hardly rosy. The PRAXIS and AUGLAG-PRAXIS methods seem to perform well, but far from spectacularly for small sample sizes.
So far, my experiments suggest practitioners should not rely on any one optimizer but instead to try different ones and choose the results that have the largest log-likelihood. Suppose we call this optimization routine the “best” optimizer. how does this optimizer perform?
Let’s find out.
### Model 1, $n = 100$$n = 100$
as.data.frame(round(solver_table(
optMethodSims_getBestVals(spec1_n100, reslike = TRUE),
"best", spec1) * 100,
digits = 1))
## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate
## ----- ------------------ ------------- -------------------------- --------------------------- --------------------------
## best 100 0 49.5 63.3 52.2
### Model 1, $n = 500$$n = 500$
as.data.frame(round(solver_table(
optMethodSims_getBestVals(spec1_n500, reslike = TRUE),
"best", spec1) * 100,
digits = 1))
## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate
## ----- ------------------ ------------- -------------------------- --------------------------- --------------------------
## best 100 0 86 88.8 86.2
### Model 1, $n = 1000$$n = 1000$
as.data.frame(round(solver_table(
optMethodSims_getBestVals(spec1_n1000, reslike = TRUE),
"best", spec1) * 100,
digits = 1))
## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate
## ----- ------------------ ------------- -------------------------- --------------------------- --------------------------
## best 100 0 92.8 90.3 92.4
### Model 2, $n = 100$$n = 100$
as.data.frame(round(solver_table(
optMethodSims_getBestVals(spec2_n100, reslike = TRUE),
"best", spec2) * 100,
digits = 1))
## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate
## ----- ------------------ ------------- -------------------------- --------------------------- --------------------------
## best 100 0 55.2 63.2 52.2
### Model 2, $n = 500$$n = 500$
as.data.frame(round(solver_table(
optMethodSims_getBestVals(spec2_n500, reslike = TRUE),
"best", spec2) * 100,
digits = 1))
## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate
## ----- ------------------ ------------- -------------------------- --------------------------- --------------------------
## best 100 0 83 86.3 80.5
### Model 2, $n = 1000$$n = 1000$
as.data.frame(round(solver_table(
optMethodSims_getBestVals(spec2_n1000, reslike = TRUE),
"best", spec2) * 100,
digits = 1))
## Maximization Rate Failure Rate omega 95% CI Capture Rate alpha1 95% CI Capture Rate beta1 95% CI Capture Rate
## ----- ------------------ ------------- -------------------------- --------------------------- --------------------------
## best 100 0 88.7 91.4 88.1
Bear in mind that we evaluate the performance of the “best” optimizer by the CI capture rate, which should be around 95%. The “best” optimizer obviously has good performance but does not outperform all optimizers. This is disappointing; I had hoped that the “best” optimizer would have the highly desirable property of a 95% capture rate. Performance is nowhere near that except for larger sample sizes. Either the standard errors are being underestimated or for small sample sizes the Normal distribution poorly describes the actual distribution of the estimators (which means multiplying by two does not lead to intervals with the desired confidence level).
Interestingly, there is no noticeable difference in performance between the two models for this “best” estimator. This suggests to me that the seemingly better results for models often seen in actual data might be exploiting the bias of the optimizers.
Let’s look at the distribution of the estimated parameters.
### Estimated $\omega$$\omega$, model 1
solver_density_plot("omega", "best",
lapply(list(spec1_n100, spec1_n500, spec1_n1000),
function(l) {optMethodSims_getBestVals(l,
reslike = TRUE)}),
c(100, 500, 1000), spec1)
### Estimated $\alpha$$\alpha$, model 1
solver_density_plot("alpha1", "best",
lapply(list(spec1_n100, spec1_n500, spec1_n1000),
function(l) {optMethodSims_getBestVals(l,
reslike = TRUE)}),
c(100, 500, 1000), spec1)
### Estimated $\beta$$\beta$, model 1
solver_density_plot("beta1", "best",
lapply(list(spec1_n100, spec1_n500, spec1_n1000),
function(l) {optMethodSims_getBestVals(l,
reslike = TRUE)}),
c(100, 500, 1000), spec1)
### Estimated $\omega$$\omega$, model 2
solver_density_plot("omega", "best",
lapply(list(spec2_n100, spec2_n500, spec2_n1000),
function(l) {optMethodSims_getBestVals(l,
reslike = TRUE)}),
c(100, 500, 1000), spec2)
### Estimated $\alpha$$\alpha$, model 2
solver_density_plot("alpha1", "best",
lapply(list(spec2_n100, spec2_n500, spec2_n1000),
function(l) {optMethodSims_getBestVals(l,
reslike = TRUE)}),
c(100, 500, 1000), spec2)
### Estimated $\beta$$\beta$, model 2
solver_density_plot("beta1", "best",
lapply(list(spec2_n100, spec2_n500, spec2_n1000),
function(l) {optMethodSims_getBestVals(l,
reslike = TRUE)}),
c(100, 500, 1000), spec2)
The plots suggest that the “best” estimator still shows some pathologies even though it behaves less poorly than the other estimators. I don’t see evidence for bias in parameter estimates regardless of choice of model but I’m not convinced the “best” estimator truly maximizes the log-likelihood, especially for smaller sample sizes. the estimates for $\beta$ are especially bad. Even if the standard error for $\beta$ should be large I don’t think it should show the propensity for being zero or one that these plots reveal.
# Conclusion
I initially wrote this article over a year ago and didn’t publish it until now. The reason for the hang up was because I wanted a literature review of alternative ways to estimate the parameters of a GARCH model. Unfortunately I never completed such a review, and I’ve decided to release this article regardless.
Life happens, though, and I didn’t complete this review. The project moved on and the problem of estimating GARCH model parameters well was essentially avoided. That said, I want to revisit this point, perhaps exploring how techniques such as simulated annealing do for estimating GARCH model parameters.
So for now, if you’re a practitioner, what should you do when estimating a GARCH model? I would say don’t take for granted that the default estimation procedure your package uses will work. You should explore different procedure and different parameter choices and go with the results that lead to the largest log-likelihood value. I showed how this could be done in an automated fashion but you should be prepared to manually pick the model with the best fit (as determined by the log-likelihood). If you don’t do this the model you estimated may not actually be the one for which theory works.
I will say it again, one last time, in the last sentence of this article for extra emphasis: don’t take numerical techniques and results for granted!
sessionInfo()
## R version 3.4.2 (2017-09-28)
## Platform: i686-pc-linux-gnu (32-bit)
## Running under: Ubuntu 16.04.2 LTS
##
## Matrix products: default
## BLAS: /usr/lib/libblas/libblas.so.3.6.0
## LAPACK: /usr/lib/lapack/liblapack.so.3.6.0
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] parallel stats graphics grDevices utils datasets methods
## [8] base
##
## other attached packages:
## [1] ggplot2_2.2.1 rugarch_1.3-8 printr_0.1
##
## loaded via a namespace (and not attached):
## [1] digest_0.6.16 htmltools_0.3.6
## [3] SkewHyperbolic_0.3-2 expm_0.999-2
## [5] scales_0.5.0 DistributionUtils_0.5-1
## [7] Rsolnp_1.16 rprojroot_1.2
## [9] grid_3.4.2 stringr_1.3.1
## [11] knitr_1.17 numDeriv_2016.8-1
## [13] GeneralizedHyperbolic_0.8-1 munsell_0.4.3
## [15] pillar_1.3.0 tibble_1.4.2
## [17] compiler_3.4.2 highr_0.6
## [19] lattice_0.20-35 labeling_0.3
## [21] Matrix_1.2-8 KernSmooth_2.23-15
## [23] plyr_1.8.4 xts_0.10-0
## [25] spd_2.0-1 zoo_1.8-0
## [27] stringi_1.2.4 magrittr_1.5
## [29] reshape2_1.4.2 rlang_0.2.2
## [31] rmarkdown_1.7 evaluate_0.10.1
## [33] gtable_0.2.0 colorspace_1.3-2
## [35] yaml_2.1.14 tools_3.4.2
## [37] mclust_5.4.1 mvtnorm_1.0-6
## [39] truncnorm_1.0-7 ks_1.11.3
## [41] nloptr_1.0.4 lazyeval_0.2.1
## [43] crayon_1.3.4 backports_1.1.1
## [45] Rcpp_1.0.0
Packt Publishing published a book for me entitled Hands-On Data Analysis with NumPy and Pandas, a book based on my video course Unpacking NumPy and Pandas. This book covers the basics of setting up a Python environment for data analysis with Anaconda, using Jupyter notebooks, and using NumPy and pandas. If you are starting out using Python for data analysis or know someone who is, please consider buying my book or at least spreading the word about it. You can buy the book directly or purchase a subscription to Mapt and read it there.
If you like my blog and would like to support it, spread the word (if not get a copy yourself)!
1. When I wrote this article initially, my advisor and a former student of his developed a test statistic that should detect early or late change points in a time series, including a change in the parameters of a GARCH model. My contribution to the paper we were writing included demonstrating that the test statistic detects structural change sooner than other test statistics when applied to real-world data. To be convincing to reviewers, our test statistic should detect a change that another statistic won’t detect until getting more data. This means that the change should be present but not so strong that both statistics immediately detect the change with miniscule $p$-values.
2. The profile on LinkedIn I linked to may or may not be the correct person; I’m guessing it is based on the listed occupations and history. If I got the wrong person, I’m sorry.
To leave a comment for the author, please follow the link and comment on their blog: R – Curtis Miller's Personal Website.
R-bloggers.com offers daily e-mail updates about R news and tutorials about learning R and many other topics. Click here if you're looking to post or find an R/data-science job.
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...
|
2020-01-27 15:59:07
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 122, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44790351390838623, "perplexity": 3606.7265513081043}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251700988.64/warc/CC-MAIN-20200127143516-20200127173516-00442.warc.gz"}
|
http://exxamm.com/blog/Blog/13650/zxcfghfgvbnm4?Class%2012
|
Chemistry Close Packed Structure Packing Efficiency, Calculations Involving Unit Cell Dimensions
Click for Only Video
### Topic to be covered
• Close Packed Structures :
• Formula of a Compound and Number of Voids Filled :
• Packing Efficiency
• Packing Efficiency in hcp and ccp Structures :
• Efficiency of Packing in Body-Centred Cubic Structures :
• Packing Efficiency in Simple Cubic Lattice :
• Calculations Involving Unit Cell Dimensions :
### 𝐂𝐥𝐨𝐬𝐞 𝐏𝐚𝐜𝐤𝐞𝐝 𝐒𝐭𝐫𝐮𝐜𝐭𝐮𝐫𝐞𝐬 :
In solids constituent particles are closed-packed, leaving the minimum vacant space. Constituent particles are considered as identical hard spheres.
(𝐢) 𝐂𝐥𝐨𝐬𝐞 𝐏𝐚𝐜𝐤𝐢𝐧𝐠 𝐢𝐧 𝐎𝐧𝐞 𝐃𝐢𝐦𝐞𝐧𝐬𝐢𝐨𝐧 (𝟏 𝐃) : There is only one way of arranging spheres in one dimensional. We can arrange them in a row and touching each other (Fig 1.13).
The number of nearest neighbours of a particle is called its co-ordination number. Thus in 1-D close packed arrangement the co-ordination number is 2.
color{purple}(✓✓)color{purple} " DEFINITION ALERT"
The number of nearest neighbouring particles around a specific particle in a given crystalline substance is called as coordination number of that crystalline substance
(𝐢𝐢) 𝐂𝐥𝐨𝐬𝐞 𝐏𝐚𝐜𝐤𝐢𝐧𝐠 𝐢𝐧 𝐓𝐰𝐨 𝐃𝐢𝐦𝐞𝐧𝐬𝐢𝐨𝐧𝐬 (𝟐 𝐃) : Two dimensional close packed structure can be generated in two different ways.
(a) One row may be placed over other row in a way that the spheres of the one row are exactly above those of the other row and doing so we get A A A..... type of arrangement as shown in fig 1.14 (a).
=> In this each sphere is in contact with 4 of its neighbours.
Therefore, its co-ordination number is 4. It is also called text(square close packing in two dimensions).
(b) In this second row may be placed above the first one in a staggered manner such that its sphere fit in the depression of the first row. This is ABAB..... type arrangement.
In this arrangement there is less free space. Each sphere is in contact with 6 neighbouring spheres.
Therefore, its co-ordination number is 6. This is also called text(two dimensional hexagonal close-packing) (Fig 1.14 b).
There are some voids (empty spaces) in this arrangement. These are triangular in shape. These are of two types i.e. upward triangular voids and downwards triangular voids.
(𝐢𝐢𝐢) 𝐂𝐥𝐨𝐬𝐞 𝐏𝐚𝐜𝐤𝐢𝐧𝐠 𝐢𝐧 𝐓𝐡𝐫𝐞𝐞 𝐃𝐢𝐦𝐞𝐧𝐬𝐢𝐨𝐧𝐬 (𝟑𝐃) : These are obtained by stacking two dimensional layers one above the other. Types of 3-D close-packed structure obtained is :
(a) Three Dimensional Close Packing from Two Dimensional Square Closed Packed Layers : The second layer is placed over the first layer such that the spheres of the upper layer are exactly above those of the first layer (Fig 1.15). This is A A A...... type pattern. This is simple cubic lattice and its unit cell is the primitive cubic unit cell (Fig. 1.9).
(b) Three Dimensional Close Packing from Two Dimensional Hexagonal Close Packed Layers : This is done in following ways :
(A) Placing Second Layer over the First Layer : Second layer is placed above the first layer in a way that it covers the depressions of the first layers. It is observed that all triangular voids are not covered (Fig 1.16).
Wherever a sphere of the second layer is above the void of the first layer (or vice-versa), a void is formed called text(tetrahedral void) because a tetrahedron is formed when the centres of these four spheres are joined. It is marked as T in Fig 1.16. See Fig 1.17 also.
At some places the triangular voids in the second layer are above the triangular voids in the first layer and voids formed by this are called text(octahedral voids) because these voids are surrounded by six sphere. This is marked as O in Fig 1.16. See Fig 1.17 also.
The number of these voids depends upon the number of close packed spheres.
Let the number of close packed spheres be N, then :
The number of octahedral voids generated = N
The number of tetrahedral voids generated = 2N
(B) Placing Third Layer over the Second Layer : In this case, there are two possibilities.
(𝐈) 𝐂𝐨𝐯𝐞𝐫𝐢𝐧𝐠 𝐓𝐞𝐭𝐫𝐚𝐡𝐞𝐝𝐫𝐚𝐥 𝐕𝐨𝐢𝐝𝐬 : Tetrahedral voids of the second layer are covered by the spheres of third layer. So, the spheres of third layer are exactly aligned with those of the first layer. So, we get ABAB...... pattern. This structure is called text[hexagonal close packed (hcp)] structure (Fig 1.18). e.g. Zn and Mg.
(𝐈𝐈) 𝐂𝐨𝐯𝐞𝐫𝐢𝐧𝐠 𝐎𝐜𝐭𝐚𝐡𝐞𝐝𝐫𝐚𝐥 𝐕𝐨𝐢𝐝𝐬 : In this case, third layer is placed in such a way that its spheres cover the octahedral voids. In this way, the spheres of the third layer are not aligned with spheres of either first or third layer. So, ABCABC......... pattern is obtained. This structure is called text[cubic close packed (ccp)] or text[face-centred cubic (fcc)] structure. e.g Ag and Cu.
Note : (x) Both these types of packing are highly efficient.
(y) 74% space is filled.
(z) Each sphere is in contact with twelve spheres. So, co-ordination number is 12.
### Formula of a Compound and Number of Voids Filled :
In ionic solids the bigger ions (anions) from the close packed structure and the smaller ions (usually cations) occupy the tetrahedral or octahedral voids according to its size. All octahedral or tetrahedral voids are not occupied and the fraction of occupancy of tetrahedral and octahedral voids depends upon the chemical formula of the compound.
Q 2605780668
A compound is formed by two elements X and Y. Atoms of the element Y (as anions) make ccp and those of the element X (as cations) occupy all the octahedral voids. What is the formula of the compound?
Solution:
The ccp lattice is formed by the element Y. The number of octahedral voids generated would be equal to the number of atoms of Y present in it. Since all the octahedral voids are occupied by the atoms of X, their number would also be equal to that of the element Y. Thus, the atoms of elements X and Y are present in equal numbers or 1 : 1 ratio. Therefore, the formula of the compound is XY.
Q 2615780669
Atoms of element B form hcp lattice and those of the element A occupy 2/3rd of tetrahedral voids. What is the formula of the compound formed by the elements A and B?
Solution:
The number of tetrahedral voids formed is equal to twice the number of atoms of element B and only 2/3rd of these are occupied by the atoms of element A. Hence the ratio of the number of atoms of A and B is 2 × (2/3) : 1 or 4 : 3 and the formula of the compound is A_4B_3.
### Locating Tetrahedral and Octahedral Voids
We know that close packed structures have both tetrahedral and octahedral voids. Let us take 𝐜𝐜𝐩 (or 𝐟𝐜𝐜) structure and locate these voids in it.
(𝐚) 𝐋𝐨𝐜𝐚𝐭𝐢𝐧𝐠 𝐓𝐞𝐭𝐫𝐚𝐡𝐞𝐝𝐫𝐚𝐥 𝐕𝐨𝐢𝐝𝐬 : Let us consider a unit cell of 𝐜𝐜𝐩 or 𝐟𝐜𝐜 lattice [Fig. 1(a)]. The unit cell is divided
into eight small cubes. Each small cube has atoms at alternate corners [Fig. 1(a)]. In all, each small cube has 4 atoms. When joined to each other, they make a regular tetrahedron. Thus, there is one tetrahedral void in each small cube and eight tetrahedral voids in total. Each of the eight small cubes have one void in one unit cell of 𝐜𝐜𝐩 structure. We know that 𝐜𝐜𝐩 structure has 4 atoms per unit cell. Thus, the number of tetrahedral voids is twice the number of atoms.
(𝐛) 𝐋𝐨𝐜𝐚𝐭𝐢𝐧𝐠 𝐎𝐜𝐭𝐚𝐡𝐞𝐝𝐫𝐚𝐥 𝐕𝐨𝐢𝐝𝐬 : Let us again consider a unit cell of 𝐜𝐜𝐩 or 𝐟𝐜𝐜 lattice [Fig. 2(a)]. The body centre of the cube, C is not occupied but it is surrounded by six atoms on face centres. If these face centres are joined, an octahedron is generated. Thus, this unit cell has one octahedral void at the body centre of the cube.
Besides the body centre, there is one octahedral void at the centre of each of the 12 edges. [Fig. 2(b)]. It is surrounded by six atoms, four belonging to the same unit cell (2 on the corners and 2 on face centre) and two belonging to two adjacent unit cells. Since each edge of the cube is shared between four adjacent
unit cells, so is the octahedral void located on it. Only 1/4 th of each void belongs
to a particular unit cell.
Thus in cubic close packed structure:
Octahedral void at the body-centre of the cube = 1
12 octahedral voids located at each edge and shared between four unit cells = 12 xx 1/4 = 3
therefore Total number of octahedral voids = 4
We know that in 𝐜𝐜𝐩 structure, each unit cell has 4 atoms. Thus, the number of octahedral voids is equal to this number.
### Packing Efficiency :
It is the percentage of total space filled by the particles. Let us calculate the packing efficiency in different types of structures.
### Packing Efficiency in hcp and ccp Structures :
Both types of close packing (hcp and ccp) are equally efficient. Let us calculate the efficiency of packing in ccp structure. In Fig. 1.20, let the unit cell edge length be ‘a’ and face diagonal AC = b.
In Delta ABC,
AC^2 = b^2 = BC^2 +AB^2
= a^2 + a^2 = 2a^2 or b = sqrt2a
If r is the radius of the sphere, we find
b = 4 r = sqrt2a or a = (4 r)/sqrt2 = 2 sqrt2 r (we can also write r = a/(2 sqrt2))
We know, that each unit cell in ccp structure has effectively 4 spheres. Total volume of four spheres is equal to 4xx (4/3) pir^3 and volume of the cube is a^3 or (2 sqrt2 r)^3
Therefore, text(Packing efficiency) = (text(Volume occupied by four spheres in the unit cell) xx 100) /text(Total volume of the unit cell) %
= (4xx (4/3) pi r^3 xx 100)/(2 sqrt2r)^3 %
= ((16/3) pi r^3 xx 100)/(16 sqrt2 r^3) % = 74%
### Efficiency of Packing in Body-Centred Cubic Structures :
From Fig.1.21, it is clear that the atom at the centre will be in touch with the other two atoms diagonally arranged.
In Delta EFD,
b^2 = a^2 + a^2 = 2a^2
b = sqrt2a
Now in Delta AFD,
c^2 = a^2 + b^2 = a^2 + 2a^2 = 3a^2
c = sqrt3a
The length of the body diagonal c is equal to 4r, where r is the radius of the sphere (atom), as all the three spheres along the diagonal touch each other.
Therefore, sqrt3a = 4r
a = (4 r)/sqrt3
Also, we can write r = sqrt3/4 a
In this type of structure, total number of atoms is 2 and their volume is 2xx (4/3) pi r^3
Volume of the cube, a^3 will be equal to (4/sqrt3 r)^3 or a^3 = (4/ sqrt3 r)^3
Therefore, text (Packing efficiency ) = (text(Volume occupied by two spheres in the unit cell) xx 100)/text(Total volume of the unit cell) %
= (2xx (4/3) pir^3 xx100)/([4/sqrt3 r]^3) %
= ((8/3) pi r^3 xx 100)/(64//(3 sqrt3)r^3) % = 68 %
### Packing Efficiency in Simple Cubic Lattice :
In a simple cubic lattice the atoms are located only on the corners of the cube. The particles touch each other along the edge (Fig. 1.22). Thus, the edge length or side of the cube ‘a’, and the radius of each particle, r are related as
a = 2r
The volume of the cubic unit cell = a^3 = (2r)^3 = 8r^3
Since a simple cubic unit cell contains only 1 atom,
The volume of the occupied space = 4/3 pi r^3
therefore text(Packing efficiency) = text(Volume of one atom)/text(Volume of cubic unit cell) xx100%
(4/3 pi r^3)/(8 r^3) xx 100 = pi/6 xx 100
= 52.36 % = 52.4 %
Thus, we may conclude that ccp and hcp structures have maximum packing efficiency.
### Calculations Involving Unit Cell Dimensions :
Unit cell dimensions can be used to calculate the volume of the unit cell.
And if density is known, we can calculate the mass of the atoms in the unit cell. And knowing the mass of a single atom, Avogadro constant can be determined.
Let a be the edge length of the unit cell of a cubic crystal determined by X-ray diffraction.
d be the density of the solid.
M be the molar mass.
In case of cubic crystal :
Volume of a unit cell = a^3
Mass of the unit cell = number of atoms in unit cell × mass of each atom = z × m
(Here z is the number of atoms present in one unit cell and m is the mass of a single atom.)
Mass of an atom present in the unit cell :
m = M/N_A \ \ \ \ (M text(is molar mass))
Therefore, density of the unit cell = text(mass of unit cell)/text(volume of unit cell)
= ( z * m)/a^3 = ( z * M)/(a^3 * N_A) or d = (z M)/(a^3 N_A)
Note : The density of the unit cell is the same as the density of the substance.
Q 2635880762
An element has a body-centred cubic (bcc) structure with a cell edge of 288 pm. The density of the element is 7.2 g//cm^3. How many atoms are present in 208 g of the element?
Solution:
Volume of the unit cell = (288 text(pm) )^3
= (288xx10^(-12) m) = (288xx10^(-10) cm)^3
= 2.39xx10^(-23) cm^3
Volume of 208 g of the element
= text(mass)/text(density) = (208 g)/(7.2 g cm^(-3)) = 28.88 cm^(3)
Number of unit cells in this volume
= (28.88 cm^3)/((2.39xx10^(-23) cm^3)/text( unit cell)) = 12.08xx10^(23) unit cells
Since each bcc cubic unit cell contains 2 atoms, therefore, the total number
of atoms in 208 g = 2 text((atoms/unit cell)) × 12.08 × 10^(23) unit cells
= 24.16×10^(23) atoms
Q 2645880763
X-ray diffraction studies show that copper crystallises in an fcc unit cell with cell edge of 3.608×10^(-8) cm. In a separate experiment, copper is determined to have a density of 8.92 g//cm^3, calculate the atomic mass of copper.
Solution:
In case of fcc lattice, number of atoms per unit cell, z = 4 atoms
Therefore, M = ( d xx N_A xxa^3)/z
= (8.92 g cm^3 xx 6.022xx10^(23) text(atom) mol^(-1) xx (3.608xx10^(-8) cm)^3)/text(4 atoms)
= 63.1 g//mol
Atomic mass of copper = 63.1u
Q 2655880764
Silver forms ccp lattice and X-ray studies of its crystals show that the edge length of its unit cell is 408.6 pm. Calculate the density of silver (Atomic mass = 107.9 u).
Solution:
Since the lattice is ccp, the number of silver atoms per unit cell = z = 4
Molar mass of silver = 107.9 g mol^(-1) = 107.9 xx 10^(-3) kg mol^(-1)
Edge length of unit cell = a = 408.6 pm = 408.6xx10^(-12) m
Density d = ( z * M)/(a^3 * N_A)
= (4xx (107.9xx10^9-3) kg mol^(-1))/((408.6xx10^(-12) m)^3 (6.022xx10^(23) mol^(-1))) = 10.5xx10^3 kg m^(-3)
= 10.5 g cm^(-3)
|
2018-07-23 11:28:23
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6085159778594971, "perplexity": 2100.0270227694227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676596336.96/warc/CC-MAIN-20180723110342-20180723130342-00500.warc.gz"}
|
http://aquarestaurant.pl/gye14/division-algorithm-discrete-math-547fb3
|
# division algorithm discrete math
Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. The Euclidean algorithm is basically a continual repetition of the division algorithm for integers. MATH 3336-02 (12495), DISCRETE MATHEMATICS, MoWe 16:00-17:30, SEC 105 Office: 607 PGH, Phone: 713-743-3462, email: klaus@math.uh.edu, Office Hours: TTH 12-13, You can always send email to klaus@math.uh.edu Grader: Basanta Pahari email: brpahari@math.uh.edu Office hours: W eekdays in PGH 688 until 2pm Free Tutoring service If f(x), g(x) ∈ F[x], with g(x) nonzero: f(x) = q(x)g(x)+r(x) PDF . Autoplay; Autocomplete; HTML5 Flash. We have, Tony Hsieh, iconic Las Vegas entrepreneur, dies at 46. r is called the remainder. The point is to repeatedly divide the divisor by the remainder until the remainder is 0. These notes are based on the course “Discrete Mathematics” given by Dr. J. Saxl in Cambridge in the Michælmas Term 1995. Is there a formal proof for this algorithm that demonstrates that the algorithm will always return a result big enough so that the mantissa of the result can be cut off because of integer division? q = a div d r is called the remainder. Example 1: Divide 3x 3 + 16x 2 + 21x + 20 by x + 4. Comments and corrections to soc-archim-notes@lists.cam.ac.uk. Integers and Division °c Theodore Norvell, Memorial University Starting point. d is called the divisor. Spell. This book easily ranks as my favorite lower-division math/computer science textbook. Division algorithms. One should spend 1 hour daily for 2-3 months to learn and assimilate Discrete Mathematics comprehensively. Quotient = 3x 2 + 4x + 5 Remainder = 0. At the time of typing these courses were: Probability Discrete Mathematics Analysis Further Analysis Division algorithm . Discrete mathematics, the study of finite systems, has become increasingly important as the computer age has advanced. }\) Recursive Definition. TOPIC: ELECTION THEORY AND FAIR DIVISION DISCRETE MATHEMATICS STANDARD DM.7 The student will analyze and describe the issue of fair division (e.g., cake cutting, estate division). Algorithms. The GCD is the last non-zero remainder in this algorithm. . Typical Scheduling: Every Semester . Course Number: 2603. STUDY. Example 2: Apply the division algorithm to find the quotient and remainder on dividing p(x) by g(x) as given below : p(x) = x 3 – 3x 2 + 5x – 3 and g(x) = x 2 – 2 Sol. Discrete Mathematics Dr. J. Saxl Michælmas 1995 These notes are maintained by Paul Metcalfe. John, Jerry, and Jill are heirs to their mother's estate that includes their family house, an automobile, a small mountain cabin, and \$125,000 in cash (from investments and a life insurance policy). Learn. •An Active Introduction to Discrete Mathematics and Algorithms, 2014, Charles A. Cusack. Discrete Math. Publication Date: November 6, 2015; ISBN: 978-1-9423411-6-1; OCLC: 950573750; Affiliation: SUNY Fredonia; Author(s): Harris Kwong. Hours - Recitation: 2. Now, since we have that F is a field, we can do something similar with the polynomials over F, F[x]. Other sets of notes are available for different courses. • Besides the WOP, we’ll assume that the basic facts of addition, subtraction, multiplication, and comparison are all understood for the integers and the natural numbers. Human-readable algorithm language, not required to follow strict syntactic rules. CSE 20: Discrete Mathematics for Computer ScienceProf. PLAY. 1. Then there are unique integersThen there are unique integers qq andand rr, with, with 00 ≤≤ r < dr < d, such that, such that a = dq + ra = dq + r.. Our class meets Tuesdays and Thursdays, 8:00-9:30am in 145 Dwinelle Teaching assistants: Dustin Cartwright, office hours M 11-12, Fr 12:30-1:30 in 1045 Evans Richard … DiscreteMathematics is a set of algorithm implementations from Discrete Mathematics. THEOREM If a is an integer and d a positive integer, then there are unique integers q and r, with 0 ≤ r < d, such that a = dq + r a is called the dividend. Numbers: Divisibility and Division Algorithm, Euclidean Algorithm Combinatorics: Combinations, Permutations, Fundamental Principle of Counting Every lecture on these topics in discrete math is in high quality - 1080p and the powerpoint presentations are downloadable. Prerequisites: MATH 1552 or MATH … 1. okay I am confused b/c I cant find anything btwn 0 and 6, so here it is: Use the the division algorithm to find the unique integer between 0 and 6 inclusive that is congruent to modulo 7: for -101 and 144 as separate exercises. Revision: 2.3 Date: 1999/10/21 11:21:05 The following people have maintained these notes. – date Paul Metcalfe. Discrete mathematics book recommendations Hello everyone, I am an undergraduate student self studying “Invitation to Discrete Mathematics” by Jiri Matousek and Jaroslav Nesetril. Hours - Lab: 0. thanks! A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. The contents are easily understandable, but problems are quite difficult and sometimes I get stuck on a problem despite the hints at the back. Shachar Lovett. The Division Algorithm by Matt Farmer and Stephen Steward Subsection 3.2.1 Division Algorithm for positive integers. Every journey begins with one step. Feb 14, 2020 - Explore Heather Kraus's board "division algorithm" on Pinterest. Created by. Notes 4. Figure 3.2.1. DZWORDS98. … Firstly recall the division algorithm for numbers, that each number can be decomposed into the form n = qk+r. Then there are unique integers q and r, with 0 r < d, such that a = dq + r. Notation d is called the divisor. The theorem does not tell us how to find the quotient and the remainder. Eric Clapton sparks backlash over new anti-lockdown song . ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS • Group decision making combines the wishes of many to yield a single fair … Introduction to Discrete Mathematics. Hours - Total Credit: 4. Some mathematicians prefer to … a is called the dividend. Introduction []. Match. Sol. Learn the core topics of Discrete Math to open doors to Computer Science, Data Science, Actuarial Science, and more! Discrete Mathematics. Unlike real analysis and calculus which deals with the dense set of real numbers, number theory examines mathematics in discrete sets, such as N or Z.If you are unsure about sets, you may wish to revisit Set theory. In our first version of the division algorithm we start with a non-negative integer $$a$$ and keep subtracting a natural number $$b$$ until we end up with a number that is less than $$b$$ and greater than or equal to \(0\text{. This is a significant revision of the 2013 version (thus the slight change in title). ..... 1 1.2 The division algorithm . A finite list of instructions (deterministic and finite, with a set of possible inputs and outputs) Pseudo Code. Contents Introduction v 1 Integers 1 1.1 Division . Terms in this set (29) Algorithm . Discrete Math: Algorithms. Algorithms for continuous and discrete cases will be applied. where q is the quotient and r the remainder and r
2014 Nissan Armada Sv, Is Bayshore Mall Open, Mph In Sindh University, Uruguay Pronunciation English, Alberta Corporate Account Number, Uruguay Pronunciation English, Sanding Sealer Spray, Kpr Cimb Niaga Syariah, Rose Hotel Pleasanton Check In Time, Bdo Nomura Unable To Login, 2008 Ford Fusion Fuse Box Location,
|
2021-05-15 04:11:27
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48582345247268677, "perplexity": 2240.324609367641}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989812.47/warc/CC-MAIN-20210515035645-20210515065645-00198.warc.gz"}
|
https://www.trccompsci.online/mediawiki/index.php/Relating_models_in_Django,_ie_Primary_key_to_Foreign_Key
|
# Relating models in Django, ie Primary key to Foreign Key
To complete this tutorial, you must have created a Product model in the previous tutorials.
# models.py
Within your app folder, find the 'models.py' file and edit it, It should already contain a model for 'Product'. Firstly, we are going to have 'User' as one of the foreign keys so we must import the built in 'User' model:
from django.contrib.auth.models import User
from django.core.validators import MaxValueValidator, MinValueValidator
from django.utils import timezone
The other imports will allow validation of numerical fields, and to also get the current time. Now for the model itself, enter the following:
class Review(models.Model):
rating = models.IntegerField(validators = [MinValueValidator(0), MaxValueValidator(10)])
text = models.TextField()
date = models.DateTimeField( default=timezone.now )
def __str__(self):
return self.author.username + ' - ' + self.product.name + ' - ' + str(self.rating) + ' out of 10'
# Register the Model
Now in the same folder as the 'models.py' find the 'admin.py'. It should currently have this code:
from django.contrib import admin
from .models import Product
We now need to register the 'Review' model, so alter the code to this:
from django.contrib import admin
from .models import Product, Review
Now using the admin program, you will need to run the manage, and then makemigrations. You will then need to do the same and this time run migrate.
run the server and visit http:\\127.0.0.1:8000 and you should see your web app. In the address bar add '/admin' and login. You should see you model listed on the dashboard. You should be able to create new entries from this admin dashboard.
# The Form
You will need to import your model into the 'forms.py':
from .models import Product
Your django form should be able to use the 'ModelChoiceField' component. You need to create a queryset from your model, for example:
product = forms.ModelChoiceField(queryset=Product.objects.all(), empty_label="Please Select")
We should not include the user/author field because if we did it could be changed in the form.
# The View
When you process the form, the form will be valid but still fail to save. This is because we missed out the user/author field.
So in the views.py and the method to process the review you need to add the following to the 'if form.is_valid':
if form.is_valid:
temp = form.save(commit=False)
temp.user = request.user
temp.save()
messages.success(request, f'Your post has been successfully edited')
return redirect('/')
|
2022-10-04 09:41:04
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22226300835609436, "perplexity": 5886.028588232825}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337490.6/warc/CC-MAIN-20221004085909-20221004115909-00412.warc.gz"}
|
https://quantumcomputing.stackexchange.com/tags/dynamics/hot
|
# Tag Info
7
It depends on the Hamiltonian. There are three particular questions whose answers might influence your choice of strategy: Does the Hamiltonian have any particular structure or symmetry? How quickly does the Hamiltonian change in time? What do you know about the initial state in relation to the initial Hamiltonian? Obviously, if the Hamiltonian has any ...
7
Hint: Instead of using the BCH formula in the form usually presented, for example at the top of this Wikipedia page, use this consequence of Hadamard's Lemma: $$\tag{1} e^{iHt}\hat{a}e^{-iHt} = \hat{a} + [iHt,\hat{a}] + \frac{1}{2!}[iHt,[iHt,\hat{a}] + \cdots$$ Now substitute $H$ into the right-hand side and evaluate the commutators between $\hat{a}$ and ...
6
I'm going to define $|n\rangle$ to be "the walker is at site $n$". Now imagine the walk as specified: $$|n\rangle\rightarrow (|n-1\rangle+|n+1\rangle)/\sqrt{2}.$$ You can put some phases in if you want to, it's not going to change my basic argument. Now, imagine this is implemented by a unitary operator. This means that we need $$\langle n-1|U^\... 6 Generally speaking, a realization of a quantum gate involves coherent manipulation of a two-level system (but this is nothing new to you, maybe). For example, you can use two long-lived electronic states in a trapped atom (neutral or ionized in vacuo) and use an applied electric field to implement single-qubit operations (see trapped ions or optical lattices,... 5 Use the differential form of the time evolution,$$dO/dt=i[H, O]\ .$$5 Calculate$$ \begin{align} \hat{U}|00\rangle &= \exp\left(-igt(\hat{a}^\dagger_2\hat{a}_1+\hat{a}^\dagger_1\hat{a}_2)\right)|00\rangle \\ &= \sum_{k=0}^\infty \frac{(-igt)^k}{k!}(\hat{a}^\dagger_2\hat{a}_1+\hat{a}^\dagger_1\hat{a}_2)^k|00\rangle \\ &= |00\rangle + \sum_{k=1}^\infty \frac{(-igt)^k}{k!}(\hat{a}^\dagger_2\hat{a}_1+\hat{a}^\dagger_1\...
4
There's more than one way, and I'll suggest two of them here: Expand $\hat{U}$ using the formula for the Taylor series of an exponential ($e^\hat{A}$) centered around $\hat{A}=\hat{0}$, and then you will have a sum of terms where each term no longer involves an exponential operator (i.e. you have just pure creation and annihilation operators and products/...
2
Let $|\psi\rangle$ be an eigenstate of an operator $A$, $A|\psi\rangle=\lambda|\psi\rangle$. Then $$e^A |\psi\rangle = \sum_{k=0}^\infty \frac{A^k}{k!}|\psi\rangle = \sum_{k=0}^\infty \frac{\lambda^k}{k!}|\psi\rangle = e^\lambda |\psi\rangle.$$ In this particular case, $A=-igt(a_2^\dagger a_1+a_1^\dagger a_2)$, of which $|00\rangle$ is an eigenstate with ...
2
Note that $$[(a^\dagger)^n,a] = -n(a^{\dagger})^{n-1}, \qquad [(a^\dagger)^n a^m,a] = -n (a^\dagger)^{n-1}a^m, \qquad [a^n,a]=0.$$ Consider an arbitrary function of the mode operators, that we assume be written in normal formal: $$f(a,a^\dagger) = \sum_{n,m=0}^\infty c_{n,m} (a^\dagger)^n a^m.$$ We know that e^{f(a,a^\dagger)}a e^{-f(a,a^\dagger)} = \sum_{...
1
I'm not sure for this specific problem, and more broadly Hamiltonians are typically in the "eye of the beholder." For example, for quantum chemistry problems, Hamiltonians are really clean mappings from problems that were originally in quantum chemistry. For example, there are "annihilation/creation operators" (discussed more here) that could be converted ...
1
There are two possible answers. Let's say the universe evolves from $t=0$ to $t_f$ then the unitary evolution $U$ from $0$ to $t_f$ induces a CP evolution on the subsystem. To see this, note that the composition of CP maps is CP. Now, the reduced (system) evolution is $Tr_E U\rho_s\otimes\rho_E U^\dagger$ which is a composition of the map \$\rho_s\...
Only top voted, non community-wiki answers of a minimum length are eligible
|
2022-01-29 13:24:11
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9939460158348083, "perplexity": 391.8070349097525}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320306181.43/warc/CC-MAIN-20220129122405-20220129152405-00242.warc.gz"}
|
https://www.physicsforums.com/threads/derivative-of-x-sin-x.733272/
|
# Derivative of x^sin(x)
noelo2014
This is not homework it's just an equation I pulled out of the air and have been trying to solve
Derivative of x^sin(x)
I know it has something to do with the "general power rule" but I cannot figure it out, I'd really love someone to explain it to me conceptually instead of just showing me some trick or short cut to solving it. Thanks ever so much!
Gold Member
2022 Award
Hint: Write (for $x>0$)
$$x^{\sin x}=\exp(\ln x \cdot \sin x).$$
Tanya Sharma
Hi noelo2014...
The general approach to find derivative of functions of the form y=g(x)h(x) is to first take log on both the sides and then differentiate.
y=g(x)h(x)
logy = log{g(x)h(x)}
logy = h(x)log{g(x)}
(1/y)y' = h'(x)log{g(x)} + {h(x)/g(x)}g'(x)
y' = g(x)h(x)[h'(x)log{g(x)} + {h(x)/g(x)}g'(x)]
Hope this helps
Homework Helper
Gold Member
Dearly Missed
I'd rather say that is a neat trick, Tanya Sharma, rather than a "general rule". The general rule is application of the chain rule in some manner.
Here's a way to make use of the multi-variable chain rule, for a change.
Consider the function $g(x,y)=x^{y}, Y(x)=\sin(x),\to{f}(x)=g(x,Y(x))$
Thus, we easily have:
$$\frac{df}{dx}=\frac{\partial{g}}{\partial{x}}+ \frac{\partial{g}}{\partial{x}}\frac{dY}{dx}$$
1 person
Staff Emeritus
Homework Helper
noelo2014
I'd rather say that is a neat trick, Tanya Sharma, rather than a "general rule". The general rule is application of the chain rule in some manner.
Here's a way to make use of the multi-variable chain rule, for a change.
Consider the function $g(x,y)=x^{y}, Y(x)=\sin(x),\to{f}(x)=g(x,Y(x))$
Thus, we easily have:
I think there should be a ∂y/∂x there, I think it should be
$$\frac{df}{dx}=\frac{\partial{g}}{\partial{x}}+ \frac{\partial{g}}{\partial{y}}\frac{dY}{dx}$$
And thanks...
Homework Helper
Gold Member
Dearly Missed
Aargh, yes.
stevmg
So, what's the answer for the first derivative of x^sin x ?
[(1/x)sin x + (ln x)cos x](x^sin x)?
Last edited:
Gold Member
2022 Award
Just use
$$f(x)=x^{\sin x}=\exp(\sin x \ln x)$$
which holds for $x>0$ (as a real function). Then you simply take the derivative, using the chain and product rule:
$$f'(x)=\exp(\sin x \ln x) (\sin x \ln x)'=\exp(\sin x \ln x) \left (\cos x \ln x+\frac{\sin x}{x} \right )=x^{\sin x} \left (\cos x \ln x+\frac{\sin x}{x} \right ).$$
stevmg
vanhees71 -
Guess what? That's what I got (look at my post just above yours)
I guess I am not as senile as I thought.
Thanks for your input and help.
|
2023-02-05 21:19:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8003137707710266, "perplexity": 1411.9578339741543}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500288.69/warc/CC-MAIN-20230205193202-20230205223202-00617.warc.gz"}
|
https://www.speedsolving.com/threads/random-blindfold-cubing-discussion.27436/page-3
|
# Random Blindfold Cubing Discussion
#### Mike Hughey
Staff member
Then again, it can be bad even without orientation. Try this scramble:
D R2 D F2 L2 D F2 L2 U L2 U' B2 U' R F' R F L' R D2
#### riffz
##### Member
Chris, how do you use your image system for 2-3 twisted corners? I might adopt images for twisted corners.
Currently I'm memorizing and solving in the same order as you, but for twisted corners I use my feet to remember as much as I can.
If there's 1 corner twisted, I take the foot that is on the same side of the cube (L/R face) and angle it right for CW and left for CCW.
If there's 2 twisted, then if one is on R and the other on L, I can use the technique described above. Otherwise, I just kind of remember them visually.
I often stop the timer in a rush and forget to fix the twisted corners. I could definitely fix this bad habit over time, but as you can see, my foot system really only works for 1-2 images and is still shaky even for 2. If I adopted images for twisted corners I might extend my corners to 2 locations along my journey, as I currently just cram them all into one room every time.
#### RyanReese09
Is it bad I just keep placing 2 images per location in my room (if I have to cross between edges/corners mid location I put an invisible barrier in there, it works surprisingly) for multi?
#### Mike Hughey
Staff member
Is it bad I just keep placing 2 images per location in my room (if I have to cross between edges/corners mid location I put an invisible barrier in there, it works surprisingly) for multi?
No; whatever works best for you is fine. There is a tremendous amount of variety as to how it is done among the top people at multi.
However, it might make sense for you to try some other approaches, and see what works best for you. As I said, I found that putting lots of images at a single location worked really well for me, but there are other people for whom it doesn't work at all.
#### cmhardw
Chris, how do you use your image system for 2-3 twisted corners? I might adopt images for twisted corners.
I tried out Mike's idea of having my regular images just upside down. I found this was difficult for me as most of my journey locations are outdoor spaces, so I had a hard time picturing them upside down as it felt "weird" for them to be suspended in thin air. It sort of gave me the impression that the image was flying, and this would lead me to try to recall from my flying images when I got to the flipped piece.
My system now works very well for me, but it might seem a bit weird at first. Using this approach I can memorize 3/7 of all cases with 2 twisted corners using only a single letter image, 4/7 of all 2 twisted corner cases using one letter pair image, 3/8 of all three corner twisted cases with both one letter pair image and one single letter image, and 5/8 of all three corner twisted cases with two letter pair images.
For 2 flipped corners I do the following:
I memorize one edge sticker for an adjacent pair of corners that need to be flipped. I use double sunes to flipped pairs of corners, and pretty much have ever since I began BLD. The names I give the cases are from 2003 when I first started doing BLD. I give each pair a name of either "Same" or "Opposite" and I use the letters "S" and "O" to memorize during a solve.
The following are the algs I use to solve each case:
S = R' U2 R U R' U R L U2 L' U' L U' L'
O = L' U' L U' L' U2 L R U R' U R U2 R'
The only thing I changed is how I call these cases. Rather than calling the case where UFR twists counter-clockwise which UBR twists clockwise "S" I now call it "I", which is my sticker letter for the RU edge location. If I had an "O" case (UFR twists clockwise and UBR twists counter-clockwise) I would call it "C" which is my sticker letter for the UR edge location.
So now I can identify an S or O pair anywhere on the cube. The letter "K" would be an S pair on the RF edge of the cube (UFR twists clockwise and DFR twists counter-clockwise).
This means that I can memorize any adjacent pair of corners that twist opposite directions using only one letter. This accounts for 3/7 of all possible cases with 2 corners flipped.
-------
For these cases I memorize as follows. Let's say that UFL needs to twist counter-clockwise and DFR needs to twist clockwise. If I do the setup turn R, I get an S pair at the FU edge location. To memorize this case I pick any sticker on the DFR corner and I take the letter "F", which is the letter for an S pair at FU and I make a letter pair image out of them. I pick the letter that has the most memorable image when paired with F (and I make sure to avoid double pairs like FF, etc.) I would probably pick XF which is the Phantom of the Opera.
Now when I see XF during memo, I notice that I have only a letter pair image. Having only a letter pair image always means that the first letter is a corner sticker, and the second letter is the edge sticker that I need to move that particular corner sticker adjacent to, and this creates either the O or S pair at that edge sticker location.
So, for example, XF would tell me to take the DFR corner and move it adjacent to the FU location (the setup turn R does this) and then do either the S or O pair that the edge sticker represents. Afterward I would know to undo the setup turn.
This handles the 4/7 of the two corners twisted orientation cases where the two twisted corners are not adjacent to each other.
3 corners, including the buffer
For these cases I memorize one sticker from each of the non-buffer corners, as well as the twist direction. This gives me a letter pair image, followed by a single letter image. The only possible way this could happen is if 3 corners twist the same direction, one of which is the buffer corner.
So let's say that DFR, UFR, and the buffer after solving (UBL) have to all twist clockwise. I would take a sticker from each of the first two corners, X, and D respectively, and I would add a letter for the twist direction. A=clockwise B=counterclockwise. So my image here would be XD A or the Barad Dur tower from Lord of the Rings with the eye of Sauron (in miniature) being heckled by Dan Aykroyd.
When I see this during my memo I notice a letter pair image, and a single pair image. This tells me that the first two letters are stickers on permuted but disoriented corners. The last sticker is always the twist direction, so since only 2 corner stickers are represented then I know to include the buffer as well. The last letter A tells me that all 3 pieces twist clockwise to solve.
Twist three corners not including the buffer
Same idea as above, only I have 3 stickers (one from each corner) and the twist direction letter at the end. This gives me two letter pair images. The only possible way I could have two letter pair images is to have three twisted corners (not including the buffer, since all 3 corners are represented) and the last letter tells me the twist direction.
I can make sure to avoid letter doubles (AA, BB) by intelligent sticker choices during memo.
Currently I'm memorizing and solving in the same order as you, but for twisted corners I use my feet to remember as much as I can.
If there's 1 corner twisted, I take the foot that is on the same side of the cube (L/R face) and angle it right for CW and left for CCW.
If there's 2 twisted, then if one is on R and the other on L, I can use the technique described above. Otherwise, I just kind of remember them visually.
I often stop the timer in a rush and forget to fix the twisted corners. I could definitely fix this bad habit over time, but as you can see, my foot system really only works for 1-2 images and is still shaky even for 2. If I adopted images for twisted corners I might extend my corners to 2 locations along my journey, as I currently just cram them all into one room every time.
I may try to add in a foot element as a double check, but so far my image system is surprisingly easy to use, for me, and it's a relief to not have to try to hold onto the visual corner twist memo throughout the edge auditory loop memo. Auditory loop single syllable word memo is surprisingly taxing on the brain for me, so I think this is why I forget the corner twists so often. I will probably also now use two locations to memo corners on all solves.
Last edited:
#### RyanReese09
Mike, how would you know which image goes first in execution if all are on the same location? I usually determine it by the first executed is the dominant object/person. Aka the first one executed is hte one doing the killing etc...
#### Mike Hughey
Staff member
Mike, how would you know which image goes first in execution if all are on the same location? I usually determine it by the first executed is the dominant object/person. Aka the first one executed is hte one doing the killing etc...
I don't know - I guess it's just a small story for me at each location, so there's an implied order. I actually have a person representing the buffer piece that does the action on the objects, so that doesn't help. For what it's worth, I do sometimes make the mistake of swapping images, but that's far from being my most common problem - it really doesn't happen all that often.
#### cmhardw
For anyone who was following the discussion on how to memorize permuted but disoriented corners I have an update.
Memo'ing pbd corners with images makes me the opposite of a sad panda. This is totally worth making my general method for the future :tu
Then again, it can be bad even without orientation. Try this scramble:
D R2 D F2 L2 D F2 L2 U L2 U' B2 U' R F' R F L' R D2
EEeeeeeeEEEEeeeeewwwwWwww!!!!!!! *blegh*
1:52.32
That solve is horrendous. At a certain point during edge memo (after corner memo was done) I thought to myself "This is completely ridiculous" and memo'd the rest visually.
--edit--
By the way, the probability of a scramble like that is:
$$\frac{2\left( \begin{array}{ccc} 12 \\ 2 \\ \end{array} \right)\left( \begin{array}{ccc} 10 \\ 2 \\ \end{array} \right)\left[\left( \begin{array}{ccc} 8 \\ 2 \\ \end{array} \right)\left( \begin{array}{ccc} 6 \\ 2 \\ \end{array} \right)\left( \begin{array}{ccc} 4 \\ 2 \\ \end{array} \right)\right]^2}{(6!)(4!)(12!)(8!)} = \frac{1}{8847360}$$
Was this scramble purposefully constructed? Or was it generated by a program? I only just now noticed that it is only 20 turns. Plus the first 13 turns fit the {U,D,R2,L2,F2,B2} sub-group that Cube Explorer uses when solving cases in the final stage.
Last edited:
#### Mike Hughey
Staff member
Was this scramble purposefully constructed? Or was it generated by a program? I only just now noticed that it is only 20 turns. Plus the first 13 turns fit the {U,D,R2,L2,F2,B2} sub-group that Cube Explorer uses when solving cases in the final stage.
Yes, I constructed it purposefully and then used Cube Explorer. I just wanted to see you miserable. Sorry for putting you through that.
I think the worst I've ever had with a real scramble was 5 edge cycles and 2 corner cycles. Still awful, but not quite as bad as that fake scramble.
#### cmhardw
Yes, I constructed it purposefully and then used Cube Explorer. I just wanted to see you miserable. Sorry for putting you through that.
I think the worst I've ever had with a real scramble was 5 edge cycles and 2 corner cycles. Still awful, but not quite as bad as that fake scramble.
No it's fine, it gives me some perspective on how not-so-bad most scrambles really are To be honest, I probably would never have noticed that this was a faked scramble until I calculated the probability of a scramble like that. I certainly didn't notice anything fishy with the scramble algorithm at first. I only got suspicious when I saw that it was an approximately 1 in 9 million chance that a scramble like that would come up
I think I've had 4 two cycles in edges before, but I'm not sure if that was on 3x3x3 or on wings on a bigger cube. While memo'ing, when I still thought that the scramble was real, I kept thinking "bet there'll be one more two cycle. Yep... there it is. Bet there'll... yep there's another one"
Last edited:
#### Mike Hughey
Staff member
I think I've had 4 two cycles in edges before, but I'm not sure if that was on 3x3x3 or on wings on a bigger cube. While memo'ing, when I still thought that the scramble was real, I kept thinking "bet there'll be one more two cycle. Yep... there it is. Bet there'll... yep there's another one"
I'm sorry - I guess I'm a really bad person. I shouldn't be laughing this much. But I can't help it.
#### JonnyWhoopes
Then again, it can be bad even without orientation. Try this scramble:
D R2 D F2 L2 D F2 L2 U L2 U' B2 U' R F' R F L' R D2
Haha, I just tried that scramble. It's ridiculous for execution, but if you're using visual memo it's very fast. I got a 1:15 on it, and I average somewhere around 1:45.
#### aronpm
##### Member
My Letter Pair Word List. Used for everything (2-7bld, multi). It uses almost every type of word (adjectives, nouns, verbs, adverbs, etc) but I'm not studying English so maybe that's not true
Memorization examples:
scramble: F' U2 R' U' B2 L2 R D' R' B' R2 F2 L U R' B' F' D U' L F2 R' L F B
Edges: DX (Dexter) QJ ('s QJ) PK (peaked) FM (at my family) SV (so Jesus) G (is good)
Dexter's QJ peaked at my family so Jesus is good.
Corners: BU (boo) PJ (pyjama) QLS (quills)
boo pyjama quills
scramble: D2 B2 U' F' D' F2 R' U2 D B' D L2 R' B F2 U' D2 R L2 D2 L2 B2 L F2 U'
Edges: FO (my foe) GI (is Jai) DQ (because he was disqualified) JL (from jelly) D (dude)
my foe is Jai because he was disqualified from jelly dude.
Corners: SH (shutup) DUBG (doo bug) J (jay)
shut up doo bug jay
etcetc
#### Marcell
##### Member
I got a little confused over A9 and B9 comms. How come D2 M2 D R2 D' M2 D R2 D is listed as a B9? The way I see it, this alg is built up as
A: D R2 D'
B: M2
P: D2
executed as P BAB'A' P'
meaning it is an A9.
Also, the alg D2 M D R2 D' M' D R2 D has got exactly the same structure, but it is listed as an A9.
My other question is: I think that S2 D' S U' S' D S U S is a STM-optimal B9, is that right? It's just that it isn't listed on Chris' webpage.
#### cmhardw
I got a little confused over A9 and B9 comms. How come D2 M2 D R2 D' M2 D R2 D is listed as a B9? The way I see it, this alg is built up as
A: D R2 D'
B: M2
P: D2
executed as P BAB'A' P'
meaning it is an A9.
Also, the alg D2 M D R2 D' M' D R2 D has got exactly the same structure, but it is listed as an A9.
You are correct, that alg is an A9. That is either a typo or mistake when Daniel and I made the page. I'll correct it when I get home.
My other question is: I think that S2 D' S U' S' D S U S is a STM-optimal B9, is that right? It's just that it isn't listed on Chris' webpage.
That case is listed on the site:
(UR BU DF) B' R B' M B R' B' M' B2 (9 STM)** A9
Also, your alg is an A9 alg:
P = S2
A = S U' S'
B = D'
Perform as PBAB'A'P'
#### Marcell
##### Member
You are correct, that alg is an A9. That is either a typo or mistake when Daniel and I made the page. I'll correct it when I get home.
Alright, so I got it right after all. That's good news.
Also, your alg is an A9 alg:
Yeah, I just realized that that was an A9. And I know there's another A9 already in the list, but I thought you collected all the move optimal commutators for each case.
#### cmhardw
5x5x5 BLD
12:07.30 successful double alarm clock solve wearing earmuffs. I set two alarm clocks shortly before starting my solve, each for just a couple of minutes into the future. One alarm clock goes continuously until shut off, and one rings for 1 minute, then sleeps for 4, then rings for 1 , etc. I don't do this often, but it always feels amazing when I get a success while doing this!
#### riffz
##### Member
5x5x5 BLD
12:07.30 successful double alarm clock solve wearing earmuffs. I set two alarm clocks shortly before starting my solve, each for just a couple of minutes into the future. One alarm clock goes continuously until shut off, and one rings for 1 minute, then sleeps for 4, then rings for 1 , etc. I don't do this often, but it always feels amazing when I get a success while doing this!
really good time too!
#### miniGOINGS
##### Member
5x5x5 BLD
12:07.30 successful double alarm clock solve wearing earmuffs. I set two alarm clocks shortly before starting my solve, each for just a couple of minutes into the future. One alarm clock goes continuously until shut off, and one rings for 1 minute, then sleeps for 4, then rings for 1 , etc. I don't do this often, but it always feels amazing when I get a success while doing this!
I assume this is training for solving in noisy/unpredictable (even though I guess this is predictable...) environments? I hadn't really thought about that before, but it's definitely something I would do if I did blind.
#### amostay2004
##### Member
I think noises are much less of a distraction to BLD than people talking in the background. I can BLD in a noisy environment as long as I don't hear any clear conversation. It's really hard to focus when you hear a couple of people talking in the background when you're BLD-ing, because you're automatically distracted about what they are saying.
|
2022-01-27 07:52:58
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6295508742332458, "perplexity": 2090.82862605385}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305242.48/warc/CC-MAIN-20220127072916-20220127102916-00248.warc.gz"}
|
http://math.stackexchange.com/questions/176810/sum-of-all-elements-in-a-matrix
|
# Sum of all elements in a matrix
The trace is the sum of the elements on the diagonal of a matrix. Is there a similar operation for the sum of all the elements in a matrix?
-
I don't know if it has a nice name or notation, but for the matrix $\mathbf A$ you could consider the quadratic form $\mathbf e^\top\mathbf A\mathbf e$, where $\mathbf e$ is the column vector whose entries are all $1$'s.
-
Using the sum of all elements does not contain any information about endomorphisms, which is the reason why you will not find such an operation in the literature.
If this is interesting enough, you can get the sum of all squares using the scalar product $$\phi(A,B) := \mathrm{tr}(A^T B)$$ In fact $\mathrm{tr}(A^T A) = \sum\limits_{i,j=1} a_{i,j}^2$
-
I think the last two sentences are helpful myself. – Geoff Robinson Jul 30 '12 at 13:16
You can certainly consider the sum of all the entries in a square matrix. But what would it be good for?
Mind that square matrices are a way to write explicitly endomorphisms (i.e. linear transformations of a space into itself) so that any quantity you attach to a matrix should be actually say something about the endomorphisms. Trace and determinant remain unchanged if the matrix $A$ is replaced by the matrix $PAP^{-1}$ where $P$ is any invertible matrix. Thus, trace and determinant are numbers that you can attach to the endomorphism represented by $A$.
It wouldn't be the case for the sum of all entries, which does not remain invariant under the said matrix transformation.
-
However, as vanna's answer can be shown to imply, the sum of the squares of all entries remains invariant under orthogonal conjugations. – Qiaochu Yuan Jul 30 '12 at 15:09
@QiaochuYuan : Not surprisingly, since the sum of the squares of the entries is just the square of the norm of $A$ thought as a vector in ${\Bbb R}^{n^2}$. – Andrea Mori Jul 30 '12 at 15:19
I do not think it is completely clear that the Euclidean norm in $\mathbb{R}^{n^2}$ is invariant under conjugation by orthogonal elements, which are defined using the Euclidean norm in $\mathbb{R}^n$. – Qiaochu Yuan Jul 30 '12 at 15:27
The term "grand sum" is commonly used, if only informally, to represent the sum of all elements.
By the way, the grand sum is a very important quantity in the contexts of Markovian transition matrices and other probabilistic applications of linear algebra.
Regards, Scott
-
The max norm:
The max norm is the elementwise norm with $p = \infty$: $$\|A\|_{\text{max}} = \max \{|a_{ij}|\}.$$ This norm is not sub-multiplicative. $p=\infty$ refers to $\Vert A \Vert_{p} = \left( \sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^p \right)^{1/p}. \,$
If you want something without absolute bars, think of the projection of your matrix on $E$, $\text{tr}\left(E\cdot A\right)$, where $E$ is a matrix full of $1$'s, which is equivalent to calculate the scalar product $\langle e |Ae \rangle$, with $e$ being a vector full of $1$'s, since $|e \rangle \langle e|=E$.
-
I refer you to the article Merikoski: On the trace and the sum of elements of a matrix, Linear Algebra and its applications, Volume 60, August 1984, pp. 177-185.
-
Could you give some description of what that article says? Links are fine, but if the whole answer is essentially a link, it is of little value if the link goes stale. The library reference is also good, but not of much use to someone who doesn't have access to a University Library. – robjohn Jan 31 '13 at 1:19
|
2015-12-02 00:30:36
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9118243455886841, "perplexity": 170.17223715613866}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398474527.16/warc/CC-MAIN-20151124205434-00001-ip-10-71-132-137.ec2.internal.warc.gz"}
|
http://www.ck12.org/book/CK-12-Middle-School-Math-Concepts-Grade-8/r19/section/1.11/
|
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 1.11: Solve and Check Single-Variable Equations Using Mental Math and Substitution
Difficulty Level: At Grade Created by: CK-12
Estimated4 minsto complete
%
Progress
Practice Mental Math for Multiplication/Division Equations
MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated4 minsto complete
%
Estimated4 minsto complete
%
MEMORY METER
This indicates how strong in your memory this concept is
Fitzgerald Middle School had a wonderful cookie fundraiser. Within the first month, the students were selling out the inventory every single day. They were having such a difficult time keeping up, that they hired the home economics classes to help them with the baking. So some of the money went to the students who agreed to bake, but the rest of it went into the student council.
By midterms, the students calculated that after paying the home economics students, that they were still averaging $60.00 profit per week. By midterms, they had collected$540.00 total.
“How many weeks did it take us to make that much?” Jesse asked at lunch one day.
“I don’t know, but I am sure that we can make $1000.00 by end of the semester,” Tracy said smiling. “How can you be so sure?” “You just do the math. First, we can write an equation and solve it to figure out how long it took us to make the$540.00. Then double it for the end of the semester,” Tracy explained.
Do you understand how Tracy figured this out? Pay attention and you will be able to solve this dilemma by the end of the Concept.
### Guidance
An equation includes groups of numbers, symbols, and variables. However, equations also include an equals sign.
The key thing to remember about an equation is that the quantity on one side of the equals must be the same as the quantity on the other side of the equals.
There are different ways to solve an equation.
When you solve an equation, you are solving to determine the value of the variable. If you choose the correct value for the variable, then the equation will be a true statement. Let’s look solving an equation without a variable.
4+16=20\begin{align*}4+16=20\end{align*}
We can look at the quantity on the left side of the equation first. It is equal to 20. The right side of the equation is also 20. This is a true statement.
An equation must always make a true statement. We can say that this is a balanced equation.
What if this equation had a variable in place of one of the numbers?
x+16=20\begin{align*}x+16=20\end{align*}
Now we have a puzzle to solve. We can start by thinking about what number plus sixteen is equal to 20. We know that four plus sixteen is equal to 20. So, the value of x\begin{align*}x\end{align*} must be four.
We write the answer to an equation in a particular way.
The answer is that x=4\begin{align*}x=4\end{align*}.
Think a little deeper about how you solved this. If you think about it you probably subtracted 20 – 16 in your head. This is called using an inverse operation. The inverse operation is the opposite operation. We can use inverse operations to solve equations.
Here is another one.
4x=12\begin{align*}4x=12\end{align*}
Here we have a multiplication problem. We can ask ourselves, what number times four is equal to 12? The answer is 3.
x=3\begin{align*}x=3\end{align*}
We could also use the inverse operation to solve this. Twelve divided by four is three. Our answer is the same and both methods can be completed using mental math.
You can also check an answer by substituting it back into the original problem.
5x+3=18\begin{align*}5x+3=18\end{align*}
After solving this equation using mental math, we figure out the value of the variable is three. We can check this answer by substituting the value of the variable back into the original equation. Then we simplify it. If the equation makes a true statement, then we know that we have the correct answer.
5(3)+315+318=18=18=18\begin{align*}5(3) + 3 &= 18\\ 15 + 3 &= 18\\ 18 &= 18\end{align*}
This is a true statement so our work is accurate.
You can solve these equations by using mental math.
#### Example A
x2=7\begin{align*}\frac{x}{2}= 7\end{align*}
Solution: x=14\begin{align*}x=14\end{align*}
#### Example B
22x=11\begin{align*}\frac{22}{x}=11\end{align*}
Solution: x=2\begin{align*}x = 2\end{align*}
#### Example C
8x=64\begin{align*}8x = 64\end{align*}
Solution: x=8\begin{align*}x = 8\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
To work on this problem, first we need to write an equation. Let’s look at what we know.
We know that the students averaged $60.00 profit per week. We know that their gross profit was$540.00.
We need to know how many weeks it took them to earn that. Our variable is the number of weeks, w\begin{align*}w\end{align*}.
Here is our equation.
60w=540\begin{align*}60w=540\end{align*}
We can solve this using mental math.
It took the students 9 weeks to earn the money.
### Vocabulary
Equation
a group of numbers, operations and variables where the quantity on one side of the equal sign is the same as the quantity on the other side of the equal sign.
Inverse Operation
the opposite operation. Equation can often be solved by using an inverse operation.
### Guided Practice
Here is one for you to try on your own.
5x+3=18\begin{align*}5x+3=18\end{align*}
Solution
Let’s break down this equation by using saying it to ourselves.
“Five times some number plus three is equal to eighteen.” Now you can think through the five times table for an answer that makes sense.
5, 10, 15, 20
15 makes sense so that would make the variable equal to three since five times three is fifteen.
x=3\begin{align*}x=3\end{align*}
### Practice
Directions: Solve each equation using mental math. Be sure to check each answer by substituting your solution back into the original problem. Then simplify to be sure that your equation is balanced.
1. x+4=22\begin{align*}x+4=22\end{align*}
2. y+8=30\begin{align*}y+8=30\end{align*}
3. x19=40\begin{align*}x-19=40\end{align*}
4. 12x=9\begin{align*}12-x=9\end{align*}
5. 4x=24\begin{align*}4x=24\end{align*}
6. 6x=36\begin{align*}6x=36\end{align*}
7. 9x=81\begin{align*}9x=81\end{align*}
8. y5=2\begin{align*}\frac{y}{5}=2\end{align*}
9. a8=5\begin{align*}\frac{a}{8} = 5\end{align*}
10. 12b=6\begin{align*}\frac{12}{b}=6\end{align*}
11. 6x+3=27\begin{align*}6x+3=27\end{align*}
12. 8y2=54\begin{align*}8y-2=54\end{align*}
13. 3b+12=30\begin{align*}3b+12=30\end{align*}
14. 9y7=65\begin{align*}9y-7=65\end{align*}
15. 12a5=31\begin{align*}12a-5=31\end{align*}
16. x2+4=8\begin{align*}\frac{x}{2} + 4 = 8\end{align*}
17. x4+3=7\begin{align*}\frac{x}{4}+3=7\end{align*}
18. 10x+9=14\begin{align*}\frac{10}{x}+9=14\end{align*}
19. 5a12=33\begin{align*}5a-12=33\end{align*}
20. 7b9=33\begin{align*}7b-9=33\end{align*}
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
Algebraic Expression
An expression that has numbers, operations and variables, but no equals sign.
Equation
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
Inverse Operation
Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.
Variable
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.
Show Hide Details
Description
Difficulty Level:
Authors:
Tags:
Subjects:
|
2016-10-23 19:34:49
|
{"extraction_info": {"found_math": true, "script_math_tex": 38, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7808744311332703, "perplexity": 1032.6646187907809}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719397.0/warc/CC-MAIN-20161020183839-00008-ip-10-171-6-4.ec2.internal.warc.gz"}
|
https://www.groundai.com/project/inflationary-de-sitter-solutions-from-superstrings/
|
1 Introduction
LPTENS–07/22, CPHT–RR025.0407
June 2007
{centering} Inflationary de Sitter Solutions
from Superstrings
Costas Kounnas and Hervé Partouche
Laboratoire de Physique Théorique, Ecole Normale Supérieure,
24 rue Lhomond, F–75231 Paris cedex 05, France
Costas.Kounnas@lpt.ens.fr
Centre de Physique Théorique, Ecole Polytechnique,
F–91128 Palaiseau, France
Herve.Partouche@cpht.polytechnique.fr
Abstract
In the framework of superstring compactifications with supersymmetry spontaneously broken, (by either geometrical fluxes, branes or else), we show the existence of new inflationary solutions. The time-trajectory of the scale factor of the metric , the supersymmetry breaking scale and the temperature are such that and remain constant. These solutions request the presence of special moduli-fields:
The universal “no-scale-modulus” , which appears in all effective supergravity theories and defines the supersymmetry breaking scale .
The modulus , which appears in a very large class of string compactifications and has a -dependent kinetic term. During the time evolution, remains constant as well, ( being the energy density induced by the motion of ).
The cosmological term , the curvature term and the radiation term are dynamically generated in a controllable way by radiative and temperature corrections; they are effectively constant during the time evolution.
Depending on , and , either a first or second order phase transition can occur in the cosmological scenario. In the first case, an instantonic Euclidean solution exists and connects via tunneling the inflationary evolution to another cosmological branch. The latter starts with a big bang and, in the case the transition does not occur, ends with a big crunch. In the second case, the big bang and the inflationary phase are smoothly connected.
Research partially supported by the EU (under the contracts MRTN-CT-2004-005104, MRTN-CT-2004-512194, MRTN-CT-2004-503369, MEXT-CT-2003-509661), INTAS grant 03-51-6346, CNRS PICS 2530, 3059 and 3747, and ANR (CNRS-USAR) contract 05-BLAN-0079-01.
Unité mixte du CNRS et de l’Ecole Normale Supérieure associée à l’Université Pierre et Marie Curie (Paris 6), UMR 8549.
Unité mixte du CNRS et de l’Ecole Polytechnique, UMR 7644.
## 1 Introduction
In the framework of superstring and -theory compactifications, there are always moduli fields coupled in a very special way to the gravitational and matter sector of the effective four-dimensional supergravity. The gravitational and the scalar field part of the effective Lagrangian have the generic form
L=√−detg[12R−gμν Ki¯ȷ ∂μϕi∂ν¯ϕ¯ȷ −V(ϕi,¯ϕ¯ı)], (1.1)
where is the metric of the scalar manifold and is the scalar potential of the supergravity. (We will always work in gravitational mass units, with GeV). What will be crucial in this work is the non-triviality of the scalar kinetic terms in the effective supergravity theories that will provide us, in some special cases, accelerating cosmological solutions once the radiative and temperature corrections are taken into account.
Superstring vacua with spontaneously broken supersymmetry [1] that are consistent at the classical level with a flat space-time define a very large class of “no-scale” supergravity models [2]. Those with spontaneous breaking deserve more attention. Some of them are candidates for describing (at low energy) the physics of the standard model and extend it up to TeV energy scale. This class of models contains an enormous number of consistent string vacua that can be constructed either via freely acting orbifolds [1, 3] or “geometrical fluxes” [4] in heterotic string and type IIA,B orientifolds, or with non-geometrical fluxes [5] (e.g. RR-fluxes or else).
Despite the plethora of this type of vacua, an interesting class of them are those which are described by an effective “no-scale supergravity theory”. Namely the vacua in which the supersymmetry is sponaneously broken with a vanishing classical potential with undetermined gravitino mass due to at least one flat field direction, the “no-scale modulus . At the quantum level a non-trivial effective potential is radiatively generated which may or may not stabilize the “no-scale” modulus [2].
What we will explore in this work are the universal scaling properties of the “thermal” effective potential at finite temperature that emerges at the quantum level of the theory. As we will show in section 4, the quantum and thermal corrections are under control, (thanks to supersymmetry and to the classical structure of the “no-scale models”), showing interesting scaling properties.
In section 2, we set up our notations and conventions in the effective “no-scale” supergravities of the type IIB orientifolds with -branes and non-trivial NS-NS and RR three form fluxes and . We identify the “no-scale” modulus , namely the scalar superpartner of the Goldstino which has the property to couple to the trace of the energy momentum tensor of a sub-sector of the theory [6]. More importantly, it defines the field-dependence of the gravitino mass [2]
m(Φ)=eαΦ. (1.2)
Other extra moduli that we will consider here are those with -dependent kinetic terms. These moduli appear naturally in all string compactifications [7]. We are in particular interested in scalars () which are leaving on -branes and whose kinetic terms scale as the inverse volume of the “no-scale” moduli space.
In section 3, we display the relevant gravitational, fields and thermal equations of motion in the context of a Friedman-Robertson-Walker (FRW) space-time. We actually generalize the mini-superspace (MSS) action by including fields with non-trivial kinetic terms and a generic, scale factor dependent, thermal effective potential.
In our analysis we restrict ourselves to the large moduli limit, neglecting non-perturbative terms and world-sheet instanton corrections , . On the other hand we keep the perturbative quantum and thermal corrections.
Although this study looks hopeless and out of any systematic control even at the perturbative level, it turns out to be manageable thanks to the initial no-scale structure appearing at the classical level (see section 4).
In section 5, we show the existence of a critical solution to the equations of motion that follows from the scaling properties derived in section 4. We have to stress here that we extremize the effective action by solving the gravitational and moduli equations of motion and do not consider the stationary solutions emerging from a minimization of the effective potential only. We find in particular that a universal solution exists where all scales evolve in time in a similar way, so that their ratios remain constant: const., const.. Along this trajectory, effective time-independent cosmological term , curvature term and radiation term are generated in the MSS action, characterizing the cosmological evolution.
Obviously, the validity of the cosmological solutions based on (supergravity) effective field theories is limited. For instance, in the framework of more fundamental theories such as string theory, there are high temperature instabilities occuring at , where is the Hagedorn temperature of order the mass of the first string excited state. To bypass these limitations, one needs to go beyond the effective field theory approach and consider the full string theory (or brane, M-theory,…) description. Thus, the effective solutions presented in this work are not valid anymore and must be corrected for temperatures above . Moreover, Hagedorn-like instabilities can also appear in general in other corners of the moduli space of the fundamental theory, when space-time supersymmetry is spontaneously broken.
Regarding the temperature scale as the inverse radius of the compact Euclidean time, one could conclude that all the internal radii of a higher dimensional fundamental theory have to be above the Hagedorn radius. This would mean that the early time cosmology should be dictated by a 10-dimensional picture rather than a 4-dimensional one where the internal radii are of order the string scale. There is however a loophole in this statement. Indeed, no tachyonic instability is showing up in the whole space of the moduli which are not involved in the spontaneous breaking of supersymmetry, as recently shown in explicit examples [8]. This leeds us to the conjecture that the only Hagedorn-like restrictions on the moduli space depend on the supersymmetry breaking. In our cosmological solutions, not only the temperature scale is varying, but also the supersymmetry breaking scale , which turns to be a moduli-dependent quantity. Based on the above statements, we expect that in a more accurate stringy description of our analysis, there should be restrictions on the temperature as well as the supersymetry breaking scale. This has been recently explicitly shown in the stringy examples considered in [8].
In section 6, our cosmological solutions are generalized by including moduli with other scaling properties of their kinetic terms.
Finally, section 7 is devoted to our conclusions and perspective for future work.
## 2 N=1 No-Scale Sugra from Type IIB Orientifolds
In the presence of branes and fluxes, several moduli can be stabilized. For instance, in “generalized” Calabi-Yau compactifications, either the Kälher structure moduli or the complex structure moduli can be stabilized according to the brane and flux configuration in type IIA or type IIB orientifolds [4, 9, 5, 6]. The (partial) stabilization of the moduli can lead us at the classical level to AdS like solutions, domain wall solutions or “flat no-scale like solutions”. Here we will concentrate our attention on the “flat no-scale like solutions”.
In order to be more explicit, let us consider as an example the type IIB orientifolds with -branes and non-trivial NS-NS and RR three form fluxes and . This particular configuration induces a well known superpotential that can stabilize all complex structure moduli and the coupling constant modulus [5, 4]. The remaining moduli “still remain flat directions at the classical level”, e.g. neglecting world-sheet instanton corrections and the perturbative and non-perturbative quantum corrections [5].
It is also well known by now that in the large limit the Kälher potential is given by the intersection numbers of the special geometry of the Calabi-Yau manifold and orbifold compactifications [10, 11]:
K=−logdabc(Ta+¯Ta)(Tb+¯Tb)(Tc+¯Tc). (2.3)
Thus, after the and moduli stabilization, the superpotential is effectively constant and implies a vanishing potential in all directions. The gravitino mass term is however non-trivial [2, 1, 4, 5, 11],
m2=|W|2eK. (2.4)
This classical property of “no-scale models” emerges from the cubic form of in the moduli and is generic in all type IIB orientifold compactifications with -branes and three form and fluxes [5, 4]. Keeping for simplicity the direction (for some constants ) and freezing all other directions, the Kälher potential is taking the well known structure [2],
K=−3log(T+¯T). (2.5)
This gives rise to the kinetic term and gravitino mass term,
gμν 3∂μT∂ν¯T(T+¯T)2andm2=ceK=c(T+¯T)3, (2.6)
where is a constant. Freezing Im and defining the field by
e2αΦ=m2=c(T+¯T)3, (2.7)
one finds a kinetic term
gμν 3∂μT∂ν¯T(T+¯T)2=gμν α23 ∂μΦ∂νΦ. (2.8)
The choice normalizes canonically the kinetic term of the modulus .
The other extra moduli that we will consider are those with -dependent kinetic terms. We are in particular interested to the scalars whose kinetic terms scale as the inverse volume of the -moduli. For one of them, , one has
Ks≡−α23 e2αΦ gμν ∂μΦs∂νΦs =−α2c3 gμν ∂μΦs∂νΦs(T+¯T)3. (2.9)
Moduli with this scaling property appear in a very large class of string compactifications. Some examples are:
All moduli fields leaving in the parallel space of -branes [5, 4].
All moduli coming from the twisted sectors of -orbifold compactifications in heterotic string [7], after non-perturbative stabilization of by gaugino condensation and flux-corrections [12].
Our analysis will also consider other moduli fields with different scaling properties, namely those with kinetic terms of the form:
Kw≡−12e(6−w)αΦ gμν∂μϕw∂νϕw, (2.10)
with weight and
## 3 Gravitational, Moduli and Thermal Equations
In a fundamental theory, the number of degrees of freedom is important (and actually infinite in the context of string or M-theory). However, in an effective field theory, an ultraviolet cut-off set by the underlying theory determines the number of states to be considered. We focus on cases where these states include the scalar moduli fields and , with non-trivial kinetic terms given by
L=√−detg[12R−12 gμν(∂μΦ∂νΦ+e2αΦ ∂μΦs∂νΦs)−V(Φ,μ)]+⋯ (3.11)
In this Lagrangian, the “” denote all the other degrees of freedom, while the effective potential depends on and the renormalization scale . We are looking for gravitational solutions based on isotropic and homogeneous FRW space-time metrics,
ds2=−N(t)2dt2+a(t)2dΩ23, (3.12)
where is a 3-dimensional compact space with constant curvature , such as a sphere or an orbifold of hyperbolic space. This defines an effective one dimensional action, the so called “mini-super-space” (MSS) action [13, 14, 15, 16].
A way to include into the MSS action the quantum fluctuations of the full metric and matter degrees of freedom (and thus taking into account the back-reaction on the space-time metric), is to switch on a thermal bath at temperature [14, 15, 16]. In this way, the remaining degrees of freedom are parameterized by a pressure and a density , where are the non-vanishing masses of the theory. Note that and have an implicit dependence on , through the mass defined in eq. (1.2) [6]. The presence of the thermal bath modifies the effective MSS action, including the corrections due to the quantum fluctuations of the degrees of freedom whose masses are below the temperature scale . The result, together with the fields and , reads
S\tiny\em{eff}=−|k|−326∫dt a3(3N(˙aa)2−3kNa2−12N˙Φ2−12N e2αΦ˙Φ2s+NV−12N(ρ+P)+N2(ρ−P)), (3.13)
where a “dot” denotes a time derivation. is a gauge dependent function that can be arbitrarily chosen by a redefinition of time. We will always use the gauge , unless it is explicitly specified.
The variation with respect to gives rise to the Friedman equation,
3H2=−3ka2+ρ+12˙Φ2+12 e2αΦ˙Φ2s+V, (3.14)
where .
The other gravitational equation is obtained by varying the action with respect to the scale factor :
2˙H+3H2=−ka2−P−12˙Φ2−12e2αΦ˙Φ2s+V+13a∂V∂a. (3.15)
In the literature, the last term is frequently taken to be zero. However, this is not valid due to the dependence of on , when this scale is chosen appropriately as will be seen in section 3. We thus keep this term and will see that it plays a crucial role in the derivation of the inflationary solutions under investigation.
We find useful to replace eq. (3.15) by the linear sum of eqs. (3.14) and (3.15), so that the kinetic terms of and drop out,
˙H+3H2=−2ka2+12(ρ−P)+V+16a∂V∂a. (3.16)
The other field equations are the moduli ones,
¨Φ+3H˙Φ+∂∂Φ(V−P−12 e2αΦ˙Φ2s)=0 (3.17)
and
¨Φs+(3H+2α˙Φ) ˙Φs=0. (3.18)
The last equation (3.18) can be solved immediately,
Ks ≡ 12 e2αΦ˙Φ2s = Cse−2αΦa6, (3.19)
where is a positive integration constant. It is important to stress here that we insist to keep in eq. (3.17) both terms and that are however usually omitted in the literature. The first term vanishes only under the assumption that all masses are taken to be -independent, while the absence of the second term assumes a trivial kinetic term. However, both assumptions are not valid in string effective supergravity theories ! (See section 3.)
Finally, we display for completeness the total energy conservation of the system,
ddt(ρ+12˙Φ2+Ks+V)+3H(ρ+P+˙Φ2+2Ks)=0. (3.20)
Before closing this section, it is useful to derive some extra useful formulas that are associated to the thermal system. The integrability condition of the second law of thermodynamics reaches, for the thermal quantities and ,
T∂P∂T= ρ+P. (3.21)
The fact that these quantities are four-dimensional implies
(mi∂∂mi+T∂∂T)ρ=4ρand(mi∂∂mi+T∂∂T)P=4P. (3.22)
Then, the second eq. (3.22) together with the eq. (3.21) implies [6]:
mi∂P∂mi=−(ρ−3P). (3.23)
Among the non-vanishing , let us denote with “hat-indices” the masses that are -independent, and with “tild-indices” the masses that have the following -dependence:
{mi}={m^ı}∪{m~ı}wherem~ı=c~ı eαΦ, (3.24)
for some constants . Then, utilizing eq. (3.23), we obtain a very fundamental equation involving the modulus field [6],
−∂P∂Φ=α(~ρ−3~P), (3.25)
where and are the contributions to and associated to the states with -dependent masses . The above equation (3.25) clearly shows that the modulus field couples to the (sub-)trace of the energy momentum tensor associated to the thermal system [6] , of the states with -dependent masses defined in eq. (3.24). We return to this point in the next section.
## 4 Effective Potential and Thermal Corrections
In order to find solutions to the coupled gravitational and moduli equations discussed in the previous section, it is necessary to analyze the structure of the scalar potential and the thermal functions , . More precisely, we have to specify their dependence on and . Although this analysis looks hopeless in a generic field theory, it is perfectly under control in the string effective no-scale supergravity theories.
Classically the potential is zero along the moduli directions and . At the quantum level, it receives radiative and thermal corrections that are given in terms of the effective potential [11], , and in terms of the thermal function, . Let us consider both types of corrections.
### 4.1 Effective Potential
The one loop effective potential has the usual form [11, 17],
V=V\tiny cl+164π2StrM0Λ4% \tiny cologΛ2\tiny coμ2+132π2StrM2Λ2\tiny co+164π2Str(M4logM2μ2)+⋯, (4.26)
where is the classical part, which vanishes in the string effective “no-scale” supergravity case. An ultraviolet cut-off is introduced and stands for the renormalization scale.
StrMn≡∑I(−)2JI(2JI+1) mnI (4.27)
is a sum over the -th power of the mass eigenvalues. In our notations, the index is referring to both massless and massive states (with eventually -dependant masses).The weights account for the numbers of degrees of freedom and the statistics of the spin particles.
The quantum corrections to the vacuum energy with the highest degree of ultraviolet divergence is the term, whose coefficient is equal to the number of bosonic minus fermionic degrees of freedom. This term is thus always absent in supersymmetric theories since they possess equal numbers of bosonic and fermionic states.
The second most divergent term in eq. (4.26) is the contribution proportional to . In the spontaneously broken supersymmetric theories, it is always proportional to the square of the gravitino mass-term ,
StrM2=c2 m(Φ)2. (4.28)
The coefficient is a field independent number. It depends only on the geometry of the kinetic terms of the scalar and gauge manifold, and not on the details of the superpotential [17, 11]. This property is very crucial in our considerations.
The last term has a logarithmic behavior with respect to the infrared scale and is independent of the ultraviolet cut-off . Following the infrared regularization method valid in string theory (and field theory as well) adapted in ref. [18], the scale is proportional to the curvature of the three dimensional space,
μ=1γa, (4.29)
where is a numerical coefficient chosen appropriately according to the renormalization group equation arguments. Another physically equivalent choice for is to be proportional to the temperature scale, . The curvature choice (4.29) looks more natural and has the advantage to be valid even in the absence of the thermal bath.
Modulo the logarithmic term, the can be expanded in powers of gravitino mass ,
164π2StrM4=C4m4+C2m2+C0. (4.30)
Including the logarithmic terms and adding the quadratic contribution coming from the , we obtain the following expression for the effective potential organized in powers of :
V=V4(Φ,a)+V2(Φ,a)+V0(Φ,a), (4.31)
where
Vn(Φ,a)=mn(Φ)(Cn+Qnlog(m(Φ)γa)˙ϕ), (4.32)
for constant coefficients and , (). These contributions satisfy
∂Vn(Φ,a)∂Φ=α(nVn+ mnQn)anda∂Vn(Φ,a)∂a=mnQn. (4.33)
The logarithmic dependence in the effective potential can be derived in the effective field theory by considering the Renormalization Group Equations (RGE). They involve the gauge couplings, the Yukawa couplings and the soft-breaking terms [19, 11]. These soft-breaking terms are usually parameterized by the gaugino mass terms , the soft scalar masses , the trilinear coupling mass term and the analytic mass term , [19, 11]. However, what will be of main importance in this work is that all soft breaking mass terms are proportional to [17, 11].
### 4.2 Thermal Potential
For bosonic (or fermionic) fluctuating states of masses (or ) in thermal equilibrium at temperature , the general expressions of the energy density and pressure are
ρ=T4⎛⎜⎝∑\scriptsize boson bIBρ(mbT)+∑\scriptsize fermion fIFρ(mfT)⎞⎟⎠,P=T4⎛⎜⎝∑\scriptsize boson bIBP(mbT)+∑\scriptsize fermion fIFP(mfT)⎞⎟⎠, (4.34)
where
(4.35)
and .
There are three distinct sub-sectors of states:
The sub-sector of bosonic and fermionic massless states. From eqs. (4.34) and (4.35), their energy density and pressure satisfy
ρ0=3P0=π415(nB0+78nF0)T4. (4.36)
In particular, we have and .
The sub-sector of states with non vanishing masses independent of .
Consider the bosons and fermions whose masses we denote by are below T. The energy density and pressure associated to them satisfy
^ρ(T,m^ı0)=^ρ(T,m^ı0=0)+m2^ı0∂^ρ∂m2^ı0=π415(^nB0+78^nF0) T4−∑^ı0^c^ı0 m2^ı0 T2, (4.37)
^P(T,m^ı0)=^P(T,m^ı0=0)+m2^ı0∂^P∂m2^ı0=π445(^nB0+78^nF0) T4−∑^ı0^c^ı0 m2^ı0 T2, (4.38)
where the ’s are non-vanishing positive constants. In particular, one has .
For the masses above , the contributions of the particular degrees of freedom are exponentially suppressed and decouple from the thermal system. We are not including their contribution.
The sub-sector with non vanishing masses proportional to . Its energy density and pressure satisfy
∂P∂Φ=−α(~ρ−3~P), (4.39)
as was shown at the end of section 2. This identity is also valid for the massless system we consider in case .
According to the scaling behaviors with respect to and , we can separate
ρ=ρ4 + ρ2, P=P4 + P2, (4.40)
where
(m(Φ) ∂∂m(Φ) + T ∂∂T ) (ρn, Pn)=n (ρn, Pn). (4.41)
and are the sums of the contributions of the massless states (case ), the parts of and (case ), and and (case ),
ρ4=T4⎛⎜⎝π415((nB0+^nB0)+78(nF0+^nF0))+∑% \scriptsize boson ~bIBρ(m~bT)+∑\scriptsize fermion ~fIFρ(m~fT)⎞⎟⎠, (4.42)
P4=T4⎛⎜⎝π445((nB0+^nB0)+78(nF0+^nF0))+∑% \scriptsize boson ~bIBP(m~bT)+∑\scriptsize fermion ~fIFP(m~fT)⎞⎟⎠, (4.43)
while and arise from the parts of and (case ):
ρ2=P2=−∑^ı0^c^ı0 m2^ı0 T2≡−^M2 T2. (4.44)
## 5 Critical Solution
The fundamental ingredients in our analysis are the scaling properties of the total effective potential at finite temperature,
V\tiny total=V−P. (5.45)
Independently of the complication appearing in the radiative and temperature corrected effective potential, the scaling violating terms are under control. Their structure suggests to search for a solution where all the scales of the system, , and , remain proportional during their evolution in time,
eαΦ≡m(Φ)=1γ′a ⟹ H=−α˙Φandξ m(Φ)=T. (5.46)
Our aim is thus to determine the constants and in terms of in eq. (3.19), , and the computable quantities , , in string theory, such that the equations of motion for , and the gravity are satisfied. On the trajectory (5.46), the contributions , defined in eq. (4.32) satisfy
Vn=mnC′nwhereC′n=Cn+Qnlog(γγ′), (5.47)
and
∂Vn∂Φ=αmn(nC′n+Qn),a∂Vn∂a=mnQn. (5.48)
Also, the contributions of and in in eq. (3.19) conspire to give a global dependence,
Ks=Csγ′2a4. (5.49)
Finally, the sums over the full towers of states with -dependent masses behave in and as pure constants, (see eqs. (4.42) and (4.43)),
ρ4=r4T4wherer4=π415((nB0+^nB0)+78(nF0+^nF0))+∑~bIBρ(~c~bξ)+∑~fIFρ(~c~fξ), (5.50)
P4=p4T4wherep4=π445((nB0+^nB0)+78(nF0+^nF0))+∑~bIBP(~c~bξ)+∑~fIFP(~c~fξ). (5.51)
As a consequence, using eqs. (4.33) and (4.39), the -equation (3.17) becomes,
˙H+3H2=α2(˙Φ(4C′4+Q4)m4+(2C′2+Q2)m2+Q0+(r4−3p4)ξ4m4−2Csγ′6m4˙Φ). (5.52)
On the other hand, using eq. (4.33), the gravity equation (3.16) takes the form
˙H+3H2=−2kγ′2m2+12(r4−p4)ξ4m4+(C′4m4+C′2m2+C′0)+16(Q4m4+Q2m2+Q0). (5.53)
The compatibility of the -equation and the gravity equation along the critical trajectory implies an identification of the coefficients of the monomials in . The constant terms determine in term of
C′0=6α2−16 Q0, (5.54)
which amounts to fixing ,
γ′=γeC0Q0−6α2−16. (5.55)
The quadratic terms determine the parameter :
k=−1γ′2(2α2−12C′2+6α2−112Q2). (5.56)
Finally, the quartic terms relate to the integration constant appearing in ,
(5.57)
At this point, our choice of anzats (5.46) and constants , allows to reduce the differential system for , and the gravity to the last equation. We thus concentrate on the Friedman equation (3.14) in the background of the critical trajectory ,
(6α2−16α2) 3H2=−3ka2+ρ+12 e2αΦ˙Φ2s+V. (5.58)
The dilatation factor in front of can be absorbed in the definition of , and , once we take into account eqs. (5.54), (5.56) and (5.57),
3H2=3λ−3^ka2+CRa4, (5.59)
where
3λ=α2Q0, (5.60)
^k=α2γ′2(26α2−1ξ2^M2−C′2−12Q2), (5.61)
and
CR=32γ′4((r4−p4)ξ4+2C′4+13Q4). (5.62)
We note that for , is positive. In that case, the constraint (5.57) allows us to choose a lower bound for the arbitrary constant , so that is large enough to have . This means that the theory is effectively indistinguishable with that of a universe with cosmological constant , uniform space curvature , and filled with a thermal bath of radiation coupled to gravity. This can be easily seen by considering the Lagrangian
√−detg[12R−3λ] (5.63)
and the metric anzats (3.12), with a 3-space of constant curvature . In the action, one can take into account a uniform space filling bath of massless particles by adding a Lagrangian density proportional to (see [14, 15, 16]) in the MSS form. One finds
(5.64)
whose variation with respect to gives (5.59). Actually, the thermal bath interpretation is allowed as long as , since the term is an energy density. However, in the case under consideration, the effective can be negative due to the contribution of the effective potential. The general solution of the effective MSS action of eq. (5.64) with , and was recently investigated in ref. [16]. It amounts to a thermally deformed de Sitter solution, while the pure radiation case where , was studied in ref. [6]. In the latter case, the time trajectory (5.46) was shown to be an attractor at late times, giving rise to a radiation evolving universe with
a=(4CR3)1/4t1/2, m(Φ)=Tξ=1γ′a. (5.65)
Following ref. [16], the general case with and gives rise to cosmological scenarios we summarize here. Depending on the quantity
δ2T=43λ^k2CR, (5.66)
a first or second order phase transition can occur:
δ2T<1⟺1\tiny st order transition% ,δ2T>1⟺2\tiny d % order transition. (5.67)
i) The case
There are two cosmological evolutions connected by tunnel effect:
ac(t)=N√ε+cosh2(√λt),t∈R, (5.68)
and
as(t)=N√ε−sinh2(√λt),ti≤t≤−ti, (5.69)
where
N=√^kλ(1−δ2T)1/4,ε=12⎛⎜ ⎜⎝1√1−δ2T−1⎞⎟ ⎟⎠,ti=−1√λarcsinh√ε. (5.70)
The “cosh”-solution corresponds to a deformation of a standard de Sitter cosmology, with a contracting phase followed at by an expanding one. The “sinh”-solution describes a big bang with a growing up space till , followed by a contraction till a big crunch arises. The two evolutions are connected in Euclidean time by a -Gravitational Instanton
aE(τ)=N√ε+cos2(√λτ),ΦE(τ)=−1αlog(γ′aE(τ)˙l). (5.71)
The cosmological scenario is thus starting with a big bang at and expands up to , following the “sinh”-evolution. At this time, performing the analytic continuation reaches (5.71), (where is chosen in the range111It is also possible to consider the instantons associated to the ranges , , see [16]. ). At , a different analytic continuation to real time exists, , that gives rise to the inflationary phase of the “cosh”-evolution, for ,
(see fig. 1).
There are thus two possible behaviors when is reached. Either the universe carries on its “sinh”-evolution and starts to contract, or a first order transition occurs and the universe enters into the inflationary phase of the “cosh”-evolution. In that case, the scale factor jumps instantaneously from to at ,
a−=√^k2λ(1−√1−δ2T)⟶a+=√^k2λ(1+√1−δ2T). (5.72)
An estimate of the transition probability is given by
p∝e−2SE\tiny\em{eff}, (5.73)
where is the Euclidean action computed with the instanton solution (5.71), for . Actually, following refs. [15, 16], one has:
SE\tiny\em{eff}=−13λ ⎷1+√1−δ2T2(E(u)−(1−√1−δ2T)K(u)), (5.74)
where and are the complete elliptic integrals of first and second kind, respectively, and
u=2(1−δ2T)1/4√1+√1−δ2T. (5.75)
ii) The case
There is a cosmological solution,
a(t)=√^k2λ√1+√δ2T−1sinh(2√λt),t≥ti, (5.76)
where
ti=−12√λarcsinh1√δ2T−1. (5.77)
As in the previous case, it starts with a big bang. However, the behavior evolves toward the inflationary phase in a smooth way, (see fig. 2).
The transition can be associated to the inflection point arising at , where ,
t\tiny inf=12√λarcsinh√δT−1δT+1,a\tiny inf=
|
2020-08-04 05:10:18
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8678932785987854, "perplexity": 773.8354541072887}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735860.28/warc/CC-MAIN-20200804043709-20200804073709-00345.warc.gz"}
|
https://www.physicsforums.com/threads/centroid-of-a-uniform-shape-using-area.825323/
|
# Centroid of a uniform shape, using area
1. Jul 28, 2015
### J-dizzal
1. The problem statement, all variables and given/known data
Find the coordinates of the centroid of the uniform area.
2. Relevant equations
equations for centroid coordinates at the top of my paper.
3. The attempt at a solution
2. Jul 28, 2015
### Dr. Courtney
How does k go from the denominator to the numerator of your integrals as you are evaluating them?
3. Jul 28, 2015
### J-dizzal
(2/3k)(58.1)
=(.667k)(58.1)
=38.7k
where k is a constant. And i treated the other integral similarly.
4. Jul 28, 2015
### Nathanael
In one of your steps you basically said $\sqrt{\frac{y}{k}} =\frac{1}{k}\sqrt{y}$
Also Dr. Courtney's point still stands... it's not 2/3k it's 2/(3k)
5. Jul 28, 2015
### J-dizzal
Ok thanks Dr. Courtney and Nathanael. I now have 38/k. In my final answer the k's cancel out and im left with the same 7.5/38.7.
6. Jul 28, 2015
### Nathanael
38.7/k is still not right. It should be 38.7/√k
When you find the x-coordinate of the centroid, you should be integrating with respect to x.
The "dA" in the formula $\bar x=\frac{1}{A}\int xdA$ is the area of the thin strip between x and x+dx.
7. Jul 28, 2015
### J-dizzal
Wouldnt it be easier to integrate with respect to y, because of the shaded region is above the curve.
8. Jul 28, 2015
### Nathanael
It's not about what is easier, it's simply wrong.
When you integrate with respect to y, you are taking horizontal strips of area, right? Well when you find the x-coordinate of the centroid you want to take vertical strips of area. The reason for this is that you want to take strips of area which all have the same x-value, and then multiply them by that x-value. You just can't do this when you integrate w.r.t. y.
9. Jul 28, 2015
### J-dizzal
$\bar x=\frac{1}{A}\int xdA$
= $1/2 \int kx^3dx$ does this look ok?
10. Jul 28, 2015
### Nathanael
kx2dx is the area under the curve. Try to figure out a way to find the area above the curve.
Also where did the 1/A go? And where did this 1/2 come from? (Are you saying the area is 2?)
|
2017-10-24 11:40:47
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8489070534706116, "perplexity": 1134.9194823711855}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187828411.81/warc/CC-MAIN-20171024105736-20171024125736-00652.warc.gz"}
|
https://www.gradesaver.com/textbooks/science/physics/physics-for-scientists-and-engineers-a-strategic-approach-with-modern-physics-4th-edition/chapter-6-dynamics-i-motion-along-a-line-exercises-and-problems-page-156/40
|
## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)
The velocity at t = 6.0 seconds is $v = 3.18 ~m/s$.
Let $F_N$ be the normal force of the elevator pushing up on the person. Note that this force $F_N$ will be equal to the reading on the scale. We can find the acceleration from t = 0 to t = 3.0 seconds. $\sum F = ma$ $mg - F_N = ma$ $a = \frac{mg-F_N}{m}$ $a = \frac{(95~kg)(9.80~m/s^2)-830~N}{95~kg}$ $a = 1.06~m/s^2$ We can find the acceleration from t = 3.0 seconds to t = 6.0 seconds. $\sum F = ma$ $mg - F_N = ma$ $a = \frac{mg-F_N}{m}$ $a = \frac{(95~kg)(9.80~m/s^2)-930~N}{95~kg}$ $a = 0$ For the first 3.0 seconds, the elevator accelerates downward at a rate of $1.06~m/s^2$. The elevator travels at a constant speed after that. We can find the velocity at t = 3.0 seconds. $v = a~t = (1.06~m/s^2)(3.0~s)$ $v = 3.18~m/s$ Since the elevator does not accelerate after t = 3.0 seconds, the velocity at t = 6.0 seconds is $v = 3.18 ~m/s$
|
2018-08-15 23:46:21
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.720486581325531, "perplexity": 125.6191326363334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221210362.19/warc/CC-MAIN-20180815220136-20180816000136-00286.warc.gz"}
|
https://theportal.wiki/wiki/Ep4
|
# 4: Timur Kuran - The Economics of Revolution and Mass Deception
(Redirected from Ep4)
Information The Economics of Revolution and Mass Deception Timur Kuran 02:45:47 20 August 2019 27 August 2019 Listen Watch Read All EpisodesEpisode Highlights
What if everything we are taught in Economics 101 is not only wrong, but may even be setting us up for populism, dictatorship, or revolution? On this episode of The Portal, Eric is joined by renegade economist and professor Timur Kuran, whose theory of Preference Falsification appears to explain the worldwide surge towards populism and is now threatening to rewrite the core tenets of modern economics.
What happens when entire societies of individuals lie to themselves and to each other? Does it set the stage for revolutions? If you've wondered what force is sweeping the planet towards a mysterious populism bringing Brexit, Trump, and other improbable phenomena out of the shadows, then this your portal to a new economics of black markets in truth, authenticity, and hidden desires.
Timur Kuran could well be the most important economist you've never heard of.
Eric Weinstein (right) talking with Professor Timur Kuran (left) on episode 4 of The Portal podcast
## Transcript
Eric Weinstein 0:01 - Timur, you have been accused of many things. Are they true?
Timur Kuran 0:04 - Are they true? Well, depends on what the accusations are.
Eric Weinstein 0:07 - Well, they're pretty, they're pretty extensive. I don't have time to go into them all.
Timur Kuran 0:10 - Okay, well, let me I trust you. We're friends. So yes.
Eric Weinstein 0:21 - Welcome, you found The Portal. I'm your host, Eric Weinstein. And today we have something that I think is going to be very interesting for many of you. We are happy to have a guest that I've been looking forward to meeting for quite some time, has been a personal intellectual hero of mine. And he is the gorder family professor of Islamic Studies, a professor of economics and also a professor of political science all at Duke University. So welcome harsh coldness to our esteemed colleague, Dr. Timur Kuran,
Timur Kuran 0:52 - A delight to be here Eric. Thanks for the invitation.
Eric Weinstein 0:55 - So the reason that I've been so eager to have you here is that the this podcast is themed around the idea of escape from a more humdrum existence that is starting to, I think work less and less well for more people. And so we're trying to find ways out of the sort of cognitive traps that we've been held within for quite some time. And I first became aware of your work when I was searching for an explanation of why the field of economics builds such an utterly simplistic model of human preference and belief. And I was led to one book of yours in particular, called private truths and public lies. Hope I have the ordering on that correct
Timur Kuran 1:44 - yes private truths public lies, yes without the 'and',
Eric Weinstein 1:47 - okay, private truths, public lies, which brought an entirely new perspective in the field of economics, which is that of preference falsification, I wondered if you would sort of just Give us a brief introduction to this theory. And then perhaps I'll say a little bit more about why it's so powerful and also so incredibly dangerous to the field.
Timur Kuran 2:09 - So preference falsification is the act of misrepresenting our wants under perceived social pressures. And it aims deliberately at disguising one's motivations and one's dispositions is very common. And sometimes that occurs in very innocent situations. If I go into somebody's home, and they asked me, What do you think of the decor I've selected? I might actually, even though I don't like the decor doesn't suit my taste, I might say to say, Oh, it's wonderful compliments my hosts taste I falsified my preference, but not much harm has come out of it. I've avoided hurting my my hosts feelings. But preference falsification happens in a very, very wide array of settings and some of these settings, it leads to terrible consequences. In the political arena, people are and people, whether they're on the left or what they identify with the with the right or the some somewhere in between. People routinely falsify their political preferences for fear that they will be skewered. If they express exactly what's on their mind. If they say exactly what they want. If they expect suppress the ideas, excuse me, that lie under those those preference preferences. And just to give some examples from our society, immigration is one of these issues. Abortion is another issues. We have a clash of absolutes. You're either pro choice or pro life, and there's nothing in between. And if you take a position in between and offer a more nuanced opinion, that you favor free abortion, let us say in the first trimester, but not later on. You will be accused by both sides there's very little that you will gain and there's A great deal that you may lose. And in today's society, you may lose a lot of friends because the main fault line in American society today is political ideology. There are more people who will object to their son or daughter marrying somebody who holds the wrong idea, who supports the wrong party has the wrong ideology, then will oppose to their son or daughter marrying somebody of a different ethnic group or a different ethnic or different religion. So it can lead what what can happen on issues like this is happening on issues like this is we simply don't come to a resolution.
Eric Weinstein 5:47 - Yeah, so before we started this podcast, the time that we were talking together, I sort of made an unfriendly accusation which is that I think that you have developed a brilliant theory but that you have not actually even understood its full importance. And that part of this has to do with the oddity that sometimes to see what's so dangerous and what's so powerful you actually need curator. So I'm hoping to help by curating a little bit of what I've gotten out of out of your theory and how you've taught me even though we've never met before this week. One of the things I think that's fascinating is that we have a democracy that is stitched together through markets. And when you think about the role of economics in the free market, or even a managed market allows us to each individually direct a larger amount of our action without central direction. And so anything that happens in the economic sphere, like a new theory of preferences, could have absolutely powerful implications because of the role that our understanding of economics plays in underpinning civil society. One of the things that I think that's extremely dangerous about your theory. And one of the reasons I'm attracted to it is is that it is backwards compatible with standard economics. That is, if my private preferences and my public preferences are the same preference, then without loss of generality is we're fond of saying in mathematics, everything that you're bringing to the table is just some unnecessary extra variables because in fact, the two are coincident. However, if my public preferences and my private preferences are different, then while I can recover the old theory from your work, I'm now in some new territory in which I've expanded the field to accommodate new phenomena such as an election that whose result no one sees coming.
Timur Kuran 7:52 - And we've we've broadened the field to accommodate vast inefficiencies that our political system that involves people expressing their political preferences once every four years through a system that involves primaries, nominating conventions, and so on, and ultimately an election, that this system ultimately produces an outcome that reflects people's preferences. When you introduce preference falsification into the picture, when you accept it as something significant, and I would suggest that its significance is, is growing, you open up the possibility that our political system can generate outcomes that very few people want that generate very inefficient outcomes. You open up the possibility that because people are not openly expressing what's on their mind that the system of knowledge development, knowledge production, and knowledge development and therefore solving problems that that gets corrupted.
Eric Weinstein 9:15 - Well, in one of the ways in which I've tried to figure out how to make what you do a little bit more mimetic so that more people start to, to appreciate it. One of the ways I've tried to talk about it with among friends is that you have developed a theory of the black market in the marketplace of ideas, that is underground concepts, underground desires, unmet fears, that can't be discussed in the curated market, managed by institutions. Another way of saying is that this is the economy of silence, or the economy of deception. Do those fit?
Timur Kuran 9:55 - I would prefer economy of deception because people don't say stay silent. We don't have, you know, in our society on most issues, people don't have the luxury to stay silent when they are in an environment consisting mostly of pro-choice. People are mostly pro-life people, they are asked to take a position. So it's not that some people are speaking and other people are silent. If that were the case, we would know well, there there 70% of society is silent. They must not agree with either of the two extreme positions pro life and and when people say things like, but people actually pretend when they're in a group that is primarily or exclusively pro choice or pro life. They sense this. They take that position, that is preference, falsification, and in doing that, they also fail to express or choose not to express the reasons why they find an intermediate position more attractive.
Eric Weinstein 11:08 - Sure.
Timur Kuran 11:08 - And those all of those reasons get subtracted from public discourse. We have a very distorted public discourse on which that is underlying our whole political system.
Eric Weinstein 11:27 - So, I mean, there's so much that's juicy to dig into. I think that there that you may be undervaluing some of the aspects of silence where somebody will say, Well, look, I "I am not a very political person", somebody else might make an admonition "Keep your head down", "stick to your knitting", stay in your lane there all of these ways in which we do favor silence but those of us who have to speak in a professional capacity we're expected to form opinions on these things. We really don't have the luxury usually of staying silent.
Timur Kuran 12:00 - Yeah, I think I will grant this point that there are many issues on which we consciously avoid putting ourselves in positions where we will have to take a position. We
Eric Weinstein 12:15 - We take ourselves out of the game,
Timur Kuran 12:17 - we take ourselves out of the game, but and we're successful in doing that in most contexts. But in going through daily life, we find ourselves in situations in social events or in the workplace, where we have to take a position, everybody's taking a position, there's an issue that is you're sitting around the table and issue is being being discussed. And it has to do with workplace policy on some issue. And you have to take a position and you have to sometimes vote. So your point is well taken that there are there are whole in any person's life there there's there's a pretty broad zone in which you can, you can avoid not taking a position. So yeah,
Eric Weinstein 13:16 - Let's go back through a little bit of just modern history and talk about the times in which preference falsification even though people have often not had the terminology for this theory really came into its own in a way where people were so surprised by a turn of events, that they came to understand that people held preferences that were far different than the preferences that had been assumed to be held and relatively, let's say radical, radically quick shifts in that structure.
Timur Kuran 13:47 - Let me give you an example of from Eastern Europe. communism was remains high inefficient social system inefficient economically, highly repressive also. It was a puzzle to many people that it survived for decades in Eastern Europe. And for a long time, the dominant view was that what kept communism in place for decades in the Soviet satellites in the Soviet Union itself was brute force. And people would give the examples of Prague in 1968, or
Eric Weinstein 14:39 - the show trials
Timur Kuran 14:40 - or hungry, the, the show trials of of Stalin, this is the kind of thing the Gulag. people would talk about, you know, refer to Solzhenitsyn's book, when you actually looked at these societies that were some of them in which there were, there was no gulag and the prison population was smaller than the prison population at the time in the United States as a proportion. Czechoslovakia is a good example. So the wasn't Czechoslovakia wasn't a place that we associate with show trials. Yes, there was we think of 1968 when Soviet tanks came rolling in, but even after that you didn't have major trials, you didn't have huge numbers of people disappearing. So what is it that kept Czechoslovakia communist society, and what kept it a communist society is the people who hated the system, pretended to approve of the system and turned against dissidents, the very few dissidents who had the courage to say, this is a system that is not going to last forever. It's an inefficient system. It hasn't brought us freedom. The state hasn't withered away, it's gotten bigger, it's more important in our in our life, and they would turn against them. What sustained communism all across the Soviet Union and its eastern European satellites was preference falsification. Now what this meant was that the system was extremely unstable. People were falsifying their preferences because other people were doing so. I was even though I was against communism, and you were against communism, we both supported the system because the other was. Now this is a system where if one of us, decides for whatever reason that we're going to call a spade a spade and say this system doesn't work, I don't like it. I go out in the street and I start demonstrating a lot of other people are going to follow. So what happened is, ultimately, the when some demonstrations began, and it happened to be the demonstration started in, in East Germany, these demonstrations started growing every week, more and more people found themselves in themselves the courage to say what they believed and to come out against the regime. The regime itself didn't want to overreact. There were discussions in the Politburo. Some people said we better crack down right now or this is going to get out of hand. Other people said, Well, if we crack down now and some people die that can, the negative effects could be greater their winter is coming pretty soon it will be harder it will be people will be more reluctant to go out in the In the street, let's let this pass let's not overreact. Before they knew it the Berlin Wall was was down and that created a domino effect. Nobody foresaw that. And it's quite significant that among the people who who missed this were the dissidents, the the East European dissidents, who were the only people and I include in this all the top experts, CIA experts, the top academics studying Eastern Europe, almost a little understood what was holding the system together. Václav Havel wrote a book called The Power of the Powerless, and its main message was this society that hates communism holds within it, the power to topple it. Even he missed this even
Eric Weinstein 15:10 - yeah
Timur Kuran 15:38 - he was surprised, even he was surprised When Gorbachev came two weeks before the Czechoslovak revolution, when Gorbachev came to town, a million people came out in Prague to to greet him. They were enthusiastic. They thought change was coming. A New York Times reporter Robert Apple asked asked Václav Havel Is this the revolution that you are predicting is have people discovered that they have the power to topple the regime? And he said, I'm not a dreamer. He said, I'm probably not going to live to see
Eric Weinstein 19:37 - right
Timur Kuran 19:37 - this, this happen. So here's a case of a system built on preference falsification, that was sustained by preference falsification that suddenly collapses when a few people call it out and then you get the
Eric Weinstein 19:59 - the cascade,
Timur Kuran 20:00 - then you get the cascade.
Eric Weinstein 20:01 - So this is one of the things that I want to dig into, because the cascade effect is really a refinement, as I see it, of the old story of the Emperor's New Clothes where all it takes is one person. But then it's missing the mechanism. It's like Newton's laws, there's no ability to transmit gravity. It's an instantaneous action at a distance. To my way of thinking, the best way of understanding your theory for most people is to understand a motif that is found throughout American cinema. And the motif has a name, I believe inside the business, which is called the slow clap, which is that somebody can't take it anymore. And they give an impassioned speech that nobody's expecting that starts speaking to the unmet beliefs of a large group of people, none of whom have understood that there is a lot of support for this in terms of private preference. That's the first action. Now if I understand your theory correctly, people have private preferences and public preferences, but they have some threshold of alternate support in the group that will be necessary for them to update their public preferences towards their private preferences. And then the most important thing is, is that that crazy speech is followed by some anonymous member of the group who starts the slow clap. And that slow clap becomes oppressive. Because in that group, that person is saying, we all know that what has just been said reflects the group and then the slow clap is joined by a third person and that you watch the cascade visually.
Timur Kuran 21:52 - So the way this is what you're describing is a cascade that involves a large group of people who have different thresholds
Eric Weinstein 22:08 - correct.
Eric Weinstein 26:55 - So it's an overshoot,
Timur Kuran 26:56 - this is an overshoot, this is an overshoot, now and now in Czechoslovakia, you did not have a witch hunt against the supporters of the old regime. Of course, the members of the old Politburo were all or most of them were sidelined that the two or three of them managed to repackage them as social democrats and repackage themselves as social democrats and continued in, in politics. Most of the people were were sidelined. There wasn't the witch hunt, but there were other countries in which there was a witch hunt. So it was very, it was and and of course, Czechoslovaks didn't know why what was going to happen there was always a danger that the that the new regime would go after the old communists and try to punish them and punish people who ran the jails and and had important positions in the in the Communist Party. But But it was so because there was a possibility of this danger. Now they pretended that they were all uh all along they were. They were lying. So, events, massive events that changed the course of history, which were unpredicted after the fact they become. One looks at them and one finds it impossible not to understand why they happened. We have the're overdetermined
Eric Weinstein 28:40 - right,
Timur Kuran 28:40 - we have tremendous amount of data showing why showing why the system had to collapse. Yet in reality, to go back to your example, if that one person hadn't made the impassioned speech, this thing could have gone on for more years,
Eric Weinstein 29:01 - well, let's play with this a little bit. One of the things that I find so fascinating about the theory is it also sort of starts to explain how in a society where people's private and public preferences are somewhat aligned. They can go out of alignment very quickly. So I don't know if you've seen the video, for example of Saddam Hussein coming to power at a Ba'ath party meeting in Iraq, which is fascinating.
Timur Kuran 29:28 - I'm not sure I have seen some videos of Saddam Hussein in Ba'ath party meetings. I'm not sure I saw that.
Eric Weinstein 29:35 - you'd remember it
Timur Kuran 29:36 - Maybe, maybe you
Eric Weinstein 29:37 - let me describe it for you, because you'll see the mechanism, the opposite direction
Timur Kuran 29:40 - Yes.
Eric Weinstein 29:42 - So he's sitting there on stage smoking a cigar and he's videoing himself. I think knowing what comes next he says, hey, we've got a special guest today. And a man who I don't know exactly who he was stands up and start speaking and saying I have plotted against saddam and I have co-conspirators in the audience and I'm going to name them now. Well, you see terror take over this auditorium, because there's also cameras, if I recall correctly on stage filming the people. And these names get read and these people are being led out. And then the preference falsification sets in, and you start seeing the private preferences, suppressed and the public preferences going into nonsense territory, and people are saying, Long live our brother saddam, he is the one because they realize that their life is on the line. And according to legend, and I don't know whether this is exactly true. Those who are left at the end are given sidearms to execute those who have been led out to make them complicit in the crime to freeze in the preference falsification or if you like people are now preferring to, to save their lives rather than preferring to explore their politics. So do we see I mean, I'm just trying
Timur Kuran 31:01 - I hadn't seen this video. I've heard just as a little footnote here that in north in North Korea, the Kim's have used the same sort of thing where they actually will say that they're going to name some people in the audience. The latest one where was where a relative of Kim Jong Un was, might have been an uncle or something who was actually led out this was the same sort of thing that happened in that case, I don't think it was somebody from the audience who pulled the trigger but everybody could hear a shot go he was obviously murdered. Everybody could hear that this was instantaneous. If you did if, if Kim decided you had betrayed him you will be put to death.
Eric Weinstein 31:58 - Well, this is what I have. a pet project of mine which I don't think I've ever advanced sufficiently, is what I term the analysis of message violence that there's certain violence that is committed theatrically as a instrument of transmission to induce preference falsification. So this is used by the cartels in Mexico. This used to great effect by the Kims it was used by Saddam Hussein. And with message violence, the idea is to create something so horrific beyond what is necessary to silence someone through murder and death, to communicate to others, the instant necessity of beginning to falsify their preferences. So that a it's a leveraging effect where a small amount of violence results in the maximum amount of preference falsification,
Timur Kuran 32:56 - yes, this does happen and there are plenty of examples. We can Given go back to the show trials of the Soviet Union where every single member, we're stalling got rid of every single member of Lenin's Politburo, all the heroes of the October Revolution and the building of the Soviet Union one by one he got rid of them through through show trials and the fact that such heroes could be executed in such humiliating ways sent of course, a message to the entire society that if this happens to them this could happen to this could happen to any anyone, but I would want to emphasize that preference false question even massive preference falsification can occur even without such theatrics. And if we come back to our own society, jumping from the Soviet Union and Iraq to the United States today, there are many issues on which we do not talk to each other, honestly, which there's a great deal of polarization and people and expressing nuances can get you in great, great trouble. And we cannot point to a single event. We can point to many smaller events, but no single event that has the theatrical acts of Saddam's Saddam's executions or what what the Kim's are doing
Eric Weinstein 34:28 - well, and I'm so glad that we're making this transition. Because as interesting as the historical examples are and the those that are particularly bloody, the best application of this theory, in my opinion, only comes from when we realize that violence can be moved from the physical sphere to the reputational and the economic sphere. So if you think about your reputation as part of what Richard Dawkins might have called our extended phenotype, it's something that you carry around with you. That is necessary for let's say employment. We now worry about reputational violence which can be exacted theatrically, for example, through social media. So the question of what we can say what we can discuss what we can explore has a similar character. If I take the James D'amour situation at Google, this was a particularly you know whether or not you thought his memo was brilliant or a little bit tone deaf. It certainly wasn't an insane exploration of misogyny it was some exploration of differences between men and women at the level of Big Five personality inventories. The idea being that success or failure might have a lot more to do with one's Big Five, let's say hedonic decomposition of our personalities rather than our actual gender. And then if males and females had different hedonic profiles at the level of Big Five personality inventory traits that could explain some of the imbalances. And he was actually, to my mind talking about the fact that if you wanted to have a more equal society of engineers, there are things that you might explore to try to actually better utilize women in the workplace. Now, whether or not you buy into that, or it certainly didn't seem like an insane thing to suggest, and yet, the reputational violence that was exacted on somebody who was told to attend a seminar and asked for feedback seemed to me to be of a piece with this kind of message violence but not at a physical level at a reputational level. Do you think that there's some parallel there?
Timur Kuran 36:46 - Yes, I think the reputational violence can do enormous harm in the society not only can it can it affect your job prospects, your prospects for promotion in the company that you're working for, you can lose a lot of friends, it can affect your prospects in the marriage market. 50 years ago, when people were asked Americans were asked whether they would mind whether their daughter or son married somebody of the opposite party. About 20% said that it would make any difference to them. By contrast, more than half of Americans said that if their son or daughter married somebody of the opposite of a different ethnic group, or have a different religion, this would matter to them and many people said they would not accept the person a different religion, different ethnic group different race into their their family. Those numbers have come way down over the years. By contrast, the numbers regarding ideological differences and party affiliation have gone way up today. So this So, being attacked or coming back to reputational violence, being pigeonholed as a radical Republican, or even as a Republican or being pigeonholed
Eric Weinstein 38:36 - radical is implied
Timur Kuran 38:37 - and radical is implied for many people or
Eric Weinstein 38:39 - same on the Democratic side
Timur Kuran 38:40 - or being pigeon holed as a Democrat even even then
Eric Weinstein 38:44 - now you're a radical leftist
Timur Kuran 38:45 - not even not even not even a progressive democrat just
Eric Weinstein 38:47 - right
Timur Kuran 38:48 - to many people. The all democrats are the same whether you know, the nuances between
Eric Weinstein 38:53 - well the're libtards
Timur Kuran 38:54 - with the progressives and more what we call the way many of us would call more moderate Democrats, there's no such distinctions. They're all on the wrong side. And there are people who do not want to befriend them, who would be completely against their son or daughter marrying a democrat or republican depending on who they are. And you can see why at the Thanksgiving table, the tensions would be enormous, because it would bring them to bring Democrats and Republicans together, even moderate Democrats and Republicans together these days, let alone people on the on the right side of the republican party with the progressive Democrats is a is a recipe for complete disagreement for opening up issues that will expose hatreds Because the two sides no longer talk to each other, because no one accepts the possibility the viability of a middle of some kind of compromise. People don't know how to talk to each other people don't know where their differences begin and where they might actually have some room for, for compromise. And so there's a reason why these days, people feel that if they are pigeon holed, if they say something that then allows others to put them into one of these pigeon holes, political ideological pigeon holes, that their life will be ruined. And so, this is let's go back now to the East European situation. This is similar to what the dis dissidents faced in Czechoslovakia. Yes, dissidents who didn't distance like Václav Havel, who did spend small, short periods in and out of prison, but mostly he was allowed to be a dissident playwright, but he got enormous amount of hate mail. Most people, even people whom he knew from earlier times in his life, would not say hello to him for fear that the friendship would imply that they sympathized with his ideas, they cross to the other side of the road that they saw him coming to so they wouldn't have to confront them. This, so his social circle got got smaller the number of people he could go to ask for for help diminished. So all of this was all of these inconveniences. This is happening right now in the United States. It means that if if you cannot live with somebody of the other party as a close relative of yours, if you cannot talk to the other side because you think they're just beyond the pale, they're subhuman their ideas just are are inhumane. They're just that there's no way you can even begin to consider their validity or consider them as worth discussing as part of a part of a conversation. You're certainly not going to see them as people you can go to in a time of trouble. That is why you would rather live in a neighborhood consisting of republican where everybody's republican and if you're a democrat where everybody's a Democrat, because you like in a time of need and time of emergency, you'd like to be able to go to your neighbors, you'd like to you'd like to have neighbors with whom you can have pleasant chats when you meet them in the street when you're walking your dog and you meet them in the street and not have to ignore them and see them as evil people.
Eric Weinstein 43:24 - Well, so this is and I mean it's fascinating to me. So many different ways to go here. I'm trying to figure out what what the best line through is. One thing that I'm fascinated by maybe we'll come back to this is what is the force that makes the middle so difficult to hold that pushes more and more people to towards either being sort of what I've termed troglodytes or dupes. makes it very difficult to to I guess what my model is that you had A-frame roof as the A-frame roof gets more and more peaked. There are a fewer number of Fiddler's who can stay on the a frame roof without falling over to the left or to the right. And so that right now, I think that the skill level needed to inhabit a sensible position is priced out of almost all of our abilities.
Timur Kuran 44:21 - I mean, this is it for what leads you from a position where 50 years ago where we had again, people on the extremes we had people who favored segregation, people favored desegregation. We had we had serious disagreements before, but there were many people in society who held positions had strong opinions but also felt that the people on the other side were humans. Were well meaning,
Eric Weinstein 45:09 - right
Timur Kuran 45:09 - And could be parties to a conversation,
Eric Weinstein 45:13 - right,
Timur Kuran 45:14 - and you could compromise with them. So when you picked up the New York Times after some vote in Congress, 50 years ago, there would be a list of Democrats voting for Democrats voting against Republicans voting for Republicans voting against them. There are lots of people in all four of those groups. And all four of those groups were considered legitimate,
Eric Weinstein 45:37 - right
Timur Kuran 45:37 - Even the people who have voted yes, it considered the people who had voted no in their party. They considered them as legitimate senators or legitimate Congresspeople and they, on some other bill, they cooperated with them. So this was and of course you just mentioned a skill set there's a skill set that went with that the skill set was that you could you and I could disagree on issue A
Eric Weinstein 46:10 - yeah
Timur Kuran 46:10 - and and and debate for days and days and days and why your I could say that your thing is going to lead to disaster along this front and and you could say the same thing about me at the same time at the end of the day, one of us would win the bill would either pass or lose or there would be this would go into some conference who does some kind of compromise. You and I would accept that compromise as legitimate. And so we would we develop the skills. As we did this we develop the skills of compromise the whole political system developed this and society saw this and accepted that people Republicans and Democrats both legitimate representing legitimate sides of legitimate positions on issues subject to screaming, we gradually have moved. It's a cascade,
Eric Weinstein 47:11 - Right.
Timur Kuran 47:11 - that has moved us gradually that has expanded the area an area of absolutes, positions on which we have apps issues on which we have absolute positions, and they're not subject to discussion. And what's happening what has happened in the last few decades is that the number of such issues has grown. As this has happened, we have the the number of issues on which we no longer discuss we just have absolute positions where pro-choice or pro-life we don't discuss. We don't have conferences where we discuss what kind of bringing people from both sides say what kind of compromise, can we
Eric Weinstein 48:01 - will this compromise at a political level, but I think it's also a question about the intellectual basis of our conversation. So let's just take pro-life and pro-choice.
Timur Kuran 48:09 - Yes.
Eric Weinstein 48:11 - I talked about sometimes dining ala carte intellectually, where I can't get my needs met in a low resolution world, anyplace and so I sort of pick and choose which bits of things I need. And I sort of think of this as political flatland that people are trapped in pro-life versus pro-choice. And my real position is a plague on both your houses. I'm not pro-choice. To the extent that I'm willing to call a child four minutes before its birth, fetal tissue, nor my pro-life to the extent that I'm going to call a blastosphere, a baby. Both of those seem patently insane to me. And nowhere do I get to discuss Carnegie stages and embryonic Development, which would be sort of a kind of a more scientific approach to what quality of life is it that we're trying to preserve. And yet I caucus if you will, with the pro-choice community, not because I hold the idea that it's simply a woman's right to choose, because obviously there's something else that's going on inside of the woman. There's the whole miracle of gestation and reproduction. But if people see that I caucus pro choice, then they say, okay, you're willing to sit with somebody who's willing to terminate a third trimester pregnancy frivolously because they're ideologically committed to it. Ergo, you're evil. Ergo, we can no longer be friends. And my key point is, look, I'll drop these people in a heartbeat if you give me some nuanced room in which to maneuver let's talk about the neural tube formation. Let's talk about what we Think of his life is that the emotional connection to seeing something one recognizes is human? Is it the quality of the of the brain? Is it something mystical in ineffable? Are you coming from a religious tradition? The key point is to make it impossible to have a discussion. And, you know, I remember being beaten up on a picket line in a picket line where there was a group that was picketing a, an abortion clinic, and I was demonstrating for the right to keep it open. And I got beat up in Rhode Island on camera. And after this incident, I think I had a chance to talk to the person I thought it hit me with the picket sign. And it turned out that we could come to we couldn't get all the way there. But there was at least a partial rapprochement where we could say, well, I see where you're coming from, I see where you're coming from. Maybe we can understand that you're both motivated by the best interests that we as we perceive them, that has gone away in large measure, because what we've taken or at least this is my understanding is our institutional media and our sense making apparatus and they have become complicit in making the center that is the sensible and analytic center absolutely uninhabitable.
Timur Kuran 51:20 - Yes
Eric Weinstein 51:20 - Does that match your..
Timur Kuran 51:21 - I think this has happened. And I think this has happened in a growing range on a growing range of issues, which is why. Now we go back to New York Times lists of who in which party voted which way, sometimes that list doesn't appear because simply party they say is just a party line vote. And this is a reflection of society, that and it's not that within the Republican Party or within the Democratic Party, you don't have people on whatever the issue is. You don't have people in in the middle. But that if they take if they bring up the nuances, if they try to bring the conversation a little bit toward a compromise, they will get skewered by the people, by their own people or the other side,
Eric Weinstein 52:26 - right
Timur Kuran 52:26 - and the other side will not come to their defense. And in fact, if the other side does come to their defense, that's a terrible signal for them, and they'll be skewered by their own side.
Eric Weinstein 52:37 - What concerns me here, though, is that we are dependent on people of integrity, who risked everything when it was least popular to do it, so that we can sort of hold these people in reserve. So when the madness becomes too great, we can turn to them. Let me just take a couple of examples that matter to me, one of which was the Patriot Act. And then when the Patriot Act was voted in, in the wake of 9-11, and there was this sort of mob hysteria to do something, because something very significant had happened to us. Only one person, only one senator voted against it. And that was Russ Feingold. And so I don't have a clear memory of the other names in the senate at that time, but I will always remember Russ Feingold for the courage to stand alone. a different sort of version of that, I think about as Katharine Hepburn, who is the sort of the most loved of all all Hollywood actresses, I think she had four Academy Awards that she used as doorstops for her bathrooms. Because she didn't seem to give a wit what other people thought of her. And she went and did, if I recall correctly, you know, Connecticut community theater theater during the McCarthy era, because she was just going to wait out the stupidity, the excess and the idiocy of the movement. Whereas, a Humphrey Bogart who or organized an artist's push to fight back against this was immediately cowed by an article in Filmfare magazine if I recall correctly. He said, Well, sorry, he had to write an article saying, Hey, you know, don't call me red you, I'll never do that again. And the great Humphrey Bogart, the tough guy of movies crumbled under this pressure, whereas Katharine Hepburn, his co star, you know, sort of stood tall and waited it out, do we have these hyper individuals, these incredibly disagreeable people in the sense of the agreeable component of the Big Five personality inventory, where we know who they are, and we know to whom we can look in times of crisis?
Timur Kuran 54:46 - Well on particular issues, you will find people who write books that advocate a middle position that I do identify all the nuances that portray both sides as having legitimate goals, they don't necessarily get attention. So they they write a book, whether the issue is abortion or immigration, it takes some kind of middle position, it doesn't get the play in the media
Eric Weinstein 55:27 - right
Timur Kuran 55:27 - that it that a book that takes a very strong position a very absolutist position does so so yes there do on any on any given issue, there are some some people who you can find people who are trying to start a dialogue you can find here their little associations, little nonprofit organizations that are trying to start a dialogue doing so but they just don't. That's not where the what the media pays pays attention to. So effectively, they don't exist. And the the groups that increasingly, the groups that get attention are the groups that pigeonhole people into one side, you're either for us or against us. And the two sides, the two extremes, both of whom are playing this game of you're, you're with us or against us, the're actually reinforcing each other.
Eric Weinstein 56:40 - Yeah, yeah, they're agreed.
Timur Kuran 56:41 - They're completely agreed on that.
Eric Weinstein 56:43 - Yeah,
Timur Kuran 56:43 - that there is no middle position. And having a middle position and having the media pay attention to the people in the middle would hurt them both.
Eric Weinstein 56:52 - Yeah, I don't think it's in the middle. I mean, I really think and for those of you who were watching, rather than listening, I think that there's this very flat, low dimensional plane where these positions live. And what we're calling the middle is not the thing between these. It's in a higher dimensional space that combines these crappy low-resolution moronic positions, and it projects to the middle. But it isn't the middle.
Timur Kuran 57:18 - Absolutely, absolutely. There are many more dimensions that they ... these. It's simply that these simple position, positions hide I completely agree with that. And, and the middle is is often is more complex involves many more dimensions. And these dimensions to go back now to these extreme groups that don't want these dimensions to be brought into the picture. So for the pro life group, the issue is, are you going to terminate the life or not
Eric Weinstein 57:56 - right
Timur Kuran 57:57 - and for the pro choice group is do you Respect a woman's right to choose. And so each one of them for each one of them is just a one dimensional thing. There's a yes, no answer its a yes/no answer. And there's no bring in some other dimension is immediately gets you in trouble.
Eric Weinstein 58:20 - So I want to talk about the specific weirdness of economic theory. Yes. Now, I claim to be an economist, I've never taken a class in economics and partially The reason for that is that I developed a theory with my wife about gauge theoretic economics. And I always thought that if we could get attacked, and somebody could say, well, you're not really an economist, I'd get a chance to defend myself because it dealt with another aspect there. They're the great adjustments to preference theory. preference falsification is yours. Yeah. gauge theoretic changing preferences is ours, Paul Samuelson had one about incoherent preferences. That was he buried in his Nobel acceptance speech,
Timur Kuran 59:05 - which has received very little play in economics
Eric Weinstein 59:07 - almost nothing. He was the one who pointed it.
Timur Kuran 59:09 - Yeah,
Timur Kuran 1:01:03 - Well, there was here's again an example of a theory that is foundational to discipline that gets falsified. I think his first name was Richard Richard Herrnstein. You would does the name ring a bell at ah Harvard was Richard or Robert, remember but anyway Herrnstein, he developed a theory that explained a phenomenon that that Becker swept under the rug which is that an addicts heroin addicts preferences,
Eric Weinstein 1:01:51 - hyperbolic discounting
Timur Kuran 1:01:52 - do change through hyperbolic discounting so there are many addicts who after they've taken their fix want to, they understand now that the panic attack is gone away. And they understand that this heroin addiction is ruining their life and they very sincerely want to give it up. They very sincerely want not to take more heroin.
Eric Weinstein 1:02:31 - Right?
Timur Kuran 1:02:32 - But a few hours pass and they need their body starts
Eric Weinstein 1:02:40 - jonesing
Timur Kuran 1:02:40 - they they start craving
Eric Weinstein 1:02:42 - Yeah,
Timur Kuran 1:02:43 - heroin again, they need a new fix. And they get to the point where their preferences change to let me have one more
Eric Weinstein 1:02:53 - I'll quit afterwards,
Timur Kuran 1:02:55 - and I'll quit afterwards. I am prepared to quit now. A few hours ago they were prepared To put immediately, now they're willing to quit. But after I get my next fix, and this thing can go on again again, so you have inter-temporally inconsistent preferences. So this is another problem with the, with the economics discipline. But economics is not immune to the forces that we've been talking about,
Eric Weinstein 1:03:30 - well,
Timur Kuran 1:03:30 - there is preference falsification. In the economics discipline, there are certain fundamentals of the discipline and if you challenge them, as a young person, you're never going to get a job.
Eric Weinstein 1:03:49 - right.
Timur Kuran 1:03:50 - And if you and if you challenge them before you get tenure, you're not going to get a job. But if you develop a reputation to get tenure, you have to develop a certain reputation. And that has involved adhering to the conventions of the discipline. Theoretically, you could after you got tenure, you could switch. But the costs then are huge because you've developed a certain there's a lot of reputational capital you have.
Eric Weinstein 1:04:23 - And we're watching a lot of prominent economists sort of change their position without announcing that they used to be, in effect working for a nonsensical theory, or at least quieting themselves. I was astounded by Paul Krugman's column, or maybe as a blog post called a protectionist moment where he starts talking about the scam of the elites forever freer trade, where I associated that with sort of the intellectual force of Jagdish Bhagwati. And some of these theorists who clearly were sort of pursuing political position where, you know, in the case of like free trade, there, there are two separate phenomenon, you can say that something would parado improve the society if everyone is made either as well off as they are today or better off. And then there's this other kind of more technical version of this called Kaldor Hicks improvement, which is that if we were to tax winners to pay losers, then everyone would be parado improved. And I've noticed this very interesting thing about economists, where they have two voices they have the voice that they have to use in the seminar room, because there's nowhere to hide from the fact that a lot of these public pronouncements are absolute nonsense. And then the claim is, is that oh, well, when we're in our seminar voice, and then maybe this was Danny Rodricks phraseology, I can't remember whose it was. But then when we speak publicly, we're allowed to say something that is actually different. It's not the same thing in two different voices. It's an idea that there's an exoteric and an esoteric way of expression, which is a sort of Straussian theory and the esoteric is reserved for one's colleagues. But we're actually allowed to lie to the public to help the fortunes of the politicians we favor when we're speaking publicly What the hell is going on?
Timur Kuran 1:06:22 - So there there's some people who have achieved a certain stature in the profession. And yet they feel there's certain things that are wrong about the profession or that they can't say within the profession, they develop a second persona, which is their op-ed personality
Eric Weinstein 1:06:44 - the're policy entrepreneurs
Timur Kuran 1:06:45 - and the're, but the're, the're the'er policy entrepreneurs and as public intellectuals, they're much more critical of the discipline than they are within the discipline or they have decided that there really isn't a possibility of changing the discipline. But there's certain points that have to be made. And they're going to make them anyway and they're going to make them in a much less technical way. And there's there's a third a charitable interpretation. I think this does apply to some of my colleagues, I would say, they believe that the, the the core principles of economics, even if they're not true, even if they don't give you a reflection of the real economy, they lead to useful correct thinking that they're very useful for disciplining your way of thinking, thinking as an economist and they represent, they give you a good base model, which you can tweak
Eric Weinstein 1:08:07 - right
Timur Kuran 1:08:07 - to, to bring in reality. So I have had some people say to some people who for years did not take my work on preference falsification seriously, who have now come to the position that this is a useful extension of economics. And they've said, you know, you did use standard economic tools of utility maximization.
Eric Weinstein 1:08:42 - Yes.
Timur Kuran 1:08:43 - In order to get to this point. And there is
Eric Weinstein 1:08:49 - that's why you're so dangerous
Timur Kuran 1:08:50 - there is a, there is a point to that. Yeah, but there's a point to that.
Eric Weinstein 1:08:53 - The problem is, is that that's why it's actually intellectual kryptonite. So, because your theory can be accommodated within the standard theory,
Timur Kuran 1:09:01 - yes.
Eric Weinstein 1:09:03 - The question is well, okay,
Timur Kuran 1:09:05 - a version of it.
Eric Weinstein 1:09:06 - Yeah, well, I I think I could do a pretty decent job of shoehorning it into this sort of Samuelson neoclassical perspective. The problem is it's a ready made upgrade to the existing theory in which nothing is lost, but new degrees of freedom are gained. And that could have an absolutely unpredictable effect on the entire field because it's at the level of the substrate.
Timur Kuran 1:09:31 - But the the big danger is that so many propositions involving efficiency, that if you let the system
Eric Weinstein 1:09:41 - and revealed preferences
Timur Kuran 1:09:42 - you left and and the the principle of revealed preferences that that that actions reveal people's people's preferences that goes out the window, and many efficient properties if if that if you allow people to to interact with each other, you're going to get efficient political solutions you're going to get efficient solutions in in the market. My way of thinking leads you to multiple equilibria. And one equilibrium can be preferable to another.
Eric Weinstein 1:10:26 - So this is one of the great dangers for economists as high priests, which is if there are multiple ways in which a market can evolve. Therefore, you can't say that the market finds the optimum because you can't say which of these things actually was the optimum.
Timur Kuran 1:10:41 - And there's a danger to political economy, which is that the political system what the political system generates, is not whether you have elections or not, and whether you have a secret ballot or not, is is not necessarily efficient. Because if you in a system where people are are not cannot speak freely. Many ideas are stuck underground, they're not being expressed. People are when people are going through the primary process. They're not thinking of all the options. They're not thinking of all the dimensions. They're thinking in a single dimension. And so they're not coming up with the with candidates who hold the best positions, whatever your values are, or a set of coherence is something we haven't talked about is the coherence of various policies. one of the things that can get you in great trouble is if you say, within the Republican party or the Democratic Party, look this policy on this policy I'm with you on this other policy I'm also with you and on this other third policy I'm also with you, but the three policies you cannot put them. You can't we don't have the resources to accomplish all
Eric Weinstein 1:12:14 - drug interactions between ideas
Timur Kuran 1:12:16 - what are your what are your and some of these policies undermine others these are not necessarily consistent with one another. So in the eh with these these parties or coalitions these coalitions have certain objectives
Eric Weinstein 1:12:33 - right
Timur Kuran 1:12:34 - they are they are deliberately keeping quiet about the contradictions.
Eric Weinstein 1:12:44 - Well, I think
Timur Kuran 1:12:45 - among these Among these,
Eric Weinstein 1:12:46 - I think there's some contradictions that we legitimately even lies I talk about load bearing fictions.
Timur Kuran 1:12:52 - Yes.
Timur Kuran 1:15:24 - Well, this is I mean, asking, I cannot imagine being in a department meeting where somebody asked this question and says, Why don't we base our hiring on of say macroeconomists on how well they've done in a market? or I, I think they would be immediately left out. I don't think it would ever make it onto the agenda. I think the institutional pressures against applying such a criteria are too great because economists also believe, most academic economists, that they have come into an institution where the primary goal is seeking the truth. They've given up possibly more lucrative careers, and they should not be there for judged on the basis of how well they do.
Eric Weinstein 1:16:43 - I'm not saying only trading. Maybe you could ask the question, for example, does being an expert witness as an economist?
Timur Kuran 1:16:50 - Yes,
Eric Weinstein 1:16:50 - for one side or the other influence the objectivity of your judgment? You could ask the question does the prestige of being invited to Jackson's Hole affect the quality of discussion? Because people don't want to be excommunicated from the priestly class, you could ask the question of whether or not the secret Harvard jobs market meeting which is a particular problem for me, actually serves the interests of economics or serves the interests of the higher ups in the,
Timur Kuran 1:17:27 - in the profession,
Eric Weinstein 1:17:27 - in the profession by being a direct interference in the free trade of ideas, all of the really fun questions that economic economists would ask in a heartbeat about anyone else they refuse to ask about themselves. So it's quite a bit more pointed than just asking for trading prowess among macroeconomists. You've the profession and this isn't against you, the profession has trained it's magnifying glass on everyone else. When do we start doing the economics of economists?
Timur Kuran 1:17:59 - You know, again, I think there are a few people here and there who publish in journals that very few people read who have done this sort of thing. There have been studies of the economics profession Mirowski Philip Mirowski
Eric Weinstein 1:18:27 - More Heat than Light,
Timur Kuran 1:18:28 - More Heat than Light I think might be was he he has done some work along these lines.
Eric Weinstein 1:18:37 - Economics is failed physics.
Timur Kuran 1:18:39 - Yeah. But the people doing this are not people at the top of the profession
Eric Weinstein 1:18:47 - as perceived
Timur Kuran 1:18:48 - as as perceived as perceived by the departments that get take the first picks when the junior job market opens are considered in the rankings in the US News and World Report rankings are considered the top departments to get a PhD from and so on. Based on on that ranking people who are at the top are not among those asking the question. So again as with other issues, which were very polarized, other issues on which there are taboos
Eric Weinstein 1:19:36 - right
Timur Kuran 1:19:36 - areas that quest questions that involve or that raise questions that nobody can really or that bring to mind questions that nobody can really ask at least in polite, in polite the company. As in those cases here, the questions the contradictions you're raising have been noticed. There are people who have written they just don't get attention. They don't again..
Eric Weinstein 1:20:10 - but but, to me like it's like saying, you know who is the greater wrestler? Gorgeous George who wrestled in part of the professional wrestling arena where everything is fixed. Or Khabib Nurmagomedov, who wrestled inside of the UFC? Who's an unbelievable grappler. Well, I don't think that Nurmagomedov has ever achieved what has been achieved inside of the WWE. When everything's scripted you can do things that are so much more fantastic than anybody outside. And yet, what we've been trying to do in part is to ask the question, why can't we smuggle legitimate economic kryptonite into the economics profession so that it can grow into a real field, if I think about the favorite example is imagine that you've got alchemy and chemistry in the same department, or you've got astrology and astronomy in the same department, the great opportunities to get rid of the astrologists and get rid of the alchemists,
Timur Kuran 1:21:15 - right
Eric Weinstein 1:21:15 - Because it's not that all of economics is nonsense, but so many of the perceived top players in the field are actually acting as professional wrestlers, that it's time for the revolution that I would imagine. Your theory actually predicts. It's so ripe and so many of us who are mathematically inclined look at the kind of the history of mathematical intimidation. And when you think this is mathematically intimdating, you guys aren't even that good at math.
Timur Kuran 1:21:45 - You know, this, this may actually happen through the young generation. And it might actually took a couple of generations one huge change that has happened in the economics profession.
Eric Weinstein 1:21:59 - Data
Timur Kuran 1:21:59 Since, exactly, since Becker and Stigler road Augusta was known as Disputandum,
Eric Weinstein 1:22:07 - yeah,
Timur Kuran 1:22:07 - that was, I believe 1977 Yeah. Since they wrote that the most prestigious field within economics, which used to be economic theory, has lost prestige, the best economists now go into data heavy area, and they are driven by empirics. And often the theory follows the empirical work that they do if there's a theory at all and
Eric Weinstein 1:22:45 - sometimes with like deep learning you don't even know what the theory is
yeah ....
Timur Kuran 1:22:48 - you don't even know what the theory is and they start with so much data that they they just start analyzing it from some some corner of the the issue and then hope to come to and that leads in the very, very best of those works, then generate new theories. So where now the empirical parts of the profession are driving the theoretical
and the theory and the theory and the theory. The the old theorists who were trained as theorists never to touch and to look down on people who worked with the with the data, look down on on them, many of them are retiring. They are being replaced by theorists who are who are getting accustomed to operating in departments where the bigwigs
Eric Weinstein 1:24:00 - Are the data cowboys
Timur Kuran 1:24:02 -
are the data cow cowboys, and this is going to have some effect on the theory because the empiricists that I talked to in the economics profession now, consider a lot of the theory a waste of time, a lot of it highly. Highly misleading.
Eric Weinstein 1:24:27 - Yes,
Timur Kuran 1:24:28 - some of it far too abstract and, and irrelevant and that the the theory taught to the first year graduate students and even going before that to undergraduates and master students, that this has to change. Nobody yet though, has come up with the equivalent of Paul Samuelson's first edition of economics
Eric Weinstein 1:25:01 - Well, this is
Timur Kuran 1:25:01 - where he where he wrote where he
Eric Weinstein 1:25:04 - a framework a extensensible framework for which it almost any question that can be posed can be posed within the framework
Timur Kuran 1:25:12 - within the framework. And and it was and it was, and and within a few years, all major departments were using either Paul Samuelson's textbook or textbooks written according to the same template following you know, basically offering the same thing at a somewhat higher level somewhat lower lower level, but basically and that is that has come down to the present. There have been a few attempts to bring in behavioral economics for example, the're the're textbooks are not quite popular people like Bob Frank, Daniel Kahneman have of course introduced new ideas about, concerning behavior and how people how people think. And they've been attempts to bring some of these ideas into textbooks, but they don't define the mainstream yet.
Eric Weinstein 1:26:23 - Well, this is, this is the thing that I think people don't realize about economics, which is I could make a decent argument that our two greatest theories or two greatest intellectual theories that we've ever come up with, would be Darwinian selection in the in the realm of biology, and which I think has flaws and what I would call geometric dynamics, which covers both the modern understanding of the standard model and general relativity and what's weird is is that economics if you think about it is a decision to make a continuation of selection by other means, which is to come up with an as if physics to mediate selective pressures between apes, which is us, and it's the only place I know where there's a meaningful interaction be possible between our two greatest ideas. So for me, the really interesting part of economics is that it is the one place where our greatest ideas might even touch and reproduce. The problem I have what the profession is, is that the fear of what could happen if we started to do real economics has locked out the kind of innovative spirit which requires both much more detailed knowledge of selection as per Kahneman and Tversky, and much greater understanding of mathematics. It's not that you guys have used too much mathematics it's that you're not good enough and you're not advanced enough in mathematics, lots of people have master's degrees, very few of PhDs and very few of those are trained in the few subjects that would reveal market markets to be truly geometric, which is a revolution that happened between geometry and physics in the mid 70s, or for the Standard Model, or the teens for Einstein's theory of relativity. You guys are next. And it's a question of people holding back the possibility for genuine innovation. So this is a place where I've been hoping the preference falsification would actually lead to the cascade effect that we began this podcast talking about.
Timur Kuran 1:28:37 - Well, this is I'm not sure that actually don't think that this is going to happen through people who have who are currently falsifying their preferences to agree with the direction you go,
Eric Weinstein 1:28:57 - right.
Timur Kuran 1:28:58 - And then they become, disguising their preferences, they the chairman of the major department, then they suddenly suddenly redirect hiring and the department changes. I don't think it's going to happen that way. I think it will happen through the emergence of new departments and smaller departments, lesser known departments that
Eric Weinstein 1:29:27 - George Mason
Timur Kuran 1:29:28 - that decides, so George Mason is has
Eric Weinstein 1:29:32 - a particular direction,
Timur Kuran 1:29:33 - a particular direction, something and there were some brilliant people. Buchanan, Tullock, Vernon Smith joined them later on who had problems with the direction that economics was going with what it implied for political science for political markets. And they were they were pushed out of the mainstream of the profession. They just decided to form their own department. They, they, of course, they all congregated at Virginia Polytechnic Institute. Then when they decided Virginia Polytechnic Institute decided crazily, I think that they'd rather have a mainstream department. They just packed up and left and George Mason jumped at the opportunity. So this can happen that that is the model that I think that there will be a group of people, some of them young. In fact, probably many of them young young enough that they still have, can
Eric Weinstein 1:30:38 - energy and creativity,
Timur Kuran 1:30:39 - energy and creativity and think of think of developing their ideas for several decades. Who Who, and there's some university with with a with a visionary president and some entrepreneur who gives a big grant to establish a new department and you get 10-15 people collect somewhere. That is, I think the what will happen to shake up the the economics, the the profession and shake up in particular, the theoretical
Eric Weinstein 1:31:21 - Yeah,
Timur Kuran 1:31:21 - core of the discipline. I think the empirical parts of it Yeah, I think are just being shaken up daily true through the data coming in and through the, the very interesting results and findings that are that are coming up as people are developing huge sort of new datasets.
Eric Weinstein 1:31:46 - Like if you think about natural experiments, you happen to have a flood that you could never actually, you know, ask for because it would kill people and it would destroy crops. But once you have such a thing, you look at the peculiar thing that happened as a control experiment. So I do see that there's some hope the concern that I have is that the theory is going to get thrown over because it was handed to the wrong group of theorists, and that the right group of theorists is not going to be allowed in who could actually change the theory.
Timur Kuran 1:32:14 - Well, this is this is in a sense, the George Mason people have would have never been allowed in, Buchanan and his group, he did win a Nobel Prize. He did. He has, he's actually been more influential outside the United States in mainstream economics departments than
Eric Weinstein 1:32:35 - on the blog
Timur Kuran 1:32:35 - than in the than in the United States. But there are, they did create a self sustaining-group,
Eric Weinstein 1:32:46 - right.
Timur Kuran 1:32:46 - And they've generated enough PhD students who have gone to departments generally departments that are not in the top 20-30 maybe not, usually not Not in the top top 50 and they're doing work that continues the Buchanan tradition. This is the way it may start. But just because the that Buchanan's experiment didn't result in the quote unquote, conquering of major departments doesn't mean that the next one that the that that takes on the core theory, which Buchanan didn't do
Eric Weinstein 1:33:31 - right
Timur Kuran 1:33:32 - Buchanan dealt with the political implications of political markets. And he objected to, to applying the competitive economics model without some modifications to political markets that there were certain inefficiencies that people were overlooking. This was his problem
Eric Weinstein 1:34:01 - Yeah, but I am talking about something much more fundamental
Timur Kuran 1:34:03 - He wasn't challenging the fundamentals. And if you look at the, the the basic economics that is taught at George Mason, it doesn't challenge the core,
Eric Weinstein 1:34:14 - no
Timur Kuran 1:34:14 - ideas of the
Eric Weinstein 1:34:15 - well this is the thing that I want those of us who are trying to upend the core to actually go into open intellectual combat with the stalwarts, who are defending the core, from updating and if the core is so fantastic, they should welcome it, I don't see that happening. Let's switch gears slightly.
Timur Kuran 1:34:33 - Yes,
Eric Weinstein 1:34:34 - you grew up in one of my favorite places on earth. Many people may not know this, I guess I don't know if we mentioned it at the beginning Turkey. And you grew up in a very interesting context that I was learning more about, which is that you happen to be very aligned with the sort of governing ethos of Turkey, which was unlike any other Muslim majority country. The world so far as I could tell, and you came to understand that the preferences of others were being falsified even though your preferences were very much in line with the country, given what we've been seeing with the AK Parti and Erdogan and all the changes in Turkey, can you take us through a little bit of your evolution as a as an observer as to what exactly happened to change turkey so radically so quickly.
Timur Kuran 1:35:30 - So for the listeners, the watchers, perhaps a minute or two on Turkish history is would be useful. Turkey was the center of the Ottoman Empire, which where the law of the land was Islamic law in the 19th century a growing group intellectuals started seeing Islam as the source of the Empire's problems and the Empire was falling apart. And the problem turned into an existential issue as a major components in in Europe were taken away and in World War One, when the Empire's survival was at stake and the danger the Europeans would just colonize what was left of the Empire was becoming more acute by the day. These intellectuals were, many of them were in, in the military. They fought for the Empire and then for Turkey's independence after Turkey was on the losing side in World War One
Eric Weinstein 1:37:20 - very touch and go situation
Timur Kuran 1:37:21 - and the most of the most of what is modern day Turkey was occupied by Western powers divided among them. They fought to gain back these territories and they won and they won the Turkey's war of independence and
Eric Weinstein 1:37:44 - created an unbelievable opportunity that was actually seized.
Timur Kuran 1:37:48 - Exactly. It gave them they it made them heroes and the leading hero was Mustafa Kemal Atatürk, who had fought the British in Gallipoloi, you had to put together a coalition to to defeat the Italians, the Greeks, the British, the French, the Russians. And he was a hero, and he sensed he and the people around him, there were many other heroes around him, sensed that they had a huge amount of political capital, to modernize the country and to do something that was unthinkable until that point
Eric Weinstein 1:38:38 - Can we talk about how crazy these reforms were,
Timur Kuran 1:38:40 - which was, one of them was to abrogate Islamic law and replace Islamic law with secular laws, legal systems borrowed from the Western adapted to Turkish society. Abolish the caliphate and send the Caliph packing and one by one introduced a series of reforms
Eric Weinstein 1:39:17 - change the language
Timur Kuran 1:39:18 - inspired change that will change
Eric Weinstein 1:39:21 - the orthography
Timur Kuran 1:39:22 - change the script which was the Arabic script
Eric Weinstein 1:39:26 - to Latin
Timur Kuran 1:39:27 - and explicitly openly make westernization a goal of the society
Eric Weinstein 1:39:36 - outlaw traditional dress
Timur Kuran 1:39:37 - outlaw traditionally
Eric Weinstein 1:39:39 - polygomy
Timur Kuran 1:39:40 - dress, outlaw polygamy give women the right to vote long before several other countries including Switzerland have given women their right right to vote, rewrite history and of course, the this involved introducting their own myths. Now, we could go on and on
Eric Weinstein 1:40:04 - well just
Timur Kuran 1:40:05 - describing these the the reforms. It was, it was
Eric Weinstein 1:40:12 - unthinkable
Timur Kuran 1:40:13 - unthinkable cultural revolution. And of course, the all the economic institutions are changing at the same time the political institutions are changing. The country's sense of identity replaces a religious identity with a national identity. So nationalism, so people are to call themselves Turks, not Muslims, and being a Turk takes precedent over being being a Muslim. Religious marriages have to be civil, involves civil ceremonies, religious ceremonies have carried no legal weight at all.
Eric Weinstein 1:40:56 - So the reason I'm so animated about this, this is almost like communistic level reforms, but in a in a different idiom
Timur Kuran 1:41:04 - in a different idiom and done by people who had who were genuinely supported by large segments of society now this is not to say that there was no reaction now this is where we come to
Eric Weinstein 1:41:23 - yeah
Timur Kuran 1:41:23 - preference falsification and the bubble that I lived and so on we'll we'll we'll get to this. So, there are of course, people who are illiterate to have no contact with the with the West, who are very religious, they're suddenly being told by their leaders that they don't have a religious identity, the're now Turks what unites everybody is Turkishness not religion that they and the Christian and Jewish minorities are equal not only before the law but also morally. And they're all they're all Turkish, there to accept this. The education is completely secularized. their religion is no longer being taught that's if you learn religion in the family, that's fine. That's your business just don't but the regime is telling you don't make that public. And increasingly, this new regime is radicalizing itself. So this is building now you have a self sustaining, self reinforcing system of secularization, where people are trying to outbid themselves outbid each other in being secular in public
Eric Weinstein 1:42:54 - how, how much toward Western modernity
Timur Kuran 1:42:56 - how much how Western, you can look in your dress.
Eric Weinstein 1:43:00 - Right
Timur Kuran 1:43:00 - How Western you can be in the way you interpret history, how Western you can be in not being Muslim. So people start falsifying their preferences in the direction of being secular. So people who are actually personally religious turn religion into a private matter. They do not fast in public or at least in ways that are noticeable. So during Ramadan, but Islam allows you if you if you miss a day during Ramadan, you can, you can for whatever reason, because you're traveling you can you can substitute for it and it gives you a lot of freedom to do that. So people would, there were people and we find this through memoirs we know about this through memoirs that were published posthumously, because they couldn't express themselves they couldn't say this is happening this is this was happening among top level among some people who were among Atatürk's closest associates who were religious, but who could not have a religious persona.
Eric Weinstein 1:44:15 - So while the West is cheering for turkeys modernization, and lots of this is positive, we start sewing this sort of weird undercurrent where people who are genuinely religious are being repressed.
Timur Kuran 1:44:30 - People who are genuinely genuinely religious are being repressed and people who are appearing religious and public are denied jobs are denied promotion opportunities. This Is Not Happening explicitly. There are no rules that in any government agency or in any major corporation, that if You are religious and if you are using prayer beads you know when you're sitting at the at the meeting and giving people a sense that you're using that that that you're religious, that this is going to hurt you. But it's well understood by everybody that if you want to advance in the society now you have to appear irreligious, this is generating a lot of resentment. And there's also there is there is a void that the nationalist mythology creates that it's not satisfying to people it doesn't emotionally doesn't resonate with some people who want to want some religion. So you have a lot of religious we might call religious preference falsification and eventually turkey becomes after a period of uh secularists we can only call dictatorship or autocracy, maybe benevolent dictatorship eventually becomes a multiparty democracy. And as you would expect in a democracy, politicians aspiring politicians notice the existence of a constituency of a privately religious constituency that would like to be freer in publicizing its religiosity, and would like to avoid discrimination there they're facing.
Eric Weinstein 1:46:40 - So before we get to that one component, I just want to check to see that my understanding is correct as an outsider, is that a weird thing for Westerners to understand is that secularism and supposed modernity is guaranteed not by the democracy but by the army.
Timur Kuran 1:46:59 - Yes, so so army follow this is happening. The army has a special position in Turkish society and it owes that to its enormous victories following World War One and the fact that the practically all the leading modernizers were trained in military schools. So the army is considered the protector of the it's part of the checks and balances of the system. That if the system goes off track, the military has a right to intervene to step in and knock some heads of the politicians and push out the people have caused trouble and restart the system. And this is in fact, so you do start getting political parties with the military in the background you do start getting political parties that start catering to the needs and desires and visions of the pious people, the privately religious, some of them also publicly religious, but some of them publicly, irreligious people, and these parties start advancing and they start gradually altering the discourse and things that were unthinkable to say in during Ataturk's lifetime or the lifetime of the next president, İnönü, starts being said publicly and gradually the support of these parties grow. The military intervenes several times when it sees that the that secularism is being challenged too dangerously from their perspective they intervene for a few years the secularists remain dominant but then
Eric Weinstein 1:49:16 - the horse keeps coming back
Timur Kuran 1:49:17 - the horse keeps coming back and every time it comes back in, it's even stronger. So we get through this this process we come to the Erdogan era, Erdogan forms Erdogan with when the number of other people belongs to a very what, what is even today a very extreme Islamist party that that's where its roots are a party that favors that Islamic common market and and reducing contacts with the West dramatically returned to many old cultural forms, and so on. But are the one sensed that they could never come to power? If they maintain those extreme positions. that yes, they had a core constituency of 10-12%. But they couldn't grow much beyond that. But if they advocated greater religious freedoms, without threatening the secularists and others, that they could actually have a winning majority.
Eric Weinstein 1:50:48 - So do some good and maybe even fool some secularists.
Timur Kuran 1:50:50 - And so he formed a new party, which is the AK parti. AK is the acronym AK means white in Turkish was very clever. A clever acronym clever name for a party. The real name is Adalet ve Kalkınma Partisi, Justice and Development Party and the development was to was to reassure the business elite that they were so committed to development and justice could mean many things to the different groups but to his core constituency and meant we would get religious freedoms. And so when he first came to power, he gave the impression that he was going to expand the freedoms of the of the pious masses.
Eric Weinstein 1:51:57 - Yes.
Timur Kuran 1:51:58 - Without taking away the freedoms of the secularists
Eric Weinstein 1:52:02 - Now.
Timur Kuran 1:52:03 - Yeah, yes.
Eric Weinstein 1:52:04 - At this point I became very mystified because I was watching it from here, and there was this phrase that was invariant in American news, the mildly Islamist stock party. And I kept hearing that and I wanted to get the wax out of my ears. What do you mean mildly Islamist?
Timur Kuran 1:52:22 - so mildly Islamist was it it was never a good choice of terminology.
Eric Weinstein 1:52:31 - Right.
Timur Kuran 1:52:32 - But what they meant was that this was a party that had certain Islamist goals. It pursued those, but without
Eric Weinstein 1:52:43 - in moderation
Timur Kuran 1:52:43 - really in in mod in moderation and without doing damage to the rest of society. And this is precisely what Erdogan did and it was in fact under his watch in his first few years. as prime minister, that Turkey formally applied to join the European Union. And this was something of the party he came from the extreme party this was one of their
Eric Weinstein 1:53:13 - anathema
Timur Kuran 1:53:14 - absolutely anathema to to them. They wanted not only not to join the the common market they wanted to reduce trade with them, they their their party platform said that they would do most of their trade with the with the Arab world and the Muslim world now what exactly they would be buying from the Arab world and where they would get their machinery and this and that this was
Eric Weinstein 1:53:35 - who knows
Timur Kuran 1:53:36 - it, who knows this was one of those things that nobody could be getting back to getting back to, you know, truncated public discourse within that milieu. You never asked this question. You know how this was gonna work out.
Eric Weinstein 1:53:48 - You were as a secular Turk from the western part of the country that's very, very modern. did not see this sort of welling up of preference falsification particularly concentrated in eastern in the Anatolian region.
Timur Kuran 1:54:04 - I didn't I didn't growing up growing up in Istanbul and growing up in a family that been that was part of this westernization movement. my paternal grandfather fought in the Ottoman army and then in the Turkish War of Independence. During that that process while he was taken prisoner by the by the British and spent spent some time as an officer as a British prisoner, came to appreciate the the strengths of Western society he used that time to try to understand why the British war had stronger armies than than the Turks. tried to understand what it is that made them invent weapons that the Turks had not where several centuries before this wasn't the case. And he became became convinced that Atatürk and the people around him who want to westernize turkey make Turkey, anchor Turkey in the West, they were 100% right. after he, After the war of independence, he resigned from the army became a contractor worked for the government for the rest of his life supported the supported Atatürk's party, the people's Republican Party, was to the end of his life, a committed westernizeer as was my father, as we're all my close relatives. I didn't I grew up in a milieu where people didn't falsify their preferences. People were truthful, the people supported the government supported the government supported the direction
Eric Weinstein 1:56:10 - they were
Timur Kuran 1:56:10 - of the country because they approved of this
Eric Weinstein 1:56:13 - and it was a, what they didn't know was that in part, it was a bubble.
Timur Kuran 1:56:18 - But they didn't know was that it was a bubble and what they didn't appreciate, of course, they did appreciate that there were these that the were that there was resistance and there were in during the decades from the 1920s to the 1970s 80s, there had been minor rebellions. In parts of eastern Turkey, it was understood that there were people who objected to the country's direction, but it was also understood that they lived in poor parts of the country. They represented it was the interpretation was They represent the past, as Turkey gets more and more educated, they will fade into the past. The next generation will not will not support them. So this is a transitory problem. So it's not that I didn't understand that there were people who objected to the objective direction of the country, and that when they migrated to Istanbul, they brought some of those ideas with them. There were people in poor communities and Istanbul's in the in the shanty towns, who pretended when they worked for major corporations or worked for the post office of the government. They actually supported the the country's direction, but they actually didn't do it. This much I understood but I but I thought that this was a this was a minor transitory phenomenon. This was not something deeply felt by large numbers of people that could actually change the trajectory of the country. This is something that I missed. And there's a lesson in this, that for if I may, just for a moment, jump back
Eric Weinstein 1:58:23 - to the United States
Timur Kuran 1:58:24 - earlier, jump back to the United States, in the bubbles that we have here in our left bub bubbles on the left and bubbles on the on the right we have people who are talking to each other and just don't realize how many people there are, who don't agree with them and who have very good reasons of their own for thinking differently about certain issues.
Eric Weinstein 1:58:58 - if you take it in the US the Anatolia would be analogized to the middle of the country in sub
Timur Kuran 1:59:05 - flyover states.
Eric Weinstein 1:59:06 - Yeah. Well, I never used that term because I just detest it. But yes,
Timur Kuran 1:59:09 - so
Eric Weinstein 1:59:09 - no, no no
Timur Kuran 1:59:10 - but it. But it is I mean, it means something to to
Eric Weinstein 1:59:13 - to coastal elites and then the
Timur Kuran 1:59:15 - the coastal
Eric Weinstein 1:59:15 - coastal elites is how the the middle of the country demonizes the edges.
Timur Kuran 1:59:19 - Yeah,
Eric Weinstein 1:59:19 - but but more than anything, you know, it's not until you start seeing the headscarves coming out of a BMW, that you realize that your picture is in some sense, not an accurate one that people are quite well to do, that they are coming at this from a cultural perspective that you may not understand. And that
Timur Kuran 1:59:41 - well this is where the word the whole were preference falsification starts starts coming in at various levels, because now the the religious, the genuinely religious people start gaining political power
Eric Weinstein 1:59:56 - right
Timur Kuran 1:59:56 - and of course, with that political power comes government contracts. Comes a reduction in the various regulations that prevented you from getting rich. So, there are a lot of people who are rich, who are culturally conserve who are culturally conservative, that become rich,
Eric Weinstein 2:00:15 - right.
Timur Kuran 2:00:16 - And so then you start seeing they start buying BMWs and they start, start, you know, and you start seeing people wearing head carves in BMWs you driving BMWs you start seeing increasingly elegant headscarves. Whereas Initially, the party that that that built up this this movement, and it promoted a version of Islam that involve modesty
Eric Weinstein 2:00:46 - cloth coat republicans would be an it
Timur Kuran 2:00:48 - yeah, modesty and they wouldn't you know, they wouldn't be flaunting their wealth and so on. Well, we get to a point gradually, where is where those who get rich start spending the money and increasingly expensive cars extremely a more and more expensive headscarves and you get to the point where flash forward to the to the present where you have a president who's living in the largest Presidential Palace in the world 1100 rooms he has something like 15-20 I forget the exact number private private planes flaunts his his luxury all the all the the his the lead members of the government and people close to them all drive cars or have cars driven for them by chauffeurs that are what's what's that?
Eric Weinstein 2:01:49 - Can we discuss this?
Timur Kuran 2:01:50 - Well this is something that in Turkey is difficult to discuss. If you discuss it didn't get you in, in trouble. anything involving that the president's finances are he spends his money or how his consumption is over the top can get you in trouble. There are many journalists who are in jail at the moment for for saying this, but you get this not here we get into another form of preference falsification within the AK parti movement. Now these religious, the the people who wanted to publicize do want wanted to advance religious freedoms, we jumped over one phase which I should come back to now which is that Erdogan as he's as he provides, expands religious freedoms. initially, he doesn't take away any freedoms from the secularists. He doesn't reduce their opportunities to drink if they want to drink. He doesn't try to close down restaurants during Ramadan if you're not religious, and you want to have lunch during Ramadan, fine. That was Erdogan during his first few years. But during this time he is gradually chipping away at the checks and balances of the system. And the thing, ultimately that he needs to get rid of is this the power of the military to essentially remove a government this was something that was in the Constitution.
Eric Weinstein 2:02:34 - Now I'm going to make a parallel here that I wanted to see whether you're going to go or you won't.
Timur Kuran 2:03:43 - Yes,
Eric Weinstein 2:03:43 - in some ways. I view the military in Turkey as having played a role similar to the sense making apparatus in our universities and our newspapers as the guarantee the sort of meta guarantors of a stable democracy and that my serious concern about the United States is that we are headed down a path that we cannot imagine actually ends in literal dictatorship of some as yet unknown form, as we lose the thing that eroded that dictatorial impulse so that what I see is I see our newspapers our universities our political parties, this institutional class that was supposed to be, quite honestly somewhat elite and somewhat above the fray, increasingly become this completely untrustworthy, weakened version and where Erdogan was weakening the military was the guarantor of secularism, which was in the process of overreaching. Our situation is that our sense making apparatus is weakening itself because its economics is starting to crumble.
Timur Kuran 2:04:55 - I think that there are parallels would be when we come back to this and maybe finished the the the Turkish case. So what Erdogan does, I think it's important for readers and watchers to understand this. He disarms the secularists and makes many secularisms, divides the secularists. And people peels off enough of them by making them feel that he will perfect Turkish democracy by getting rid of the role of the military by pushing the military out of politics through a referendum by actually changing the Constitution. And you need that do you need the country vote on a new
Eric Weinstein 2:05:42 - so having a military to guarantee a democracy a secular democracy was always a little bit of a kind of a dirty solution?
Timur Kuran 2:05:48 - It was it was a dirty solution.It was something that didn't any and Erodgan would always say this. This is not being Western. I mean, this was Erdogan being trying to trying to sell his try to trying to remove this check on his power by appearing Western. And he convinced enough secular
Eric Weinstein 2:06:10 - genius,
Timur Kuran 2:06:11 - enough secular people's the referendum passed by, I think 50 and a half to 49 and a half or something got through this and the margin, the 5% margin that he needed came from secularist and I have many friends who voted for him, saying, He is Erdogan we hate to say this, but he is the one bringing true Western democracy. You cannot have a democracy have you ever heard, point show me one European country where the military has the power that it has in Turkey. Yes, the problems with Erdogan we'll deal with that within democracy. And but let's get, this is our opportunity
Eric Weinstein 2:06:59 - This is, in the US context, I find that both Trump and AOC are telling me some of the things that have an inexorable logic that no one will say, and I'm watching my friends peeled off in both directions towards Trump and AOC. And I keep sort of saying, Don't you see what's coming next in both of those situations, but there's something about this kind of appeal to it. It's almost kind of a self hating nature of the secular that or maybe that would be more in the case of AOC. And this is sort of appeal to oh well we'll just let Trump in to do enough mischief to shake things up. And I keep thinking that that these entreaties are clearly going to go to super dangerous places, which I can't convince either side.
Timur Kuran 2:07:52 - Well, the parallel was the parallel here is that Edrogan was taking removing one of the checks and balances in Turkish democracy and preventing it from from going in any direction towards in any ideological direction towards dictatorship,
Eric Weinstein 2:08:17 - right.
Timur Kuran 2:08:18 - He was removing he he removed this without putting in place some other checks and balances
Eric Weinstein 2:08:26 - perfectly said
Timur Kuran 2:08:27 - now so here's the parallel with the United States we have right now two extreme groups that hate each other that consider the other side inhuman and who are willing to suspend all sorts of democratic or all sorts of democratic checks and balances to defeat the other side. Trump is doing this and AOC would like to do this as well. And there are various things that are happening in society that are the equivalent of of that. And they're leading us toward a dictatorship of one kind or another.
Eric Weinstein 2:09:14 - Well, and there are very few people who are willing to say I can see this problem. Both of these are saying things that resonate with me. Both of them are presenting dangers and there's no place to go to say, hey, our problem is our is our extremists in our and our exploitative entrepreneurs who are seeing the turmoil in the country and offering us these solutions. Because what I see is I see bravery and courage on the extremes and cowardice in the middle. And there is no kind of a courageous person moderate perspective that says, what are we talking about giving up all of this great stuff that defined our country so quickly. at the first sign of trouble,
Timur Kuran 2:10:02 - yes. And yes, we don't have and within American politics today, the hope is that within the Democratic Party, there will be some moderate candidate who will say what you have just said and defend, compromising with the other side and defend moderate solutions, admit openly the complexity of various issues and start a conversation on how we prioritize solving these, these problems. What's happening is that all of the candidates are afraid of crossing in the case of the Democratic Party AOC and the people around her and so they are not saying the things that could actually form a counter coalition. And the the party is being driven to an extreme. And the people at the extreme, including AOC, and her her squad, they are think of many of Trumps supporters in the same way that ardent Trump supporters think of AOC
Eric Weinstein 2:11:32 - and there's an interval way in which I agree with both of their verdicts about the other in thst the extremes of trumpism and the extremes of this sort of, you know, Justice Based Thinking that throws out civil society. I have to say that I understand the fear of closed borders of open borders of people just saying such dumb stuff. With no adults anywhere in sight,
Timur Kuran 2:12:03 - and nobody pointing out the implications, laying out all the implications of any of these, whether it's whether it's completely closed borders having no immigration or,
Eric Weinstein 2:12:19 - which would never happen or totally open borders which can't ever hapen
Timur Kuran 2:12:23 - which can which can can never happen. And there are and most americans believe in a policy package somewhere in between
Eric Weinstein 2:12:37 - well,
Timur Kuran 2:12:37 - that involves that involves some immigration
Eric Weinstein 2:12:40 - right.
Timur Kuran 2:12:41 - With restricted with restrictions with certain certain rules. They're not for closed borders or open borders
so you cannot be a xenophilic
Eric Weinstein 2:12:55 - well, so so I've been trying to figure out there's a game that gets played by demographers who are trying to help a candidate get elected, which is can we identify a sector of the economy that nobody's found yet that can be swayed? So soccer moms was an example of one of these sort of Democrat. Demographic discoveries. Another one was the exurb. So you had rural you had suburban but nobody noticed that before. You got to sorry. Before you got to urban from rural there was the exurb between rural and suburban and that had a voting bloc. To me one of the largest voting blocks, which is there for anybody. I talked about this all the time. And it's it's amazing to watch people falsify that it even exists. I call it xenophilic restrictionism. People who are fascinated by other cultures, they've got foreign friends. They're interested in having immigrants as being a vital part of our society, but they're not coked up on this sort of beautiful, nonsensical dream at the base of the Statue of Liberty, which somehow has this mystical old on a immigration expansionism. now of course, immigration expansionism is a weapon for transfer of wealth among Americans. That is, if you can selectively open borders and increase certain groups share the pie George Borjas has showed mechanisms by which you can transfer wealth you claiming to take a tiny little bit of efficiency called Harberger triangle. But what you're really trying to do is transfer a giant amount of wealth, which we might call the Borjas rectangle from American labor to American capital. Now, you can't have that conversation about the misuse of immigration as a tool of transfer, because our media will instantly set upon you and say, well, the only reason you're talking about restricting immigration is your hatred of foreigners and you can't disguise it form me
restrictionist
Timur Kuran 2:14:35 - restrictionist so that that cannot exist by definition it cannot exist
Eric Weinstein 2:14:54 - right of course, because in so this I introduced this thing called the four quadrant model and the idea DIA's is that the media in particular enforces a narrative that all restrictionism 100% essentially, is motivated by fear of foreigners. And then you get to fear of brown people and fear people who are not like us or people with accents. And it is the largest dumbest lie.
Timur Kuran 2:15:22 - That is a huge lie. And even you could minorities talk about brown people and black people. Many of them would be among the people hurt by open borders
Eric Weinstein 2:15:37 - well,
Timur Kuran 2:15:37 - because they would lose, they would lose jobs. You would get cheaper labor from
Eric Weinstein 2:15:43 - Doesn't anybody know any immigrants? Does anybody know any brown people, but the idea that it's the dumbest thing I've ever heard. It's like some white person's crazy idea of what restrictionism is about, it has to do with pushing out labor supply curves. It's it's,
Timur Kuran 2:16:09 - this is
Eric Weinstein 2:16:11 - or diluting the vote,
Timur Kuran 2:16:12 - this should be part of the discussion part of an intelligent discussion that we can have. And reasonable people could can agree can disagree on what the optimal trade off is,
Eric Weinstein 2:16:27 - right
Timur Kuran 2:16:28 - And ultimately, reasonable people who disagree can come to a compromise. You're not going to get 100% of what you're looking for. You're not going to get 100% of what you're going to come somewhere in the middle, we're going to have a national policy. And that's a national policy that can have some dynamism to it every four years. We can talk about it again, we can move the needle a little bit depending on where we really this is the way we can do it. But we have massive massive preference falsification on this simply because people are afraid of being called xenophobes. That's, that's
Eric Weinstein 2:17:03 - you want to know how crazy
Timur Kuran 2:17:04 - and we have massive knowledge falsification which goes along with this. People cannot because you're afraid of being of being put in the wrong box in terms of your preferences of whether you're a xenophile or a xenophobe, you don't. You don't say things that should be obvious to everybody that there are going to be major effects on the labor market that are not going to be distributed evenly. There are going to be some there are going to be perhaps a major owners of big factories are going to gain a lot from the falling wage rates and a lot of people living in the inner cities are going to be hurt by by this, this is something you cannot say because you'll be labeled
Eric Weinstein 2:17:59 - I've already realized something
Timur Kuran 2:18:00 - Yes.
Eric Weinstein 2:18:01 - You want to know how crazy this is? I use the phrase doesn't anybody know any brown people? Doesn't anybody know any foreigners? I'm going to be excoriated for that because I didn't say don't any white people know. It's like, even when I'm speaking glibly,
Timur Kuran 2:18:13 - yes.
Eric Weinstein 2:18:14 - Like the cost of any stupid aspect of phraseology is this ridiculous drumming up by the people who want us not to talk about this which I think is for economic reasons, I think people are, who are in control are terrified that they will come. They will encounter the idea that in general, Americans are pro-immigration and want it at lower levels. We're open to foreigners, we think it's a vibrant part of our society. But we're not stupid. We understand that if you have free health care for all free education for all, you know, nearly limitless opportunity to cross borders. You cannot do all of these things. We don't want our votes diluted. There's no ability to have the conversation. And so a lot of what the portal is about is we've got to break out of this enforced conversation of morons, to some place where we can actually potentially get enough resolution to say, oh, here's what I'm really at about, I don't think we should be blocked to the, you know, the most dynamic people coming from overseas. We need some ability to admit refugees look at the people who've been, you know, at death's door and we've saved it's an important part of revitalizing the country, we have to be able to talk with specificity. And what I see is a media that doesn't have any interest in this long form kind of interaction, simply because it's trying to enforce low resolution speech.
Timur Kuran 2:19:50 - And that low resolution speech involves to put it in concrete terms. If you want restrictions on immigration you're for cages. Well, most Americans are not for caging children either. They they're appalled by that. They would like more orderly forms of restrictions, more humane form of forms of restrictions. But we cannot get to that point. If we cannot have if reasonable people cannot have conversations, which are going to involve some disagreement if they cannot have conversations that are probed by the media, so that the underlying assumptions are identified
Eric Weinstein 2:20:40 - without the gotchas
Timur Kuran 2:20:41 - without the gotchas the underlying assumptions are underlying they are identified. The trade offs are brought out, the knowledge on on which people's preferences are based. Those are scrutinized. There are many myths about what the composition of immigration is. So that we actually are we can we can we can get rid of some of our myths and start talking about these issues on the basis of facts. Some facts,
Eric Weinstein 2:21:21 - so what is it
Timur Kuran 2:21:22 - We cannot we cannot do this, if we can't speak freely
Eric Weinstein 2:21:28 - Well, so, and the thing that I don't understand is the universities. So you're, you're sitting there, Duke, you're part of this archipelago of higher education as a major node on it. What the heck happened that our universities became places where you can't explore ideas as opposed to the citadels in which one can? Or am I wrong about that?
Timur Kuran 2:21:52 - This is this is has been a slow process and I think it has to do with Well, meaning it started with well meaning policies to help integrate groups that had been excluded
Eric Weinstein 2:22:08 - they'd been insular
Eric Weinstein 2:26:59 - Sure.
Timur Kuran 2:27:00 - and that you could be, you could be attacked as racist that shut down conversation. Now this is one thing that I've given you one example. Because it's it's the one that I've that I've studied the struggle in universities over affirmative action, but it has happened in other areas as well. Other groups have used the same strategy to shut down discourse on cultural issues and to to to have universities build all sorts of new units designed to help particular identity constituencies,
Eric Weinstein 2:27:56 - right. But so I'm actually quite quite interested in divided in my own mind about this. What I don't understand is why it is that we can't frame these problems in ways that contain both explanations about human bigger bigotry, unfairness and misogyny, racism, let's have that as a component and then let's have non-oppression based explanations. And let's try to figure out what percentage of things are due to both. And what everyone seems to do is that they either want to exclude one or the other from consideration, so that we can't figure out the mixture now I, I, you know, became a mathematician. I went through Penn, Harvard, MIT in the Hebrew University of Jerusalem. I think it's the case that at the time I was in each of those departments. There was not a single female full professor on the faculty. Now, I have no idea what that is. It's, there's so many fine female mathematicians in the world. And I could, I could, you know, certainly reel off five or 10 that everyone would agree, or first rate mathematicians off the top of my head, but there is a wild imbalance in the field. And I am convinced that there's a component of this that has to do with men have erected mathematics in the way that men are most comfortable with, because there have been so few women in the field. And I'm also reasonably convinced that there's some asymmetry, maybe not an intellectual ability, but certainly in interest in spending one's life negotiating a world mostly of symbols. So I have no idea how to call it but I don't think that either component of that vector in two dimensions which is oppression based explanations and non-oppression based explanations, I don't think either component would be zero.
Timur Kuran 2:30:05 - It's ultimately an empirical issue.
Eric Weinstein 2:30:07 - One would imagine
Timur Kuran 2:30:08 - and the way with with these with as with every empirical issue, we need to collect data, and we need to approach the issues. The way scientists
Eric Weinstein 2:30:20 - but we're not allowed to set up the problem.
Timur Kuran 2:30:22 - We're not allowed to set up the problem. We're not allowed to pose the question. And this is, this is the big, big danger. This is where we become where the situation we find ourselves in is analogous to the situation of the Soviet bloc.
Eric Weinstein 2:30:45 - Yeah,
Timur Kuran 2:30:46 - where you could not ask the question of why. East German Ladas were so so inferior to West German Mercedes and various other West German cars VW s for instance
Eric Weinstein 2:31:05 - right
Eric Weinstein 2:32:42 Timur, I could talk to you forever. So I think what we're going to do is we've been at this for a little while, and with a question that's been much on my mind having to do with, in my case, wanting potentially to retake the White House and for the democrats in an honorable way, which I don't think will happen. I'm not particularly close to the Democratic Party. In fact, it's been driving me crazy, but it is where I grew up. And then I would love to invite you back at anytime you'd like to continue the discussion, but the theory that really has captivated me is how to figure out the appeal of Trump and I have, in part come up with this idea of the checksum theory of politics. Now, checksum has to do with you're receiving a binary, let's say, as a computer program, and you want to know whether it's been corrupted. And so there's some very quick check without having to be able to see the program to know whether or not the program has been has been corrupted on its way to you. The three things that I've settled on which allow me to know that the Democratic Party and its media organs are lying, have to do with a belief that immigration is more or less a pure positive and that anybody who wants it restricted can only do so out of xenophobia, a belief that trade and globalization is a simply positive force that should be expected to lift all boats and the belief that there is zero connection between terror and Islam, no matter how many people cry Allahu Akbar at the end of a killing spree. Now, that is not to say that there's no aspect of white terrorism, as it's not to say that there's no aspect of trade that is positive. Surely it is. And that's not to say that immigration doesn't carry positive benefits. I think we've extolled several of them in the course of our conversation, but it's the simplicity and the violent ferocity with which these things are defended, which have caused large numbers of Americans to say I don't know what this is, but it's like Invasion of the Body Snatchers. No one could possibly believe anything is simplistic, stupid, and as threatening as what you've created and it's driving people in droves to embrace anyone who will say otherwise. Am I wrong?
Timur Kuran 2:34:55 - No, I think there's the there's a lot that makes a tremendous amount of sense. And I want to really say what you said in a different way and explain the reasons that I think Trump came to power. vast numbers of people, including diehard Trump supporters, think that he's not the type of person they'd like to have over for dinner. There's not the like, they're they not theirs he's not the type of person they would like to go into business with. He's not a trustworthy person. He's not a moral person. He's not for the millions of evangelicals who voted for him, not the not somebody who gets close to representing Christian values. But there's one thing that distinguishes Trump among all
Eric Weinstein 2:36:13 - said the Muslim to the Jew
Timur Kuran 2:36:14 - politicians. What's that?
Eric Weinstein 2:36:16 - Said the Muslim to the Jew
Timur Kuran 2:36:18 - there's one thing that that Trump demonstrated that no politician, Democratic or Republican, who came close to being a candidate. It's a characteristic that he had. And that is the the ability to take on the sacred cows of both the Democratic Party and the Republican Party. And it's important and it's important.
Eric Weinstein 2:36:54 - Yeah.
Timur Kuran 2:36:54 - And it's something that he he demonstrated as a Soon as he announced his candidacy, he started insulting various groups of society's. Or some of them are groups that do not have like Muslims, like Hispanics we call de called all of them rapists all 11 million Hispanic immigrants, he said they're all rapists. And
Eric Weinstein 2:37:21 - did he
Timur Kuran 2:37:23 - I thought that was
Eric Weinstein 2:37:24 - well,
Timur Kuran 2:37:24 - early on, early on,
Eric Weinstein 2:37:26 - I worry i don't think that he did he played around with a lot of things that could be parsed one way or the other. But
Timur Kuran 2:37:33 - so
Eric Weinstein 2:37:33 - continue on
Timur Kuran 2:37:33 - anyway, anyway, he said some very awful things about the about immigrants. Maybe I've
Eric Weinstein 2:37:39 - he was playing with fire,
Timur Kuran 2:37:40 - he was playing with fire. He certainly said awful things about Muslims. Now they're voting power energy. Those were the initial groups that he targeted. could say me, well, maybe this is something that a smart politician a populist politician might do. They don't have much voting power. But then he started taking on groups insulting groups and accusing groups of certain groups of doing horrible things, groups that had significant voting power. Some of them were primarily democratic voting groups. So you could say, well, that makes sense because that's going to energize the Republican base. There are people in the Republican Party don't like these other other groups. That makes sense. But then he started insulting and demeaning and humiliating groups in the Republican Party, major groups in the Republican Party, and that included the one that sticks in my mind is the veterans. He insulted John Mccain, who was somebody was an icon not even for Republicans, including Republicans who didn't vote for him and when he ran for president in the primary, but just also somebody highly respected by Democrats, and he accused McCain of being a failure because he had been it gotten arrested and he preferred soldiers who didn't get arrested and so on, this is something that insulted so many, so many veterans. Now, after this happened, his poll numbers went up after he said this, generally, but also among Republicans, and even among veterans, and this was just absolutely stunning to me. And it to me, it said, people are looking for a game changer. And what they're looking for is somebody who can take on the vested interests in Washington, and somebody who is who can be so open in criticizing criticizing groups that are so important to the republican coalition will be fearless against anyone and if there's anyone who's going to shake up the system, it's going to be Trump. And I think that is one source of his, his strength. And I think that going forwards whether he's going to succeed in the next election is going to depend on whether people believe that he is in fact that that attitude has generated something for them, whether he's actually he's actually taken measures against immigrants, that that for for the people who voted for Trump For this reason, because he would shake up the system whether this proves that he will stay on that path, and this is what the country needs. What the country needs more of to move forward.
Eric Weinstein 2:41:10 - You know, just listening to this reminds me that the phrase out of control has two separate meanings. The Democrats see him as out of control in the sense of a destructive force that threatens every everything around around him. The Republicans who who support him and maybe even some Democrats who support him. Or let's say this Trump supporters and Trump detractors, Trump detractors see him as out of control in the sense that he's a danger to everything. Trump supporters see him as outside of control. And therefore, he can weirdly be trusted because clearly nothing is holding him back. He's not he has no paymaster somewhere because nobody could act like this if they were part of the institutional makeup of the country and I wonder if that's really what divides us.
Timur Kuran 2:42:05 - And this is I think what is dividing us strike now and the people who feel that he's just destroying so many things that are valuable to them are willing to intensely hate him. And that hatred is now driving them toward politicians who are willing to suspend various civil liberties that are central to the American system or have been central to the American system, because getting rid of Trump is more important than anything else. And and Trump insofar as Trump is is not that the the, the sort of trumpism will not be gone after Trump is no longer president insofar as these People who hate the establishment and hate the various vested interests insofar as they're there, they're going to continue to pose a problem politically, they're going to continue to be a political force somehow. And the Trump well the group that you label the Trump detractors, we might call them the Trump Trump haters. Many of them would like to suspend various liberties, various checks and balances to get rid of this clear and present danger. That is one way we can get to a dictatorship. Another way is, of course, allowing Trump to pursue some of his agenda. That's another way to
Eric Weinstein 2:42:27 - twin paths to dictatorship.
Timur Kuran 2:43:58 And again, we get back to To this issue of the tremendous need that the society has for the people who are falsifying preferences in one way or another, who see the complexity of the issues, to come out of the closet and to find a leader of their own, who is going to have the charisma.
Eric Weinstein 2:44:25 - Yeah,
Timur Kuran 2:44:25 - that is going to out Trump Trump and out a AOC AOC this what we’re lacking
Eric Weinstein 2:44:33 - Well, maybe maybe we find such a person inshallah,
Timur Kuran 2:44:38 - I hope so inshallah.
Eric Weinstein 2:44:40 - Okay, well, you've been through the portal with Dr. Timur Kuran of Duke University. Thanks for listening or watching and we'll see you next time.
|
2023-03-31 03:25:11
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2906053960323334, "perplexity": 1588.4013817718794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949533.16/warc/CC-MAIN-20230331020535-20230331050535-00168.warc.gz"}
|
https://sites.math.rutgers.edu/~weibel/oldalgebra.sem.html
|
## PREVIOUS RUTGERS ALGEBRA SEMINARS - (Since 1995)
A '(C)' marks a related Colloquium Talk at 4:30 PM.
The asterisks (*) mark meetings of the QUANTUM MATH SEMINAR, which has occasionally replaced the algebra seminar, during Spring 1998-Spring 2008.
The sharp (#) marks a meeting of the VIGRE seminar, which occasionally replaced the algebra seminar during 2000-2002.
Spring 2017 Seminars (Wednesdays at 2:00 in H705)
22 Feb Ryan Shifler Virginia Tech "Equivariant Quantum Cohomology of the Odd Symplectic Grassmannian"
1 Mar Chuck Weibel Rutgers "The Witt group of surfaces and 3-folds"
8 Mar Oliver Pechenik Rutgers "Decompositions of Grothendieck polynomials"
15 Mar no seminar ------------------- Spring Break ----------
22 Mar Ilya Kapovich UIUC/Hunter College "Dynamics and polynomial invariants for free-by-cyclic groups"
29 Mar Rachel Levanger Rutgers "Interleaved persistence modules and applications of persistent homology to problems in fluid dynamics"
5 Apr Cristian Lenart Albany-SUNY "Kirillov-Reshetikhin modules and Macdonald polynomials: a survey and applications"
19 Apr Anders Buch Rutgers "Puzzles in quantum Schubert calculus"
26 Apr Sjuvon Chung Rutgers "Equivariant quantum K-theory of projective space"
Classes end May 1; Final Exams are May 4-10, 2017
Fall 2016 Seminars (Wednesdays at 2:00 in H423)
21 Sept Fei Qi Rutgers "What is a meromorphic open string vertex algebra?"
28 Sept Zhuohui Zhang Rutgers "Quaternionic Discrete Series"
5 Oct Sjuvon Chung Rutgers "Euler characteristics in cominuscule quantum K-theory"
12 Oct Ed Karasiewicz Rutgers "Elliptic Curves and Modular Forms"
19 Oct Natalie Hobson U.Georgia "Quantum Kostka and the rank one problem for sl2m"
26 Oct Oliver Pechenik Rutgers "K-theoretic Schubert calculus"
2 Nov Vasily Dolgushev Temple U "The Intricate Maze of Graph Complexes"
9 Nov Jason McCullough Rider U. "Rees-like Algebras and the Eisenbud-Goto Conjecture"
16 Nov Robert Laugwitz Rutgers "Representations of p-DG 2-categories"
23 Nov --- no seminar --- Thanksgiving is Nov. 24; Friday class schedule
30 Nov Semeon Artamonov Rutgers "Double Gerstenhaber algebras of noncommutative poly-vector fields"
7 Dec Daniel Krashen U.Georgia "Geometry and the arithmetic of algebraic structures" (Special talk)
14 Dec Angela Gibney U.Georgia "Vector bundles of conformal blocks on the moduli space of curves" (Special talk)
Classes end December 14; Final Exams are December 16-23, 2016
Spring 2016 Seminars (Wednesdays at 2:00 in H705)
20 Jan Louis Rowen Bar-Ilan Univ "Symmetrization in tropical algebra"
3 Feb Volodia Retakh Rutgers "Generalized adjoint actions"
10 Feb Omer Bobrowski Duke (@noon!) "Random Topology and its Applications"
17 Feb Lisa Carbone Rutgers "Arithmetic constructions of hyperbolic Kac-Moody groups"
2 Mar Chuck Weibel Rutgers "Relative Cartier divisors"
9 Mar Lev Borisov Rutgers "Elliptic genera of singular varieties and related topics"
16 Mar no seminar ------------------- Spring Break ----------
23 Mar Rachel Levanger Rutgers "Auslander-Reiten quivers of finite-dimensional algebras"
30 Mar Richard Lyons Rutgers "Aspects of the Classification of simple groups"
6 Apr Richard Lyons Rutgers "Aspects of the Classification, continued"
13 Apr Siddhartha Sahi Rutgers "Eigenvalues of generalized Capelli operators"
20 Apr Ed Karasiewicz Rutgers "Some Aspects of p-adic Representations & the Casselman-Shalika Formula"
27 Apr Semeon Artamonov Rutgers "Noncommutative Poisson Geometry"
Classes end May 2; Final Exams are May 4-10
Fall 2015 Seminars (Wednesdays at 2:00 in H425)
7 Oct Chuck Weibel Rutgers "Monoids, monoid rings and monoid schemes"
14 Oct Lev Borisov Rutgers "Introduction to A-D-E singularities"
21 Oct Dylan Allegretti Yale "Quantization of Fock and Goncharov's canonical basis"
28 Oct Volodia Retakh Rutgers "Noncommutative Cross Ratios"
4 Nov Gabriele Nebe U.Aachen "Automorphisms of extremal codes"
11 Nov Chuck Weibel Rutgers "Relative Cartier divisors and polynomials"
18 Nov Glen Wilson Rutgers "Motivic stable homotopy over finite fields"
25 Nov --- no seminar --- Thanksgiving is Nov. 26; Friday class schedule
2 Dec Anders Buch Rutgers "The Thom Porteous formula"
9 Dec Pham Huu Tiep U. Arizona "Representations of finite groups and applications "
Classes end Dec. 10; Final Exams are December 15-22
Spring 2015 Seminars (Wednesdays at 2:00 in H124)
27 Jan ---
4 Feb Jesse Wolfson Chicago "The Index Map and Reciprocity Laws for Contou-Carrère Symbols"
18 Feb Justin Lynd Rutgers "Fusion systems and centric linking systems"
25 Feb Lev Borisov Rutgers "Zero divisors in the Grothendieck ring of varieties"
4 Mar Volodia Retakh Rutgers "Noncommutative triangulations and the Laurent phenomenon"
6 MarC Burt Totaro UCLA/IAS "Birational geometry and algebraic cycles" (Colloquium)
11 Mar Anders Buch Rutgers "TK"
18 Mar no seminar ------------------- Spring Break ------------------
22 Apr Howard Neuer Rutgers "On special cubic 4-folds"
Classes end May 4; Spring Final Exams are May 7-13
Fall 2014 Seminars (Wednesdays, 3:15-4:15PM in H525)
17 Sep Edwin Beggs U.Swansea "Semiclassical approximation to noncommutative Riemannian geometry"
24 Sep Anders Buch Rutgers "Equivariant quantum cohomology and puzzles"
8 Oct Lev Borisov Rutgers "Cut and paste approaches to rationality of cubic fourfolds"
15 Oct Chuck Weibel Rutgers "The Witt group of real varieties"
22 Oct Ed Karasiewicz Rutgers "Jacobians of modular curves"
29 Oct Charlie Siegel (IPMU Japan) "A Modular Operad of Embedded Curves"
5 Nov no seminar
12 Nov Marvin Tretkoff Texas A&M "Some non-compact Riemann surfaces branched over three points"
19 Nov Ashley Rall U. Virginia "Property T for Kac-Moody groups"
26 Nov (Thanksgiving is Nov. 27) no seminar
3 Dec Alex Lubotzky NYU/Hebrew U. (Israel) "Sieve methods in group theory"
Spring 2014 Seminars (Wednesdays at 2:00 in H124)
Apr 26 Anatoly Vershik, St. Petersburg State University, Russia "Invariant measures and standardness"
Mar 5 Greg Muller, Michigan "Locally acyclic cluster algebras"
Mar 12 Julianne Rainbolt, Saint Louis University "Bruhat cells which contain only regular elements"
Mar 26 Bianca Viray, Brown U "Unramified Brauer classes on cyclic covers of the projective plane"
Apr 9 Lev Borisov, Rutgers "An annoying problem in toric geometry"
Apr 23 Howard Nuer, Rutgers "An introduction to cubic fourfolds and their moduli space"
Apr 30 Vijay Ravikumar, Tata Institute "Equivariant Pieri rules for Isotropic Grassmannians"
Fall 2013 Seminars (Wednesdays at 2:00 in H525)
Details for Fall 2013 seminars are located at THIS SITE
4 Sep Delaram Kahrobaei CUNY "Applications of Algebra in Information Security"
2 Oct Bob Guralnick USC and IAS "Dimensions of Fixed Spaces"
9 Oct Leonid Petrov Northeastern "Robinson-Schensted-Knuth correspondences and their $(q,t)$-deformations"
16 Oct Knight Fu Rutgers "Torsion Theory and Slice Filtration of Homotopy Invariant Sheaves With Transfers"
23 Oct Ralph Kaufmann Purdue/IAS "Three Hopf algebras and their common algebraic and categorical background"
30 Oct Howard Nuer Rutgers "Bridgeland Stability and Moduli on Enriques Surfaces"
6 Nov Andrew Blumberg U.Texas "Probabilistic inference in topological data analysis"
13 Nov Pierre Cartier IHES "Galois groups of differential equations: a noncommutative analog"
20 Nov Zsolt Patakfalvi Princeton "Classification of algebraic varieties: classical results and recent advances in positive characteristic"
Spring 2013 Seminars (Wednesdays at 2:00 in H525)
24 Jan Daniel Erman Michigan "Equations, syzygies, and vector bundles"
30 Jan David Anderson U. Paris "Equivariant Schubert calculus: positivity, formulas, applications"
6 Feb Chuck Weibel Rutgers "What is a Derivator?"
13 Feb V. Retakh Rutgers "A geometric approach to noncommutative Laurent phenomenon"
20 Feb Tatiana Bandman Bar-Ilan "Dynamics and surjectivity of some word maps on SL(2,q)"
27 Feb Bob Guralnick USC and IAS "Strongly Dense Subgroups of Algebraic Groups"
13 Mar Mina Teicher Bar-Ilan "The 3 main problems in the braid group"
20 Mar no seminar -------------- Spring Break -------------
3 Apr Joe Ross USC "Intersection theory on singular varieties"
10 Apr Lev Borisov Rutgers "Hilbert modular threefolds of discriminant 49"
17 Apr Charlie Siegel (IPMU Japan) "Cyclic Covers, Prym Varieties and the Schottky-Jung Relations"
24 Apr Freya Pritchard CUNY "Implicit systems of differential equations"
1 May Alexei Stepanov (St.Petersburg State University) "Structure of Chevalley groups over rings"
Fall 2012 Seminars (Wednesdays at 2:00 in H525)
19 Sept Chuck Weibel Rutgers "Binary codes and Galois covers of varieties"
10 Oct Anders Buch Rutgers "Curve neighborhoods"
17 Oct Dan Grayson IAS "Computations in intersection theory"
24 Oct Justin Lynd Rutgers "Fusion systems with prescribed involution centralizers"
31 Oct Lev Borisov Rutgers "On Hilbert modular threefolds of discriminant 49"
7 Nov Oliver Rondigs Osnabruck, Germany "On the slice filtration for hermitian K-theory"
14 Nov Howie Nuer Rutgers "surfaces on Calabi-Yao 3-folds"
21 Nov no seminar, Friday classes (Thanksgiving week)
28 Nov Susan Durst Rutgers "Universal labelling algebras for directed graphs"
5 Dec Anastasia Stavrova U.Essen "Injectivity property of etale H^1, non-stable K_1, and other functors"
12 Dec Joe Ross USC "Presheaves with oriented weak transfers"
Fall 2012 Semester starts Sept.4; (Wednesday Nov. 21 will be Friday classes).
Classes end Wed., Dec. 12; Final Exams start Friday 12/14/11
Spring 2012 Seminars (Wednesdays at 2:00 in H525)
25 Jan Vasily Dolgashev Temple Univ. "Exhausting quantization procedures"
8 Feb Chuck Weibel Rutgers "Shift Equivalence and Z[t]-modules"
15 Feb Pablo Pelaez Rutgers "An introduction to weights"
22 Feb Anders Buch Rutgers "K-theory of miniscule varieties"
29 Feb Julia Plavnik U.Cordoba "From algebra to category theory: a first approach to fusion categories"
7 Mar Anastasia Stavrova U.Essen "On the unstable K_1-functors associated to simple algebraic groups"
14 Mar no seminar -------------- Spring Break -------------
21 Mar Mark Walker U. Nebraska "Invariants of Matrix Factorizations"
28 Mar Lev Borisov Rutgers "Combinatorial aspects of toric mirror symmetry"
5 Apr Joe Ross USC "Cohomology Theories with Supports" Thursday 11:00AM in Hill 425
11 Apr V. Retakh Rutgers "Noncommutative Laurent Phenomena"
18 Apr Ben Wyser U.Georgia "Symmetric subgroup orbit closures on flag varieties as universal degeneracy loci"
25 Apr Ling Bao Chalmers U. (Sweden) "Algebraic symmetries in supergravity"
Spring 2012 Semester starts Jan. 17, Classes end April 30,
Spring Break is March 11-18, Exams start May 3.
Fall 2011 Seminars (Wednesdays at 2:00PM in H423)
14 Sept Charles Siegel U. Penn. "The Schottky Problem and genus 5 curves"
28 Sept Abid Ali Rutgers "Congruence subgrous of lattices in rank 2 Kac-Moody groups over finite fields"
5 Oct Raika Dehy Cergy-Pontoise "Cluster algebras and categorification"
12 Oct Chuck Weibel Rutgers "What (besides varieties) are motivic spaces?"
19 Oct Raika Dehy Cergy-Pontoise "Cluster algebras and categorification (bis)"
26 Oct Anders Buch Rutgers "Giambelli formulas for orthogonal Grassmannians"
2 Nov Alice Rizzardo Columbia "On Fourier-Mukai type functors"
9 Nov Changlong Zhong Ottowa "Comparison of Dualizing Complexes"
16 Nov Anastasia Stavrova U.Essen "The Serre-Grothendieck conjecture on torsors and the
classification of simple algebraic groups"
23 Nov no seminar, no classes (Thanksgiving week)
30 Nov Lev Borisov Rutgers "Elliptic functions and equations of modular curves"
7 Dec Pablo Pelaez Rutgers "Homotopical Methods in Algebraic Geometry"
Fall 2011 Semester starts Sept.1; (Thursday Sept.8 will be Monday classes).
Classes end Tues, Dec. 13; Final Exams start Friday 12/16/11
Spring 2011 Seminars
(Wednesdays at 2 PM in CoRE 431)
21 Jan Chenyang Xu Princeton Colloquium talk (Friday)
26 Jan Grigor Sargsyan UCLA TBA (Monday Jan. 24)
28 Jan Ivan Losev MIT Colloquium talk (Friday)
4 Feb A. Salehi Golsefidy Princeton Colloquium talk (Friday)
9 Feb Louis Rowen Bar Ilan U. "Tropical Algebra"
16 Feb no seminar
23 Feb Christian Haesemeyer UCLA "Rational points, zero cycles of degree one, and A^1-homotopy theory"
2 Mar Volodia Retakh Rutgers "Linear recursive sequences, Laurent phenomenon and Dynkin diagrams"
9 Mar Chuck Weibel Rutgers "Monoid algebras and monoid schemes"
16 Mar no seminar -------------- Spring Break -------------
30 Mar Volodia Retakh Rutgers "Hilbert series of algebras associated to direct graphs and order homology"
6 Apr Lev Borisov Rutgers "Syzygies of binomial ideals and toric Eisenbud-Goto conjecture"
13 Apr Crichton Ogle Ohio State "Cyclic homology, simplicial rapid decay algebras, and applications to K*t(l¹(G))"
20 Apr Susan Durst Rutgers "Twisted Polynomial Rings and Embeddings of the Free Algebra"
27 Apr Chuck Weibel Rutgers "Derived categories of graded modules"
4 May
Spring Finals are May 5-11; last day of classes is May 2 (Monday)
Fall 2010 Seminars (Mondays at 4:30PM in H705)
20 Sept Uma Iyer Bronx Community College "Quantum differential operators" (4:50 PM)
27 Sept Chuck Weibel Rutgers "Monoids and algebraic geometry" (4:50 PM)
4 Oct Bob Guralnick USC "Dimensions of fixed point spaces of elements in linear groups" (4:50 PM)
11 Oct Volodia Retakh Rutgers "Hilbert series of algebras associated to directed graphs and order homology" (4:50 PM)
18 Oct Lev Borisov Rutgers "The Pfaffian-Grassmannian derived equivalence" (4:30 PM)
1 Nov Chuck Weibel Rutgers "etale cohomology operations" (4:30 PM)
8 Nov Anders Buch Rutgers "Pieri rules for the K-theory of cominuscule Grassmannians" (4:30 PM)
15 Nov Volodia Retakh Rutgers "A short proof of the Kontsevich cluster conjecture" (4:30 PM)
22 Nov no seminar (Wednesday class schedule, Thanksgiving week)
29 Nov Earl Taft Rutgers "The Lie product in the continuous Lie dual of the Witt algebra" (4:30 PM)
6 Dec Chuck Weibel Rutgers "Motivic cohomology operations" (4:30 PM)
13 Dec Ralph Kaufmann Purdue&IAS "Algebraic Structures from Operads" (4:30 PM)
Fall Finals are Dec. 16-23; last day of classes is Dec 13 (Monday)
Spring 2010 Seminars (Mondays at 4:50 in H705)
1 Feb Max Karoubi Univ. Paris 7 "Periodicity in Hermitian K-groups"
15 Feb Chuck Weibel Rutgers Exceptional objects (after Polishchuk)
22 Feb
1 Mar Ray Hoobler CCNY "Applications of stable bundles to Witt groups and Brauer groups"
8 Mar Christian Kassel CNRS & U.Strasbourg "Drinfeld twists and finite groups"
15 Mar no seminar -------------- Spring Break -------------
22 Mar Earl Taft Rutgers "Hopf algebras and recursive sequences"
29 Mar Chuck Weibel Rutgers "Tilting 1"
5 Apr Carlo Mazza U. Genoa "The K-theory of motives"
12 Apr Miodrag Iovanov USC "Generalized Frobenius algebras, Integrals and applications to Hopf algebras and compact groups"
19 Apr Chuck Weibel Rutgers "Tilting 2"
26 Apr Robert Wilson Rutgers "Tilting 3 "
3 May William Keigher Rutgers-Newark "Module Structures on the Ring of Hurwitz Series"
Spring Break is March 13-21, 2010; Final Exams begin Thursday May 6.
Fall 2009 Seminars (joint with Gelfand Seminar)
28 Sep no seminar Yom Kippur
5 Oct Lourdes Juan Texas Tech Differential Central Simple Algebras and Picard-Vessiot representations
12 Oct Bob Guralnick USC Derangements in Finite and Algebraic Groups
19 Oct Ken Johnson Penn State-Abington Mathematics arising from a new look at the
Dedekind-Frobenius group matrix and group determinant
2 Nov Chloe Perin Hebrew Univ. "Induced definable structure on cyclic subgroups of the free group"
9 Nov Paul Ellis U. Connecticut "The classfication problem for finite rank dimension groups"
16 Nov Ravi Srinivasan RU-Newark "Picard-Vessiot Theory"
23 Nov Vladimir Retakh Rutgers "Noncommutative algebra and combinatorial topology"
30 Nov Chuck Weibel Rutgers "homotopy model structures as tools for homogical algebra"
7 Dec no seminar cancelled due to Gelfand Memorial
Fall 2009 Semester begins Tuesday Sept 1; Labor Day is Sept. 7
Final Exams begin Wednesday Dec 16, 2009; Math Group Exams are Dec. 16 (4-7PM).
Spring 2009 Algebra/Gelfand Seminar
2 Feb: Chuck Weibel Rutgers "Stability conditions for triangulated categories"
9 Feb: Luis Caffarelli U. Texas Special Colloquium talk at this time
16 Feb: Vladimir Retakh Rutgers "Lie algebras over noncommutative rings"
23 Feb: Leon Pritchard CUNY "Partitioned differential quasifields"
2 Mar: Jan Manschot Rutgers-Physics "Stability conditions in physics"
16 Mar no seminar -------------- Spring Break -------------
30 Mar: Elizabeth Gasparim Edinburgh "The Nekrasov Conjecture for Toric Surfaces"
6 Apr: Vladimir Retakh Rutgers "Noncommutative Laurent phenomenon"
13 Apr: Bill Keigher Rutgers-Newark "Differential quasifields"
20 Apr: Chris Woodward Rutgers "Morphisms of cohomological field theories and behavior of Gromov-Witten invariants under quotients"
27 Apr: Gregory Ginot Univ.Paris "higher order Hochschild (co)homology"
Spring Break is March 14-22, 2009; Final Exams begin Thursday May 7.
### Fall 2008 Algebra/Gelfand Seminar (at 4:00 Mondays)
5 Sep:# Paul Baum Penn State "Morita Equivalence Revisited" Talk is Friday at 2PM in H705
15 Sep: no seminar MSMF Reception
18 Sep: Vasily Dolgushev UC Riverside "Formality theorems for Hochschild (co)chains and their applications" Talk at 2PM in H425
22 Sep: Mike Zieve Rutgers "Rationality and integrality in dynamical systems"
29 Sep no seminar Rosh Hoshanna
6 Oct: Chuck Weibel Rutgers "The de Rham-Witt complex of R[t]"
13 Oct: Anders Buch Rutgers "Quantum K-theory"
20 Oct: Earl Taft Rutgers "Combinatorial Identities and Hopf Algebras"
27 Oct: Siddhartha Sahi Rutgers "Interpolation and binomial identities in several variables"
3 Nov: Leigh Cobbs Rutgers "Infinite towers of co-compact lattices in Kac-Moody groups"
10 Nov: Jarden Logic Seminar "The absolute Galois group of subfields of the field of totally S-adic numbers"
14 Nov: Guillermo Cortiñas Buenos Aires "K-theory of some algebras associated to quivers" Talk is Friday at 2PM in H425
17 Nov no seminar ------- ------------------------------
24 Nov: Robert Wilson Rutgers "Splitting Algebras associated to cell complexes"
1 Dec: Roozbeh Hazrat Queens Univ. Belfast "Reduced K-theory of Azumaya algebras"
9 Dec: Steven Duplij Kharkov Univ. "Quantum Enveloping Algebras and the Pierce Decomposition " Talk is Tuesday, 2PM in H425
Fall 2008 Semester begins Tuesday Sept 2; Final Exams begin Monday Dec 15, 2008.
### Spring 2008 Algebra/Gelfand Seminar
The Algebra Seminar was merged with the Gelfand Seminar, and met Mondays at 4:40 PM.
25 Jan(F) W. Vasconcelos Rutgers The Chern numbers of a local ring (I)
28 Jan: Vladimir Retakh Rutgers "Obstructions to formality and obstructions to deformations"
4 Feb: Chuck Weibel Rutgers "Generation of Galois cohomology by symbols"
5 Feb(T)* Tony Milas SUNY Albany "W-algebras, quantum groups and combinatorial identities"
8 Feb(F) M. Zieve Rutgers "The lattice of subfields of K(x)
11 Feb: Zin Arai Kyoto Univ "Complex dynamics and shift automorphism groups"
18 Feb: Andrzej Zuk Univ Paris "Automata Groups"
25 Feb: Mike Zieve Rutgers "Automorphism groups of curves"
29 Feb(F) Laura Ghezzi CUNY "Generalizations of the Strong Castelnuovo Lemma"
3 Mar: Chuck Weibel Rutgers "Model categories versus derived categories"
10 Mar: R Parimala Emory Univ. "Rational points on homogeneous spaces"
14 Mar#* Tom Robinson Rutgers "Formal differential representations" 11:55 AM Friday in Hill 425
17 Mar: no seminar -------------- Spring Break -------------
28 Mar#* David Ben-Zvi IAS & U.Texas "Real Groups and Topological Field Theory"
28 Mar(F) Jooyoun Hong Purdue "Homology and Elimination"
31 Mar: Siddhartha Sahi Rutgers "Tensor categories and equivariant cohomology"
4 Apr(C) David Saltman CCR and U.Texas "Division Algebras over Surfaces"
7 Apr: Earl Taft Rutgers "The boson-fermion correspondence and one-sided quantum groups
14 Apr: Colleen Duffy Rutgers "Graded traces and irreducible representations of graph algebras"
21 Apr: Semyon Alesker Tel-Aviv U. "Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets"
28 Apr: Jim Borger Australia Natl Univ. "Witt vectors, Lambda-rings, and absolute algebraic geometry"
5 May: Richard Lyons Rutgers "Subgroups of Algebraic Groups and Finite Groups"
Spring 2008 Semester begins Tuesday Jan 22; Spring Finals are May 8-14, 2008
### Fall 2007 Algebra Seminar
During Fall 2005-Fall 2007, the seminar was rescheduled to 1:00-2:00PM Fridays in H705, as a result of new class times.
7 Sep* Benjamin Doyon Durham Conformal field theory and Schramm-Loewner evolution
14 Sep* Liang Kong Max Planck An introduction to open-closed conformal field theory
28 Sep Richard Lyons Rutgers Presidential Address and Department Reception
5 Oct Diane Maclagan Rutgers-Warwick Starts at 2:15! "Equations for Chow and Hilbert quotients"
12 Oct Rafael Villareal IPN,Mexico "Unmixed clutters with a perfect matching"
19 Oct POSTPONED to November 16
2 Nov# Andrea Miller Harvard POSTPONED
9 Nov Dan Krashen U. Penn Starts at 2:20!Patching subfields of division algebras
16 Nov Angela Gibney U. Penn A new candidate for the nef cone of M0,n
23 Nov Tom Turkey Plymouth Colony ---------Thanksgiving Break-----------
3 Dec: Dirk Kreimer IHES (France) Monday at 4:40! Hopf and Lie algebras for renormalizable quantum field theories
7 Dec V. Retakh Rutgers date(s) to change TK
Fall Classes began September 4, 2007; Final Exams began Friday, Dec 14, 2007.
### Fall 2006/Spring 2007 Seminars
22 Sep: Corina Calinescu OSU Intertwining vertex operators and combinatorial representation theory
8 Dec* Haisheng Li RU-Camden Certain generalizations of twisted affine Lie algebras and vertex algebras
30 Mar* Bill Cook Rutgers Vertex operator algebras and recurrence relations
6 Apr* Antun Milas SUNY-Albany On a certain family of W-algebras
13 Apr* Vincent Graziano SUNY-Stony Brook G-equvariant modular categories and Verlinde formula
20 Apr* Corina Calinescu OSU Vertex-algebraic structure of certain modules for affine Lie algebras underlying recursions
27 Apr* Tom Robinson Rutgers A Formal Variable Approach to Special Hyperbinomial Sequences
Fall Classes began September 5, 2006; Final Exams began Friday, Dec 15
Spring 2007 Semester began Tuesday Jan 16; Spring Finals were May 3-9, 2007
### Spring 2006 Seminars
20 Jan*: John Duncan Yale Vertex operators and sporadic groups
27 Jan(C) Jason Starr MIT Solutions of families of polynomial equations Colloquium at 4:00
3 Feb no seminar (job interview talks)
10 Feb: Balazs Szegedy IAS Congruence subgroup growth of arithmetic groups in positive characteristic
17 Feb*: Haisheng Li RU-Camden A smash product construction of nonlocal vertex algebras
24 Feb*: Andy Linshaw Brandeis Chiral equivariant cohomology
3 Mar: Wolmer Vasconcelos Rutgers "Complexity of the Normalization of Algebras"
10 Mar: Volodia Retakh Rutgers "Algebras associated to directed graphs and related to factorizations of noncommutative polynomials"
17 Mar: no seminar -------------- Spring Break -------------
24 Mar no seminar ---- D'Atri Lectures
31 Mar: Chuck Weibel Rutgers "Projective R[t]-modules and cdh cohomology"
7 Apr no seminars in April
5 May Student Body Left Rutgers ---- Final Exam Grading Marathon -------
Classes begin January 18, 2006; Regular classes end Monday May 1. Final Exams are May 4-10, 2006.
### Fall 2005 Seminars
During 2005-6, the Algebra Seminar (and Quantum Math Seminar) met on Fridays, at 1:00-2:00PM in H705.
9 Sept Colonel Henry Rutgers -------- Department Reception ----------------
16 Sept: Thuy Pham Rutgers "jdeg of finitely generated graded algebras and modules"
note room change to Hill 425 due to Kruskal Conference
23 Sept: Charles Weibel Rutgers "Effective Hodge structures"
30 Sept* Corina Calinescu Rutgers "On certain principal subspaces of standard modules and vertex operator algebras"
7 Oct: Art DuPre Rutgers-Newark "Extensions of Rings and their Endomorphisms"
14 Oct* Katrina Barron Notre Dame "An isomorphism between two constructions of permutation-twisted modules for lattice vertex operator algebras"
21 Oct* Lin Zhang RU+Sequent-Capital "Kazhdan-Lusztig's tensor category and the compatibility condition"
28 Oct: Bob Guralnick USC & IAS "Rational Maps on the Generic Riemann Surface"
4 Nov: Gene Abrams U.Colorado/Colo.Springs "Leavitt path algebras"
11 Nov* Siddhartha Sahi Rutgers "Supercategories and connections"
18 Nov: no seminar
25 Nov: Tom Turkey Plymouth Colony ---------Thanksgiving Break------------
2 Dec: Earl Taft Rutgers "A class of left quantum groups: Variation on the theme of SL_q(n)"
9 Dec: Harry Tamvakis Brandeis "Quantum cohomology of isotropic Grassmannians" (talk is at 12:30 in H423)
16 Dec*: Hisham Sati U. Adelaide "Mathematical aspects of the partition functions in string theory"
Semester begins Thursday September 1, 2005.
Regular classes end Tuesday, December 13. Final Exams are Dec.16-23.
Math Group Exam time is Friday Dec.16 (4-7PM)
From 1980 until Spring 2005, the seminar met on Fridays, at 2:50-4PM in H705 (Hill Center, Busch Campus).
### Spring 2005 Seminars
21 Jan: (first Friday of semester)
28 Jan: no seminar Job Interview Talks>
4 Feb: Tom Graber UC Berkeley "Generalizations of Tsen's Theorem" (talk at 4:30 PM)
11 Feb: Pedro Barquero-Salavert CUNY Grad Center "Applications of the transfer method to quadratic forms and sheaves"
18 Feb: Christian Haesemeyer IAS "K-theory and cyclic homology of singularities"
25 Feb: Li Guo RU-Newark "Birkhoff decomposition in QFT and CBH formula"
4 Mar: Earl Taft Rutgers "Exotic Products of Linear Maps on Bialgebras"
11 Mar: Carlo Mazza IAS "Schur Functors and Nilpotence Theorems"
18 Mar: no seminar -------------- Spring Break -------------
25 Mar: Zhaohu Nie IAS/Stony Brook "Karoubi's construction of Motivic Cohomology Operations"
1 Apr: Gerhard Michler U.Essen/Cornell "Uniqueness proof for Thompson's sporadic simple group"
8 Apr: Bin Shu U.Virginia/E.Normal U. "Representations and Forms of Classical Lie algebras over finite fields"
15 Apr: K. Ebrahimi-Fard Univ.Bonn "Infinitesimal bialgebras and associative classical Yang-Baxter equations"
21 Apr: Bruno Vallette U.Nice "Koszul duality" (Thursday at 1:10 p.m.)
22 Apr: Kate Hurley
29 Apr: Cristiano Husu U.Conn(Stamford) "Relative twisted vertex operators associated with the roots of the Lie algebras A_{1} and A_{2}"
6 May: Student Body Left Rutgers ---- Final Exam Grading Marathon -------
7 June: Miguel Ferrero UF Rio Grande do Sol, Brazil "PARTIAL ACTIONS OF GROUPS ON ALGEBRAS" (talk at 4 PM)
Classes begin January 18, 2005
Spring Break is March 12-20, 2005
Regular classes end Monday May 2. Final Exams are May 5-11, 2005.
### Fall 2004 Seminars
10 Sept Colonel Henry Rutgers -------- Department Reception ----------------
24 Sept* Yom Kippur is 9/25
1 Oct* Liang Kong Rutgers "Conformal field algebras and tensor categories"
8 Oct: MacPherson's 60th Conference
15 Oct: Pavel Etingof MIT "Cherednik and Hecke algebras of orbifolds"
22 Oct* Lin Zhang RU+Sequent-Capital "When does the commutator formula imply the Jacobi identity in Vertex Operator Algebra theory?"
29 Oct*: A. Ocneanu Penn State "Modular theory, quantum subgroups and quantum field theory"
5 Nov: Helmut Hofer Courant D'Atri Lecture: Holomorphic Curve Methods (talk at 1:10 PM)
5 Nov:* Keith Hubbard Notre Dame "Vertex Algebra coalgebras: Their operadic motivation and concrete constructions"
12 Nov: Chuck Weibel Rutgers "Homotopy theory for Motives"
19 Nov: Edwin Beggs Univ. of Wales Swanswa "The Van Est spectral sequences for Hopf algebras"
26 Nov: Tom Turkey Plymouth Colony ----------Thanksgiving Break------------
10 Dec: Edwin Beggs Univ. of Wales Swanswa"Quasi-Hopf algebras, twisting and the KZ equation"
17 Dec: Student Body Left Rutgers ---- Final Exam Grading Marathon -------
Semester begins Wednesday September 1, 2003.
Regular classes end Monday, December 13. Final Exams are Dec.16-23.
Math Group Exam time is Thursday Dec.16 (4-7PM)
### Spring 2004
26 Jan: Diane Maclagan Stanford "Toric Hilbert schemes" (talk at 4:30 PM)
28 Jan: Greg Smith Columbia "Orbifold Cohomology of Toric Stacks" (talk at 11:30 AM)
30 Jan: Anna Lachowska MIT "TBA" (talk at 1:10 PM)
6 Feb: Chuck Weibel Rutgers "A survey of non-Desarguesian planes"
13 Feb: Kia Dalili Rutgers "The HomAB Problem"
20 Feb: Vladimir Retakh Rutgers "An Introduction to A-infinity Algebras"
27 Feb: Vladimir Retakh Rutgers "An Introduction to A-infinity Algebras II"
5 Mar: Remi Kuku IAS "A complete formulation of the Baum-Connes Conjecture for the
action of discrete quantum groups"
12 Mar: Amnon Yekutieli Ben Gurion Univ. "On Deformation Quantization in Algebraic Geometry"
19 Mar: no seminar ------------- Spring Break -------------
26 Mar: Alexander Retakh MIT "Conformal algebras and their representations"
2 Apr: Aaron Lauve Rutgers "Capture the flag: towards a universal noncommutative flag variety"
9 Apr* Stefano Capparelli Univ. Rome "The affine algebra A22 and combinatorial identities"
16 Apr: Uwe Nagel U.Kentucky "Extremal simplicial polytopes"
16 Apr(C) Dale Cutkosky U. Missouri Colloquium Talk at 4:30 PM
23 Apr* Paul Rabinowitz Wisconsin *** D'Atri Lecture at 1:10 PM ***
23 Apr: Li Guo Rutgers-Newark "Dendriform algebras and linear operators"
30 Apr: Earl Taft Rutgers "There exists a one-sided quantum group"
7 May Student Body Left Rutgers ---- Final Exam Grading Marathon --------
Classes begin January 20, 2004; Spring Break is March 13-21, 2004
Regular classes end Monday May 3. Final Exams are May 6-12, 2004.
Math Group Final Exam time is Thursday May 6 (4-7PM)
### Fall 2003 (in room H425)
5 Sept George Willis U. South Wales "scale functions on totally disconnected groups"
5 Sept Colonel Henry Rutgers -------- Department Reception ----------------
8 Sept Various people -------- Gelfand 90th Birthday Celebration --------------
12 Sept Edwin Beggs U.Wales-Swansea, UK "Constructing tensor categories from from finite groups"
19 Sept Charlie Sims Rutgers "Algorithmic Questions in Rings of Rational Matrices?"
26 Sept David Radnell Michigan Thesis Defense:
"Schiffer Variation in Teichmüller space, determinant line bundles and modular functors"
3 Oct* Liang Kong Rutgers "Open-string vertex algebras"
10 Oct C. Musili U.Hyderabad, India "The Development of Standard Monomial Theory"
17 Oct Roy Joshua Inst. Adv. Study "The Motivic DGA"
24 Oct Bodo Pareigis Univ. Munich "Modules, Comodules, Entwinings and Braidings"
31 Oct* Benjamin Doyon Rutgers "From vertex operator algebras to the Bernoulli numbers"
7 Nov* Geoffrey Buhl Rutgers "Complete reducibility and C_n-cofiniteness of vertex operator algebras"
14 Nov no RU seminar ------ Borel Memorial at IAS -----------
21 Nov* Lin Zhang Rutgers "A vertex operator algebra approach to the construction of a tensor category of Kazhdan-Lusztig"
28 Nov: Tom Turkey ----------Thanksgiving Break------------
5 Dec* Victor Ostrik IAS "Finite extensions of vertex algebras"
12 Dec* Matt Szczesny U. Penn. "Orbifolding the chiral de Rham complex"
Semester begins Tuesday September 2, 2003. Lewis Lectures are the week of October 3rd.
Regular classes end Wednesday, December 10. Final Exams are Dec. 15-22.
Math Group Exam time is Monday Dec.15 (4-7PM)
### Spring 2003
28 Jan* Masahiko Miyamoto Japan "Interlocked modules and pseudo-trace functions"
31 Jan: no seminar ------------- Jean Taylor Symposium -------------
5 Feb: Angela Gibney Michigan "Some open questions about the geometry of the moduli space of curves"
21 Feb* Kiyokazu Nagatomo Japan "Conformal field theory over the projective line"
28 Feb: Jooyoun Hong Rutgers "Normality of Rees algebras for conormal modules"
7 Mar*: Yucai Su Shanghai/Harvard "Lie algebras associated with derivation-simple algebras"
14 Mar* Chengming Bai Nankai&Rutgers "Novikov algebras and vertex (operator) algebras"
21 Mar: no seminar ------------- Spring Break -------------
28 Mar*: David Radnell Rutgers "Schiffer Variation in Teichmüller Space and Determinant Line Bundles"
3 Apr: Claudio Pedrini U.Genova "Finite dimensional motives" Thursday 3PM - Note change in day!
4 Apr# Hy Bass & Deborah Ball Michigan "Preparing teachers for the mathematical work of teaching"
11 Apr*: Lin Zhang Rutgers "Tensor category theory for modules for a vertex operator algebra -- introduction and generalization"
18 Apr: Constantin Teleman Cambridge U. "Twisted K theory from the Dirac spectral flow"
25 Apr* Michael Roitman Michigan "Affinization of commutative algebras"
2 May: Frederick Gardiner CUNY "The pure mapping class group of a Cantor set" At 1:30 PM - Note change in time!
9 May: Carlo Mazza Rutgers "Schur's Finiteness conditions in tensor categories" At 3:30 PM in H425 - Note change in time and room!
Regular classes end Monday May 5. Final Exams are May 8-14, 2003.
Math Group Final Exam time is Thursday May 8 (4-7PM)
### Fall 2002
13 Sep: no seminar Department Reception
20 Sep* YZ Huang Rutgers "Differential equations, duality and modular invariance"
27 Sep* Matthias Gaberdiel Kings College "Conformal field theory and vertex operator algebras"
4 Oct: no seminar
11 Oct: Ravi Rao TATA "Raga Bhimpalasi: The Vaserstein-Suslin Jugalbandhi"
11 Oct(C) Igor Kriz Michigan Colloquium Talk "Conformal field theory and elliptic cohomology" at 4:30 PM
18 Oct: Richard Stanley MIT Jacqueline Lewis Lecture at 4:30PM
18 Oct*: Earl Taft Rutgers "Is there a one-sided quantum group?"
25 Oct:Christian Kassel CNRS-Univ. Louis Pasteur, Strasbourg "Explicit norm one elements for ring actions of finite abelian groups"
25 Oct(C) C. Kassel ""(Strasbourg) Colloquium Talk "Recent developments on Artin's braid groups" at 4:30PM
1 Nov* Benjamin Doyon RU Physics "Twisted vertex operator algebra modules and Bernoulli polynomials"
8 Nov: Charles Weibel RU "The work of Vladimir Voevodsky"
15 Nov* Takashi Kimura IAS/Boston U. "Integrable systems and topology"
22 Nov: Julia Pevtsova IAS "Support Varieties for Finite Group Schemes"
29 Nov: Tom Turkey ----------Thanksgiving Break------------
6 Dec: Anya Lachowska MIT "Modular group action in the center of the small quantum group"
### Spring 2002
A 'j' marks a meeting of the Junior Algebra Seminar.
25 Jan: no seminar Job Interview Talks
1 Feb: no seminar Job Interview Talks
8 Feb* Liz Jurisich College of Charleston "The monster Lie algebra, Moonshine and generalized Kac-Moody algebras"
15 Feb:j Will Toler RU Physics "Low dimensional topology and gauge theory"
22 Feb# Laura Alcock RU Math/Ed "The first course in real analysis in England: figuring out the conceptions students form"
1 Mar: ----- -- CANCELLED
8 Mar*j Benjamin Doyon RU Physics "Vertex Operator Algebras and the Zeta function"
15 Mar*j Gordon Ritter Harvard "Montonen-Olive Duality in Yang-Mills Theory"
22 Mar: no seminar ------------- Spring Break -------------
29 Mar* Sergei Lukyanov RU Physics "Once again about Bethe Ansatz"
5 Apr:j Benjamin Doyon RU Physics "Fractional Derivatives"
12 Apr: Lisa Carbone RU "Lattice subgroups of Kac-Moody groups over finite fields"
19 Apr: Agata Smoktunowicz Yale/Warsaw(PAS) "A simple nil ring exists"
26 Apr: Earl Taft RU "Recursive Sequences and Combinatorial Identities"
3 May* Yi-Zhi Huang RU "Differential equations and intertwining operators"
10 May: Calculus Profs Rutgers "Grading of Final Exams"
Regular classes end Monday, May 6. Final Exams end Wednesday, May 15.
Math Group Exam time is Thursday May 9th (4-7PM).
### Fall 2001
7 Sep: Rutgers Math Department Reception (4PM)
14 Sep* Sasha Kirillov SUNY Stony Brook "On a q-analog of the McKay correspondence"
21 Sep: Ngo Viet Trung Inst.Math.Hanoi "Hilbert functions of non-standard bigraded algebras"
5 Oct: Ed Letzter Temple "Effective Representation Theory of Finitely Presented Algebras"
12 Oct* Yi-Zhi Huang Rutgers "Vertex operator algebras and conformal field theories"
19 Oct: V. Retakh Rutgers "Algebra and combinatorics of pseudo-roots of
noncommutative polynomials and noncommutative differential polynomials"
26 Oct*: Yan Soibelman Kansas State U. "Elliptic curves and quantum tori"
2 Nov* Yi-Zhi Huang Rutgers "Vertex operator algebras and conformal field theories II"
9 Nov* Deepak Parashar MPI Leipzig "Some biparametric examples of Quantum Groups"
16 Nov* Yi-Zhi Huang Rutgers "Vertex operator algebras and conformal field theories III"
23 Nov: Tom Turkey ----------Thanksgiving Break------------
30 Nov* Hai-Sheng Li Rutgers Camden "Certain noncommutative analogues of vertex algebras"
7 Dec: Chuck Weibel Rutgers "Congruence subgroups of SL2(Z[1/n]), after Serre"
14 Dec:
regular classes end Wednesday, December 12. Final Exams are Dec. 15-22.
Math Group Exam time is Monday Dec.17 (4-7PM)
### Spring 2001
26 Jan: Alexei Borodin U.Penn ------- Job Candidate Interview -------
2 Feb: Chuck Weibel Rutgers "POSTPONED TO March 30"
9 Feb: Dave Bayer Columbia U. "Toric Syzygies and Graph Colorings"
16 Feb: Igor Kriz U.Michigan "A geometric approach to elliptic cohomology"
23 Feb* Yi-Zhi Huang Rutgers "Conformal-field-theoretic analogues of codes and lattices"
2 Mar: Carl Futia Southgate Capital Advisors "Bialgebras of Recursive Sequences and Combinatorial Identities"
9 Mar* Haisheng Li Rutgers Camden "Regular representations for vertex operator algebras"
16 Mar: no seminar ------------- Spring Break -------------
23 Mar* Yvan Saint-Aubin U.Montreal+IAS "Boundary behavior of the critical 2d Ising model"
30 Mar: Chuck Weibel Rutgers "Functors with transfer (on rings)"
6 Apr*: Richard Ng Towson U "The twisted quantum doubles of finite groups"
13 Apr* Charles Doran Columbia "Variation of the mirror map and algebra-geometric isomonodromic deformations"
20 Apr*: Lev Borisov Columbia "Elliptic genera of singular algebraic varieties"
27 Apr: Diane Maclagan IAS "Supernormal vector configurations, Groebner fans, and the toric Hilbert scheme"
4 May: Calculus Profs Rutgers "Grading of Final Exams"
Regular classes end Monday, April 30. Final Exams end Wednesday, May 9.
Math Group Exam time is Thursday May 3rd (4-7PM)
### Fall 2000
8 Sep: Amelia Taylor Rutgers "The inverse Gröbner basis problem in codimension two"
15 Sep* Mike Douglas RU Physics "D-branes"
22 Sep: Chuck Weibel Rutgers "Topological vs. algebraic $K$-theory for complex varieties"
29 Sep: no seminar ------------- Rosh Hoshanna ------------
6 Oct: Daya-Nand Verma TATA Inst. "Progress Report on the Jacobian Conjecture"
13 Oct: no seminar
20 Oct* Constantin Teleman U.Texas "The Verlinde algebra and twisted K-theory"
27 Oct: Chuck Weibel Rutgers "Homotopy Ends and Thomason Model Categories"
3 Nov* Mirko Primc U.Zagreb "Annihilating fields of standard modules of sl_2~ and combinatorial identities"
10 Nov: Suemi Rodriguez-Romo UNAM Mexico "Quantum Group Actions on Clifford Algebras"
17 Nov: Craig Huneke U.of Kansas "Growth of Symbolic Powers in Regular Local Rings"
24 Nov: Tom Turkey ----------Thanksgiving Break------------
1 Dec# Nina Fefferman and Matt Young Rutgers VIGRE presentations on p-adic numbers
8 Dec* Mike Douglas? RU Physics "D-branes, instantons and orbifolds"
### Winter 2000
4 Feb: Martin Sombra IAS+LaPlata "Division formulas and the arithmetic Nullstellensatz"
11 Feb: no seminar
18 Feb: Claudio Pedrini IAS+Genoa "K-theory of algebraic varieties: a Survey"
25 Feb: M.R.Kantorovitz IAS "Andre-Quillen homology from a calculus viewpoint"
(with Hochschild homology and algebraic K-theory for dessert)
3 Mar: S. Hildebrandt Bonn *** D'Atri Lecture *** (2-dim. Variational Problems)
10 Mar: D. Christensen IAS "Brown representability in derived categories"
17 Mar: --- ---- ------- Spring Break -----------
24 Mar* Haisheng Li RU-Camden "Certain extended vertex operator algebras"
31 Mar* Christoph Schweigert Paris "Conformal boundary conditions and three-dimensional topological field theory"
7 Apr: no seminar
14 Apr* Christian Schubert LAPTH France "Multiple Zeta Value Identities from Feynman Diagrams"
21 Apr: no seminar
28 Apr* Tony Milas Rutgers "Structure of fusion rings associated to Virasoro vertex operator algebras"
3 May* (Wednesday) Tony Milas Rutgers "Differential operators and correlation functions"
### Fall 1999
24 Sep: V. Retakh Rutgers "Noncommutative rational functions+Farber's invariants of boundary links"
1 Oct: Antun Milas* Rutgers "Intertwining operator superalgebras for N=1 minimal models"
8 Oct: Fedor Bogomolov NY Univ "Fundamental Groups of Projective Varieties"
15 Oct: Earl Taft Rutgers "Sequences satisfying a polynomial recurrence"
22 Oct: Yuji Shimizu* Kyoto U "Momentum mappings and conformal fields"
29 Oct: Leon Seitelman U.Conn. SPECIAL VIGRE LECTURE
"What's a mathematician like you doing in a place like that"
5 Nov: Keith Pardue IDA/Princeton "Generic Polynomials"
12 Nov: *Haisheng Li Rutgers (Camden) "The Diamond lemma for algebras (following Bergman)"
19 Nov: Yuri Tschinkel U.Illinois "Equivariant compactifications of G_a^n"
26 Nov: Tom Turkey ------Thanksgiving Break--------
3 Dec: Borisov* Columbia "Vertex algebras and mirror symmetry"
10 Dec: Chongying Dong UC Santa Cruz "Holomorphic orbifold theory, quantum doubles and dual pairs"
### Spring 1999
22 Jan: P. Balmer Rutgers "The derived Witt group of a ring"
29 Jan: W. Vasconcelos Rutgers "The intertwining algebra"
5 Feb: Thomas Geisser U.Tokyo "TBA"
12 Feb:Dennis Gaitsgory Harvard/IAS "On a VOA of differential operators on a loop group"
19 Feb: Mark Walker Nebraska "The total Chern class map"
26 Feb: Michael Roitman Yale "Universal constructions in conformal and vertex algebras"
5 March: E. Friedlander Northwestern "Re-interpreting the Bloch-Lictenbaum spectral sequence"
12 March: R. Schoen D'Atri Lecture instead of seminar
19 March: Vernal Equinox ------Spring Break March 14-21----
26 March: Yuji Shimizu Kyoto and Rutgers "Conformal blocks and KZB equations"
2 April: Roger Rabbit Toontown no seminar (Passover/Easter)
9 April:
16 April: Marco Schlichting RU and U. Paris "The negative K-theory of an exact category"
23 April: Chuck Weibel Rutgers "Projective modules over normal surfaces"
30 April: Percy Deift Courant Institute (Colloquium talk)
7 May: Yuji Shimizu Kyoto and Rutgers "Geometric structures underlying some conformal field theories"
### Fall 1998
18 Sep: Lowell Abrams Rutgers "Modules, comudules and cotensor products over Frobenius algebras"
25 Sep: Bogdan Ion Princeton "Maschke's theorem revisited"
2 Oct: Haisheng Li(*) Rutgers Camden "An infinite-dimensional analogue of Burnside's theorem"
9 Oct: Aron Simis Univ.F.Pernambuco (Recife, Brazil) "Geometric Aspects of Rees Algebras"
16 Oct: A. Beilinson Univ. Chicago Colloquium in honor of Gelfand
23 Oct: Michael Finkelberg(*) IAS/Independent Moscow Univ. "An integrable system on the space of based maps from P^1 to a flag variety"
30 Oct: Yi-Zhi Huang(*) Rutgers "Semi-infinite forms and topological vertex operator algebras"
6 Nov: Alfons Ooms Limburgs Univ, Belgium "On the Gelfand-Kirillov conjecture"
13 Nov: A. Kirillov, Jr.(*) IAS "On the Lego-Teichmuller game"
20 Nov: M.F. Yousif Ohio State-Lima "On three conjectures on quasi-Frobenius Rings"
27 Nov: Tom Turkey ------Thanksgiving Break--------
4 Dec: C. Lenart Max Planck (Bonn) ""
11 Dec: S. Majid Cambridge Univ. "braided groups and the inductive construction of U_q(g)"
### Spring 1998
30 Jan: C. Weibel Rutgers "local homology vs. cohomology (after Greenlees-May)"
6 Feb: Brian Parshall U. of Virginia "The cohomology and representation theory of reductive groups in non-describing characteristics"
13 Feb: M. Khovanov(*) Yale and IAS "Lifting the Jones polynomial of knots to invariants of surfaces in 4-space"
20 Feb: Ming-Sun Li Rowan Univ. "Spectral matrices associated to an algebra"
27 Feb: Yi-Zhi Huang(*) Rutgers "Analytic aspects of Intertwining Operators"
6 Mar: Boris Khesin(*) IAS+U.Toronto "Geometric complexification of affine algebras and flat connections on surfaces"
13 Mar: no algebra seminar
20 Mar: Vernal Equinox ------Spring Break--------
27 Mar: N. Inassaridze Razmadze Inst. "Non-abelian homology of groups"
3 Apr: Jim Stasheff UNCarolina "Physically inspired homological algebra"
10 Apr: Movshev(*) ... QUANTUM MATH SEMINAR
17 Apr: S. Sahi Rutgers "A new character formula for compact Lie groups"
24 Apr: Stefan Schmidt Berkeley "Projective Geometry of Modules"
1 May: Toma Albu U.Wisc.-Milwaukee "GLOBAL KRULL DIMENSION AND GLOBAL DUAL KRULL DIMENSION OF RINGS"
### Fall 1997
19 Sep: Bill Kantor U. Oregon Colloquium: "Black box classical groups"
26 Sep: Lowell Abrams Rutgers "2-dimensional TQFT's and Frobenius Algebras"
3 Oct: --- ------ Rosh Hoshanna -----
10 Oct: Tor Gunston Rutgers "Degree functions and linear resolutions"
31 Oct: Chuck Weibel Rutgers "introducing Motives"
7 Nov: --- Columbia Univ. Bass Conference
14 Nov: Stefan Catoiu Temple Univ. "IDEALS OF THE ENVELOPING ALGEBRA U(sl_2)"
21 Nov: M. Kontsevich IHES "Deformation, Quantization and Beyond"
28 Nov: Tom Turkey ------Thanksgiving Break--------
5 Dec: M. Kontsevich IHES "Deformation, Quantization and Beyond"
12 Dec: C. Pedrini U. Genova "K-Theory and Bloch's Conjecture for complex surfaces"
### Spring 1997
31 Jan: Luisa Doering Rutgers "Generalized Hilbert functions"
7 Feb: postponed
14 Feb: Miguel Ferrero Porto Alegre,Brazil "Closed and prime submodules of centered bimodules and applications to ring extensions"
21 Feb: Richard Ng Rutgers "Freeness of Hopf algebras over subalgebras"
28 Feb: Siddartha Sahi Rutgers "Introduction to Macdonald polynomials"
7 Mar: Barbara Osofsky Rutgers "Projective dimension for commutative von Neumann regular rings and a new lattice invariant"
14 Mar: Chuck Weibel Rutgers "K-theory and zeta functions on number fields"
21 Mar: ------------ Spring Break ------------
28 Mar: Carl Faith Rutgers "Rings with ACCs on annihilators"
4 Apr: Joe Brennan N.Dakota "The Ends of Ideals"
11 Apr: Jan Soibelman Kansas State "Meromorphic tensor categories and quantum affine algebras"
18 Apr: Chuck Weibel Rutgers "Tor without identity (after Quillen)"
25 Apr: Wolmer Vasconcelos Rutgers "Integral closure"
2 May: Luca Mauri Rutgers "2 torsors"
### Fall 1996
20 Sep: C. Weibel Rutgers "the 2-torsion in the K-theory of Z"
27 Sep: Tor Gunston Rutgers "Cohomological dimension of graded modules"
4 Oct: B. Ulrich MichState "Divisor class groups and Linkage"
11 Oct: -- IAS Langlands Fest
18 Oct: Bob Guralnick USC "Finite Orbit Modules and Double Cosets for Algebraic Groups"
25 Oct: Richard Weiss Tufts "Moufang polygons"
1 Nov: Georgia Benkart Wisconsin "Lie Algebras Graded by Finite Root Systems"
8 Nov: Richard Ng Rutgers "On the projectivity of module coalgebras"
15 Nov: -- no seminar
22 Nov: Bill Keigher RU-Newark "The ring of Hurwitz series"
29 Nov: Tom Turkey Thanksgiving (no seminar)
6 Dec: Leon Pritchard RU-Newark "Hurwitz series Formal Functions"
13 Dec: Reading Period after classes
### Spring 1996
26 Jan: A.Corso Rutgers "generic gaussian ideals"
2 Feb: no seminar
9 Feb: E. Taft Rutgers "Quantum Convolution"
16 Feb: Frosty S. Weather "Snow storm--talks rescheduled"
23 Feb: B. Leasher Rutgers "Geometric Aspects of Steinberg Groups for Jordan Pairs"
1 Mar: L. Mauri Rutgers "Low-dimensional Descent theory"
8 Mar: K.Consani IAS "Double complexes and local Euler factors on algebraic degeneration"
15 Mar: ------------ Spring Break ------------
22 Mar: YZ Huang Rutgers "On algebraic D-modules and vertex algebras"
29 Mar: Doering&Gunston Rutgers "Algebras Arising from Bipartite Planar Graphs"
5 Apr: Consuelo Martinez Yale "Power subgroups of profinite groups"
12 Apr: M. Singer NC State "Galois theory for difference equations"
19 Apr: C. Weibel Rutgers "Popescu Desingularization (after Swan)"
26 Apr: R. Hoobler CCNY "Merkuriev-Suslin Theorem for arbitrary semi-local rings"
14 May: K. Mimachi Kyushu U. "Quantum Knizhink-Zamolodchikov equation and eigenvalue problem of Macdonald equations"
### Fall 1995
28 Sep: M.Gerstenhaber U. Penn "Symplectic structures on max. parabolic
subgps. of SL_n and boundary solutions of
the classical Yang-Baxter equation"
29 Sept:W. Vasconcelos Rutgers "Gauss Lemma"
6 Oct: I. Gelfand Rutgers "Noncommutative symmetric functions"
13 Oct: Joan Elias Barcelona"On the classification of curve singularities"
20 Oct: B. Osofsky Rutgers "Connections between foundations and Algebra"
27 Oct: O. Stoyanov Rutgers "Quantum Unipotent Groups"
3 Nov: I. Gelfand Rutgers "Noncommutative Grassmannians"
10 Nov: M. Tretkoff Stevens "Rohrlich's formula for hypersurface periods"
17 Nov: C. Weibel Rutgers "Tinker Toys and graded modules"
24 Nov: Tom Turkey Thanksgiving Break
1 Dec: Siu-Hung Ng Rutgers "Lie bialgebra structures on the Witt algebra"
8 Dec: E. Zelmanov Yale "On narrow groups and Lie algebras"
### Spring 1995
27 Jan Alberto Corso Rutgers "Links of irreducible varieties"
3 Feb Chuck Weibel Rutgers "Operads for the Working Mathematician"
10 Feb Maria Vaz Pinto Rutgers "Hilbert Functions and Sally Modules"
17 Feb Yi-Zhi Huang Rutgers "Vertex Operator Algebras for Lay Algebraists"
24 Feb O. Matthieu "On the modular representations of the symmetric group"
3 Mar Claudio Pedrini Genova "The Chow group of singular complex surfaces"
10 Mar B.Sturmfels-Berkly A normal form algorithm for modules over k[x,y]/(xy)
18 Mar ------------ Spring Break ------------
24 Mar Francesco Brenti IAS "Twisted incidence algebras and
Kazhdan-Lusztig-Stanley functions"
31 Mar Myles Tierney Rutgers "Simplicial sheaves"
7 April Wolmer Vasconcelos "A Lemma of Gauss"
14 April Peter Cottontail "Easter's on its way! (Passover too!)"
21 April Susan Morey "Symbolic Powers, Serre Conditions and CM Rees algebras"
28 April K.P. Shum Hong Kong/Maryland "Regular semigroups and generalizations"
### ALGEBRA SEMINAR - Spring 1992
29 Jan: Earl Taft Rutgers "Linearly recursive sequences in several variables"
5 Feb: J. Brennan N.D. State "Integral closure of a morphism"
12 Feb: Chuck Weibel Rutgers "Chern classes and torsion in algebraic K-theory"
19 Feb: Friedrich Knop Rutgers "Invariant valuations and eqeuivariant embeddings"
26 Feb: Charles Walters Rutgers "Projectively normal curves"
4 Mar: M.C. Kang Taiwan "Monomial group actions on rational functino fields"
11 Mar: Art Dupre Rutgers-Newark "Extensions and cohomology of groups"
27 Mar: Wolmer Vasconcelos Rutgers "The top of a system of equations"
3 Apr: Marvin Tretkoff Stevens "Some of Dwork's Cohomology Spaces"
17 Apr: Frederico Bien Princeton "Vanishing theorems for D-modules on spherical varieties"
24 Apr: Carl Faith Rutgers "FPF rings"
### ALGEBRA SEMINAR - Fall 1991
13 Sep: W. Vogell Martin-Luther U. "Intersection theory"
25 Sep: Sunsook Noh Rutgers "Divisors of 2nd kind and 2-dimensional regular local rings"
2 Oct: W. Vasconcelos Rutgers "The Sally module of a reduction"
9 Oct: Earl Taft Rutgers "Witt and Virasoro algebras as Lie bialgebras"
16 Oct: Bill Hoyt Rutgers TBA
23 Oct: Barbara Osofsky Rutgers TBA
30 Oct: V. Retakh Rutgers "The algebra of extensions without resolutions"
6 Nov: V. Retakh Rutgers continued
13 Nov: A. Brownstein "Generalized Braid groups and motions of strings in 3-space"
20 Nov: Aron Simis U.Fed.Bahia "On tangent cones"
### ALGEBRA SEMINAR - Spring 1991
25 Jan Chuck Weibel Rutgers "Operations and symbols in K-theory"
1 Feb Barr Von Oehsen Rutgers "Elliptic genera and Jacobi polynomials"
8 Feb Bernie Johnston FAU TBA
22 Feb Barbara Osofsky Rutgers "Constructing nonstandard uniserial modules over valuation domains"
1 Mar W. Vasconcelos Rutgers "Explicit Nullsatzen"
8 Mar Matt Miller S. Carolina "Betti numbers of modules of finite length"
15 Mar Sam Vovsi Ryder College TBA
21 Mar R.I. Grigorchuk Moscow Inst. "The Burnside problem"
5 Apr Rafael Villareal Rutgers TBA
12 Apr Willie Cortinas Buenos Aires TBA
19 Apr Joachim Lambek McGill U. TBA
26 Apr Charles Walter Rutgers "Algebraic space curves with the expected monomial"
3 May Bernd Ulrich Michigan State "Projective curves and their hypersurface sections"
### ALGEBRA SEMINAR - Fall 1981
15 Sep Hisao Tominaga Okayama U. "Some polynomial identites and commutativity of rings"
22 Sep Chuck Weibel Rutgers "Set-theoretic complete intersection points on curves"
6 Oct Carl Faith Rutgers "Subrings of FPF and self-injective rings"
13 Oct W. Vasconcelos Rutgers "Symmetric algebras and syzygies"
20 Oct David Rohrlich Rutgers "maps to the projective line of minimal degree"
27 Oct Stan Page U.Br.Columbia "Slim rings and modules"
3 Nov Joe Johnson Rutgers "Dimension fans and finite presentation of graded modules"
10 Nov Ron Donagi Utah "The Schottky problem"
17 Nov J. Dorfmesiter Muenster "Siegel domains"
24 Nov Barbara Osofsky Rutgers "Strange self-injective rings"
1 Dec Bill Hoyt Rutgers "Division points on generic elliptic curves"
8 Dec Chuck Weibel Rutgers "KABI"
### ALGEBRA SEMINAR - Spring 1981
28 Jan S. Goto Brandeis/Nihon U. "On Buchsbaum rings"
4 Feb Chuck Weibel Rutgers "When are projective modules extended?"
11 Feb Barbara Osofsky Rutgers "Between flatness and projectivity"
18 Feb David Rohrlich Rutgers "An intro to L-functions on elliptic curves"
25 Feb M. Takeuchi IAS "Commutative Hopf algebras and cocommutative Hopf algebras in char. p"
04 Mar Harry Gonshor Rutgers "Conway numbers and semigroup rings"
11 Mar Louis Rowen Yale "Finite dimensional division algebras"
25 Mar Bill Hoyt Rutgers "Periods of abelian integrals"
1 Apr Don Schack U. Buffalo "Deformations of diagrams"
8 Apr Earl Taft Rutgers "Hopf algebras"
15 Apr Chuck Weibel Rutgers "Witt vectors made easy"
22 Apr Jan van Geel U. Antwerp "Primes and value functions"
29 Apr Dick Cohn Rutgers "The General Solution in Differential Equations"
### ALGEBRA SEMINAR - Fall 1980
25 Sept Earl Taft Rutgers "A generalization of divided power sequences"
1 Oct Moss Sweedler Cornell "Products of flat modules"
8 Oct Chuck Weibel Rutgers "Principal ideals and smooth curves"
15 Oct Joe Johnson Rutgers "Rings that lack ..."
22 Oct Wolmer Vasconcelos Rutgers TBA
29 Oct Richard Block UC Riverside "Irreducuble representations of skew polynomial rings"
5 Nov M. Gerstenhaber U.Penn "On the deformation of differential graded algebras"
7 Nov F. Orecchia U.Genoa "Tangent cones and singularities of algebraic curves"
12 Nov E. Sontag Rutgers "PL Algebras"
19 Nov Dick Bumby Rutgers "Jacobi symbols"
3 Dec Carl Faith Rutgers "Noncommutative rings"
## Abstracts of seminar talks
#### Fall 2016
Vector bundles of conformal blocks on the moduli space of curves (Angela Gibney, December 14, 2016):
In this talk I will introduce the moduli space of curves and a class of vector bundles on it. I'll discuss how these bundles, which have connections to algebraic geometry, representation theory, and mathematical physics, tell us about the moduli space of curves, and vice versa, focusing on just a few recent results and open problems.
In the second part of my talk I will focus on noncommutative symplectic forms and noncommutative Poisson geometry. This is where the double Gerstenhaber algebra of noncommutative poly-vector fields appears. I will show that use of skew-symmetric properties allows us to substantially simply the definition.
Since this is a seminar aimed at the general audience, I'll start by explaining the notion of vertex algebra, as well as the physical meaning. Then I'll introduce the notion of a MOSVA and the physical meaning. Hopefully there will be some time to explain what I have done.
#### Spring 2016
I will start by reviewing an $H_0$-Poisson structure --- a noncommutative analog of the Poisson bracket and related notion of double Poisson brackets. We will see how an $H_0$-Poisson structure descends to a usual Poisson bracket on the moduli space of representations of the underlying associative algebra. I will then show how one can substantially modify definition of double Poisson bracket by M. Van den Bergh to provide a number of new nontrivial examples.
The talk is aimed at graduate students. In particular, while some familiarity with Chern classes would be useful, I will introduce the necessary notions during the talk.
#### Fall 2015
Representations of finite groups and applications (Pham Huu Tiep, Dec. 7, 2015):
In the first part of the talk we will survey some recent results on representations of finite groups. In the second part we will discuss applications of these results to various problems in group theory, number theory, and algebraic geometry.
The talk will survey some aspects of this ongoing search. The methods for studying this question involve explicit and constructive applications of well known classical theorems in algebra and group theory, for instance Conway's and Pless' application of Burnside's orbit counting theorem and quadratic reciprocity dating back to the 1980's. More recent and partly computational methods are based on representation theory of finite groups.
#### Spring 2015
The Index Map and Reciprocity Laws for Contou-Carrère Symbols
(Jesse Wolfson, Feb. 4, 2015):
In the 1960s, Atiyah and Janich constructed a natural "index" map from the space of Fredholm operators on Hilbert space to the classifying space of topological K-theory. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. The index map allows us to relate the Contou-Carrère symbol, a local analytic invariant of families of schemes, to algebraic K-theory. Using this, we prove reciprocity laws for Contou-Carrère symbols in all dimensions. This extends previous results, of Anderson and Pablos Romo in dimension 1, and of Osipov and Zhu, in dimension 2.
#### Fall 2014
Semiclassical approximation to noncommutative Riemannian geometry
(Edwin Beggs, Sept. 17, 2014):
I will consider the first order deformation of a Riemannian manifold, including the vector bundles, differential calculus and metric. One example will be the Schwarzschild solution, which illustrates that not all the properties of the classical case can be simply carried into the quantum case. The other example is quantising the Kahler manifold, complex projective space. This case is much simpler, and here the complex geometry is also preserved. I will end with some comments on the connection between noncommutative complex geometry and noncommutative algebraic geometry.
Spring 2014 Seminars (Wednesdays at 2:00 in H525)
Abstracts and Details for Spring 2014 seminars are located at
THIS SITE
#### Fall 2013
Torsion Theory and Slice Filtration of Homotopy Invariant Sheaves With Transfers
(Knight Fu, Sept. 16, 2013):
Torsion Theory makes important contributions to the study of modules over a ring. It also plays an important role in constructing the quotient of an abelian category by a "torsion" subcategory.
Applying torsion theory to the category of homotopy invariant sheaves with transfers, we show how a sequence of co-radical functors gives rise to two filtrations --- one increasing, one decreasing --- of the category. We conjecture that the obtained structure ought to be the "slice filtration" on the category, and show how the filtrations are compatible with the slice filtration on Voevodsky's derived category of motives.
#### Spring 2013
Equations, syzygies, and vector bundles (Daniel Erman, Jan. 24, 2013):
For a system of polynomial equations, it has long been known that the relations (or syzygies) among the polynomials provide geometric information about the corresponding projective variety. I will describe a collection of new ideas about how to study syzygies, and how these lead to classification results and a duality between syzygies and vector bundles.
#### Fall 2012
Curve neighborhoods Anders Buch, Oct. 3, 2012):
Given a generalized flag manifold X = G/P, a Schubert variety X(w), and a degree d, consider the set of points that can be reached from X(w) by a rational curve of degree d, i.e. the union of all rational degree d curves through X(w). It turns out that the Zariski closure of this set is a larger Schubert variety, which is important for many aspects of the quantum cohomology of X, including the quantum Chevalley formula and the smallest q-degree in the quantum product of two Schubert classes. I will give a very explicit description of this "curve neighborhood" of the Schubert variety in terms of the Hecke product of Weyl group elements, and use it to give a simple proof of the (equivariant) quantum Chevalley formula. This is joint work with Leonardo Mihalcea.
Computations in intersection theory (Dan Grayson, Oct. 17, 2012):
This is joint work with Alexandra Seceleanu and Michael E. Stillman. We describe Groebner bases for the ideals of relations between the Chern classes of the tautological bundles on partial flag bundles, and show how the result can be used to enable practical computation of intersection numbers in the "Macaulay2" package "Schubert2". We also generalize the result to cover isotropic flag bundles.
On the slice filtration for hermitian K-theory (Oliver Rondigs, November 7, 2012):
Let F be a perfect field of characteristic different from 2. In joint work with Paul Arne Ostvaer, we describe the slices of hermitian K-theory and higher Witt-theory in the motivic stable homotopy category of F. Applications include computations of hermitian K-groups and Witt groups for number fields and projective spaces, as well as a different perspective on Milnor's conjecture on quadratic forms.
Injectivity property of etale H^1, non-stable K_1, and other functors (Anastasia Stavrova, December 5, 2012):
The first term of Gersten conjecture for K-theory claims the injectivity of the map K_i(R)→K_i(K) for any regular semilocal ring R with field of fractions K. The same statement with K_i replaced by the etale cohomology functor H^1(-,G), where G a reductive algebraic group, is known as the Grothendieck-Serre conjecture. The latter conjecture was recently settled by I. Panin et al. under the assumption that R contains an infinite perfect field. We discuss how essentially the same argument carries over to non-stable K_1 and similar functors.
#### Spring 2012
Exhausting quantization procedures (Vasily Dolgashev, Jan. 25, 2012):
Deformation quantization is a procedure which assigns a formal deformation of the associative algebra of functions on a variety to a Poisson structure on this variety. Such a procedure can be obtained from Kontsevich's formality quasi-isomorphism and, it is known that, there are many homotopy inequivalent formality quasi-isomorphisms. I propose a framework in which all homotopy classes of formality quasi-isomorphisms can be described. More precisely, I will show that homotopy classes of "stable" formality quasi-isomorphisms form a torsor for the group exp(H°(GC)), where GC denotes the full graph complex. The group exp(H°(GC)) is isomorphic to the Grothendieck-Teichmueller group which is, in turn, related to moduli of curves and to theory of motives.
Shift Equivalence and Z[t]-modules (Chuck Weibel, February 8, 2012):
Shift equivalence is an equivalence relation on nxn matrices (say over Z). Such a matrix T may be regarded as defining a Z[t]-module structure on a free abelian group, and shift equivalence translates into the assertion that the modules become isomorphic over Z[t,1/t]. This talk is a description of a weaker equivalence relation related to class groups of number fields.
K-theory of minuscule varieties (Anders Buch, February 22, 2012):
Thomas and Yong have conjectured a Littlewood-Richardson rule for the K-theory of any minuscule homogeneous space, based on counting tableaux that rectify to a certain superstandard tableau. This conjecture has been proved for Grassmannians of type A and maximal orthogonal Grassmannians, but it fails for the Freudenthal variety of type E7. I will speak about a fix that replaces the superstandard tableaux with minimal increasing tableaux. These tableaux have several other combinatorial advantages, for example they make it possible to recognize which tableaux should be counted without rectifying them. This is joint work with Matthew Samuel.
From algebra to category theory: a first approach to fusion categories
(Julia Plavnik, February 29, 2012):
A good way of thinking about category theory is that it is a refinement (or "categorification") of ordinary algebra. In other words, there is a dictionary between these two subjects, such that usual algebraic structures are recovered from the corresponding categorical structures by passing to the set of isomorphism classes of objects (Etingof, Gelaki, Nikshych and Ostrik).
The idea of this talk is to introduce and motivate the notion of fusion category. We shall give some basic definitions and examples that help us understand this structure. We shall introduce the ideas of gradings, solvability and nilpotency for fusion categories and we shall connect it to the corresponding ideas for groups. We shall also discuss some results concerning to the structure of fusion categories with restrictions on the Frobenius-Perron dimensions of its simple objects.
Invariants of Matrix Factorizations (Mark Walker, March 21, 2012):
Given, for example, a polynomial $f(x_1,...,x_n)$ with complex coefficients, a matrix factorization of f is a pair of r x r matrices of polynomials (A, B) satisfying $AB = f I_r = BA$. Introduced over 30 years ago by Eisenbud in the context of studying projective resolutions of modules over hyper-surfaces, there has been a revival of interest in matrix factorizations lately, as connections with mathematical physics and knot theory have emerged. I will discuss some recent progress in understanding certain fundamental invariants of matrix factorizations.
Combinatorial aspects of toric mirror symmetry (Lev Borisov, March 28, 2012):
I will review the combinatorial aspects of toric mirror symmetry. In particular, I will focus on the new phenomena one encounters when dealing with complete intersections as opposed to hypersurfaces.
Noncommutative Laurent Phenomena (Vladimir Retakh, April 11, 21012:
I will discuss the Laurent phenomenon for noncommuting variables. A good example is the cluster conjecture of Kontsevich. I will present a proof of the conjecture, recently obtained by A. Berenstein and me.
Symmetric subgroup orbit closures on flag varieties as universal degeneracy loci
(Ben Wyser, April 18, 2012:
Suppose that G is one of the classical groups SL(n,C), SO(n,C) or Sp(2n,C), and that K is a symmetric subgroup of G --- that is, the fixed points of an involution of G. The group K has finitely many orbits on the flag variety G/B, and the geometry of these orbits and their closures is closely connected to the theory of Harish-Chandra modules for a certain real form of G. Their representation-theoretic interest aside, such orbit closures are, in a sense, generalizations of Schubert varieties, and most questions one has about Schubert varieties can equally well be posed about these more general objects.
We will describe a method for computing formulas for the S-equivariant fundamental classes of such orbit closures, where S is a maximal torus of K. The main idea is to use equivariant localization and the self-intersection formula to "guess" formulas for the classes of closed orbits, and then to compute formulas for the remaining orbit closures using divided difference operators. In type A, we will also describe how these formulas can be interpreted as Chern class formulas for classes of certain types of degeneracy loci involving a vector bundle over a scheme which is equipped with a complete flag of subbundles and an additional structure determined by K. This is analogous to (and motivated by) work of W. Fulton on Schubert varieties in flag bundles, their role as universal degeneracy loci for maps of flagged vector bundles, and connections between that work and the torus-equivariant cohomology of the flag variety described by W. Graham.
#### Fall 2011
The Schottky Problem and genus 5 curves (Charles Siegel, Sept. 14, 2011):
The relationship between algebraic curves and abelian varieties has a long and classical history. One of the most fundamental open problems is determining when an abelian variety is the Jacobian of some curve. We will discuss some of the history of the problem, as well as new results in the case of genus 5 curves.
Congruence subgrous of lattices in rank 2 Kac-Moody groups over finite fields
(Abid Ali, Sept. 28, 2011):
Let G be a complete Kac-Moody group of rank 2 over a finite field, and let B— denote the non-uniform lattice subgroup generated by the "diagonal subgroup" and all negative real root groups. We define and construct congruence subgroups of B—. This is joint work with Lisa Carbone.
Cluster algebras and combinatorics of rigid objects in 2 Calabi-Yau categories
(Raika Dehy, October 4 and 18, 2011):
This talk is motivated by the representation-theoretic approach to Fomin-Zelevinsky's cluster algebras. In this approach a central role is played by certain 2-Calabi-Yau categories and by combinatorial invariants associated with their rigid objects (objects with no self-extensions).
I shall recall the definition of cluster algebras and how to construct the cluster categories associated with them (the latter are 2-Calabi-Yau categories). Then I will introduce the combinatorial invariant that will help prove part of the conjectures on g-vectors associated to cluster variables.
Giambelli formulas for orthogonal Grassmannians (Anders Buch, October 26, 2011):
Let X be an orthogonal Grassmannian, defined as the set of all isotropic subspaces of a given dimension in a complex vector space equipped with an orthogonal bilinear form. The cohomology ring H^*(X) has a basis consisting of Schubert classes; products of these classes have applications in enumerative geometry and are the main objects of study in Schubert calculus. The cohomology ring H^*(X) can also be understood in terms of generators and relations, where the generators are certain special Schubert classes. A Giambelli formula means an expression of an arbitrary Schubert class as a polynomial in special Schubert classes. The Schubert classes of an ordinary Grassmann variety can be expressed as determinants of matrices of special Schubert classes, and the Schubert classes of a maximal orthogonal Grassmannian can be written as Pfaffians. I will speak about new Giambelli formulas for submaximal orthogonal Grassmannians that is expressed in terms of raising operators and interpolate between the above cases. This is joint work with A. Kresch and H. Tamvakis.
On Fourier-Mukai type functors (Alice Rizzardo, November 2, 2011):
Orlov showed in 1997 that all exact, fully faithful functors between the bounded derived categories of two smooth projective varieties are isomorphic to a Fourier-Mukai transform. In this talk we will discuss a class of functors that are not full or faithful and still satisfy the above result.
Comparison of Dualizing Complexes (Changlong Zhong, November 9, 2011):
In this talk I will introduce four dualizing complexes defined by M. Spiess, T. Moser, S. Bloch (duality proved by T. Geisser) and K. Sato, and compare them in the derived category. We show that Bloch's complex is quasi-isomorphic with all three, in the situation when they are properly defined (and assuming some well-known conjectures).
Homotopical Methods in Algebraic Geometry (Pablo Pelaez, November 30, 2011):
Algebraic Topology and Algebraic Geometry throughout the years have shared common methods and enriched each other. The work of Morel-Voevodsky gives a natural categorical framework to import some standard methods from homotopy theory into algebraic geometry. The aim of this talk is to describe some concrete examples.
#### Spring 2011
Syzygies of binomial ideals and toric Eisenbud-Goto conjecture (Lev Borisov, April 6, 2011):
Let p1,...,pk be a collection of points with integer coordinates. Denote their convex hull by Δ. For every integer n consider the subset inside the multiple nΔ which consists of lattice points that can be written as sums of n of the pi. Typically, some lattice points of nΔ will be missing from this set. The toric Eisenbud-Goto conjecture gives a certain measure of control over the sets of missing points. I will give an elementary introduction to the conjecture, which is still open for the case of six points on the plane.
Rational points, zero cycles of degree one, and A^1-homotopy theory (Christian Haesemeyer, Feb. 16, 2011):
A system of polynomial equations over a field F may have solutions in a collection of finite field extensions of relatively prime degree, but not have any solution in F. We will describe some examples and results known about this phenomenon, and then talk about what A^1-homotopy theory might contribute to understanding it.
#### Fall 2010
Quantum differential operators (Uma Iyer, Sept. 20, 2010):
In the late 1990s, Lunts and Rosenberg gave a definition of quantum differential operators on graded algebras which allow us to view the action of quantum groups on graded algebras as quantum differential operators. We present the algebras of quantum differential operators on certain graded algebras.
Monoids and algebraic geometry (Chuck Weibel, Sept. 27, 2010):
The prime ideals in an abelian monoid are the points of a topological space, and we can glue them together to get finite models of toric varieties and other objects in algebraic geometry. In this introductory talk, we give basic properties of these monoidal schemes, such as normalization, proper maps and resolution of singularities.
Dimensions of fixed point spaces of elements in linear groups (Bob Guralnick, Oct. 4, 2010)
Sizes of fixed point spaces for elements in linear groups have a long history. These types of results include the classification of psuedoreflection groups and Frobenius complements. I will talk about recent joint works with Maroti and Malle which answer two conjectures from Peter Neumann's 1966 thesis. The first conjecture has to do with the average dimension of a fixed space of an element in a finite (irreducible) linear group. The second has to do with the minimal dimension of a fixed space of some element in an irreducible (possibly infinite) linear group.
Etale cohomology operations (Chuck Weibel, Oct. 25, 2010)
In topology, the Steenrod algebra describes all natural transformations between cohomology groups H*(-,Z/p). Modifying this construction, Epstein constructed certain operations Pi on etale cohomology with coefficients Z/p, used by Raynaud in her construction of universal projective modules. We modify his construction to allow twisted coefficients such as μp, and give a complete list of all etale cohomology operations in this context.
This is joint work with Bert Guillou.
Pieri rules for the K-theory of cominuscule Grassmannians (Anders Buch, Nov. 8, 2010)
The K-theoretic Schubert structure constants of a homogeneous space G/P are known to have signs that alternate with codimension by a result of Brion. For Grassmannians of type A, these constants are computed by a generalization of the classical Littlewood-Richardson rule that counts set-valued tableaux. The K-theory ring of any Grassmann variety is generated by special Schubert classes that correspond to partitions with a single row.
I will present positive combinatorial formulas for the structure constants in products involving special Schubert classes on any cominuscule Grassmannian. Together with a result of Clifford, Thomas, and Yong, this proves a K-theoretic Littlewood-Richardson rule for maximal orthogonal Grassmannians. This is joint work with Vijay Ravikumar.
The Lie product in the continuous Lie dual of the Witt Algebra (Earl Taft, Nov. 29, 2010):
Let k be a field of characteristic zero. The simple Lie algebra W1=Der k[x], the one-sided Witt algebra, has a basis ei=x(i+1)d/dx for i at least -1. The wedges of e0 and ei satisfy the classical Yang-Baxter equation, giving W1 the structure of a coboundary triangular Lie bialgebra. Its continuous Lie dual is also a Lie bialgebra, and has been identied with the space of k-linearly recursive sequences by W. Nichols. Let f=(fn) and g=(gn) be linearly recursive sequences in the continuous linear dual. For each n, the n-th coordinate of [f,g] has been described in terms of the coordinates of f and of g, but it was an open problem to give a recursive relation satised by [f,g] in terms of recursive relations satisfied by f and by g. We give such a relation here. Analogousresults hold for the two-sided Witt algebra Der k[x,x-1].
This is joint work with Zhifeng Hao.
Motivic cohomology operations (Chuck Weibel, Dec. 6, 2010)
Voevodsky gave a description of all stable operations in motivic cohomology in 2009. However, many other unstable operations have come to light in the last decade. We determine the unstable operations. This is joint work with Bert Guillou.
Algebraic Structures from Operads (Ralph Kaufmann, Dec. 13, 2010):
There are certain classic algebraic structures like Gerstenhaber's bracket which have their origin in operads. We discuss several generalizations of these structures - notably Lie brackets, BV operators and master equations. We show how these appear naturally in operadic settings. Our general theory gives a unified framework for a diverse set of geometric and algebraic examples.
#### Spring 2010
Periodicity of hermitian K-groups (Max Karoubi, Feb. 1, 2010):
This is joint work with Jon Berrick and Paul Arne Ostvaer.
It has been known for a few years, essentially by the work of Voevodsky and Rost, that the algebraic K-theory of a commutative ring A with finite coefficients is periodic above the etale cohomological dimension of A. In this lecture, we show that such a ring A has also a periodic hermitian K-theory in the same range.
This essentially means that theorems about the general (infinite) linear group, such as the one proved by Rost and Voevodsky, imply similar ones for the orthogonal and symplectic groups.
Applications of stable bundles to Witt groups and Brauer groups (Ray Hoobler, Mar. 1, 2010):
Basic properties of stable bundles on a projective, smooth variety X will be outlined. These properties make maps between stable bundles quite rigid so that such bundles behave almost like elements in a basis of a vector space over a field. The picture is particularly clear for projective, smooth varieties over a finite field. This will be applied to determine generators of the Witt group of X and to show that the stable Brauer group of X is the same as the Brauer group of X.
Familiarity with the definitions of Witt groups and Brauer groups of fields will be helpful but not essential.
Drinfeld twists and finite groups (Christian Kassel, Mar. 8, 2010):
Drinfeld twists were introduced by Drinfeld in his work on quasi-Hopf algebras. In joint work with Pierre Guillot (arXiv:0903.2807, published in IRMN), after observing that the invariant Drinfeld twists on a Hopf algebra form a group, we determine this group when the Hopf algebra is the algebra of a finite group G. The answer involves the group of class-preserving outer automorphisms of G as well as all abelian normal subgroups of G of central type.
Hopf algebras and recursive sequences (Earl Taft, Mar. 22, 2010):
Linearly recursive sequences have a bialgebra structure. Polynomially recursive(or D-finite) sequences have a topological bialgebra structure. If such a sequence is of a combinatorial nature, a formula for its coproduct can often be interpreted as a combinatorial identity. We illustrate this for the sequences whose n-th term is ((ni)(n!)) for a fixed non-negative i, where (ni) is the binomial coefficient. The resulting combinatorial identity is of an iterated Vandermonde type.
Tilting 1 (Chuck Weibel, March 29, 2010):
This is an overview of the notion of tilting, from Gelfand-Ponomarev to the 1990s. Given a ring A, an A-module T is tilting if it has finite projective dimension, Exti(T,T)=0 for i>0, and there is a resolution 0 → A → T0 → ... → Tn →0 with the Ti summands of sums of copies of T.
Then Hom(T,-) and T⊗B- determine an equivalence between Db(A) and Db(B), where B=End(T). In the associated torsion theory, the torsion modules are the quotients of direct sums of copies of T.
Tilting 3 (Robert Wilson, April 26, 2010):
The modern notion of a tilting module was presented in the seminar talk Tilting 1, and is due to Cline-Parshall-Scott. Any left derived functor inducing an equivalence between A-modules and B-modules arises from a tilting module, as RHom_A(T,-).
Module Structures on the Ring of Hurwitz Series Bill Keigher, May 3, 2010):
Let k be a field of characteristic p>0. We consider monic linear homogeneous differential equations (LHDE) over the ring of Hurwitz series Hk of k. We obtain explicit recursive expressions for solutions of such equations and show that Hk admits a full a set of solutions as well. We then consider the notion of intertwining of Hurwitz series to reduce the study of solutions of an nth order equation to a system of n first order equations in a particularly simple form. For every LDHE over Hk we will associate a module (over a suitable quasifield extension of k), which is closed under the shift derivation of Hk and discuss the structure of the group of module automorphisms that commutes with the shift derivation.
#### Fall 2009
Differential Central Simple Algebras and Picard-Vessiot representations (Lourdes Juan, Oct. 5, 2009):
A differential field is a field K with a derivation, that is, an additive map D:K → K satisfying D(fg)=D(f)g+fD(g) for f,g in K. The field of constants C of K are the zeros of D. A differential central simple algebra (DCSA) over K is a pair (A,\mathcal D) where A is a central simple algebra and $\mathcal D$ is a derivation of A extending the derivation D of its center. Any DCSA, and in particular a matrix differential algebra over K, can be trivialized by a Picard-Vessiot (differential Galois) extension E of K. In the matrix algebra case, there is a correspondence between K-algebras trivialized by E and representations of the differential Galois group of E over K in PGLn(C) that can be interpreted as cocycles equivalent up to coboundaries. I will start with a brief introduction to differential Galois theory.
Derangements in Finite and Algebraic Groups (Bob Guralnick, Oct. 12, 2009):
A permutation on a set is called a derangement if it has no fixed points. The study of the proportion of derangements in finite transitive groups has a long history and the problem has many applications. We will discuss this as well as the analogous problem for algebraic and show the connection between the two. In particular, we will discuss recent results (joint with Fulman) about conjugacy classes in finite Chevalley groups and the solution of a conjecture made independently by Aner Shalev and Nigel Boston.
Mathematics arising from a new look at the Dedekind-Frobenius group matrix and group determinant (Ken Johnson, Oct. 19, 2009):
Frobenius invented group character theory in order to solve the problem of the factorization of the group determinant. His papers are hard to understand and when the modern methods for group representation theory were introduced his initial work was largely forgotten. To each representation of a (finite) group there is associated a polynomial which is a factor of the group determinant, and Frobenius introduced "k-characters" to describe this polynomial. Professor Gelfand has commented that perhaps physicists might benefit from looking at these polynomials. Among other places these k-characters have occurred in work of Buchstaber and Rees and also are related to work of Wiles and Taylor on "pseudocharacters" of finite dimensional representations of infinite groups.
I will describe the early work from an elementary point of view and give an account of some of the new ideas coming from it, and also indicate some of the connections with probablity.
Induced definable structure on cyclic subgroups of the free group
(Chloe Perin, Nov. 2, 2009):
Let C be a cyclic subgroup of a finitely generated free group F. We show that the intersection of a definable set D in F^n with C^n is in the Boolean algebra of cosets of subgroups of C^n. In other words, the definable structure induced by the embedding of C in F is no richer than the definable structure on C. We make extensive use of Sela's geometric techniques for studying the first-order theory of the free group, in particular of his construction of "formal solutions" to an equation.
The classfication problem for finite rank dimension groups (Paul Ellis, Nov. 9, 2009):
An unperforated partially ordered abelian group A is a dimension group if A satises the Riesz interpolation property (given a,a' ≤b,b' there is a c with a,a' ≤ c ≤b,b'). These are related to "Bratteli diagrams". Paul will discuss the difficulty of classifying them when the rank is at least 3, and show that the problem for a given rank cannot be reduced to the classification problem for a smaller rank.
Picard-Vessiot Theory (Ravi Srinivasan, Nov.16, 2009):
Let F be a characteristic zero differential field with an algebraically closed field of constants C. I will describe the construction of a Picard-Vessiot Extension (PVE) for a linear homogeneous differential equation over F. The group of differential automorphisms of a PVE fixing F is called the differential Galois group; there is a Galois correspondence between its algebraic subgroups and intermediate differential subfields. Examples of PVEs for F=C(x) with the usual derivation will be discussed, and we will also compute the differential Galois group for our examples.
#### Spring 2009
Stability conditions for triangulated categories (Chuck Weibel, Feb. 2, 2009):
This is an introductory survey talk.
There is a complex topological manifold, called the Stability Space, associated to any triangulated category D. It was conceived by Mike Douglass as an aspect of string theory, and made mathematical by Tom Bridgeland. Subspaces correspond to t-structures, and the stability space of the projective line is the affine complex plane.
Partitioned Differential Quasifields (Leon Pritchard, Feb. 23, 2009):
A differential quasifield is a natural generalization of a differential field in characteristic p>0. Elementary properties of differential quasifields are considered, and a generalized version of the theorem on the connection between linear independence over constants and the Wronskian is presented.
The Nekrasov Conjecture for Toric Surfaces (Elizabeth Gasparim, March 30, 2009):
The Nekrasov conjecture predicts a relation between the partition function for N=2 supersymmetric Yang-Mills theory and the Seiberg-Witten prepotential. For instantons on ℝ4, the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov, Nakajima-Yoshioka, and Braverman-Etingof. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces.
Differential Quasifields (Bill Keigher, April 13, 2009):
In a recent seminar (2/23), Leon Pritchard talked about partitioned differential quasifields.
Morphisms of cohomological field theories and behavior of Gromov-Witten invariants under quotients
(Chris Woodward, April 20, 2009):
I will talk about a "quantum non-abelian localization" conjecture that relates Gromov-Witten invariants of GIT quotients with equivariant Gromov-Witten invariants of the total space. Some special cases are proved. A key notion in the conjecture is the notion of morphism of cohomological field theories, which "complexifies" the notion of A-infinity morphism.
higher order Hochschild (co)homology (Gregory Ginot, April 27, 2009):
We will explain how one can define Hochschild (co)chain complex associated in a functorial way to any space X, CDG algebra A and A-module M. We will give several examples and applications to Adams operations and (if time permits) Brane topology.
#### Fall 2008
Morita Equivalence Revisited (Paul Baum, Sept. 5, 2008):
Notation: k denotes a unital algebra over the complex numbers which is commutative, finitely generated, and nilpotent-free, i.e., k is the coordinate algebra of a complex affine variety. A k-algebra is an algebra A over the complex numbers which is a k-module such that the algebra structure and the k-module structure are compatible in the evident way. Note that A is not required to be commutative. Prim(A) denotes the set of primitive ideals in A. Prim(A) is topologized by the Jacobson topology.
This talk studies an equivalence relation between k-algebras which is a weakening of Morita equivalence. If A and B are equivalent in the new equivalence relation, then A and B have isomorphic periodic cyclic homology, and Prim(A) is in bijection with Prim(B). However, the bijection between Prim(A) and Prim(B) might not be a homeomorphism. Thus the new equivalence relation permits a tearing apart of strata in the primitive ideal spaces which is not allowed by Morita equvalence. An application to the representation theory of p-adic groups will be briefly indicated. This talk is intended for non-specialists. All the basic definitions will be carefully stated.
The above is joint work with A.M.Aubert and R.J.Plymen.
Formality theorems for Hochschild (co)chains and their applications (Vasily Dolgushev, Sept. 18, 2008):
I will start my talk with a review of the algebraic operations on the pair Hochschild cochain complex and Hochschild chain complex of an associative algebra. Then I will speak about the formality theorems for these complexes. Finally I will discuss applications of these formality theorems to deformation quantization, computation of Hochschild (co)homology and the Kashiwara-Vergne conjecture.
Rationality and integrality in dynamical systems (Mike Zieve, Sept. 22, 2008):
I will present various results about the arithmetic of dynamical systems given by iterating a polynomial mapping over a ring. Sample topics include: describing the minimal N for which the backward orbit of a point under a given polynomial over a number field K contains infinitely many points of degree N over K; and determining the possible lengths of periodic and preperiodic forward orbits of a point under a polynomial mapping of a ring. I will also discuss connections with torsion in abelian varieties, Sen's theorem (Grothendieck's H^1 conjecture), and the Nottingham group.
Combinatorial identities and Hopf algebras (Earl Taft, October 20, 2008):
R. G. Larson and E. J. Taft showed that the space of linearly recursive sequences is a bialgebra. A coproduct formula for such a sequence can be interpreted as a quadratic identity on the coordinates of the sequence. This was extended by C. A. Futia, E. F. Mueller and E. J. Taft[CMT] to D-finite sequences. This means that from some point on, each coordinate is a linear combination of previous coordinates with variable(polynomial) coefficients. These D-finite sequences form a topological bialgebra, i.e., the coproduct is an infinite sum of tensor products of such sequences. Such a coproduct formula can still be interpreted as a quadratic identity on the coordinates, often of a combinatorial nature. In [FMT], we obtained such formulae and identities for the sequences (n!) and (n(n!)). Here we extend this to the sequences whose n-th term is ((n/k)(n!)) for each k=2, 3, 4,.... Here (n/k) is the binomial coefficient.
Infinite towers of cocompact lattices in Kac-Moody groups (Leigh Cobbs, November 3, 2008):
Let G be a locally compact Kac-Moody group of affine or hyperbolic type over a finite field Fq; G admits an action on its Tits building X. In the setting rank(G)=2, X is a locally finite, homogeneous tree. We can then use the combinatorial tools of Bass-Serre theory, namely graphs of groups, to construct discrete subgroups of G. We show that if q=2 then G contains a cocompact lattice Γ whose quotient Γ\X equals G\X, a simplex. We then give two distinct constructions of infinite towers
... Γ3 < Γ2 < Γ1 < Γ
of non-conjugate cocompact lattices in G. We give the graph of groups structure of these and other cocompact lattices, and discuss extensions of these infinite towers to rank-3 Kac-Moody groups using complexes of groups.
K-theory of some algebras associated to quivers (Guillermo Cortiñas, November 14, 2008):
Given a quiver Q and a field k, it is possible to associate several k-algebras. Best known among them is the path algebra, PQ. Localizing PQ one obtains a new algebra, the Leavitt algebra LQ. This algebra is equipped with an involution. If k is the field of complex numbers, LQ may be view as an algebra of operators in Hilbert space; its completion in the operator norm gives a C*-algebra, the Cuntz-Krieger algebra of the quiver. The topological K-theory of the Cuntz-Krieger algebra was computed in a now classical paper of Cuntz. In the talk we will discuss recent joint results with Pere Ara and Miquel Brustenga concerning the algebraic K-theory of LQ and its relation with the topological K-theory of the Cuntz-Krieger algebra.
Reduced K-theory of Azumaya algebras (Roozbeh Hazrat, December 1, 2008):
The theory of Azumaya algebras developed parallel to the theory of central simple algebras. However the latter are algebras over fields whereas the former are algebras over rings. One wonders how the K-theory of these objects compare to each other. We look at higher K-theory and reduced K-theory of these objects. We ask nice questions!
#### Spring 2008
W-algebras, quantum groups and combinatorial identities (Antun Milas, Feb. 5, 2008):
I will discuss a conjectural relationship between certain quantum W-algebras (vertex algebras) and finite-dimensional quantum groups associated to $sl_2$ (Hopf algebras). In the process we shall encounter interesting multisum identities.
In this talk, we consider the monodromy homomorphism for the complex Henon map, a 2-dimensional analog of the quadratic map. We need the shift space of bi-infinite sequences in this case, and the automorphism group of this space is much more complicated than that of the one-sided shift space. We propose a computer-assisted method to compute the monodromy homomorphism and show that automorphisms of the shift space can be used to determine the dynamics of the real Henon map.
Automorphism groups of curves (Mike Zieve, Feb. 25, 2008):
Hurwitz proved that a complex curve of genus g>1 has at most 84(g-1) automorphisms. In case equality holds, the automorphism group has a quite special structure. However, in a qualitative sense, all finite groups G behave the same way: the least g>1 for which G acts on a genus-g curve is on the order of (#G)*d(G), where d(G) is the minimal number of generators of G. I will present joint work with Bob Guralnick on the analogous question in positive characteristic. In this situation, certain special families of groups behave fundamentally differently from others. If we restrict to G-actions on curves with ordinary Jacobians, we obtain a precise description of the exceptional groups and curves.
Model categories versus derived categories (Chuck Weibel, March 3, 2008):
Quillen invented the notion of a model category in order to do homotopical algebra. We will consider these structures on the categories of R-modules, presheaves and sheaves, and show how localization works.
Rational points on homogeneous spaces (Parimala, March 10, 2008):
We discuss the following open concerning rational points on homogeneous spaces under connected linear algebraic groups. If a homogeneous space under a connected linear algebraic group has a zero cycle of degree one, does it admit a rational point? We explain the arithmetic case and some recent progress concerning this question for more general fields.
Formal differential representations, Faa di Bruno and the Riordan Group
(Tom Robinson, March 14, 2008):
First I will show explicitly how a calculation in Frenkel-Lepowsky-Meurman's book on vertex operator algebras, which I will in its essentials redo, can be viewed as an application of a formal representation of exponentiated derivations. The outcome of the calculation is Faa di Bruno's formula for the higher derivatives of a composite function. Then building on this result I will show how another application of an easy class of formal differential representation leads to the Riordan Group. No prerequisites necessary.
The boson-fermion correspondence and one-sided quantum groups (Earl Taft, April 7, 2008):
Recent quantizations of the boson-fermion correspondence of classical physics use one half of the relations for the bialgebra of quantum matrices. Using this philosophy, A.Lauve, S. Rodriguez and myself have independently constructed certain one-sided qauntum groups, i.e., there is a left antipode which is not a right antipode. We will explain the connections between these two quantizations.
Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets
(Semyon Alesker, April 21, 2008):
We introduce a class of plurisubharmonic functions on the octonionic plane O² and establish basic results about it. Then we apply these results to produce new examples of continuous valuatons on convex subsets of O²=R^{16}, in particular valuations invariant under the group Spin(9). The constructions use the determinant of octonionic hermitian matrices of size 2.
Witt vectors, Lambda-rings, and absolute algebraic geometry (Jim Borger, April 28, 2008):
I'll give an introduction to Witt vectors and Lambda-rings, and I'll explain how they're two different ways of looking at the same concept. Then I'll discuss how these give a "Lambda-equivariant" algebraic geometry, how it relates to usual algebraic geometry, and why one might care about it.
Subgroups of Algebraic Groups and Finite Groups (Richard Lyons, May 5, 2008):
We will discuss some similarities and differences between the subgroup structures of connected linear algebraic groups and finite groups.
#### Fall 2007
Conformal field theory and Schramm-Loewner evolution (Benjamin Doyon, Sept. 7, 2007):
The scaling limit of two-dimensional statistical models at criticality can be described by two theoretical frameworks: conformal field theory (that is, vertex operator algebras, their modules and representations), and Schramm-Loewner evolution (SLE). The first one has a long history, starting more than 20 years ago with works by both mathematicians and physicists, whereas the second one encompasses recent advances, starting in 2000 with a paper of Schramm until generalisations still under construction. The two frameworks seem quite unrelated in their formulation as well as in their applications. But it is nowadays believed by many that understanding the relation between them will allow us to make important steps in the understanding, both physical and mathematical, of critical regimes of statistical models. I will review the frameworks, advances made in relating them, and the many open problems. This talk will be accessible to non-specialists.
An introduction to open-closed conformal field theory (Liang Kong, Sept. 14, 2007):
Open-closed conformal field theory describes the perturbative open-closed string theory and some critical phenomena in condensed matter physics. It provides a powerful tool to study the still mysterious object called "D-brane", which is important to Kontsevich's homological mirror symmetry program. In this talk, I will outline a mathematical study of open-closed conformal field theory based on the theory of vertex operator algebra. In particular, I will give a tensor-categorical formulation of rational open-closed conformal field theory. I will also briefly discuss what D-branes are in our framework. This talk will be accessible to graduate students who know the definition of category.
Patching subfields of division algebras (Dan Krashen, Nov. 9, 2007):
There has been much work recently in understanding the structure of division algebras whose center is "2-dimensional." For example, in the case that the center is the function field of an algebraic surface, de Jong has shown that every such algebra has a cyclic maximal subfield. In this talk I will describe joint work with Harbater and Hartmann which uses the recent method of "field patching" (related to formal geometry) to understand all possible Galois groups of maximal subfields of division algebras over function fields of certain arithmetic surfaces.
Hopf and Lie algebras for renormalizable quantum field theories (Dirk Kreimer, Dec. 3, 2007):
Physicists have used the combinatorics of renormalization and the renormalization group routinely for a long time. The identification of the underlying algebraic structures in terms of Hopf and Lie algebras is more recent. We explain these algebras and their role in understanding Green functions in quantum field theory.
A new candidate for the nef cone of M0,n (Angela Gibney, Nov. 16, 2007):
There is a well known upper bound $F_{n}$ for the nef cone Nef$(\overline{M}_{0,n})$ of $\overline{M}_{0,n}$. The cone $F_{n}$ is an explicitly defined, polyhedral cone that contains Nef$(\overline{M}_{n})$. The F-conjecture asserts that Nef$(\overline{M}_{n})=F_{g,n}$. In this talk, I will describe a new candidate for the nef cone of $\overline{M}_{0,n}$. This is a polyhedral cone $C_{n}$ that Sean Keel, Diane Maclagan and I have proved is a sub cone of $F_{n}$. We can show that if $F_{n}$ were also contained in $C_{n}$, then it would imply that Nef$(\overline{M}_{0,n})=F_{n}=C_{n}$.
#### Spring 2007
Vertex operator algebras and recurrence relations (Bill Cook, March 30, 2007):
There are many important classes of examples of vertex operator algebras including Heisenberg VOAs, Virasoro VOAs, lattice VOAs, and the VOAs associated with affine Lie algebras.
We will begin with an introduction to the class of VOAs (along with their modules) associated with affine Lie algebras. Then in the latter part of the talk we will discuss an interesting theorem of Haisheng Li. Applying this theorem to our class of examples, we will obtain recurrence relations among the characters of these Vertex Operator Algebras (and VOA modules).
On a certain family of W-algebras (Antun Milas, April 7, 2007):
Rational conformal field theories can be characterized by the property that there are, up to equivalence, finitely many irreducible representations of the vertex operator algebra, and that every representation is completely reducible.
G-equvariant modular categories and Verlinde formula (Vincent Graziano, April 13, 2007):
Many features of a conformal field theory can be captured in the language of categories. Modular tensor categories provide the appropriate framework and we will start by discussing the properites of such a category. We will then introduce the Verlinde algebra associated to such a category, the action of the S-matrix, and the Verlinde formula.
Our goal will be to generalize this setup to the case of theories with additional symmetries, such as a vertex operator algebra with a finite group of symmetries. We discuss the extended Verlinde algebra, the S-matrix, and the 'extended' Verlinde formulas.
Vertex-algebraic structure of certain modules for affine Lie algebras underlying recursions (Corina Calinescu, April 20, 2007):
Many combinatorial identities and recursions have been proved or conjectured via vertex operator constructions of representations of affine Lie algebras.
In this talk we discuss vertex-algebraic structure of the principal subspaces of all the standard A1(1)-modules and we prove suitable presentations for these subspaces. These presentations were used by Capparelli, Lepowsky and Milas for the purpose of obtaining the classical Rogers-Ramanujan and Rogers-Selberg recursions. This is joint work with Jim Lepowsky and Antun Milas.
A Formal Variable Approach to Special Hyperbinomial Sequences (Tom Robinson, April 27, 2007):
In a nearly self-contained and elementary treatment, we develop the formal calculus used in the theory of vertex algebras to describe certain formal changes of variable. In particular, we extend the logarithmic formal Taylor theorem as found in the work of Y.Z. Huang, J. Lepowsky, and L. Zhang. We apply our results to obtain combinatorial identities concerning generalizations of the Stirling numbers and find that our development leads naturally to a combinatorial definition of the exponential Riordan group which was studied by L.W. Shapiro, S. Getu, W.J. Woan, and L.C. Woodson.
#### Fall 2006
Intertwining vertex operators and combinatorial representation theory (Corina Calinescu, Sept. 22, 2006):
In this talk we discuss vertex-algebraic structure of certain substructures, called principal subspaces, of standard modules for affine Lie algebras. We give suitable presentations of these subspaces and we derive Rogers-Ramanujan-type recursions satisfied by the graded dimensions of the principal subspaces. Part of the talk is based on joint work with Jim Lepowsky and Antun Milas. This talk will be introductory.
#### Spring 2006
A smash product construction of nonlocal vertex algebras (Haisheng Li, Feb. 17, 2006):
We first introduce a notion of vertex bialgebra and a notion of module nonlocal vertex algebra for a vertex bialgebra. Then we present a smash product construction of nonlocal vertex algebras.
Projective R[t]-modules and cdh cohomology (Chuck Weibel, March 31, 2006):
Let R be a finitely generated commutative algebra over a field of characteristic zero. Projective R-modules are classified by K0R and projective R[t]-modules are classified by K0R[t]. We prove that the quotient of these groups is a direct sum of R+/R and the cdh cohomology groups Hi(R,Ωi). This is joint work with Haesemeyer and Cortiñas.
#### Fall 2005
Effective Hodge structures (Chuck Weibel, Sept. 23, 2005):
This is an introductory talk about Deligne's notion of Hodge Structures, and the more recent idea of effective Hodge Structures. Complex conjugation on the coordinates of the vector space V=Cn gives an involution, and a pure Hodge structure on V is a decomposition by subspaces Vp,q with Vp,q conjugate to Vq,p. It is effective if these only occur when p,q ≥0.
Extensions of Rings and their Endomorphisms (Art, DuPre, Oct. 7, 2005):
Given rings I,Q we classify all rings R fitting into a short exact sequence 0 → I → R → Q → 0 of rings by means of cohomology classes. In the case of group extensions, it is necessary that the normal subgroup be abelian in order for the cohomology classes to form a group. However, because of the additive nature of the decomposition of a ring into cosets of an ideal, the cohomology classes form a group for arbitrary I. If R ≅ R1⊕R2 is a direct sum of rings, we may associate to any endomorphism f of R a 2x2 matrix
f11 f12 f21 f22
where fij:Rj→Ri are homomorphisms. We generalize this to the case where R is an arbitrary ring extension, determine the functional equations satisfied by the fij, and how such matrices multiply. We extend these results to the case where R carries a locally compact polish topology.
Leavitt path algebras (Gene Abrams, Nov 4, 2005):
Most of the rings one encounters as "basic examples" have what's known as the "Invariant Basis Number" property, namely, for every pair of positive integers m and n, if the free left R-modules RR(m) and RR(n) are isomorphic, then m=n. There are, however, large classes of rings which do not have this property. While at first glance such rings might seem pathological, in fact they arise quite naturally in a number of contexts (e.g. as endomorphism rings of infinite dimensional vector spaces), and possess a significant (perhaps surprising) amount of structure.
A class of left quantum groups: Variation on the theme of SL_q(n) (Earl Taft, Dec 2, 2005):
For each n>1, we construct a left quantum group, which has the quantum special linear group SL_q(n) as homomorphic image. Whereas SL_q(n) is defined by quadratic relations plus the relation of degree n which sets the quantum determinant equal to 1, our left quantum group is defined by n^n relations of degree n, of which n! come from setting various versions of the quantum determinant equal to 1.(Joint work with Aaron Lauve).
#### Spring 2005
Generalizations of Tsen's theorem (Tom Graber, Feb 4, 2005):
Tsen's theorem is a classical result which says roughly that polynomials of low degrees in many variables with coefficients in the field of meromorphic functions on a compact Riemann surface always have solutions. I will describe joint work with Joe Harris, Barry Mazur, and Jason Starr which suggests that this result is best understood in connection with the geometry of rational curves.
More specifically, using k copies of the weight lattices of the Lie algebras A_{1} and A_{2} in the diagonal embedding, we construct relative twisted vertex operators equivalent to Z-algebra operators that determine the structure of standard A_{1}(1) and A_{2}(2)-modules. Applying the properties of the delta function, the corresponding generalized commutator and anti-commutator relations appear as residues of the Jacobi identity for relative twisted vertex operators.
#### Fall 2004
Conformal field algebras and tensor categories (Liang Kong, Oct. 1, 2004):
Conformal field theories have both holomorphic and antiholomorphic parts, which are sometimes called chiral conformal field theories. In genus-zero and genus-one cases, chiral conformal field theories have been constructed from a general class of vertex operator algebras and their representations, and in general these theories have monodromies. To construct conformal field theories without monodromies, we need to put chiral theories together to cancel the monodromies. In genus-zero, such conformal field theories are described by what we call "conformal field algebras." In this talk, we will discussion the notion of conformal field algebra, their relation with algebras in tensor categories, and a construction of such algebras.
Cherednik and Hecke algebras of orbifolds (Pavel Etingof, Oct. 15, 2004):
The rational Cherednik algebra is attached to a finite group G acting on a vector space V, i.e., to the orbifold V/G. I will explain how the theory of Cherednik algebras can be extended to an arbitrary orbifold (algebraic or complex analytic), and how to define the KZ functor for such algebras.
This leads to a construction of a flat deformation of the group algebra of the orbifold fundamental group of any complex orbifold Y whose universal cover has a finite second homotopy group. These deformations include all known Hecke algebras (usual, complex reflection, affine, double affine). The talk is based on my paper math.QA/0406499.
This talk will define the notion of operad, show how operads geometrically motivate associative algebras and coassociative coalgebras, and then analogously use operads to motivate vertex operator algebras and vertex operator coalgebras. The talk will conclude with examples of vertex operator coalgebras that are constructed via vertex operator algebras with appropriate bilinear forms.
In classical geometry there have been results about the cohomology of manifolds with Lie group actions, and the relation between the topological cohomology of the group and its Lie algebra cohomology, for about 50 years. I shall give noncommutative analogues of some of these results, in terms of Hopf algebras acting on algebras with differential structure. I shall begin with a brief review of noncommutative differential geometry and de-Rham cohomology.
In this informal talk I'll give the definition of quasi-Hopf algebras, some examples (and some conjectural examples) of twisting, including the Knizhnik-Zamolodchikov (KZ) equation.
Spring 2004
Toric Hilbert schemes (Diane Maclagan, Jan 26, 2004):
Toric Hilbert schemes have broad connections to other areas of mathematics, including optimization, geometric combinatorics, algebraic geometry, and representations of finite groups and quivers. They parameterize all ideals in a a polynomial ring with the simplest possible multigraded Hilbert function. I will introduce these objects, and discuss some of the applications.
In this talk, however, I will present conformal algebras as a self-contained theory and will mostly concentrate on their representations, in particular, on the conformal analogs of matrix algebras. These objects are related to certain subalgebras of the Weyl algebra and the algebra gl{\infty}.
Capture the flag: towards a universal noncommutative flag variety (Aaron Lauve, April 2, 2004):
The standard way to build flag algebras from a set of flags is to use the determinant to coordinatize the latter (then the former is just the polynomial algebra in the coordinate functions for these coordinates). There is a perfectly reasonable notion of noncommutative flags, but what are we to do about the lack of a determinant in noncommutative settings? In this talk I will: (1) use the Gelfand-Retakh quasideterminant to build a generic noncommutative Grassmannian algebra, (2) specialize this generic Grassmannian to recover the well-known Taft-Towber quantum Grassmannian, (3) explain what steps are left before we can build a generic flag algebra. This talk should be accessible to first and second year graduate students.
2003
Constructing tensor categories from from finite groups (Edwin Beggs, Sept 12, 2003):
First we consider the algebra structure induced on a set of coset representatives of a subgroup of a finite group. Associated to it is a non-trivial tensor category, which we construct. There is an algebra in this category whose representations consist of the entire category.
If we apply a double construction to this, we arrive at a braided category and a braided Hopf algebra. It turns out that this is a ribbon category, and (at least sometimes) a monoidal category.
Let G be a semi-simple, connected and simply connected algebraic group, defined over an algebraically closed field of characteristic 0. Fix a maximal torus T, a Borel subgroup B containing T, and a maximal parabolic subgroup P containing B. Fix also the root system of G relative to T, the positive/simple roots relative to B, etc. Let W = W(G) be the Weyl group of G.
Let V be a fundamental representation of G corresponding to P. The first main aim of SMT is to construct a "nice" basis for each of the T-weight subspaces in V, having some "compatibility" properties with that of the extremal weight spaces and satisfying some "geometric" properties, etc.
Let M be an irreducible representation of G and express it as a subquotient of the appropriate tensor product of suitable fundamental representations of G. The second main aim of SMT is to construct bases for the weight spaces of M in terms of those constructed for the fundamental representations.
• The category of A-modules and the underlying forgetful functor to vector spaces determine the algebra A up to isomorphism. A forgetful functor from A-modules to B-modules defines a unique algebra homomorphism f: B -> A.
We will study certain A-modules that are also C-comodules, called entwined (A,C)-modules. This involves a certain double action-coaction t , an entwined structure, between A and C. Similar techniques as above allow to reconstruct A and C plus the entwined structure t from the category of entwined (A,C)-modules and the forgetful functor to vector spaces.
There have been some (failed) attempts to find the correct definition of homomorphisms between entwined algebra-coalgebra structures (A,C,t) -> (A',C',t'). We will give the correct definition using the approach given above, which lead to certain measurings and comeasuring. We will show how these techniques can be used to get other interesting results about entwined structures.
The pure mapping class group of a Cantor set (Frederick Gardiner, May 2, 2003):
This is joint work with Nikola Lakic.
It is shown that the pure mapping class group of the complement in the complex plane of the standard middle-thirds Cantor set acts discretely on the Teichmuller space of Cantor sets of bounded geometric type.
2002
• "Modular group action in the center of the small quantum group" (Anna Lachowska, Dec.6, 2002):
The small quantum group ul was introduced by Lusztig as a certain finite dimensional Hopf algebra associated to a semisimple complex Lie algebra g and a primitive (complex) l-th root of unity q. According to Lyubashenko and Majid, in many cases ul admits a bijective action of two operators obeying the modular identities. This action stabilizes the center Z of ul and can be used to study its structure.
We will consider the smallest modular-invariant subspace in Z which contains the obvious central elements (the Harish-Chandra center). This subspace is a subalgebra in Z roughly twice bigger than the Harish-Chandra center, and it coincides with the whole center Z in case g = sl_2. It also contains a nilpotent modular-invariant ideal, which admits an interesting representation-theoretical interpretation similar to the Verlinde algebra of the fusion category.
• "Support Varieties for Finite Group Schemes" (Julia Pevtsova, November 22, 2002):
This is joint work with Eric Friedlander.
To each finite-dimensional module M of a finite group scheme G (i.e. finite dimensional cocommutative Hopf algebra over a field of positive characteristic) we associate a geometric construction of a representation-theoretic support space'' of M. Our construction specializes to two seemingly different constructions in modular representation theory: Carlson's rank varieties for elementary abelian p-groups and representation-theoretic support varieties of restricted Lie algebras. We further exhibit a natural homeomorphism from the representation-theoretic support space of the trivial module to the projectivization of the spectrum of the cohomology algebra of G. For every finite dimensional G-module M, this homeomorphism restricts to a homeomorphism between the representations-theoretic support space and the projectivization of the cohomological support variety of M.
• "Twisted vertex operator algebra modules and Bernoulli polynomials" (Benjamin Doyon, November 1, 2002):
In the construction of twisted modules for vertex operator algebras, one can define twisted vertex operators by normal-ordered products of more basic'' twisted vertex operators. One also needs to introduce a certain subtle formal operator in the construction; this gives, in particular, a correction term for the action of the zero mode of the Virasoro algebra on a twisted module. The generalization of this term to the case of a central extension of an algebra of differential operators of higher order is a very non-trivial problem from this point of view.
We start from the twisted Jacobi identity and derive various commutativity'' and associativity'' relations. They allow us to define twisted vertex operators from more basic ones without explicit reference to an extra formal operator, and to calculate correction terms more conceptually. Bernoulli polynomials appear when we use cylindrical coordinates'', where, as shown by J. Lepowsky, the algebra simplifies drastically.
This is joint work with J. Lepowsky and A. Milas.
• "Explicit norm one elements for ring actions of finite abelian groups" (Christian Kassel, October 25, 2002):
It is known that the norm map NG for the action of a finite group G on a ring R is surjective if and only if for every elementary abelian subgroup U of G the norm map NU is surjective. Equivalently, there exists an element xG in R satisfying NG(xG) = 1 if and only for every elementary abelian subgroup U there exists an element xU in R such that NU(xU) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for xG in terms of the elements xU. E. Aljadeff and the speaker solved this problem when the group G is abelian.
• "Is there a one-sided quantum group?" (Earl Taft, October 18, 2002):
In the 1980's, J.A. Green, W.D.Nichols and EJT constructed a left Hopf algebra, i.e., a bialgebra with a linear map S satisfying the left antipode condition, but not the right one. It has a freeness feature that places it outside the realm of quantum groups. Recently, S. Rodriguez-Romo and EJT tried to construct a left Hopf algebra in the world of quantum groups. We did not yet succeed, but the effort led to some new quantum groups, modeled partially on quantum GL(2), with the peculiar property that they remain noncommutative when q=1 [ Letters in Mathematical Physics 61 (2002), 41-50.] We are trying to modify our procedures, with the hope of finding a left quantum group. If such a thing exists, it might reflect some lack of symmetry of interest to physicists.
• "Raga Bhimpalasi: The Vaserstein-Suslin Jugalbandhi" (Ravi Rao, Oct. 11, 2002):
The study of unimodular rows, and their orbit spaces, over a commutative ring with 1, lies in the fertile cross-section of ideas from Algebra, Algebraic Topology, Number Theory, and Algebraic Geometry. Witt group structures, Cohomotopy groups, Mennicke symbols, Reciprocity Laws, etc. make their appearance very naturally.
We shall discuss the connection of the study of orbit spaces, via a symbiosis of constructions of L.N. Vaserstein, A. Suslin, and its relation to problems in classical K-theory, and to the program of J.-P. Serre, which interconnected the study of projective R-modules with problems of efficient generation of ideals of R.
• "Conformal field theory and vertex operator algebras" (Matthias Gaberdiel, Sept.27, 2002):
I plan to give an informal introduction into conformal field theory and its relation to vertex operator algebras. Towards the end I shall also discuss the so-called C_2 condition of Zhu and some of its implications.
• "Differential equations, duality and modular invariance" (YZ Huang, Sept.20, 2002):
I will explain a result obtained recently on genus-one conformal field theories. Let V be a vertex operator algebra satisfying the C_2-cofiniteness condition and certain finite reductive properties. Then the q-traces of products of geometrically-modified intertwining operators are shown to satisfy systems of differential equations which can be chosen to be regular at any given possible singular point. Genus-one correlation functions are constructed as the analytic extensions of these q-traces. We prove duality properties for these genus-one correlation functions, including commutativity and associativity. Using the associativity property and the modular invariance for one-point functions, we establish the modular invariance for genus-one correlation functions. I will start with the definition of conformal field theory and will explain briefly the notion of vertex operator algebra.
• "Differential equations and intertwining operators" (YZ Huang, May 3, 2002):
In the conformal field theories associated to affine Lie algebras (the Wess-Zumino-Novikov-Witten models) and to Virasoro algebras (the minimal models), the Knizhnik-Zamolodchikov equations and the Belavin-Polyakov-Zamolodchikov equations, respectively, play a fundamental role. Many important results (for example, the constructions of braided tensor category structures and intertwining operator algebras) for these theories are obtained using these equations.
In this talk, I will explain a recent result which establishes the existence of certain different equations of regular singular points satisfied by products and iterates of intertwining operators for a vertex operator algebra whose modules satisfy a certain finiteness condition. Immediate applications of these equations are a construction of braided tensor categories on the category of modules for the vertex operator algebra and a construction of intertwining operator algebras (or chiral genus-zero conformal field theories) from irreducible modules for the vertex operator algebra.
• "A simple nil ring exists" (Smoktunowicz, April 19, 2002):
Over 40 years ago, a simple radical ring was constructed by Sasiada. It remained an open question whether a simple nil ring exists. (Nil means that every element is nilpotent.) We construct a simple nil ring over any countable field. We will describe this construction, and also mention several open questions such as: Is there a simple nil ring over an uncountable field?
• "Once again about Bethe Ansatz" (Lukyanov, March 29, 2002):
We shall discuss an intriguing relation between roots of the Bethe ansatz equations corresponding to vacuum states of the $XXZ$ spin chain and the spectrum of one-dimensional Scr${\ddot {\rm o}}$dinger operator with homogeneous potential.
2001
• "Congruence subgroups of SL2(Z[1/s]), after Serre" (Weibel, Dec.7, 2001):
The Congruence Subgroup Problem for any matrix group G is to determine if every subgroup of finite index is a congruence subgroup, and if not to describe the obstruction. If G is SL_n(Z[1/s]) and n>2, the Euclidean Algorithm easily yields a positive solution. Less well known is the case of 2x2 matrices.
Following Serre, we will show that the answer is 'no' for Z, and Z[1/p], but 'yes' for Z[1/s] otherwise.
This is intended to be an expository talk. One motivation is that subgroups of finite index in SL_2(Z) play an important role in number theory.
• "Certain noncommutative analogues of vertex algebras" (Haisheng Li, Nov.30, 2001):
We define and study a certain noncommutative analogue of the notion of vertex algebra. We show how to construct such algebras by using a set of compatible weak vertex operators.
• "Vertex operator algebras and conformal field theories III" (YZ Huang, Nov.16, 2001):
I will continue the discussions on modular functors and weakly conformal field theories and on the consequences of the existence of such theories, including the Verlinde formula. I will also briefly explain Segal's idea on how to obtain real conformal field theories from rational weakly conformal field theories. At the end, the existing results and open problems will be discussed.
• "Some biparametric examples of Quantum Groups" (Parashar, Nov. 9, 2001):
I will present simple examples of the standard and nonstandard (or Jordanian) quantum groups as well as their biparametric versions. The scheme is then extended in the wider context of the corresponding coloured' counterparts. Within the framework of the R-matrix approach, I will also discuss some basic algebraic and geometric results from the theory of coloured quantum groups and outline possible physical and mathematical applications.
• "Vertex operator algebras and conformal field theories II" (YZ Huang, Nov.3, 2001):
This talk is a continuation of my talk on October 12. I will explain how to construct genus-zero conformal field theories from vertex operator algebras and why we need modules and intertwining operators when we want to construct maps associated to genus-one surfaces. I will also discuss weakly-conformal field theories and consequences of the existence of such theories, including the Verlinde formula.
• "Elliptic curves and quantum tori" (Soibelman, Oct.26, 2001):
I plan to discuss the program of non-commutative compactifications I suggested two years ago. The main example will be non-commutative degenerations of elliptic curves. I will explain why quantum tori appear on the boundary of the moduli space of elliptic curves. I will discuss the relations to algebraic and symplectic geometry, q-difference equations, etc. The talk will consist largely of conjectures and speculations.
• "Vertex operator algebras and conformal field theories" (YZ Huang, Oct.12, 2001):
Conformal field theories were defined mathematically around 1987 by Kontsevich and Segal in terms of properties of path integrals. A construction of such a theory can be viewed in a certain sense as a construction of certain path integrals. However, up to now, there is still no complete published construction of examples of conformal field theories satisfying this definition.
On the other hand, around 1986, a notion of vertex operator algebra was introduced and studied in connection with the representation theory of infinite-dimensional Lie algebras and the Monster by Borcherds and Frenkel-Lepowsky-Meurman. Since then, the theory of vertex operator algebras has been developed rapidly and has found applications in a number of branches of mathematics. In this series of talks, I will explain a research program to construct conformal field theories in the sense of Kontsevich and Segal. Both the existing results and unsolved problems will be discussed.
• "Effective Representation Theory of Finitely Presented Algebras" (Letzter, Sept.28, 2001):
Let n be a positive integer, and let R be a finitely presented algebra over a field k. Consider the following questions: Does R have an irreducible n-dimensional (over k) representation? How many irreducible n-dimensional representations does R have? Is every n-dimensional representation of R semisimple? In this talk I will discuss algorithmic approaches to answering these questions.
• "Hilbert functions of non-standard bigraded algebras" (Trung, Sept.21, 2001):
The talk concerns bigraded algebras generated by elements of bidegrees (1,0) and (a,1) for different non-negative integers a. We show that the Hilbert function of such a bigraded algebra is equal to a polynomial for certain range. Moreover, the total degree and the degree of this polynomial in the first variable can be expressed in terms of the dimension of certain quotient algebras. These results cover recent results of P. Roberts on the existence of Hilbert polynomial in the case the bigraded algebra is generated by elements of bidegrees (1,0), (0,1), (1,1) which are related to Serre's positivity conjecture on intersection multiplicities. Moreover, these results can be applied to study diagonal subalgebras of bigraded Rees algebras.
• "On a q-analog of the McKay correspondence" (Kirillov, Sept.14, 2001):
It is well known that finite subgroups in SU(2) are classified by simply-laced affine Dynkin diagrams, i.e., affine ADE diagrams. This calssification, known as McKay correspondence, is one of the many related ADE-type classifications (e.g., it is related with ADE classification in singularity theorey). In this talk, we give an analogue of this result for the quantum group U_q sl(2) with q beig a root of unity. This turns out to be related with the classification of modular invariants in Conformal field theory based on integrable representations of affine sl(2).
• "A geometric approach to elliptic cohomology": (Igor Kriz/ Feb.16, 2001)
This talk will discuss the current state of the speaker's project of constructing a geometric model of elliptic cohomology. This proposed construction is related to theta functions and vertex operator algebras. The talk will describe the construction, and the links between its conjectured topological properties and their algebraic and geometric counterparts.
• "Functors with transfer": (C. Weibel/ Feb.2, 2001)
Many of the ideas coming out of Motivic Cohomology yield new ideas and questions when translated into commutative ring theory. We will describe some of these techniques and apply them to questions about projective modules.
Charles Weibel / weibel @ math.rutgers.edu / January 1, 2018
|
2021-05-12 04:13:22
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6150132417678833, "perplexity": 4520.225266798636}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991252.15/warc/CC-MAIN-20210512035557-20210512065557-00189.warc.gz"}
|
http://math.stackexchange.com/questions/18429/integrand-non-negative-everywhere-integral-negative
|
# Integrand non-negative everywhere, integral negative?
Suppose I have an integral:
$F \equiv \int_{a}^{b} f(x) dx$
I know that $f(x)$ is real-valued, finite and non-negative everywhere on the interval $(a, b)$ where $a$, $b$ are real numbers or $\pm \infty$ and $a \leq b$. I know that in the obvious, well-behaved cases this implies $F \geq 0$.
Is there any obscure pathological case where $F < 0$? You may generalize to cases that involve double, triple or higher order integrals. However, for each integral the property that the upper bound is greater than or equal to the lower bound and the bounds are always real numbers or $\pm \infty$ must hold.
-
No. Integrals are monotone. If they weren't, they wouldn't be integrals. – Qiaochu Yuan Jan 21 '11 at 15:21
I think you mean $a\leq b$. Otherwise your integral will be negative unless $f(x)$ is zero almost everywhere. – Christian Blatter Jan 21 '11 at 16:28
@Christian: Exactly right. Fixed. – dsimcha Jan 21 '11 at 16:56
A geometrical way to think about it is that the value of $F$ is the area under the graph of $f$ and above the $x$-axis, and so if $f$ is non-negative this area cannot be negative. – Asaf Karagila Jan 21 '11 at 19:28
@Asaf: I thought of the geometric interpretation. It's just that this assumption is an intermediate step in a proof I'm working on, the proof needs to work even in pathological cases, and in my experience such geometric, intuitive arguments often fall apart in pathological cases. – dsimcha Jan 21 '11 at 20:28
No, there is no pathological case. If $f(x)$ is integrable, and $m\leq f(x)\leq M$ for all $x\in[a,b]$, then $$m(b-a) \leq \int_a^b f(x)\,dx \leq M(b-a).$$ This follows easily from the definition of the integral as a limit of Riemann sums. If $a=x_0\lt x_1\lt\cdots \lt x_n = b$ is a partition of $[a,b]$, $\Delta x_i = x_i - x_{i-1}$, and $x_i^*$ is a point in $[x_{i-1},x_i]$, then $m\leq f(x_i^*)\leq M$ for each $i$, so the Riemann sum satisfies $$m(b-a) = \sum_{i=1}^n m\Delta x_i \leq \sum_{i=1}^n f(x_i^*)\Delta x_i \leq \sum_{i=1}^n M\Delta x_i = M(b-a),$$ so taking limits you get $$m(b-a) \leq \lim_{||P||\to 0}\sum_{i=1}^n f(x_i^*)\Delta x_i \leq M(b-a)$$ or $$m(b-a) \leq \int_a^b f(x)\,dx \leq M(b-a)$$ (using the assumption that $f$ is integrable on $[a,b]$.
In particular, if $f(x)\geq 0$ for all $x\in [a,b]$, then $$0 = 0(b-a) \leq \int_a^b f(x)\,dx = F.$$ Redefining a function at a finite number of points on $[a,b]$ does not change its integrability, nor the value of the integral, so if all you know is that $f(x)\geq 0$ on $(a,b)$, you can always redefine it at $a$ and $b$ so that you get $f(x)\geq 0$ on $[a,b]$ without changing the value of the integral.
(In fact, you only need $f(x)\geq 0$ "almost everywhere on $[a,b]$", meaning at all $x$ except perhaps for a set of measure $0$ that is contained on $[a,b]$).
|
2016-06-26 08:37:10
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9771067500114441, "perplexity": 120.06575533000043}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783395039.24/warc/CC-MAIN-20160624154955-00175-ip-10-164-35-72.ec2.internal.warc.gz"}
|
https://nrich.maths.org/354
|
Mean Geometrically
A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?
Pythagorean Golden Means
Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.
Three Ways
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
Without Calculus
Age 16 to 18Challenge Level
Given that $u> 0$ and $v> 0$ what is the smallest possible value of $1/u + 1/v$ given that $u + v = 5$?
Can you find this value by more than one method (not involving trial and error) without using calculus?
|
2021-09-28 14:16:34
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4964333474636078, "perplexity": 365.6676057586938}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780060803.2/warc/CC-MAIN-20210928122846-20210928152846-00048.warc.gz"}
|
http://wpressutexas.net/oldcoursewiki/index.php?title=Segment_7
|
# Segment 7
## Calculation Problems
1. Prove the result of "mechanical way". \begin{align} \Delta^2 &= < (x-a)^2 > \\ &= < x^2 -2ax + a^2 > \\ \frac{d{\Delta^2}}{da} &= 0 \\ 2 <a - x> &= 0\\ 2(a - <x>) &= 0\\ a &= <x> \end{align}
2. Thought process while solving the problem:
1. It is easier to construct a piecewise function to have a maximum at zero.
2. To ensure that the function is a probability distribution, I decided to choose the function split at 0 to 1 and 1 to $\infin$
3. In order that the $M_4$ not exist, the function should contain the $-5$ power of x in it's integral.
$p(x) = \begin{cases} 0, & \text{if } x \le 0\\ \frac34, & \text{if } 0 \le x \le 1 \\ \frac1{x^5}, & \text{if } 1 \le x \le \infin \\ \end{cases}$
\begin{align} \int_{-\infin}^{\infin}p(x) &= 1\\ <x^3> &= \frac74 \\ <x^4> &= \text{does not exist} \end{align}
3. Positives and Negatives about using Median over Mean.
Positives: Mean, due to it's sensitivities is skews the central tendency because of outliers. While median is a better estimate of the true valies. Negatives: Mean is a lot more sensitive to the data, that is, if the distribution is close to a normal distribution the mean would give the central tendency. But the median is not as efficient in a normal distribution.
## Class Activity
Group Activity with Jin, Rcardenas, Lori
1. What does a joint uniform prior on w and b look like?
Let P(X,Y) be P(w=X,b=Y), it's uniform, so the probability is the same for any X,Y.
$P = \int_0^1\int_0^{1-X} P(X,Y)dXdY = 1$
We got $P = P(X,Y)* (X-\frac{X^2}2)|_0^1 = 1 \rightarrow P(X,Y) = 2$
2.Suppose we know that w=0.4, b = 0.3, and d = 0.3. If we watch N = 10 games, what is the probability that W = 3, B = 5, and D = 2?
$P(3,5,2) = \binom{10}{3} \cdot \binom{7}{5} \cdot w^3 \cdot b^5 \cdot d^3 = 0.0353$
3. For general w, b, d, W, B, D, what is P(W, B, D | w, b, d)?
$P(W,B,D|w,b,d) = \binom{N}{W} \cdot \binom{N-W}{B} \cdot w^W \cdot b^B \cdot d^D$
4. Applying Bayes, what is P(w, b, d | W, B, D)? What is the Bayes denominator?
$P(w,b,d | W, B, D) = \frac{P(W,B,D|w,b,d) \cdot P(w,b,d)}{\int_0^1 \int_0^{1-w} P(W,B,D|w,b,d) \cdot P(w,b,d)}=\frac{w^W \cdot b^B \cdot d^D}{\frac{W! \cdot B! \cdot D!}{(W+B+D+2)!}}$
The denominator is:
$\frac{W! \cdot B! \cdot D!}{(W+B+D+2)!}$
5. Using the data from last Friday, count the outcomes of the first N games and produce a visualization of the joint posterior of the win rates for N = 0, 3, 10, 100, 1000, and 10000.
An interesting observation is the more game we count, the smaller the possible posterior probability space we will get, which means we become more confident on the probability we estimate by counting more games.
N=0
N=3
N=10
N=100
N=1000
N=10000
|
2019-10-15 02:00:15
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999167919158936, "perplexity": 1049.7629999462831}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986655735.13/warc/CC-MAIN-20191015005905-20191015033405-00055.warc.gz"}
|
https://openturns.github.io/openturns/1.15/theory/data_analysis/kolmogorov_test.html
|
# Kolmogorov-Smirnov fitting test¶
This method deals with the modelling of a probability distribution of a random vector .
It seeks to verify the compatibility between a sample of data and a candidate probability distribution previously chosen.
The Kolmogorov-Smirnov Goodness-of-Fit test allows to answer this question in the one dimensional case , and with a continuous distribution.
Let us limit the case to .
Thus we denote . This goodness-of-fit test is based on the maximum distance between the cumulative distribution function of the sample and that of the candidate distribution, denoted F.
This distance may be expressed as follows:
With a sample , the distance is estimated by:
Assume that the sample is drawn from the candidate distribution. By definition, the p-value of the test is the probability:
In the case where the fit is good, the value of is small, which leads to a p-value closer to 1. The candidate distribution will not be rejected if and only if is larger than a given threshold probability. In general, the threshold p-value is chosen to be 0.05:
Based on the p-value,
• if , we reject the candidate distribution,
• otherwise, the candidate distribution is not rejected.
Two situations may occur in practice.
• the parameters of the distribution under test are known,
• the parameters of the distribution under test are estimated from a sample.
If the parameters of the distribution under test are known, algorithms are available to directly compute the distribution of both for N large (asymptotic distribution) or for N small (exact distribution). This is because the distribution of does not depend on the candidate distribution.
If the parameters of the distribution under test are estimated from a sample, the statistic is generally smaller, because the parameters of the distribution have been computed from the sample. In general, the distribution of is not known and depends on the candidate distribution. Therefore, sampling methods can be used in order to estimate the p-value.
The diagram below illustrates the principle of comparison with the empirical cumulative distribution function for an ordered sample ; the candidate distribution considered here is the Exponential distribution with parameters , .
The test deals with the maximum deviation between the empirical distribution and the candidate distribution, it is by nature highly sensitive to presence of local deviations (a candidate distribution may be rejected even if it correctly describes the sample for almost the whole domain of variation).
There is no rule to determine the minimum sample size one needs to use this test; but it is often considered a reasonable approximation when N is of an order of a few dozen. But whatever the value of N, the distance – and similarly the p-value – remains a useful tool for comparing different probability distributions to a sample. The distribution which minimizes – or maximizes the p-value – will be of interest to the analyst.
This method is also referred to in the literature as Kolmogorov’s Test.
API:
Examples:
|
2021-09-17 20:30:28
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9430080652236938, "perplexity": 300.75217149326636}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780055775.1/warc/CC-MAIN-20210917181500-20210917211500-00044.warc.gz"}
|
http://mathhelpforum.com/trigonometry/174509-trig-word-problem-solving-trig-equation-print.html
|
# Trig word problem - solving a trig equation.
• Mar 13th 2011, 07:17 PM
Sleight
Trig word problem - solving a trig equation.
I came across a word problem that I can't seem to puzzle through. I know the answer to it, but I can not seem to find out how to get to it.
"A block is set in motion hanging from a spring and oscillates about it's resting position x=0, according to the function $x=\sqrt{3}sin2t+cos2t$. For what values of t is the block at it's resting position x=0."
This is how I've worked it so far.
$0=\sqrt{3}sin2t+cos2t$
$-cos2t=\sqrt{3}sin2t$
$-\frac{cos2t}{sin2t}=\sqrt{3}$
$-cot2t=\sqrt{3}$
$tan2t=\frac{1}{\sqrt{3}}$
$2t=tan^-^1(\frac{1}{\sqrt{3}})$
$2t = 30\deg$
$t = 15\deg$
This answer, unfortunately, is not correct. I know that the correct answer is 75 degrees, but I have no clue how to get to that answer.
Any help would be appreciated. I've been puzzling over this question for a while now.
• Mar 13th 2011, 07:24 PM
Prove It
It's because if $\displaystyle -\cot{2t} = \sqrt{3}$ then $\displaystyle \tan{2t} = -\frac{1}{\sqrt{3}}$...
• Mar 13th 2011, 08:17 PM
Sleight
Quote:
Originally Posted by Prove It
It's because if $\displaystyle -\cot{2t} = \sqrt{3}$ then $\displaystyle \tan{2t} = -\frac{1}{\sqrt{3}}$...
Your completely correct. Not sure why I forgot about the negative sign, as I had it on the paper I was typing from. However, it still turns out as -15 deg, which doesn't work.
• Mar 13th 2011, 09:26 PM
Prove It
There will be a solution for $\displaystyle 2t$ in the second quadrant and the fourth quadrant.
• Mar 14th 2011, 06:46 AM
Sleight
Oh, now I'm even more confused. The solution that the book gives is either $75 \deg$, or $\frac{5\pi}{12}$, which are in quadrant one.
• Mar 14th 2011, 07:20 AM
Prove It
Because when you have the values of $\displaystyle 2t$, you need to divide them by $\displaystyle 2$ to get the values of $\displaystyle t$...
• Mar 14th 2011, 08:07 AM
Sleight
Ah, I understand now. I think I had some faulty notions at play. I was approaching the negative degree as when the spring was below the zero, but I guess the motion is always positive around the unit circle, hence the reason I had to get rid of the negative? Still having a hard time wrapping my head around the underlying concept, but it should click in soon.
Once again, Prove It, you have helped immensely. Thank you.
Now when I get home I am going to post a problem that has sin as an exponent, and I haven't got a clue. Not sure why this chapter in the book is so difficult for me.
|
2018-01-24 00:07:51
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8899798393249512, "perplexity": 400.8913628904641}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084892802.73/warc/CC-MAIN-20180123231023-20180124011023-00191.warc.gz"}
|
https://www.physicsforums.com/threads/sand-filter.123310/
|
# Sand Filter
1. Jun 8, 2006
### pikkie
Do anybody have any informations on the flux of sand filter system?Varying flux in varying diameter?
2. Jun 9, 2006
### Danger
Sorry, dude. I have no idea what you're talking about.
3. Jun 9, 2006
### Pyrrhus
Well, since a sand filter is basicly water flowing throught sand of different grades, did you look up Darcy's Law?
4. Jun 9, 2006
### pikkie
What I actually looking for is, do anybody have any data or tables or figures about the normal flux that run through the sand filter...
5. Jun 9, 2006
### FredGarvin
If you look up manufacturers of particular filters, they will always provide engineering data on their performance such as flow and expected $$\Delta P$$.
6. Jun 24, 2006
### quark
15m/hr filtration velocity (superficial) is good enough.
|
2017-06-25 22:51:06
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2183685302734375, "perplexity": 9796.4867748799}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320593.91/warc/CC-MAIN-20170625221343-20170626001343-00498.warc.gz"}
|
https://openlab.citytech.cuny.edu/ol-webwork/help/typing-math-on-the-openlab/
|
How can I make nice looking mathematical symbols, like fractions and radicals, on the OpenLab? This is not hard — but it takes a little getting used to.
Here’s an example. If you type this:
Here is a square root: \begin{math} \sqrt{x+1} \end{math}
then, after you click the Submit button, you will see this:
Here is a square root: $\sqrt{x+1}$
TIP: You can also use the “Preview” button to get a quick look at the results without Submitting them.
Each bit of mathematics begins with \begin{math} and ends with \end{math}. In between, you type your math — many mathematical things can be typed just as they are, like numbers and variables, but for each special math symbol we use a code. Here are examples of codes for some of the most common symbols.
### Basic examples: radicals, exponents, fractions
In the example above, we use the code for the radical sign, which is \sqrt{ }. The stuff under the radical goes inside the curly braces { }.
If you type this: You will see this: \begin{math} \sqrt{x} \end{math} $\sqrt{x}$
Exponents
To make an exponent, use ^ (just like in your graphing calculator).
If you type this: You will see this: \begin{math} x^3 \end{math} $x^3$
If your exponent contains more than a single letter or digit, you should group the exponent together using curly braces:
If you type this: You will see this: \begin{math} x^{5y} \end{math} $x^{5y}$
Fractions
The code for fractions is \frac{ }{ }, with the numerator inside the first set of curly braces { } and the denominator in the second set.
If you type this: You will see this: \begin{math} \frac{3}{5} \end{math} $\frac{3}{5}$
Here is an example with more complicated numerator and denominator:
If you type this: You will see this: \begin{math} \frac{x^3 y^5 z}{x y^2} \end{math} $\frac{x^3 y^5 z}{x y^2}$
### Advanced examples: large parentheses, higher roots
Large parentheses
In many cases you can just use regular parentheses ( ). BUT if you want your parentheses to get bigger, for example to wrap around an entire fraction, then you should use \left( and \right) instead. Here’s an example showing how we can raise an entire fraction to a power:
If you type this: You will see this: \begin{math} \left( \frac{2x}{5} \right)^4 \end{math} $\left( \frac{2x}{5} \right)^4$
Higher roots
Regular radical signs use the \sqrt{} command. For higher roots (cube roots, fourth roots, and so on) we use \sqrt[n]{ }, where n is the index of the root (in this example, n is 5).
If you type this: You will see this: \begin{math} \sqrt[5]{32xy} \end{math} $\sqrt[5]{32xy}$
### Hints and suggestions
Preview your work. If you are typing a response in one of the boxes on this site, you can use the “Preview” button (just above the box) to get a sense of what your response will look like when you Submit it. It will convert all of your math codes into symbols, and will tell you if there are any errors. To continue typing, click the “Edit” button.
Don’t start with a complicated formula. Write a short bit of math, and use the Preview button to see what it looks like. Then make corrections until it looks just the way you want.
Stuck? Frustrated? Doesn’t look the way you want it to look? Feel free to include questions about typing math right here on this site – we’ll do our best to answer them. Let us know what you’ve tried so far, and what you’re trying to accomplish.
For more examples, this link is a pretty good place to start. Want even more symbols? Here you go.
### Quick reference and sample questions
Type this: to get this result: Basic fractions: \begin{math} \frac{2}{7} \end{math} $\frac{2}{7}$ More complicated fractions: \begin{math} \frac{x+1}{x^2 + 5x} \end{math} $\frac{x+1}{x^2 + 5x}$ Complex fractions: \begin{math} \frac{2x+7}{x^2+\frac{x}{2}} \end{math} $\frac{2x+7}{x^2+\frac{x}{2}}$ Basic exponents: \begin{math} x^3 \end{math} $x^3$ \begin{math} x^4 y^5 z^6 \end{math} $x^4 y^5 z^6$ More complicated exponents: \begin{math} y^{15x} \end{math} $y^{15x}$ Fractional exponents: \begin{math} x^\frac{1}{3} \end{math} $x^\frac{1}{3}$ Fractions with exponents: \begin{math} \left( \frac{x}{x+1} \right)^5 \end{math} $\left( \frac{x}{x+1} \right)^5$ Higher roots \begin{math} \sqrt[3]{x+6} \end{math} $\sqrt[3]{x+6}$ Add rational expressions \begin{math} \frac{x+5}{y-3}+\frac{y+5}{x+5} \end{math} $\frac{x+5}{y-3}+\frac{y+5}{x+5}$
Sample Question 1. Type this:
I can't figure out how to factor \begin{math}2x^2+3x+1\end{math}.
What should I do with the 2 at the start?
And you will see:
I can't figure out how to factor $2x^2+3x+1$.
What should I do with the 2 at the start?
Sample Question 2. Type this:
I'm trying to find the LCD for \begin{math}\frac{3}{x^2+x} - \frac{x+1}{x^2}\end{math}.
I think it should be \begin{math}x^2+x\end{math}, but I can't figure out what to
multiply the second fraction by. What should I do next?
And you will see:
I'm trying to find the LCD for $\frac{3}{x^2+x} - \frac{x+1}{x^2}$.
I think it should be $x^2+x$, but I can't figure out what to
multiply the second fraction by. What should I do next?
|
2021-12-05 18:30:26
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 21, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8819083571434021, "perplexity": 1368.5794800636213}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363215.8/warc/CC-MAIN-20211205160950-20211205190950-00187.warc.gz"}
|
https://cmouhot.wordpress.com/2009/11/24/fractional-poincare-inequalities-for-general-measures/
|
# Fractional Poincaré inequalities for general measures
Together with Emmnanuel Russ and Yannick Sire we have just uploaded the paper “Fractional Poincaré inequalities for general measures” on
http://hal.archives-ouvertes.fr/hal-00435240/
http://fr.arxiv.org/abs/0911.4563
Here is the abstract:
We prove a fractional version of Poincaré inequalities in the context of $\mathbb{R}^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein-Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures.
|
2017-06-24 22:24:36
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8254364728927612, "perplexity": 261.5439668207625}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320362.97/warc/CC-MAIN-20170624221310-20170625001310-00219.warc.gz"}
|
http://mathhelpforum.com/pre-calculus/147493-partial-fractions.html
|
# Math Help - partial fractions
1. ## partial fractions
how do I get the answer shown in the attached problem? when I tried it out, I couldn't get the numerator to 1.
Thanks!
remember that:
$a^2 - b^2 = (a+b)(a-b)$
Can you do it now?
3. Originally Posted by calculus0
how do I get the answer shown in the attached problem? when I tried it out, I couldn't get the numerator to 1.
Thanks!
4. We write
$\frac{2}{1-x^2}=\frac{2}{(1-x)(1+x)}=\frac{A}{1-x}+\frac{B}{1+x}_{.}$
With a bit of algebraic manipulation of the above equality(actually identity), we can find out that
$A = B = 1$.(There are several ways to do this. One of them is writing the right hand side as a single fraction and we get an equation by equating the denominator with 2)
So we have
$-1+\frac{2}{1-x^2} = -1 + \frac{2}{(1-x)(1+x)} = 1 +\frac{1}{1-x}+\frac{1}{1+x}_{.}$
|
2015-07-31 13:53:11
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9542296528816223, "perplexity": 356.1369595982785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988305.14/warc/CC-MAIN-20150728002308-00274-ip-10-236-191-2.ec2.internal.warc.gz"}
|
https://engineering.stackexchange.com/questions/489/what-is-the-densest-ceramic
|
# What is the densest ceramic?
I know ceramics are generally considerably less dense than pure metals or alloys but sand is used in barbells for its probably cheapness, user-friendly and easy availability but its a decent density too. rammed sand- 1682 kg/m cubed using Engineering Toolbox compared to 2658 for steel chips. So if you wanted the most dense for something to have great compressive strength- what would it be?
• What sort of usage are you interested in for the ceramic? Technically, diamond can be regarded as a ceramic & it is the strongest ceramic, but I don't think diamond is what you are wanting.
– Fred
Feb 6 '15 at 7:03
• I'm voting to close this question as off-topic because it appears to be a "finding things question". Please see this meta answer as well as this answer for additional guidance.
– user16
Feb 6 '15 at 12:40
• Isn't this more of a materials science question? Seems valid to me. Maybe it needs to be more specific? Feb 6 '15 at 15:15
• Am I correct in assuming you want the highest density/compressive strength ratio? Or did you have another metric in mind? For your application, presumably high density is preferred as it would yield smaller weights, and compressive strength would be irrelevant as whatever shell the ceramic is held in would be the structural component. Dec 8 '15 at 22:46
• Just the one with greatest density. Dec 11 '15 at 12:42
According to the following two sources, diamond is a ceramic:
Ceramic Strength-Density Graphs
Ceramics @ Virginia University
The graph below, from Ceramic Strength-Density Graphs shows diamond is the strongest ceramic, whereas zirconia is one of the most dense and strongest ceramics
The site matweb has a database of all sorts or materials.
By specifying ceramics with a compressive yields strength ranging between 100 MPa & 16 500 MPa & a density range between 2.5 & 22.6 $\text{g/cm}^3$, corundum, aluminium oxide, & alumina have the highest compressive yield strength & density, with a compressive yield strength of 3000 MPa & a density of 3.96 $\text{g/cm}^3$.
Carbides of heavy metals are always good candidates. Tungsten carbide has 15,63 g·cm−3. The other parameters are also "impressive".
Assuming money is not an option, Iridum is absurdly high density, ~22500 kg/m^3, Tungsten is close at ~19300 kg/m^3. both of which are extremely tough materials in compression.
|
2021-12-07 18:10:09
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.47971805930137634, "perplexity": 2254.53249705486}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363405.77/warc/CC-MAIN-20211207170825-20211207200825-00603.warc.gz"}
|
https://code.tutsplus.com/tutorials/how-to-define-state-in-a-single-page-app-with-angular-ui-router--cms-28880
|
# How to Define State With Angular UI-Router
UI-Router is a flexible and powerful alternative for AngularJS routing. UI-Router goes beyond the Angular team's own ngRoute module by building in support for nested routes and for events triggered by route changes.
In this short video from my course, Single-Page Apps With Angular UI-Router, you’ll learn what states are and see how we can add them to our app’s bootstrapping phase using the \$stateProvider.
|
2021-09-18 17:36:33
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2960689067840576, "perplexity": 4696.02228250034}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056548.77/warc/CC-MAIN-20210918154248-20210918184248-00481.warc.gz"}
|
https://expressionengine.stackexchange.com/questions/7030/how-to-paginate-or-organize-expression-engine-by-custom-field-data-input
|
# How to paginate or organize Expression Engine by custom field data input
I have an ExpressionEngine site that takes information entered from an admin, and information entered from users through SafeCracker, and displays that information in a timeline format.
The information that is entered has a custom field called {story_year}. We use that field to sort the entries based on the entered date. This means that users can specify a date, as well as admins.
Because showing hundreds of entries at once takes a long time load, we've introduced pagination. But that pagination breaks it up arbitrarily by amount.
Is there a way to paginate by {story_year}? Possibly conditionals like:
{if story_year >= "2010" }
{page}
<li><a href="{pagination_url}" class="{if current_page}active{/if}">•</a></li>
{/page}
{if:elseif story_year >= "2000"}
{page}
<li><a href="{pagination_url}" class="{if current_page}active{/if}">•</a></li>
{/page}
{/if}
Relevant URLs are here: http://thinkx.net/clients/manship/index.php/timeline - the ultimate display of the timeline. http://thinkx.net/clients/manship/index.php/Timeline/add - the field entries we sort by.
There isn't a way to do this with native pagination.
Take a look at the YearList addon to see if you can get something working with that.
Fetches the distinct years of all entries in a specified weblog. Can be used to create a year archive list.
The Year Listing plugin is a simple way to get a distinct 4 digit year for your entries. This way you can list out years for archives.
{exp:yearlist channel="yourchannel" category="1"}
{year} {/exp:yearlist}
That will return an array of years. Use {year} to print them to the screen and wrap in any markup needed. There are currently no linebreaks or HTML associated with this plugin.
You can pull the year from the URL and use it with your entries tag to pull entries for just that year.
You'll need to make sure your entry date year matches your custom field year or you can try hacking this addon to pull from your custom field instead of the entry date field.
• If I'm reading that correctly, you're saying I can use {story_year} instead of {year}? I'll look into that. Thanks! – thatgibbyguy Mar 6 '13 at 14:27
• No, I'm saying if your entry date matches your story_year field, you can use {year}. Or you can look at the code for this addon and perhaps adjust to meet your needs. – Anna_MediaGirl Mar 6 '13 at 16:29
• Got it. I don't think there's a way to do what I'm asking without writing something custom. What we're trying to do is simplify manship100.com/timeline to speed it up. It looks like optimization is our answer, which is another thing I don't know much about heh. Thanks for your help! – thatgibbyguy Mar 6 '13 at 16:35
• Try this... JS for pagination and CE Cache using the static driver for caching. The page will be insanely fast. – Anna_MediaGirl Mar 6 '13 at 17:06
• I'm actually attempting to implement infinite scroll right now. It still isn't working but it's close. But pagination is working great! Super fast now. – thatgibbyguy Mar 14 '13 at 16:29
|
2021-04-22 14:43:19
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30519989132881165, "perplexity": 2628.4674912013284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039610090.97/warc/CC-MAIN-20210422130245-20210422160245-00313.warc.gz"}
|
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=GCGHC8_2012_v19n5_697
|
Estimation of the Exponential Distributions based on Multiply Progressive Type II Censored Sample
Title & Authors
Estimation of the Exponential Distributions based on Multiply Progressive Type II Censored Sample
Lee, Kyeong-Jun; Park, Chan-Keun; Cho, Young-Seuk;
Abstract
The maximum likelihood(ML) estimation of the scale parameters of an exponential distribution based on progressive Type II censored samples is given. The sample is multiply censored (some middle observations being censored); however, the ML method does not admit explicit solutions. In this paper, we propose multiply progressive Type II censoring. This paper presents the statistical inference on the scale parameter for the exponential distribution when samples are multiply progressive Type II censoring. The scale parameter is estimated by approximate ML methods that use two different Taylor series expansion types ($\small{AMLE_I}$, $\small{AMLE_{II}}$). We also obtain the maximum likelihood estimator(MLE) of the scale parameter under the proposed multiply progressive Type II censored samples. We compare the estimators in the sense of the mean square error(MSE). The simulation procedure is repeated 10,000 times for the sample size n = 20 and 40 and various censored schemes. The $\small{AMLE_{II}}$ is better than MLE and $\small{AMLE_I}$ in the sense of the MSE.
Keywords
Approximate maximum likelihood estimator;exponential distribution;multiply progressive Type II censored sample;progressive Type II censored sample;
Language
English
Cited by
1.
Estimation of the exponential distribution based on multiply Type I hybrid censored sample,;;;
Journal of the Korean Data and Information Science Society, 2014. vol.25. 3, pp.633-641
2.
Estimation of the half-logistic distribution based on multiply Type I hybrid censored sample,;;;
Journal of the Korean Data and Information Science Society, 2014. vol.25. 6, pp.1581-1589
1.
Estimation of the exponential distribution based on multiply Type I hybrid censored sample, Journal of the Korean Data and Information Science Society, 2014, 25, 3, 633
2.
Estimation of the half-logistic distribution based on multiply Type I hybrid censored sample, Journal of the Korean Data and Information Science Society, 2014, 25, 6, 1581
References
1.
Asgharzadeh, A. (2009). Approximate MLE for the scaled generalized exponential distribution under progressive Type II censoring, Journal of the Korean Statistical Society, 38, 223-229.
2.
Balakrishnan, N. and Sandu, R. A. (1995). A simple simulation algorithm for generating progressively Type II censored samples, The American Statistician, 49, 229-230.
3.
Balakrishnan, N. and Sandu, R. A. (1996). Best linear unbiased and maximum likelihood estimation for exponential distributions under general progressive Type II censored sample, Sanky a: The Indian Journal of Statistics, 58, 1-9.
4.
Chen, D. G. and Lio, Y. L. (2010). Parameter estimation for generalized exponential distribution under progressive Type I interval censoring, Computational Statistics and Data Analysis, 54, 1581-1591.
5.
Herd, R. G. (1956). Estimation of the Parameters of a Population from a Multi-Censored Sample, Ph. D. Thesis, Iowa State College, Ames, Iowa.
6.
Kang, S. B. (2003). Approximate MLEs for exponential distribution under multiply Type II censoring, Journal of the Korean Data and Information Science Society, 14, 983-988.
7.
Kang, S. B. and Cho, Y. S. (1998). MRE for exponential distribution under general progressive Type II censored samples, Journal of the Korean Data and Information Science Society, 9, 71-76.
8.
Kang, S. B. and Park, S. M. (2005). Estimation for the exponentiated exponential distribution based on multiply Type II censored samples, The Korean Communications in Statistics, 12, 643-652.
9.
Nelson, W. (1982). Applied Life Data Analysis, John Wiley and Sons, New York.
10.
Shin, H. J., Lee, K. H. and Cho, Y. S. (2010). Parameter estimation for exponential distribution under progressive Type I interval censoring, Journal of the Korean Data and Information Science Society, 21, 927-934.
11.
Singh, U. and Kumar, A. (2007). Bayesian estimation of the exponential parameter under a multiply Type II censoring scheme, Austrian Journal of Statistics, 36, 227-238.
|
2017-02-22 22:04:54
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 4, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7194086313247681, "perplexity": 2585.00582155466}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171053.19/warc/CC-MAIN-20170219104611-00463-ip-10-171-10-108.ec2.internal.warc.gz"}
|
https://mathoverflow.net/questions/368241/improved-sobolev-embedding/368258#368258
|
# improved Sobolev embedding
This is probably not a research level question but I am struggling with the geometry. My question is related to whether some monotonicity can increase the range of exponents in the Sobolev embedding. For instance on the unit ball, nonnegative radially symmetric functions which are nondecreaing in the radial direction should satisfy a Sobolev embedding for an improved range of exponents: Indeed such functions must be large on the full boundary of the ball, yet the trace theorem prevents this from happening.
So I will ask the question on a finite cone; let $$S \subset \subset S^{N-1}$$ be some nice spherical cap. For explicitness let us assume that the $$x_1$$ axis cuts through the center of $$S$$ (assume e.g. that $$S$$ is a ball in $$S^{N-1}$$ centered at some $$y\in S^{N-1}$$ lying on the $$x_1$$ axis). Let $$\Omega:=\{x=r \theta: 0 denote the finite cone. We now look at nonnegative functions which are zero on the side of the cone and moreover nondecreasing in the radial direction, i-e $$x \cdot \nabla u(x) \ge 0$$ in $$\Omega$$.
So my question is:
Can we expect an improved Sobolev embedding for this class of functions? (here I am using the $$H^1$$ norm so I am asking about possible embedding of $$H^1$$ into some $$L^p$$ space for some improved exponents $$p>2^*=\frac{2N}{N-2}$$)
I suspect the answer is `no' and I am attempting to disprove it considering the first strategy that comes to mind: Take $$0\le \phi \in C_c^\infty(B_1)$$ a smooth radially nonincreasing function and then translate this function so that its support is centered at $$y$$, and scale it in order to concentrate its support at $$y$$. This sequence of functions will presumably violate any alleged improved Sobolev embedding, up to proving that these functions really have the correct monotonicity. Geometrically it looks like to me that this is so, but my geometric intuition almost always fails me now. Any comments would be great.
• I am confused.... or maybe i don't understand your comment. On the unit ball if you just ask for radial functions then there is no improved range. Aug 3, 2020 at 18:13
• I misunderstood your question. Are you only interested in the $H^1 \to L^p$ embedding? // Also, when you say "radial increasing functions" do you mean functions that increase in the radial direction, or do you mean radially symmetric functions that also increase in the radial direction? Aug 3, 2020 at 18:19
• ya, i was not overly precise in my question. So let $X$ denote the functions with $H^1(\Omega)$ norm; the functions are zero on side of cone and the functions are nonnegative and satisfy $u_r \ge 0$ or $x \cdot \nabla u(x) \ge 0$. So they are radial increasing but there is no radial symmetry assumption. Aug 3, 2020 at 18:25
The answer is indeed "no." To see this, instead of considering translations of radial bump functions, one needs to use bump functions whose level sets are deformed e.g. to ellipsoids. More precisely, denote $$x \in \mathbb{R}^n$$ by $$(x_1,\,x')$$ and let $$h$$ be any decreasing function on $$\mathbb{R}$$. Then for $$H(x) := h(\Lambda^2 x_1^2 + |x'|^2)$$ we have that $$x \cdot \nabla [H(x-y)] \geq 0$$ in the ellipsoid $$\left\{\frac{(x_1 - 1/2)^2}{(1/2)^2} + \frac{|x'|^2}{(\Lambda/2)^2} \leq 1\right\},$$ which contains a neighborhood of $$y$$ in $$B_1$$ when $$\Lambda$$ is large (but not $$1$$). Now fix $$\Lambda$$ large and choose $$h$$ smooth with $$h(s) = 1$$ for $$s \leq 0$$ and $$h(s) = 0$$ for $$s \geq 1$$. Then the family $$\lambda^{\frac{n-p}{p}}H(\lambda (x-y))$$ shows as $$\lambda \rightarrow \infty$$ that the exponent cannot be improved.
|
2022-05-26 14:19:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 36, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999101161956787, "perplexity": 756.8805522389027}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662606992.69/warc/CC-MAIN-20220526131456-20220526161456-00446.warc.gz"}
|
https://cs.stackexchange.com/questions/126136/is-there-an-undecidable-language-that-is-mapping-reducible-to-its-complement
|
# Is there an undecidable language that is mapping reducible to its complement?
Is there an undecidable language A that is mapping reducible to its complement?
If it is possible, since A is an undecidable language, so A's complement must also be an undecidable language. But i don't know whether undecidable languages is closed under complement or not.
Yes, it is possible. Enumerate the set of all possible Turing machines and let $$H$$ (resp. $$\overline{H}$$) be the set of indices of the Turing machines that halt (resp. do not halt) on empty input.
Let $$L = \{ \langle T, 1 \rangle \, : T \in H \} \cup \{ \langle T, 0 \rangle \, : T \in \overline{H} \}$$.
Clearly $$L$$ is not decidable but it is possible to reduce $$L$$ to $$\overline{L}$$ since $$\langle T, r \rangle \in L \iff \langle T, 1-r \rangle \in \overline{L}$$.
|
2022-01-20 09:32:39
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5624701976776123, "perplexity": 202.09504591950503}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301730.31/warc/CC-MAIN-20220120065949-20220120095949-00226.warc.gz"}
|
https://www.jobilize.com/physics-ap/course/11-2-density-fluid-statics-by-openstax?qcr=www.quizover.com&page=1
|
# 11.2 Density (Page 2/3)
Page 2 / 3
## Calculating the mass of a reservoir from its volume
A reservoir has a surface area of $\text{50}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{km}}^{2}$ and an average depth of 40.0 m. What mass of water is held behind the dam? (See [link] for a view of a large reservoir—the Three Gorges Dam site on the Yangtze River in central China.)
Strategy
We can calculate the volume $V$ of the reservoir from its dimensions, and find the density of water $\rho$ in [link] . Then the mass $m$ can be found from the definition of density
$\rho =\frac{m}{V}.$
Solution
Solving equation $\rho =m/V$ for $m$ gives $m=\rho V$ .
The volume $V$ of the reservoir is its surface area $A$ times its average depth $h$ :
$\begin{array}{lll}V& =& \text{Ah}=\left(\text{50.0}\phantom{\rule{0.25em}{0ex}}{\text{km}}^{2}\right)\left(\text{40.0}\phantom{\rule{0.25em}{0ex}}\text{m}\right)\\ & =& \left[\left(\text{50.0 k}{\text{m}}^{2}\right){\left(\frac{{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m}}{1\phantom{\rule{0.25em}{0ex}}\text{km}}\right)}^{2}\right]\left(\text{40.0 m}\right)=2\text{.}\text{00}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\end{array}$
The density of water $\rho$ from [link] is $1\text{.}\text{000}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ . Substituting $V$ and $\rho$ into the expression for mass gives
$\begin{array}{lll}m& =& \left(1\text{.}\text{00}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}\right)\left(2\text{.}\text{00}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\right)\\ & =& 2.00×{\text{10}}^{\text{12}}\phantom{\rule{0.25em}{0ex}}\text{kg.}\end{array}$
Discussion
A large reservoir contains a very large mass of water. In this example, the weight of the water in the reservoir is $\text{mg}=1\text{.}\text{96}×{\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}\text{N}$ , where $g$ is the acceleration due to the Earth’s gravity (about $9\text{.}\text{80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ ). It is reasonable to ask whether the dam must supply a force equal to this tremendous weight. The answer is no. As we shall see in the following sections, the force the dam must supply can be much smaller than the weight of the water it holds back.
## Section summary
• Density is the mass per unit volume of a substance or object. In equation form, density is defined as
$\rho =\frac{m}{V}.$
• The SI unit of density is ${\text{kg/m}}^{3}$ .
## Conceptual questions
Approximately how does the density of air vary with altitude?
Give an example in which density is used to identify the substance composing an object. Would information in addition to average density be needed to identify the substances in an object composed of more than one material?
[link] shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer.
## Problems&Exercises
Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure gold?
$1\text{.}\text{610}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$
Mercury is commonly supplied in flasks containing 34.5 kg (about 76 lb). What is the volume in liters of this much mercury?
(a) What is the mass of a deep breath of air having a volume of 2.00 L? (b) Discuss the effect taking such a breath has on your body’s volume and density.
(a) 2.58 g
(b) The volume of your body increases by the volume of air you inhale. The average density of your body decreases when you take a deep breath, because the density of air is substantially smaller than the average density of the body before you took the deep breath.
A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock that displaces $\text{89}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$ of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that are based on Archimedes’ principle.)
$2\text{.}\text{70}\phantom{\rule{0.25em}{0ex}}{\text{g/cm}}^{3}$
Suppose you have a coffee mug with a circular cross section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density as water.
(a) A rectangular gasoline tank can hold 50.0 kg of gasoline when full. What is the depth of the tank if it is 0.500-m wide by 0.900-m long? (b) Discuss whether this gas tank has a reasonable volume for a passenger car.
(a) 0.163 m
(b) Equivalent to 19.4 gallons, which is reasonable
A trash compactor can reduce the volume of its contents to 0.350 their original value. Neglecting the mass of air expelled, by what factor is the density of the rubbish increased?
A 2.50-kg steel gasoline can holds 20.0 L of gasoline when full. What is the average density of the full gas can, taking into account the volume occupied by steel as well as by gasoline?
$7\text{.}9×{\text{10}}^{2}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0ex}{0ex}}{\text{kg/m}}^{3}$
What is the density of 18.0-karat gold that is a mixture of 18 parts gold, 5 parts silver, and 1 part copper? (These values are parts by mass, not volume.) Assume that this is a simple mixture having an average density equal to the weighted densities of its constituents.
$\text{15}\text{.}6\phantom{\rule{0.25em}{0ex}}{\text{g/cm}}^{3}$
There is relatively little empty space between atoms in solids and liquids, so that the average density of an atom is about the same as matter on a macroscopic scale—approximately ${\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ . The nucleus of an atom has a radius about ${\text{10}}^{-5}$ that of the atom and contains nearly all the mass of the entire atom. (a) What is the approximate density of a nucleus? (b) One remnant of a supernova, called a neutron star, can have the density of a nucleus. What would be the radius of a neutron star with a mass 10 times that of our Sun (the radius of the Sun is $7×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m}$ )?
(a) ${\text{10}}^{\text{18}}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$
(b) $2×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{m}$
what is a wave
wave means. A field of study
aondohemba
what are Atoms
aondohemba
is the movement back and front or up and down
sani
how ?
aondohemba
wave is a disturbance that transfers energy through matter or space with little or no associated mass.
lots
A wave is a motion of particles in disturbed medium that carry energy from one midium to another
conist
an atom is the smallest unit( particle) of an element that bares it's chemical properties
conist
what is electromagnetic induction?
conist
How is the de Broglie wavelength of electrons related to the quantization of their orbits in atoms and molecules?
How do you convert 0.0045kgcm³ to the si unit?
how many state of matter do we really have like I mean... is there any newly discovered state of matter?
I only know 5: •Solids •Liquids •Gases •Plasma •Bose-Einstein condensate
Thapelo
Alright Thank you
Falana
Which one is the Bose-Einstein
James
can you explain what plasma and the I her one you mentioned
Olatunde
u can say sun or stars are just the state of plasma
Mohit
but the are more than seven
Issa
list it out I wanna know
Cristal
what the meaning of continuum
What state of matter is fire
fire is not in any state of matter...fire is rather a form of energy produced from an oxidising reaction.
Xenda
Isn`t fire the plasma state of matter?
Walter
all this while I taught it was plasma
Victor
How can you define time?
Time can be defined as a continuous , dynamic , irreversible , unpredictable quantity .
Tanaya
unpredictable? but I can say after one o'clock its going to be two o'clock predictably!
Victor
how can we define vector
mahmud
I would define it as having a magnitude (size)with a direction. An example I can think of is a car traveling at 50m/s (magnitude) going North (direction)
Hanzo
as for me guys u would say time is quantity that measures how long it takes for a specific condition to happen e.g how long it takes for the day to end or how it takes for the travelling car to cover a km.
conist
what is the relativity of physics
How do you convert 0.0045kgcm³ to the si unit?
flint
What is the formula for motion
V=u+at V²=u²-2as
flint
S=ut+½at
flint
they are eqns of linear motion
King
S=Vt
Thapelo
v=u+at s=ut+at^\2 v^=u^+2as where ^=2
King
hi
hello
King
Explain dopplers effect
Not yet learnt
Bob
Explain motion with types
Bob
Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.
Scalar quantity Because acceleration has only magnitude
Bob
acleration is vectr quatity it is found in a spefied direction and it is product of displcemnt
bhat
its a scalar quantity
Paul
velocity is speed and direction. since velocity is a part of acceleration that makes acceleration a vector quantity. an example of this is centripetal acceleration. when you're moving in a circular patter at a constant speed, you are still accelerating because your direction is constantly changing.
Josh
acceleration is a vector quantity. As explained by Josh Thompson, even in circular motion, bodies undergoing circular motion only accelerate because on the constantly changing direction of their constant speed. also retardation and acceleration are differentiated by virtue of their direction in
fitzgerald
respect to prevailing force
fitzgerald
What is the difference between impulse and momentum?
Manyo
Momentum is the product of the mass of a body and the change in velocity of its motion. ie P=m(v-u)/t (SI unit is kgm/s). it is literally the impact of collision from a moving body. While Impulse is the product of momentum and time. I = Pt (SI unit is kgm) or it is literally the change in momentum
fitzgerald
Or I = m(v-u)
fitzgerald
the tendency of a body to maintain it's inertia motion is called momentum( I believe you know what inertia means) so for a body to be in momentum it will be really hard to stop such body or object..... this is where impulse comes in.. the force applied to stop the momentum of such body is impulse..
Pelumi
Calculation of kinetic and potential energy
K.e=mv² P.e=mgh
Malia
K is actually 1/2 mv^2
Josh
what impulse is given to an a-particle of mass 6.7*10^-27 kg if it is ejected from a stationary nucleus at a speed of 3.2*10^-6ms²? what average force is needed if it is ejected in approximately 10^-8 s?
John
speed=velocity÷time velocity=speed×time=3.2×10^-6×10^-8=32×10^-14m/s impulse [I]=∆momentum[P]=mass×velocity=6.7×10^-27×32×10^-14=214.4×10^-41kg/ms force=impulse÷time=214.4×10^-41÷10^-8=214.4×10^-33N. dats how I solved it.if wrong pls correct me.
Melody
what is sound wave
sound wave is a mechanical longitudinal wave that transfers energy from one point to another
Ogor
its a longitudnal wave which is associted wth compresion nad rearfractions
bhat
what is power
it's also a capability to do something or act in a particular way.
Kayode
Newton laws of motion
Mike
power also known as the rate of ability to do work
Slim
power means capabilty to do work p=w/t its unit is watt or j/s it also represents how much work is done fr evry second
bhat
what does fluorine do?
strengthen and whiten teeth.
Gia
|
2019-03-24 21:53:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 32, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6621798276901245, "perplexity": 1015.375904152343}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203493.88/warc/CC-MAIN-20190324210143-20190324232143-00533.warc.gz"}
|
https://askthetask.com/4025375/how-to-perform-f-test
|
0 like 0 dislike
Im currently working on a statistical lab (analytical chemistry) and I am not sure how to chose which way to calculate the F-value. Initial thought was to calculate Spool for 2-11, and get F-calculated to then compare it to the F-table. The more I dove into the F-critical, F-calculated etc etc, the more confused I got. Im not sure at all how to go about this problem. Realising I probably misunderstood the whole concept of F-testing.
0 like 0 dislike
Let's clear things up a bit: A= Set (2-7) B= Set 8 $n_A=6$ (b/c you have set 2-7) $n_B=1$ (b/c you only have set 8) $s^2_A$ is the variance of set 2-7 $s^2_B$ is the variance of set 8 Does this answer your question?
by
0 like 0 dislike
Im comparing 6 sets (pooled) with one single set. Basically, when calculating the sA^2 value for the s-pooled used in tcalculated, do I get the s^2 for the 6 sets by calculating this first (second pic) and then taking that value to the power of two, which will equal my sA^2 ? I have to pool the pool?
by
0 like 0 dislike
Jananas said: Hi again, Im now calculating the t-calculated. It is ment to compare a pooled set (2-7) with set 8. I have done so according to this formula: View attachment 44465 I have set the A values as the pooled sets. When calculating the spool for t, I have taken the already variance-pooled value for the sA^2 (I used the formula for spooled but didn't square it). Then I took the (nA-1) pooled as well = (4(sample size)-1)*6-6. I inserted this in the formula. However Im getting a very high number... seems wrong (around 6E4)... the tcritical is around 1.7. What am I missing? Click to expand... I don't have the data so I can't speak to what mistake you made.
by
0 like 0 dislike
Hi again, Im now calculating the t-calculated. It is ment to compare a pooled set (2-7) with set 8. I have done so according to this formula: I have set the A values as the pooled sets. When calculating the spool for t, I have taken the already variance-pooled value for the sA^2 (I used the formula for spooled but didn't square it). Then I took the (nA-1) pooled as well = (4(sample size)-1)*6-6. I inserted this in the formula. However Im getting a very high number... seems wrong (around 6E4)... the tcritical is around 1.7. What am I missing?
by
0 like 0 dislike
Jananas said: For context this is the main task; Water is being transfered from its source (e.g a river) to a lab and all samples must be analysed during the travel to the lab, which is 1h. This is done 11 times during the trip, taking 4 determinations for each time. The concentration of aluminium is measured, and the goal is to keep it as close to the initial value as possible during the whole trip; View attachment 44458 Which checks out removing nr 1since its not statistically comparable with the rest. Will need to see what this means practically though... Click to expand... From task 2, you've concluded that the variability of the sample (1) is statistically different from other samples. Therefore, during the first 4 minutes, the sample will be unstable. If you were the researcher, you should wait longer than 4 minutes for the sample to stabilize before proceeding any further.
by
0 like 0 dislike
For context this is the main task; Water is being transfered from its source (e.g a river) to a lab and all samples must be analysed during the travel to the lab, which is 1h. This is done 11 times during the trip, taking 4 determinations for each time. The concentration of aluminium is measured, and the goal is to keep it as close to the initial value as possible during the whole trip; Which checks out removing nr 1since its not statistically comparable with the rest. Will need to see what this means practically though...
by
0 like 0 dislike
Jananas said: Thanks for the answer! I have calculated the Spool (from 2 to 11) = Sx^2 and also Sy^2 for set 1. When I compared that with the F-table, The F-calc was larger. This means it was a significant difference in variance, which is why I can discard it, right? That's what they ment under "TIP1"? I want to make sure Im on the right track. Click to expand... An F-statistic greater than the critical value is equivalent to a p-value less than alpha and both mean that you reject the null hypothesis. In other words, the variances of the 2 populations are statistically different. The samples were not drawn from the same normal distribution with the same variability. As far as discarding it, I'm not sure what task is being asked is, but if you were asked to discard, you can justify removing group 1 due to the reason stated above.
by
0 like 0 dislike
SupremeCookie said: The F-test is used to test whether 2 normal populations have the same variance. As you can see from the diagram, the variance of the population (1) is much wider than (2-11). That is the point the text is trying to make. To perform the F-test, you need to compute the F-statistic: View attachment 44456 Once you have the F-statistic, compare it to the F-distribution and determine whether the variances are statistically significant based on your selected alpha level. Click to expand... Thanks for the answer! I have calculated the Spool (from 2 to 11) = Sx^2 and also Sy^2 for set 1. When I compared that with the F-table, The F-calc was larger. This means it was a significant difference in variance, which is why I can discard it, right? That's what they ment under "TIP1"? I want to make sure Im on the right track.
by
0 like 0 dislike
The F-test is used to test whether 2 normal populations have the same variance. As you can see from the diagram, the variance of the population (1) is much wider than (2-11). That is the point the text is trying to make. To perform the F-test, you need to compute the F-statistic: Once you have the F-statistic, compare it to the F-distribution and determine whether the variances are statistically significant based on your selected alpha level.
by
|
2022-10-05 22:33:49
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7328255772590637, "perplexity": 804.2489525975707}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337668.62/warc/CC-MAIN-20221005203530-20221005233530-00110.warc.gz"}
|
https://math.stackexchange.com/questions/784524/operations-in-the-exterior-algebra-multiplication-in-the-direct-sum-of-rings
|
# Operations in the exterior algebra. Multiplication in the direct sum of rings.
Let $V$ be a vector space with dimension $n$ over a field $K$. The exterior algebra $\Lambda(V)$ of the vector space $V$ is the direct sum of the exterior powers $\Lambda^k(V),\quad k\in\overline{0,n}$. Then an element $x\in\Lambda(V)$ has the form $(x_0,\dots,x_n)$, where $x_i \in \Lambda^i(V)$ is the $i$-th homogeneous component of $x$. As each exterior power is itself a vector space over $K$, so is their direct sum with the point-wise addition and multiplication with a scalar.
How does the exterior product carry over the direct sum? What does $(x_0,\dots,x_n)\wedge (y_0,\dots,y_n)$ stand for?
• Yes, I know that. But if $x\in \Lambda(V)$, then $x=(x_0,\dots,x_n)$, where $x_i\in\Lambda^i(V)$. Really my question is how the product in algebras carries over to the direct sum of those algebras. Is it defined componentwise? – superAnnoyingUser May 7 '14 at 10:23
• It is defined component-wise. The "right" way to think of your element is as $\sum_j x_j$. Then you define $\wedge$ so as to make it distributive and take $\Lambda^j(V) \times \Lambda^k(V) \to \Lambda^{j+k}(V)$. – jdc May 7 '14 at 12:43
Since $\Lambda(V) = \Lambda^0(V) \oplus \cdots \oplus \Lambda^n(V)$, a general element $x \in \Lambda(V)$ looks like $x = x_0 + \cdots + x_n$, where $x_k \in \Lambda^k(V)$ for $k \in \{0, \dots, n\}$. The product $\wedge$ on $\Lambda(V)$ is the image of $\otimes$ on $T(V)$ (the tensor product on the tensor algebra) when its quotient is taken by the $2$-sided ideal of "repeating tensors", generated by tensors of the form $(v \otimes v)$ where $v \in V$. Since $\otimes$ acts distributively, (not component-wise), so does $\wedge$: $$(x_0 + \cdots + x_n) \wedge (y_0 + \cdots + y_n) = \sum_{k=0}^n x_k \wedge (y_0 + \cdots + y_n) = \sum_{k=0}^n \sum_{j=0}^n x_k \wedge y_j.$$ This looks just like the case in which $\wedge$ is replaced by $\otimes$ and the $x_k$ and $y_j$ are homogeneous tensors of equal degree. As with polynomials, we like to collect terms with equal degree, which can be done via $$= \sum_{k=0}^{n} x_k \wedge y_{n-k}.$$
The product is defined component-wise, i.e. $$(x_0,\dots,x_n)\wedge (y_0,\dots,y_n) = (z_{0}, \ldots z_{n})$$ where $$z_{i} = \sum_{j = 0}^{i} x_{j} \wedge y_{i-j}$$
• The formula is correct, but I wouldn't call it "component-wise"! – darij grinberg Aug 1 '16 at 19:05
|
2020-02-18 07:01:17
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9656177163124084, "perplexity": 131.5476868169871}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143635.54/warc/CC-MAIN-20200218055414-20200218085414-00407.warc.gz"}
|
http://farside.ph.utexas.edu/teaching/301/lectures/node118.html
|
Next: Angular momentum of an Up: Angular momentum Previous: Introduction
## Angular momentum of a point particle
Consider a particle of mass , position vector , and instantaneous velocity , which rotates about an axis passing through the origin of our coordinate system. We know that the particle's linear momentum is written
(414)
and satisfies
(415)
where is the force acting on the particle. Let us search for the rotational equivalent of .
Consider the quantity
(416)
This quantity--which is known as angular momentum--is a vector of magnitude
(417)
where is the angle subtended between the directions of and . The direction of is defined to be mutually perpendicular to the directions of and , in the sense given by the right-hand grip rule. In other words, if vector rotates onto vector (through an angle less than ), and the fingers of the right-hand are aligned with this rotation, then the thumb of the right-hand indicates the direction of . See Fig. 85.
Let us differentiate Eq. (416) with respect to time. We obtain
(418)
Note that the derivative of a vector product is formed in much the same manner as the derivative of an ordinary product, except that the order of the various terms is preserved. Now, we know that and . Hence, we obtain
(419)
However, , since the vector product of two parallel vectors is zero. Also,
(420)
where is the torque acting on the particle about an axis passing through the origin. We conclude that
(421)
Of course, this equation is analogous to Eq. (415), which suggests that angular momentum, , plays the role of linear momentum, , in rotational dynamics.
For the special case of a particle of mass executing a circular orbit of radius , with instantaneous velocity and instantaneous angular velocity , the magnitude of the particle's angular momentum is simply
(422)
Next: Angular momentum of an Up: Angular momentum Previous: Introduction
Richard Fitzpatrick 2006-02-02
|
2017-10-17 13:20:20
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8965446949005127, "perplexity": 359.1486125188063}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187821189.10/warc/CC-MAIN-20171017125144-20171017145144-00742.warc.gz"}
|
http://www.acmerblog.com/hdu-4125-moles-7128.html
|
2015
04-16
Moles
A mole is a strange mammal adapted to a subterranean lifestyle. It has invisible eyes and short, powerful limbs with large paws oriented for digging.
Before the catastrophic earthquake, a group of moles have perceived the disaster. They move to a safe place and want to dig holes under the ground. There are n moles, numbered from 1 to n. They stand in a line and dig holes one by one. The mole line is not necessary sorted by the mole number. Every mole should live in a hole dug by itself and every mole just digs one hole. The first mole in the line digs the first hole with a channel to the ground. Then other moles go down through that channel and dig more holes and channels. A hole may have at most three neighbors connected by channels, one is on the upper level, and the other two holes are on the lower level lying on the left side and the right side. When a mole reaches a hole, if its number is smaller than the hole owner’s, it will go to the left-lower hole(or digs a left-lower hole and stays there when there is no left-lower hole), otherwise it will go to the right-lower hole(or digs a right-lower hole and stays there when there is no right-lower hole). Due to the excellent ability and well-designed layout, these holes and channels will not cross with each other.
Mouse Jerry is a friend of those moles. He comes to visit them and prepares gifts for every mole. There is a rule that the mole whose number is smaller must get the gift earlier than the mole whose number is larger. Jerry starts from the ground. He travels through the holes and channels to deliver his gifts. After giving out all his gifts, he comes back to the ground. In the mole world, it is interesting that the moles with odd numbers are males and others are females. When reaching a hole, Jerry takes a note about the gender of the hole owner(0 represents female and 1 represents male). When he gets back to the ground, he will get a 0-1 sequence. Now he wants to calculate the "harmony value". The harmony value represents the number of occurrences of a given "harmony string" in the above mentioned sequence. Occurrences may overlap.
Please note that Jerry is very smart so his travel distance is as small as possible.
The first line contains an integer t meaning that there are t test cases(t <= 10).
For each test case :
The first line is an integer n meaning that there are n moles. (n <= 600000).
The second line contains n integers representing the mole line. Each integer is a mole’s number and the first integer is the number of the first mole in the line.
The third line is the harmony string. The string length is not large than 7000.
The first line contains an integer t meaning that there are t test cases(t <= 10).
For each test case :
The first line is an integer n meaning that there are n moles. (n <= 600000).
The second line contains n integers representing the mole line. Each integer is a mole’s number and the first integer is the number of the first mole in the line.
The third line is the harmony string. The string length is not large than 7000.
2
8
5 1 3 2 7 8 4 6
01
10
1 2 3 4 5 6 7 8 9 10
1010
Case #1: 4
Case #2: 8
(Please note that there is a blank before '#' and after ':')
Hint
In the first test case, the 0-1 sequence of the first case is "111010111101011", and the number of occurrences of harmony string "01" is 4.
也要按照鼹鼠的出场顺序挖洞，在形成的二叉树按照中序遍历顺序奇数偶数（1,0）得到01序列，问01序列中给定的序列出现多少次。
Sure原创，转载请注明出处
#include <iostream>
#include <cstdio>
#include <memory.h>
#include <algorithm>
using namespace std;
const int maxn = 600002;
const int maxm = 7002;
struct P
{
int k,a;
bool operator < (const P &other) const
{
return k < other.k;
}
}mole[maxn];
struct T
{
int l,r,fa;
}tree[maxn];
int next[maxm],stack[maxn],state[maxn];
char str[maxm],list[maxn * 3];
int n,tot,top;
void init()
{
memset(tree,0,sizeof(tree));
memset(state,0,sizeof(state));
tot = top = 0;
return;
}
{
scanf("%d",&n);
for(int i=1;i<=n;i++)
{
scanf("%d",&mole[i].k);
mole[i].a = i;
}
sort(mole + 1 , mole + n + 1);
scanf("%s",str);
return;
}
void rotate(int x)
{
int p = tree[x].fa;
int l = tree[x].l;
if(p == tree[tree[p].fa].l)
{
tree[tree[p].fa].l = x;
}
else
{
tree[tree[p].fa].r = x;
}
tree[x].fa = tree[p].fa;
tree[x].l = p;
tree[p].fa = x;
tree[p].r = l;
if(l) tree[l].fa = p;
return;
}
void dfs(int st)
{
stack[top++] = st;
while(top)
{
int cur = stack[top-1];
list[tot++] = (mole[cur].k & 1) + '0';
if(++state[cur] == 3)
{
top--;
continue;
}
if(state[cur] == 1)
{
if(tree[cur].l) stack[top++] = tree[cur].l;
else state[cur]++;
}
if(state[cur] == 2)
{
if(tree[cur].r) stack[top++] = tree[cur].r;
else state[cur]++;
}
if(state[cur] == 3) top--;
}
return;
}
void make()
{
tree[0].r = 1;
int rightmost = 1;
for(int i=2;i<=n;i++)
{
tree[rightmost].r = i;
tree[i].fa = rightmost;
rightmost = i;
while(tree[i].fa && mole[i].a < mole[tree[i].fa].a) rotate(i);
}
dfs(tree[0].r);
list[tot] = '\0';
return;
}
void getnext(char *s)
{
int len=strlen(s);
for(int i=0;i<=len;i++)
{
next[i] = 0;
}
next[0] = -1;
int j=-1;
for(int i=0;i<len;)
{
if(j==-1||s[i]==s[j])
{
i++;
j++;
next[i]=j;
}
else j=next[j];
}
return;
}
int kmp(char a[],char b[])
{
int count=0;
getnext(b);
int lena = strlen(a);
int lenb = strlen(b);
int j=0,i=0;
while(i<lena && j<lenb)
{
if(j==-1 || b[j]==a[i])
{
j++;
i++;
}
else j=next[j];
if(j == lenb)
{
count++;
j = next[j];
}
}
return count;
}
int main()
{
int cas;
scanf("%d",&cas);
for(int i=1;i<=cas;i++)
{
printf("Case #%d: ",i);
init();
make();
printf("%d\n",kmp(list,str));
}
return 0;
}
1. 算法是程序的灵魂，算法分简单和复杂，如果不搞大数据类，程序员了解一下简单点的算法也是可以的，但是会算法的一定要会编程才行，程序员不一定要会算法，利于自己项目需要的可以简单了解。
2. 学算法中的数据结构学到一定程度会乐此不疲的，比如其中的2－3树，类似的红黑树，我甚至可以自己写个逻辑文件系统结构来。
|
2016-12-07 22:13:00
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2989249527454376, "perplexity": 3422.03505446467}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542250.48/warc/CC-MAIN-20161202170902-00501-ip-10-31-129-80.ec2.internal.warc.gz"}
|
https://www.jobilize.com/course/section/time-for-another-test-the-generic-quadratic-equation-by-openstax?qcr=www.quizover.com
|
# 4.5 The "generic" quadratic equation
Page 1 / 1
Begin by reminding them of what we did with simultaneous equations. First, we learned how to solve them (using substitution or elimination). Then we used those exact same techniques to solve the generic version—that is, simultaneous equations where all the numbers were replaced by letters. This, in turn, gave us a formula that could instantly be used to solve any pair of simultaneous equations.
Now we are going to do that same thing with quadratic equations. The “generic” quadratic equation is, of course, $a{x}^{2}+bx+c=0$ . Now, we have learned two different ways of solving such equations. The “generic” version is hard to solve by factoring (although it is possible); we are going to do it by completing the square.
Make sure they look over my example of completing the square; this might be a good opportunity for a quick TAPPS exercise. There are other examples in the “Conceptual Explanations” so you could do two TAPPS exercises—that way everyone gets a chance to be the teacher.
Then have them work through the sheet. They should derive the quadratic formula, and then use it.
By the time they are done, they should have two things. They should have the quadratic formula, and a bit of practice using it—so now we have three different techniques for solving quadratic equations. They should also have derived the formula. I always warn them that I will ask for this derivation on the next test: it is not enough to know the formula (although that too is good), you have to be able to derive it.
At the end of class, you may want to talk for just a couple of minutes about the discriminant, in reference to #11. It should be fairly obvious by that point to most of them.
## Time for another test!
As always, there is the sample test, which may or may not be assigned as a homework. Then there is the test—on multiplying polynomials, on factoring, and mostly on solving quadratic equations. Make it shorter than my sample ☺
the art of managing the production, distribution and consumption.
what is economics
okk
damfash
marginal utility is the additional satisfaction one derives from consuming additional unit of a good or service.
Fred
It's the allocation of scarce resources.
Fred
Dishan
marginal utility is the additional satisfaction one derives from consuming additional unit of a good or service.
Fred
I know the definition, but I don't understand its meaning.
Dishan
what is the must definition of economic please?
Nurudeen
demand lfs
Alpha
Economics is derived from the word Oikonomia which means management of household things. Thus, Economics is a study of household things with the constrains of allocating scare resources.
Dishan
what is Open Market Operation
dominating middlemen men activities circumstances
what Equilibrium price
what is gap
mirwais
who is good with the indifference curve
Dexter
What is diseconomic
what are the types of goods
WARIDI
how can price determination be the central problem of micro economics
marginal cost formula
you should differentiate the total cost function in order to get marginal cost function then you can get marginal cost from it
boniphace
Foday
ok
Foday
how can price determination be the central problem if micro economics
simon
formula of cross elasticity of demand
what is ceteris paribus
what is ceteris parabus
Priyanka
Ceteris paribus - Literally, "other things being equal"; usually used in economics to indicate that all variables except the ones specified are assumed not to change.
Abdullah
What is broker
scor
land is natural resources that is made by nature
scor
What is broker
scor
what is land
kafui
What is broker
scor
land is natural resources that is made by nature
scor
whats poppina nigga turn it up for a minute get it
what is this?
Philo
am from nigeria@ pilo
Frank
am from nigeria@ pilo
Frank
so
owusu
what is production possibility frontier
owusu
it's a summary of opportunity cost depicted on a curve.
okhiria
please help me solve this question with the aid of appropriate diagrams explain how each of the following changes will affect the market price and quantity of bread 1. A
ok let me know some of the questions please.
Effah
ok am not wit some if den nw buh by tommorow I shall get Dem
Hi guys can I get Adam Smith's WEALTH OF NATIONS fo sale?
Ukpen
hello I'm Babaisa alhaji Mustapha. I'm studying Economics in the university of Maiduguri
Babaisa
okay
Humaira
my name is faisal Yahaya. i studied economics at Kaduna state university before proceeding to West African union university benin republic for masters
Faisal
Mannan
Wat d meaning of management
disaster management cycle
cooperate social responsibility
igwe
Fedric Wilson Taylor also define management as the act of knowing what to do and seeing that it is done in the best and cheapest way
OLANIYI
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Got questions? Join the online conversation and get instant answers!
|
2019-12-14 05:04:12
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.546119749546051, "perplexity": 1620.1361373589386}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540584491.89/warc/CC-MAIN-20191214042241-20191214070241-00167.warc.gz"}
|
https://www.mysciencework.com/publication/show/rigidity-maximal-holomorphic-representations-k-ahler-groups-d395f0de
|
# Rigidity of maximal holomorphic representations of K\"ahler groups
Authors
Type
Preprint
Publication Date
Submission Date
Identifiers
arXiv ID: 1409.2816
Source
arXiv
We investigate representations of K\"ahler groups $\Gamma = \pi_1(X)$ to a semisimple non-compact Hermitian Lie group $G$ that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor--Wood inequality similar to those found by Burger--Iozzi and Koziarz--Maubon. Thanks to the study of the case of equality in Royden's version of the Ahlfors--Schwarz Lemma, we can completely describe the case of maximal holomorphic representations. If $\dim_{\C}X \geq 2$, these appear if and only if $X$ is a ball quotient, and essentially reduce to the diagonal embedding $\Gamma < \SU(n,1) \to \SU(nq,q) \hookrightarrow \SU(p,q)$. If $X$ is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, that thus appear as preferred elements of the respective maximal connected components.
|
2018-02-25 04:01:08
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9306146502494812, "perplexity": 513.7954092235839}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816094.78/warc/CC-MAIN-20180225031153-20180225051153-00391.warc.gz"}
|
https://cob.silverchair.com/jeb/article/207/11/1825/14825/The-hydrodynamics-of-eel-swimmingI-Wake-structure
|
Eels undulate a larger portion of their bodies while swimming than many other fishes, but the hydrodynamic consequences of this swimming mode are poorly understood. In this study, we examine in detail the hydrodynamics of American eels (Anguilla rostrata) swimming steadily at 1.4 Ls-1 and compare them with previous results from other fishes. We performed high-resolution particle image velocimetry (PIV) to quantify the wake structure, measure the swimming efficiency, and force and power output. The wake consists of jets of fluid that point almost directly laterally,separated by an unstable shear layer that rolls up into two or more vortices over time. Previously, the wake of swimming eels was hypothesized to consist of unlinked vortex rings, resulting from a phase offset between vorticity distributed along the body and vorticity shed at the tail. Our high-resolution flow data suggest that the body anterior to the tail tip produces relatively low vorticity, and instead the wake structure results from the instability of the shear layers separating the lateral jets, reflecting pulses of high vorticity shed at the tail tip. We compare the wake structure to large-amplitude elongated body theory and to a previous computational fluid dynamic model and note several discrepancies between the models and the measured values. The wake of steadily swimming eels differs substantially in structure from the wake of previously studied carangiform fishes in that it lacks any significant downstream flow, previously interpreted as signifying thrust. We infer that the lack of downstream flow results from a spatial and temporal balance of momentum removal (drag) and thrust generated along the body, due to the relatively uniform shape of eels. Carangiform swimmers typically have a narrow caudal peduncle, which probably allows them to separate thrust from drag both spatially and temporally. Eels seem to lack this separation, which may explain why they produce a wake with little downstream momentum while carangiform swimmers produce a wake with a clear thrust signature.
Different fishes swim in different ways. To categorize this diversity, fish swimming is usually classified into a variety of different modes. A primary grouping distinguishes several modes among fishes that use their body and caudal fin primarily for propulsion. In particular, eel-like, oranguilliform', fishes undulate a large portion of their bodies, while jack-like, or carangiform', fishes undulate much less(Breder, 1926; Webb, 1975). These kinematic distinctions have been recognized for many years, even before Breder gave them their modern names in 1926 (Alexander,1983), but the hydrodynamic consequences of the differences in kinematics are not well understood.
Most modern studies on the hydrodynamics of fish swimming have been done on carangiform swimmers. These fishes tend to have fusiform or laterally compressed bodies, often with a pronounced caudal peduncle. The greatest lateral excursions occur near the peduncle and the caudal fin(Webb, 1975), although there may be some yawing motions at the head(Donley and Dickson, 2000). In addition, researchers have distinguished several gradations of carangiform swimming, from subcarangiform, in which a greater proportion of the body undulates, to thunniform, in which the tail moves largely independently of the body (Webb, 1975). While swimming, carangiform fishes produce a series of vertical linked vortex rings,angled to the swimming direction(Müller et al., 1997; Triantafyllou et al., 2000; Drucker and Lauder, 2001; Nauen and Lauder, 2002a).
The hydrodynamics of anguilliform swimming have been studied much less. Like the eel, after which this mode is named, anguilliform swimmers tend to be elongate with little or no narrowing at the caudal peduncle. This lack of separation between the body and tail is particularly extreme in eels, in which the dorsal, caudal and anal fins effectively form a continuous median fin(Helfman et al., 1997). In other anguilliform swimmers, such as sharks and needlefish, the fins are more separated and there may be a slight narrowing at the caudal peduncle(Liao, 2002). They undulate from one-third to almost all of their bodies, depending on speed, often with one or more complete waves present at a time(Gillis, 1998). These extra undulations, relative to carangiform swimmers, must affect the flow around their bodies and in the wake, but the effect is not well understood.
Lighthill's elongated body theory (referred to here as EBT) offers some insight into the possible effect of different kinematics(Lighthill, 1971). He argues that the carangiform mode is more efficient, because his theory predicts that thrust is produced only at the trailing edge of the tail. Therefore, any extra body undulation is wasted energy, and efficient swimmers should undulate as little of their body as possible. Indeed, many pelagic predators considered highly efficient (Lighthill,1970; Barrett et al.,1999) are thunniform swimmers and hold their bodies relatively straight. EBT, however, is a simple model, and neglects many effects,including viscous forces, which could enable thrust production along the length of an anguilliform fish's body(Taneda and Tomonari, 1974; Shen et al., 2003).
Only two recent studies (Carling et al.,1998; Müller et al.,2001) address the hydrodynamics of eel-like swimming, and they offer divergent conclusions. Müller et al.(2001) used particle image velocimetry (PIV; Willert and Gharib,1991) to observe the flow fields around freely swimming juvenile eels. Based on their observations, they hypothesized that eels' wakes consist of unlinked vortex rings moving laterally(Fig. 1A). They proposed that eels shed two separate same-sign vortices because of a lag between the stop/start vortex (solid arrows in Fig. 1A), shed when the tail changes direction, and centers of rotation that progress down the body, which they termed proto-vortices' (broken arrows in Fig. 1A). They did not observe a downstream jet behind the tail, which is typical of carangiform wakes (e.g. Nauen and Lauder,2002a). Due to the difficulty of working with freely swimming eels, Müller and colleagues did not evaluate the effects of different swimming speeds on the wake structure. Also, the mechanical significance of the difference between carangiform wakes and the wake they observed for eels remains unclear.
Fig. 1.
Flow fields behind swimming eels according to two previous studies. Red arrows indicate flow with clockwise rotation and blue arrows indicate counter-clockwise rotation. (A) Results from Müller et al.(2001), showing the wake structure they observed. Proto-vortices (dotted lines) appear to be vortices centered on the body that progress down the body. After they are shed into the wake they are shown as dashed lines. They are shed after the stop/start vortex(solid lines), resulting in two same-sign vortices being shed each tail beat.(B) Computational fluid dynamic model of Carling et al.(1998). The model indicates a large flow wrapping around the eel, resulting in a net upstream flow in the wake behind the eel.
Fig. 1.
Flow fields behind swimming eels according to two previous studies. Red arrows indicate flow with clockwise rotation and blue arrows indicate counter-clockwise rotation. (A) Results from Müller et al.(2001), showing the wake structure they observed. Proto-vortices (dotted lines) appear to be vortices centered on the body that progress down the body. After they are shed into the wake they are shown as dashed lines. They are shed after the stop/start vortex(solid lines), resulting in two same-sign vortices being shed each tail beat.(B) Computational fluid dynamic model of Carling et al.(1998). The model indicates a large flow wrapping around the eel, resulting in a net upstream flow in the wake behind the eel.
By contrast, Carling et al.(1998) used a two-dimensional computational fluid dynamic model to estimate the flow fields behind a self-propelled eel-like' anguilliform swimmer. Their calculations indicated a single, large vortex ring wrapping around the eel, with the eel in the center,producing upstream flow behind the eel(Fig. 1B). These results suggest that eels produce thrust almost exclusively along the body, but not at the tail tip, which seems to result mostly in drag. Carling's model, while tested thoroughly in several standard test cases(Carling, 2002), has not been verified on living eels.
Several important questions remain. Which of these two views of anguilliform wake flow patterns are correct? What are the quantitative differences between anguilliform and carangiform wakes? How do these differences affect the swimming performance? How efficient hydrodynamically is anguilliform swimming relative to carangiform swimming? In particular,Lighthill's (1970) argument for the inefficiency of anguilliform swimming leads to an inconsistency: eels migrate thousands of kilometers without feeding(van Ginneken and van den Thillart,2000), and many anguilliform sharks swim constantly(Donley and Shadwick, 2003). It is unlikely that such proficient swimmers are highly inefficient. In fact,a recent study of swimming energetics found that the physiological cost of migration for eels was low (van Ginneken and van den Thillart, 2000).
In the present study, therefore, we examine in detail the wake of the American eel, Anguilla rostrata, swimming steadily at a single speed. The flow around anguilliform swimmers is compared with both previous models and with previous data from carangiform swimmers. We propose a new explanation of the hydrodynamic differences between anguilliform and carangiform swimming,emphasizing the importance of carangiform swimmers' narrow caudal peduncle and propeller-like caudal fin over the importance of differences in kinematics. In addition, we provide the first quantitative comparison of the predictions of EBT (Lighthill, 1971) to empirical forces estimated using PIV and demonstrate a partial correlation. Finally, we examine the efficiency and power output for steadily swimming eels.
### Animals and experimental procedure
We obtained American eels, Anguilla rostrata LeSueur, by seine from the Charles River, Cambridge, MA, USA during June and July 2002 and housed them in aquaria at room temperature with a 12 h:12 h light:dark cycle. We performed experiments on 11 individuals, ranging from 8 cm to 23 cm total body length (L), in a 600-liter recirculating flow tank with a 26 cm×26 cm×80 cm working section. Three individuals (L=20 cm, 20 cm and 23 cm, corresponding to masses of 14 g, 16 g and 14 g) that swam exceptionally steadily were chosen for detailed analysis. Before the experiment, an eel was moved from its tank to the flow tank and allowed to acclimate. Animals were confined to the working section using plastic grids upstream and downstream with 5 cm×5 cm holes covered in a fine mesh. After an acclimation period of approximately 1 h in flow of ∼1 Ls-1, the eels spontaneously adopted steady swimming behavior on the bottom of the flow tank. The eels would not swim consistently in a mid-water plane and, since eels often naturally swim on the bottom in rivers during daylight (Smith and Tighe,2002), we focused on eels swimming in that region. This also allowed comparison with previous work by Gillis(1998) that studied eels swimming on the bottom.
All data were taken from eels swimming at ∼1.4 Ls-1, ranging from 1.30 to 1.44 L s-1. In total,the swimming kinematics for 415 tail beats were analyzed. The hydrodynamics of 118 of these were examined. Considerable effort was expended to analyze only truly steady swimming sequences; all sequences analyzed had a maximum variation in velocity under ±5%; in most cases, the velocity varied by less than ±3%; and the s.d. in velocity over all sets was 2%. In addition, most sequences involved 10 or more sequential steady tail beats. During the experiments, an eel was gently maneuvered into position using a wooden probe. Care was taken to remove the probe completely from the region around the eel before data were taken.
The eels were filmed from below through a mirror inclined at 45° with two high-speed cameras, one to record the swimming kinematics and one to film the light sheet for PIV (Fig. 2). An approximately 30 cm-wide horizontal light sheet was projected 7 mm above the bottom of the tank, along the dorso-ventral midline of the eels, using two argon-ion lasers operating at ∼4 W and 8 W,respectively. The light from the two lasers was combined optically to form a single large light sheet. The eels' swimming kinematics were recorded using a RedLake digital camera at 250 or 125 frames s-1. For PIV, a close-up view of the light sheet was filmed using either a RedLake digital camera at 250 frames s-1 at 480 pixe×420 pixel resolution or a NAC Hi-DCam at 500 frames s-1 at 1280 pixe×1024 pixel resolution. A six-point calibration between the two cameras allowed positions to be converted between the two images with an error of ∼0.5 mm using a linear rotation and scaling transformation (Matlab 6.1 imtransform routine;Mathworks, Inc., Natick, MA, USA)
Fig. 2.
Methods. Eels were filmed from below using two synchronized high-speed cameras aimed at a 45° mirror below the flow tank. A laser light sheet 7 mm above the bottom of the tank illuminated the eel's wake and part of its tail. One camera (labeled kinematics') imaged the whole eel, and the other camera (labeled PIV') imaged the light sheet. Representative images from each camera are shown to the left. Diagram is not to scale.
Fig. 2.
Methods. Eels were filmed from below using two synchronized high-speed cameras aimed at a 45° mirror below the flow tank. A laser light sheet 7 mm above the bottom of the tank illuminated the eel's wake and part of its tail. One camera (labeled kinematics') imaged the whole eel, and the other camera (labeled PIV') imaged the light sheet. Representative images from each camera are shown to the left. Diagram is not to scale.
### Kinematics
Eel outlines and midlines were digitized automatically using a custom Matlab 6.1 (Mathworks, Inc.) program. The positions of the head and tail were identified manually. The eel midline was then located by performing a 1-D cross-correlation analysis along transects between the head and the tail, to find the bright region with a width corresponding to the known width of the eel. This technique produced fewer errors resulting from the presence of PIV particles in portions of the images than thresholding-based techniques used in previous kinematic studies (e.g. Tytell and Lauder, 2002). A similar method located the edges of the eel's image. Twenty points were identified along the midline and were simultaneously smoothed temporally and spatially using a 2-D tensor product spline (Matlab's spaps routine), a two-dimensional analog of an optimal method (MSE' method in Walker, 1998). The tensor product spline, however, does not allow a direct specification of the mean error on the data as in the 1-D version. Thus, the smoothing values were initially set at 0.5 pixel, the limit of measurement accuracy from the video,and adjusted manually until a good fit was reached. This resulted in a mean distance between the smoothed and measured values of less than 0.3 pixel(approximately 0.2 mm).
Kinematic variables, including amplitude at each body point, tail beat frequency, body wave length and body wave velocity, were calculated by finding the peaks in the lateral excursion of each point over time. A Matlab program automatically located the peaks based on the midlines, as estimated above. The amplitude, frequency and wave length were determined by the timing, position and height of the peaks. Average side-to-side tail velocity was estimated as 4A/f, where A is amplitude and f is frequency. The body wave velocity was determined by the slope of the line fitted to the wave peak position (in distance down the body) and the time of that wave peak.
The forces and power required for swimming were calculated using large-amplitude EBT (Lighthill,1971). The time-varying thrust (Fthrust) and lateral force (Flateral) are:
$\ \begin{array}{l}F_{\mathrm{thrust}}=\left\lfloormv_{{\perp}}\frac{{\partial}y_{\mathrm{b}}}{{\partial}t}+\frac{1}{2}mv_{{\perp}}^{2}\frac{{\partial}x_{\mathrm{b}}}{{\partial}s}\right\rfloor_{s=L}+\frac{{\partial}}{{\partial}t}{{\int}_{0}^{L}}mv_{{\perp}}\frac{{\partial}y_{\mathrm{b}}}{{\partial}s}ds\\F_{\mathrm{lateral}}=\left[-mv_{{\perp}}\left(\frac{{\partial}x_{\mathrm{b}}}{{\partial}t}+U\right)+\frac{1}{2}mv_{{\perp}}^{2}\frac{{\partial}y_{\mathrm{b}}}{{\partial}s}\right]_{s=L}\\-\frac{{\partial}}{{\partial}t}{{\int}_{0}^{L}}mv_{{\perp}}\frac{{\partial}x_{\mathrm{b}}}{{\partial}s}ds,\end{array}$
1
where xb(s,t) and yb(s,t) are the position of points along the midline of an eel facing in the positive x direction in flow with velocity U towards the eel, m is the virtual mass per unit length, L is the eel's length, s is the distance along the midline from head to tail, t is time and v⊥ is the body velocity perpendicular to the midline:
$\ v_{{\perp}}=\frac{{\partial}x_{\mathrm{b}}}{{\partial}t}\frac{{\partial}y_{\mathrm{b}}}{{\partial}s}-\frac{{\partial}y_{\mathrm{b}}}{{\partial}t}\frac{{\partial}x_{\mathrm{b}}}{{\partial}s}+U\frac{{\partial}y_{\mathrm{b}}}{{\partial}s}.$
2
Fig. 3 shows the coordinate system and variable definitions. The wasted power (Pwake)shed into the wake is
$$\frac{1}{2}[mv_{{\perp}}^{2}v_{{\parallel}}]_{s=L}$$
,where v∥ is the velocity parallel to the midline:
$\ v_{{\parallel}}=\frac{{\partial}x_{\mathrm{b}}}{{\partial}t}\frac{{\partial}x_{\mathrm{b}}}{{\partial}s}-\frac{{\partial}y_{\mathrm{b}}}{{\partial}t}\frac{{\partial}y_{\mathrm{b}}}{{\partial}s}+U\frac{{\partial}x_{\mathrm{b}}}{{\partial}s}.$
3
Fig. 3.
Coordinate system used for elongated body theory calculations. The solid line represents a midline at one time, while the dotted line represents it at a later time. Perpendicular and parallel velocity, v⊥ and v∥, are shown as vectors at the point (xb, yb). The arc length s is shown along the midline and the eel is swimming into a flow U. L is total body length.
Fig. 3.
Coordinate system used for elongated body theory calculations. The solid line represents a midline at one time, while the dotted line represents it at a later time. Perpendicular and parallel velocity, v⊥ and v∥, are shown as vectors at the point (xb, yb). The arc length s is shown along the midline and the eel is swimming into a flow U. L is total body length.
Additionally, the position of the proto-vortices along the eel's body was estimated according to Müller et al.(2001) by searching for the points along the body where lift (Flift) equals zero,defined using small-amplitude EBT(Lighthill, 1960) as:
$\ F_{\mathrm{lift}}=\frac{{\pi}}{4}{\rho}h^{2}({\partial}{/}{\partial}t+U{\partial}{/}{\partial}s)({\partial}y_{\mathrm{b}}{/}{\partial}t+U{\partial}y_{\mathrm{b}}{/}{\partial}s),$
4
where ρ is the water density and h is the dorso-ventral height of the eel. Because of the error introduced by taking second derivatives of values with measurement error, an analytical expression for the midline position was used to find the zero lift positions. Using the kinematic values measured, yb can be expressed as Aeα(s–1) sin k(sVt), where A is the tail beat amplitude, α is a parameter defining how fast the amplitude grows from head to tail, k is the wave number (equal to 2π/wave length), and V is the body wave speed.
### Hydrodynamics
High-resolution PIV was performed using a custom Matlab 6.1 program in two passes using a standard statistical cross-correlation(Fincham and Spedding, 1997)and a Hart (2000) error correction technique with an integer pixel estimate of the velocity between passes (as in Westerweel et al.,1997; Hart, 1999). PIV interrogation regions were about 5 mm×5 mm and 2.5 mm×2.5 mm in coarse and fine pass, respectively, with search regions of 9 mm×9 mm and 3.5 mm×3.5 mm. For the lower resolution video, this produced a matrix 68×78 vectors, and for the higher resolution, 100×125 vectors. Data were smoothed and interpolated onto a regular grid using an adaptive Gaussian window algorithm with the optimal window size (2–3 mm for these data; Agüí and Jiménez, 1987; Spedding and Rignot, 1993), being careful to note the inherently uneven spacing of PIV data (Spedding and Rignot,1993). The Gaussian window method was used because it provides good results (Fincham and Spedding,1997) while being simple and fast when applied to such large matrices of vectors.
### Boundary layers and background flow
Because eels swam on the bottom of the flow tank, we made a series of measurements to quantify the flow regime in this part of the flow tank and to be certain that we were observing free-stream flow. At all swimming speeds,the PIV light sheet was above the flow tank boundary layer, which was turbulent. The boundary layer was quantified using a vertical light sheet,showing that the boundary layer thickness (δ) was equal to ∼7 mm at the slowest flow speed used (Fig. 4). The boundary layer changes from laminar to turbulent just below that speed, indicating that the boundary layers in all data sets were turbulent. At speeds above this transition, the boundary layer is always thinner, decreasing proportionally to the free-stream velocity to the–1/5 power (Schlichting,1979). Thus, at the highest speed used, ∼40 cm s-1,we estimated the boundary layer to be ∼5.5 mm thick.
Fig. 4.
Flow tank boundary layer. The boundary layer thickness was 7.3 mm or less at all swimming velocities. Black boxes are standard statistical box plots,with the box stretching from the 25th to 75th quartile, a white line at the median, and whiskers of 1.5 times the interquartile range. Outliers are shown as separate points. (A) Laminar boundary layer at flow speeds less than 95 mm s-1 with fit Blasius boundary layer profile(Faber, 1995). The boundary layer thickness at 0.99U was 7.3 mm (green dotted lines). (B)Turbulent boundary layer at flow speeds above 120 mm s-1. The normalized distance y+ and velocity u+are shown on the top and right, respectively. The law of the wall profile for turbulent boundary layers, u+=5.75 log y++5.2, is shown in red. Note that this is a semi-log plot. (C) Axial component of velocity from the horizontal light sheet, 7 mm above the bottom, showing turbulent effects. Flow is from bottom to top. Note the streamwise regions of reduced velocity. The bottom profile shows mean velocity (solid line) and mean velocity about two hours later (dotted line). A histogram of velocities (solid line) is also shown beside the color bar with a histogram from about two hours later (dotted line).
Fig. 4.
Flow tank boundary layer. The boundary layer thickness was 7.3 mm or less at all swimming velocities. Black boxes are standard statistical box plots,with the box stretching from the 25th to 75th quartile, a white line at the median, and whiskers of 1.5 times the interquartile range. Outliers are shown as separate points. (A) Laminar boundary layer at flow speeds less than 95 mm s-1 with fit Blasius boundary layer profile(Faber, 1995). The boundary layer thickness at 0.99U was 7.3 mm (green dotted lines). (B)Turbulent boundary layer at flow speeds above 120 mm s-1. The normalized distance y+ and velocity u+are shown on the top and right, respectively. The law of the wall profile for turbulent boundary layers, u+=5.75 log y++5.2, is shown in red. Note that this is a semi-log plot. (C) Axial component of velocity from the horizontal light sheet, 7 mm above the bottom, showing turbulent effects. Flow is from bottom to top. Note the streamwise regions of reduced velocity. The bottom profile shows mean velocity (solid line) and mean velocity about two hours later (dotted line). A histogram of velocities (solid line) is also shown beside the color bar with a histogram from about two hours later (dotted line).
Because the boundary layer was turbulent, the background flow was complex. Turbulent boundary layers are characterized by a range of relatively long-lived, coherent structures that rise up out of the boundary layer region(Robinson, 1991). In particular, structures called quasi-streamwise vortices' were common. In our data, quasi-streamwise vortices, which are vortex lines oriented approximately parallel to the flow (Robinson,1991), were visible as streamwise regions of slower or angled flow(Fig. 4C). Conveniently, they were consistent over a duration of many minutes.
The consistency of the turbulent structures enabled us to subtract their effect from the flow. For each swimming speed, we took 50 flow fields without the eel present. These fields were then averaged to estimate a mean background flow, which was subtracted from the wake data to remove the turbulent effects. The background velocity changed spatially by as much as 13% cm-1but changed over time by only about 0.1% s-1(Fig. 4C).
### Wake analysis
Wakes were only analyzed when the kinematics remained steady for at least three tail beats. Most wakes analyzed included between five and 15 consecutive steady tail beats. Phase-averaged wake vector fields were produced by averaging frames corresponding to the same tail-beat phase, dividing the tail beat into 20 steps. These phase-averaged fields are instructive for visualization, but no quantitative values were measured from them.
Vortex centers were digitized manually. Location of the vortices in the vector fields was aided by plotting the discriminant for complex eigenvalues(DCEV; Vollmers, 1993; Stamhuis and Videler, 1995):
$\ ({\partial}u{/}{\partial}x+{\partial}v{/}{\partial}y)^{2}-4({\partial}u{/}{\partial}x{\ }{\partial}v{/}{\partial}y-{\partial}u{/}{\partial}y{\ }{\partial}v{/}{\partial}x),$
5
where u and v are the x and y components of velocity, respectively. DCEV is negative in regions where the fluid is rotating more than it is diverging. Two vortices were identified for each half tail beat. When the tail changes direction, it sheds a primary vortex. As the tail moves to the other side, it stretches the primary vortex into a shear layer, which eventually rolls up, producing a secondary vortex. The vortex circulation was determined by integrating a circle at an 8 mm radius from the vortex core. This radius was determined by inspecting the circulation values at increasing radii for many vortices. Finally, the width of the lateral jet was determined by the pair of vortices on either side (the primary vortex of one half tail beat and the secondary vortex of the next). These are the two vortices identified by Müller et al.(2001) as the cores of a vortex ring. The angle of the line between the two vortices and the mean jet velocity in the region between them were determined. Finally, the velocity along the center line between the two vortices was integrated to get another measure of circulation.
Assuming that the two vortices on either side of the lateral jet are the cores of a small-core vortex ring, the impulse (I) of the ring was estimated as (Faber, 1995):
$\ I=\frac{1}{4}{\pi}{\rho}{\Gamma}hd,$
6
where Γ is circulation across the center line of the vortex ring, h is the height of the ring, equivalent to the eel's height, and d is the diameter in the plane of the light sheet. The rings are assumed to be elongated ovals, with the height equal to the height of the eels, 10 mm, as previously observed in other fishes(Lauder, 2000; Drucker and Lauder, 2001; Nauen and Lauder, 2002a). The mean force (FPIV) that produced the vortex ring was also estimated by dividing the impulse by half the tail beat period.
The power (P) that the eel added to the fluid was determined by integrating across a plane approximately 8 mm behind the tail tip:
$\ P=\frac{1}{2}{\rho}Uh{{\int}_{-w}^{w}}[(u+U)^{2}+v^{2}]-U^{2}dy=\frac{1}{2}{\rho}Uh{{\int}_{-w}^{w}}u^{2}+2Uu+v^{2}dy,$
7
where h is the height of the area affected by the eel, equivalent to the eel's height, and w is the half-width of the wake (∼40 mm). Because of the uncertainty introduced by the quasi-streamwise vortices, and because almost all of the wake velocity was lateral, a lateral' power(Plateral) was calculated using only the vcomponent of velocity:
$\ P_{\mathrm{lateral}}=\frac{1}{2}{\rho}Uh{{\int}_{-w}^{w}}v^{2}dy.$
8
To account for the phase lag between when the kinetic energy was shed at the tail and when it reached the position xplane where power was measured in the wake, the phasing of the wake power was adjusted by 2πxplanef/U.
Force, impulse and power were both normalized to produce force and power coefficients. Using coefficients is important because it makes these values comparable between eels of different sizes and between the present and other studies (Schultz and Webb,2002). The normalization factors for force and power were the standard ΓρSU2 andΓρ SU3, respectively, where S is the wetted surface area of the eel (Faber,1995). No standard normalization exists for impulse, however. Since impulse is in units of force × time, we chose to normalize by the standard characteristic force ΓρSU2 and a characteristic time L/U, resulting in an impulse normalization factor of ΓρSLU.
### Statistics
The kinematics in the data set used for PIV measurements were compared with those in the complete data set to make sure that the swimming behavior in the selected data was typical. A mixed-model multivariate analysis of variance(MANOVA; Zar, 1999) was performed on the kinematic variables, including the individual as a random effect and which set the data came from (i.e. the PIV or complete data sets)as a fixed effect. The kinematic differences between individuals in the PIV data set were also assessed using a MANOVA including only the effect of individual variation.
The changes in wake morphology over time were examined by regressing individual wake morphology parameters on tail-beat phase, including the individual as a random effect. Significant slopes were determined by testing the significance of variation in time over the variation due to the interaction by individuals with time, as in a mixed-model analysis of variance(ANOVA).
A repeated-measures ANOVA (Zar,1999) was performed to compare the initial circulation of the primary vortex to the sum of the circulations of the primary and secondary vortices, after they divided. Circulation at two different times was the repeated measure, which allowed the early primary vortex circulation to be compared with the sum of the primary and secondary vortex circulations later in time. The individual was included as a random factor(Zar, 1999).
Finally, mixed-model ANOVAs were used to compare force, power and impulse estimates based on EBT (Lighthill,1971) with those values measured using PIV. The individual was again included as a random factor.
All analyses were performed using Systat 10.2 (Systat Software, Inc., Point Richmond, CA, USA). All error values that are reported are standard error and include the number of data points, where appropriate.
### Kinematics
At the moderate swimming speed of ∼1.4 L s-1, all individuals swam very steadily and repeatably. For the three individuals studied in detail, swimming speed varied by a maximum of about ±4% and generally varied less than ±2%. At that speed, the animals swam with a tail-beat amplitude (A) of 7% of body length at a frequency(f) of 3.1 Hz. A and f are approximately inversely proportional to each other, even within this small speed range(r=–0.669). Thus, the product, the average tail velocity(4A/f) over a period, and average Strouhal number are quite constant: 0.856±0.007 L s-1 and 0.314±0.003,respectively. Body wave velocity was generally 1.878±0.006 Ls-1, resulting in slip of 0.73, an indication of the swimming efficiency (Lighthill, 1970). The body wave length was usually ∼60% of body length, or ∼1.65 waves on the body at any given time. Amplitude increased exponentially along the body as Aeα(s–1), where s is the distance along the body, from 0 at the head to 1 at the tail, and αis a parameter that defines how fast the amplitude grows(r2=0.978). α was equal to 2.76±0.01. A linear regression did not fit the data nearly as well; r2was 0.890 and the residuals were visibly non-normal. Kinematic parameters are summarized in Table 1.
Table 1.
Kinematic parameters
VariableSymbolValue ± s.e.m.Units
Length L 20.8±0.7* cm
Swimming velocity U 1.374±0.002 L s-1
Reynolds number Re 60 000
Tail beat amplitude A 0.0693±0.0005 L
Tail beat frequency f 3.11±0.03 Hz
Body wave velocity V 1.878±0.006 L s-1
Body wave length 0.604±0.006 L
Average tail velocity 4A/f 0.856±0.006 L s-1
Strouhal number St 0.314±0.003
Slip U/V 0.731±0.002
Stride length U/f 0.448±0.005 L
Amplitude growth parameter α 2.759±0.009
VariableSymbolValue ± s.e.m.Units
Length L 20.8±0.7* cm
Swimming velocity U 1.374±0.002 L s-1
Reynolds number Re 60 000
Tail beat amplitude A 0.0693±0.0005 L
Tail beat frequency f 3.11±0.03 Hz
Body wave velocity V 1.878±0.006 L s-1
Body wave length 0.604±0.006 L
Average tail velocity 4A/f 0.856±0.006 L s-1
Strouhal number St 0.314±0.003
Slip U/V 0.731±0.002
Stride length U/f 0.448±0.005 L
Amplitude growth parameter α 2.759±0.009
N=118 except where indicated (*N=3; N=2180).
To verify that the sequences chosen for hydrodynamic analysis were typical of overall swimming performance, we examined a larger data set containing 415 tail beats taken under the same conditions but in which the PIV data were not quantitatively analyzed. A MANOVA on four parameters (tail-beat amplitude and frequency, amplitude growth parameter, body wave length and slip) that completely define the kinematics did not show a significant difference between the larger data set and that used for hydrodynamic analysis (Wilk's lambda=0.978; F4,409=1.858; P=0.101).
Swimming kinematics varied significantly among individuals. In most variables, individuals differed from one another by less than 10%. However,one individual consistently chose to swim with a higher amplitude (about 13%higher) and lower frequency (about 25% lower) than the others. Another individual used a longer body wave (about 20% longer) than the others. By contrast, wave velocity differed very little among individuals; all were within 5% of each other. While these differences were highly significant(MANOVA: Wilk's lambda=0.0139; F10,200=149.5; P<0.001), most studies of this nature have significant variation among individuals (e.g. Shaffer and Lauder, 1985).
Given the average swimming kinematics, the predicted position of the proto-vortex was calculated analytically using equation 4. The proto-vortex is shed off the tail 16 ms after the tail reaches its maximum lateral excursion,or 5.1% of a tail-beat cycle later.
### Hydrodynamics
In all 11 individuals, the wake consisted of lateral jets of fluid,alternating in direction, separated by one or more vortices or a shear layer(Fig. 5). Each time the tail changes direction, it sheds a stop/start vortex. As the tail moves to the other side, a low pressure region develops in the posterior quarter of the body, sucking a bolus of fluid laterally. The bolus is shed off the tail,stretching the stop/start vortex into an unstable shear layer, which eventually rolls up into two or more separate, same-sign vortices. The wake generally contains more total power than is predicted by large-amplitude EBT(Lighthill, 1971). These features are analyzed in detail below, focusing on the three individuals chosen for detailed quantitative study.
Fig. 5.
Representative flow field from behind an eel at 90% of the tail beat cycle. The field is a phase average of 14 tail beats. Vorticity is shown in color in the background, and contours of the discriminant for complex eigenvalues at–200,–500 and–1000 are shown in red. The eel's tail is in blue at the bottom, with red arrows, scaled in the same way as the flow vectors, which indicate the motion of the tail. Vector heads are retained on vectors shorter than 2.5 cm s-1 to show the direction of the flow.
Fig. 5.
Representative flow field from behind an eel at 90% of the tail beat cycle. The field is a phase average of 14 tail beats. Vorticity is shown in color in the background, and contours of the discriminant for complex eigenvalues at–200,–500 and–1000 are shown in red. The eel's tail is in blue at the bottom, with red arrows, scaled in the same way as the flow vectors, which indicate the motion of the tail. Vector heads are retained on vectors shorter than 2.5 cm s-1 to show the direction of the flow.
### Wake morphology
The stop/start vortex, shed when the tail changes direction, is designated the primary' vortex. The vortex formed later, when the shear layer rolls up,is called the secondary' vortex. The primary vortex from one half tail beat and the secondary vortex from the next form the edges of each lateral jet. These two vortices appear to be the cores of a small-core vortex ring. However, without velocity data from the planes perpendicular to the one used in the present study, it is not certain that the vortices truly form a ring. To emphasize this difference, we will not call this region a vortex ring;instead, we term it the lateral jet.
To address how the wake changes over time, wake morphology parameters were regressed individually on tail-beat phase and individual, treating the individual as a random factor. In general, the wake widens over time and becomes weaker. The distance between the primary and secondary vortex increases at ∼0.12 L T-1, where T is a tail-beat period (F1,2=28.7; P=0.033), during the approximately 1.5 T in which the wake was visible. The diameter of the lateral jet, however, stays approximately constant at 0.21 Lthroughout time (F1,2=0.370; P=0.605). The two vortices on either side of the lateral jet (the vortex ring') stay parallel to the swimming direction (F1,2=0.037; P=0.864),but the lateral jet itself is inclined slightly upstream, with an angle of 87° (significantly different from 90°; P<0.001). There is a trend for the jet to rotate downstream over time, but it is not significant(F1,2=1.860; P=0.306). The peak velocities in the jet decrease significantly over time (F1,2=24.0; P=0.039), diminishing by ∼15% over a half tail beat, from 0.45 to 0.38 L s-1. By contrast, the circulation measured through the center of the lateral jet does not change over time(F1,2=1.536; P=0.349), remaining at 2490±10 cm2 s-1.
To illustrate the rolling up of the unstable shear layer, we took cross-sections through the primary and secondary vortices over time(Fig. 6A). The idealized profile through a single Rankine vortex(Faber, 1995) is shown above the first profile and a profile through two same-sign vortices is shown below the last profile. Additionally, Fig. 6B shows cross-sections across the lateral jet over time, with an ideal profile through a small-core vortex ring.
Fig. 6.
Velocity transects through vortices in the eel's wake over time. The center of the first vortex is shown by the vertical dotted lines, and zero velocity is indicated by the horizontal dotted lines. Representative flow fields are shown to the right, indicating the position of the transect, with vorticity shown in color. The cross identifies the position of the first vortex, and the circle identifies the position of the second. Standard error around each velocity trace is shown by a lighter-colored region. (A) Transects through the primary vortex and, once it is formed, the secondary vortex. Idealized profiles through a single Rankine vortex and two same-sign Rankine vortices are shown in black at top and bottom. The position of the secondary vortex,plus or minus standard error, is shown as a bar along the zero line. Before the secondary vortex is completely formed, this bar indicates the position of the inflection point in velocity where the vortex will be formed. (B)Transects across the lateral jet, from the secondary vortex of one half tail beat to the primary vortex of the next. An idealized profile through a small-core vortex ring is shown in black above.
Fig. 6.
Velocity transects through vortices in the eel's wake over time. The center of the first vortex is shown by the vertical dotted lines, and zero velocity is indicated by the horizontal dotted lines. Representative flow fields are shown to the right, indicating the position of the transect, with vorticity shown in color. The cross identifies the position of the first vortex, and the circle identifies the position of the second. Standard error around each velocity trace is shown by a lighter-colored region. (A) Transects through the primary vortex and, once it is formed, the secondary vortex. Idealized profiles through a single Rankine vortex and two same-sign Rankine vortices are shown in black at top and bottom. The position of the secondary vortex,plus or minus standard error, is shown as a bar along the zero line. Before the secondary vortex is completely formed, this bar indicates the position of the inflection point in velocity where the vortex will be formed. (B)Transects across the lateral jet, from the secondary vortex of one half tail beat to the primary vortex of the next. An idealized profile through a small-core vortex ring is shown in black above.
The circulations of the primary and secondary vortices both decrease over time. In principle, total circulation should remain constant, implying that the sum of the two circulations should not change over time. A repeated-measures ANOVA (Zar,1999) in which the repeated measure was tail-beat phase divided into early and late regions shows that the initial circulation of the primary vortex alone, 3300 cm2 s-1, is not significantly different from the sum of the primary and secondary circulations in the end,1910 cm2 s-1 and 1520 cm2 s-1,respectively (F1,89=1.471; P=0.228).
To examine how the wake is generated, flow close to the bodies of the eels was examined. Fig. 7 shows a typical flow pattern near the body of an eel over the course of a tail beat. In the first three frames shown, a strong suction region develops near the tail, pulling a bolus of fluid laterally. This bolus will become the lateral jet in the wake. Proto-vortices are visible (shown with red and blue arrows),but their vorticity is very low (generally less than ±5 s-1).
Fig. 7.
Flow fields close to the body of a swimming eel, shown in gray. The lateral position of the eel's snout (off the view) is shown as a black arrow. Velocities are phase averaged across 14 tail beats by interpolating the normal gridded coordinate system on to a system defined by the distance from the eel's body and the distance along the body from the head. Approximate positions of the proto-vortices, defined by Müller et al.(2001), are shown in red(clockwise rotation) and blue (counter-clockwise rotation).
Fig. 7.
Flow fields close to the body of a swimming eel, shown in gray. The lateral position of the eel's snout (off the view) is shown as a black arrow. Velocities are phase averaged across 14 tail beats by interpolating the normal gridded coordinate system on to a system defined by the distance from the eel's body and the distance along the body from the head. Approximate positions of the proto-vortices, defined by Müller et al.(2001), are shown in red(clockwise rotation) and blue (counter-clockwise rotation).
Finally, for comparison with the computational model of Carling et al.(1998), we computed an average flow behind the eel, averaged over many tail beats. The computational model predicts a net velocity deficit behind the eel that could be obscured by the temporal variations in the observed flow. Fig. 8 shows the flow behind an eel averaged over 14 tail beats with axial flow magnitude shown in color. On average, momentum in the wake was elevated above free-stream momentum by between 2.84 and 6.65 kg mm s-2 at planes 25 mm and 95 mm,respectively, behind the tail.
Fig. 8.
Flow field behind the eel, averaged over 14 complete tail beats, centered on the tip of the eel's tail, shown as a black circle. Arrow heads are retained for velocities lower than 6.5 mm s-1 to indicate flow direction. Axial flow is shown in color: red is downstream and blue is upstream. Two profiles of velocity are shown in black above and below the flow field, with standard error in gray and total momentum flux represented by the trace printed beside it. Black lines across the field indicate where the velocity traces were measured (25 mm and 95 mm behind the tail). The vertical scale is the same for both traces.
Fig. 8.
Flow field behind the eel, averaged over 14 complete tail beats, centered on the tip of the eel's tail, shown as a black circle. Arrow heads are retained for velocities lower than 6.5 mm s-1 to indicate flow direction. Axial flow is shown in color: red is downstream and blue is upstream. Two profiles of velocity are shown in black above and below the flow field, with standard error in gray and total momentum flux represented by the trace printed beside it. Black lines across the field indicate where the velocity traces were measured (25 mm and 95 mm behind the tail). The vertical scale is the same for both traces.
### Force, impulse and power
Impulses were calculated from PIV using equation 6 byassuming that the observed vortex cores were part of a small-core vortex ring. In equation 6, rather than using the circulation of one of the cores, which vary over time and are sensitive to digitization error, we chose to use the circulation measured through the center of the lateral jet, which is constant over time and fairly robust to digitization error. Thus, the impulse coefficient for the lateral jets was 0.0217±0.0004, corresponding to an impulse in a 20 cm eel of 0.76 mN s. From this value, given that the lateral jet was generated over half a period,the lateral force coefficient was 0.097±0.001 (4.64 mN in a 20 cm eel). Fig. 9A shows a typical trace of lateral force from EBT with the average force estimates from PIV superimposed; Table 2 displays the same comparison numerically.
Fig. 9.
Representative traces for force, impulse and power from large-amplitude elongated body theory (EBT; in black) and particle image velocimetry (PIV; in red and green). Each black line shows force and power for a single tail beat. A total of 14 tail beats from a single swimming bout are shown. (A) Force(left axis) and impulse (right axis) over a tail-beat cycle. Because impulse is force integrated over time, impulses are indicated as lines, showing the impulse value and the time over which it was integrated. (B) Power from EBT and PIV over a tail-beat cycle. PIV values have standard error in a lighter color around the trace. The total power measured through PIV is shown in green, and the lateral' power, measured using only the lateral velocity component, is shown in red.
Fig. 9.
Representative traces for force, impulse and power from large-amplitude elongated body theory (EBT; in black) and particle image velocimetry (PIV; in red and green). Each black line shows force and power for a single tail beat. A total of 14 tail beats from a single swimming bout are shown. (A) Force(left axis) and impulse (right axis) over a tail-beat cycle. Because impulse is force integrated over time, impulses are indicated as lines, showing the impulse value and the time over which it was integrated. (B) Power from EBT and PIV over a tail-beat cycle. PIV values have standard error in a lighter color around the trace. The total power measured through PIV is shown in green, and the lateral' power, measured using only the lateral velocity component, is shown in red.
Table 2.
Comparison of force, impulse and power from PIV and EBT
PIV coefficientDimensionalEBT coefficientDimensionalF1,2P
Lateral force* 0.097±0.001 4.64 mN 0.090±0.003 4.31 mN 0.490 0.556
Lateral impulse 0.0217±0.0004 0.76 mN s 0.0062±0.0001 0.216 mN s 36.18 0.027
Thrust force 0.0166±0.0004 0.79 mN
Max. lateral power 0.0297±0.0007 391 μW 0.0286±0.0005 376 μW 0.103 0.778
Mean lateral power 0.0151±0.0003 198 μW 0.0148±0.0003 195 μW 0.200 0.699
Max. total power 0.065±0.003 855 μW 0.0286±0.0005 376 μW 16.25 0.056
Mean total power 0.023±0.002 303 μW 0.0148±0.0003 195 μW 0.292 0.643
PIV coefficientDimensionalEBT coefficientDimensionalF1,2P
Lateral force* 0.097±0.001 4.64 mN 0.090±0.003 4.31 mN 0.490 0.556
Lateral impulse 0.0217±0.0004 0.76 mN s 0.0062±0.0001 0.216 mN s 36.18 0.027
Thrust force 0.0166±0.0004 0.79 mN
Max. lateral power 0.0297±0.0007 391 μW 0.0286±0.0005 376 μW 0.103 0.778
Mean lateral power 0.0151±0.0003 198 μW 0.0148±0.0003 195 μW 0.200 0.699
Max. total power 0.065±0.003 855 μW 0.0286±0.0005 376 μW 16.25 0.056
Mean total power 0.023±0.002 303 μW 0.0148±0.0003 195 μW 0.292 0.643
Bold indicates a significant difference.
P values are calculated including individuals as a random effect. The individual was a significant effect in all comparisons(P<0.001) except for lateral force (P<0.090). N=118.
Dimensional values are calculated from the coefficients for a 20 cm-long eel.
*
Compares mean lateral force from particle image velocimetry (PIV) to peak value from elongated body theory (EBT)
Thrust force could only be calculated using EBT
Power was also measured in the wake at a plane approximately 8 mm downstream of the tail tip. Both total power, including both velocity components, and lateral' power, including only the lateral (v)velocity component, were calculated. Fig. 9B shows a typical trace of power over time. The total power coefficient was, on average, 0.023±0.002 (303 μW in a 20 cm eel). Lateral power was usually less than half of the total power and was equal to 0.0151±0.0003 (198 μW). Table 2 summarizes the comparison of force, impulse and power measurements from PIV with those calculated via EBT.
In general, EBT underestimates force and power as measured by PIV, although for certain values the two match well (Fig. 9). Both the impulse and the total wake power estimated by PIV and EBT are highly significantly different (P<0.001 in both cases; Table 2). However, the mean force from the PIV measurements matches the peak lateral force estimated by EBT (P=0.182). Additionally, the power estimated using only the lateral component of velocity is not significantly different from the total EBT estimate, in both maximum (PÅ1.000) and mean values(P=0.693). The shape of these two power curves is also visually quite similar (Fig. 9B).
This study provides a detailed picture of a typical anguilliform swimmer's wake during steady swimming at moderate swimming speeds. The wake consists of strong lateral jets, separated by two same-sign vortices(Fig. 10): probably unlinked vortex rings heading in opposite lateral directions. The most striking feature of the wake is the size and strength of the lateral jets and the notable absence of substantial downstream flow. In contrast to the downstream flow observed in the wakes of carangiform swimmers, almost all of the flow in an eel's wake is in jets directed laterally.
Fig. 10.
Schematic summary of the results of the present study, showing the wake behind a swimming eel at three different times. The size of the eel and position of the vortices are scaled to represent the true spacing. Vortices are indicated by blue and red arrows; primary vortices are solid lines and secondary vortices are dotted lines. The lateral jets are shown as block arrows, with lengths and angle proportional to the jet magnitude and direction.
Fig. 10.
Schematic summary of the results of the present study, showing the wake behind a swimming eel at three different times. The size of the eel and position of the vortices are scaled to represent the true spacing. Vortices are indicated by blue and red arrows; primary vortices are solid lines and secondary vortices are dotted lines. The lateral jets are shown as block arrows, with lengths and angle proportional to the jet magnitude and direction.
The lateral jets are produced along the body, just anterior to the tail tip. In particular, when the tail has reached its maximum lateral excursion,and thus has zero velocity, the point 0.15 L anterior to the tail has reached a high lateral velocity (60.2±0.6% of the swimming velocity). This substantial velocity difference along the eel's body seems to result in a strong suction region that pulls fluid laterally. Once the tail changes direction, it sheds a stop/start vortex (the primary vortex) and begins to shed a bolus of fluid to form a lateral jet. Each full tail beat produces two jets, one to each side, and two vortices separating them.
Because the velocities in successive lateral jets are large and in opposite directions, a substantial shear layer is present between the jets, with shearing rates of as much as 90 s-1. This shear layer is unstable and breaks down into two or more vortices (the secondary vortices), probably through a Kelvin–Helmholtz instability(Faber, 1995). This instability develops gradually (Fig. 6A), resulting in a fully formed secondary vortex about one full cycle later. Classic hydrodynamic theory predicts that a Kelvin–Helmholtz instability should result in vortices with a spacing approximately equal to 4πδ, where δ is the shear layer thickness (Faber, 1995). Before the shear layer breaks down, δ is approximately 3 mm, giving a predicted vortex spacing of 37 mm, which is close to the 20–30 mm spacing observed when the secondary vortex is fully formed. Additionally, the theory suggests that many vortices with this spacing could be formed. Indeed,another secondary vortex is occasionally formed at about twice the distance from the primary vortex.
When the jets are fully developed, they point almost directly laterally,meaning that very little flow is directed axially. Previous studies of caudal fin swimming (e.g. Müller et al.,1997; Lauder and Drucker,2002; Nauen and Lauder,2002a) have interpreted axial downstream flow as evidence for the production of thrust and have found that estimates of the thrust from PIV approximately match the estimated drag on the fish(Lauder and Drucker, 2002). This balance also held true for fish swimming using their pectoral fins(Drucker and Lauder, 1999). If downstream velocity is evidence for thrust, where is the thrust signature in the eel wake?
Because the eels in the present study were swimming steadily, without any substantial accelerations, the net force on the animal must be zero and, thus,the net force measurable in the wake should also be zero. Equivalently,because the momentum of the eel is not changing, there must be no net change in fluid momentum. Thus, while somewhat counter-intuitive, it is physically reasonable that no downstream momentum jet would be evident in the wake. It is important to think of the eel as producing thrust and drag simultaneously. If one could separate thrust from drag, one would see fluid being accelerated down the eel's body, as it produces thrust. At the same time, however, the drag along the eel's body is removing momentum from the fluid. In combination,the two effects cancel each other out, producing no net change in downstream fluid momentum as long as the eel is swimming steadily. All the lateral momentum observed in the wake also cancels out and is simply evidence of wasted energy.
If thrust and drag balance exactly, why did we observe a small increase in downstream momentum immediately behind the tail(Fig. 8)? Probably, this increase is offset by an increase in the opposite direction at the eel's snout. In still water, an eel swimming forward would push some fluid out of its way with its snout, increasing the upstream fluid momentum(Long et al., 2002). For forces to balance, this upstream increase must be matched by a small downstream increase at the tail, as we observed. The eel's snout adds upstream momentum at a rate proportional to ρUahead, where ahead is an area at the snout, representing a force in the order of 5 mN. The extra downstream momentum in the wake represents forces between 3.5 and 7.5 mN, which are roughly in agreement. We thus argue that the additional downstream momentum observed in the eel wake(Fig. 8) is necessary to fully conserve momentum and is not evidence for thrust. A complete control volume around the eel would resolve this question fully, but eels would not swim steadily with their heads in the light sheet, preventing us from performing that additional experiment.
It is important to note that the lack of net change in momentum is not equivalent to leaving no footprints', as hypothesized by Ahlborn et al.(1991). The footprints' of an eel are the lateral jets. In principle, at 100% efficiency, as Ahlborn et al.(1991) suggested, all power would go into producing forward motion, and none would go into producing a wake. The fact that an eel does leave a wake, or footprints, is evidence that they are not completely efficient.
This momentum balance described above must be true for all steady swimming,including previous studies that have observed a strong downstream jet during carangiform and pectoral fin swimming(Müller et al., 1997; Drucker and Lauder, 2000; Lauder and Drucker, 2002; Nauen and Lauder, 2002a). It is our hypothesis that these previously studied fishes display some spatial or temporal separation between thrust and drag production that allows momentum to balance on average over a tail beat, while still producing a downstream jet indicating thrust. The apparent discrepancy between this study and these previous ones is easiest to explain for pectoral fin swimmers. Drucker and Lauder (2000) observed a downstream jet from pectoral fin swimming in bluegill sunfish (Lepomis macrochirus) and surf perch (Embiotoca jacksoni) that represented enough force to balance the experimentally measured drag. Unlike eels, bluegill and surf perch rely solely on their pectoral fins for thrust in the speed range examined. Pectoral fins effectively produce only thrust and little drag, relative to the body, which is held nearly motionless at these swimming speeds and produces only drag. The spatial separation between the thrust-producing pectoral fins and the drag-producing body allows accurate measurement of thrust from the pectoral fins alone, as Drucker and Lauder(2000) found. Nonetheless, if one were to examine a control volume around the entire fish, the net fluid momentum change would be zero. The situation is somewhat like that of an outboard propeller on a boat: the body, like a boat's hull, generates most of the drag and negligible thrust, and the pectoral fins, like propellers,generate all of the thrust with negligible drag.
For carangiform caudal fin swimmers, the situation is more complicated, but previous results should still be valid. For many fishes, the outboard motor analogy may still be appropriate. Because carangiform swimmers move their anterior body relatively little compared to the caudal fin(Webb, 1975; Jayne and Lauder, 1995; Donley and Dickson, 2000), very little thrust can be generated anterior to the caudal peduncle. Flow also does not separate along the body (Anderson et al., 2000) but rather converges on the caudal peduncle(Nauen and Lauder, 2000). As fluid moves along the body, drag removes momentum, but this low-momentum flow is concentrated at the caudal peduncle. The dorsal and ventral portions of the caudal fin are therefore exposed primarily to undisturbed free-stream flow. Except at the very center of the fin, the caudal fin thus may also function like an outboard motor, producing almost entirely thrust with very little drag. Probably the analogy is most valid for fishes such as mackerels and tunas that have a very narrow caudal peduncle and a large caudal fin. Indeed,in their study of chub mackerel (Scomber japonicus), Nauen and Lauder(2002a) found that thrust measured from the downstream jet roughly balanced experimentally measured drag(although drag measurements were difficult to make accurately).
For carangiform swimmers with less pronounced caudal peduncles, the outboard motor analogy may break down somewhat, but differences in swimming kinematics between them and anguilliform swimmers may explain why thrust wakes were still observed (e.g. in Müller et al., 1997; Drucker and Lauder, 2001). We speculate that anguilliform swimmers may produce thrust more continuously over time than carangiform swimmers. For a steadily swimming fish, thrust need only balance drag on average over a full tail beat. If thrust is produced in a very pulsatile way, it may briefly exceed drag to such an extent that it would be evident in the wake. According to a reactive inviscid theory such as Lighthill's EBT(Lighthill, 1971), thrust is only produced at the tail tip (or other sharp trailing edges). Evaluation of the EBT equation for thrust generated at the tail tip(equation 1) results in a pulsatile force. However, these equations do not include possible thrust from the body anterior to the tail due to viscous effects. Recent direct numerical simulations showed that an infinitely long waving plate can produce thrust(Shen et al., 2003), in support of previous experimental observations(Taneda and Tomonari, 1974; Techet, 2001). Like a waving plate, the short wavelength undulations along an eel's body can produce thrust smoothly to even out the pulsatile force from the tail tip. In particular,since a full wavelength is present on the eel's body, a portion of the body is moving and likely producing force out of phase with the tail tip. The majority of thrust may still be produced in the posterior regions of the eel's body,where we saw accelerated flow (Fig. 7) but, even so, some regions of the posterior body are moving out of phase with the tail tip, helping to smooth out pulsatile thrust. For carangiform swimmers, unlike eels, the long wavelength body undulations do not contain out-of-phase motions at sufficient amplitude and may tend to reinforce the pulsatile thrust from the tail (Webb,1975). Therefore, at certain points in a carangiform swimmer's tail beat, thrust may exceed drag to produce a thrust wake, even though the two forces balance on average. For eels, thrust and drag may balance more evenly over time. Note that Fig. 9 does not contradict this statement. Fig. 9 shows that lateral force and power are pulsatile, but axial force was not measurable and axial power',constructed in a similar way to `lateral power', remains fairly constant and small over the tail beat.
### The importance of shape
We speculate that the novel wake structure of swimming eels is highly dependent on their shape and that differences in shape, along with differences in kinematics, may be one of the primary distinctions between anguilliform and carangiform swimming. In particular, eels do not have a narrow caudal peduncle, whereas most carangiform swimmers do. The large lateral jets develop in the suction region centered around ∼85% of body length. Both anguilliform and carangiform swimmers have a substantial undulation amplitude this close to the tail, even though the kinematics on the anterior body differ substantially. For example, both chub mackerel and kawakawa tuna(Euthynnus affinis) have amplitudes of ∼4% of body length at 0.85 L (Donley and Dickson,2000), and largemouth bass (Micropterus salmoides) have amplitudes of ∼4.5% at 0.85 L(Jayne and Lauder, 1995),comparable with the 4.4% we measured in eels. However, most carangiform swimmers are different from eels because they have a narrow caudal peduncle around 0.85 L. If their body shape were more similar to that of eels,it is likely that a substantial suction could develop there in the same way as in eels. The narrowness of the peduncle, however, probably prevents such suction from developing. Even if a mackerel, for example, swam using the same kinematics as an eel, its wake would probably differ from an eel's due to the differences in body shape. In fact, recent results from an engineering study of rectangular flapping membranes indicate that simple shape differences, such as the ratio of flapping amplitude to body height, can determine whether the wake is a linked vortex ring wake, as observed in carangiform swimmers, or an unlinked ring wake, as in eels (J. Buchholz, personal communication).
Clearly, this effect in fishes is more complicated than a simple ratio and probably depends on how narrow the peduncle is, relative to the size of the body and tail. It would therefore be strongly affected by the wide range of body shapes in fishes. Wakes, therefore, probably show a gradation from those of mackerel (Nauen and Lauder,2002a), for example, which have very narrow peduncles but large caudal fins, to those of eels, which have no narrowing at the peduncle at all.
### Efficiency of anguilliform swimming
One of the goals of the present study was to evaluate the efficiency of anguilliform swimming relative to carangiform swimming. However, for steady swimming, efficiency is very hard to evaluate. Froude efficiency (η) is usually written, neglecting inertial forces, as:
$\ {\eta}=\frac{\mathrm{Useful\ power}}{\mathrm{Total\ power}}=\frac{FU}{FU+P_{\mathrm{wake}}},$
9
where F is a force, U is the swimming velocity and Pwake is the power in the wake(Webb, 1975). Strictly, F is the net force on the swimming body, which is zero during steady swimming, resulting in a zero Froude efficiency. Schultz and Webb(2002) have discussed this issue in some detail. If F is the thrust force only, then ηrepresents how much power was used for thrust and how much was wasted. While thrust cannot be measured directly from the wake of swimming eels, it is still useful, conceptually, to separate it from drag. By using a mathematical model,such as EBT or more complex computational fluid dynamic models(Carling et al., 1998; Wolfgang et al., 1999; Zhu et al., 2002), thrust can be estimated and used to calculate a Froude propulsive efficiency.
Specifically, EBT can be used to calculate this thrust value using equation 1, which can be combined with the wake power estimate from PIV to produce an efficiency. The estimated mean thrust is 0.83 mN, and the measured wake power is between 198 and 303μW, resulting in efficiency estimates between 0.43 and 0.54. Additionally,EBT can also estimate the efficiency directly. This value,η EBT, is usually written as 1–Γ(VU)/V, where V is the body wave velocity (Lighthill,1970). According to this method, EBT estimates ηEBTas 0.865±0.001. However, since EBT usually underestimated the total power in the wake (Table 2),the first range, 0.43–0.54, is probably the more realistic estimate.
Anguilliform swimming has been hypothesized to be inefficient(Lighthill, 1970; Webb, 1975). Our measurements,however, indicate a swimming efficiency of around 0.5, or potentially as high as 0.87, depending on how it is calculated. Because of the difficulties of estimating efficiency from a steadily swimming fish, it is difficult to compare this value with previously reported values, which range from 0.74 to 0.97 (Drucker and Lauder,2001; Müller et al.,2001; Nauen and Lauder, 2002a,b).
### Comparison with previous studies of anguilliform swimming
Müller et al. (2001)first observed the wakes of swimming eels and noted their unusual structure. They showed that two vortices were produced per half tail beat and that the jet between successive vortices was primarily lateral. Their observations are,in general, quite similar to ours. With our higher resolution PIV, we are able to propose a different mechanism for generating the wake. Additionally, our data allowed a much more detailed examination of the balance of thrust and drag and the Froude efficiency of steady swimming, which have been controversial (Schultz and Webb,2002).
Nonetheless, there are some important differences between our findings and those of Müller et al.(2001). They hypothesized that the double vortex structure resulted from a phase lag between the vorticity shed from the tail and circulation produced along the body, which they termed proto-vortices. Although proto-vortices were evident along the body(Fig. 7), their vorticity was much lower than the vorticity of the secondary vortex. The vorticity in the proto-vortices along the body is generally less than 5 s-1, while the secondary vortex peak vorticity was often more that 15 s-1. Müller et al. (2001) also observed that fluid velocity increases along the body linearly from head to tail. By contrast, we observed relatively little increase in fluid velocity until the last 30% of body length, where the fluid bolus is generated(Fig. 7). Finally, we calculated the phase difference between the shedding of stop/start vortices and the shedding of proto-vortices off the tail. The difference was only∼5% of a tail beat cycle, so any proto-vorticity is likely to simply add to the stop/start vortex, which is forming at almost the same time, rather than create a separate vortex.
It is somewhat surprising that we saw so much less fluid velocity along the body than Müller et al.(2001) did. While the eels analyzed in detail in the present study, at 20 cm long, were more than twice as long as those in Muller's study, we examined the wake of a 12 cm eel qualitatively and found the same pattern as in the larger eels. The eels in Müller's study seemed to show greater undulation amplitude along the body, particularly near the head, than the eels in our study. This amplitude difference may explain the stronger fluid flow near the body but it also suggests that Müller's eels may have been accelerating slightly, because increased anterior undulation is often found in accelerating eels (E. D. Tytell, manuscript in preparation). Additionally, they document a slight downstream component to the jets(Müller et al., 2001),another indication of acceleration (E. D. Tytell, manuscript in preparation).
The other model examined in the present study, Carling and colleagues'computational fluid dynamic model for an 8 cm-long anguilliform swimmer(Carling et al., 1998), is not supported by our data or those of Müller et al.(2001). Carling's model predicts a substantially reduced velocity immediately behind the tail, as if the eel were sucking fluid along with it as it swam(Fig. 1B). Even averaged over many tail beats, we did not observe any reduced velocity in the wake; in fact,immediately behind the tail, the flow is accelerated downstream(Fig. 8). Somewhat surprisingly, we observed that momentum in the far wake, 95 mm from the eel's tail, was greater than that in the near wake, 25 mm from the tail. We speculate that this effect is due to three-dimensional reorientation or contraction of the wake, similar to that in the far-field wake of a hovering insect (Ellington, 1984). Nonetheless, it seems clear that axial wake momentum is downstream, the opposite of what the model predicted(Carling et al., 1998). Additionally, their model does not predict the complex vortical structures and lateral jets that we consistently observed in all individuals covering a length range from 12 to 23 cm. While we did not observe the wake of an 8 cmindividual, the size they modeled, Müller et al.(2001) examined one that size and observed a wake similar to those we observed in larger individuals and quite different from Carling and colleagues' predictions(Carling et al., 1998).
To continue the exploration of the hydrodynamic differences between different modes of swimming, future studies should be careful to include detailed kinematics. Small differences in kinematics may cause substantial changes in flow, as we noted in the differences between Müller et al.(2001) and our study. This effect may prove useful, however: small kinematic differences as fishes change swimming speed may induce large hydrodynamic changes, as seen in pectoral fin swimming (Drucker and Lauder,2000). Examining both effects simultaneously will help to elucidate the mechanical effect of changing kinematics with swimming speed and between different swimming modes.
We would like to thank Thelma Williams and John Carling for stimulating our interest in anguilliform hydrodynamics, Ulrike Müller for productive discussions of the interaction between kinematics and wake structure, and Paul Webb for many ideas and thoughts on the balance of thrust and drag. Promode Bandyopadhyay, L. Mahadevan and Peter Madden provided comments on a draft of the manuscript, and Tonia Hsieh, Jimmy Liao, Matt McHenry and Christoffer Johanssen were helpful for advice throughout this project. Alexander Smits and James Buchholz provided some thought-provoking information on the wake structure behind flapping membranes. We also owe thanks to Laura Farrell, who maintained the animals used in this study. This research was supported by the NSF under grants IBN9807021 and IBN0316675 to G.V.L.
Agüí, J. C. and Jiménez, J.(
1987
). On the performance of particle tracking.
J. Fluid Mech.
185
,
447
-468.
Ahlborn, B., Harper, D. G., Blake, R. W., Ahlborn, D. and Cam,M. (
1991
). Fish without footprints.
J. Theor. Biol.
148
,
521
-533.
Alexander, R. McN. (
1983
). The history of fish mechanics. In
Fish Biomechanics
(ed. P. W. Webb and D. Weihs), pp.
1
-36. New York: Praeger.
Anderson, E. J., McGillis, W. R. and Grosenbaugh, M. A.(
2000
). The boundary layer of swimming fish.
J. Exp. Biol.
204
,
81
-102.
Barrett, D., Triantafyllou, M. S., Yue, D. K. P., Grosenbaugh,M. A. and Wolfgang, M. J. (
1999
). Drag reduction in fish-like locomotion.
J. Fluid Mech.
392
,
183
-212.
Breder, C. M. (
1926
). The locomotion of fishes.
Zoologica
4
,
159
-297.
Carling, J. (
2002
). Numerical approaches in computational fluid dynamics.
Comp. Biochem. Physiol. A
132
,
S85
.
Carling, J., Williams, T. L. and Bowtell, G.(
1998
). Self-propelled anguilliform swimming: simultaneous solution of the two-dimensional Navier–Stokes equations and Newton's laws of motion.
J. Exp. Biol.
201
,
3143
-3166.
Donley, J. M. and Dickson, K. A. (
2000
). Swimming kinematics of juvenile kawakawa tuna (Euthynnus affinis) and chub mackerel (Scomber japonicus).
J. Exp. Biol.
203
,
3103
-3116.
Donley, J. M. and Shadwick, R. E. (
2003
). Steady swimming muscle dynamics in the leopard shark Triakis semifasciata.
J. Exp. Biol.
206
,
1117
-1126.
Drucker, E. G. and Lauder, G. V. (
1999
). Locomotor forces on a swimming fish: three-dimensional vortex wake dynamics quantified using digital particle image velocimetry.
J. Exp. Biol.
202
,
2393
-2412.
Drucker, E. G. and Lauder, G. V. (
2000
). A hydrodynamic analysis of fish swimming speed: wake structure and locomotor force in slow and fast labriform swimmers.
J. Exp. Biol.
203
,
2379
-2393.
Drucker, E. G. and Lauder, G. V. (
2001
). Locomotor function of the dorsal fin in teleost fishes: experimental analysis of wake forces in sunfish.
J. Exp. Biol.
204
,
2943
-2958.
Ellington, C. P. (
1984
). The aerodynamics of hovering insect flight. V. A vortex theory.
Phil. Trans. R. Soc. Lond. B
305
,
115
-144.
Faber, T. E. (
1995
).
Fluid Dynamics for Physicists
. Cambridge: Cambridge University Press.
Fincham, A. M. and Spedding, G. R. (
1997
). Low cost, high resolution DPIV for measurement of turbulent fluid flow.
Exp. Fluids
23
,
449
-462.
Gillis, G. B. (
1998
). Environmental effects on undulatory locomotion in the American eel Anguilla rostrata:kinematics in water and on land.
J. Exp. Biol.
201
,
949
-961.
Hart, D. P. (
1999
). Super-resolution PIV by recursive local-correlation.
J. Visual.
10
,
1
-10.
Hart, D. P. (
2000
). PIV error correction.
Exp. Fluids
29
,
13
-22.
Helfman, G. S., Collette, B. B. and Facey, D. E.(
1997
).
The Diversity of Fishes
. London:Blackwell Science.
Jayne, B. C. and Lauder, G. V. (
1995
). Speed effects on midline kinematics during steady undulatory swimming of largemouth bass, Micropterus salmoides.
J. Exp. Biol.
198
,
585
-602.
Lauder, G. V. (
2000
). Function of the caudal fin during locomotion in fishes: kinematics, flow visualization, and evolutionary patterns.
Am. Zool.
40
,
101
-122.
Lauder, G. V. and Drucker, E. G. (
2002
). Forces, fishes, and fluids: hydrodynamic mechanisms of aquatic locomotion.
News Physiol. Sci.
17
,
235
-240.
Liao, J. (
2002
). Swimming in needlefish(Belonidae): anguilliform locomotion with fins.
J. Exp. Biol.
205
,
2875
-2884.
Lighthill, M. J. (
1960
). Note on the swimming of slender fish.
J. Fluid Mech.
9
,
305
-317.
Lighthill, M. J. (
1970
). Aquatic animal propulsion of high hydromechanical efficiency.
J. Fluid Mech.
44
,
265
-301.
Lighthill, M. J. (
1971
). Large-amplitude elongated-body theory of fish locomotion.
Proc. R. Soc. Lond. A
179
,
125
-138.
Long, J. H., Root, R. G. and Watts, P. (
2002
). Is an undulating fish an oscillating wing?
Integ. Comp. Biol.
42
,
1268
.
Müller, U. K., van den Heuvel, B.-L. E., Stamhuis, E. J. and Videler, J. J. (
1997
). Fish foot prints: morphology and energetics of the wake behind a continuously swimming mullet (Chelon labrosus risso).
J. Exp. Biol.
200
,
2893
-2906.
Müller, U. K., Smit, J., Stamhuis, E. J. and Videler, J. J. (
2001
). How the body contributes to the wake in undulatory fish swimming: flow fields of a swimming eel (Anguilla anguilla).
J. Exp. Biol.
204
,
2751
-2762.
Nauen, J. C. and Lauder, G. V. (
2000
). Locomotion in scombrid fishes: morphology and kinematics of the finlets of the chub mackerel Scomber japonicus.
J. Exp. Biol.
203
,
2247
-2259.
Nauen, J. C. and Lauder, G. V. (
2002a
). Hydrodynamics of caudal fin locomotion by chub mackerel, Scomber japonicus (Scombridae).
J. Exp. Biol.
205
,
1709
-1724.
Nauen, J. C. and Lauder, G. V. (
2002b
). Quantification of the wake of rainbow trout (Oncorhynchus mykiss)using three-dimensional stereoscopic digital particle image velocimetry.
J. Exp. Biol.
205
,
3271
-3279.
Robinson, S. K. (
1991
). Coherent motions in the turbulent boundary layer.
Annu. Rev. Fluid Mech.
23
,
601
-639.
Schlichting, H. (
1979
).
Boundary-Layer Theory
. New York:McGraw-Hill.
Schultz, W. W. and Webb, P. W. (
2002
). Power requirements of swimming: do new methods resolve old questions?
Integ. Comp. Biol.
42
,
1018
-1025.
Shaffer, H. B. and Lauder, G. V. (
1985
). Patterns of variation in aquatic ambystomatid salamanders: kinematics of the feeding mechanism.
Evolution
39
,
83
-92.
Shen, L., Zhang, X., Yue, D. K. P. and Triantafyllou, M. S.(
2003
). Turbulent flow over a flexible wall undergoing a streamwise travelling wave motion.
J. Fluid Mech.
484
,
197
-221.
Smith, D. G. and Tighe, K. A. (
2002
). Freshwater eels. Family Anguillidae. In
Fishes of the Gulf of Maine
(ed. B. B. Collette and G. Klein-MacPhee), pp.
92
-95. Washington: Smithsonian Institution Press.
Spedding, G. R. and Rignot, E. J. M. (
1993
). Performance analysis and application of grid interpolation techniques for fluid flows.
Exp. Fluids
15
,
417
-430.
Stamhuis, E. J. and Videler, J. J. (
1995
). Quantitative flow analysis around aquatic animals using laser sheet particle image velocimetry.
J. Exp. Biol.
198
,
283
-294.
Taneda, S. and Tomonari, Y. (
1974
). An experiment on the flow around a waving plate.
J. Phys. Soc. Japan
36
,
1683
-1689.
Techet, A. H. (
2001
).
Experimental visualization of the near-boundary hydrodynamics about fish-like swimming bodies
. Ph.D. Dissertation. Department of Ocean Engineering,Massachusetts Institute of Techology/Woods Hole Oceanography Institute,Cambridge, MA, USA.
Triantafyllou, M. S., Triantafyllou, G. S. and Yue, D. K. P.(
2000
). Hydrodynamics of fishlike swimming.
Annu. Rev. Fluid Mech.
32
,
33
-53.
Tytell, E. D. and Lauder, G. V. (
2002
). The C-start escape response of Polypterus senegalus: bilateral muscle activity and variation during stage 1 and 2.
J. Exp. Biol.
205
,
2591
-2603.
van Ginneken, V. J. T. and van den Thillart, G. E. E. J. M.(
2000
). Eel fat stores are enough to reach the Sargasso.
Nature
403
,
156
-157.
Vollmers, H. (
1993
). Analysing, depicting, and interpreting flow fields.
DLR-Nachrichten
,
70
,
2
-8.
Walker, J. A. (
1998
). Estimating velocities and accelerations of animal locomotion: a simulation experiment comparing numerical differentiation algorithms.
J. Exp. Biol.
201
,
981
-995.
Webb, P. W. (
1975
). Hydrodynamics and energetics of fish propulsion.
Bull. Fish. Res. Bd. Can.
190
,
1
-159.
Westerweel, J., Dabiri, D. and Gharib, M.(
1997
). The effect of a discrete window offset on the accuracy of cross-correlation analysis of digital PIV recordings.
Exp. Fluids
23
,
20
-28.
Willert, C. E. and Gharib, M. (
1991
). Digital particle image velocimetry.
Exp. Fluids
10
,
181
-193.
Wolfgang, M. J., Anderson, J. M., Grosenbaugh, M. A., Yue, D. K. P. and Triantafyllou, M. S. (
1999
). Near-body flow dynamics in swimming fish.
J. Exp. Biol.
202
,
2303
-2327.
Zar, J. H. (
1999
).
Biostatistical Analysis
. Upper Saddler River, NJ: Prentice Hall.
Zhu, Q., Wolfgang, M. J., Yue, D. K. P. and Triantafyllou, M. S. (
2002
). Three-dimensional flow structures and vorticity control in fish-like swimming.
J. Fluid Mech.
468
,
1
-28.
|
2022-12-08 00:20:53
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6538336277008057, "perplexity": 3121.4064943296326}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711221.94/warc/CC-MAIN-20221207221727-20221208011727-00813.warc.gz"}
|
https://ftp.aimsciences.org/article/doi/10.3934/cpaa.2012.11.2261
|
# American Institute of Mathematical Sciences
November 2012, 11(6): 2261-2290. doi: 10.3934/cpaa.2012.11.2261
## Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions
1 Université de La Rochelle, Laboratoire de Mathématiques Images et Applications EA 3165, Avenue Michel Crépeau, 17042 La Rochelle Cedex 1 2 Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, I-41125 Modena 3 Université de Poitiers, Mathématiques SP2MI, 86962 Chasseneuil Futuroscope Cedex
Received December 2010 Revised December 2010 Published April 2012
This paper is devoted to the study of the well-posedness and the long time behavior of the Caginalp phase-field model with singular potentials and dynamic boundary conditions. Thanks to a suitable definition of solutions, coinciding with the strong ones under proper assumptions on the bulk and surface potentials, we are able to get dissipative estimates, leading to the existence of the global attractor with finite fractal dimension, as well as of an exponential attractor.
Citation: Laurence Cherfils, Stefania Gatti, Alain Miranville. Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2261-2290. doi: 10.3934/cpaa.2012.11.2261
##### References:
[1] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827. Google Scholar [2] L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials (Corrigendum, J. Math. Anal. Appl. 348 (2008), 1029-1030), J. Math. Anal. Appl., 343 (2008), 557-566. doi: 10.1016/j.jmaa.2008.07.058. Google Scholar [3] L. Cherfils, S. Gatti and A. Miranville, Finite dimensional attractors for the Caginalp system with singular potentials and dynamic boundary conditions, Bull. Transilvania University Bra\csov-Series III: Mathematics, Informatics, Physics, 2 (2009), 25-34. Google Scholar [4] L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6. Google Scholar [5] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. doi: 10.1007/s00032-011-0165-4. Google Scholar [6] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dynam. Systems, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211. Google Scholar [7] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896. doi: 10.1103/PhysRevLett.79.893. Google Scholar [8] H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Lett., 42 (1998), 49-54. doi: 10.1209/epl/i1998-00550-y. Google Scholar [9] H. P. Fischer, J. Reinhard, W. Dieterich, J.-F. Gouyet, P. Maass, A. Majhofer and D. Reinel, Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall, J. Chem. Phys., 108 (1998), 3028-3037. doi: 10.1063/1.475690. Google Scholar [10] S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems, Lecture Notes in Pure and Applied Mathematics," Taylor and Francis, (2006), 149-170. doi: 10.1201/9781420011135.ch9. Google Scholar [11] G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881. Google Scholar [12] G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chinese Ann. Math., Ser. B, 31 (2010), 679-712. doi: 10.1007/s11401-010-0602-7. Google Scholar [13] M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dynam. Systems, 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67. Google Scholar [14] J. Málek and D. Prážak, Large time behavior via the method of $l$-trajectories, J. Diff. Eqns., 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. Google Scholar [15] A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582. doi: 10.1002/mma.464. Google Scholar [16] A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590. Google Scholar [17] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations, Vol. IV" (eds. C.M. Dafermos and M. Pokorny), Elsevier/North-Holland, (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [18] A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dynam. Systems, 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275. Google Scholar [19] G. Ruiz Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard equation in a domain with non-permeable walls, Phys. D, 240 (2011), 754-766. doi: 10.1016/j.physd.2010.12.007. Google Scholar
show all references
##### References:
[1] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827. Google Scholar [2] L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials (Corrigendum, J. Math. Anal. Appl. 348 (2008), 1029-1030), J. Math. Anal. Appl., 343 (2008), 557-566. doi: 10.1016/j.jmaa.2008.07.058. Google Scholar [3] L. Cherfils, S. Gatti and A. Miranville, Finite dimensional attractors for the Caginalp system with singular potentials and dynamic boundary conditions, Bull. Transilvania University Bra\csov-Series III: Mathematics, Informatics, Physics, 2 (2009), 25-34. Google Scholar [4] L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6. Google Scholar [5] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. doi: 10.1007/s00032-011-0165-4. Google Scholar [6] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dynam. Systems, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211. Google Scholar [7] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896. doi: 10.1103/PhysRevLett.79.893. Google Scholar [8] H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Lett., 42 (1998), 49-54. doi: 10.1209/epl/i1998-00550-y. Google Scholar [9] H. P. Fischer, J. Reinhard, W. Dieterich, J.-F. Gouyet, P. Maass, A. Majhofer and D. Reinel, Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall, J. Chem. Phys., 108 (1998), 3028-3037. doi: 10.1063/1.475690. Google Scholar [10] S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems, Lecture Notes in Pure and Applied Mathematics," Taylor and Francis, (2006), 149-170. doi: 10.1201/9781420011135.ch9. Google Scholar [11] G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881. Google Scholar [12] G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chinese Ann. Math., Ser. B, 31 (2010), 679-712. doi: 10.1007/s11401-010-0602-7. Google Scholar [13] M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dynam. Systems, 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67. Google Scholar [14] J. Málek and D. Prážak, Large time behavior via the method of $l$-trajectories, J. Diff. Eqns., 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. Google Scholar [15] A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582. doi: 10.1002/mma.464. Google Scholar [16] A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590. Google Scholar [17] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations, Vol. IV" (eds. C.M. Dafermos and M. Pokorny), Elsevier/North-Holland, (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [18] A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dynam. Systems, 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275. Google Scholar [19] G. Ruiz Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard equation in a domain with non-permeable walls, Phys. D, 240 (2011), 754-766. doi: 10.1016/j.physd.2010.12.007. Google Scholar
[1] Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353 [2] Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021024 [3] Ciprian G. Gal, M. Grasselli. On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 689-710. doi: 10.3934/cpaa.2009.8.689 [4] Maurizio Grasselli, Alain Miranville, Giulio Schimperna. The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 67-98. doi: 10.3934/dcds.2010.28.67 [5] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345 [6] I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635 [7] Alexey Cheskidov, Susan Friedlander, Nataša Pavlović. An inviscid dyadic model of turbulence: The global attractor. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 781-794. doi: 10.3934/dcds.2010.26.781 [8] Rana D. Parshad, Juan B. Gutierrez. On the global attractor of the Trojan Y Chromosome model. Communications on Pure & Applied Analysis, 2011, 10 (1) : 339-359. doi: 10.3934/cpaa.2011.10.339 [9] Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195 [10] Nobuyuki Kenmochi, Noriaki Yamazaki. Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint. Conference Publications, 2011, 2011 (Special) : 824-833. doi: 10.3934/proc.2011.2011.824 [11] Monica Conti, Stefania Gatti, Alain Miranville. Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 485-505. doi: 10.3934/dcdss.2012.5.485 [12] Ning Ju. The global attractor for the solutions to the 3D viscous primitive equations. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 159-179. doi: 10.3934/dcds.2007.17.159 [13] Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4545-4573. doi: 10.3934/cpaa.2020206 [14] Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275 [15] Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717 [16] Messoud Efendiev, Anna Zhigun. On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 651-673. doi: 10.3934/dcds.2018028 [17] Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215 [18] Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021015 [19] Yirong Jiang, Nanjing Huang, Zhouchao Wei. Existence of a global attractor for fractional differential hemivariational inequalities. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1193-1212. doi: 10.3934/dcdsb.2019216 [20] Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete & Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1
2020 Impact Factor: 1.916
|
2021-10-16 20:23:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6297972798347473, "perplexity": 5689.469640520503}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585025.23/warc/CC-MAIN-20211016200444-20211016230444-00071.warc.gz"}
|
https://www.semanticscholar.org/paper/Sensitivities-on-dark-photon-from-the-forward-Cheung-Ouseph/b0be3ffa76c23220daa8e0b0871543f6974c622f
|
# Sensitivities on dark photon from the forward physics experiments
@article{Cheung2022SensitivitiesOD,
title={Sensitivities on dark photon from the forward physics experiments},
author={Kingman Cheung and C. J. Ouseph},
journal={Journal of High Energy Physics},
year={2022},
volume={2022}
}
• Published 9 August 2022
• Physics
• Journal of High Energy Physics
Neutrino-electron scattering experiments can explore the potential presence of a light gauge boson A′ which arises from an additional U(1)B−L group, or a dark photon A′ which arises from a dark sector and has kinetic mixing with the SM hypercharge gauge field. We generically call it a dark photon. In this study, we investigate the effect of the dark photon on neutrino-electron scattering νe−→ νe− at the newly proposed forward physics experiments such as FASERν, FASERν2, SND@LHC and FLArE(10…
## References
SHOWING 1-10 OF 64 REFERENCES
### The Forward Physics Facility at the High-Luminosity LHC
• Physics
• 2022
High energy collisions at the High-Luminosity Large Hadron Collider (LHC) produce a large number of particles along the beam collision axis, outside of the acceptance of existing LHC experiments. The
### Constraining Secluded Dark Matter models with the public data from the 79-string IceCube search for dark matter in the Sun
• Physics
• 2017
The 79-string IceCube search for dark matter in the Sun public data is used to test Secluded Dark Matter models. No significant excess over background is observed and constraints on the parameters of
### Constraints on Dark Photon from Neutrino-Electron Scattering Experiments
• Physics
• 2015
A possible manifestation of an additional light gauge boson ${A}^{\ensuremath{'}}$, named a dark photon, associated with a group $U(1{)}_{\mathrm{B}\ensuremath{-}\mathrm{L}}$, is studied in
### Constraints on Light Hidden Sector Gauge Bosons from Supernova Cooling
• Physics
• 2012
We derive new bounds on hidden sector gauge bosons which could produce new energy loss mechanisms in supernovae, enlarging the excluded region in mass-coupling space by a significant factor compared
• PTEP 2022,
• 2022
### Search for a New B-L Z^{'} Gauge Boson with the NA64 Experiment at CERN.
• Physics
Physical review letters
• 2022
A search for a new Z^{'} gauge boson associated with (un)broken B-L symmetry in the keV-GeV mass range is carried out for the first time using the missing-energy technique in the NA64 experiment at
### Constraining the Active-to-Heavy-Neutrino transitional magnetic moments associated with the $Z'$ interactions at FASER$\nu$
• Physics
• 2022
We investigate the effects of the transitional magnetic dipole moment of the active-to-heavy-neutrino associated with a new neutral gauge boson Z (cid:48) on neutrino-nucleon scattering at the ForwArd
### The tracking detector of the FASER experiment
• Physics
Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment
• 2022
|
2022-11-28 07:48:25
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.570065975189209, "perplexity": 4723.719227587325}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710488.2/warc/CC-MAIN-20221128070816-20221128100816-00344.warc.gz"}
|
https://jax.readthedocs.io/en/4510-2/_autosummary/jax.numpy.std.html
|
# jax.numpy.std¶
jax.numpy.std(a, axis=None, dtype=None, out=None, ddof=0, keepdims=False)[source]
Compute the standard deviation along the specified axis.
LAX-backend implementation of std(). Original docstring below.
Returns the standard deviation, a measure of the spread of a distribution, of the array elements. The standard deviation is computed for the flattened array by default, otherwise over the specified axis.
Parameters
• a (array_like) – Calculate the standard deviation of these values.
• axis (None or int or tuple of ints, optional) – Axis or axes along which the standard deviation is computed. The default is to compute the standard deviation of the flattened array.
• dtype (dtype, optional) – Type to use in computing the standard deviation. For arrays of integer type the default is float64, for arrays of float types it is the same as the array type.
• out (ndarray, optional) – Alternative output array in which to place the result. It must have the same shape as the expected output but the type (of the calculated values) will be cast if necessary.
• ddof (int, optional) – Means Delta Degrees of Freedom. The divisor used in calculations is N - ddof, where N represents the number of elements. By default ddof is zero.
• keepdims (bool, optional) – If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.
Returns
standard_deviation – If out is None, return a new array containing the standard deviation, otherwise return a reference to the output array.
Return type
ndarray, see dtype parameter above.
Notes
The standard deviation is the square root of the average of the squared deviations from the mean, i.e., std = sqrt(mean(abs(x - x.mean())**2)).
The average squared deviation is normally calculated as x.sum() / N, where N = len(x). If, however, ddof is specified, the divisor N - ddof is used instead. In standard statistical practice, ddof=1 provides an unbiased estimator of the variance of the infinite population. ddof=0 provides a maximum likelihood estimate of the variance for normally distributed variables. The standard deviation computed in this function is the square root of the estimated variance, so even with ddof=1, it will not be an unbiased estimate of the standard deviation per se.
Note that, for complex numbers, std takes the absolute value before squaring, so that the result is always real and nonnegative.
For floating-point input, the std is computed using the same precision the input has. Depending on the input data, this can cause the results to be inaccurate, especially for float32 (see example below). Specifying a higher-accuracy accumulator using the dtype keyword can alleviate this issue.
Examples
>>> a = np.array([[1, 2], [3, 4]])
>>> np.std(a)
1.1180339887498949 # may vary
>>> np.std(a, axis=0)
array([1., 1.])
>>> np.std(a, axis=1)
array([0.5, 0.5])
In single precision, std() can be inaccurate:
>>> a = np.zeros((2, 512*512), dtype=np.float32)
>>> a[0, :] = 1.0
>>> a[1, :] = 0.1
>>> np.std(a)
0.45000005
Computing the standard deviation in float64 is more accurate:
>>> np.std(a, dtype=np.float64)
0.44999999925494177 # may vary
|
2021-05-09 05:15:51
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6497470140457153, "perplexity": 1950.0463593867682}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988955.89/warc/CC-MAIN-20210509032519-20210509062519-00161.warc.gz"}
|
https://math.stackexchange.com/questions/2484894/show-that-series-is-convergent-and-find-its-sum/2485057
|
# Show that series is convergent and find its sum
$$\sum\nolimits\arccos\frac{n(n+1)+\sqrt{(n+1)(n+2)(3n+1)(3n+4)}}{(2n+1)(2n+3)}, n\in\mathbb{N}^{*}$$
I need help proving that this series is convergent and calculating its sum.
What I've done so far: $$\lim_{n\to\infty}\arccos\frac{n(n+1)+\sqrt{(n+1)(n+2)(3n+1)(3n+4)}}{(2n+1)(2n+3)}=0$$ The result shows that the series may be convergent, but I don't know how to continue. WolframAlpha shows that the series is convergent (by the comparison test), but I have no idea to what other convergent series could I compare it.
Thank you!
As a thumb rule, if the terms of a convergent series are really ugly and you know in advance that such a series has a nice closed form, then the given series is a telescopic one, very likely. For any $n\geq 1$ we have
$$\small\cos\left(\arccos\frac{n}{2n+1}-\arccos\frac{n+1}{2n+3}\right)\\ =\small \frac{n(n+1)}{(2n+1)(2n+3)}+\small\sqrt{\small\left(1-\frac{n^2}{(2n+1)^2}\right)\left(1-\frac{(n+1)^2}{(2n+3)^2}\right)}$$ hence $$\sum_{n=1}^N \arccos\frac{n(n+1)+\sqrt{(n+1)(3n+1)(n+2)(3n+4)}}{(2n+1)(2n+3)}$$ is simply given by $\arccos\frac{1}{3}-\arccos\frac{N+1}{2N+3}$. I guess you can compute the limit as $N\to +\infty$.
It is given by $\arctan(2\sqrt{2})-\frac{\pi}{3}=\frac{\pi}{6}-\arctan\left(\frac{1}{2\sqrt 2}\right)\approx 0.183761866\approx \frac{43}{234}$.
• You forgot to mention "By creative telescoping", which is what I find to be your trademark. – Simply Beautiful Art Oct 23 '17 at 0:51
To know the answer, first you must know some results:
the propose (1) is sum of angle of cosine:
$$\arccos\left(x_1\right)-\arccos\left(x_2\right)=\arccos\left(x_1\cdot x_2+\sqrt{\left(1-x_1^2\right)\left(1-x_2^2\right)}\ \right)\quad, x_1<x_2$$
hence
$$\arccos\left(x_1\right)-\arccos\left(x_2\right)=\arccos\left(x_1\cdot x_2+\sqrt{\left(1-x_1\right)\left(1+x_1\right)\left(1-x_2\right)\left(1+x_2\right)}\right)\tag{1}$$
So compare the expression (1) with expression (2):
$$\arccos\left(\frac{n(n+1)+\sqrt{(n+1)(n+2)(3n+1)(3n+4)}}{(2n+1)(2n+3)}\ \right)\tag{2}$$
hence can be easily deduced:
$$x_1=\frac{n}{2n+1}\ \ ,\ \ \ x_2=\frac{n+1}{2n+3}\tag{3}$$
where $x_1<x_2$.
so using (3) in (1) obtain:
$$S(k)=\sum_{n=1}^k\arccos\left(\frac{n(n+1)+\sqrt{(n+1)(n+2)(3n+1)(3n+4)}}{(2n+1)(2n+3)}\ \right)$$
$$S(k)=\sum_{n=1}^k\left(\arccos\left(\frac{n}{2n+1}\right)-\arccos\left(\frac{n+1}{2n+3}\right)\right)$$
$$S(k)=\sum _{n=1}^{k }\arccos\left(\frac{n}{2n+1}\right)-\sum _{n=1}^{k }\arccos\left(\frac{n+1}{2n+3}\right)$$
it is easy to see that this is a telescoping series.
So.
$$S(k)=\arccos\left(\frac{1}{3}\right)-\arccos\left(\frac{k+1}{2k+1}\right)$$
hence using limits I have:
$$\lim_{k\to\infty}S(k)=\lim_{k\to\infty}\left(\arccos\left(\frac{1}{3}\right)-\arccos\left(\frac{k+1}{2k+1}\right)\ \right)$$
$$\lim_{k\to\infty}S(k)=\arccos\left(\frac{1}{3}\right)-\lim_{k\to\infty}\left(\arccos\left(\frac{k+1}{2k+1}\right)\ \right)$$
$$\lim_{k\to\infty}S(k)=\arccos\left(\frac{1}{3}\right)-\arccos\left(\lim_{k\to\infty}\frac{k+1}{2k+1}\right)\$$
$$\lim_{k\to\infty}S(k)= \arccos\left(\frac{1}{3}\right)-\arccos\left(\frac{1}{2}\right)$$
Finally:
$$\sum_{n=1}^{\infty }\arccos\left(\frac{n(n+1)+\sqrt{(n+1)(n+2)(3n+1)(3n+4)}}{(2n+1)(2n+3)}\ \right)=\arccos\left(\frac{1}{3}\right)-\arccos\left(\frac{1}{2}\right)$$
that which is the same:
$$\sum_{n=1}^{\infty }\arccos\left(\frac{n(n+1)+\sqrt{(n+1)(n+2)(3n+1)(3n+4)}}{(2n+1)(2n+3)}\ \right)=\arccos\left(\frac{1+2\sqrt{6}}{6}\right)$$
This is the best I could do:
first I substituted $n$ with $\dfrac{1}{x}$ to try a MacLaurin expansion and I got
$$\arccos\frac{\sqrt{8 x^4+42 x^3+67 x^2+42 x+9}+x+1}{3 x^2+8 x+4}$$
then with the help of Mathematica I found that this is
$\dfrac{x^2}{2 \sqrt{3}}+O(x^3)$
resetting $x=\dfrac{1}{n}$ I got
$\dfrac{1}{(2\sqrt{3} )n^2}$ which converges
Actually making a table of the values of $a_n$ and $\dfrac{1}{(2\sqrt{3} )n^2}$ you can see that they are very close.
No clue about founding the sum. Just an approximate value $0.18$
|
2020-02-22 00:34:51
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9258601665496826, "perplexity": 244.84048859209352}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145621.28/warc/CC-MAIN-20200221233354-20200222023354-00542.warc.gz"}
|
https://www.mail-archive.com/wien@zeus.theochem.tuwien.ac.at/msg12792.html
|
# Re: [Wien] f orbital under an external magnetic field
Dear Martin Pieper，
Thank you for your comments!
Actually, I intend to demonstrate that the energy difference between the
ground state of Er^3+ (S=3/2; L=6; J=15/2) and the excited state (S=3/2;
L=0; J=3/2) can be tuned by the external magnetic field, With the magnetic
filed and the crystal field, the excited state splits into four states,
|+3/2>, |+1/2>, |-1/2>, and |-3/2>. For the 45 Tesla magnetic field, the
delta energy between the |+3/2> and |-3/2> is over 10 meV. Since we can not
directly get the excited state in wien2k, even by forcing the occupation
number, the calculation will still be trick.
However, because the spin quantum number of the two states is the same
(S=3/2), there is no spin flip from the ground state to the excited state.
In this case, we can estimate the energy difference between the ground
state and the excited state by calculating the energy difference between
the occupied states of f electron in minority spin of the ground state and
the unoccupied counterparts in minority spin of the ground state. The
energy difference should become smaller with increasing the magnetic field,
which can be attributed to the lower in energy of the |-3/2> state relative
to the |+/-3/2> state with no magnetic field.
Since the energy shift is in the magnitude of meV, we can not seen this
shift from the dos calculation due to the smear of the dos. Since the f
band is usually very local and the band is very flat, so I checked the
eigenvalues of the 7 f-electron at the Gamma point and try to show the
energy shift from the variations of the eigenvalues. However, the results
show that there is only an energy shift from the 0 T to 4 T. When the
magnetic filed is increasing, the eigenvalues are almost the same as that
of 4 T.
This most probably is the old problem of the energy zero in disguise.
This may be the problem. But I have calculated all the energy differences
between the 3 unoccupied and 4 occupied states of f electron in minority
spin, the 12 (3*4) values are keep the same trend while the magnetic filed
is varied and they are all flat. For the different f states, they get
different J and the energy shifts (g_J*\mu_B*J*B) induced by the magnetic
filed should be also different. So I am confused. It should be noted that
the energy difference is independent to the energy zero.
Best,
Bin
On Thu, Aug 6, 2015 at 7:23 PM, pieper <pie...@ifp.tuwien.ac.at> wrote:
> As an afterthought:
>
> This most probably is the old problem of the energy zero in disguise. The
> Zeeman interaction you estimated and as accounted for in Wien2k is
> basically g*\mu_B*S*B. It gives you the energy difference between a moment
> pointing up and one pointing down. However, it has a vanishing trace, the
> zero is at B=0 and the center stays there.
>
> Best regards,
>
> Martin Pieper
>
>
> ---
> Dr. Martin Pieper
> Karl-Franzens University
> Institute of Physics
> Universitätsplatz 5
> A-8010 Graz
> Austria
> Tel.: +43-(0)316-380-8564
>
>
> Am 06.08.2015 04:55, schrieb Bin Shao:
>
>> Dear all,
>>
>> I made calculations of a compound with Er^3+(4f^11 5d^0 6s^0, ground
>> state S=3/2, L=6, J=15/2) doping under an external magnetic field. I
>> got the corresponding occupation of Er^3+ with 7 electrons in majority
>> spin and 4 electrons in minority spin. With soc including, I got
>> eigenvalues at Gamma point of the Er^3+ under the magnetic field from
>> 4 Tesla to 45 Tesla. However, the picture indicates that the
>> eigenvalues with the different magnetic fields almost keep the same as
>> that of 4 T. Why? According to a simple estimation, the magnetic field
>> of 45 T will introduce an energy shift about 10 meV, that would
>> definitely be seen from the figure.
>>
>> Any comments will be appreciated. Thank you in advance!
>>
>> Best regards,
>>
>> Bin
>>
>>
>> _______________________________________________
>> Wien mailing list
>> Wien@zeus.theochem.tuwien.ac.at
>> http://zeus.theochem.tuwien.ac.at/mailman/listinfo/wien
>> SEARCH the MAILING-LIST at:
>> http://www.mail-archive.com/wien@zeus.theochem.tuwien.ac.at/index.html
>>
> _______________________________________________
> Wien mailing list
> Wien@zeus.theochem.tuwien.ac.at
> http://zeus.theochem.tuwien.ac.at/mailman/listinfo/wien
> SEARCH the MAILING-LIST at:
> http://www.mail-archive.com/wien@zeus.theochem.tuwien.ac.at/index.html
>
--
Bin Shao
Postdoc
Department of Physics, Tsinghua University
Beijing 100084, P. R. China
Email: binshao1...@gmail.com
_______________________________________________
Wien mailing list
Wien@zeus.theochem.tuwien.ac.at
http://zeus.theochem.tuwien.ac.at/mailman/listinfo/wien
SEARCH the MAILING-LIST at:
http://www.mail-archive.com/wien@zeus.theochem.tuwien.ac.at/index.html
|
2019-10-18 16:57:13
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6973992586135864, "perplexity": 3169.371593906296}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986684226.55/warc/CC-MAIN-20191018154409-20191018181909-00275.warc.gz"}
|
http://amsacta.unibo.it/2768/
|
# No embedding of the automorphisms of a topological space into a compact metric space endows them with a composition that passes to the limit
Frosini, Patrizio ; Landi , Claudia (2010) No embedding of the automorphisms of a topological space into a compact metric space endows them with a composition that passes to the limit. DOI 10.6092/unibo/amsacta/2768.
Full text available as:
Preview
PDF
## Abstract
The Hausdorff distance, the Gromov-Hausdorff, the Fréchet and the natural pseudo-distances are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as $inf_{\rho} F(\rho)$ where $F$ is a suitable functional and $\rho$ varies in a set of correspondences containing the set of homeomorphisms. Our main result states that the set of homeomorphisms cannot be enlarged to a metric space $K$, in such a way that the composition in $K$ (extending the composition of homeomorphisms) passes to the limit and, at the same time, $K$ is compact.
Abstract
Document type
Monograph (Technical Report)
Creators
CreatorsAffiliationORCID
Frosini, Patrizio
Landi , Claudia
Keywords
Space of homeomorphisms, correspondence, compact metric space
Subjects
DOI
Deposit date
06 May 2010 10:12
|
2022-05-16 15:11:18
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6478462219238281, "perplexity": 675.5860708684464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510138.6/warc/CC-MAIN-20220516140911-20220516170911-00318.warc.gz"}
|
https://www.gktoday.in/question/the-length-of-a-rectangular-garden-is-12-metres-an
|
The length of a rectangular garden is 12 metres and its breadth is 5 metres. Find the length of the diagonal of a square garden having the same area as that of the rectangular garden :
[A] $8\sqrt{15} m$
[B] $13 m$
[C] $2\sqrt{30} m$
[D] $\sqrt{13} m$
|
2018-10-16 15:18:06
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 11, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5386325120925903, "perplexity": 248.57806273616438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583510754.1/warc/CC-MAIN-20181016134654-20181016160154-00290.warc.gz"}
|
https://www.physicsforums.com/threads/thermodynamics-reversibility-and-heat-addition.216580/
|
# Thermodynamics - reversibility and heat addition
1. Feb 19, 2008
### Haftred
I understand entropy is a state function, insofar as we deny the existence of irreversible cycles. However, for a said change of state, the heat transferred as a result of a reversible change is greater than that for an irreversible change. This is simply a reiteration of the Clausius inequality, as because entropy is a state function, a change dS is greater for a dq/T if dq is irreversible. However, it seems to me that it does not logically make sense for more heat having to be added for a reversible change. What is it about reversibility that requires more heat to be added to change states? I understand that entropy works out if the former statement is true; however, I guess I just don't understand how logically more heat is needed to effect a reversible change of state rather than an irreversible one. Where does the extra heat go if the final states are the same?
- Thanks
Last edited: Feb 19, 2008
|
2018-01-23 06:44:31
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8454962968826294, "perplexity": 374.1487990250193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891750.87/warc/CC-MAIN-20180123052242-20180123072242-00714.warc.gz"}
|
https://byjus.com/cube-calculator/
|
# Cube Calculator
Enter the Number :
Cube of the Number :
The Cube Calculator an online tool which shows Cube for the given input. Byju's Cube Calculator is a tool
which makes calculations very simple and interesting. If an input is given then it can easily show the result for the given number.
Oleum is
|
2018-11-18 10:18:34
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8338850140571594, "perplexity": 2412.888443688734}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039744348.50/warc/CC-MAIN-20181118093845-20181118115845-00176.warc.gz"}
|
https://www.thestudentroom.co.uk/showthread.php?t=4109525
|
# Waves/harmonic question
Watch
#1
Paper 1, Question 7.3
I have no idea what they are doing in that one, so any help will be appreciated (and in as much simplicity as possible )
0
4 years ago
#2
(Original post by 123chem)
Paper 1, Question 7.3
I have no idea what they are doing in that one, so any help will be appreciated (and in as much simplicity as possible )
Will walk you through it as a simpleish derivation:
the relationship between the speed of a wave, it's frequency and wavelength is:
from Newton's second law and some approximations (not going to walk you through this bit as it is complicated so just take the result as a given) the speed of a wave on a string is related to the tension,mass and length of a string by:
combining these gives us:
We can rearrange this to get:
we will now define where is the distance between the two fixed points (in this case X and Y). This relates the length between the points and the FUNDAMENTAL wavelength. Since we are using the fundamental wavelength, it also follows that we must use the fundamental frequency. (take it as a given that the tension remains the same regardless which mode we are in).
We can see from the diagram that we are in mode 3. The distance XY is given as 0.66m. We can calculate the fundamental frequency as:
substituting in our relation between the fundamental wavelength and the distance XY into our earlier relation we also find that:
we know the values of , , and and can plug these into our equation to get:
0
X
new posts
Back
to top
Latest
My Feed
### Oops, nobody has postedin the last few hours.
Why not re-start the conversation?
see more
### See more of what you like onThe Student Room
You can personalise what you see on TSR. Tell us a little about yourself to get started.
### Poll
Join the discussion
Yes, I'd look at higher ranking universities than my current choices (106)
43.98%
Yes, I'd look for a course or uni that is a better fit for me (39)
16.18%
No, I'd stick with my current uni choice (91)
37.76%
Something else (let us know in the thread below!) (5)
2.07%
|
2021-04-14 13:48:11
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8811481595039368, "perplexity": 813.486119903441}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077818.23/warc/CC-MAIN-20210414125133-20210414155133-00244.warc.gz"}
|
http://math.stackexchange.com/questions/317604/fxy-fx1-y-forall-x-y-in-0-1-find-all-f0-1-rightarrow-mathbbr
|
# $f(xy)=f(x(1-y)) \forall x,y\in (0,1)$, Find all $f:(0,1)\rightarrow \mathbb{R}$
Find all functions $f:(0,1)\rightarrow \mathbb{R}$ such that $f(xy)=f(x(1-y))$ for all $x,y\in (0,1)$.
-
For any $x \in (0,\frac12)$, $$f(x) = f\left((\frac{x}{x+\frac12})(x + \frac12)\right) = f\left((1 - \frac{x}{x+\frac12})(x + \frac12)\right) = f(\frac12)\tag{*}$$ For any $x \in (\frac12,1)$, $$f(x) = f\left((\frac{2x}{1+x})(\frac{1+x}{2})\right) = f\left((1 - \frac{2x}{1+x})(\frac{1+x}{2})\right) = f\left(\frac{1-x}{2}\right) \underbrace{= f(\frac12)}_{\text{by (*)}}$$
This implies $f(x) = f(\frac12)$ is a constant over $(0,1)$.
-
We have $\Delta = \{(xy,x(1-y)) | x,y \in (0,1) \} = \{(s,t) | s,t \in (0,1), s+t<1 \}$, and we have $f(a) = f(b)$ for all $(a,b) \in \Delta$.
Hence $f$ must be a constant.
To see this, note that $(0,\frac{1}{2})^2 \in \Delta$, hence if $x,y \in (0, \frac{1}{2})$, then $f(x) = f(y)$, ie, $f$ is constant on $(0, \frac{1}{2})$.
Note also that if $\alpha \in (0,1)$, then we also have $S_\alpha = (0,\alpha)\times (0,1-\alpha) \subset \Delta$. If $(a,b) \in S_\alpha$, we have $\min(a,b) < \min(\alpha, 1-\alpha) = \frac{1}{2}$, from which it follows that $f(a) = f(\min(a,b))$, hence $f$ is constant on $(0,\alpha)$. Since $\alpha \in (0,1)$ is arbitrary, we see that $f$ is constant on $(0,1)$.
-
Take analytical continuation of a function $f$ to the domain $[0,1]$ and that satisfies the same functional equation for $f$, so $g(x) = y(x), \forall x \in (0,1)$. Consider $y = 0$ in $g(xy) = g(x-xy)$, you'll get that $g(x) = g(0)$ which is a constant, so is $f$ then.
You cannot take $y=0$, though. – Andres Caicedo Mar 1 at 6:41
@copper.hat I mean extension of $f: \mathbb (0,1) \rightarrow\mathbb R$ to a $g$ s.t. $g: [0,1] \rightarrow\mathbb R$ and $g(x) = f(x)$ for $\forall x \in (0.1)$, and moreover satisfying same functional extension but for $\forall x \in [0,1]$. – Kaster Mar 1 at 7:27
|
2013-12-19 07:11:26
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9933394193649292, "perplexity": 97.50737976740342}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345762220/warc/CC-MAIN-20131218054922-00054-ip-10-33-133-15.ec2.internal.warc.gz"}
|
http://openstudy.com/updates/4e461ad10b8b3609c724d7ca
|
## Mimi_x3 4 years ago Can you please help me with this maths question 1. The cubic polynomial P(x)= x^3 +rs^2 +sx+t , where r,s and t are real numbers has three real zeroes,1,a, and -a (a) find the value of s + t
1. myininaya
is the polynomial written correctly?
2. Mimi_x3
woops my bad sorry P(x) = x^3 +rx^2 +sx +t
3. helpme1234
Colorful pi can you help me?
4. myininaya
let me try to help him and then i will look at your question
5. Mimi_x3
him ? are yu referring to me ?
6. myininaya
is this algebra?
7. Mimi_x3
noo , its a polynomial question
8. myininaya
so what class is it then?
9. Mimi_x3
class ? what do yu mean ? Its from the topic polynomials
10. myininaya
$P(x)=(x-1)(x-a)(x+a)=x^3+rs^2+sx+t$
11. Mimi_x3
12. myininaya
so I guess you are doing it for fun? $P(x)=(x-1)(x^2-a^2)=x^3+rs^2+sx+t$ no i just thought u would like to see the steps
13. Mimi_x3
no , im not doing it for fun , its my maths assignment LOL
14. myininaya
$P(x)=x(x^2-a^2)-1(x^2-a^2)=x^3+rsx^2+sx+t$
15. myininaya
so the class is called polynomials lol?
16. Mimi_x3
what does yu mean by class ?
17. myininaya
$P(x)=x^3-a^2x-x^2+a^2=x^3+rsx^2+sx+t$ people go to class to learn and get assignments lol
18. Mimi_x3
no the class is called maths LOL
19. myininaya
$P(x)=x^3-x^2-a^2x+a^2=x^3+rsx^2+sx+t$
20. Mimi_x3
2 unit maths to be specific
21. myininaya
ok we are almost done with this problem can you guess what to do next?
22. myininaya
we want both sides to be the same
23. Mimi_x3
collecting like terms ?
24. myininaya
ok the the coefficient of x^2 on left hand side is -1 the coefficient of x^2 on the other sides is rs so what can you say about rs and -1?
25. myininaya
they must be equal
26. myininaya
so we need that for the other terms as well
27. myininaya
we have -1=rs -a^2=s a^2=t
28. polpak
I seriously think I'm being trolled here.. but just to ease my tired brain.. $\sqrt{25} \ne -5$
29. myininaya
you are right polpak
30. Mimi_x3
wait , dnt yu factorise it further to (x^2-a^2)(x-1)
31. myininaya
that =5
32. myininaya
factor?
33. polpak
Someone was trying to tell me that it equals both 5 and -5
34. myininaya
ok the objective is to find s+t
35. polpak
But I can only really prove it by going to the definition. Which they don't accept.
36. Mimi_x3
37. polpak
Sorry for bothering you mimi, but myin rarely watches group chat (with good reason)
38. myininaya
what do you think s+t is based on what i gave you above? hint we don't need that first equation rs=-1
39. Mimi_x3
yu talking to me or polpak ? LOL
40. myininaya
you
41. Mimi_x3
lols , tbh im lost , i dont get it xDD
42. myininaya
do you understand when I wrote P as P(x)=(x-1)(x-a)(x+a)?
43. Mimi_x3
yeah , i dont understand the part -1 = rs -a^2 = s a^2 = t where did yu get it from ?
44. myininaya
both sides had to be equal we were looking at $P(x)=x^3-a^2x-x^2+a^2=x^3+rsx^2+sx+t$
45. myininaya
look at the coefficients of each term on both sides we want each term to be the same as it is on the other side
46. myininaya
so if we look at the x^3's they are good they both have coefficient 1 what is the coefficient of x^2 on left hand side ? what is the coefficient of x^2 on right hand side?
47. Mimi_x3
ohh i get it rs = -1
48. myininaya
yes now what is the coefficient of x on the left hand side? what is the coefficient of x on the right hand side?
49. Mimi_x3
x on the left hand side: 1 i think x on the right hand side 1 i think
50. myininaya
no what is the number in front of x on boths?
51. myininaya
both sides*
52. Mimi_x3
there's no other number
53. myininaya
-a^2? s?
54. Mimi_x3
lols , its a letter not a number
55. myininaya
they are numbers
56. Mimi_x3
they are ? arent they are called pronumerals ?
57. Mimi_x3
are they*
58. myininaya
never heard of this term
59. Mimi_x3
ohh ok
60. myininaya
when you are doing math and you don't see a word and you see a letter it usually represents a number unless it is the word a or i
61. myininaya
becauce a and i can be words
62. Mimi_x3
ohh ok , i never knew that they are numbers xD
63. myininaya
but usually the book will make it clear to you what they are
64. myininaya
yep
65. Mimi_x3
ok , so x= a^3
66. myininaya
a^3 is nowhere to be found we aren;t solving equation for x
67. Mimi_x3
ok , so x= a^3
68. Mimi_x3
ohh ok
69. myininaya
ok so look again at the polynomial what is the coefficent of x on both sides?
70. Mimi_x3
ok , so x= a^3
71. Mimi_x3
ok , so x= a^3
72. Mimi_x3
x= s
73. myininaya
ok no $P(x)=x^3-a^2x-x^2+a^2=x^3+rsx^2+sx+t$ ^ ^ | | -a^2 = s
74. Mimi_x3
ok , so x= a^3
75. Mimi_x3
ohh lols , sorry im so slow of understanding it
76. myininaya
ok do you see the terms that don't an x in em? there terms are called constants and they must be equal of that we have the same polynomial on both sides
77. myininaya
god my english is getting bad
78. Mimi_x3
ok , so x= a^3
79. Mimi_x3
lols , probably getting annoyed of me not getting it xD
80. myininaya
no its just late here i been up for along time today
81. myininaya
a long*
82. polpak
Ditto!
83. myininaya
so what are the constants on both sides?
84. myininaya
the terms that don't have a x in them
85. Mimi_x3
ok , so x= a^3
86. Mimi_x3
t
87. Mimi_x3
ok , so x= a^3
88. myininaya
t and what is it on the other side?
89. myininaya
remember you are looking for non-x terms
90. Mimi_x3
ok , so x= a^3
91. Mimi_x3
a
92. myininaya
a or a^2?
93. Mimi_x3
a
94. myininaya
$P(x)=x^3-a^2x-x^2+a^2=x^3+rsx^2+sx+t$ ^ ^ | | a^2 = t
95. Mimi_x3
96. myininaya
so what is s again? and what is t again?
97. Mimi_x3
s = 1 t = a^2
98. myininaya
s is not 1
99. Mimi_x3
a^3
100. myininaya
not a^3 scroll up we have it above
101. Mimi_x3
-a^2
102. Mimi_x3
lolss , im so~ slow
103. myininaya
YES! :)
104. Mimi_x3
lols , finally i get it
105. myininaya
ok now s+t is -a^2+a^2 right?
106. Mimi_x3
yeah
107. myininaya
but what is -a^2+a^2?
108. Mimi_x3
so substitute it in s and t
109. Mimi_x3
idk
110. myininaya
-5+5 is
111. myininaya
-6+6 is -24+24 is -999+999 is
112. Mimi_x3
where did 5 come from ?
113. myininaya
114. Mimi_x3
0 ?
115. myininaya
yes! :)
116. myininaya
and now you are done
117. Mimi_x3
really ? omg ! yay i got it right xD
118. Mimi_x3
thank you very mcuh for helping (:
119. myininaya
np
|
2015-08-30 07:59:15
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7355080842971802, "perplexity": 6864.901124319505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644064951.43/warc/CC-MAIN-20150827025424-00056-ip-10-171-96-226.ec2.internal.warc.gz"}
|
https://space.stackexchange.com/questions/33698/is-software-available-for-writing-two-line-element-files-from-keplerian-elements
|
# Is software available for writing two line element files from Keplerian elements?
I am interested in writing a TLE file from existing Keplerian elements. Is software available on the web to do this?
• @JCRM - The reference in your answer does not provide an answer this question. That reference refers to Keplerian orbital elements, which are distinct from the mean orbital elements used in TLEs and the software that propagates them. Jan 24, 2019 at 10:21
• absolutely, that translates state vectors to keplerian elements, not keplerian elements to SGP4 elements. Thanks for pointing that out @DavidHammen
– user20636
Jan 24, 2019 at 13:18
Keplerian orbital elements and the mean orbital elements used in the simplified perturbation models are rather different beasts. Keplerian elements assume a spherically symmetric central body and no perturbations from other bodies, atmosphere, etc. The simplified perturbation models account for those perturbations (non-spherical Earth, third body accelerations, atmospheric drag, radiation pressure) in a simplified manner. The one thing the two concepts do have in common is that the computational cost of computing position and velocity at some point in time is independent of the time difference between epoch and the point in time; neither approach uses numerical integration.
The way two line elements are created is via a number of measurements of some indicator of satellite state that are spread over time. The numerical values in the TLE are adjusted via an orbit determination process so as to minimize in a least squares sense a weighted sum of the squared errors between measurement and simplified perturbation model prediction. An inverse mapping from measurement (or from Cartesian states, or from Keplerian elements) to mean elements is not needed.
That said, people have tried to do this. An outline of an algorithm:
1. Compute Cartesian state (position and velocity) from your Keplerian orbital elements.
2. Transform that Cartesian state to the True Equator Mean Equinox frame. As far as I can tell, the US Air Force is the only entity that uses TEME. Your Keplerian elements are almost certainly in some other ECI frame (there are a bunch of them).
3. Form an initial guess of the TLE set by
• Setting the epoch time to the time of the Keplerian elements.
• Setting the TLE orbital elements to the TEME Keplerian elements computed from the TEME position and velocity.
• Setting B* to zero or to some informed guess.
4. Repeat until converged:
1. Compute Cartesian state from the TLE set.
2. Compute the differences between the target position and velocity versus the position and velocity computed from the TLE set.
3. Combine the error vectors to form a scalar error value. For example, $$||\Delta \boldsymbol{x}||^2/||\boldsymbol{x}||^2 + ||\Delta \boldsymbol{v}||^2/||\boldsymbol{v}||^2$$.
4. Exit the loop if the error is sufficiently small.
5. Estimate the Jacobian between the TLE elements and the Cartesian state.
6. Use the Jacobian from step 4.5 and the error vectors from step 4.2 to compute an updated TLE set. Care might be needed here, particularly for nearly circular or nearly equatorial orbits. You might want to use singular value decomposition or ANOVA so as to only attack the statistically significant elements.
• "The numerical values in the TLE are adjusted via an orbit determination process so as to minimize in a least squares sense a weighted sum of the squared errors between measurement and simplified perturbation model prediction." I love these sentences, you pack so much into each one! I used to think that predicted trajectories from "master integrators" using advanced models that combined recent plus historical observational data where then "down-fitted" to SGP4 -> TLEs, rather than the TLEs coming directly from observations (cont.)
– uhoh
Jan 25, 2019 at 1:47
• (cont.) But these eccentricity blips 1, 2 support the TLEs being fitted to only recent data.
– uhoh
Jan 25, 2019 at 1:49
• @uhoh - It could go that way, precision orbit determination and from that develop a TLE, but the TLE generation would still be via an orbit determination process. It's also important to keep in mind that NORAD / whichever TLA agency produces TLEs nowadays has to do this for thousands of objects in space based on who knows how many observations per object per day. Performing precision orbit determination on all of them would be a daunting computing task, particular when only a small fraction of those objects warrant high precision scrutiny. Jan 25, 2019 at 4:16
|
2022-12-04 23:11:25
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5779273509979248, "perplexity": 1244.7523635861}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710980.82/warc/CC-MAIN-20221204204504-20221204234504-00566.warc.gz"}
|
https://www.physicsforums.com/threads/limit-question.133664/
|
# Limit question
1. ### vabamyyr
66
I have a question:
what is lim (n--->infinity)= 1/(3+(-1)^n))? My opinion that this limit does not exist.
2. ### arildno
12,015
"Do not opine, PROVE!"
Apocryphal quote from Euclid.
3. ### CRGreathouse
3,682
$$\lim_{n\rightarrow\infty}\frac{1}{3+(-1)^n}$$
perhaps? The equals sign in your post is confusing me. If so, are you familiar with the lim sup and lim inf? That would give you an easy direct proof: if lim sup = lim inf, that's the limit; otherwise, the limit does not exist.
4. ### vabamyyr
66
i have dealt with sup but not with inf but i will look them up. Thx anyway.
5. ### manoochehr
6
manooch
if n∈Z (Z=Integer) then we have two answer for equation
if n∈R (R=Real) then equation is undefined
for example: (-1)^1/2 does not exist.
6. ### d_leet
It certainly does, it just isn't real.
686
8. ### manoochehr
6
thank you for help me
9. ### manoochehr
6
thank you for conduce:tongue:
Accordingly this sequence isn't convergent
|
2015-04-18 20:56:55
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9217089414596558, "perplexity": 4737.185624229826}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246636213.71/warc/CC-MAIN-20150417045716-00286-ip-10-235-10-82.ec2.internal.warc.gz"}
|
http://upstatenypostcardclub.com/vegan-cake-qnrpyu/7e7cd2-latex-font-size
|
# latex font size
Should you require a different font size for your document, use the extsizes package. To change the fond typeface of the entire document, a simple line must be added to the preamble: The line \usepackage{tgbonum} establishes the font family TeX Gyre Bonum, whose font package name is tgbonum, as the default font for this document. All the best, Tom. sty}, \file{preview-latex. Is there any way to say “whatever the parent grouping’s font size is, make this group’s one size {larger|smaller}”, akin to CSS’s “font-size: larger/smaller”? Package Roman Math Sans serif Typewriter (adsbygoogle = window.adsbygoogle || []).push({}); Need help with your thesis or book project? : In most cases, the available font sizes for the standard classes are sufficient. Stefan Will change it in the post. … but this doesn’t work. Viewed 4k times 10. The first statement in the document declares this is a Beamer slideshow: \documentclass{beamer} The first command after the preamble, \frame{\titlepage}, generates the title page. And again, it does not show the actual font size. How are {\large You} today? Thanks for your question! I am writing my thesis and its format in Latex. Anyway, the font size is hard coded in the class file (first few lines of code and line 233 onwards). I am a big fan of latex myself. Hope it helps, In case using this particular font for your thesis is a university requirement, they should have a license and be able to tell you how to install it on your system. Have recently started using Latex. A minimal working example of a simple beamerpresentation is presented below. So perhaps the solution for your doubt is to use. Changing the font size in LaTeX can be done on two levels, either affecting the whole document or parts/elements of it. There are four styles used in typesetting math formulas which affect the size and certain formatting parameters (notably the placement of sub and superscripts on variable size symbols). Common examples include Times, Courier, and Helvetica. Hi, I’m dealing with a monospace type 1 font that is not part of CTAN, I used the fontinst instructions and they work well but it is a little too big when used inline with another font. (In Hangul office, it is intuitively possible to adjust this option). 1 Times New Roman in Latex. Thank you Tom, wonderful resource that led me to solve the problem I had. Is any solution for that in order to change font size..? I did not find this option. Briefly, the Gulliver font is proprietary and no similar free font exists. For arbitrary sizes, have a look at Increase font size, standard sizes are explained here: LaTeX font sizes. « MIT Information Systems & Technology website. After compilation, a two-page PDF file will be produced. Perhaps you can post the code as a comment below. This might also answer Michael Peters to whom you recommended the luximono package. 21\DeclareMathSizes{\@xxpt}{\@xxpt}{\@xivpt}{\@xpt} 22\DeclareMathSizes{\@xxvpt}{\@xxvpt}{\@xviipt}{\@xiipt} 23\DeclareMathSizes{\@xxxpt}{\@xxxpt}{\@xxpt}{\@xivpt} 24\DeclareMathSizes{\@xxxvipt}{\@xxxvipt}{\@xxvpt}{\@xviipt}} 4.4 EC Font Declaration Patch. The line height is assigned at the end of the paragraph. Changing the font size locally. Sorry for the “bad” news. not all fonts allow all sizes. Let’s discuss. The standard classes, article, report and book support 3 different font sizes, 10pt, 11pt, 12pt (by default 10pt). If it’s not too long and may be useful to other people, I’ll be happy to publish your code. D. Thesis in Gulliver font of latex. 1. See the example in Rob’s comment below. This should not happen. I’ve been using TeX for 35 years, and LaTeX for the last 20 or so, creating lecture notes with Beamer. Could you please add a short explanation on *why* this fixes the spacing? Now why \par does the trick, I’m not entirely sure. It allows for the following font sizes: 8pt, 9pt, 10pt, 11pt, 12pt, 14pt, 17pt, 20pt. In case you really need to use lucimono fonts, take a look at the luximono style file and how the scaling is done. How to Write a Thematic Essay. Entire document To change the basic font size used all the way through your document, put either "11pt" or "12pt" in your \documentclass line. The default font size for Latex is 10pt. 0. And manually ending the paragraph seems to also adjust \baselineskip when the font size is changed, whereas standard paragraph ending (blank line) does not. Would you mind pointing me to the place where I can get the cls-file for this class, please? Programmming. I wasn’t aware of that issue. That’s more like it! If you need to write a great thematic The Bibliography Latex Font Size Small essay, you’re on the right way. The anyfontsize package solves that beautifully. And finally I want to set font size either explicitly or using LaTeX syntax: \Huge \huge \LARGE \Large \large 0 Comments. In some standard templates like sage even though using \documentclass[12pt]{sage} font size won’t change. jetpack_widget_social_icons ul{display:block;margin:0 0 1. The reference to italic shape is odd:. For example, italic', oblique' and upright' are all font shapes. When using another font type, such as the Adobe Times Roman equivalent available in the PSNFSS package (see example below), however, you can benefit from that font size. This page may contain information about the author, institution, event, logo, and so on. Sorry for the delay in accepting it. See the article above for more details. II seems ridiculous to change the math font size in big projects manually! Using a different font size on a global level will affect all normal-sized text as well as the size of headings, footnotes, etc. Ask Question Asked 4 years, 10 months ago. Do you know the procedure to upload it. This how to explains how you can add captions for Microsoft Word tables like you see on tutorial screen shots, text books with diagrams and so forth. Hi Rob. See this post for a short introduction. If he wants Times, then he could just add \usepackage{times}. Thanks for your question. For example, if you wanted to just make a small part of your text in a different font, you would use something like: Or, if you wanted to put a larger region in a different size, you'd use something like: Thank you for your feedback. The purpose is of \aMacro is so that I can change those certain text ‘automatically’ (instead of manually) if I happen to want another size. Best wishes, Tom. 5. By changing the font size locally, however, a single word, a few lines of text, a large table or a heading throughout the document may be modified. LaTeX Font Info: External font cmex10' loaded for size (Font) 6> on input line 25. SQL Based Databases. Thanks a lot for this article and especially for the hint to use \par to adjust line spacing. Great to hear, thanks! To change the size of a font use a new font size parameter. But, personally, I dislike this font. In the example, {\huge huge font size} declares that the text inside the braces must be formatted in a huge font size. Here is alternative, more flexible approach. A large font size and simple layout will increase the visibility of the important features of the title page. For example, if you had: but you wanted to use 12pt type (10pt is the default), you would change it to: To change just a part of your paper into a different font size, you can use some of the sizing environments. d. thesis in the font Gulliver. Wow! But the actual size is not absolute. You can find an informative discussion on the Gulliver font here. With this line you define the type of document. Font Sizes \tiny \scriptsize \footnotesize \small \normalsize \large \Large \LARGE \huge \Huge Thanks for your comment! I have modified the theme I want to upload it to net so that it might useful to new theme seeker. Otherwise, you might want to take a look at the LaTeX font catalogue and pick another font you like. Post was not sent - check your email addresses! Therefore, you can’t and shouldn’t change the font size if you are submitting to a SAGE journal. The font Times New Roman or Times Roman is probably one of the best known fonts. Text can be added to Jupyter Notebooks using Markdown cells. LaTeX knows several font size modifier-commands (from biggest to smallest): A table of the exact font sizes in points can be found on wikibooks. I greatly appreciate it. Thank You. You just saved my life with the “anyfontsize”! The font family code pcr, is unique to the font package. PSNFSS provides the default Type 1 fonts listed, excluding Computer Modern (CM), Utopia, Fourier, and Euler. Cool, thanks Rob! Table 1, adapted from the PSNFSS documentation, summarizes the commonly used Latex font packages. Active 3 years, 4 months ago. Beamer even offers smaller size commands: \Tiny and \TINY. My university has given the reference format which is made by Hancom office (this program is only for Hangul in South korea). You want to use the command \DeclareMathSizes as explained in the answer to this question. Thanks, Tom. Code chunk font size in Rmarkdown with knitr and latex. LaTeX: Changing the Font Size Latex provides 10 different font sizes. Show Hide all comments. Please see the documentation for more details. Note: The figure is scaled and therefore does not show the actual font size. I want to write my ph. s w ap r e f e r en c e Toggle navigation. Are you referring to Rob’s comment? The paper types with their dimensions are given below: letterpaper (11 x 8.5 in) legalpaper (14 x 8.5 in) a5paper (5.8 x 8.3 in) a4paper (8.3 x 11.7 in) executivepaper (10.5 x 7.25 in) b5paper (25 x 17.6 cm) I want to write my Ph. LaTeX knows several font size modifier-commands (from biggest to smallest): \Huge \huge \LARGE \Large \large \normalsize (default) \small \footnotesize \scriptsize \tiny. Keep up the good work. The font family you intend to use has to be available as a package. You can download and install the luximono font as described here. The baseline-skip should be set to roughly 1.2x the font size. It seems you asked the very same question about a year ago. I’ve frequently found the available font sizes limiting. To change the basic font size used all the way through your document, put either "11pt" or "12pt" in your \documentclass line. To adjust the line width, use the geometry package. The two arguments to \fontsize are the actual font size and the size of the baseline-skip. Helvetica, Times, or Arial are frequently used fonts. There are some font typefaces that support only a limited number of characters; these fonts usually denote some special sets. The following example shows font size 50pt/5pt and compares them with \Huge and \tiny. Font size control of LateX previews in Org files. 33 “Erroneous nesting of equation structures” in using “\begin{align}” in a multi-line equation in rmarkdown to knit+pandoc pdf. The command \par ends the paragraph, that’s for sure. I am writing my thesis. I noticed some fonts have a scale option that can be passed to their usepackage command that corrects this issue but I have no clue how to go about adjusting my .sty for that. Also, note that certain commands may overrule the font size commands. To change the fond typeface of the entire document, a simple line must be added to the preamble: The line \usepackage{tgbonum} establishes the font family TeX Gyre Bonum, whose font package name is tgbonum, as the default font for this document. Formula-specific options (fleqn and leqno) Landscape print mode (landscape) Single- and double-sided documents (oneside, twoside) Titlepage behavior (notitlepage, titlepage) Chapter opening page (openright, openany) Font size. Change font size in LaTex, and how the corresponding font size is being applied, upon changing normalsize for the document class. But, I have a problem. How do I change length of the text which have a fixed height. Apparently, TeX reads the whole paragraph first for optimal space adjustment between words. Can ’ t make things too small/big you will need to look for an answer: -.! Line you define the type of document a newspaper hope it helps Tom. Window.Adsbygoogle || [ ] ).push ( { } ) ; need help your... Available font sizes in the document class and commands for the font size right place look! Essay, you ’ re on the right place to look to find out how to add a below. Line 25 very same question about a year ago any clue on where I can change the of! Titlepage, and the size of all equations globally sizes limiting names, and how the font! Codes can be latex font size on two levels, either affecting the whole document parts/elements... I had may produce the same add a short explanation on * why * this fixes the spacing even... { } ) ; need help with your thesis or book project document... Decide what the base font size either explicitly or using LaTeX syntax: \Huge \Huge \large \large \large Comments... So I usually do n't mix different font sizes for the last 20 or so I! About the author, institution, event, logo, and Helvetica place to for... Net so that it might useful to new theme seeker the trick, I ’ m not an on. List of fonts some of these commands after changing it to LaTeX.. Anyone know help me the list above, \Huge and \tiny page is the default type fonts... The place where I need to look for an answer: - ) the between. Are declared in advance example in Rob ’ s comment below ).push ( latex font size! Not entirely sure to publish your code a comprehensive list of fonts,! Too long and may be useful to new theme seeker submitting to a sage template! Controlled with font size in Rmarkdown with knitr and LaTeX for the document size is being applied, changing... Adjust line spacing depends on the Gulliver font is proprietary and no similar free font exists for this for. The journal guidelines would have been good enough write a great thematic the Bibliography LaTeX font sizes limiting order change... And commands for the “ anyfontsize ” package infos Formatting ⇒ change of font locally!: \tiny and \tiny ’ t change otherwise, you can set the normal font for... Smaller size commands { Times } equations globally and don ’ t too! Good work for the following example shows font size to look to find out how to add a comment.. Mean, allover fontsize is the titlepage, and Helvetica you might want to do it: Hello, \aMacro. Table Topic is solved them with \Huge and \ssmall line you define the type of.... For that in order to change parts of your TeX file ve found! Offers smaller size commands: \tiny and \tiny size and font family code pcr, is unique the. Too small/big … Several fonts require the addition of the issue you above. Letters for use in mathematical formulas ( typically used for a serif theme! Information about the author, institution, event, logo latex font size and how the corresponding font size LaTeX! The hint to use \par to adjust line spacing make changes to font... Adapted from the PSNFSS documentation, summarizes the commonly used LaTeX font packages wonderful that., Courier, and Euler document class and commands for the document korea ) line height is assigned the... And finally I want to take a look at Increase font size big. If it ’ s for sure size should be the right way this should be the first line your... In Rmarkdown with knitr and LaTeX } to the class file ( first few lines of code and 233. Different font sizes: 10pt ( default ) 11pt then he could add., Courier, and how the scaling is done set font size 50pt/5pt and compares them with and! Know help me whenever you use a new font size with one of the paragraph, ’... “ the TeXbook ” by Donald E. Knuth would be the right place look. - of typefaces, or Arial are frequently used fonts the good work the first line of your file! The class-file for sage font sizes: 10pt ( default ) 11pt be happy to publish your code book! Non-Standard latex font size type to change the font size publish your code, sometimes the defines... Found on wikibooks whom you recommended the luximono package size depending on what I put in the following shows... Usually do n't mix different font size in multi-page table Topic is solved here: LaTeX font in!, event, logo, and the size of all equations globally macro to modify the font size Rmarkdown. Upright ' are all font shapes creating lecture notes with beamer on * why * fixes! It is intuitively possible to adjust line spacing sage journal template in the macro theme seeker seems ridiculous to the! \Tiny and \tiny scaled and therefore does not show the actual font size to 13 in report?. Fonts listed, excluding Computer Modern ( CM ), which is made by office... Serif Typewriter LaTeX forum ⇒ text Formatting ⇒ change of font size resource that led me to the class (... On two levels, either affecting the whole document or parts/elements of it either explicitly or using LaTeX syntax \Huge. Using LaTeX syntax: \Huge \Huge \large \large 0 Comments affecting the whole document or parts/elements a. For sage w ap r e f e r en c e Toggle navigation \usepackage { }! Only works with a non-standard font type }, thank you Tom, resource. As described here OML encoding TeX math italic and defines: Asked 4 years, 10 months.. Latex, and the second one contains sample content size.. creating lecture notes with beamer of font latex font size LaTeX! Am changing it to net so that it might useful to new theme seeker article especially... My life with the “ anyfontsize ” package infos have the same paragraph moresize! This post your blog can not share posts by email another font you.. About a year ago be able to provide a font or set of known! Originally developed for a thesis type in LaTeX is controlled with font of! – larger than others set to roughly 1.2x the font size locally and keep up the good!... Are the actual font size Small essay, you need to decide what the base font size explicitly!: the case is important in these commands after changing it to net so that it might useful other! Same size font class-file for sage latex font size lecture notes with beamer in Overleaf changing the font size available to size... Ll be happy to publish your code sage } font size in LaTeX, and Euler using \documentclass [ ]... File ( first few lines of code and line 233 onwards ) the! A minimal working example of the text which have a look at the LaTeX font sizes:,... For parts/elements of it posts by email as \large to change font to! You mean the modified template own purposes, you need to look an!, using one of these commands after changing it to net so that it useful... Fontsize is the default type 1 fonts listed, excluding Computer Modern CM. Look for an answer: - ) it allows for the font size won ’ make. Is any solution for your doubt is to use lucimono fonts, take a look at LaTeX! Important in these commands your document, you have to make changes to the list,... Pick another font you like shouldn ’ t on CTAN a comment below intend to use the command as... This context and pick another font you like Roman math Sans serif Typewriter LaTeX forum ⇒ Formatting. Seem to be distributed with TeXLive and wasn ’ t and shouldn t! Perhaps the solution for that in order to change the font size commands: and... Typefaces that support only a limited number of characters ; these fonts usually denote special. Include Times, or Arial are frequently used fonts latex font size seems ridiculous change. I am writing my thesis in normal font, please use in mathematical formulas ( typically used for thesis. Base font size thing into a macro to modify the font size to. And keep up the good work … Several fonts require the addition of issue... And shouldn ’ t change the font family code pcr, is unique to the font size in,! Out how to set font size parameter South korea ) the Bibliography LaTeX font catalogue pick... Fontsize is the default type 1 fonts listed, excluding Computer Modern ( CM ), which makes and! Rmarkdown with knitr and LaTeX fonts listed, excluding Computer Modern ( CM ), which is made Hancom... Compares them with \Huge and \ssmall and its format in LaTeX \aMacro fine }, thank.. A simple beamerpresentation is presented below changes to the list above, \Huge and \tiny increasing size, they:! On what I put in the macro the math font size with one of the should... Large font size in LaTeX can be added to Jupyter Notebooks using Markdown.. Not sent - check your email addresses adjustment between words ( CM ), which is made by office. Might useful to new theme seeker Rob ’ s not too long and may be to... Document or parts/elements of a simple beamerpresentation is presented below modify the font to be distributed with TeXLive and ’!
|
2021-07-27 21:46:45
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.888217031955719, "perplexity": 2751.598915997926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153491.18/warc/CC-MAIN-20210727202227-20210727232227-00688.warc.gz"}
|
http://mathhelpforum.com/pre-calculus/186907-location-roots-problem.html
|
# Thread: Location of roots problem.
1. ## Location of roots problem.
Let
4x^2-4(α-2)x+α-2=0
(α is real number)
be a quadratic equation, then find the value of α for which both the roots are positive.
the conditions will be
1) Discriminant D≥0
by this condition i got α (-∞,2][3,∞)
2) f(0) greater than or equal to 0
by this we get α (2,∞)
3) now should i use -b/2a(point exactly between both roots)
and equate as -b/2a greater than 0
if-3rd point is right then what will be the final answer
α (?,?)union(?,?)
2. ## Re: Location of roots problem.
Originally Posted by sumedh
Let $4x^2-4(α-2)xα-2=0 (α\epsilon R)$
What does this mean?
3. ## Re: Location of roots problem.
sorry the alpha are not seen α=alpha
4x^2-4(α-2)x+α-2=0
where alpha is real.
4. ## Re: Location of roots problem.
What you've done is fine. The condition that the equation has two roots can be satisfied by $D\geq 0$ and so
$\alpha \in ]-\infty, 2]\cup [3,+\infty[$
To continue I would say, let $\delta, \gamma$ be the two positive roots of the function and therefore:
$\delta=\frac{-b+\sqrt{D}}{2a}=\frac{4(\alpha-2)+\sqrt{16(\alpha-2)^2-16(\alpha-2)}}{8}\geq 0$
$\gamma =\frac{-b-\sqrt{D}}{2a}=\frac{4(\alpha-2)-\sqrt{16(\alpha-2)^2-16(\alpha-2)}}{8}\geq 0$
Now, You have to solve two irrational inequality's in function of $\alpha$ and so you'll find out the values of $\alpha$ wherefore $\delta\geq 0$ and $\gamma \geq 0$
This is the way I would try to solve this question.
5. ## Re: Location of roots problem.
But in a different way:-
This method is based on the graph of quadratic equation that is a parabola
In third point ' 3)...... '
As (-b/2a ,-D/4a) is the vertex of parabola
it's x component lies exactly in the middle of the two roots( in case of one roots they coincide)
So it(-b/2a) will be greater than the smallest root,
and hence it will be greater than zero.
As zero is the point on x axis from where positive numbers start,
so taking it as reference point.
We get that -b/2a will be greater than 0
equating this we will get an interval (2,∞).
From this and other two interval in question
(-∞,2][3,∞)-------------(1)
(2,∞) ------------(2)
We get final interval as
[3,∞)
To continue I would say, let \delta, \gamma be the two positive roots of the function and therefore:
\delta=\frac{-b+\sqrt{D}}{2a}=\frac{4(\alpha-2)+\sqrt{16(\alpha-2)^2-16(\alpha-2)}}{8}\geq 0
\gamma =\frac{-b-\sqrt{D}}{2a}=\frac{4(\alpha-2)-\sqrt{16(\alpha-2)^2-16(\alpha-2)}}{8}\geq 0
Now, You have to solve two irrational inequality's in function of \alpha and so you'll find out the values of \alpha wherefore \delta\geq 0 and \gamma \geq 0
I will try this
6. ## Re: Location of roots problem.
Originally Posted by sumedh
Let
$4x^2-4(a-2)x+a-2=0$
(a is real number)
be a quadratic equation, then find the value of $a$ for which both the roots are positive.
the conditions will be
1) Discriminant $D\ge 0$ by this condition i got $a$: $(-\infty,2][3,\infty)$
Now Descartes rule of signs tells us under the conditions in force here that neither of the roots are negative iff $(a-2)\ge 0$, or rather $a\ge 2$, so we are now restricted to $a \in [3,\infty)$.
CB
7. ## Re: Location of roots problem.
Thank you very much
|
2016-09-25 11:05:33
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 18, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.917477011680603, "perplexity": 1326.7872665882671}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738660181.83/warc/CC-MAIN-20160924173740-00227-ip-10-143-35-109.ec2.internal.warc.gz"}
|
https://stats.stackexchange.com/questions/504855/are-stationary-markov-chains-iid-random-variables
|
# Are stationary markov chains iid random variables?
Let $$\{X_t\}_{t=1}^{\infty}$$ be a Markov Chain.
An initial marginal distribution $$\pi^T$$ for a markov chain is a stationary distribution if $$\pi^TP = \pi^T$$.
My understanding of this is that if the initial marginal distribution is stationary, then the marginal distributions of the chain is the same at all time points after the starting time. Does this mean that $$X_1,X_2,X_3 \ldots$$ are all iid Random Variables?
• No, the $X_i$'s are identically distributed under stationarity, all with the same distribution $\pi$ but they are dependent from one another. They are thus "did" rather than "iid". – Xi'an Jan 14 at 10:27
|
2021-03-02 17:48:51
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9532082676887512, "perplexity": 231.90432913492566}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178364027.59/warc/CC-MAIN-20210302160319-20210302190319-00122.warc.gz"}
|
https://math.stackexchange.com/tags/perfect-numbers/info
|
The Stack Overflow podcast is back! Listen to an interview with our new CEO.
# About
Tag Info
Questions about or involving perfect numbers which are positive integers that are equal to the sum of their proper positive divisors.
A positive integer $n$ is said to be a perfect number if it is equal to the sum of its proper positive divisors.
The smallest example of a perfect number is $6$ as it has positive proper divisors $1$, $2$, $3$, and $1 + 2 + 3 = 6$.
More generally, $2^{p-1}(2^p-1)$ is perfect whenever $2^p - 1$ is a prime (called a Mersenne prime); the case above corresponds to $p = 2$. Furthermore, every even perfect number is of this form.
It is currently unknown whether there are infinitely many perfect numbers or whether any odd perfect numbers exist.
|
2019-10-22 02:09:01
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4610344171524048, "perplexity": 177.9293706580928}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987795403.76/warc/CC-MAIN-20191022004128-20191022031628-00402.warc.gz"}
|
https://www.tutorialspoint.com/obtain-all-other-zeroes-of-3x-4-plus-6x-3-2x-2-10x-5-if-two-of-its-zeroes-are-sqrt-frac-5-3-and-sqrt-frac-5-3
|
# Obtain all other zeroes of $3x^4 + 6x^3 - 2x^2 - 10x - 5$, if two of its zeroes are $\sqrt{\frac{5}{3}}$ and $-\sqrt{\frac{5}{3}}$.
#### Complete Python Prime Pack for 2023
9 Courses 2 eBooks
#### Artificial Intelligence & Machine Learning Prime Pack
6 Courses 1 eBooks
#### Java Prime Pack 2023
8 Courses 2 eBooks
Given:
$3x^4 + 6x^3 - 2x^2 - 10x - 5$ and the two of its zeroes are $\sqrt{\frac{5}{3}}$ and $-\sqrt{\frac{5}{3}}$.
To do:
We have to find all the other zeroes
Solution:
If $\sqrt{\frac{5}{3}}$ and $-\sqrt{\frac{5}{3}}$ are zeros of the given polynomial then $(x+\sqrt{\frac{5}{3}})(x-\sqrt{\frac{5}{3}})$ is a factor of it.
This implies,
$(x+\sqrt{\frac{5}{3}})(x-\sqrt{\frac{5}{3}})=x^2-(\sqrt{\frac{5}{3}})^2=x^2-(\frac{5}{3})$
Therefore,
Dividend$f(x)\ =\ 3x^4\ + \ 6x^3\ –\ 2x^2\ -\ 10x\ -\ 5$
Divisor$=x^2-(\frac{5}{3})$
$3x^2-5$)$3x^4+6x^3-2x^2-10x-5$($x^2+2x+1$
$3x^4-5x^2$
---------------------
$6x^3+3x^2-10x-5$
$6x^3-10x$
---------------------
$3x^2-5$
$3x^2-5$
------------
$0$
------------
Quotient$=x^2+2x+1$
$f(x)=(x^2-\frac{5}{3})(x^2+2x+1)$
To find the other zeros put $x^2+2x+1=0$.
$x^2+x+x+1=0$
$x(x+1)+1(x+1)=0$
$(x+1)(x+1)=0$
$x+1=0$ and $x+1=0$
$x=-1$ and $x=-1$
All the zeros of $f(x)$ are $-1$, $-1$, $-\sqrt{\frac{5}{3}}$ and $\sqrt{\frac{5}{3}}$.
Updated on 10-Oct-2022 13:19:38
|
2022-12-04 18:33:46
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30432814359664917, "perplexity": 3297.500077709937}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710978.15/warc/CC-MAIN-20221204172438-20221204202438-00826.warc.gz"}
|
https://git.rockbox.org/cgit/rockbox.git/tree/manual/getting_started/mpio_install.tex?id=332433eb3d
|
summaryrefslogtreecommitdiffstats log msg author committer range
blob: 6810f8f0604349581b14348d606d2de19023e373 (plain)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 % $Id$ % Installing the bootloader is the trickiest part of the installation. As explained above, we cannot distribute the bootloader directly, and thus need to patch a compatible version of the MPIO firmware, which can be downloaded as described above. \begin{enumerate} \item Download official Rockbox bootloader for MPIO \playertype{} from \opt{mpiohd200}{ \url {http://download.rockbox.org/bootloader/mpio/hd200}}% \opt{mpiohd300}{ \url {http://download.rockbox.org/bootloader/mpio/hd300}} and save it to your desktop. The archive contains three files: bootloader.mpio, bootloader.map and rockbox-info.txt. The first file is actual bootloader, two others can be used for debugging and are irrelevant for end user. \item Download mkmpioboot utility from \url{http://download.rockbox.org/bootloader/mpio/mkmpioboot} \item Process previously downloaded official firmware to include rockbox bootloader. Open terminal window and type the following command: \begin{code}[firstline=\opt{mpiohd200}{1}\opt{mpiohd300}{2}, lastline=\opt{mpiohd200}{1}\opt{mpiohd300}{2}] mkmpioboot HD200_UPG.SYS bootloader.mpio HD200_UPG.rb mkmpioboot HD300_UPG.SYS bootloader.mpio HD300_UPG.rb \end{code} \item Copy \opt{mpiohd200}{\fname{HD200\_UPG.rb}}% \opt{mpiohd300}{\fname{HD300\_UPG.rb}} to the SYSTEM folder of your \dap{} and rename back to \opt{mpiohd200}{\fname{HD200\_UPG.SYS}}\opt{mpiohd300}{\fname{HD300\_UPG.SYS}} \item Safe eject your \dap{} \item Connect Wall charger and turn on the device. This should trigger firmware upgrade process which will install rockbox bootloader to the flash memory of the player. \end{enumerate}
|
2021-03-08 06:38:07
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9627423286437988, "perplexity": 4791.623041482115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178381989.92/warc/CC-MAIN-20210308052217-20210308082217-00088.warc.gz"}
|
http://tex.stackexchange.com/tags/vim/hot
|
# Tag Info
25
It would help to know which, if any, LaTeX plugin you're using for vim. (E.g., the latex-suite, vim-auctex, latex-box, etc.) Next, as far as viewer choice, the only widely used open source PDF viewer for Linux which currently supports SyncTeX well out of the box is Okular. That's probably your best choice. There are instructions fo setting up SyncTeX with ...
23
Disclaimer: I usually edit .tex files in Vim, but I don't use the Vim-LaTeX suite. I wouldn't say the following suggestions are tricks per se - they are provided by third-party plugins - but they actually help me with my usual TeX workflow: snipMate Created by Michael Sanders From the manual: snipMate.vim aims to be an unobtrusive, concise vim script ...
22
I suppose you already have Vim installed in your operating system. Usually, the installation is very straightforward for every operating system. For Macs, we have MacVim, and AFAIK two options are available: Getting the correct MacVim version for your operating system in the project website, unzip the archive file and drag MacVim.app to your Applications ...
22
The Vim-LaTex / LaTeX-Suite for Vim adds these markers automatically and by intension. You can jump to the next such marker using CTRL+J, which removes this marker. The idea is to speed things up by allowing you to jump to the end of the group or environment which was just added. This is also useful for templates where you can add <+name+> markers ...
21
I have the following function in my $VIM/ftplugin/context.vim file to format ConTeXt paragraphs (same as LaTeX: the environments are enclosed in \start... and \stop... instead of \begin{...} and \end{...}. It should be easy to adapt this to LaTeX (In fact, I think that I copied it originally from someone who had written it for LaTeX and adapted it to ... 21 I would highly recommend the vim latex-suite, which you can get either from http://vim-latex.sourceforge.net/index.php?subject=manual&title=Manu or (on an Ubuntu machine) using sudo apt-get install vim-latexsuite sudo vim-addons -w install latex-suite It provides many shortcuts. If I were going to type your first summation ... 16 If you write "... this was written by \ref{foo}" the following output is possible: ... this was written by [1] which looks ugly in fact of the linebreak. This is the reason why vim is very nice to you and told you that you should write: "... this was written by~\ref{foo}". Then your output is at least: ... this was written by [1] So it is not a ... 14 The following hopefully answers your "abbreviated questions": vim suggests inserting \@ before . in ...GRIP/ABP. So, do it! The reason here is that GRIP/ABP or any capitalized word before a period is usually an abbreviation. And, in some instances, abbreviations have periods, while some don't. To treat the end-of-abbreviation period as an end-of-sentence ... 12 Your question is a little confusing. What you mean is compiling a LaTeX document manually (into a PDF). You might want to adjust your title. Creating one would be the process of writing the document. This can be done in the command line using pdflatex <filename>. In VIM you could just use ESC:!pdflatex % (% can be used instead of the current filename) ... 11 With the vim-latexsuite spell-checking works fine. Since it makes typing latex files a lot easier and faster I can only recommend it to every LaTeX-writing vim user. 11 This is a solution for evince, thanks to José Aliste who wrote gedit-synctex-plugin: Preamble Download these files deflate them to ~/bin (or something within$PATH) Backward Search (Evince → Editor) Adopt the first line of »~/bin/evince« (EDITORCMD) to your needs. (run »evince_backward_search« to get help for possible entries) Compile your .tex File ...
11
7
The following solution only applies to paragraph formatting, it will properly work depending on the LaTeX styling settings. Another possible solution would be to set a hard wrap of 80 characters. http://vimcasts.org/episodes/hard-wrapping-text/ formatoptions: t - Auto-wrap text using textwidth c - Auto-wrap comments using textwidth, ...
7
I know it's against policy to answer commenting on other answers, but in this case -- given I lack the 50 reputation needed to comment directly on the relevant answer -- I think it's worth it. The function provided by Aditya functions perfectly for LaTeX, mutatis mutandis, except for one detail: often after beginning an environment, or a section, the very ...
7
A quite long comment... I don't think your question is specific to Vim or any other editor. Whether you are given a code completion, auto-environment closing tool or not it is quite important to reduce the mental load while writing your code. A short-term loss can be a long-term benefit if you can spend extra seconds to write-up with a better indentation. I ...
Only top voted, non community-wiki answers of a minimum length are eligible
|
2015-11-27 10:11:56
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7410597801208496, "perplexity": 4302.265462711836}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398448506.69/warc/CC-MAIN-20151124205408-00186-ip-10-71-132-137.ec2.internal.warc.gz"}
|
https://www.gradesaver.com/textbooks/math/algebra/college-algebra-10th-edition/chapter-4-section-4-5-inequalities-involving-quadratic-functions-4-5-assess-your-understanding-page-314/1
|
## College Algebra (10th Edition)
$x>-3$
$-3x-2<7$ $-3x<9$ If we divide an inequality by a negative number, the inequality changes direction: $x>-3$
|
2018-07-18 05:17:21
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9609018564224243, "perplexity": 710.0840806188779}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590051.20/warc/CC-MAIN-20180718041450-20180718061450-00434.warc.gz"}
|
https://math.stackexchange.com/questions/2610023/convergence-of-probability-measures-and-respective-distribution-functions
|
# Convergence of Probability Measures and Respective Distribution Functions
Suppose $\{P_n\}$ and P are probability measures on the real line with corresponding distribution functions $\{F_n\}$ and $F$, respectively.
Prove that $P_n$ converges weakly to P if and only $$\lim_{n \rightarrow \infty} F_n(x) = F(x)$$ at every point x where F is continuous.
We define weak convergence as: Let S be a metric space with its Borel $\sigma$-algebra $\Sigma$. We say that a bounded sequence of positive probability measures $P_n$ on $(S, \Sigma)$, $n = 1, 2, ...,$ converges weakly to the finite positive measure P, and write: $P_n \Rightarrow P$
Could we somehow use the fact that the probability measure for an interval $(a,b)$ = $F(b)-F(a)$ to answer this question?
• What is your definition of weak convergence again? I get the double arrow notation, but not your actual working definition. – kimchi lover Jan 18 '18 at 3:28
• Indeed, you are missing a condition that defines the convergence. – cgrudz Jan 18 '18 at 3:29
• Weak convergence of measures. Essentially what is given to us by Portmanteau theorem. Here is a link: en.wikipedia.org/wiki/… – Francois Wassert Jan 18 '18 at 3:50
• Your question can only make sense as a request to prove part of the Portmanteau theorem, which (in effect) asserts the equivalence of many alternative definitions of weak convergence. So you have to be more precise about exactly what we are assuming "weak convergence" means. Under one definition, what you want to prove is the definition of weak convergence. – kimchi lover Jan 18 '18 at 13:16
|
2019-06-20 11:26:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9525527954101562, "perplexity": 202.6459733611629}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999210.22/warc/CC-MAIN-20190620105329-20190620131329-00199.warc.gz"}
|
https://www.semanticscholar.org/paper/Co-rotating-vortices-with-N-fold-symmetry-for-the-Godard-Cadillac-Gravejat/65fd8340009ca026301aa8e4c9ea76e4ae41ccfd
|
• Corpus ID: 223953348
# Co-rotating vortices with N fold symmetry for the inviscid surface quasi-geostrophic equation
@article{GodardCadillac2020CorotatingVW,
title={Co-rotating vortices with N fold symmetry for the inviscid surface quasi-geostrophic equation},
author={Ludovic Godard-Cadillac and Philippe Gravejat and Didier Smets},
journal={arXiv: Analysis of PDEs},
year={2020}
}
• Published 15 October 2020
• Mathematics, Physics
• arXiv: Analysis of PDEs
We provide a variational construction of special solutions to the generalized surface quasi-geostrophic equations. These solutions take the form of N vortex patches with N-fold symmetry , which are steady in a uniformly rotating frame. Moreover, we investigate their asymptotic properties when the size of the corresponding patches vanishes. In this limit, we prove these solutions to be a desingularization of N Dirac masses with the same intensity, located on the N vertices of a regular polygon…
Global solutions for the generalized SQG equation and rearrangements
• Mathematics
• 2021
In this paper, we study the existence of rotating and traveling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation. The solutions are obtained by maximization of the energy
Existence and stability of smooth traveling circular pairs for the generalized surface quasi-geostrophic equation
• Mathematics
• 2021
Abstract. In this paper, we construct smooth travelling counter-rotating vortex pairs with circular supports for the generalized surface quasi-geostrophic equation. These vortex pairs are analogues
Existence of co-rotating and travelling vortex patches with doubly connected components for active scalar equations
• Mathematics
• 2021
Abstract. By applying implicit function theorem on contour dynamics, we prove the existence of co-rotating and travelling patch solutions for both Euler and the generalized surface quasi-geostrophic
Vortex collapses for the Euler and Quasi-Geostrophic models
This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics
Multipole vortex patch equilibria for active scalar equations
• Mathematics
• 2021
. We study how a general steady configuration of finitely-many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady
Existence and regularity of co-rotating and travelling global solutions for the generalized SQG equation
• Mathematics
• 2021
By studying the linearization of contour dynamics equation and using implicit function theorem, we prove the existence of co-rotating and travelling global solutions for the gSQG equation, which
H\"older regularity for collapses of point vortices
• Mathematics
• 2021
The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of α models. In these
On the global classical solutions for the generalized SQG equation
• Mathematics
Journal of Functional Analysis
• 2022
## References
SHOWING 1-10 OF 22 REFERENCES
Steady symmetric vortex pairs and rearrangements
We prove an existence theorem for a steady planar flow of an ideal fluid, containing a bounded symmetric pair of vortices, and approaching a uniform flow at infinity. The data prescribed are the
On the V-states for the Generalized Quasi-Geostrophic Equations
• Mathematics
• 2015
We prove the existence of the V-states for the generalized inviscid SQG equations with $${\alpha \in ]0, 1[.}$$α∈]0,1[. These structures are special rotating simply connected patches with m-fold
Smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation
• Mathematics, Physics
• 2017
We construct families of smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation (SQG). These solutions can be viewed as the equivalents for this equation of the vortex
A Simple Energy Pump for the Surface Quasi-geostrophic Equation
• Mathematics, Physics
• 2012
We consider the question of growth of high order Sobolev norms of solutions of the conservative surface quasi-geostrophic equation. We show that if s>0 is large then for every given A there exists
Desingularization of Vortices for the Euler Equation
• Mathematics
• 2010
We study the existence of stationary classical solutions of the incompressible Euler equation in the planes that approximate singular stationary solutions of this equation. The construction is
Surface quasi-geostrophic dynamics
• Physics, Environmental Science
• 1995
The dynamics of quasi-geostrophic flow with uniform potential vorticity reduces to the evolution of buoyancy, or potential temperature, on horizontal boundaries. There is a formal resemblance to
Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar
• Physics, Environmental Science
• 1994
The formation of strong and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied through the symbiotic interaction of mathematical theory and numerical
Mathematical Theory of Incompressible Nonviscous Fluids
• Mathematics
• 1993
This book deals with fluid dynamics of incompressible non-viscous fluids. The main goal is to present an argument of large interest for physics, and applications in a rigorous logical and
Existence of Corotating and Counter-Rotating Vortex Pairs for Active Scalar Equations
• Mathematics, Physics
• 2016
In this paper, we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the $${{\rm (SQG)}_{\alpha}}$$(SQG)α equations with {\alpha \in
|
2022-07-03 00:05:53
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7794532179832458, "perplexity": 2197.173081751951}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104205534.63/warc/CC-MAIN-20220702222819-20220703012819-00432.warc.gz"}
|
https://rebelsky.cs.grinnell.edu/~rebelsky/Courses/CSC207/2019S/01/readings/list-adts
|
Summary
We consider ways to think about lists, which are among the simplest collections of values.
## Introduction
While abstract data types (ADTs) serve a variety of purposes, most ADTs are used to store collections of values. What distinguishes ADTs is how the ADT organizes and provides access to the elements in the collection. We’ll also see other issues, such as whether the ADT is homogeneous or heterogeneous, mutable or immutable, and dynamic or static. But most of these additional design issues are secondary to the primary design question: How does the ADT organize and access the elements?
Lists are conceptually among the simplest abstract data types. In essence, a list is a collection of values that you can visit one by one, with the order in which the elements are visited is controlled by the client.
How do clients control the order in which elements are visited? Typically, the instructions to add elements (and to remove elements, if removal is permitted) allow clients to clearly specify where in the list each new element goes.
## What are we listing? The “type” of a list
We now have a definition of list that suggests two primary operations: Clients should be able to add elements to lists (with control over positioning of elements) and clients should be able to visit elements of the list. Figuring out how to express each of those operations is an interesting design issue, one that we will get to in a moment. However, before looking at the details of these operations, let’s consider a few of the design issues we raised above.
What types does the list store? Before you learned about writing generic data types, you probably would have picked a type: “our lists will store strings” or “our lists will store integers”. You might also have thought about heterogeneous lists: “our lists will store any type of values”. And, as you’ve seen, allowing collections to store multiple types of values can be useful. And that utility should lead us to design heterogeneous lists.
However, heterogeneity can cause us to lose important benefits of using Java. In particular, many programmers use Java because Java provides compile-time type checking, and they know that compile-time type checking helps catch a lot of potential program bugs. If our lists our heterogeneous, we need to do run-time type checking. Hence, Java provides the generics that you’ve seen before. If we parameterize our lists with the type of value they store, we can still write generic code, but we can support homogeneous lists.
public interface MyList<T> {
...
} // interface MyList<T>
What if our client wants heterogeneous lists? That’s one of the nice things about Java’s generics: A list of Object values is heterogeneous as any Java value is either already an object or can be boxed into an object.
Because the homogeneous/heterogeneous question is so nicely solved by Java’s generics, we won’t return to that design question again. (You should, however, revisit these issues if you’re working in other languages or if other design decisions prevent you from using generics.)
Of course, the question of whether lists should be homogeneous or heterogeneous is not the only question you should ask. Let’s move on to a few more.
## How should lists change? Exploring Lisp Lists
A natural next question in the design of our list ADT might be Should lists be mutable or immutable? It may be strange to think about immutable lists. However, there are many situations in which it is convenient to make lists immutable. You may want to ensure that the same sequence is used in every situation. You may find that making lists immutable improves certain core operations. You may just know that mutable structures lead to complexity in program design and analysis.
It is certainly possible to think about lists as immutable structures. In fact, Lisp, one of the earliest programming languages, provides lists that many programmers treat as immutable. (Lisp lists are mutable; the latest versions of Scheme, a popular descendant of Lisp, provides both mutable and immutable lists.)
Let’s start by exploring the immutable model in a little more depth. Basic Lisp lists are built from a simple recursive definition of list.
• The empty list is a list.
• Adding an element to the front of a list produces a new list.
How does that typically translate into methods?
• We need a constructor to build empty lists. We might call this empty or we might just use a zero-parameter constructor. (It’s hard to specify constructors in interfaces, so we might settle for empty.) We might also make a design decision that null represents the empty list, although that will likely make our code less object-oriented.
• We need a method to create a new value to the front of a list. We might have a method prepend(T val) or we might have a two parameter constructor. Once again, it’s hard to specify constructors in interfaces, so we’ll stick with the method.
• We need a way to get the first element in a list. Traditionally, the operation is called car, but we’ll use the clearer head.
• We need a way to step through the list. The tradition in Lisp is to have a method that returns everything but the front of the list. Traditionally, the operation is called cdr, but we’ll use the clearer tail.
• We need a way to determine if a list is empty. We’ll use isEmpty.
Putting that all together, we get the following interface.
/**
*/
public interface LispList<T> {
/**
* Create the empty list.
*/
public LispList<T> empty();
/**
* Create a new list by prepending a new element to the front of
* this list.
*
* @param val a value
* @return lst list
* @post
* @post
* lst.tail() == this
*/
public LispList<T> prepend(T value);
/**
* Get the first element of the list.
*/
/**
* Get a list that contains all but the head of this list.
*/
public LispList<T> tail();
/**
* Determine if the list is empty.
*/
public boolean isEmpty();
} // interface LispList<T>
With these methods, it’s relatively straightforward to iterate through the elements of a list. Here’s a simple procedure that prints the elements of a list, one by one.
/**
* Print all the elements in a list.
*/
public static <T> void printList(PrintWriter pen, LispList<T> lst) {
while (!lst.isEmpty()) {
lst = lst.tail();
} // while
} // printList(PrintWriter, LispList<T>)
Of course, in addition to iterating lists, we need to provide a way for clients to control the order of elements in the list. And they can do so by building the list from back to front. Rearranging the elements involves building new lists, but it’s not that hard. For example, if we have the list [a, b, c] and want to replace the b with some new value, we might write something like the following:
newlst = lst.tail().tail().prepend(newval).prepend(lst.head());
Are there other methods we could include in the interface? Certainly. We might want methods that get the _i_th element of a list, that reverse a list, that extract sublists, that replace elements of the list, and so on and so forth. However, we are striving to start with minimalist interfaces, so we’ll start with the five basic methods.
While Lisp lists are conceptually simple, they also have some significant drawbacks. For example, there are many problems in which you want to change the elements of the list without building a new list. For example, we might be concerned with the storage requirements of our lists. In addition, Lisp Lists are closely tied to a particular implementation, one involving simple structures that link together the front of the list and the rest of the list. In practice, we might like the freedom to choose between implementations. Hence, while Lisp lists were a useful detour, we will continue our exploration by designing an ADT for mutable lists.
## Categories of methods
We’ve now made two major design decisions: Our list ADT will use generics so that we can support homogeneous lists of various types and our list ADT will support mutation. These decisions, along with our overall philosophy that lists are iterable collections, suggest four basic categories of methods.
• We need methods that the client can use to add elements to the list.
• We need methods that the client can use to remove elements from the list. (We might also choose to make these methods optional.)
• We might want methods that the client can use to replace elements of the list. Why not just use the methods to add and remove elements?
Because it might be much more efficient to replace elements. (Again, we might choose to make this methods optional.)
• We need methods to iterate through the elements of the list.
For the first two categories of methods, we might just allow people to work at the front and back of the list, generalizing the design of Lisp lists, although in a more mutable form. But it is clearly more useful to indicate positions in the list. That is, we might say that we want to remove an element at a particular position, or to add an element at, before, or after that position.
But how should we represent positions? There are a variety of approaches that designers use. We’ll consider each, and then explore Java’s standard technique.
## Positions - numeric and generalized
Most programmers start by thinking of positions as numbers. “I want to be able to remove the element at position 5.” In some ways, that design works well. Numbers are easy for people to understand, and most programmers are used to the numeric positions in arrays.
But there are also some significant disadvantages to using numeric positions. First, the semantics can be difficult. For example, what does it mean to remove the element at position 5? Do we end up with nothing there? Does everything shift left? Can we only remove at a position when it’s the beginning or end? What happens to the other positions? And so on and so forth. As importantly, using numbers can bias the implementation: There are implementations of lists, such as Lisp Lists, in which using numbers as positions leads to some inefficient implementations.
It’s also good practice to look at ways to generalize things. Hence, rather than saying “positions are integers”, we can say “we use positions”, and allow implementors to decide what form of position is best. If we choose this approach, we might first define a Position interface.
public interface Position {
} // interface Position
Now, in our MutableList interface, we can use these values.
public interface MutableList<T> {
...
/**
* Get the value at a particular position.
*/
public T get(Position pos);
/**
* Remove the element at a particular position.
*
* @pre the position is valid
*/
public void remove(Position pos);
/**
* Determine if a position is valid.
*/
public void isValid(Position pos);
...
} // interface Mu<T>tableList
How do we use positions? That is, how do we get a position in the middle of the list? One option is to have the list interface provide mechanisms for dealing with positions.
public interface MutableList<T> {
...
/**
* The front of the list
*/
public Position front();
/**
* Get the position that immediately follows pos.
*
* @pre pos is not at the end of the list.
*/
public Position next(Position pos);
/**
* Determine if a position is at the end of the list.
*/
public boolean atEnd(Position pos);
...
} // interfaceMutableList<T>
We now have enough methods that we can iterate lists, as well as mutate them.
public static <T> void printList(PrintWriter pen, MutableList<T> lst) {
Position here = lst.front();
while (!lst.atEnd(here)) {
pen.println(lst.get(here));
here = list.next(here);
} // while
} // printList(PrintWriter, MutableList<T>)
## Lists with a current element
Some designers (including the designers of some textbooks) decide that rather than having a separate position type, they’ll just keep track of the “current” element of the list.
public interface MutableListWithCurrent<T> {
...
/**
* Get the current element.
*/
public T get();
/**
* Advance to the next element.
*/
public void next();
/**
* Reset to the beginning of the list.
*/
public void reset();
/**
* Determine if we've reached the end of the list.
*/
public boolean atEnd();
...
} // interface MutableListWithCurrent<T>
With this interface, it’s equally easy to iterate lists.
public static <T> void printList(PrintWriter pen, MutableListWithCurrent<T> lst) {
lst.reset();
while (!lst.atEnd()) {
pen.println(lst.get());
lst.next();
} // while
} // printList(PrintWriter, MutableListWithCurrent<T>
It all sounds great, doesn’t it? But, as Joseph Bergin suggests in Lists with Current Considered Harmful, it’s not a very good design. For example, if we have more than one subprogram that’s interacting with a list, each might have its own notion of the current position. And, if we’re sorting a list in place, we will almost certainly need to keep track of positions. Hence, our lists will not have a current element.
## List cursors
You may have found the position interface a bit puzzling. After all, why are we having one object (the list) do all the work with another object (the position). Wouldn’t it make more sense to have the object that we’re getting information about do all the work? Alternately, might we generalize the notion of “current”.
I’ve found it useful to think of a “cursor” that we move through the list. Once we create a cursor, we can get the value at the cursor and move the cursor, and we leave the list implicit.
public interface ListCursor<T> {
/**
* Get the current element.
*
* @pre
* This cursor is valid.
*/
public T get();
/**
* Advance to the next element.
*
* @pre
* The cursor is not at the end of the list.
*/
public void next();
/**
* Determine if the cursor is valid.
*/
public boolean isValid();
/**
* Determine if the cursor is at the end of the list.
*/
public boolean atEnd();
} // interface ListCursor<T>
public interface BidirectionalListCursor<T> {
/**
* Retreat to the previous element.
*
* @pre
* The cursor is not at the beginning of the list.
*/
public void prev();
/**
* Determine if the cursor is at the beginning of the list.
*/
public boolean atFront();
} // interface BidirectionalListCursor<T>
public interface MutableList<T> {
...
/**
* Get a cursor for the front of the list.
*/
public ListCursor<T> front();
...
} // MutableList<T>
Once again, it’s easy to iterate using this design.
public static <T> void printList(PrintWriter pen, MutableList<T> lst) {
Cursor here = lst.front();
while (!here.atEnd()) {
pen.println(here.get());
here.next();
} // while
} // printList(PrintWriter, MutableList<T>)
How do we use these cursors for adding or removing elements? Here’s a case in which we might make the cursor a parameter to the method.
public interface MutableList<T> {
...
/**
* Add an element immediately after the cursor.
*
* @pre
* The cursor was created by this list.
* @pre
* The cursor remains valid.
*/
public void addAfter(T val, ListCursor<T> cursor);
...
} // MutableList<T>
All seems well and good, doesn’t it? However, given your experience with the other designs above, you’re probably waiting for a criticism. Believe it or not; I don’t have one. When I design my own list interfaces, I tend to include some form of cursor. Of course, there are still a host of design decisions: Do we allow cursor to retreat? What methods do we support for adding and removing elements? And so on and so forth.
## Iterators
While cursors provide a wonderful strategy for iterating lists, and one that I recommend, it’s also useful to know what the designers of Java came up with. In Java, clients iterate lists with objects that are in the class java.util.Iterator. Iterators are much like cursors, in the sense that you can build multiple iterators for a list, that you use them to get and advance, and that you can use them to add and remove elements. The differences are in the particular decisions.
First, Java’s iterators combine our get and next method. That is, when you call next, you get the next unvisited value and you advance beyond that value.
Second, Java’s iterators use hasNext to indicate whether or not we’ve reached the end of the list. (Hey, it’s just a name.)
Third, interfaces and classes that provide iterators traditionally do so with an iterator method and indicate that they implement the Iterable<T> interface.
Given those design decisions, iteration is easy.
public static <T> void printList(PrintWriter pen, MutableList<T> lst) {
Iterator<T> it = lst.iterator();
while (lst.hasNext()) {
pen.println(lst.next());
} // while
} // printList(PrintWriter, MutableList<T>)
In fact, this pattern is so common that Java provides a syntax for iterating any class that implements Iterator. One can use for (<varname>variable</varname> : <varname>collection</varname>) to set variable to each element of
collection
in turn. For example,
public static <T> void printList(PrintWriter pen, MutableList<T> lst) {
for (T val : lst) {
pen.println(val);
} // for
} // printList(PrintWriter, MutableList<T>)
It’s almost not worth writing the function any more.
But iterators differ from cursors in other important ways, too. You may recall that we made cursors parameters to list methods that mutate the list. Iterators, on the other hand, expect that you will use the iterator to mutate the underlying list.
For reasons that I don’t completely understand, iterators provide only an optional remove method which removes the value most recently returned by next
package java.util;
public interface Iterator<T> {
...
/**
* Remove the last element returned by next.
*/
public void remove()
throws UnsupportedOperationException, IllegalStateException;
} // interface Iterator<T>
That’s right. The implementor can indicate that the remove method is not available by throwing an exception. Clearly, whoever designed this interface was not sold on compile-time type checking.
What happens if we decide to call remove twice in a row, without a call to next in between? The semantics of remove are such that such a sequence of calls is considered invalid, and hence should be avoided.
What if we want to add a value? It turns out that the Java designers didn’t think addition was as important as removal. Hence, add is part of a subinterface called ListIterator. The add method adds a value immediately before the last element visited by next. List iterators also provide a prev that allows us to back up in the list. Finally, for no clear reason, list iterators also provides two methods that grab the index of the next or previous element, nextIndex and prevIndex.
## Putting it together
Where are we? We’ve considered a wide variety of design issues that one might consider while designing a list ADT. We ended up deciding that most of the work in a list can be done through the ListIterator interface. Putting it all together, this is what we seem to get.
/**
* Very simple lists.
*/
public interface SimpleList<T>
extends Iterable<T>
{
public Iterator<T> iterator();
public ListIterator<T> listIterator();
} // interface SimpleList<T>
Where are all the details? They’re in the ListIterator interface. We’ll consider the details in the next reading.
|
2020-11-26 04:53:17
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2281985580921173, "perplexity": 1879.2068754114428}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141186414.7/warc/CC-MAIN-20201126030729-20201126060729-00366.warc.gz"}
|
https://tex.stackexchange.com/questions/343570/documentclassarticle-show-authors-position
|
# \documentclass{article}: show author's position [duplicate]
My Latex code is:
\documentclass{article}
\begin{document}
\author{Author}
\title{Title}
\maketitle
Document's text.
\end{document}
The output is:
How can I show the author's position underneath the author's name, e.g. Professor at ...?
• The standard article class has no further fields. But you can write something like \author{Author\\Some University}. – gernot Dec 11 '16 at 23:00
• @gernot Do you want to make that an answer? Or do we have a good duplicate that suits? – Johannes_B Jan 8 '17 at 13:52
• @Johannes_B I decided to do both since none of the other posts deals with the simple case of a single author; but there are similar posts. Thanks for reviewing the post. – gernot Jan 8 '17 at 17:37
The standard article class has no further fields. But you can put additional information into the \author command, like
\author{Author\\Some University}
which will result in
\documentclass{article}
\begin{document}
\author{Author\\Some University}
\title{Title}
\maketitle
Document's text.
\end{document}
For more complicated arrangements of multiple authors with mixed affiliations consider to use the package authblk.
|
2020-06-04 02:03:38
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6292296648025513, "perplexity": 2579.7105048506833}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347436828.65/warc/CC-MAIN-20200604001115-20200604031115-00227.warc.gz"}
|
https://homework.cpm.org/category/CON_FOUND/textbook/a2c/chapter/7/lesson/7.2.1/problem/7-97
|
### Home > A2C > Chapter 7 > Lesson 7.2.1 > Problem7-97
7-97.
Simplify each expression below. If you are stuck, the ideas in problem 7-74 should be helpful.
1. $\frac { x } { 1 - \frac { 1 } { x } }$
Use a Giant One to eliminate the denominator of the fraction in the denominator.
Distribute:
$\frac{x}{x} \cdot \frac{x}{1-\frac{1}{x}}$
$\frac{1}{x} \cdot x = 1$
$\frac{x^2}{x-1}$
1. $\frac { \frac { 1 } { a } + \frac { 1 } { b } } { \frac { 1 } { b } - a }$
Eliminate all the denominators of the smaller fractions. What will the Giant One look like?
Distribute:
$\frac{ab}{ab} \cdot \frac{ \frac{1}{a} - \frac{1}{b} }{ \frac{1}{b} -a}$
|
2021-10-20 19:23:10
|
{"extraction_info": {"found_math": true, "script_math_tex": 6, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8881582617759705, "perplexity": 2039.5294732982545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585348.66/warc/CC-MAIN-20211020183354-20211020213354-00160.warc.gz"}
|
http://math.stackexchange.com/questions/268071/schr%c3%b6der-functional-equation
|
# Schröder functional equation
I have the following Schröder functional equation:
$f(h(s))=c.f(s)$
where $f,h: ℂ→ℂ$, here $f$ is not analytic and $h$ is analytic and $c∈ℝ$.
My question is: How we can solve this equation (the form of $f$ and its domain of definition). We can take $h$ as $h(s)=1-s$ or $h(s)=s-1$ and for both cases we can take $c=-1$.
Some motivations are available here:
http://en.wikipedia.org/wiki/Schr%C3%B6der%27s_equation
-
I solve such questions using the Carleman-matrix-concept (which you also find mentioned in the wikipedia article). Carleman-matrices contain in its rows the coefficients of a function in its power series representation, and of its powers. So the Carleman-matrix, say C contains the coefficients of $f(x)^0$ (which is the constant 1), $f(x)$, $f(x)^2$ ... in its rows.
In your case where $f(x)=1 - 1 x$ we have $$C= \begin{bmatrix} 1&.&.&. \\ 1&-1&.&. \\ 1&-2&1&. \\ 1&-3&3&1 \\ \end{bmatrix} \qquad \text{ for } \qquad \begin{array} {} f(x)^0 &=&1 \\ f(x)&=&1-x \\ f(x)^2&=&1-2x+x^2 \\ f(x)^3&=&1-3x+3x^2-x^3 \\ \end{array}$$ such that with a "vandermonde"-type column-vector $V(x) = [1,x,x^2,x^3,...]$ of appropriate size we shall have $$C \cdot V(x) = V(f(x))$$
Then if you find a diagonalization of C such that $M^{-1}\cdot D \cdot M = C$ where $D$ is diagonal and $M$ and $M^{-1}$ are triangular, then $M$ in its second row contains the coefficient of the Schröder-function and that in $M^{-1}$ in its second row its inverse. In our case we find that $$M^{-1} =\begin{bmatrix} 1 & . & . & . \\ 1/2 & 1/2 & . & . \\ 1/6 & 1/2 & 1/3 & . \\ 0 & 1/4 & 1/2 & 1/4 \end{bmatrix}$$ , with $D=\operatorname{diag}([1,-1,1,-1])$ and $$M = \begin{bmatrix} 1 & . & . & . \\ -1 & 2 & . & . \\ 1 & -3 & 3 & . \\ -1 & 4 & -6 & 4 \end{bmatrix}$$ is a possible solution. (Here $M^{-1}$ can be recognized as the set of coefficients of integrals of the bernoulli-polynomials when we extend the dimension of the matrix infinitely)
In general, the eigenvector-matrix $M$ can be understood as limit of the n'th power of $C$ scaled by the reciprocal of the n'th power of $f'(0)$ when n goes to infinity, and so the Schröder-function as limit of the n'th iterate of $f(x)$ divided by $f'(0)^n$ where $n \to \infty \qquad$ - but can furtherly be scaled by an arbitrary constant factor $\gamma \ne 0$
Remark: Your example which requires only matrix size of $n \times n= 2 \times 2$ is much easier, but then one wouldn't see the general principle (and the relation to the bernoulli-polynomials, so I used the bigger matrix size with n=4 here.
@ Gottfried Helms : This is a great answer. Thank you very much. But the function $f$ is not analytic. – ZE1 Dec 31 '12 at 11:13
@rh1: hmm, what do you mean? Unfortunately I mixed your given $h(s)$ by my usual $f(x)$ and so this overlaps your $f(s)$, sorry perhaps I should adapt this. By my example, the function $f(s)$ in your sense is $f(s)=-1+2s$ and $f^{-1}(s)=1/2(1+s)$ so $h(s)=f^{-1}(c \cdot f(s))= 1/2(1+(-1(-1+2s)))=1/2(2-2s)=1-s$ for any complex $s$. The $m$'th iterates are then expressible simply by $m$'th powers of $c$ - and, taking care of the problem of non-unique solutions of fractional or complex powers of $c=-1$ one can define even fractional- and complex-valued iterations heights $m$ – Gottfried Helms Dec 31 '12 at 12:55
|
2014-09-30 22:27:57
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9675365686416626, "perplexity": 249.48687952485358}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663167.8/warc/CC-MAIN-20140930004103-00061-ip-10-234-18-248.ec2.internal.warc.gz"}
|
http://scikit-learn.org/dev/modules/generated/sklearn.covariance.ledoit_wolf.html
|
# sklearn.covariance.ledoit_wolf¶
sklearn.covariance.ledoit_wolf(X, assume_centered=False, block_size=1000)[source]
Estimates the shrunk Ledoit-Wolf covariance matrix.
Read more in the User Guide.
Parameters: X : array-like, shape (n_samples, n_features) Data from which to compute the covariance estimate assume_centered : boolean, default=False If True, data are not centered before computation. Useful to work with data whose mean is significantly equal to zero but is not exactly zero. If False, data are centered before computation. block_size : int, default=1000 Size of the blocks into which the covariance matrix will be split. This is purely a memory optimization and does not affect results. shrunk_cov : array-like, shape (n_features, n_features) Shrunk covariance. shrinkage : float Coefficient in the convex combination used for the computation of the shrunk estimate.
Notes
The regularized (shrunk) covariance is:
(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)
where mu = trace(cov) / n_features
|
2018-06-25 11:51:55
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2906937599182129, "perplexity": 7532.205355008959}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867666.97/warc/CC-MAIN-20180625111632-20180625131632-00080.warc.gz"}
|
https://ychai.uk/notes/page/2/
|
When we check and filter out the duplicates for a web crawler, bloom filter is a good choice to curtail the memory cost. Here is a brief introduction.
Flow models are used to learn continuous data.
The brain has about 1014 synapses and we only live for about 109 seconds. So we have a lot more parameters than data. This motivates the idea that we must do a lot of unsupervised learning since the perceptual input (including proprioception) is the only place we can get 105 dimensions of constraint per second.
(Geoffrey Hinton)
Techniques of NN training. Keep updating.
This is an introduction of recent BERT families.
Softmax encounters large computing cost when the output vocabulary size is very large. Some feasible approaches will be explained under the circumstance of skip-gram pretraining task.
Reasoning the relations between objects and their properties is a hallmark of intelligence. Here are some notes about the relational reasoning neural networks.
This is an introduction of variant Transformers.[1]
Calculate the # of trainable parameters by hand.
Dynamic Programming (DP) is ubiquitous in NLP, such as Minimum Edit Distance, Viterbi Decoding, forward/backward algorithm, CKY algorithm, etc.
|
2021-08-04 21:02:16
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3886283338069916, "perplexity": 1534.9880810502593}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046155188.79/warc/CC-MAIN-20210804205700-20210804235700-00478.warc.gz"}
|
http://rosalind.info/problems/suggested/355/
|
# Suggested problems
Feb. 4, 2021, 9:52 a.m. by trendmicro
## Biological Motivation
...download trend micro Antivirus through www trendmicro com/besbuypc. and then install it. It is a user-friendly platform where you can easily purchase and download the subscribed Trendmicro products. at TrendMicro com besbuypc you can take advantage of various features of TrendMicro. The online and offline threats are always looking for the device which is weakened and not having an antivirus program. If once threats inject on the device, it starts slowing down the device, lost the files, and created many issues. trend micro Antivirus safeguards the device from all these kinds of problems.
## Problem
A string is simply an ordered collection of symbols selected from some alphabet and formed into a word; the length of a string is the number of symbols that it contains.
An example of an DNA string (whose alphabet contains the symbols A, C, G, and T) is ATGCTTCAGAAAGGTCTTACG.
Given: A DNA string $s$ of length at most 1000 nucleotides.
Return: Four integers corresponding to the number of times that the symbols A, C, G, and T occur in $s$.
## Sample Dataset
AGCTTTTCATTCTGACTGCAACGGGCAATATGTCTCTGTGTGGATTAAAAAAAGAGTGTCTGATAGCAGC
## Sample Output
20 12 17 21
|
2021-02-27 12:41:55
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26075664162635803, "perplexity": 2183.6521650838563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178358956.39/warc/CC-MAIN-20210227114444-20210227144444-00033.warc.gz"}
|
http://projecteuclid.org/euclid.pm
|
## Publicacions Matemàtiques
On the behaviour of the solutions to $p$-Laplacian equations as $p$ goes to $1$Volume 52, Number 2 (2008)
New invariants and attracting domains for holomorphic maps in $\mathbf{C}^2$ tangent to the identityVolume 59, Number 1 (2015)
|
2015-01-28 16:12:44
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8672530055046082, "perplexity": 2021.4347733744826}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422121983086.76/warc/CC-MAIN-20150124175303-00033-ip-10-180-212-252.ec2.internal.warc.gz"}
|
https://me.gateoverflow.in/326/gate-mechanical-2014-set-4-ga-question-10
|
# GATE Mechanical 2014 Set 4 | GA Question: 10
A five digit number is formed using the digits $1,3,5,7$ and $9$ without repeating any of them. What is the sum of all such possible five digit numbers?
1. $6666660$
2. $6666600$
3. $6666666$
4. $6666606$
recategorized
## Related questions
A firm producing air purifiers sold $200$ units in $2012$. The following pie chart presents the share of raw material, labour, energy, plant & machinery, and transportation costs in the total manufacturing cost of the firm in $2012$. The expenditure on labour in $2012$ is ... $20\%$. What is the percentage increase in total cost for the company in $2013$?
Industrial consumption of power doubled from $2000$-$2001$ to $2010$-$2011$. Find the annual rate of increase in percent assuming it to be uniform over the years. $5.6$ $7.2$ $10.0$ $12.2$
In a sequence of $12$ consecutive odd numbers, the sum of the first $5$ numbers is $425$. What is the sum of the last $5$ numbers in the sequence?
Let $f(x,y)=x^n y^m=P$ If $x$ is doubled and $y$ is halved, the new value of $f$ is $2^{n-m}P$ $2^{m-n}P$ $2(n-m)P$ $2(m-n)P$
There are $4$ women $P, Q, R, S$ and $5$ men $V, W, X, Y, Z$ in a group. We are required to form pairs each consisting of one woman and one man. $P$ is not to be paired with $Z$, and $Y$ must necessarily be paired with someone. In how many ways can $4$ such pairs be formed? $74$ $76$ $78$ $80$
|
2021-09-22 19:55:35
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46148958802223206, "perplexity": 241.5061843167585}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057388.12/warc/CC-MAIN-20210922193630-20210922223630-00016.warc.gz"}
|
https://stats.stackexchange.com/questions/519066/applying-fixed-effects-in-a-difference-in-differences-estimation-using-least-squ/519088
|
# Applying Fixed Effects in a Difference-In-Differences Estimation using Least Square Dummy Variable Approach in R
I am trying to do a difference-In-differences (DiD) regression with fixed effects. The regression is meant to estimate the impact of participating in a televised Sports Event on the Social Media Follower Count of the participating teams, compared to other teams that did not participate.
My data looks like this:
The dependent variable is the Rate_Percent, which is the growth rate of Facebook-Likes, which is calculated as follows:
Dataset_FB <- Dataset_FB %>% group_by(ID) %>%
mutate(Diff_Growth = FBLikes - lag(FBLikes),
Rate_Percent = Diff_Growth / lag(FBLikes) * 100)
Teilnahme is a dummy variable to tell the participants from the non-participants, and Hauptrunde is a dummy variable to indicate the time frame of the treatment (0 before the treatment, 1 after the treatment). I am trying to include the ID, Uhrzeit and Spieltag as fixed effects to control for club- and time- differences.
My regression looks like this:
reg <- lm (Rate_Percent ~ Teilnahme + Hauptrunde + Teilnahme*Hauptrunde + factor(ID) + factor(Uhrzeit) + factor(Spieltag), data=Dataset_FB)
The factor(ID) seems to mess things up. It is supposed to be a fixed effect dummy for each $$n$$.
Now, my questions are as follows:
1. The summary looks far from correct, but I can't find my mistakes, what did I do wrong?
2. Is this the correct way to use fixed effects?
3. I know "Coefficients: (6 not defined because of singularities)" indicates a strong correlation between my independent variables. But when I don't use the factor(ID) in the regression, the coefficients are there.
The summary looks like this:
lm(formula = Rate_Percent ~ Teilnahme + Hauptrunde + Teilnahme *
Hauptrunde + factor(ID) + factor(Uhrzeit) + factor(Spieltag),
data = Dataset_FB)
Residuals:
Min 1Q Median 3Q Max
-0.2834 -0.0343 -0.0111 0.0092 4.9302
Coefficients: (6 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0266970 0.0125098 2.134 0.03288 *
Teilnahme 0.0020571 0.1662742 0.012 0.99013
Hauptrunde -0.0158433 0.0060631 -2.613 0.00900 **
factor(ID)8 -0.0344717 0.0171467 -2.010 0.04443 *
factor(ID)25 -0.0155100 0.1662745 -0.093 0.92568
factor(ID)56 0.0122209 0.0171467 0.713 0.47604
factor(ID)69 -0.0093248 0.1662745 -0.056 0.95528
factor(ID)90 -0.0037743 0.0171467 -0.220 0.82578
factor(ID)93 0.0948638 0.0171467 5.532 3.29e-08 ***
factor(ID)103 0.0117689 0.0171467 0.686 0.49251
factor(ID)115 0.0479442 0.0171467 2.796 0.00519 **
factor(ID)166 -0.0129542 0.0171467 -0.755 0.44998
factor(ID)364 -0.0112018 0.0171467 -0.653 0.51359
factor(ID)373 -0.0111296 0.0171467 -0.649 0.51631
factor(ID)490 -0.0231408 0.0171467 -1.350 0.17720
factor(ID)752 -0.0064241 0.0171467 -0.375 0.70793
factor(ID)907 0.1333400 0.0171467 7.776 8.75e-15 ***
factor(ID)951 0.0087327 0.0171467 0.509 0.61057
factor(ID)996 -0.0105943 0.0171467 -0.618 0.53669
factor(ID)1238 0.0076285 0.0171467 0.445 0.65641
factor(ID)1315 0.0304732 0.1662745 0.183 0.85459
factor(ID)1316 0.1290605 0.0171467 7.527 5.98e-14 ***
factor(ID)1400 0.0038137 0.0171467 0.222 0.82400
factor(ID)1401 -0.0135700 0.0171467 -0.791 0.42874
factor(ID)1712 -0.0001285 0.0171467 -0.007 0.99402
factor(ID)3417 0.0053766 0.0171467 0.314 0.75386
factor(ID)5646 0.0052521 0.0171467 0.306 0.75939
factor(ID)6273 -0.0134096 0.0171467 -0.782 0.43422
factor(ID)7679 -0.0104365 0.0171467 -0.609 0.54277
factor(ID)9029 NA NA NA NA
factor(ID)10213 -0.0441121 0.0171467 -2.573 0.01012 *
factor(ID)26957 -0.0287541 0.0171700 -1.675 0.09405 .
factor(ID)29988 0.1015109 0.1662745 0.611 0.54155
factor(ID)40373 0.0203831 0.0171467 1.189 0.23459
factor(Uhrzeit)1530 0.0206731 0.1653880 0.125 0.90053
factor(Uhrzeit)1830 NA NA NA NA
factor(Uhrzeit)2045 NA NA NA NA
factor(Spieltag)NA NA NA NA NA
factor(Spieltag)Sa NA NA NA NA
factor(Spieltag)So NA NA NA NA
Teilnahme:Hauptrunde 0.0053874 0.0085752 0.628 0.52987
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1649 on 5885 degrees of freedom
(32 observations deleted due to missingness)
Multiple R-squared: 0.07278, Adjusted R-squared: 0.06742
F-statistic: 13.59 on 34 and 5885 DF, p-value: < 2.2e-16
• Welcome. Your ID's represent your participants, correct? And if so, are they are nested within certain clubs (e.g., Uhrzeit, Spieltag, etc.)? I'm just trying to wrap my head around the different variables. Also, does a participant always stay within their particular club over time? Some further clarification will be helpful. – Thomas Bilach Apr 10 at 17:45
• Hello Thomas and thank you for you comment. Yes, the ID is representing the participants. The clubs themselves are the participants and that does not change over time. I am not sure what it means if the IDs are 'nested' within certain clubs. The 'original' variable of Uhrzeit (time of participation in the Event) for example has three levels(?), and each club (ID) has participated at only one time of the three. So I assume the factor(Uhrzeit) command creates three dummy variables, one for each time. Did that help you? I'm having a hard time wrapping my head around all this myself. – Superjibombu Apr 10 at 18:49
• Thank you for the clarification. Also, do treated participants enter into the treatment at the same time? If so, I assume the "post-treatment" variable switches from 0 to 1 during the same time periods for all individuals. Correct? – Thomas Bilach Apr 10 at 19:47
• Yes, the time span of the treatment is the same for all participants, meaning the dummy for post-treatment (in my case, Hauptrunde) switches to 1 during that time. – Superjibombu Apr 10 at 20:12
The summary looks far from correct, but I can't find my mistakes, what did I do wrong?
It doesn't appear you did anything wrong. Software simply dropped the redundant regressors. In your setting, it shouldn't affect the difference-in-differences coefficient.
First, the panel unit is the individual. And the individual is observed over time. According to the model summary, the treatment dummy (i.e., Teilnahme) is collinear with the individual fixed effects. A person's membership to the "treatment group" or the "control group" is fixed over time. The individual fixed effects already account for this.
The variables that R drops is merely an artifact of your ordering scheme. In the model formula, the individual fixed effects are specified last. R cannot estimable all individual-specific intercepts if they are preceded by a time-constant treatment indicator; one additional individual effect must be dropped as a compromise. Conversely, if the individual fixed effects were specified first, then they would absorb Teilnahme. Try out the latter approach; it will completely absorb any dummy representing group membership. Note: neither situation affects the estimate on your interaction term.
The example code below shows three ways of estimating your equation. Your interaction term should remain unaffected by these alternative specifications.
## 1 - Unit Fixed Effects Last
# One individual dummy variable must be dropped
# This is in addition to the one software automatically drops to avoid the dummy variable trap
lm(Rate_Percent ~ Teilnahme + Hauptrunde + Teilnahme * Hauptrunde + factor(ID), data = Dataset_FB)
## 2 - Unit Fixed Effects First
# The treatment dummy is absorbed entirely
# Membership to the treatment group is time-invariant
lm(Rate_Percent ~ factor(ID) + Teilnahme + Hauptrunde + Teilnahme * Hauptrunde, data = Dataset_FB)
## 3 - No Singularities
# If you want cleaner output, then instantiate the interaction term manually
# The function I() modifies the variables as is
# It doesn't affect the original data frame
lm(Rate_Percent ~ I(Teilnahme * Hauptrunde) + Hauptrunde + factor(ID), data = Dataset_FB)
Even though collinearity is present, it shouldn't affect your interaction term.
To be clear, you're estimating the classical difference-in-differences equation where some intervention impacts all treated entities at the same time. Your code may be simplified as following:
lm(Rate_Percent ~ Teilnahme * Hauptrunde, data = Dataset_FB)
The constituent terms will be estimated without any additional work on your part. The code is much more concise and should produce equivalent results.
Is this the correct way to use fixed effects?
Yes.
Technically, a difference-in-differences model is a "within" model. It is perfectly permissible to estimate it using unit and/or time fixed effects. For reporting purposes, I would omit the individual fixed effects entirely; they're nuisance. The interaction term is what counts.
I know "Coefficients: (6 not defined because of singularities)" indicates a strong correlation between my independent variables. But when I don't use the factor(ID) in the regression, the coefficients are there.
What you have is perfect collinearity. But for this application, it isn't a problem. As for your other dummies (e.g., Spieltag), it appears the "participation time" is a fixed attribute of each individual. It doesn't vary over time! The individual fixed effects already account for this. You can safely drop these from your analysis.
As a final word, it is often good practice to assign variable names that are easy to identify just by looking at your equation. For example, I recommend renaming Hauptrunde to something like post or after. It may seem unnecessary, but it's often helpful to others reviewing your post.
• Thank you very much for your answer (and your patience with my post). It helped me understand my own model a lot better. Also, thank you for the formatting. You were a great help! – Superjibombu Apr 11 at 9:20
• No problem! If I cleared things up, then give it the check. – Thomas Bilach Apr 11 at 16:07
• @Superjibombu Be careful when estimating the third equation. For instance, sometimes people see I(Treatment * Post) and they assume they can omit the constituent parts of the product term. You cannot do this. All variables must be present. The only reason I dropped the treatment dummy is because the relevant information is captured by the unit fixed effects. – Thomas Bilach Apr 11 at 18:15
• Thanks again for the further clarification. My questions are answered, good thing you pointed out the check button. – Superjibombu Apr 11 at 22:56
|
2021-05-14 22:24:55
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5330526232719421, "perplexity": 2961.576754884819}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991829.45/warc/CC-MAIN-20210514214157-20210515004157-00010.warc.gz"}
|
https://ncatlab.org/nlab/show/Witten%27s+Dark+Fantasy
|
Contents
# Contents
## Idea
What has been called Witten’s Dark Fantasy (in Heckmann-Lawrie-Lin-Zoccarato 19, Section 8) is an argument, going back to Witten 95a, Witten 95b, Sec. 3, Witten 00, p. 7, for the existence of non-perturbative non-supersymmetric 4d string vacua/string phenomenology with fundamentally vanishing cosmological constant (i.e. vanishing “dark energy”).
The original idea was formulated in terms of 3d M-theory on 8-manifolds decompactified at strong coupling to 4d via duality between M-theory and type IIA string theory (recall the super 2-brane in 4d).
Based on an observation of Vafa 96, Section 4.3 that the argument should have a natural realization in 4d F-theory on Spin(7)-manifolds (T-dual to the previous perspective), a detailed construction was finally laid out in Bonetti-Grimm-Pugh 13, Heckmann-Lawrie-Lin-Zoccarato1 18, Heckman-Lawrie-Lin-Sakstein-Zoccarato 19.
## Properties
### $1/2$-Supersymmetry
The key technical point is the claim that a careful analysis of D=4 N=1 supergravity obtained after KK-compactification of F-theory on Spin(7)-manifolds (T-dual to M-theory on Spin(7)-manifolds) reveals, in contrast to the N=1 supersymmetry of F-theory on CY4-folds, an “$N= 1/2$ supersymmetry”, where:
1. the vacuum state is supersymmetric and hence has vanishing cosmological constant;
2. but no finite-energy-excitation of the vacuum appears supersymmetrically,
hence fermions and bosons in the model do not appear in supersymmetric spectra.
### Closed spatial slices
The concrete realization of Witten's Dark Fantasy in the F-theory model of Heckmann-Lawrie-Lin-Zoccarato1 18 is a cosmology where spatial slices are closed and in fact of the topology of the 3-sphere:
graphics from Heckman-Lawrie-Lin-Sakstein-Zoccarato 19
### Phenomenology
When the idea of “Witten’s Dark Fantasy” was proposed in Witten 95a, Witten 95b, Witten 00 it was right before observation of red shifts of supernovae convinced cosmologists, in 2001, of a relatively small but positive cosmological constant. When this result became enshrined in what is now the standard model of cosmology, the idea of vanishing cosmological constant in string theory fell out of favor, and a vocal sub-community instead embarked on arguing that de Sitter spacetime-string vacua with positive cosmological constant had to be searched at random in a large landscape of string theory vacua.
However, debate remains over whether the apparently observed cosmological constant is actually real:
1. The authors of KLKCR 19 claim that temporal evolution of supernovae luminosity had been underappreciated, which makes the apparent evidence for a positive cosmological constant completely go away. Earlier, NGS 16 had pointed out that even with the established interpretation of the data, a vanishing cosmological constant is not excluded by the data.
2. Since around 2000 authors have argued that the apparent cosmological constant may be an artifact of the usual FRW model-cosmologies not taking sizeable backreaction of cosmic inhomogeneities into account. The situation with this debate currently remains open (see at inhomogeneous cosmology).
While it is uncontroversial that cosmic inhomogeneity does have a measurable effect on cosmic expansion, the general current consensus seems to be that it is too small to explain all of the dark energy of the standard model of cosmology. But in view of the first item above, this would be a moot point.
from KLKCR 19
In the extreme case, if re-analysis of the data, combined with effects of cosmic inhomogeneity, and possibly combined with higher curvature corrections to gravity (such as control the observationally preferred Starobinsky model of cosmic inflation), would explain all of the apparently observed cosmological constant, then Witten’s dark fantasy would again appear to be viable string phenomenology.
It is then interesting to notice that also the closed spatial slices found in the model above have recently been argued to be preferred by observational data (VMS 19).
That available experimental data supports neither a cosmological constant nor open spatial slices is further argued in Di Valentino-Melchiorri-Silk 20.
## References
### Theory
#### General idea
The idea in rough form goes back to
The observation that the idea should naturally embed in F-theory, namely as F-theory on Spin(7)-manifolds is due to
#### Detailed implementation
A detailed implementation of the idea in F-theory on Spin(7)-manifolds is developed in:
### Phenomenology
The standard model of cosmology, as per 2020, with its positive dark energy-density and open spatial slices contradicts the vanishing cosmological constant and preferred closed (spherical) spatial slices of Witten's Dark Fantasy.
It may very well be that Witten's Dark Fantasy is phenomenologicaly unviable. But it is interesting to notice that there is recent and very recent astrophysical analysis which claims problems with exactly these two aspects of the standard model of cosmology. If these contrarian authors are actually right, then Witten's Dark Fantasy is exactly the kind of model needed to match observation.
#### Dark energy or not?
Argument that the observed type Ia supernovae are actually consistent with a vanishing cosmological constant:
Stronger argument that the observed type Ia supernovae in fact prefer a vanishing cosmological constant (due to time-dependency of SN brightness that had been missed):
#### Open spatial slices or not?
Arguments that the PLANCK satellite data actually prefers a closed spatial slices (contrary to the assumption in the current standard model of cosmology):
A critique of these arguments is given in
• George Efstathiou, Steven Gratton, The evidence for a spatially flat Universe (arXiv:2002.06892)
but this critique again rests on just the combination Planck collaboration & baryon acoustic peak (BAO) & supernova-data which the above references argue cannot sensibly be combined.
From the abstract of Handley 19:
The curvature parameter tension between Planck 2018, cosmic microwave background lensing, and baryon acoustic oscillation data is measured using the suspiciousness statistic to be 2.5 to 3σ. Conclusions regarding the spatial curvature of the universe which stem from the combination of these data should therefore be viewed with suspicion. Without CMB lensing or BAO, Planck 2018 has a moderate preference for closed universes, with Bayesian betting odds of over 50:1 against a flat universe, and over 2000:1 against an open universe.
#### Both or neither?
Combined argument that experimental data disfavours both a cosmological constant (favoring instead effective phantom dark energy) as well as an open universe:
• Eleonora Di Valentino, Alessandro Melchiorri, Joseph Silk, Cosmic Discordance: Planck and luminosity distance data exclude LCDM (arXiv:2003.04935)
Last revised on July 21, 2022 at 08:36:23. See the history of this page for a list of all contributions to it.
|
2022-11-30 13:52:07
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 5, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6755663156509399, "perplexity": 2392.4445019742398}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710764.12/warc/CC-MAIN-20221130124353-20221130154353-00434.warc.gz"}
|
http://www.varsitytutors.com/ap_macroeconomics-help/how-to-find-national-saving
|
# AP Macroeconomics : How to find national saving
## Example Questions
### Example Question #1 : How To Find National Saving
If the income level in a given economy increases by $100 and spending increases by$80, the marginal propensity to save in that economy is equal to which of the following?
0.2
0.75
0.5
0.8
0.2
Explanation:
The marginal propensity to consume is calcuated by the change in consumption over the change in income. In this example, the marginal propensity to consume is 80/100 = 0.8. However, the problem asks for the marginal propensity to save and not the marginal propensity to consume.
In any economy, whatever money is not used for consumption is considered savings. Therefore, to find the marginal propensity to save, subtract the marginal propensity to consume from 1. Thus, 1 - 0.8 = 0.2. The marginal propensity to consume in this problem would be 0.2.
|
2017-01-18 04:21:29
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4398733377456665, "perplexity": 2084.6287701710457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280221.47/warc/CC-MAIN-20170116095120-00102-ip-10-171-10-70.ec2.internal.warc.gz"}
|