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# Does anyone actually use the term ensemble interpretation ?

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Gold Member

## Main Question or Discussion Point

Does anyone actually use the term "ensemble interpretation"?

People have been asking questions about the "ensemble interpretation" here lately. It's mentioned on the Wikipedia page "interpretations of quantum mechanics" and has its own page. But is that term actually used in books and articles, or was it invented by the guy who wrote those Wikipedia entries? I don't think Ballentine gives it a name. Isham just describes this position as "anti-realist".

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Matterwave
Gold Member

I once heard a talk by a Professor who used this term, so I think it is used by more than just the guy who made the wiki article.

Demystifier

I think even Balentine does not use this term in his textbook.

What many people call the "ensemble interpretation" I always thought was a part of good old Copenhagen. I had never heard a teacher or read a book where they used "ensemble interpretation" before. I had heard of the Born statistical interpretation of the wavefunction, but thought that was just a part of the normal Copenhagen interpretation.

Staff Emeritus
Gold Member

That's one of the reasons why I'm asking. I think the "ensemble interpretation" is what the Copenhagen interpretation was until it was redefined by people who had misunderstood it.

Demystifier

That's one of the reasons why I'm asking. I think the "ensemble interpretation" is what the Copenhagen interpretation was until it was redefined by people who had misunderstood it.
The Copenhagen interpretation carries its name due to Bohr, right?
And I think that the ensemble interpretation is NOT (or at least not completely) the interpretation advocated by Bohr.

Having this in mind, perhaps it would be better to say that ensemble interpretation is the orthodox interpretation.

Staff Emeritus
Gold Member

The Copenhagen interpretation carries its name due to Bohr, right?
Bohr and Heisenberg apparently, but I don't think the two of them really agreed about these things.

And I think that the ensemble interpretation is NOT (or at least not completely) the interpretation advocated by Bohr.
Maybe not, but I find it hard to find anything in the ensemble interpretation that I think Bohr would disagree with.

Regarding what Bohr actually advocated, I came across this article by Asher Peres, which I think you will find interesting too considering that you started a thread about Copenhagen a few months ago. (I hope Dmitry67 will read it too). The interestesting stuff starts on page 6:

There seems to be at least as many different Copenhagen
interpretations as people who use that term, probably there are more. For example, in
two classic articles on the foundations of quantum mechanics, Ballentine (1970) and Stapp
(1972) give diametrically opposite definitions of “Copenhagen.”
...
I shall now explain my own Copenhagen interpretation.
It relies on articles written by Niels Bohr. Whether or not you agree with Bohr, he is the
definitive authority for deciding what is genuine Copenhagen.
Peres's description of the CI sounds very much like how I've been describing the ensemble interpretation. There is however one thing that looks pretty odd today. I'll let Peres say it:
It is remarkable that Bohr never considered the measuring process as a dynamical
interaction between an apparatus and the system under observation. Measurement had
to be understood as a primitive notion.
By the way, the article by Ballentine that Peres mentioned is called "The statistical interpretation of quantum mechanics", so I guess that's what Ballentine calls it.

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I posted this in another thread.

http://www.dipankarhome.com/ENSEMBLE%20INTERPRETATIONS.pdf [Broken]
This paper reviews various meanings of probability and ensemble interpretations proposed since Einstein and up to Ballentine.
In the book Compendium of Quantum Physics, Ballentine himself writes an entry on ensemble in quantum mechanics, and he cites this paper as the secondary source, if I remember correctly. So the paper shouldn't be too outdated.

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Fredrik, regarding the Peres paper you linked, here is what Jan Faye has to say about Popper and Bohr in the accepted philosophical reference on CI (http://plato.stanford.edu/entries/qm-copenhagen/): [Broken]
Bohr was definitely neither a subjectivist nor a positivist philosopher[*], as Karl Popper (1967) and Mario Bunge (1967) have claimed. He explicitly rejected the idea that the experimental outcome is due to the observer.
Faye works at Copenhagen University and is probably the leading Bohr scholar. He's published on the history of Bohr's philosophical development etc, so he's also a good source if you're looking for more.

Popper was one of the leading opponents of CI who came to define the version of it we have today. Strange how we let its opponents define it. It's no wonder that it's gone from being orthodox to somewhat taboo.

Bohr collected much of his philosophy on QM in Atomic Theory and the Description of Nature, which is very readable. Unfortunately, Bohr's own views shifted slightly through time (although less than would be implied by the changes in his terminology). So you have to keep in mind the dates of each anthologized paper. I think his Essays on Atomic Physics and Human Knowledge are also worth reading. They contain some of his later thoughts, but are spread over a longer period and are not as clear about his overall interpretation.

Also, Bohr and Heisenberg definitely disagreed. Heisenberg's Physics and Philosophy started to steal some of the thunder from Bohr. It's more dense, describes things differently, and starts off in a more observer-centric direction. I like Bohr's explanations much more than Heisenberg's, and honestly I didn't read Heisenberg as closely.

*Being a "positivist philosopher" and having positivist philosophies are different things .

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Since you mentioned Bohr and got me reading back through my thesis, here are some more thoughts . Regarding Bohr and subjectivity (Bohr quote from ATDN):
To the confusion of this last point, Bohr did refer to the subjectivity of quantum observers. His subjectivity, however, applies no more to humans than to any type of physical system. Subjectivity, for Bohr, is not a property restricted to minds. His views on this topic are demonstrated by his discussion of relativity. He writes, “The theory of relativity reminds us of the subjective … character of all physical phenomena, a character which depends essentially upon the state of motion of the observer.”
Bohr speaks about relatively a lot and derives much of his philosophy from it without even needing quantum weirdness. Jan Faye talks about this in the SEP article too.

The defining characteristic of Bohr philosophy is not, I find, epistemological subjectivity or observer-dependence, but is illustrated in the following quote from ATDN:
[Quantum mechanics] may be regarded as a natural generalization of the classical mechanics with which in beauty and self-consistency it may well be compared. This goal has not been attained, still, without a renunciation of the causal space-time mode of description that characterizes the classical physical theories which have experienced such a profound clarification through the theory of relativity.
Bohr does not reject causality outright or give measurement or observers any special status they didn't already have. What he does do is deny that we can comprehend or model what an electron does between physical interactions. For Bohr, classical objects and particles exist, and that's all there is to it. But we cannot describe or visualize what they are doing when they aren't in the process of interacting with something (being measured). Discussions of phenomena are only intelligible when specifying the conditions of measurement, so it's simply meaningless to ask about what an electron is doing when it isn't being measured - it's like asking whether apples are red when no one is looking at them.

The sensation of redness requires both an apple and proper measurement conditions that will allow for the redness to manifest itself. Redness requires an apple, proper lighting, and a person who is looking at the apple. An apple is neither red nor not red when a person isn't looking or there isn't proper lighting. At the same time, red apples are real entities. Likewise, electrons with spin are real and basic entities - just don't ask about them without specifying the experimental conditions that will determine the properties they will manifest.

Phenomena and visualizability are also very technical terms in Bohr's writing.

As a side note, one other interesting feature of Bohr's interpretation is that it restores many of our naive/natural ontological beliefs. Redness, a property that we naively consider to be real, was shunned from the realm of intrinsic objective properties by classical physics and demoted to being an extrinsic or even subjective property. If you look at how Bohr treats atomic properties, however, they undergo this same sort of purging from the intrinsic and objective. Bohr denies there is such a thing as an intrinsic, persistent, objective property. Tables and chairs, colors and sounds, meaning, mass, and spin, are now all just as real, as our naive intuitions would have us believe (if not, admittedly, as real as we naively might want them to be).

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I was always wondering, how Bohr draw a line between quantum and classical. He did it once when he developed the first naive model of hydrogen, but I am asking - did he say anything about it later, when QM was fully developed theory.

I was always wondering, how Bohr draw a line between quantum and classical. He did it once when he developed the first naive model of hydrogen, but I am asking - did he say anything about it later, when QM was fully developed theory.
Sure - classical is the old, obsolete, approximate model, and quantum is the correct version of things .

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Sure - classical is the old obsolete approximate model, and quantum is the correct version of things .
If what you are saying is true, then Bohr was not a proponent of Copenhagen Int. like Newton was not a proponent of Newtonian mechanics (I've heard he believed in particle-wave duality for light)

If what you are saying is true, then Bohr was not a proponent of Copenhagen Int. like Newton was not a proponent of Newtonian mechanics (I've heard he believed in particle-wave duality for light)
I'd say that's a fair statement. I don't know if I've ever heard a version of the CI that really matched Bohr's philosophy.

There was no demarcation between classical and quantum (or macro and micro) behavior for Bohr. Electrons and turtles both can only manifest their properties in a context-dependent way, depending on the conditions of their interactions with other systems. There's no need to differentiate, since Bohr's electrons were essentially classical except in how they behave causally and how we can talk about them between interactions.

I think it would probably be fair to say that all interactions were classical for Bohr. It's that time between interactions, that we can't talk about, that screws things up. But when two electrons collide, they do so classically, exchanging momentum and changing directions. You just can't infer anything about the next causal step they will take. It doesn't even make sense to say they exist when they aren't interacting with something. You'll never hear Bohr put it that way though, because to do so would be to talk about an electron as a thing that can exist or not independent of the context of its interactions - something he denies we can do.

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electrons with spin are real and basic entities
"Spin" is real? As far as I know, if "spin" exists (Of course it does in QM) "spinor" must be
used because "spin" is a very strange thing which I can't understand as a real particle.

"Spin" is real? As far as I know, if "spin" exists (Of course it does in QM) "spinor" must be
used because "spin" is a very strange thing which I can't understand as a real particle.
Easy. Spin is the property of electrons that causes them to go one direction or another in a magnetic field. There isn't really anything else to understand. There is no quantum version of spin that you have to understand for Bohr, since quantum causality is inherently "unvisualizable." From http://plato.stanford.edu/entries/qm-copenhagen/: [Broken]
Moreover, there is no further evidence in Bohr's writings indicating that Bohr would attribute intrinsic and measurement-independent state properties to atomic objects (though quite unintelligible and inaccessible to us) in addition to the classical ones being manifested in measurement (Faye 1991).
It's certainly arguable whether or not something without intrinsic properties can be considered "real," but for Bohr there was simply no other option. This gets back to the re-equalization of the ontological statuses of different types of properties though. You can reject Bohr's terminology, but within his system it can't be denied that spin is just as real as mass, temperature, color, or any other property, and vice versa.

I'm not familiar enough with the ensemble interpretation anymore to say how all of this compares, but feel free to enlighten me .

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Staff Emeritus
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It's certainly arguable whether or not something without intrinsic properties can be considered "real," but for Bohr there was simply no other option. This gets back to the re-equalization of the ontological statuses of different types of properties though. You can reject Bohr's terminology, but within his system it can't be denied that spin is just as real as mass, temperature, color, or any other property, and vice versa.

I'm not familiar enough with the ensemble interpretation anymore to say how all of this compares, but feel free to enlighten me .
I think the idea is that a system can be said to have a certain property (say a momentum in a specified range) if and only if QM assigns probability 1 to the possibility that a measurement will give us a result that's consistent with that. I think this goes back at least to EPR, who (if I remember correctly) called such a property an "element of physical reality".

Personally, I don't think such a definition adds anything of value to the theory.

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I think the idea is that a system can be said to have a certain property (say a momentum in a specified range) if and only if QM assigns probability 1 to the possibility that a measurement will give us a result that's consistent with that. I think this goes back at least to EPR, who (if I remember correctly) called such a property an "element of physical reality".

Personally, I don't think such a definition adds anything of value to the theory.
EPR:
If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity.
The problem with this criterion for Bohr is that systems only manifest real properties during their "disturbance" or interaction. Bohr requires disturbance for the existence of properties, while EPR say something is only real if we know what it is without any interaction or disturbance. If I remember correctly, and I'm not as familiar with this, but I think a lot of what Einstein and Bohr argued about was whether or not we in principle could come up with a theory that would allow properties to meet the EPR criterion. Bohr said it was impossible and Einstein said QM was simply incomplete. Nothing in Bohr's version of things would ever meet the EPR criterion though. Actually, I don't think Bohm even gives you a property that can meet the EPR criterion.

Einstein wants a visualizable causal picture, and Bohr says we can't have one.

I'm not quite sure where I stand, since I want it all like Einstein but I don't think his vision is quite possible. Ah well. It feels a little arrogant to say that in principle we can never come up with something better, so the current statistical theory must be complete. The more QM is studied though, the more I think we might not be able to come up with a better completely context-independent model for things.

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In "Veiled Reality", Bernard d'Espagnat, 1995, he devotes a section titled "The (So-Called) Ensemble Interpretations".

For this he cites two alternative namings: "statistical" and "stochastic" theory.

Staff Emeritus
Gold Member

In "Veiled Reality", Bernard d'Espagnat, 1995, he devotes a section titled "The (So-Called) Ensemble Interpretations".
Peres uses the term "ensemble interpretations" too, in the article I linked to in #7. And so does this article:
I posted this in another thread.
http://www.dipankarhome.com/ENSEMBLE%20INTERPRETATIONS.pdf [Broken]
This one has a lot of interesting information actually. Section 1 starts with this:

Ensemble interpretations of quantum theory contend that the wave function describes an ensemble of identically prepared systems. They are thus in contrast to “orthodox” or “Copenhagen” interpretations, in which the wave function provides as complete a description as is possible of an individual system.
That seems like a reasonable way to define the difference between the two, but I have to say that I don't understand how this is a difference if the only way to reveal those properties is to perform a large number of measurements on an ensemble of identically prepared systems. I think this means that what really distinguishes ensemble interpretations from Copenhagen interpretations is a different interpretation of probability. The fact that the author spends 15 pages discussing interpretations of probability seems to support that idea.

My own opinion is that all this stuff about interpretations of probability is nonsense, or at least unscientific. Science leads directly to the relative frequency view of probability. I'll explain why. Every time we define a probability measure (which is a perfectly well-defined mathematical concept) and claim that it associates probabilities with things in the real world, we're leaving the domain of pure mathematics and entering the domain of science, or pseudo-science (if we fail to meet the requirements of a "theory", which I'm about to define). I define a theory as a set of statements that associates a probability with each possible result of each experiment in some set of experiments. (I also require that this set is finite, logically consistent, and doesn't contain any statements that can be removed without changing the assignment of probabilities). It's the only definition that makes any sense to me, considering the obvious fact that the only thing experiments can tell us is how accurate a theory's predictions are. How do we find out how accurate the predictions are? By performing repeated measurements on an ensemble.

I define science to be the process of finding new theories and performing experiments to test the accuracy of their predictions. So the relative frequency view of probability is built into the definition of science, or at least into my definition of science. So anyone who supports any other interpretation of probability either has a very different definition of science than me, or is ignorant of the fact that any attempt to apply mathematics to the real world is either science or pseudo-science.

At this point you may be thinking "What about probabilities assigned to single events?". If the method you use to assign such a probability can't be used to assign probabilities to other single events, the assignment is meaningless, as the method fails to define a theory. If it can assign probabilities to many other single events, then the theory can be (statistically) falsified by repeated experiments.

You may also be thinking "What about the existence of the N→∞ limit?" (The average result of a series of N measurements is often claimed to have some specific value in the limit N→∞). This question is irrelevant, since the axioms of a theory specify relationships between well-defined mathematical quantities and things in the real world that are defined operationally and can therefore never be perfectly well-defined. For example a "clock" is defined by a description in plain English of what that word means. Since the theory isn't well-defined anyway, it doesn't make sense to require that the N→∞ limit must be.

I'll quote a few more interesting passages from that article:

For both orthodox and ensemble cases, we have used the plural — “interpretations” — to emphasise that each class contains several variants, and, particularly on the ensemble side, it will be a major task to distinguish between them. The most important division will be between what we have previously called [1] PIV ensemble interpretations — ensemble interpretations with pre-assigned initial values for dynamical variables, and what Gibbins [2, p. 76] has called minimal ensemble interpretations, which carry no such superstructure.

They use the term "PIV" a lot, and it helps to know what they mean by it. Section 5.1 explains it in more detail:

...the PIV assumption. the idea that all observables have values prior to any measurement.

The premeasurement value is available to become the measuremental result (though, of course. PIV theorists are not obliged to require that either the initial or any repeated measurement actually gives the PIV).

In section 4.4, they're quoting Ballentine as saying some things about PIV's in the ensemble intepretation that I find really weird:

His stated position is, “the Statistical Interpretation. . . is completely open with respect to hidden variables. It does not demand them, but it makes the search for them entirely reasonable.”

OK, that one isn't weird by itself, because if (for example) the hidden variables aren't observables, there's no conflict with Bell's theorem. But these statements look like they would very much be in conflict with Bell:

For example, he states [3, p. 361], “a momentum eigenstate. . . represents the ensemble whose members are single electrons each having the same momentum, but distributed uniformly over all positions”. Also on p. 361 of ref. [3], he says, “the Statistical Interpretation considers a particle to always be at some position in space, each position being realised with relative frequency u/i(r)~2in an ensemble of similarly prepared experiments”. Later [3, p. 379] he states, “there is no conflict with quantum theory in thinking of a particle as having definite (but, in general, unknown) values of both position and momentum”.

I'm very surprised by this. Could it be that in 1970, when this was written, Ballentine still didn't understand Bell's theorem? (Bell's theorem was published in 1964).

And finally, after reading parts of this article, I have come to the following conclusions about the terminology used by the authors of the article and the people they mention. Stapp's definition of the Copenhagen interpretation is essentially equivalent to what Ballentine calls the statistical interpretation and what the authors call an ensemble interpretation. (Ballentine claims that Stapp has redefined the CI in an attempt to save it). Murdoch on the other hand, defines "the ensemble interpretation" to be an interpretation in which the state vector represents the "state of knowledge of an object". Murdoch defines the statistical interpretation essentially the same way as Ballentine.

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Thanks for the recap Fredrik. I agree with what you've said about probability. Regarding ensemble vs Bohr, I see some subtle differences. Bohr would disagree with PIV or hidden-variable-compatible ensemble interpretations. In order to make his radical claim that objects could both be real and not have any intrinsic objective properties, he required the completeness of his extrinsic-only model.* Completeness implies that there are no hidden, more basic, intrinsic variables to learn about.

I think that Bohr's view was probably similar to a "state of knowledge of an object" view, with one qualification. Bohr didn't allow talk of context-independent properties. He might say that the wave function represents the state of knowledge about what properties an object may manifest when measured in a certain way. He wouldn't say that an electron, a, has a 50% chance of currently being spin up. He might say that a has a 50% chance of being spin up when a measurement occurs. It didn't make sense for him to talk about observables independent of observation.

*Conceptually it was probably the reverse of this. A denial of intrinsicality and visualizability would have led to the thesis that QM is complete, and probably not vice versa as stated.

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dextercioby
Homework Helper

But these statements look like they would very much be in conflict with Bell:

For example, he states [3, p. 361], “a momentum eigenstate. . . represents the ensemble whose members are single electrons each having the same momentum, but distributed uniformly over all positions”. Also on p. 361 of ref. [3], he says, “the Statistical Interpretation considers a particle to always be at some position in space, each position being realised with relative frequency u/i(r)~2in an ensemble of similarly prepared experiments”. Later [3, p. 379] he states, “there is no conflict with quantum theory in thinking of a particle as having definite (but, in general, unknown) values of both position and momentum”.

I'm very surprised by this. Could it be that in 1970, when this was written, Ballentine still didn't understand Bell's theorem? (Bell's theorem was published in 1964).
Can you, please, be more specific ? What does Bell's theorem have to do with the quotes from Ballentine ? Was Ballentine addressing Bell's theorem in the quoted lines and I'm not getting it, or ?

I think that Bohr's view was probably similar to a "state of knowledge of an object" view, with one qualification. Bohr didn't allow talk of context-independent properties. He might say that the wave function represents the state of knowledge about what properties an object may manifest when measured in a certain way. He wouldn't say that an electron, a, has a 50% chance of currently being spin up. He might say that a has a 50% chance of being spin up when a measurement occurs. It didn't make sense for him to talk about observables independent of observation.
In section 5 of the Home and Whitaker paper they mention something similar:
Before this, though, we should clarify one point. A similar, though less extreme, position to PIVs is where observables do not necessarily “have” values prior to measurement, but the value that any observable will take, if it is measured, is fixed. This would, of course, usually be described as a hidden-variables theory, which one might wish to be local and/or deterministic.
The difference still is that for Bohr there is no preset property of the system, since there are no real properties independent of measurement. You still wouldn't get hidden variables. In this case Bohr is still closer to a subjective knowledge interpretation.

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Can you, please, be more specific ? What does Bell's theorem have to do with the quotes from Ballentine ? Was Ballentine addressing Bell's theorem in the quoted lines and I'm not getting it, or ?
If I haven't misunderstood something fundamental, what he said is exactly the sort of thing that's proved wrong by Bell inequality violations. He said that each member of the ensemble can have a well-defined position and a well-defined momentum at the same time, even though the position and momentum operators don't commute. If he's right about this, then the same thing should hold for any pair of operators that don't commute, in particular any two of the operators $S_x, S_y, S_z$ for a spin-1/2 particle. (I don't think it's really necessary to change the discussion to be about spin components instead of position and momentum, but it will be much easier for me to explain my point if we do).

If there are observables corresponding to two of these operators that have well-defined values at all times, then by rotational invariance, there must exist an observable with a well-defined value for each operator of the form $$\vec a\cdot\vec S$$, where $\vec a$ is a unit vector. But the assumption that this operator corresponds to an observable that has a well-defined value $a_n$ in ensemble member n, which will be the result of a measurement of $$\vec a\cdot\vec S$$ on ensemble member n, leads directly to a Bell inequality called the CHSH inequality, which is violated by QM. See pages 215, 216 in Isham's book for a derivation.

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Fredrik,
I wasn't sure I followed your very interesting argument about how science leads directly to the relative frequency view of probability.
My own opinion is that all this stuff about interpretations of probability is nonsense, or at least unscientific. Science leads directly to the relative frequency view of probability.
...
I define a theory as a set of statements that associates a probability with each possible result of each experiment in some set of experiments..... It's the only definition that makes any sense to me, considering the obvious fact that the only thing experiments can tell us is how accurate a theory's predictions are. How do we find out how accurate the predictions are? By performing repeated measurements on an ensemble.

I define science to be the process of finding new theories and performing experiments to test the accuracy of their predictions. So the relative frequency view of probability is built into the definition of science, or at least into my definition of science.
I understand the relative frequency view of probability to be the view that probabilities *just are* relative frequencies of actual events - at least, when one is arguing about what probability is, I think this is how the view is understood. I find the view attractive as it would makes probabilities wholly unproblematic, if correct. But I was unable to see how this view followed from your view about science and scientific theories.

Someone (call him P) who believed that probabilities were not just relative frequencies, but were something more fundamental or primitive, irreducible properties of objects or events or whatever (not defending this view note!) could, it seems to me, agree what you say. Yes, theories are just assignments of probabilities to possible results of experiments. Yes, we discover probabilities through repeated experiments and testing, just as you say. But this is just because, given the probabilities, certain actual relative frequencies are the *most likely*. The probabilities are thus inferred on the basis of the frequencies, but not identified with them. The greater the number of experiments done where the proportion holds, the more likely it is that the probabilities of events match that proportion. So we should change and update our beliefs about the probabilities in a rational and scientific way. There's nothing non-empirical going on here and nothing which goes against your definition of science, as far as I can see here.

I didn't understand your point about single events. Normally, single events are taken to be a challenge for the view that probabilities just are relative frequencies. The interpretational worry is that: Identifying probabilities with actual relative frequencies can seem too strong. A fair coin *can* keep landing heads. The more tosses we do, the less rational it is to believe it's fair. But it still may be that way. We'd have to be awfully unlucky - but that can happen. But if probabilities just are actual relative frequencies, then if the coin lands heads every time, we forced to say that its chances are 100 percent. That seems wrong. The relationship between frequency and probability does not appear to be one of entailment; the frequencies are evidence for probabilities,not logical proof. On a single event, the problem is very acute - the chance is either 1 or 0 - but if we believe QM, say, we may have very good reason for thinking that the probability is actually between these two values.

You seemed to say that, unless you could assign probabilities to other single events, you weren't being scientific. But well established theories do assign probabilities to events irrespective of how often they occur - imagine some very complex collection of quantum particles in some very peculiar and unusual arrangement - theoretically, the S' equation will tell us what the likelihood of P happening will be - but maybe the universe only throws up one of these things (yeah, it's big - but there's got to be some arrangement of quantum things it only occasionally throws up). QM assigns a clear probability - if we could solve the equation, this is the probability we *ought* to believe - but it's not the relative frequency.


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# Nonlocal mean curvature flow

The nonlocal mean curvature flow refers to an evolution equation for surfaces for which the normal velocity equals its nonlocal mean curvature.

This flow was fist constructed by Caffarelli and Souganidis [1].

## References

1. Caffarelli, Luis A.; Souganidis, P. E. (2010), "Convergence of nonlocal threshold dynamics approximations to front propagation", Archive for Rational Mechanics and Analysis (Berlin, New York: Springer-Verlag) 195 (1): 1–23, ISSN 0003-9527

format: plain


text:
# Upper Closure of Subset is Subset of Upper Closure

## Theorem

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $X, Y$ be subsets of $S$.

Then

$X \subseteq Y \implies X^\succeq \subseteq Y^\succeq$

where $X^\succeq$ denotes the upper closure of $X$.

## Proof

Let $X \subseteq Y$.

Let $x \in X^\succeq$.

By definition of upper closure of subset:

$\exists y \in X: y \preceq x$

By definition of subset:

$y \in Y$

Thus by definition of upper closure of subset:

$x \in Y^\succeq$

$\blacksquare$


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## Lines and Angles Class 9 MCQs Questions with Answers

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In ΔABC, ∠A = 50° and the external bisectors of ∠B and ∠C meet at O as shown in figure. The measure of ∠BOC is

(a) 40°
(b) 65°
(c) 115°
(d) 140°

Question 2.
In figure the value of x is

(a) 120°
(b) 130°
(c) 110°
(d) 100°

Question 3.
An exterior angle of a triangle is 80° and the interior opposite angles are in the ratio 1 : 3, measure of interior opposite angles are
(a) 30°, 90°
(b) 40°, 120°
(c) 20°, 60°
(d) 30°, 60°

Question 4.
In ΔABC, the bisectors of ∠ABC and ∠BCA intersect each other at O. The measure of ∠BOC is
(a) 90° + ∠A
(b) 90° + $$\frac{∠A}{2}$$
(c) 180 – ∠A
(d) 90° – $$\frac{∠A}{2}$$

Answer: (b) 90° + $$\frac{∠A}{2}$$

Question 5.
In figure if ∠A + ∠B + ∠C + ∠D + ∠E + ∠F = k right angles, then find value of k.

(а) 2
(b) 3
(c) 4
(d) 5

Question 6.
In the given figure, the measure of ∠ABC is

(a) 80°
(b) 20°
(c) 100°
(d) 60°

Question 7.
The angle of a triangle are in the ratio 5 : 3 : 7, the triangle is
(а) an acute-angled triangle
(b) an obtuse angled triangle
(c) an right angled triangle
(d) an isosceles triangle.

Question 8.
In figure l1 || l2, the value of x is

(a) 80°
(b) 100°
(c) 110°
(d) 70°

Question 9.
If one angle of triangle is equal to the sum of the other two, then the triangle is
(a) an isosceles triangle
(b) an obtuse-angled triangle
(c) an equilateral triangle
(d) a right triangle

Question 10.
One of the angles of a triangle is 75°. If the difference of other two is 35°, then the largest angle of other two angles has a measure
(a) 80°
(b) 75°
(c) 70°
(d) 135°

Question 11.
An exterior angle of a triangle is 105° and its two interior opposite angles are equal. Each of these equal angles is
(a) 37 $$\frac{1°}{2}$$
(b) 52 $$\frac{1°}{2}$$
(c) 72 $$\frac{1°}{2}$$
(d) 75°

Answer: (b) 52 $$\frac{1°}{2}$$

Question 12.
In Figure measure of ∠ABC is

(a) 60°
(b) 70°
(c) 80°
(d) 50°

Question 13.
In Figure AB || CD, the value of x is

(a) 35°
(b) 40°
(c) 60°
(d) 75°

Question 14.
In the figure, PS ⊥ l, RQ ⊥ l, the degree measure of y is

(a) 55°
(b) 90°
(c) 80°
(d) 135°

Question 15.
Each angle of an equilateral triangle is
(a) 50°
(b) 90°
(c) 54°
(d) 60°

Question 16.
Given lines l1 || l2 and l2 || l3 in figure. The value of x is

(a) 40°
(b) 140°
(c) 50°
(d) 80°

Question 17.
In figure, find the value of y.

(a) 28°
(b) 40°
(c) 140°
(d) 56°

Question 18.
In the given figure, if AOB is a straight line, then ∠BOC is

(a) 80°
(b) 70°
(c) 60°
(d) 20°


text:
# Error executing Application from BootLoader in STM32L475 (IoT Board)

The bootloader should jump to the Application after some verification. For now lets assume it straightaway jumps there. I am working on B-L475E-IOT01A2 device, which uses ARM M4 Cortex. I am using STM32Cube IDE for writing the code. Here is the code(main() code) for implementing it.

int  boot_main(void)
{

pFunction appEntry;
uint32_t appStack;

/* Check if firmware update required */
if(checkFirmwareUpdate())
{

/* Perform the update */
performFirmwareUpdate();

}

/* Get the application stack pointer (First entry in the application vector table) */

/* Get the application entry point (Second entry in the application vector table) */
appEntry = (pFunction) *(__IO uint32_t*) (APPLICATION_ADDRESS + 4);

/* Reconfigure vector table offset register to match the application
location */

/* Set the application stack pointer */
__set_MSP(appStack);

/* Start the application */
appEntry();

while(1);

}

I have also set the correct memory addresses in both the linker script files. The Application I am using here is FREERTOS. Here are the Linker Script Files:-

/* Entry Point */
ENTRY(Boot_Reset_Handler)

/* Highest address of the user mode stack */
_estack = 0x20017fff;   /* end of "RAM" Ram type memory */

_Min_Heap_Size = 0x200 ;    /* required amount of heap  */
_Min_Stack_Size = 0x400 ;   /* required amount of stack */

/* Memories definition */
MEMORY
{
FLASH   (rx)    : ORIGIN = 0x08000000,  LENGTH = 44K

RAM2    (rwx)   : ORIGIN = 0x10000000,  LENGTH = 32K
RAM (rwx)   : ORIGIN = 0x20000000,  LENGTH = 96K
}<br/>

Application

/* Entry Point */
ENTRY(Reset_Handler)

/* Highest address of the user mode stack */
_estack = 0x20018000;    /* end of RAM */
/* Generate a link error if heap and stack don't fit into RAM */
_Min_Heap_Size = 0x0000;     /* required amount of heap */
_Min_Stack_Size = 0x800; /* required amount of stack */

/* Specify the memory areas */
MEMORY
{
RAM (xrw)       : ORIGIN = 0x20000000, LENGTH = 96K
RAM2 (xrw)      : ORIGIN = 0x10000000, LENGTH = 32K
FLASH (rx)      : ORIGIN = 0x0800B000, LENGTH = 459K    /* Use only the
first bank */
FLASH_UC (r)    : ORIGIN = 0x0807DC00, LENGTH = 9K      /* Fixed-location
area */
}

STM32 has only 1MB of flash memory and the main memory is organized into two banks each of 512KB. So I am just using the first bank(512KB).
The Application contains some lines of code which print some values to the screen. So whenever I try running the BootLoader I can't see anything being printed.(There is no problem with the Application. It is running smoothly)

Can anyone point out what I am missing in my code? Or is there some problem in memory allocation?

Thanks

• Please don't format as code stuff which isn't code. – Dmitry Grigoryev Jul 8 at 13:32
• There isn't really enough information here to have an answerable question - you need to debug it and figure out what is actually happening. Beware that a lot of startup code assumes a processor at reset state and not one that is already "used" - sometimes it is best to set a flag, reset, and have some assembly before the C startup branch to the target. There are existing questions on this subject here which you should read. – Chris Stratton Jul 8 at 13:50
• @ChrisStratton's advice is spot on. Don't bother trying to jump between firmwares (be it the STM32 bootloader or your applications). If you ever succeed, it'll be out of luck and you will loose hours to understand the next time it starts failing. To safely exit from bootloader mode, you need to reset. To safely switch between application firmwares, you need to reset. – user72833 Jul 8 at 14:23
• _estack = 0x20017fff; is not a valid stack address, it should be divisible by 4. (8 according to some ARM docs, but STM32s don't seem to care). – berendi - protesting Jul 8 at 14:31
• Verify that appStack and appEntry do indeed contain the expected values, i.e. they weren't overwritten by flashing the bootloader after the app. If both are correct, step into appEntry() in instruction debug mode to see where it lands. – berendi - protesting Jul 8 at 14:35


text:
rivet is hosted by Hepforge, IPPP Durham

Rivet analyses reference

ATLAS_2016_I1502620_W_MU

W and Z inclusive cross sections at 7 TeV
Experiment: ATLAS (LHC)
Inspire ID: 1502620
Status: VALIDATED
Authors:
• Jan Kretzschmar
• Christian Gutschow
References:
Beams: p+ p+
Beam energies: (3500.0, 3500.0) GeV
Run details:
• p + p -> W + X ( W -> mu nu )

High-precision measurements by the ATLAS Collaboration are presented of inclusive $W^+\to\ell^+$, $W^-\to\ell^-$, $Z/\gamma^\ast\to\ell\ell$ ($\ell=e,\mu$) Drell-Yan production cross sections at the LHC. The data were collected in proton-proton collisions at $\sqrt{s}=7$ TeV with an integrated luminosity of 4.6 fb${}^{-1}$. Differential $W^+$ and $W^-$ cross sections are measured in a lepton pseudorapidity range $|\eta_\ell| < 2.5$. Differential $Z/\gamma^\ast$ cross sections are measured as a function of the absolute dilepton rapidity, for $|y_{\ell\ell}| < 3.6$, for three intervals of dilepton mass, $m_{\ell\ell}$, extending from 46 to 150 GeV. The integrated and differential electron- and muon-channel cross sections are combined and compared to theoretical predictions using recent sets of parton distribution functions. The data, together with the final inclusive $e^\pm p$ scattering cross-section data from H1 and ZEUS, are interpreted in a next-to-next-to-leading-order QCD analysis, and a new set of parton distribution functions, ATLAS-epWZ16, is obtained. The ratio of strange-to-light sea-quark densities in the proton is determined more accurately than in previous determinations based on collider data only, and is established to be close to unity in the sensitivity range of the data. A new measurement of the CKM matrix element $|V_{cs}|$ is also provided. N.B.: Use ATLAS_2016_I1502620_W_EL and ATLAS_2016_I1502620_W_MU to specify the decay channel directly. Note that when merging multiple yoda files, the $W$ charge asymmetry $A_l = (W^+ - W^-) / (W^+ + W^-)$ may need to be recalculated from the individual histogram. A post-processing script is provided.


text:
sch25 {CR2} R Documentation

## Data from 25 schools (based on the NELS dataset)

### Description

For examining the association between amount homework done per week and math outcome.

### Usage

data(sch25)

### Format

A data frame with 546 rows and 8 variables:

schid

The school identifier (the grouping variable)

ses

Student-level socioeconomic status

byhomewk

Total amount of time the student spent on homework per week. 1 = None, 2 = Less than one hour, 3 = 1 hour, 4 = 2 hours, 5 = 3 hours, 6 = 4-6 hours, 7 = 7 - 9 hours, 8 = 10 or more

math

Mathematics score.

male

Dummy coded gender, 1 = male, 0 = female

minority

Dummy coded minority status, 1 = yes, 0 = no

mses

Aggregated socioeconomic status at the school level

mhmwk

Aggregated time spent on homework at the school level

### Source

[Package CR2 version 0.2.1 Index]


text:
# Harry R. Schwartz

Software engineer, nominal scientist, gentleman of the internet. Member, ←Hotline Webring→.

### Opening Your Last Rails Migration in Emacs

Published 12 Mar 2014. Tags: emacs, rails.

I was pairing with a Vim user the other day, and I noticed that he had a nifty shortcut (from rails.vim?) to open up the last Rails migration.

Emacs being Emacs, I decided to implement it myself.

(defun visit-last-dired-file ()
"Open the last file in an open dired buffer."
(interactive)
(end-of-buffer)
(previous-line)
(dired-find-file))

(defun visit-last-migration ()
"Open the last file in 'db/migrate/'."
(interactive)
(dired (expand-file-name "db/migrate" (projectile-project-root)))
(visit-last-dired-file)
(kill-buffer "migrate"))

Just run that with M-x visit-last-migration and it’ll summon up your last migration. I haven’t bothered binding it to a key, since I don’t really need it that often.

It’s a bit of a hack, and it depends on projectile-mode, so if you have an alternate solution I’d sure appreciate an email or pull request!


text:
# Colliding beams vs static target

In this problem I need to find the threshold energy of a positron-electron collision that creates a Z boson (the reverse of the following picture; Z mass 92 GeV) in two distinct cases: collision ring and static target.

For the collision ring, where proton and electron have equal but opposite momentum, I found a kinetic energy of approximately 46 GeV for each particle. However, in the static target case, where the electron is at rest, I found a beam energy of $8.6\times 10^{6}$ GeV.

This seems way too high, but I checked my calculations a couple times and I don't see any mistake. Is this a reasonable value?

• This problem (and ones similar to it) is given a lot for a reason. I haven't (and won't) check you figures, but this effect occurs in non-relativistic regimes to a degree as well and you may be applying intuition that is based on a Newtonian understanding. Think about what the velocity addition law does for you in this case; or about the difference in speed between the beam and the CoM in the fixed target version of the experiment. May 5 '16 at 0:57
• Wouldn't you usually say that $\sqrt{s}$ = 92 GeV? Then for a fixed target experiment we know that $\sqrt{s}=E_{\text{CM}}\approx\sqrt{E_{\text{proton}}}$. From that, we would conclude that $E_{\text{proton}}=s=8464$GeV. That would be 8.4 TeV which is realistic for such a configuration, but not realizable with current technology i think. May 18 '16 at 20:51


text:
# How do you find the percent change given before:75 and after:60?

Dec 2, 2016

This was a -20% change.

#### Explanation:

The formula for calculating the percent change is:

$p = \frac{N - O}{O} \cdot 100$ where $p$ is the percent change, $N$ is the New Value and $O$ is the Old Value.

We can substitute the values provided in the problem and solve for $p$:

$p = \frac{60 - 75}{75} \cdot 100$

$p = - \frac{15}{75} \cdot 100$

$p = - \frac{1500}{75}$

$p = - 20$


text:
# Thread: [SOLVED] Show that the following vectors are two different bases for the following pl

1. ## [SOLVED] Show that the following vectors are two different bases for the following pl

Show that

$\begin{pmatrix} 4\\0\\1 \end{pmatrix}$ , $\begin{pmatrix} 2\\1\\0 \end{pmatrix}$
and
$\begin{pmatrix} 2\\-1\\1 \end{pmatrix}$ , $\begin{pmatrix} 0\\2\\-1 \end{pmatrix}$

are two different bases for the plane

$x-2y-4z=0$

Thanks in advance!

2. Originally Posted by JCS007
Show that

(4 0 1) $.^T$ , (2 1 0) $.^T$ and
(2 -1 1) $.^T$ , (0 2 -1) $.^T$

are two different bases for the plane

$x-2y-4z=0$

Thanks in advance!
Consider an arbitrary vector written in terms of the first basis:
$V = a \left ( \begin{matrix} 4 \\ 0 \\ 1 \end{matrix} \right ) + b \left ( \begin{matrix} 2 \\ 1 \\ 0 \end{matrix} \right )$

Now show that $V_x - 2V_y - 4V_z = 0$ for all a and b.

The proof for the other basis is similar.

-Dan


text:
Precalculus (6th Edition) Blitzer

$constraints$, $corner$
The limiting conditions, or restrictions, are also called "constraints". Once we obtain the graph of inequalities, the maximum and minimum values can be found at one or more "corners" of the solution region. Thus the answers are $constraints$ and $corner$.


text:
A few helpers to get specific things from the cache repository

getUserTags(cacheRepo, shownCache, cacheId, concatenated = TRUE)

getCacheId(cacheRepo, shownCache, artifact)

getArtifact(cacheRepo, shownCache, cacheId)

## Arguments

cacheRepo A repository used for storing cached objects. This is optional if Cache is used inside a SpaDES module. Primary way of supplying cacheRepo; the data.table obj resulting from showCache, i.e., it will override cacheRepo. If this and cacheRepo are missing, then it will default to getOption('reproducible.cachePath') A character vector of o cacheId values to use in the cache Logical. If TRUE, the returned userTags will be concatenated tagKey:tagValue Character vector of artifact values in the artifact column of showCache

## Value

getCacheId returns the cacheId values for 1 or more artifacts in the cache.

getArtifact returns the artifact value for 1 or more entries in the cache, by cacheId.


text:
Simulating or generating datasets for testing modelling methodologies

I'm interested to know what a good practice is when generating/simulating data for testing or comparing modelling methods. I'm focusing on linear models and measuring prediction accuracy and measuring parameter estimate accuracy.

I'm looking at prediction accuracy (PSE) in terms of cross validated squared prediction error and parameter estimate accuracy in terms of $\|\beta-\hat{\beta}\|$ (MSE).

I have attempted a bunch of simulations using what I believe to be a 'sound' method, but I am not happy with the results. I'm guessing I'm overlooking something. The issue is that adjacent parameter estimates are highly correlated with each other and I cannot figure out why. Another issue is that while I can generate datasets that show a large difference in parameter estimate accuracy, the prediction performance will be very similar. I am not sure why this happens, I would expect the methods with improved MSE to have much improved PSE over the other methods. Here's a typical example: http://ohyur.com/random/typical.png

I will outline what I have attempted.

• Set the true parameter estimates, $\beta$
• Set correlation structure for the predictor variables
• Generate the predictors with specified correlation structure, $X$. I've tried using mvrnorm() in R to generate N(0,1) columns and mvrunif() function to generate U(-1,1) columns. Both with the specified correlation structure.
• Generate the response using $Y=\mu + X\beta+\epsilon$, where $\epsilon$ is $N(0,\sigma$).

To me, that sounds like a reasonably method of generating data for a linear model.

Here is an example of what the generated data might look like: http://ohyur.com/random/data.png ($\mu=3,$ $\beta=(3,2,-3,5,2,-3,5,5)$, $\sigma=15$, $\text{cor}(x_i,x_j)=0.9^{|i-j|}$

Now in my simulations, in each replication I'm generating data according to the above steps. Then fitting some models, estimating the CV PSE and calculating the MSE and storing the results. Here is a boxplot of the results for 3 different modelling methods over 100 replications. http://ohyur.com/random/psemse.png

Now, most interestingly to me, Here is a pairwise plot of the estimated parameters for a full OLS model. Why are the adjacent predictors correlated? http://ohyur.com/random/estimates.png

One thing I'm coverned about is that in the real world, some predictors are more correlated with the response than others, and not just due to effect size. This isn't really accounted for in my setup. But I'm really just after the simplest possible 'sound' method.

Any insight/comments or recommendations on how to properly generate data for testing liner models would be greatly appreciated.

• I will try checking out the hat matrix now. I tried an example with fairly uncorrelated predictors ($\rho_{ij}=0.25^{|i-j|}$) and got the following: The data: ohyur.com/random/lowdata.png The estimates: ohyur.com/random/lowestimates.png . So even while the pairwise correlations in the other data are much stronger than these, it is still only the adjacent predictors that are correlated. – dcl May 22 '12 at 2:59


text:
## Measurement of the Malusʼ law

### Experiment number : 4298

• #### Goal of experiment

The goal of this experiment is to quantitatively verify the validity of Malus’ law.

• #### Theory

This experiment will focus on a situation in which linearly polarised light is incident on and passes through a generally rotated linear polarising filter. Malus’ law gives an answer to the question about what will be the intensity of the transmitted light depending on the rotation of the polarising filter. If the plane in which the polarising filter passes is rotated relative to the plane of the incident polarised light by an angle $$\varphi$$, then the following applies:

$I(\varphi)=I_0\cos^2\varphi,$

where $$I_0$$ is the intensity of the incident light and $$I$$ is the intensity of the transmitted light.

A well-known special case of Malus’ law is the situation where the plane passed through the polarising filter is oriented perpendicular to the plane of polarisation of the incident light. In such a case, $$\varphi=90^\circ$$ and hence $$I=0$$; the polarising filter, therefore transmits no light. Such a situation is qualitatively investigated in the Linearly Polarised Light LCD Monitor experiment Linearly polarized light of LCD.

An illustration of what happens to the intensity of the transmitted light when the angle $$\varphi$$ changes is provided by an animation created in GeoGebra by Chris Hamper:

.
• #### Equipment

Light source, one ordinary and one rotating polarising filter, luxmeter.

Choosing a luxmeter: Measurements can be made with either a luxmeter, which is strongly directional, or a luxmeter, which collects light from the entire half-space. In our sample case, the measurement was made with a Vernier LS-BTA directional luxmeter, but very similar results were also obtained with a Voltcraft BL-10L luxmeter (Figure 2), which collects light from the entire half-space.

Selection of light source: It is essential for the measurement that the light source provides a constant, unvarying luminous flux. For this reason, it is not appropriate to use, for example, an incandescent light bulb powered by ordinary (flat, pencil) batteries – these discharge over time and the light output from the bulb decreases measurably. Similarly, you cannot use AC power supplies that provide a variable voltage and the connected light source does not shine continuously, but actually flickers at a high frequency (this is a common feature of mains-connected luminaires – see for example this experiment). However the same feature can also be observed in luminaires connected to some DC sources which imperfectly smooth the input mains voltage to the output DC voltage. In other words, it is essential to pay considerable attention to the choice of source and to test it beforehand.

For the purposes of our measurements, it worked very well to connect a 6 V bulb to an accumulator, where the voltage drop is negligible within minutes. Similarly, a high-brightness diode can be used.

• #### Procedure

1. Arrange the light source, the two polarising filters and the luxmeter in a row – Figure 3. Place both filters and luxmeter as close together as possible to minimise the amount of light entering the luxmeter from the surroundings (daylight, unwanted reflections from walls, etc.).

1. Turn on the light source and set the rotating filter so that the illumination intensity measured by the luxmeter is maximum (i.e., both filters transmit the same plane of polarisation).

2. Now rotate the movable filter gradually (for example, in 5° steps) and note the measured intensity value for each position.

3. Make measurements over a rotation range from 0° to 90° or up to 180°. Then plot the obtained values on a graph.

• #### Example of results

A sample of the measured data is given in the table below and the graph in Figure 4. The obtained values are plotted with a function of the form $$I(\varphi)=I_0\cos^2\varphi+K$$, which indicates a reasonable match between the measured data and the theoretically predicted trend.

Table of measured data:

 rotation angle $$\varphi$$ (°) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 78 85 90 intensity I (lx) 545 538 528 506 485 446 411 373 334 296 256 220 180 148 125 105 94 85 80

• #### Technical notes

• In the graph in Fig. 4, it can be noticed that even for an angle of 90°, the measured illuminance does not drop to zero as the relationship in the Theory section predicted. This is because not only light passed through the polarising filters but also ambient light or reflected light from the bulb enters the luxmeter − for example, in Figure 5 it can be seen that some of the light from the bulb is reflected into the luxmeter from the adjacent wall.

• The above effect, i.e. the trapping of light entering the luxmeter outside the polarising filters, can be eliminated in several ways:

• darkening the room

• placing the apparatus away from walls that would reflect the light

• placing the polarising filters and the luxmeter as close together as possible

These recommendations are more important to follow when working with a full-space luxmeter; a directional luxmeter is less prone to unwanted interception of ambient or reflected light.

• #### Methodical notes

It should be emphasised that Malus’ law applies to linearly polarised light. If we let natural, i.e. unpolarised, light fall on the filter, we always measure half the intensity behind the filter than in front of it, regardless of the rotation of the filter.


text:
# Exclamation Marks Everywhere!

Discrete Mathematics Level 5

$\large a! \cdot b! \cdot c! \cdot d! \cdot e! = 400!$

Find the number of ordered quintuples of non-negative integers $$(a,b,c,d,e)$$ satisfying the above equation.

×


text:
# Modeling the Earth's Magnetosphere Using Spacecraft Magnetometer Data

This is a webpage by N. A. Tsyganenko, dedicated to empirical modeling of the Earth's magnetospheric magnetic field. Here one can find the latest studies and download source codes of data-based models, developed by the author of this web page (and his co-authors) during the last 30 years.

For convenience, the Fortran source codes are grouped into categories 'Magnetic field', 'Plasma', and 'Current sheet' in the sidebar on the left.

PREVIOUS VERSION OF THIS WEB-PAGE

## Empirical modeling

Data-based (or empirical) modeling of the geomagnetic field started as a discipline as early as in the first half of XIX century, when Gauss developed mathematical foundations of the modeling of Earth's main magnetic field and obtained first estimates of its spherical harmonic coefficients, using then available ground-based data. That approach, based on the potential (current-free) nature of the main field outside Earth, is still at the core of modern IGRF models.

With the advent of space era and understanding the crucial role of the geomagnetic field in the dynamics of the Earth's upper atmosphere and radiation belts, a need was realized to extend the models from low to high altitudes, eventually including the entire magnetosphere, an integral part of our space environment. Modeling the magnetic field in that region is much more difficult, mostly because the magnetic field from external sources (currents in the magnetospheric plasma) rapidly outweighs the main field with growing distance from Earth. The external field is not current-free and, hence, it is no longer possible to conveniently represent it by a scalar potential, uniquely defined by observations at a surface, as was the case with the main field. Rather, vector measurements of the magnetic field should now be made throughout the entire 3D modeling region, making it necessary to accumulate large amounts of space magnetometer data taken in a wide range of geocentric distances.

This task turns out to be even more complicated due to the fact that, unlike the main geomagnetic field that varies on a timescale of thousands of years, the Earth's magnetosphere is a very dynamical system, whose configuration depends on many internal and external factors. The first factor is orientation of the Earth's magnetic axis with respect to the direction of the incoming solar wind flow, which varies with time because of (i) Earth's diurnal rotation and its yearly orbital motion around Sun, and (ii) frequent "side gusts" of the solar wind. The animation on the left below shows how the magnetospheric field varies in response to the diurnal wobbling of the geodipole. The background color coding displays the distribution of the scalar difference DB between the total model magnetic field and that of the Earth's dipole alone. Yellow and red colors correspond to the negative values of DB (depressed field inside the ring current, in the dayside polar cusps, and in the plasma sheet of the magnetotail). Black and blue colors indicate a compressed field (in the subsolar region on the dayside and in the magnetotail lobes on the nightside).

Another important factor is the state of the solar wind, in particular, the orientation and strength of the interplanetary magnetic field (IMF), "carried" to the Earth's orbit from Sun due to the high electrical conductivity of the solar wind plasma. Interaction between the terrestrial and interplanetary fields becomes much more effective when the interplanetary magnetic field turns antiparallel to the Earth's field on the dayside boundary of the magnetosphere. In this case, geomagnetic and interplanetary field lines connect across the magnetospheric boundary, which greatly enhances the transfer of the solar wind mass, energy, and electric field inside the magnetosphere. As a result, the magnetospheric field and plasma become involved in a convection, as illustrated in the second animation below (right):

In actuality, that kind of steady convection is rarely realized. The solar wind is far from being a stationary flow: periods with a stable ram pressure are often interrupted by strong "gusts"; in addition, the interplanetary magnetic field often fluctuates both in magnitude and orientation. This results in dramatic dynamical changes of the entire magnetospheric configuration, which culminate in magnetospheric storms, accompanied by an explosive conversion of large amounts of the solar wind energy into the kinetic energy of charged particles in the near-Earth space, manifested in polar auroral phenomena and ionospheric disturbances. The third animation below (left panel) illustrates the dynamical changes of the global magnetic field in the course of a disturbance: a temporary compression of the magnetosphere by enhanced flow of the solar wind is followed by a tailward stretching of the field lines. Eventually, the increase of the tail magnetic field results in a sudden collapse of the nightside field (a substorm) and a gradual recovery of the magnetosphere to its pre-storm configuration.

## Space weather

Space weather is a modern field of space research, focused on the solar activity and its impact upon the near-Earth environment, spacecraft hardware, and humans. It includes investigation and prediction of solar flares, coronal mass ejections (CME), sunspots, magnetic storms, particle precipitation into the Earth atmosphere, and associated ionospheric phenomena. The flow of plasma from the Sun, known as the solar wind, is the principal factor determining the space weather in our planetary system. This is why it is very important to know in advance its principal characteristics: particle density, bulk velocity, the strength and direction of the Interplanetary Magnetic Field (IMF). The NASA Advanced Composition Explorer (ACE) satellite (operating since 1997) and the recently (2015) launched Deep Space Climate Ovservatory (DSCOVR) mission reside at the L1 libration point (1,500,000 km sunward from Earth) and provide continuous flow of information about the solar wind state nearly one hour in advance. Their real-time data are provided online by the National Oceanic and Atmospheric Administration (NOAA) Space Weather Prediction Center (SWPC).

The orientation of the IMF vector is a crucial factor that determines the state of the near-Earth space environment. Southward IMF ($B_z<0$ in GSM coordinates) combined with a high-speed solar wind result in magnetic storms, which may damage the spacecraft equipment, affect the navigation systems, aircraft and satellite operation. It also results in the so-called "geomagnetically induced electric currents" (GIC) in the ground infrastructures, such as pipelines and electric power grids. For example, on September 1-2, 1859, one of the largest recorded geomagnetic storms caused the failure of the telegraph systems all over Europe and North America. The intensity of geomagnetic storms is quantified by the Dst index, derived from the disturbance of the horizontal H-component of the magnetic field at low and middle latitudes. A rapid decrease of the Dst to low negative values (Dst$<-50$ nT) manifests the development of a storm. Anyone interested in the current state of space weather can check it at the real-time Dst trend on the webpage of the World Data Center for Geomagnetism (Kyoto).

Here is a very interesting and helpful resource SpaceWeather.com for everyone with even a modest background in space physics. The site provides real-time space weather conditions such as the solar wind parameters, sunspot numbers, solar flares, solar images from SDO (Solar Dynamics Observatory) instruments and NOAA 24 and 48 hour forecasts of the flare and geomagnetic storm probabilities. Besides the space environment data, the website offers up-to-date news on spacecraft scientific missions, breathtaking pictures of auroral displays, stunning atmospheric optical phenomena, noctilucent clouds, and much more interesting information.


text:
# How do you find the domain and range of x + sqrt( x-1 )?

May 27, 2017

See below.

#### Explanation:

Since a square root cannot have a negative under the radical (or else it would be imaginary),

$x \setminus \ge q 1$.

When $x = 1$, the range (or $y$) is $\left(1\right) + \sqrt{1 - 1} = 1$. Thus, the range of $y$ is $y \ge q 1$

Thus, in interval notation, the domain of the function is $\left[1 , \infty\right)$, and the range is $\left[1 , \infty\right)$


text:
main-content

## International Journal of Data Science and Analytics OnlineFirst articles

02-08-2022 | Regular Paper

### Attention-like feature explanation for tabular data

A new method for local and global explanation of the machine learning black-box model predictions by tabular data is proposed. It is implemented as a system called AFEX (Attention-like Feature EXplanation) and consisting of two main parts. The …

Authors:
Andrei V. Konstantinov, Lev V. Utkin

Open Access 30-07-2022 | Regular Paper

### Optimizing graph layout by t-SNE perplexity estimation

Perplexity is one of the key parameters of dimensionality reduction algorithm of t-distributed stochastic neighbor embedding (t-SNE). In this paper, we investigated the relationship of t-SNE perplexity and graph layout evaluation metrics including …

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Chun Xiao, Seokhee Hong, Weidong Huang

Open Access 25-07-2022 | Regular Paper

### Data-driven versus a domain-led approach to k-means clustering on an open heart failure dataset

Domain-driven data mining of health care data poses unique challenges. The aim of this paper is to explore the advantages and the challenges of a ‘domain-led approach’ versus a data-driven approach to a k-means clustering experiment. For the …

Authors:
A. Jasinska-Piadlo, R. Bond, P. Biglarbeigi, R. Brisk, P. Campbell, F. Browne, D. McEneaneny

24-07-2022 | Regular Paper

### Semantic enhanced Markov model for sequential E-commerce product recommendation

To model sequential relationships between items, Markov Models build a transition probability matrix $$\mathbf {P}$$ P of size $$n \times n$$ n × n , where n represents number of states (items) and each matrix entry $$p_{(i,j)}$$ p ( i , j ) …

Authors:
Mahreen Nasir, C. I. Ezeife

20-07-2022 | Regular Paper

### PARAS: a parameter space-driven approach for complete association rule mining

To enable efficient association rule mining, existing techniques prestore intermediate results as itemsets. However, the actual rule generation is still performed at query-time. The response time thus tends to remain unacceptably long for …

Authors:
Xika Lin, Abhishek Mukherji, Elke A. Rundensteiner, Matthew O. Ward

Open Access 16-07-2022 | Regular Paper

### Bounding open space risk with decoupling autoencoders in open set recognition

One-vs-Rest (OVR) classification aims to distinguish a single class of interest (COI) from other classes. The concept of novelty detection and robustness to dataset shift becomes crucial in OVR when the scope of the rest class is extended from the …

Authors:
Max Lübbering, Michael Gebauer, Rajkumar Ramamurthy, Christian Bauckhage, Rafet Sifa

14-07-2022 | Regular Paper

### ScholarRec: a scholars’ recommender system that combines scholastic influence and social collaborations in academic social networks

Identifying and recommending influential scholars is one of the leading applications of scholarly data analytic. The existing methods to identify influential scholars focus on scholastic influence or social collaborations. In the former approach …

Authors:
Mitali Desai, Rupa G. Mehta, Dipti P. Rana

11-07-2022 | Regular Paper

### Domain-specific text dictionaries for text analytics

We investigate the use of sentiment dictionaries to estimate sentiment for large document collections. Our goal in this paper is a semiautomatic method for extending a general sentiment dictionary for a specific target domain in a way that …

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Andrea Villanes, Christopher G. Healey

Open Access 10-07-2022 | Regular Paper

### Multilingual hope speech detection in English and Dravidian languages

Recent work on language technology has aimed to identify negative language such as hate speech and cyberbullying as well as improve offensive language detection to mediate social media platforms. Most of these systems rely on using machine …

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Bharathi Raja Chakravarthi

18-06-2022 | Regular Paper

A predictive model that is trained with non-randomly selected samples can offer biased predictions for the population. This paper discusses when non-random selection is a problem. For the applications in which it is a problem, this paper presents …

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Jonathan Cook

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### Data-driven analytics of COVID-19 ‘infodemic’

The rampant of COVID-19 infodemic has almost been simultaneous with the outbreak of the pandemic. Many concerted efforts are made to mitigate its negative effect to information credibility and data legitimacy. Existing work mainly focuses on …

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Minyu Wan, Qi Su, Rong Xiang, Chu-Ren Huang

10-06-2022 | Review

### A survey on event and subevent detection from microblog data towards crisis management

Social media data analysis is a popular research domain since the last decade. Detecting the events and sub-events from social media posts that require special attention is one of the key research problem in this domain with wide range of …

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Shatadru Roy Chowdhury, Srinka Basu, Ujjwal Maulik

06-06-2022 | Regular Paper

### Exo-SIR: an epidemiological model to analyze the impact of exogenous spread of infection

Epidemics like Covid-19 and Ebola have impacted people’s lives significantly. The impact of mobility of people across the countries or states in the spread of epidemics has been significant. The spread of disease due to factors local to the …

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Nirmal Kumar Sivaraman, Manas Gaur, Shivansh Baijal, Sakthi Balan Muthiah, Amit Sheth

Open Access 27-05-2022 | Original Paper

### COVID-19 and 5G conspiracy theories: long term observation of a digital wildfire

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Johannes Langguth, Petra Filkuková, Stefan Brenner, Daniel Thilo Schroeder, Konstantin Pogorelov

26-05-2022 | Regular Paper

### Using big data and federated learning for generating energy efficiency recommendations

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Arnab Dey, Andrew Heger, Darin England

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### An iterative topic model filtering framework for short and noisy user-generated data: analyzing conspiracy theories on twitter

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### Explainability of the COVID-19 epidemiological model with nonnegative tensor factorization

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Till D. Frank


text:
## Chemistry and Chemical Reactivity (9th Edition)

*graph is attached Equation of line => $y= 248x+0.002$ At Absorbance=$0.635$ Concentration = $2.55*10^{-3}$$\frac{g}{L} = 2.55*10^{-3}$$\frac{mg}{mL}$
To get equation of line: Input the data into a Linear regression which will give you the equation of the line To get Concentration at Absorbance of $0.635$: The equation of the line is for a Absorbance vs Concentration $(g/L)$ Therefore we can input the $0.635$ into the equation of the line as $y$ and then we can solve for $x$ So... $0.635= 248x+.002$ $x= 2.55*10^{-3}\frac{g}{L}$ To get Concentration in $\frac{mg}{mL}$ $\frac{2.55*10^{-3}g}{L}*\frac{1000mg}{1g}*\frac{1L}{1000mL}=2.55*10^{-3}\frac{mg}{mL}$ Thus, Concentration = $2.55*10^{-3}$$\frac{g}{L} = 2.55*10^{-3}$$\frac{mg}{mL}$


text:
## Forecasting emissions through Kaya identity using Neural ODEs (Proposals Track)

Pierre Browne (Imperial College London)

### Abstract

Starting from Kaya identity, we used a Neural ODE model to predict the evolution of several indicators related to carbon emissions, on a country-level : population, GDP per capita, energy intensity of GDP, carbon intensity of energy. We compared the model with a baseline statistical model - VAR - and obtained good performances. We conclude that this machine-learning approach can be used to produce a wide range of results and give relevant insight to policymakers.


text:
CryptoDB

Paper: In How Many Ways Can You Write Rijndael?

Authors: Elad Barkan Eli Biham URL: http://eprint.iacr.org/2002/157 Search ePrint Search Google In this paper we ask the question what happens if we replace all the constants in Rijndael, including the replacement of the irreducible polynomial, the coefficients of the MixColumn operation, the affine transformation in the S box, etc. We show that such replacements can create new dual ciphers, which are equivalent to the original in all aspects. We present several such dual ciphers of Rijndael, such as the square of Rijndael, and dual ciphers with the irreducible polynomial replaced by primitive polynomials. We also describe another family of dual ciphers consisting of the logarithms of Rijndael. We then discuss self-dual ciphers, and extend our results to other ciphers.
BibTeX
@misc{eprint-2002-11680,
title={In How Many Ways Can You Write Rijndael?},
booktitle={IACR Eprint archive},
keywords={secret-key cryptography / AES, Galois Field, Dual Cipher, Self Dual, Logarithm},
url={http://eprint.iacr.org/2002/157},
note={Asiacrypt 2002. barkan@cs.technion.ac.il 11976 received 16 Oct 2002},
author={Elad Barkan and Eli Biham},
year=2002
}


text:
# How to prove that nth differences of a sequence of nth powers would be a sequence of n!

Given an infinite sequence of numbers, first differences denote a sequence of numbers that are pairwise differences, second differences denote a new sequence of pairwise differences of this sequence, and so on.

1  2  4  7 11 16 22 29
1  2  3  4  5  6  7    -- first differences
1  1  1  1  1  1      -- second differences
0  0  0  0  0        -- third differences

Sequences of nth powers of numbers exhibit an interesting property. Take a sequence of squares: (1,4,9,16,25,…).

• First differences: (3,5,7,9,11…)
• Second differences: (2,2,2,2,2…)

Take a sequence of cubes: (1,8,27,64,125…)

• First differences: (7,19,37,61,91…)
• Second differences: (12,18,24,30,36…)
• Third differences: (6,6,6,6,6…)

Take a sequence of powers of 4: (1,16,81,256,625,…)

• First differences: (15,65,175,369,671,…)
• Second differences: (50,110,194,302,434,…)
• Third differences: (60,84,108,132,156,…)
• Fourth differences: (24,24,24,24,24,…)

1, 2, 6, 24… that’s clearly a factorial! So a sequence of powers of n will have a sequence of repeating n! as its nth differences. How do I prove this relation holds? Bonus points if you give just enough theoretical information for a non-mathematician, but leave the proof as an exercise.

Note: It actually doesn’t matter if our sequence includes negative numbers too, sequence (…,-2,-1,0,1,2,…) still has (…,1,1,1,1,1,…) as its first differences.

#### Solutions Collecting From Web of "How to prove that nth differences of a sequence of nth powers would be a sequence of n!"

The difference sequence of any polynomial $Cx^n + \cdots$ (where here and elsewhere, $\cdots$ hides terms of lower degree) is of the form $Cnx^{n-1} + \cdots$, since $(x+1)^n – x^n = nx^{n-1} + \cdots$ (just open the binomial). If we start with $x^n + \cdots$ then we get $nx^{n-1} + \cdots$, then $n(n-1)x^{n-2} + \cdots$, then $n(n-1)(n-2)x^{n-3} + \cdots$, and eventually, after $n$ times, $n(n-1)(n-2)\cdots 1 x^0 = n!$. You can prove this by induction.


text:
# Tag Info

9

I would use \text like this: \documentclass{article} \usepackage{amsmath} \begin{document} $$\text{All x are y; all y are z; therefore, all x are z.}$$ \end{document}

8

This does what you want, though I find it cumbersome and not really informative. \documentclass{book} \usepackage{amsthm,xpatch} \makeatletter \let\qed@empty\openbox % <--- change here, if desired \def\@begintheorem#1#2[#3]{% \deferred@thm@head{% \the\thm@headfont\thm@indent \@ifempty{#1} {\let\thmname\@gobble} ...

7

If we use the amsthm package then we can do this by hijacking the \qedsymbol command and hacking the way that the theorem environments are constructed internally. This comes down to adding some code to \@begintheorem to overwrite \qedsymbol so that it becomes a boxed version of the last theorem number. There are two issues with the code below. The first ...

6

You can store the "message" in the main vertical list about the end of previous environment using very specific value of penalty. And you can read this value at the start of your environment. \newenvironment{env} {\ifnum\lastpenalty=12345 \vskip-\bigskipamount \penalty0 \fi \addvspace{\bigskipamount}\bfseries} ...

6

\documentclass{article} \newenvironment{example}{\par\hrule\bfseries}{\par\hrule} \makeatletter \def\CloseEverything{% \expandafter\end\expandafter{\@currenvir}\CloseEverything } \makeatother \begin{document} \begin{example} \begin{enumerate} \item blah \item blah \CloseEverything

5

Like this? \documentclass{article} \usepackage{enumitem,showframe} %% showframe only for demo \newlist{mylist}{itemize}{1} \setlist[mylist]{leftmargin=*} \begin{document} \begin{mylist}[{label=\makebox[3cm][l]{\bfseries Title}}] \item title 1 \\ title 2 \item Title \end{mylist} \begin{mylist}[label={\makebox[3cm][l]{\bfseries How}}] ...

5

For lists I'd recommend you use the enumitem package and use the newlist facility that is built in. If you have many of these you could use \newcommand{\DefineNewList}[3]{% % #1 = name % #2 = max depth % #3 = label \newlist{#1}{enumerate}{#2} \setlist[#1]{label*=#3} }% but I don't see much value in that. Code: ...

4

After a small fight with it (first time using prop) I think this works, although it's a bit of a mix of things, and may it would be better to think of a better “user interface”. First, you have an environment cryptolist which is like an enumerate except that \item{…} does have an argument. cryptolist has an optional argument which is the name of the list ...

4

Loop Space's answer is good, but has the disadvantage of clobbering the optional argument, so attribution can't be specified any more. I'd suggest using a mandatory argument. \documentclass{article} \usepackage{amsthm} \swapnumbers % optional, of course \newtheorem{thm}{Theorem}[section] % the main one \newtheorem{lemma}[thm]{Lemma} % other statement ...

4

The symbol must be inserted at the end of the environment, 'flushed' right, so use \hfill\myqedsymbol where \myqedsymbol is a macro wrapper for the particular symbol (\triangle in this case) \documentclass{article} \usepackage{amsthm} \usepackage{hyperref} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \theoremstyle{definition} ...

4

If you define your environment as \newenvironment{myenvironment}[1]{\taggedblock{#1}}{\endtaggedblock} then things work: Note: I have seen this trick used when wrapping environment around tabular but not sure why it is needed. Code: \documentclass[10pt,a4paper]{article} \usepackage[latin1]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} ...

3

You could use the same mechanism as section titles: \documentclass{article} \usepackage{amsthm} \usepackage{lipsum} \newtheorem{theorem}{Theorem} \makeatletter \newcommand{\proofpart}[2]{% \par \addvspace{\medskipamount}% \noindent\emph{Part #1: #2}\par\nobreak \addvspace{\smallskipamount}% \@afterheading } \makeatother \begin{document} ...

3

Your problem is not the square brackets: it's the definition of the matrix environment. If you do \show\matrix you get > \matrix=macro: #1->\null \,\vcenter {\normalbaselines \m@th \ialign {\hfil $##$\hfil &&\quad \ hfil $##$\hfil \crcr \mathstrut \crcr \noalign {\kern -\baselineskip } #1\crcr \mathstrut \crcr \noalign {\kern -\baselineskip ...

3

Why define \foo? May be I missed something but you can inglude \csname inside the definitions \newenvironment{variables} {\paragraph{Variables :} \csname alignat*\endcsname{2}} {\csname endalignat*\endcsname}

3

You can't do \begin{textbf} or \begin{emph} and you shouldn't do \begin{Large} either. You want \newcommand{\session}[1]{% \begin{center}\Large\itshape\bfseries \stepcounter{sessioncounter}% Session \value{sessioncounter}: #1% \end{center}% } in the preamble and use \session{Hello world} in the document.

3

Your definition matches the format of an environment, not a command. This is how environments are defined: \newenvironment{example}{<starting commands>}{<ending commands>} They are then used like this: \begin{example} <text> \end{example} But I think you want a command that takes one argument, as in this example. Also, ...

3

This is adapted from what I have used in the past (not sure where I got it from): Code: \documentclass{article} \usepackage{amsmath} \newcommand*{\QED}[1]{% \ifmmode% Check for math mode. \tag*{\fbox{#1}}% \else% {\rightskip\fill\parfillskip-\rightskip% \linepenalty100% \exhyphenpenalty0% ...

3

The idea is to add a conditional that's true when a proof starts and change the code accordingly: if the conditional is false, we're not nesting, so we set \qed@current to \qed@empty, otherwise we use the same mechanism as in the other answer. \documentclass{book} \usepackage{amsthm,xpatch} \makeatletter \let\qed@empty\openbox % <--- change here, if ...

2

I think it's better to use the \excludecomment macro provided by the comment package. This way you don't need to start and end comment environments from inside macros or environments of your own. \documentclass{article} \usepackage{ifthen} \usepackage{comment} \specialcomment{solution}{\textbf{Solution}\quad}{} \newboolean{iflecturer} ...

2

Here is one possibility, by redefining the way \descriptionlabel works at the beginning of every induction environment: \documentclass{article} \newenvironment{induction}[1] {\renewcommand{\descriptionlabel}[1]{\hspace\labelsep\normalfont\bfseries$#1$-induction, $##1$}% \begin{description}} {\end{description}} \begin{document} ...

2

You can obtain what you want and automatic numbering of items within enumerate-like environments, with enumitem: \documentclass{article} \usepackage[utf8]{inputenc} \usepackage{enumitem} \newlist{induction}{enumerate}{4}%% up to 4 levels for this new list \setlist[induction]{wide=0pt, leftmargin =3em} \setlist[induction, ...

2

\csname endalignat*\endcsname expands to the command \endalignat* that you can't normally type. So \foo here is a helper macro that makes the environment definition that you intended except that the command names are passed as arguments #1 and #2, then \expandafter is used to make the command names socthe construction is the same as ...

1

Not everything that's possible with word processing programs is necessarily good typography. Here's a version with exaggerated letter spacing. \documentclass{article} \usepackage{enumitem} \usepackage{showframe} % just for the example \newlist{mylist}{itemize}{1} \setlist[mylist]{ leftmargin=4cm,% adjust to suit labelwidth=4cm,% the same as leftmargin ...

1

Using the idea of @Harish Kumar i tried to implement something like this \documentclass{article} \usepackage{enumitem} \newlength{\fwidth} % Width of the label column \setlength{\fwidth}{2.5cm} % Fix width of label \newlist{mlist}{itemize}{1} \setlist[mlist]{leftmargin=*,labelsep=0pt} \newenvironment{mylist}[2][\fwidth]% ...

1

You can use the environ package instead of comment to discard the content of an environment. \documentclass{article} \usepackage{etoolbox} \usepackage{environ} \newtoggle{lecture} \NewEnviron{myitemize}{% \iftoggle{lecture} {\begin{itemize}\BODY\end{itemize}} {}% } \begin{document} \toggletrue{lecture} The environment is visible ...

1

1

\documentclass{article} \usepackage{enumerate} \newcommand\newEnum[1]{% \newenvironment{enum#1}{\enumerate[#1.]}{\endenumerate}} \newEnum{1} \newEnum{a} \begin{document} \begin{enumerate} \item foo \item bar \end{enumerate} \begin{enum1} \item foo \item bar \end{enum1} \begin{enuma} \item foo \item bar \end{enuma} \end{document} However, using ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Publications with A. Kolesnikov

#### All publications by Alexei S. Kolesnikov and S. Shelah

Click the Sh:-number to get to the papers' detail page (which may include pdf's of the paper).
Can't find an Sh:-number (in particular, an "F-number")? You can try your luck here.
number title
Sh:927 Baldwin, J. T., Kolesnikov, A. S., & Shelah, S. (2009). The amalgamation spectrum. J. Symbolic Logic, 74(3), 914–928.


text:
# Can recursion be replaced with a finite set of higher-order functions? [closed]

I am wondering if there is some proof that all recursive algorithms can be rewritten to use some known set of higher-order functions instead of recursion. I'm talking about functions like fold, map, filter, etc.

If recursion can always be replaced, I'm also interested in knowing whether there is some mechanical translation, or if it's something that can't be automated.

Apologies if this question is ill-formed. I'm no PLT expert.

• Are you looking for Kleene normal form? – François G. Dorais Dec 9 '16 at 1:41
• You could always use the Y combinator which is sufficient to simulate all recursion... – cody Dec 9 '16 at 3:00
• Does it matter that fold, map, filter, etc. are themselves defined by (structural) recursion? And would you consider it cheating if I suggested that all you need is fix : (t -> t) -> t which computes fixed points? – Andrej Bauer Dec 9 '16 at 7:10
• Can we migrate this to cs.computerscience.com? – Andrej Bauer Dec 9 '16 at 7:15
• I am not sure about Kleene normal form, but the others were correct to guess that I was lacking basic knowledge about fixed-point combinators. Incidentally, I don't care whether or not the higher order functions are defined by structural recursion. But I'm going to go away and read more before thinking about this. Thanks guys. – nepp Dec 9 '16 at 16:42

You may replace recursion with computation of fixed points, so all you need is a fixed-point operator

fix : (a -> a) -> a

which satisfies

fix f = f (fix f)

Then, given a recursive definition of f : a -> b,

f = Φ(f)

we may define f as

f = fix (λg.Φ(g))

Here fix has type ((a -> b) -> (a -> b)) -> (a -> b).

You may consider this to be cheating, but the question is not really research-level precisely because the connection between recursion and fixed points is a basic piece of knowledge. I am guessing that maybe you did not know about the connection.

• Thank you! I studied this about 8 years ago, and since I haven't looked at he subject since I had indeed forgotten the connection. Maybe I'll return when I've figured out what I'm really trying to ask, but this answer is definitely the starting point I was looking for. – nepp Dec 9 '16 at 16:35


text:
# Confusion understanding the torque in FBD?

I am confused by the sign and direction torques and forces in linked/attached photo especially that the gravitational force acting on the counter weight on left is having positive sign in torque equation below and gravitational force acting on the propeller actuator on the right has negative sign in below torque equation?

Convention is to use the right hand rule. Your right thumb points in the direction of the torque (moment) vector produced by the rotation associated with the curl of your fingers. Thumb pointing out of the page is positive torque due to a counter clockwise rotation.

But as already noted it’s really arbitrary. As long as you are consistent when setting the sum of the moments equal to zero, the results will be the same.

Hope this helps.

It looks as though the torques about $$O$$ are being considered.

A convention has to be decided on and with no further information it looks as though anticlockwise torques are defined to be positive and clockwise torques are negative.

Or

You can choose arbitrary direction for the sum of the torques.

$$\sum \tau_i=0$$

The effect of torque is rotation. Although the forces are in same direction, but since they are acting on opposite ends about O, their effect of rotation is opposite. So if torque of one is positive, the other has to be negative. Usually, CCW is chosen as positive.


text:
Punctuate

Posted on: May 31st, 2013 by

Counterintuitive probability

Posted on: May 31st, 2013 by
2

1) Jack has 2 kids. One of them is a girl. What is the probability that the other kid is also a girl? 2) Jack has 2 kids. At least one of them is a girl. What is the probability that both the kids are girls? via EE quals questions.

Which way is the bus travelling?

Posted on: May 30th, 2013 by

From the picture below, can you tell whether the bus is travelling to the left or to the right? Preschoolers can get this puzzle. Can you? via Thomas Cover and via this link

Weight of watermelon

Posted on: May 29th, 2013 by
1

A counter-intuitive puzzle: A watermelon weighs 1kg. It contains 99% water. You let some water in the watermelon evaporate until the watermelon contains 98% water. What is the weight of the watermelon after the evaporation? Try to guess the answer before solving for it. via Counter-intuitive Conundrums

Rational approximations

Posted on: May 28th, 2013 by

Given positive integers h,k and a real $\epsilon$>0 show that there exists positive integers m and n so that

. -- via the whiteboard on 2nd floor of Packard.

A Prime numbers puzzle

Posted on: May 27th, 2013 by
1

Let p(n) denote the number of primes less than n. Show that there are infinitely many n for which n is divisible by p(n). via the white board on 2nd floor of Packard.

Probability and maximum entropy

Posted on: May 26th, 2013 by

Information theorists may already be familiar with this, but it's an interesting puzzle for non-information-theorists. Here are two questions: First an easy one: Suppose you roll a coin 1000 times. What is the conditional probability that the first roll is heads given that you observe 700 heads out of the 1000 rolls? Next a hard one: Suppose you roll a die (faces numbered 1 through 6) 1000 Continue reading the story "Probability and maximum entropy"

How to hunt a zombie

Posted on: May 25th, 2013 by
1

A zombie moves along a 1 dimensional line. It starts at position x, and at each time step, it moves y steps to the right. x and y are integers unknown to you. (If y is negative, it means the zombie moves -y steps to the left.) After each time step, you are allowed to examine any integer position and ask if the zombie is Continue reading the story "How to hunt a zombie"

Find the range

Posted on: May 24th, 2013 by
Recently, a chinese mathematician made progress towards the twin prime conjecture. For an informal introduction, read it in the news here. I cannot resist being formal, so let me give a puzzle and explain the conjecture as well: Let $p_n$ be the sequence of prime numbers and $q_n=p_{n+1}-p_n$. Exercise: Show that $\limsup_{n\rightarrow \infty} q_n = \infty$. Now the twin prime conjecture states that $\liminf_{n\rightarrow \infty} q_n = 2$. We Continue reading the story "Twin prime conjecture."


text:
# Equivalent definitions of disconnected sets

My teacher works with two definitions of disconnected sets in $\mathbb{R^n}$:

1) $\Omega \subset \mathbb{R^n}$ is disconnected iff there exists open sets $A,B$ such that: 1) $A\cap \Omega \neq \varnothing$ ; $B\cap \Omega \neq \varnothing$ 2) $A\cap B = \varnothing$ ; 3)$\Omega \subseteq A\cup B$

2)$\Omega \subset \mathbb{R^n}$ is disconnected iff there exists relatively open sets $A´,B´$ to $\Omega$ such that: 1) $A´ \neq \varnothing$ ; $B´\neq \varnothing$ 2) $A´\cap B´ = \varnothing$ ; 3) $\Omega = A´\cup B´$

I was trying to prove that this definitions are equivalent, but I was only able to prove that Definition 1) implies definition 2) But I´m having a hard time proving that definition 2) implies definition 1)

My attempt was like this: We know that $A´$ and $B´$ are relatively open to $\Omega$ so $\exists$ open sets $A$ and $B$ such that:

$A´=A\cap \Omega \neq \varnothing$ and $B´=B\cap \Omega\neq \varnothing$ but I don´t know how to prove that this sets $A$ and $B$ are disjoint and $\Omega \subseteq A\cup B$

Any suggestions would be highly appreciated

• Consider using A' - closure B' and B' - closure A' . – William Elliot Sep 18 '17 at 7:04

then there are open disjoint U,V with A subset U $\cup$ V and not empty
U $\cap$ A, V $\cap$ A. Proof upon request.


text:
## Trigonometry 7th Edition

$18^{\circ}$ $48'$
To convert $18.8^{\circ}$ to degrees and minutes, let's break it as- $18.8^{\circ}$ = $18^{\circ}$ + $0.8^{\circ}$ Now on multiplying 0.8 by 60, we have the number of minutes equivalent to 0.8° Therefore $18.8^{\circ}$ = $18^{\circ}$ + $0.8(60')$ $18.8^{\circ}$ = $18^{\circ}$ + $48'$ $18.8^{\circ}$ = $18^{\circ}$ $48'$