text
stringlengths 100
957k
| meta
stringclasses 1
value |
|---|---|
Limiting two-Higgs-doublet models
Journal of High Energy Physics, Nov 2014
We update the constraints on two-Higgs-doublet models (2HDMs) focusing on the parameter space relevant to explain the present muon g −2 anomaly, Δa μ , in four different types of models, type I, II, “lepton specific” (or X) and “flipped” (or Y). We show that the strong constraints provided by the electroweak precision data on the mass of the pseudoscalar Higgs, whose contribution may account for Δa μ , are evaded in regions where the charged scalar is degenerate with the heavy neutral one and the mixing angles α and β satisfy the Standard Model limit β − α ≈ π/2. We combine theoretical constraints from vacuum stability and perturbativity with direct and indirect bounds arising from collider and B physics. Possible future constraints from the electron g −2 are also considered. If the 126 GeV resonance discovered at the LHC is interpreted as the light CP-even Higgs boson of the 2HDM, we find that only models of type X can satisfy all the considered theoretical and experimental constraints.
This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP11%282014%29058.pdf
Alessandro Broggio, Eung Jin Chun, Massimo Passera. Limiting two-Higgs-doublet models, Journal of High Energy Physics, 2014, 58, DOI: 10.1007/JHEP11(2014)058
|
{}
|
# Norms and determinants of quaternionic line bundles
Research paper by Johannes Cleven
Indexed on: 01 Jul '98Published on: 01 Jul '98Published in: Archiv der Mathematik
#### Abstract
We investigate the relationship between the determinants det (P) and norms N ( P ) of right modules P of rank one over associative composition algebras C. We show: $${\rm det} (P) \cong N(P) \otimes N(P)$$ if C is a quaternion algebra, and $${\rm det}\,(P) \cong N(P) \otimes {\rm det}\, (C)$$ if C is a quadratic étale algebra.
|
{}
|
What techniques exist to show that a problem is not NP-complete? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:31:49Z http://mathoverflow.net/feeds/question/9221 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9221/what-techniques-exist-to-show-that-a-problem-is-not-np-complete What techniques exist to show that a problem is not NP-complete? Qiaochu Yuan 2009-12-18T01:44:23Z 2009-12-18T09:30:55Z <p>The standard way to show that a problem <strong>is</strong> NP-complete is to show that another problem known to be NP-complete reduces to it. That much is clear. Given a problem in NP, what's known about how to show that it is <strong>not</strong> NP-complete? </p> <p>(My real question is likely to be inappropriate for this site for one or more reasons; I'm curious about what a proof that factoring isn't NP-complete would look like.)</p> http://mathoverflow.net/questions/9221/what-techniques-exist-to-show-that-a-problem-is-not-np-complete/9225#9225 Answer by Greg Kuperberg for What techniques exist to show that a problem is not NP-complete? Greg Kuperberg 2009-12-18T02:06:24Z 2009-12-18T03:13:55Z <p>There are much stronger versions of the P vs. NP conjecture that complexity theorists often take as axioms, and that imply that many problems are not NP-complete. The most standard class of assumptions is the conjecture that the NP hierarchy does not collapse. You can define NP as the analysis of polynomially bounded solitaire games with complete information. For instance, generalized Sudoku is such a game and it is known to be NP-complete. The nth level of the polynomial hierachy $\Sigma^n P$ and $\Pi^n P$ can be similarly defined by games with $n$ half moves. For instance, suppose that in generalized Sudoku, there is an initial board, and I can first try to make you lose by filling in some of the squares, with restrictions. (Like say only the "red" squares, and only with certain numbers.) After that you can move. Then whether I can win is a natural problem in $\Sigma^2 P$.</p> <p>In particular, the assertion that the polynomial hierarchy PH does not collapse implies that NP does not equal co-NP. If a problem is both in NP and co-NP, it cannot be NP-complete without collapse. (Proof: If solitaire were equivalent to co-solitaire, than a game with two half moves would also be equivalent to solitaire.) A good near example is the graph isomorphism problem, which is in NP and co-AM. AM is like NP but with randomness; it is the model in which Arthur gets an adaptive proof in response to a randomized question and becomes statistically convinced. AM is not quite NP, but it is conjectured to be the same. So if you put two standard conjectures together, graph isomorphism is not NP-complete either. <strong>Edit:</strong> Ryan and Harrison both point out that <a href="http://portal.acm.org/citation.cfm?id=31202" rel="nofollow">Boppana, Håstad, and Zachos</a> proved that if NP is contained in co-AM, then PH is contained in $\Sigma^2 P$. I.e., the hierarchy would collapse at the second level whether or not AM = NP. In particular this applies to graph isomorphism.</p> <p>Problems in BQP, such as factoring, are strongly suspected not to be NP-complete either, but it is an open problem to show that that would imply that the polynomial hierarchy collapses. However, decision-based problems from factoring, such as whether a number is square-free, are known to be in both NP and co-NP. This was known earlier, but it follows particularly quickly from the fact that primality is in P, since that certifies a factorization. <strong>Addendum:</strong> Certification of factorization is equivalent to <a href="http://en.wikipedia.org/wiki/Primality%5Fcertificate" rel="nofollow">certification of primality</a>, which as David pointed was first proved by Vaughan Pratt.</p> http://mathoverflow.net/questions/9221/what-techniques-exist-to-show-that-a-problem-is-not-np-complete/9226#9226 Answer by David Eppstein for What techniques exist to show that a problem is not NP-complete? David Eppstein 2009-12-18T02:10:59Z 2009-12-18T02:10:59Z <p>The only sure way to show that a decision problem is not NP-complete is to prove that its answer is yes for all instances or no for all instances. Everything else depends on the assumption that P ≠ NP, because if P = NP then every nontrivial decision problem is NP-hard.</p> <p>That said, standard ways of convincing theoretical computer scientists that a problem is unlikely to be NP-complete are to find a polynomial-time algorithm for it, or to show that the complementary problem belongs to NP. Famously, for testing whether a number is composite (clearly in NP: just exhibit a factorization), Vaughan Pratt's paper "Every prime has a succinct certificate" (SIAM J. Comput. 1975) showed that it was unlikely to be NP-complete long before it was shown to be in P.</p> <p>Actually, now that I think about it, there is another way of proving that a problem is not NP-complete: prove that it's not in NP, for instance by showing that it's complete for a higher complexity class like NEXP that is known not to collapse down to NP.</p> http://mathoverflow.net/questions/9221/what-techniques-exist-to-show-that-a-problem-is-not-np-complete/9228#9228 Answer by Ryan O'Donnell for What techniques exist to show that a problem is not NP-complete? Ryan O'Donnell 2009-12-18T02:15:19Z 2009-12-18T02:15:19Z <p>There are several results along these lines known, all of which use one technique: You show that if the problem is NP-complete, then some very strongly believed complexity hypothesis fails. In the following explanations, an asterisk* means "unless an extremely strongly-believed complexity hypothesis (e.g., P $\neq$ NP) fails".</p> <p>Factoring is known to be not NP-complete.* One has to be slightly careful with Factoring for a technical reason: in its most natural version ("Given a number, factor it") it is not a "decision problem". A standard decision problem version is: "Given n, L, and U, is there a prime factor of n between L and U?" This is easily seen to be in NP -- a witness is just a factor of n between L and U. On the other hand, this problem is not* NP-complete because it is also in coNP: there is a witness that proves n has no prime factor between L and U, namely a prime factorization of n. So if Factoring were NP-complete then all problems in NP would be in coNP; i.e., NP=coNP. So the asterisk in this paragraph refers to the assumption NP $\neq$ coNP, which is extremely strongly believed.</p> <p>The Graph-Isomorphism problem is a more interesting example. Telling if two given graphs are isomorphic is obviously in NP (the witness is the isomorphism). But in addition, Graph-<em>Nonisomorphism</em> is "almost" in NP as well. Specifically, it is in the class AM, which is essentially "randomized NP". There is a constant-round randomized "interactive proof" that two graphs are not isomorphic. (Basically, if you put the two graphs behind your back, randomly relabel them, then show them to a prover and the prover can always tell you which is which, then you become convinced the graphs must be non-isomorphic.) From this it follows that Graph-Isomorphism is not NP-complete.* Because if it were, Graph-Nonisomorphism would be coNP-complete, and then all coNP problems would be in AM. And this is known to imply that "the polynomial time hierarchy collapses" (the Boppana-Hastad-Zachos Theorem), which is very widely believed not to happen.</p> <p>(By the way, I assumed you were mostly interested in problems that aren't NP-complete because they are (presumably) too easy. In the other direction, there are also problems that shouldn't be NP-complete because they are too hard; e.g., $\Sigma_2$-complete problems like, "Given a Disjunctive Normal Form formula $\phi$ and a number $s$, is there an equivalent DNF formula of size at most $s$?")</p> http://mathoverflow.net/questions/9221/what-techniques-exist-to-show-that-a-problem-is-not-np-complete/9229#9229 Answer by Harrison Brown for What techniques exist to show that a problem is not NP-complete? Harrison Brown 2009-12-18T02:19:28Z 2009-12-18T02:24:51Z <p>Okay, so here is what we <em>can</em> do, and this I'mma put in an answer, because it's something like an answer. If we assume something stronger than P \neq NP, like that the polynomial hierarchy doesn't collapse, then we can prove that some things aren't NP-complete.</p> <p>The best example of such a result (actually, it's the first one I thought of, although in retrospect I've seen the ones other people mentioned too) is that if graph isomorphism is NP-complete, the polynomial hierarchy collapses to the second level. I can't seem to find an online version of the paper in which this is proved (it's by Boppana et al.) but I believe the argument goes more or less as follows: Graph isomorphism is in NP, easily. Graph isomorphism is also in co-AM; if you don't know what this means or haven't seen the proof, basically it means that if you have two graphs and I tell you they're <em>not</em> isomorphic, you can check that I'm telling the truth probabilistically. How: ou secretly choose one of the two graphs, randomly permute its vertices, and send it to me; I tell you which one it is. If the graphs are isomorphic, I won't be able to give you the right answer more than half the time, but if they aren't isomorphic, I can always give you the right answer. (We're assuming here that I have unbounded computational power.)</p> <p>So we apply a derandomization result, to show that co-AM is contained in some level of the polynomial hierarchy (I think the second level), and then we're done.</p> <p>As far as I know this is the only method that gives these kinds of conditional results. The reason it works is that PH is robust against things like randomization, but of course we have to assume stronger conjectures in complexity theory, so there's definitely a tradeoff.</p> <p>Here's something else I just thought of: Is there anything known about what a conditional proof that a certain problem isn't NP-hard <em>can't</em> be? (Along the lines of relativization, etc.) It's not as straightforward as relativization, since all we want is a conditional proof, but it seems like an interesting question.</p> http://mathoverflow.net/questions/9221/what-techniques-exist-to-show-that-a-problem-is-not-np-complete/9253#9253 Answer by Ryan Williams for What techniques exist to show that a problem is not NP-complete? Ryan Williams 2009-12-18T09:30:55Z 2009-12-18T09:30:55Z <p>Here's one more pointer. Ladner's theorem states that if $P \neq NP$ then there is a problem in $NP$ which is not $NP$-complete. </p> <blockquote> <p>Richard E. Ladner: On the Structure of Polynomial Time Reducibility. J. ACM 22(1): 155-171 (1975)</p> </blockquote> <p>Unfortunately the known proofs of this result produce a fairly unnatural problem. The zombie-metaphorical explanation is you take the head and torso of an $NP$-complete set and surgically stick on arms and legs of a polynomial time computable set. (If that ain't unnatural, I don't know what is.)</p> <p>See <a href="http://oldblog.computationalcomplexity.org/media/ladner.pdf" rel="nofollow">this writeup</a> for two proofs.</p>
|
{}
|
## Discussion Forum
### Bavmorda's Hat - DMS Discussion
Bavmorda's Hat - DMS Discussion
From the week of October 30 to November 3 we chose Wednesday's problem (that you can find here):
Bavmorda, the witch, is out shopping for fabric. She needs to make a new hat for herself, as her old one got caught in a cauldron explosion. To build her hat she will need to create a cone with no base and will add to it a ring of fabric to make the flap. She needs the cone to have a diameter of $5$ inches to fit her head comfortably, and she wants the ring around the hat to be $4$ inches wide. If the hat has to be $10$ inches tall, how many square inches of fabric should Bavmorda buy?
Recall that the formula for surface area of a cone with no base is $\pi rs$, where $r$ is the radius of the base and $s$ is the length from the edge of the circle to the tip of the cone. Use $\pi=3.14$ and round your answer to the nearest tenth if necessary.
25% of the students who tried the problem got it right on their first attempt.
The problem wants us to find how much fabric should Bavmorda buy, that is, we want to find the surface area of the cone (with no base) and the surface area of the ring that goes around the hat.
We are provided with a formula for the surface area of the cone portion of the hat, however, we cannot use it right away as we do not know the length from the edge of the circle to the tip of the cone. A common mistake was to use $s = 10$ inches in the formula. To figure out missing information it is useful to make a diagram. In this problem a diagram of the side view of the hat helps:
We can see then it would be possible to find the missing information by using the Pythagorean Theorem. Afterwards we just need to be extremely careful plugging in the values of $r$ and $s$ in the formula. What else do we need to do to finish the problem?
Share your thoughts and questions below!
|
{}
|
Home > Department of Mathematics > Seminars > Mini-Conference on Infinite Combinatorics 2012
Department of Mathematics
Columbia House
London School of Economics
Houghton Street
London WC2A 2AE, UK
Email: maths.info@lse.ac.uk
Tel: +44(0)207 955 7732/7925
Connect with LSE Maths:
Click here for information on how to get to LSE using public transport and maps of the campus
# Mini-Conference on Infinite Combinatorics 2012
Page Contents >
## Conference theme
Steinhaus's sum-theorem for large' sets is at the heart of several strands of infinite combinatorics and of related finitary combinatorics, particularly Fraïssé direct and inverse limits. These strands include additive combinatorics, e.g. Ruziewicz-van der Waerden theorems, automatic continuity theorems (for measurable homomorphisms), and reduction of continuous to sequential properties. Probability theory is an end-user (e.g. regular variation, Kingman's discrete skeleton and extreme value theories). Generalizations of Steinhaus' result range from continuous groups to discrete semigroups involving notions such as amenability at 1', idempotent ultrafilters, and recently shift-compactness of group actions and refinement topologies. (The latter notions may be viewed as strengthened-form Baire Category theorems and can explain en route some of the measure-category duality via density topologies.) A model-theoretic approach to group action relies on generic and mutually generic automorphisms.
This Mini-Conference in Infinite Combinatorics was aimed at bringing together mathematicians researching in fields where a common underpinning may be seen to be a theorem of Stainhaus. The organiser (Prof. Adam Ostaszewski) came to this field several years ago, motivated by research questions on regular variation arising in Probability Theory, and established the connection with several fields: infinite combinatorics (Ramsey Theory), general topology, group theory and descriptive set theory. Speakers were invited to represent these different aspects and the organiser was very pleased to create a platform for further research.
Where possible, slides and recordings of the day's presentations are made available below.
## Speakers
Nick Bingham, Imperial College / LSE
Title: Topological Regular Variation
Abstract:
This talk aims to motivate first the classical theory of regular variation (see e.g. the book of that title, Bingham-Goldie-Teugels, 1987/89), and its more recent topological counterpart (see e.g. the 17 Bingham-Ostaszewski papers, 2008-11, all on MathSciNet, plus Ostaszewski papers, most on Math-SciNet). The contents are illustrated by the section headings:
1. Why regular variation?
2. Extreme-value theory (EVT)
3. Regular variation; BGT
4. Bitopology [Euclidean and density topologies]
5. Measure-category duality
6. Analytic sets
7. Groups and actions
8. Innite dombinatorics
9. Logical assumptions
Title: Steinhaus’ Theorem and its descendants
Abstract:
The classical real-line Steinhaus sum-theorem that A+A contains an open set when A is large in the sense of measure or category, has generated a corpus of work which will be recalled and continues to inspire, among them results on automatic continuity in group theory and open mapping principles in analysis. Such results may be deduced from the notion of shift-compactness associated with group action. In the presence of almost completeness this property follows from the separation of points from closed nowhere dense sets by appropriate group members. Variant forms of separation may be deduced by passing to a refinement topology; thus the density topology permits measure results to follow from category results. Analyticity provides a useful mild form of completeness. Mention will be made of the relationship between this approach and the use of generic automorphisms in the sense of Truss.
Dona Strauss, Hull
Title: Chains of idempotents in beta-N (Joint work with N. Hindman and Y. Zelenyuk)
Abstract:
The properties of idempotents in $\beta S$ play a significant role in combinatorics. They have often provided surprisingly short proofs of important theorems - Hindman's Theorem and the Hales-Jewett Theorem are examples - and they have suggested new theorems. I shall describe what we know about idempotents in $\beta N$ and talk about a new result obtained by N. Hindman, Y. Zelenyuk and myself about the existence of decreasing chains of idempotents in $\beta N$. We showed that, for every non-minimal idempotent $p$ in $\beta N$ and every countable ordinal $\alpha$, there is a decreasing chain of idempotents in $\beta N$ indexed by $\alpha$, with $p$ as the maximum element. Whether there are any uncountable chains of idempotents in $\beta N$ is an open question.
An audio recording (with accompanying screenshots) of this presentation is available here; users can watch the video on a full screen by clicking on the button to the left of the volume control.
Sławomir Solecki, University of Illinois at Urbana-Champaign
Title: An abstract approach to Ramsey theory with applications
Abstract:
We give an abstract approach to finite Ramsey theory and prove a general Ramsey-type theorem. We deduce from it a self-dual Ramsey theorem, which is a new result naturally generalizing both the classical Ramsey theorem and the dual Ramsey theorem of Graham and Rothschild. In fact, we recover the pure finite Ramsey theory from our general Ramsey-type result in the sense that the classical Ramsey theorem, the Hales-Jewett theorem (with Shelah's bounds), the Graham-Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and the new self-dual Ramsey theorem are all obtained as iterative applications of the general result.
An audio recording (with accompanying screenshots) of this presentation is available here; users can watch the video on a full screen by clicking on the button to the left of the volume control.
Title: Sparse Pairwise Sums
(no slides available)
Abstract: Hindman's celebrated theorem states that, whenever the natural numbers are finitely coloured, some colour class contains all the finite sums from a sequence. What would a sparse' version of this be? One obvious question would be: does there exist a subset S of the natural numbers that is so rich that whenever it is finitely coloured then some colour class contains all the singleton and pairwise sums from a sequence, and yet so sparse that it does not contain all the finite sums from any sequence?
This is the simplest possible' sparse infinitary question, and it has remained open through the years. Amusingly, a published conjecture of Nesetril and Rodl would imply that it is true while a published conjecture of Hindman would imply that it is false.
In this talk, we will look at some background and recent developments on this problem.
An audio recording of this presentation is available here.
Support from the London Mathematical Society and from the British Combinatorial Committee is gratefully acknowledged.
Share:|||
Hugo Steinhaus
|
{}
|
# Reflection in CPP
It this cursed? Is this totally valid? Only way to know is to use it in prod.
Recently at work, we came across an issue that felt easy, but in cpp is not. We have a collection of classes that need to define a method to serialize themselves to json, and we wanted to be sure that all fields in the class were accounted for. Of course, we know better than to trust ourselves, so we were trying to think of automated ways to enforce this property.Unfortunately, our codebase is in C++, and since there’s no mechanism for reflection, it’s simply impossible. In the end, we decided not to enforce it in any way, but it got my mind churning - surely someone else has solved this problem right?
And they have - far better than the method I’m about to detail below, but there’s projects like tsmp which are built for exactly this. tsmp is implemented as a clang tool, which is cool, but I felt like this could be done with my all time favorite C/C++ feature - the preprocessor. So I played around with some sample programs, and it’s actually not that bad!
Before I show my solution to this, you’ll need to understand a relatively common C/C++ preprocessor technique - x lists.
## X Lists
X Lists allows one to define some ordered collection of data, and then operate on it multiple times. It’s basically a way of having a list of syntax fragments and then running for-each over it. The trick is to refer to an undefined macro X which becomes a user-suppliable callback.
For example, let’s say I want to build some kind of interactive CLI interface that controls a counter. There’s three commands:
• quit - closes the program
• inc - increments the counter
• get - prints the current counter value
To implement this, I could use an x-list like the following example
#define CommandList \
X(quit) \
X(inc) \
X(get)
Now, for each of my commands, I define a callback:
size_t counter = 0;
void handle_quit() {
exit(0);
}
void handle_inc() {
counter++;
}
void handle_get() {
printf("The counter is %zu\n", counter);
}
Finally, we get input from the user:
int main() {
while (true) {
// TODO get the appropriate callback
}
}
Now we could write some if statements comparing every line to each of our commands, but that’s tedious (and error-prone). Instead, we can use our X list!
int main() {
while (true) {
if (false) {}
#define X(name) else if (line == #name) { handle_##name(); }
CommandList;
#undef X
}
}
Which expands to:
int main() {
while (true) {
if (false) {}
else if (line == "quit") { handle_quit(); }
else if (line == "inc") { handle_inc(); }
else if (line == "get") { handle_get(); }
}
}
What if we also want to introduce a new command - “help” that prints out unique help text for each option? Well, we can go back and modify our X list:
#define CommandList \
X(quit, "closes the program") \
X(inc, "increments the counter") \
X(get, "prints the current counter value") \
X(help, "list available commands")
And then our main function, to ignore the descriptions:
...
if (false) {}
#define X(name, _) else if (line == #name) { handle_##name(); }
...
And all that’s left now is to add our new callback. Since help needs to list all commands, we can use another X macro!
void handle_help() {
printf("Counter - a simple CLI counter program\n");
printf(" available commands:\n");
#define X(name, desc) printf(" %s - %s", name, desc);
CommandList;
#undef X
}
Which will print:
Counter - a simple CLI counter program
available commands:
quit - closes the program
inc - increments the counter
get - prints the current counter value
help - list available commands
While this could have been done without X lists, it certainly made things easier. For larger programs where you have many more options, the benefits of this approach becomes more apparent.
## Using X Lists to implement Reflection
While thinking about c++’s lack of reflection, I realized that defining types with X lists would give us exactly what we need. We can store the name of a struct field and it’s type, and then retrieve that in other places where we need to iterate over all the members. Let’s look at a small example where we print out the name of every field:
#define DefineField(name, type...) type name;
#define FooFields \
X(x, int) \
X(y, std::string) \
X(z, float)
struct Foo {
// Variadic argument for type in case the expression is some complex template
#define X(name, type...) DefineField(name, type);
FooFields
#undef X
};
Now, to print out every field, we can simply do:
void printFooFields(const Foo& f) {
#define X(name, _) std::cout << #name << " " << f.name << std::endl;
FooFields;
#undef X
}
This way when ever you update the list of members of Foo, you don’t need to remember to update printFooFields
Using this approach, I’ve made a bigger example where I implement a pretty printer for structs, click here to see it!
## Conclusion
This is a viable way to implement reflection in C++, and it’s honestly not even that cursed. However, it does restrict how you define and add member variables to your structs, which I could see becoming messy. I’m not sure I want to introduce this to our codebase at work, but I could see myself doing this for smaller projects and being perfectly happy with it.
Written on February 23, 2023
|
{}
|
## Topology of real algebraic sets.(English)Zbl 0808.14045
Mathematical Sciences Research Institute Publications. 25. New York: Springer-Verlag. x, 249 p. (1992).
The book is devoted to the following fundamental problem of real algebraic geometry: to classify topologically real algebraic sets, to give a topological characterization of all topological spaces which are homeomorphic to real algebraic sets. Main results in this direction achieved at the moment are presented in a comprehensive and self- contained setting. Namely, a complete topological characterization of nonsingular algebraic sets, of algebraic sets with only isolated singularities and of algebraic sets of dimension $$\leq 3$$ is given, as well as the affirmative solution to the Nash conjecture that every smooth manifold in $$\mathbb{R}^ n$$ is smoothly isotopic to a close nonsingular algebraic set. A detailed introduction to the methods used in this topic is done. In particular, the theory of resolution towers is developed, which is a far generalization of resolution of singularities adapted to the problem.
### MSC:
14P25 Topology of real algebraic varieties 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 32C05 Real-analytic manifolds, real-analytic spaces 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 57N80 Stratifications in topological manifolds 58-02 Research exposition (monographs, survey articles) pertaining to global analysis 58A35 Stratified sets
|
{}
|
$$\require{cancel}$$
# 9.4: Summary
• Contributed by Wendell Potter and David Webb et al.
• Physics at UC Davis
1. Become familiar with the idea of wavefronts and rays.
2. Geometric optics is the approximation that rays always travel in straight lines. This approximation is good provided that the wavelength is much smaller than anything it encounters (i.e. we are neglecting diffraction). The geometric optics approximation allows us to perform ray-tracings to locate images.
3. When a wave encounters an interface between two media, some of the wave's energy can reflect into the original medium while the rest can be transmitted into the new medium. Because the two media have different allowed wave speeds, the transmitted wave is typically deformed, a phenomenon called refraction.
4. The law of reflection states that $$\theta_{inc} = \theta_{ref}$$, where both angles are measured from the normal of the reflecting surface.
5. Objects with rough surfaces have normals that change over their surface. As a result, light incident on rough surfaces is reflected in all directions. This is calleddiffuse reflection.
6. Each non-absorbing material has a refractive index that describes how quickly light travels through it. The higher the refractive index, the slower light travels in that medium. The refractive index in a medium is defined as $$n_{medium} = c/v_{medium}$$, where $$v_{medium}$$ is the speed of the wave in the medium and $$c$$ is the speed of light in vacuum.
7. To find the direction that light bends when refracted, we use Snell’s law $$n_1 \sin \theta_1 = n_2 \sin \theta_2$$. Both $$\theta_1$$ and $$\theta_2$$ are measured from the normal of the interface between the two media.
8. If Snell’s law cannot be satisfied then none of the wave can be transmitted; instead it is all reflected. This phenomenon is called total internal reflection. Total internal reflection can only occur when the light is coming from a faster medium and reaches the boundary between media.
9. Our eyes can only track back the rays that reach our eyes, and so if rays appear to be coming from somewhere then our brain thinks there is an object there. If there is no object there, the object our brain thinks it sees is called an image.
10. Images come in two types: real and virtual.
• A real image is where the light rays actually come to a point and then spread out again. This sort of image can be placed on a screen.
• A virtual image is an image where the light rays do not cross, but our brain traces back the rays and is tricked into thinking that they cross.
11. For thin lenses or particular curved mirrors there is a focal length $$f$$. The relationship between the object distance $$o$$ and image distance $$i$$ is $\dfrac{1}{o} + \dfrac{1}{i} = \dfrac{1}{f}$which is known as the thin lens equation. If $$i$$ is positive,this is a real image; if $$i$$ is negative this is a virtual image.
12. The image of an object is typically a different size than the object. We use the magnification $$m = −i/o$$ to describe the change in size; for example $$m=2$$ means the image is twice as big as the original object. If the magnification is negative then this means the image is inverted.
# Derivations
In this chapter we simply presented Snell's Law, magnification, and the thin lens equation as true instead of deriving these results from what we know. We present the derivations here for the interested reader. While not necessary to learn and apply these equations, understanding these derivations will deepen your understanding of the concepts they reflect.
## Snell's Law
Let us draw both the peaks (as wavefronts) and the rays of light together for light that is traveling from air into water.
Look at the distance between the wavefronts on the boundary, shown as a bold line between the two indicated normals (dashed lines). Let us call this distance $$h$$ for hypotenuse, because it is the hypotenuse of both right-angled triangles indicated in the water and in the air. We know that the distance between the wavefronts (which makes up the opposite side of these triangles) is given by the wavelength in that medium. Writing this out for the triangle in the water we have
$\sin \theta_w = \dfrac{\lambda_w}{h} \implies h = \dfrac{\lambda_w}{\sin \theta_w}$
For the triangle in the air we have a similar relationship:
$\sin \theta_a = \dfrac{\lambda_a}{h} \implies h = \dfrac{\lambda_a}{\sin \theta_a}$
Because we know that the hypotenuse is the same in both of these equations, we are lead to conclude that
$\dfrac{\lambda_w}{\sin \theta_w} = \dfrac{\lambda_a}{\sin \theta_a}$
By multiplying this equation by the frequency $$f$$ and recalling that $$v_{wave} = f \lambda$$ we have
$\dfrac{f \lambda_w}{\sin \theta_w} = \dfrac{f \lambda_a}{\sin \theta_a}$
$\dfrac{v_w}{\sin \theta_w} = \dfrac{v_a}{\sin \theta_a}$
Finally we recall that $$v_a = c/n_a$$ (and a similar result for water) we have
$\dfrac{c}{n_w \sin \theta_w} = \dfrac{c}{n_a \sin \theta_a}$
This results holds if and only if
$n_w \sin \theta_w = n_a \sin \theta_a$
is true. This has been Snell's Law.
## Magnification
Consider an object with height $$h_o$$ and a converging lens that produces a real image with height $$h_i$$. Examine at the principal ray that goes through the center. Because it is a straight line, the gradient (slope) does not change.
We see that the gradient on the left hand side is
$\text{gradient }= \dfrac{\Delta y}{\Delta x} = \dfrac{-h_o}{o}$
We can use the information on the right-hand side to calculate the gradient we get
$\text{gradient }= \dfrac{\Delta y}{\Delta x} = \dfrac{h_i}{i}$
Because this ray does not bend, we know these gradients are the same. Therefore: $\dfrac{-h_o}{o} = \dfrac{h_i}{i}$ Rearranging this equation we have
$m \equiv \dfrac{h_i}{h_o} = -\dfrac{i}{o}$
## The Thin Lens Equation
We will prove the thin lens equation for a converging lens that produces a real image. The other cases can be shown in a similar manner. Examine the ray that enters the lens parallel to the optical axis and refracts through the focal point.
We know that there are two ways of calculating the gradient of the ray that passes through the focal point. The first manipulates the fact that the incoming ray has the same height as the object, and drops to the optical axis within a focal length
$\text{gradient} = \dfrac{\Delta y}{\Delta x} = -\dfrac{h_o}{f}$
The second way of calculating the gradient uses the fact that the height of the ray drops to the location of the image in the distance $$i$$. Because $$h_i < 0$$ we should be slightly careful with the sign of $$\Delta y$$
$\Delta y = y_f - y_i = h_i - h_o \implies \text{gradient} = \dfrac{h_i-h_o}{i}$
We can get rid of $$h_i$$ by applying our equation for the magnification:
$h_i = m h_o = -\dfrac{i}{o}h_o$
Writing out our gradient again we obtain
$\text{gradient} = \dfrac{-\frac{i}{o} h_o - h_o}{i} = - \left( \dfrac{1}{o} + \dfrac{1}{i} \right) h_o$
As a straight line has a constant gradient, the segment we use should not matter. Therefore these two expressions for the gradient must be equal:
$- \left( \dfrac{1}{o} + \dfrac{1}{i} \right) h_o = - \dfrac{h_o}{f}$
Cancelling the $$-h_o$$ from both sides leaves the thin lens equation
$\dfrac{1}{o} + \dfrac{1}{i} = \dfrac{1}{f}$
|
{}
|
## effect of rotational speed on ballmills
#### The effect of ball milling time and rotational speed on ...
Oct 30, 2014 The effect of manufacturing parameters such as the effect of ball milling time and rotational speed on the final composite was analyzed by scanning electron microscopy (SEM), differential scanning calorimetry (DSC), particle size distribution, and contact angle measurements.
#### effect of speed in ball milling
effect of speed in ball milling. Oct , ball milling as a mixing technique for uhmwpe based composites is not a new approach but yet, the effect of time, rotational speed, loading of milling jar, and type of ball mill has not been reported properly for uhmwpe composites with and wt uhmwpemwcnts were manufactured with different rotational speed and mixing times
#### Effects of rotational direction and rotation-to-revolution ...
Jul 01, 2002 The rotational direction of a pot in a planetary ball mill and its speed ratio against revolution of a disk were studied in terms of their effects on the specific impact energy of balls calculated from the simulation on the basis of the Discrete Element
#### Ball Mill Critical Speed
A Ball Mill Critical Speed (actually ball, rod, AG or SAG) is the speed at which the centrifugal forces equal gravitational forces at the mill shell’s inside surface and no balls will fall from its position onto the shell. The imagery below helps explain what goes on inside a mill as speed varies. Use our online formula The mill speed is typically defined as the percent of the Theoretical ...
#### Mill Speed - Critical Speed - Paul O. Abbe
Mill Speed - Critical Speed. Mill Speed . No matter how large or small a mill, ball mill, ceramic lined mill, pebble mill, jar mill or laboratory jar rolling mill, its rotational speed is important to proper and efficient mill operation. Too low a speed and little energy is imparted on the product.
#### how to calculate ball mill rotational speed
The "Critical Speed" for a grinding mill is defined as the rotational speed where centrifugal forces equal gravitational forces at the mill shell's inside surface. This is the rotational speed where balls will not fall away from the mill's shell. Result #1: This mill would need to spin at RPM to be at 100% critical speed
#### What it is the optimun speed for a ball mill ...
Oct 19, 2006 Calculation of optimum speed depends on knowing the media diameter and jar I.D. also. For instance, if your jar had in inside diameter of 90 mm and your milling media was 12.7 mm diameter lead balls, the optimum rotation would be 98 RPM. Optimum RPM= .65 x Critical speed (cascading action of the media stops)
#### How can I determine the best RPM for Dry Ball Milling ...
Because you want the grinding balls to experience a free-fall motion, i.e. cataracting motion, I would recommend you consider a rotational speed between 65 and 85 % of the critical speed of the mill.
#### Factors Affecting Ball Mill Grinding Efficiency
The following are factors that have been investigated and applied in conventional ball milling in order to maximize grinding efficiency: a) Mill Geometry and Speed – Bond (1954) observed grinding efficiency to be a function of ball mill diameter, and established empirical relationships for recommended media size and mill speed that take this factor into account. As well, mills with different ...
#### Effects of rotation speed and outlet opening on particle ...
Sep 01, 2016 4.3. Effect of the rotation speed. Rotation speed of the roller shaft is regarded as an important operational factor in the vertical rice mill . To quantify its effect, the rotation speed of roller shaft is varied from 1200 to 1600 rpm while keeping the outlet opening unchanged.
#### Factors Affecting Ball Mill Grinding Efficiency
The following are factors that have been investigated and applied in conventional ball milling in order to maximize grinding efficiency: a) Mill Geometry and Speed – Bond (1954) observed grinding efficiency to be a function of ball mill diameter, and established empirical relationships for recommended media size and mill speed that take this factor into account. As well, mills with
#### Ball mill - Wikipedia
A ball mill is a type of grinder used to grind or blend materials for use in mineral dressing processes, paints, pyrotechnics, ceramics, and selective laser sintering.It works on the principle of impact and attrition: size reduction is done by impact as the balls drop from near the top of the shell. A ball mill consists of a hollow cylindrical shell rotating about its axis. T
#### rotational speed for ball mill
rotational speed for ball mill. Oct , ball milling as a mixing technique for uhmwpe based composites is not a new approach but yet, the effect of time, rotational speed, loading of milling jar, and type of ball mill has not been reported properly for uhmwpe composites with and wt uhmwpemwcnts were manufactured with different rotational speed and mixing times
#### rotating soeed of a 4 ton ball mill
Jan 11, 2020 dem modeling of ball mills with experimental validation: influence of .25 jul 2016 . despite the study of ball mills by the discrete element method for . i.e. the percentage with respect to the rotation speed for which its charge.calculate ball mill grinding capacity - 911 metallurgist17 mar 2017 . the sizing of ball mills and ball milling circuits from laboratory
#### how speed of the ball mill affect grinding efficiency
Grinding in Ball Mills: Modeling and Process Control - Cybernetics ... Grinding in ball mills is an important technological process applied to reduce the ... ware rate, influence on the particle breakage rate and energy efficiency of the ... The speed of rotation of the mill determines three basic types of operation. Get Price
#### (PDF) Spindle Rotational Speed Effect on Milling Process at Low Cutting Speed
The spindle rotational speed fluctuates during milling due to intermittent cutting forces applied to the spindle, but the speed effect when machining with a relatively large cutter at low cutting ...
#### Fritsch Pulverisette - 7 Premium Line - Sample Preparation - Milling - Planetary Mills - Planetary Micro Ball Mills ...
FRITSCH Planetary Ball Mills – high-performance all-rounder in routine laboratory work. The Planetary Micro Mill PULVERISETTE 7 premium line with 2 grinding stations is designed for a broad range of applications and ideally suited for loss-free grinding down to a final fineness of 100 nm of hard, medium-hard and brittle materials.
#### Effect of rotation speed and welding speed on Friction Stir Welding
Apr 01, 2018 Aluminum AA1100 is the most widely used grade of Aluminium due to its excellent corrosion resistance, high ductility and reflective finish, the selected material was welded with Friction Stir Welding (FSW) process on a CNC machine, using a combination of different tool rotation speed (1500 rpm, 2500 rpm, 3500 rpm) and welding speed (10 mm/min, 30 mm/min,
#### (PDF) Influence of rotational speed on the cyclic fatigue of Mtwo instruments
To evaluate the effect of rotational speed on cyclic fatigue of Mtwo nickel-titanium files. A total of 120 new Mtwo rotary instruments sizes 10, 0.04 taper; 15, 0.05 taper; 20, 0.05 taper; and 25 ...
#### What Impacts Does the Rotation Speed Have on Ceramic Ball
For example, the rotation speed of ceramic ball mill has a direct impact on the stationary condition of the internal grinding media and the effect of grinding operation. When the rotation speed is low, the steel balls acting as the grinding media will only be raised to a low height and then roll down from the top under the effect of their own ...
#### Maximum Rotation Speed In Impact Crushing Machines
rotational speed influence on hammer crusher production. maximum rotation speed in impact crushing machines . ... to speed up main shaft speed and VSI Crusher machine is one of the ... Know More; zimbabwe rotation impact crusher prices. Impact Crusher Operation and Maintenance Tips: Impact crusher is a crushing machine of high rotation speed ...
#### Relative rotation speeds of the planets - Our Planet
Dec 04, 2019 Rotation periods and speeds (at the equator) of Solar System planets. Planet – Rotation Period – Revolution Period – Rotation speed at the equator – Mean orbital velocity around Sun. Mercury – 58.6 days – 87.97 days – 10.83 km/h (6.73 mph) –
#### Analysis of Grinding Rate Constant on a Stirred Ball Mill Using Discrete Element Method Simulation
The results of discrete element method simulation were compared with actual grinding experimental results. The grinding rate constant K can be expressed as K=a exp(bn), where n is the rotation speed. To investigate the correlation between K and the simulation results, a new factor, the calculated force, was defined as F cal =average force acting on a ball coordination
#### Rotational Speed - Physics Video by Brightstorm
The other speed an object has is rotational speed, rotational speed is the number of rotations per time. So let's look at an example, something you'll often be asked is let's say we have 2 points on the record and the record is spinning in certain speed like 33 revolutions per minute okay.
#### how speed of the ball mill affect grinding efficiency
Grinding in Ball Mills: Modeling and Process Control - Cybernetics ... Grinding in ball mills is an important technological process applied to reduce the ... ware rate, influence on the particle breakage rate and energy efficiency of the ... The speed of rotation of the mill determines three basic types of operation. Get Price
#### Effects of rotational speeds on the performance of a ...
The centrifugal pumps usually work at various rotational speeds. The variation in the rotational speeds will affect the internal flow, the external performance, and the anti-cavitation performance of the pump. In order to improve the anti-cavitation performance of the centrifugal pumps, variable-pitch inducers are placed upstream of the impeller.
#### Rotational speed - Wikipedia
Rotational speed (or speed of revolution) of an object rotating around an axis is the number of turns of the object divided by time, specified as revolutions per minute (rpm), cycles per second (cps), radians per second (rad/s), etc. The symbol for rotational speed is ...
#### (PDF) The effects of screw rotation speed on viscosity ...
The effects of screw rotation speed on viscosity, mooney scorch time and dieswell of hot-feed blending rubber-compound prefared by gs/w 250 extruder machine May
#### Speeds and feeds - Wikipedia
Cutting speed. Cutting speed may be defined as the rate at the workpiece surface, irrespective of the machining operation used. A cutting speed for mild steel of 100 ft/min is the same whether it is the speed of the cutter passing over the workpiece, such as in a turning operation, or the speed of the cutter moving past a workpiece, such as in a milling operation.
#### 10.8 Work and Power for Rotational Motion - University ...
The rotational axis is fixed, so the vector r → r → moves in a circle of radius r, and the vector d s → d s → is perpendicular to r →. r →. Figure 10.39 A rigid body rotates through an angle d θ d θ from A to B by the action of an external force F → F → applied to point P .
#### Effects of Modification Techniques on Mechanical ...
Mar 07, 2017 Speed of rotation is a parameter that controls the rate at which centrifugal casting process affects cooling. Some studies have reported that mould rotational speed range of 1200–1500 RPM as the optimum and other relevant processing parameters are pre-heating and pouring temperatures . The effect of CCT on casting can be classified into three ...
#### Effect of Rotation Speed of a Friction Pair on ...
Abstract: In order to analyze the effect of rotation speed of a friction pair on transmission characteristics of a rotating oil film, a 3D computational model of an oil film between a friction pair is built; pressure, temperature, velocity, and torque transferred by the rotating oil film are investigated at rotation speeds of 1000r/min, 3000r/min, and 4000r/m using the computational fluid ...
#### Rotation Angle and Angular Velocity Physics
Note that the speed of a point on the rim of the tire is the same as the speed v of the car. See Figure 4. So the faster the car moves, the faster the tire spins—large v means a large ω, because v=rω. Similarly, a larger-radius tire rotating at the same angular velocity (ω) will produce a greater linear speed
#### physics - How would fast rotation affect gravity ...
Centrifugal acceleration is $\omega^2 r$, so in order to have a centrifugal acceleration of $1/2 g$ at the equator you will need $\omega = \sqrt {g / 2r}$, giving $\omega = 0.000875 \text{ rad/s}$; the planet rotates about 12 times faster than Earth, completing a rotation in just a little under 2 hours.
#### Effect of Rotation Speed on Microstructure and Mechanical ...
In the present study, 1.86-mm-thick steel plates (UNS S32205) were friction-stir-welded at various rotation speeds of 300 to 600 rpm and a constant welding speed of 100 mmmin −1 . The effect of rotation speed on the microstructure and mechanical properties of the welds was researched. The welding temperature was recorded during friction stir welding (FSW), and the microstructure and ...
#### Influence of blade rotation on the lightning stroke ...
where v is the line speed (m/s), w is the rotation speed (r/min), and l is the length of the blade (m). The blade tip speed of an actual 2‐MW wind turbine during nominal operation is approximately 60 m/s. Therefore, the speed (w) of the fan model during simulation is modeled at 450 r/min.However, if the natural wind speed is low, the wind turbine may not reach its rated speed.
#### Effect of the Rotation Speed during Friction Stir Welding ...
Dec 27, 2017 Microstructure and corrosion resistance of a friction stir welded SAF 2707 hyper duplex stainless‐steel are investigated. Friction stir welding is performed at a constant welding speed of 100 mm min −1 and rotation speeds of 200, 300, and 500 rpm using a tungsten‐rhenium‐based tool. Microstructure evolution of the joint is analyzed by metallographic microscope, scanning electron ...
#### The effect of rotation speed on the power consumption and ...
(2020). The effect of rotation speed on the power consumption and cutting accuracy of guided circular saw: Experimental measurement and analysis of saw critical and flutter speeds. Wood Material Science Engineering: Vol. 15, No. 3, pp. 140-146.
|
{}
|
# A condition for a function to be constant
I need to proof this result:
Let $\alpha >1$ and $c\in\mathbb{R}$. If $f:U\subset\mathbb{R}^m\rightarrow\mathbb{R}^n$, U open, satisfies $|f(x)-f(y)|\leq c|x-y|^\alpha$ for every $x$, $y$ $\in U$, then $f$ is constant in every component of $U$.
I just didn't have any idea on how to start it, I'm doing my first multivariable analysis course now!
-
Could you please explicitly state what your question is. – Sasha Mar 25 '12 at 13:51
I need to prove this result. I have no idea how to start solving this. – Marra Mar 25 '12 at 13:54
Ok, I'll do it! – Marra Mar 25 '12 at 14:12
I retracted that downvote and I'll delete the comment. – user21436 Mar 25 '12 at 14:52
We show that $f$ is locally constant. Let $x_0\in U$ that I assume open, and let $r$ such that $B(x_0,r)\subset U$. Then for $y\in B(x_0,r)$ and $n\geq 1$ \begin{align*} |f(x_0)-f(y)|&\leq \sum_{k=0}^{n-1}\left|f\left(x_0+\frac{k+1}n(y-x_0)\right)-f\left(x_0+\frac kn(y-x_0)\right)\right|\\ &\leq \sum_{k=0}^{n-1}\left|x_0+\frac{k+1}n(y-x_0)-\left(x_0+\frac kn(y-x_0)\right)\right|^{\alpha}\\ &=\sum_{k=0}^{n-1}n^{-\alpha}|y-x_0|^{\alpha}\\ &\leq r^{\alpha}n^{1-\alpha} \end{align*} and since $n$ is arbitrary, $f(x_0)=f(y)$ for all $y\in B(x_0,r)$.
I think you meant that $r$ is arbitrary. – Marra Mar 25 '12 at 13:58
@GustavoMarra No, that's $n$ since after I take the limit $n\to \infty$. – Davide Giraudo Mar 25 '12 at 13:59
But $n^{1-\alpha}$ does not converge to zero when $n\rightarrow\infty$. – Marra Mar 25 '12 at 14:06
After bounding, we don't have a sum anymore, and since $1-\alpha<0$, $n^{1-\alpha}$ does converge to $0$. – Davide Giraudo Mar 25 '12 at 14:07
|
{}
|
# Nekrasov integral equation
A non-linear integral equation of the form
$$\phi(x)=\lambda\int\limits_a^b[\phi(y)+R(\lambda,y,\phi(y))]K(x,y)dy,\tag{*}$$
where $R$ and $K$ are known functions, $K$ being symmetric, $\phi$ is the unknown function, and $\lambda$ is a numerical parameter. Integral equations of this type were obtained by A.I. Nekrasov (see [1]) in the solution of problems arising in the theory of waves on the surface of a fluid. Under certain conditions Nekrasov has constructed a solution of \ref{*} in the form of a series in powers of a small parameter; its convergence has been proved by the method of majorants.
Sometimes an equation of the type \ref{*} is called a Hammerstein equation, although Nekrasov [2] published his investigations before A. Hammerstein [3].
#### References
[1] A.I. Nekrasov, "Collected works" , 1 , Moscow (1961) (In Russian) [2] A.I. Nekrasov, Izv. Ivanovo-Vozn. Politekhn. Inst. , 6 (1922) pp. 155–171 [3] A. Hammerstein, "Nichtlineare Integralgleichungen nebst Anwendungen" Acta Math. , 54 (1930) pp. 117–176
|
{}
|
291KiB, 1280x853, 1205278630850.jpg
No.10175641
Who's this faggot supposed to be? I don't remember anyone in the series that looks like him.
|
{}
|
# GROMACS Tutorial 6 -- Excess Chemical Potential of Methane using Test Particle Insertion
In this tutorial, we'll be using test particle insertion (TPI) to calculate the excess chemical potential of methane solvation in water. Most users are unaware that GROMACS has a built-in method for running TPI. This tutorial will not be a comprehensive discussion on the statistical mechanics of TPI, but will address issues when needed. The user is encouraged to seek out scientific resources regarding this method.
TPI involves perturbing some states to some other, very similar states. We will be taking bulk water and inserting a methane particle and measuring the potential energy change from this. There is a statistical mechanical relationship between this change in potential energy and the excess chemical potential. For us, state A is the bulk water system, and state B is the water system with a methane.
With GROMACS you need to run state A as a normal MD simulation. We already did this for our case of bulk water in Tutorial 1. We'll reuse the output trajectory files for inserting the methane.
## Setup
### Create water system
Follow Tutorial 1 to run a system containing TIP4PEW water.
### Add test particle to topology file
Our original topology file just had water. In the new topology file, we simply need to add 1 test particle, and it needs to be the last molecule in the system. We'll use opls_066 for the particle's atom type which is OPLS's united atom methane. Here's what my final topology file looks like (the number of waters will be different for your system):
#include "oplsaa.ff/forcefield.itp"
#include "oplsaa.ff/tip4pew.itp"
[ moleculetype ]
; Name nrexcl
Methane 3
[ atoms ]
; nr type resnr residue atom cgnr charge mass
1 opls_066 1 CH4 C 1 0 16.043
[ System ]
Methane in water
[ Molecules ]
SOL 395
Methane 1
### Add test particle to gro file
You also need to add the test particle to the gro file. Simply edit conf.gro (or any of the other .gro files uses) and add a line at the end containing the test particle's position (right before the box coordinates). The line I added looks like this:
396CH4 C 1581 0.000 0.000 0.000
The actual position doesn't matter; GROMACSS just wants a placeholder for the test particle. Additionally, you need to add 1 to the total number of particles in the system on the second line of the .gro file.
### Parameter files
We only need one parameter file for TPI. Simply copy prd.mdp from your bulk water simulation and change integrator to tpi. You should change nsteps to the number of insertions per frame that you want to attempt. I chose 100000 steps for my simulation. You will also need to change cutoff-scheme to group, since Verlet has not been implemented for TPI.
## Simulation
For the simulation, we are just rerunning the bulk water simulation using the saved trajectory file (which was named prd.xtc in the first tutorial). To do this first run grompp:
$gmx grompp -f mdp/tpi.mdp -o tpi.tpr -po tpi.mdp -pp tpi.top -c conf.gro Now use the -rerun flag with mdrun: $ gmx mdrun mdrun -s tpi.tpr -o tpi.trr -x tpi.xtc -c tpi.gro -e tpi.edr -g tpi.log -rerun prd.xtc
## Analysis
The log file, named tpi.log in this case, contains a line with the average volume and the average excess chemical potential. My two lines looked like this:
<V> = 1.18704e+01 nm^3
<mu> = 8.81230e+00 kJ/mol
<mu> is output in kJ/mol, but if we convert it to kcal/mol we get 2.106 kcal/mol. This is in line with our results from the free energy of solvation done in Tutorial 4 using the lambda-coupling method where I got 2.289 kcal/mol. The difference can be attributed to the usage of an all-atom model with the free energy of solvation simulations and a united-atom model in this case.
## Summary
In this tutorial, we looked at how to use GROMACS to perform test particle insertion in order to get the excess chemical potential of a united-atom OPLS methane.
Author: Wes Barnett, Vedran Miletić
|
{}
|
# Tietze's extension theorem for contractible manifolds
I've read that the Tietze's extension theorem was still valid for continuous applications from a closed subspace of a normal topological space to a contractible topological manifold (understood as Hausdorf and 2nd countable).
But I can't find any clear reference for this result.
What I have found is that the theorem generalize to applications from a normal space to an Absolute Retract (but for which family ? Normal spaces ? Metric spaces ? both ?), that manifolds are ANR (Abolute Neighbourhood Retract, once again for which family of spaces ?), and that Contractible ANR implies AR. Is this correct ?
Is there a direct proof somewhere that Tietze generalizes to contractible manifolds ?
PS: it is not a research-level question, but I can't get any answer on MathStackExchange... sorry.
Let us say that a topological space $Y$ is a normal absolute extensor if each continuous map $f:Z\to Y$ defined on a closed subspace $Z$ of a normal topological space $X$ extends to a continuous function $\bar f:X\to Y$. By the Tietze Theorem, the real line is a normal absolute extensor and so is the countable product $\mathbb R^\omega$ of real lines and any retract of $\mathbb R^\omega$.
Since each Polish (= separable completely metrizable) space admits a closed embedding into $\mathbb R^\omega$, we conclude that Polish AR's are normal absolute extensors. This fact is explicitely written as Theorem 16.1(d) in the old book Theory of Rectracts'' of S.-T. Hu.
Now it remains to observe that each contractible topological manifold $M$ is a Polish contractible ANR and hence a Polish AR. So, $M$ is a normal absolute extensor.
|
{}
|
# Question
With each hybrid automobile model, prepare a summary that does the following:
1. Lists the statistically significant independent variables (use 95% level of confidence),
2. Interprets the directional of the relationship of each statistically significant independent variable with respect to the preference for the hybrid model concerned,
3. Identifies or distinguishes the relative importance of each of the statistically significant independent variables, and
4. Assesses the strength of the statistically significant independent variables as they join to predict the preferences for the hybrid model concerned.
Sales0
Views22
|
{}
|
# Linebreaks in long character strings
I would like to print a number of 256-bit long hashes in hexadecimal (so 64 characters, I'm skipping the usual "0x" without white spaces or punctuation) in line and using a monospaced font.
So, I first went for \texttt, which does not hyphenate my strings. I saw some questions with interesting answers, such as How to get long \texttt sections to break and Automatic linebreak on specific character but those both define linebreak for a specific character and I would find ugly to use their trick for each of the 16 characters of hexadecimal without some sort of natural loop.
Is there a way to define a command such that the text inside will be typed as texttt but break on any character or, even better, a standard way to typeset long hexadecimal strings that takes this issue into account?
MWE:
\documentclass{article}
\newcommand{\hash}[1]{\texttt{#1}}%In a perfect world, this would be changed to allow linebreaks anywhere in #1
\begin{document}
SHA-256 is a hash function with a 256-bit long output: \hash{d270f747a8743f11aef93c10e9cb6932cc0b862464c1133dc0f8889088740d15}
\end{document}
## 4 Answers
There is already a package for this:
\documentclass{article}
\usepackage{seqsplit}
\newcommand{\hash}[1]{{\ttfamily\seqsplit{#1}}}
\begin{document}
SHA-256 is a hash function with a 256-bit long output:
\hash{d270f747a8743f11aef93c10e9cb6932cc0b862464c1133dc0f8889088740d15}
\end{document}
• David's answer is nice because it shows how stuff works but I have to admit - yours is even more elegant. – Anab Aug 9 '16 at 19:51
• @Anab disaster! :-) – David Carlisle Aug 9 '16 at 20:10
You can add a penalty after each character
\documentclass{article}
\newcommand{\hash}[1]{\texttt{\zz#1\zz}}%In a perfect world, this would be changed to allow linebreaks anywhere in #1
\def\zz#1{%
\ifx\zz#1\else
#1\linebreak[1]\expandafter\zz
\fi}
\begin{document}
SHA-256 is a hash function with a 256-bit long output: \hash{d270f747a8743f11aef93c10e9cb6932cc0b862464c1133dc0f8889088740d15}
\end{document}
• Thanks, this does exactly what I wanted! I am just a bit curious as to how that works: is it a recursion that does nothing if #1 is empty (stopping condition) or else suggests a line break at the current position? – Anab Aug 9 '16 at 18:39
• @Anab it recurses through each character adding a potential breakpoint stopping when it sees the \zz that is placed at the end. – David Carlisle Aug 9 '16 at 18:44
• Thanks, that's what I thought was going on but my TeX was not nearly good enough for me to be sure of it without asking. :) – Anab Aug 9 '16 at 18:49
• @DavidCarlisle Nice. But this \hash{\char"02C6} (or \hash{\char"02C6{}}) breaks it. Is there a way to modify your code to cope with hexidecimal/octal/substitution-denoted characters? – Jonathan Komar Mar 22 '18 at 12:18
• @JonathanKomar ^^^^02c6 is the hex representation of the character, it is a single character token that can be used anywhere a character can be used eg \^^^^02c6 would be a command token with that name. \char"02C6 is completely different it is 6 tokens invoking a non-expandable sequence of instructions which if used in a typesetting context will access the the glyph from the font with number hex 02C6. – David Carlisle Mar 22 '18 at 13:00
Configure it as a URL. EDITED to do it inside a group as \hexdump{}, so that \url remains unaffected by the redefinitions.
\documentclass{article}
\usepackage{url,lipsum}
\urlstyle{rm}
\newcommand\hexdump[1]{%
\begingroup\urlstyle{tt}%
\def\UrlBreaks{%
\do\1\do\2\do\3\do\4\do\5\do\6\do\7\do\8\do\9\do\0\do\A\do\B\do\C\do\D\do\E\do\F}%
\url{#1}\endgroup%
}
\textwidth3.34in
\parskip 1ex
\begin{document}
\noindent\hexdump{5A0FF349ABC5A0FF349ABC5A0FF349ABC5A0FF349ABC5A0FF349ABC5A0FF349ABC5A0FF349ABC5%
A0FF349ABC5A0FF349ABC5A0FF349ABC5A0FF349ABC5A0FF349ABC5A0FF349ABC5A0FF349ABC}
\lipsum[1]
URLs should be unaffected: \url{www.xyz.com}
\end{document}
• Thanks, but I have two small issues with that: though it does the job and listing all the characters is not that long, it still require to do it, and modifies the url command that I use at several other occasions in my document (which, granted, was not specified in my question). – Anab Aug 9 '16 at 18:46
• @Anab I can (and will) resolve the 2nd issue easily enough. I'm not sure what you mean by the first issue. – Steven B. Segletes Aug 9 '16 at 18:56
• What I mean is that, compared to David Carlisle's answer, that I accepted, yours require to a long list of \do (line 5) which, though not that hard to do for 16 characters, feels a bit blunt to me. That's however just a feeling and I've done way uglier at several occasions when I couldn't find an elegant solution quickly enough. ;) – Anab Aug 9 '16 at 19:21
• @Anab I agree David's answer is very elegant. Nonetheless, I have revised to not interfere with underlying \url configuration. – Steven B. Segletes Aug 9 '16 at 19:31
• I'm sticking with the accepted answer for its elegance but I like seeing different ways to do stuff so you got my upvote now that it the rest of my document unaffected. – Anab Aug 9 '16 at 19:40
# Solution Making TeX Treat Characters like Words
This works, but has the limitation that denoted characters do not work. e.g. using hexidecimal (\char"02C6) or octal notation.
\documentclass{article}
\def\hash#1{\xscan#1\relax}% calls xscan which looks ahead one token, #1
\def\xscan{\afterassignment\xxscan\let\token= }% assign single token to \token and call \xxscan
\def\xxscan{%
\ttfamily% set font style
\ifx\token\relax\ttfamily\else%test for end-of-line or end of group and switch to ttfamily
\ifcat\token\space%
\token% token is catcode 10
\spaceskip=.5em% remove glue from space for fixed-width space
\xspaceskip=.5em% remove glue from space for fixed-width space
\else%
\token\hskip 0pt plus 1sp minus 1sp % add glue to any non-catcode 10 (space)
\fi
\expandafter\xscan% feed next token to \xscan, which is effectively a recursive call
\fi}
\begin{document}
SHA-256 is a hash function with a 256-bit long output: \hash{d270f747a8743f11aef93c10e9cb6932cc0b862464c1133dc0f8889088740d15}
%SHA-256 is a hash function with a 256-bit long output: \hash{d270f\char"02C6\relax747a8743f11aef93c10e9cb6932cc0b862464c1133dc0f8889088740d15}% \char" denoted char syntax not supported
\end{document}
# Solution Using Intercharacter Tokens
\XeTeXinterchartokenstate=1 % turn them on
\XeTeXinterchartoks 0 0 = {\penalty0\relax}#1} % use insert token
This is my preferred solution if using TeX Live 2017 and using xelatex. This has the advantage that it supports the original capabilities of \texttt-it should-like denoted characters (see comments under David Carlisle's answer).
\documentclass{article}
\newcommand{\hash}[1]{\texttt{\XeTeXinterchartokenstate=1\XeTeXinterchartoks 0 0 = {\penalty0\relax}#1}}%In a perfect world, this would be changed to allow linebreaks anywhere in #1
\begin{document}
SHA-256 is a hash function with a 256-bit long output: \hash{d270f747a8743f11aef93c10e9cb6932cc0b862464c1133dc0f8889088740d15}
SHA-256 is a hash function with a 256-bit long output: \hash{d270f\char"02C6\relax747a8743f11aef93c10e9cb6932cc0b862464c1133dc0f8889088740d15}
\end{document}
|
{}
|
# Qfunc and erfc in cvx
Can I use qfunc or erfc in cvx? Whenever I use it, it gives an error "Undefined function ‘erfc’ for input arguments of type ‘cvx’."
Is there any way to manually define the function?
No, there is no way to define either of these functions, sorry. The underlying solvers simply cannot handle them. There is, however, a function `log_normcdf` that you may find useful, that uses a polynomial approximation that is reasonably good for `-4 <= x <= 4`.
A full list of the supported functions is in the documentation.
A. I just wonder why we cannot do that. First of all the q-function can be expressed through the exponential functions as in the Craig expression
Thus since CVX can handle the exponential function, we can do the same for q-function as well.
I think that a few code lines defining the q-function based on the exponential functions can do the trick. Is it true?
B. I mean we can do it like that (my MatLab code)
stepsize = 0.001;
ang = 0:stepsize:(pi/2);
[~,sizea] = size(ang);
Now to compute q function at a value x, we can just do
q_approx = stepsize * 1/pi * sum(exp(-x^2./(2*sin(ang).^2)));
We know that exp(-y) is concave, and the sum of concave functions are concave. So q_approx is concave. Note that exp can be expressed in CVX. Thus we can certain add q_approx to the core of CVX as well. Just need to define a precision parameter to specify stepsize.
No. Remember, CVX cannot support arbitrary combinations of the functions in its function list. All combinations must satisfy the disciplined convex programming ruleset. The Q function cannot be represented in this way.
I mean we can do it like that (my MatLab code): stepsize = 0.001; ang = 0:stepsize:(pi/2); [~,sizea] = size(ang); Now to compute q function at a value x, we can just do: q_approx = stepsize * 1/pi * sum(exp(-x^2./(2*sin(ang).^2)));
The sum of exp(-y) is concave -> can add q func to the core of CVX
I’m afraid you’re just going to have to take my word for it. It can’t be done. You can, however, approximate the function with a convex polynomial, if you like, and use the `polyval` function.
|
{}
|
## Cryptology ePrint Archive: Report 2020/557
On the sensitivity of some APN permutations to swapping points
Lilya Budaghyan and Nikolay Kaleyski and Constanza Riera and Pantelimon Stanica
Abstract: We define a set called the pAPN-spectrum of an $(n,n)$-function $F$, which measures how close $F$ is to being an APN function, and investigate how the size of the pAPN-spectrum changes when two of the outputs of a given $F$ are swapped. We completely characterize the behavior of the pAPN-spectrum under swapping outputs when $F(x) = x^{2^n-2}$ is the inverse function over $\mathbb{F}_{2^n}$. We also investigate this behavior for functions from the Gold and Welch monomial APN families, and experimentally determine the size of the pAPN-spectrum after swapping outputs for representatives from all infinite monomial APN families up to dimension $n = 10$.
Category / Keywords: foundations / Boolean function, almost perfect nonlinear (APN), partial APN
Date: received 12 May 2020
Contact author: nikolay kaleyski at uib no
Available format(s): PDF | BibTeX Citation
Short URL: ia.cr/2020/557
[ Cryptology ePrint archive ]
|
{}
|
# Math Help - Reciprocal Graphs: Asymptotes
1. ## Reciprocal Graphs: Asymptotes
Hey guys! I'm having some trouble with quadratic asymptotes. The linear ones I understand just fine.
Where I'm having trouble is finding the points to plot so I can sketch my asymptote on the Cartesian plane.
I would appreciate it if you could help me
Thank You!
*EDIT! I'm looking for the Hyperbola, not the asymptotes!!
2. Originally Posted by Nocholas
Hey guys! I'm having some trouble with quadratic asymptotes. The linear ones I understand just fine.
Where I'm having trouble is finding the points to plot so I can sketch my asymptote on the Cartesian plane.
I would appreciate it if you could help me
Thank You!
Do you have a particular example that we can look at?
--Chris
3. yea I made one up that I can't solve. y=2x^2+9x+6
I have my parabola. The asymptotes are the x-intercepts
I need to sketch the hyperbola by using points.
4. anyone? hyperbola?
|
{}
|
It has been only a couple of days since Apple announced and released Swift. It has possibly been the most important and interesting announcement of Apple in the last few years and started an amount of discussions around the web.
With the hours passing, we've found out that there was at least another Swift in the history of programming languages (with a similar icon) and that Swift is under development since 2010 and its father is Chris Lattner, creator of clang and LLVM.
As we can read from his page, it's no surprise that people have found similarities with Rust, Haskell, Ruby, Python, Julia, and many other language. And the more I play with Swift, the happier I am of what they did. I really hope that the language will be released with an open source license as LLVM, clang and in some sense Objective-C.
But let's come to the hearth of this post. I find abstract types and monadic construct very powerful and fun to code. And the type system in Swift, desite not being as flexible and powerful as Haskell's, gives a lot of freedom in this directon. Optionals fall exactly in this context and correspond very closely to Haskell's Maybe class.
As you can read in Apple Inc. “The Swift Programming Language.” ebook, the optional type in Swift is nothing but synctactic sugar for the following enum
/* Reimplement the Swift standard library's optional type */
enum OptionalValue<T> {
case None
case Some(T)
}
var possibleInteger: OptionalValue<Int> = .None
possibleInteger = .Some(100)
WhereT represents a generic type, nil is indeed syntactic sugar for OptionalValue<T>.None and ? the same for OptionalValue<T>.Some(T).
You can then use a switch and swift pattern matching capabilities to manage the error
switch possibleInteger {
case .Some(let value) :
/* just type 'value' here to see '100' appearing in playground REPL */
println("Value is: \(value)")
case .None :
println("Error: no value.")
}
I am not going in the realm of Monads (yet) and try to explain them, but I think that the either type could be very useful in Swift. As for the optional, we can easily define it as
enum Either<T1, T2> {
case Left(T1)
case Right(T2)
}
As in Haskell, Either is parameterized by two types, not one. A value of the Either type either contains a value of type T1 or of type T2. With it, we can discriminate between two possibilities and using Swift pattern matching we can write a nice, clean code.
Either can be used as a generalization of optional types in which Left not only encodes failure but is accompanied by an error message (so often T1 will be just String). Then Right encodes success and the accompanying value.
For example
var possibleInteger: Either<String,Int> = .Left("Not a number")
var possibleInteger2: Either<String,Int> = .Right(3)
switch possibleInteger1 {
case .Left(let errorText) :
println("Error: \(errorText)")
case .Right(let value) :
println("The value is: \(value)")
}
func safeDiv(x:Float, y:Float) -> Either<String,Float> {
if y == 0 {
return Either<String, Float>.Left("Error: division by zero")
} else {
return Either<String, Float>.Right(x/y)
}
}
In some sense you still have a kind of assert without having to stop the execution and raise and exception. This let's you control certain kind of errors or expected behaviors with much ease.
Additionally, it would be pretty easy to use generics and typealiases to shorten sensibly the notation and clean the code above.
|
{}
|
# Category:Stabilizers
This category contains results about Stabilizers.
Let $G$ be a group.
Let $X$ be a set.
Let $*: G \times X \to X$ be a group action.
For each $x \in X$, the stabilizer of $x$ by $G$ is defined as:
$\Stab x := \set {g \in G: g * x = x}$
where $*$ denotes the group action.
## Subcategories
This category has only the following subcategory.
|
{}
|
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 09 Dec 2019, 06:00
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# All the terms of a certain sequence x1,x2,........xnx1,x2,........xn a
Author Message
Manager
Joined: 05 Oct 2016
Posts: 89
Location: United States (OH)
GPA: 3.58
All the terms of a certain sequence x1,x2,........xnx1,x2,........xn a [#permalink]
### Show Tags
26 Jan 2018, 13:06
2
00:00
Difficulty:
95% (hard)
Question Stats:
36% (02:56) correct 64% (02:36) wrong based on 25 sessions
### HideShow timer Statistics
All the terms of a certain sequence x1,x2,........xn are positive integers. The nth term (n>1) of the sequence is given by the formula:
xn =xn-1 + 1 (If xn-1 is even)
xn =xn-1 + 3 (If xn-1 is odd)
What is the value of x1 + x6?
(1) The second term of the sequence is 3
(2) Two of the first three terms of the sequence are even and add up to 8
--== Message from the GMAT Club Team ==--
THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.
If you would like to discuss this question please re-post it in the respective forum. Thank you!
To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.
Intern
Joined: 09 Jan 2018
Posts: 3
Re: All the terms of a certain sequence x1,x2,........xnx1,x2,........xn a [#permalink]
### Show Tags
26 Jan 2018, 14:11
X2 = x1 + 3, x3 = x2+2 = x1+5, x4 = x3+3 = x2 + 4, x5 = x4+1 = x2+5, x6 = x5+3 = x2+8.
(1) x2 = 3...x1 = 0, x6 = 11. Works.
(2) x1&x2 = even.. Not possible as x2 = x1+odd = odd
x2&x3 = even, X2 = 3,x3 = 5..not possible as both not even
x3 and x1 not possible as both can't be even at the same time. thus not (2)
Sent from my ONEPLUS A5000 using GMAT Club Forum mobile app
Manager
Joined: 05 Oct 2016
Posts: 89
Location: United States (OH)
GPA: 3.58
Re: All the terms of a certain sequence x1,x2,........xnx1,x2,........xn a [#permalink]
### Show Tags
27 Jan 2018, 10:57
DTUguy wrote:
X2 = x1 + 3, x3 = x2+2 = x1+5, x4 = x3+3 = x2 + 4, x5 = x4+1 = x2+5, x6 = x5+3 = x2+8.
(1) x2 = 3...x1 = 0, x6 = 11. Works.
(2) x1&x2 = even.. Not possible as x2 = x1+odd = odd
x2&x3 = even, X2 = 3,x3 = 5..not possible as both not even
x3 and x1 not possible as both can't be even at the same time. thus not (2)
Sent from my ONEPLUS A5000 using GMAT Club Forum mobile app
I am sorry but it's still not clear to me
Retired Moderator
Joined: 25 Feb 2013
Posts: 1159
Location: India
GPA: 3.82
Re: All the terms of a certain sequence x1,x2,........xnx1,x2,........xn a [#permalink]
### Show Tags
27 Jan 2018, 11:10
SandhyAvinash wrote:
All the terms of a certain sequence x1,x2,........xn are positive integers. The nth term (n>1) of the sequence is given by the formula:
xn =xn-1 + 1 (If xn-1 is even)
xn =xn-1 + 3 (If xn-1 is odd)
What is the value of x1 + x6?
(1) The second term of the sequence is 3
(2) Two of the first three terms of the sequence are even and add up to 8
Odd numbers are being added to $$x_{n-1}$$, so if $$x_{n-1}$$ is even then $$x_n$$ is odd and if $$x_{n-1}$$ is odd then $$x_n$$ is even
to know the value of $$x_1+x_6$$ we only need value of $$x_1$$ remaining can be found out through the formula given in the question stem
Statement 1: implies $$x_2=x_n=3=odd$$, so $$x_{n-1}=x_1=even$$. we can substitute the value of $$x_2$$ in the equation $$x_n=x_{n-1}+1$$ to get $$x_1$$. Sufficient
Statement 2: the sequence will have alternate odd and even numbers, so if out of 1st three two are even, this implies $$x_1$$ is even, $$x_2$$ is odd and $$x_3$$ is even
given $$x_1+x_3=8 =>x_1+x_2+3=8$$
or $$x_1+x_1+1+3=8$$, so $$x_1$$ can be calculated. Sufficient
Option D
--== Message from the GMAT Club Team ==--
THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.
If you would like to discuss this question please re-post it in the respective forum. Thank you!
To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.
Non-Human User
Joined: 09 Sep 2013
Posts: 13730
Re: All the terms of a certain sequence x1,x2,........xnx1,x2,........xn a [#permalink]
### Show Tags
28 Nov 2019, 01:13
Hello from the GMAT Club BumpBot!
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: All the terms of a certain sequence x1,x2,........xnx1,x2,........xn a [#permalink] 28 Nov 2019, 01:13
Display posts from previous: Sort by
|
{}
|
Journal article
### Study of forward Z+jet production in pp collisions at $\sqrt{s} = 7$ TeV
Abstract:
A measurement of the $Z(\rightarrow\mu^+\mu^-)$+jet production cross-section in $pp$ collisions at a centre-of-mass energy $\sqrt{s} = 7$ TeV is presented. The analysis is based on an integrated luminosity of $1.0\,\text{fb}^{-1}$ recorded by the LHCb experiment. Results are shown with two jet transverse momentum thresholds, 10 and 20 GeV, for both the overall cross-section within the fiducial volume, and for six differential cross-section measurements. The fiducial volume requires that both ...
Publication status:
Published
Peer review status:
Peer reviewed
### Access Document
Files:
• (Version of record, pdf, 645.1KB)
Publisher copy:
10.1007/JHEP01(2014)033
### Authors
Publisher:
Springer Berlin Heidelberg Publisher's website
Journal:
Journal of High Energy Physics Journal website
Volume:
1
Issue:
1
Article number:
033
Publication date:
2014-01-01
DOI:
EISSN:
1029-8479
ISSN:
1029-8479
Source identifiers:
435922
Keywords:
Pubs id:
pubs:435922
UUID:
uuid:4f6c6478-909f-4ca3-bffd-ea6abc38fb50
Local pid:
pubs:435922
Deposit date:
2014-08-20
|
{}
|
# How do you express f(theta)=sin^2(theta)-3cot^2(theta)+csc^4theta in terms of non-exponential trigonometric functions?
Feb 2, 2016
$f \left(\theta\right) = \frac{1 - \cos \left(2 \theta\right)}{2} + \frac{5 + 3 \cos \left(4 \theta\right)}{3 - 4 \cos \left(2 \theta\right) + \cos \left(4 \theta\right)}$
#### Explanation:
Let me try to do that.
I hope that I have understood your question correctly - if anybody thinks that I have misinterpreted something, please let me know!
I will use the following identities:
$\csc \left(\theta\right) = \frac{1}{\sin} \left(\theta\right)$, $\textcolor{w h i t e}{\times \times} \cot \left(\theta\right) = \cos \frac{\theta}{\sin} \left(\theta\right)$
${\sin}^{2} \left(\theta\right) = \frac{1 - \cos \left(2 \theta\right)}{2}$
${\cos}^{2} \left(\theta\right) = \frac{1 + \cos \left(2 \theta\right)}{2}$
=======================
First of all, let me show the ${\sin}^{2} \left(\theta\right)$ and ${\cos}^{2} \left(\theta\right)$ identities:
1a) Prove $\text{ } {\sin}^{2} \left(\theta\right) = \frac{1 - \cos \left(2 \theta\right)}{2}$
I will use the identities
$\left[1\right] \textcolor{w h i t e}{\times x} \cos \left(x + y\right) = \cos \left(x\right) \cos \left(y\right) - \sin \left(x\right) \sin \left(y\right)$
$\left[2\right] \textcolor{w h i t e}{\times x} {\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1$
Thus, it holds
$\cos \left(2 \theta\right) = \cos \left(\theta + \theta\right)$
$\textcolor{w h i t e}{\times \times x} \stackrel{\left[1\right]}{=} \cos \left(\theta\right) \cos \left(\theta\right) - \sin \left(\theta\right) \sin \left(\theta\right)$
$\textcolor{w h i t e}{\times \times x} = {\cos}^{2} \left(\theta\right) - {\sin}^{2} \left(\theta\right)$
$\textcolor{w h i t e}{\times \times x} \stackrel{\left[2\right]}{=} \left(1 - {\sin}^{2} \left(\theta\right)\right) - {\sin}^{2} \left(\theta\right)$
$\textcolor{w h i t e}{\times \times x} = 1 - 2 {\sin}^{2} \theta$
$\textcolor{w h i t e}{\times}$
$\iff \cos \left(2 \theta\right) - 1 = - 2 {\sin}^{2} \left(\theta\right)$
$\iff 1 - \cos \left(2 \theta\right) = 2 {\sin}^{2} \left(\theta\right)$
$\iff \frac{1 - \cos \left(2 \theta\right)}{2} = {\sin}^{2} \left(\theta\right)$
=======================
1b) Prove $\text{ } {\cos}^{2} \left(\theta\right) = \frac{1 + \cos \left(2 \theta\right)}{2}$
The proof is very similar to the one above:
$\cos \left(2 \theta\right) \stackrel{\left[1\right]}{=} \cos \left(\theta\right) \cos \left(\theta\right) - \sin \left(\theta\right) \sin \left(\theta\right)$
$\textcolor{w h i t e}{\times \times x} = {\cos}^{2} \left(\theta\right) - {\sin}^{2} \left(\theta\right)$
$\textcolor{w h i t e}{\times \times x} \stackrel{\left[2\right]}{=} {\cos}^{2} \left(\theta\right) - \left(1 - {\cos}^{2} \left(\theta\right)\right)$
$\textcolor{w h i t e}{\times \times x} = 2 {\cos}^{2} \left(\theta\right) - 1$
$\iff \cos \left(2 \theta\right) + 1 = 2 {\cos}^{2} \left(\theta\right)$
$\iff \frac{\cos \left(2 \theta\right) + 1}{2} = {\cos}^{2} \left(\theta\right)$
=======================
2) Apply the identities
So, now I will apply the identities and simplify:
${\sin}^{2} \left(\theta\right) - 3 {\cot}^{2} \left(\theta\right) + {\csc}^{4} \left(\theta\right)$
$= {\sin}^{2} \left(\theta\right) - 3 {\left[\cos \frac{\theta}{\sin} \left(\theta\right)\right]}^{2} + \frac{1}{\sin} ^ 4 \left(\theta\right)$
$= {\sin}^{2} \left(\theta\right) - \frac{3 {\cos}^{2} \left(\theta\right)}{\sin} ^ 2 \left(\theta\right) + \frac{1}{{\sin}^{2} \left(\theta\right)} ^ 2$
... combine the two latter fractions into one by determining the least common denominator...
$= {\sin}^{2} \left(\theta\right) - \frac{3 {\cos}^{2} \left(\theta\right) \cdot {\sin}^{2} \left(\theta\right)}{{\sin}^{2} \left(\theta\right) \cdot {\sin}^{2} \left(\theta\right)} + \frac{1}{{\sin}^{2} \left(\theta\right)} ^ 2$
$= {\sin}^{2} \left(\theta\right) + \frac{- 3 {\cos}^{2} \left(\theta\right) {\sin}^{2} \left(\theta\right) + 1}{{\sin}^{2} \left(\theta\right)} ^ 2$
... apply the ${\sin}^{2} \left(\theta\right)$ and ${\cos}^{2} \left(\theta\right)$ identities...
$= \frac{1 - \cos \left(2 \theta\right)}{2} + \frac{- 3 \cdot \frac{1 + \cos \left(2 \theta\right)}{2} \cdot \frac{1 - \cos \left(2 \theta\right)}{2} + 1}{{\left[\frac{1 - \cos \left(2 \theta\right)}{2}\right]}^{2}}$
$= \frac{1 - \cos \left(2 \theta\right)}{2} + \frac{- 3 \cdot \frac{\left(1 + \cos \left(2 \theta\right)\right) \cdot \left(1 - \cos \left(2 \theta\right)\right)}{4} + \frac{4}{4}}{{\left(1 - \cos \left(2 \theta\right)\right)}^{2} / 4}$
... get rid of the double fractions by factoring $\frac{1}{4}$ both in the numerator and the denominator and then canceling...
$= \frac{1 - \cos \left(2 \theta\right)}{2} + \frac{\cancel{\frac{1}{4}} \cdot \left(- 3 \left(1 + \cos \left(2 \theta\right)\right) \cdot \left(1 - \cos \left(2 \theta\right)\right) + 4\right)}{\cancel{\frac{1}{4}} \cdot {\left(1 - \cos \left(2 \theta\right)\right)}^{2}}$
$= \frac{1 - \cos \left(2 \theta\right)}{2} + \frac{- 3 \left(1 + \cos \left(2 \theta\right)\right) \cdot \left(1 - \cos \left(2 \theta\right)\right) + 4}{1 - \cos \left(2 \theta\right)} ^ 2$
... use the formulas $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$ and ${\left(a - b\right)}^{2} = {a}^{2} - 2 a b + {b}^{2}$...
$= \frac{1 - \cos \left(2 \theta\right)}{2} + \frac{- 3 \left({1}^{2} - {\cos}^{2} \left(2 \theta\right)\right) + 4}{{1}^{2} - 2 \cos \left(2 \theta\right) + {\cos}^{2} \left(2 \theta\right)}$
$= \frac{1 - \cos \left(2 \theta\right)}{2} + \frac{1 + 3 {\cos}^{2} \left(2 \theta\right)}{1 - 2 \cos \left(2 \theta\right) + {\cos}^{2} \left(2 \theta\right)}$
... apply the ${\cos}^{2} \left(\theta\right)$ identity again...
$= \frac{1 - \cos \left(2 \theta\right)}{2} + \frac{1 + 3 \cdot \frac{1 + \cos \left(4 \theta\right)}{2}}{1 - 2 \cos \left(2 \theta\right) + \frac{1 + \cos \left(4 \theta\right)}{2}}$
... again, get rid of the double fractions by factoring $\frac{1}{2}$ both in the numerator and the denominator and then canceling...
$= \frac{1 - \cos \left(2 \theta\right)}{2} + \frac{\cancel{\frac{1}{2}} \left(2 + 3 \left(1 + \cos \left(4 \theta\right)\right)\right)}{\cancel{\frac{1}{2}} \left(2 - 4 \cos \left(2 \theta\right) + \left(1 + \cos \left(4 \theta\right)\right)\right)}$
$= \frac{1 - \cos \left(2 \theta\right)}{2} + \frac{2 + 3 \left(1 + \cos \left(4 \theta\right)\right)}{2 - 4 \cos \left(2 \theta\right) + \left(1 + \cos \left(4 \theta\right)\right)}$
$= \frac{1 - \cos \left(2 \theta\right)}{2} + \frac{5 + 3 \cos \left(4 \theta\right)}{3 - 4 \cos \left(2 \theta\right) + \cos \left(4 \theta\right)}$
This is now a non-exponential expression that uses only the $\cos$ function.
Hope that this helped! :-)
|
{}
|
# How do you find the vertex of y=x^2+10x+21?
Mar 26, 2018
$\text{vertex} = \left(- 5 , - 4\right)$
#### Explanation:
$x = - \frac{b}{2 a}$
$x = - \frac{10}{2 \left(1\right)}$
$x = - 5$
Sub $- 5$ into the equation
$y = {\left(- 5\right)}^{2} + 10 \left(- 5\right) + 21$
$y = - 4$
The formula $- \frac{b}{2 a}$ is used to find the axis of symmetry which is
always the $x$ value of the vertex. Once you find the $x$ value of the vertex, you simply substitute that value into the quadratic equation and find the $y$ value, which in this case, is the vertex.
Mar 26, 2018
(-5,-4)
#### Explanation:
You have to use the quadratic formula $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2} a$
which becomes
$x = - \frac{b}{2 a} \pm \left(\frac{\sqrt{{b}^{2} - 4 a c}}{2 a}\right)$
We know that $- \frac{b}{2 a}$ is constant and that the other part is plussing and minusing from it
So it is the vertex and as $a = 1 b = 10 c = 21$ i.e. just the coefficents of all the terms in sequence.
The vertex must be $- \frac{10}{2 \cdot 1}$ so the x co-ordinate of the vertex is $- 5$
Plug in $f \left(- 5\right)$ and you get the y co-ordinate
$f \left(- 5\right) = {\left(- 5\right)}^{2} + 10 \left(- 5\right) + 21$ becomes $f \left(- 5\right) = 25 - 50 + 21$
so $f \left(- 5\right) = - 4$
so the co-ordiantes of the vertex are (-5,-4)
|
{}
|
# need an LCD display
Status
Not open for further replies.
#### MrDEB
##### Well-Known Member
getting really lost as to which one to use for a PIC project.
some have a controller some don't say.
I get impression one with Hitachi HD44780 is pretty much standard?
found a moduel over at goldmine
Hantronix HDM64GS24Y 240 x 64 Graphics LCD Display Module-The Electronic Goldmine
that is only $8.95 everywhere else a 240x64 display is upwards of$100
planning on displaying muzzle velocity and muzzle energy in an air cannon building a chronograph using an 18F452 PIC, 2 ir LEDs and 2 ir sensors
maybe throw in a wind speed measurement?
working on code as well as finding parts.
any input or suggestions would be helpful.
#### Mr RB
##### Well-Known Member
That's a very nice display for \$8.95. Shame it needs an external 100v 400Hz EL backlight supply but you can get little modular EL suplies cheap now anyway.
#### arunb
##### Member
see here www.xmocular.com, they have a good range, I am currently using the 240x128 type LCD
Status
Not open for further replies.
Replies
1
Views
1K
Replies
21
Views
4K
Replies
0
Views
1K
Replies
7
Views
1K
Replies
2
Views
3K
|
{}
|
## Thursday, November 18, 2010
### Myths & Legends: November 18-24
For more information and links to the actual javascript code, see the Myths & Legends Reference Page.
Heracles and the Belt of Hippolyte. To find out more about the labors of Heracles, see this Wikipedia article: link; for information about the image: image source.
Helen and Paris. To find out more about Helen and Paris, the prince of Troy, see this Wikipedia article: link; for information about the image: image source.
Leda and the Swan. To find out more about Leda and Zeus disguised as a swan, see this Wikipedia article: link; for information about the image: image source. There's also a post here.
Polyxena at the Well. To find out more about Achilles and Polyxena, see this Wikipedia article: link; for information about the image: image source. There's also a post here.
Helen and Paris. To find out more about Helen and Paris, the prince of Troy, see this Wikipedia article: link; for information about the image: image source.
|
{}
|
# What is $D_i^h$?
I'm looking at solution seven in here.
They write $v=-D_i^{-h}D_i^h u$ $0<|h|<<1$
What is $D_i^h$? They use $\nabla$ for the gradient, so I don't think it is the $i^{th}$ component of the gradient. I don't think it is $\partial_i$, as in taking $\frac{\partial}{\partial x_i}$ since then I wouldn't know what taking $\partial_i^h$ would possibly mean, since $0<h<1$.
Any ideas?
• It's a discrete difference approximation to the $i^{\text{th}}$ partial derivative of $u$. Jul 17 '17 at 4:43
It's a discrete difference approximation to the $i^{\text{th}}$ partial derivative of $u$.
See $5.8$ of Evans.
|
{}
|
## Monday, December 26, 2022
### Selecting Dispersed Points
Fellow blogger Erwin Kalvelagen posted a comparison of two binary programming models, one quadratically constrained and one linearly constrained, for the problem of selecting a maximal number of points from a finite set subject to the requirement that no two selected points be closer than a specified distance. The models were an answer to a question posted on Computational Science Stack Exchange. Not surprisingly, the linearized version tended to solve faster than the quadratic version.
In Erwin's linear model, the constraints take the form $$\underline{D}(x_i + x_j - 1) \le d_{i,j} \quad (1)$$ where $d_{i,j}$ is the distance between points $i$ and $j$, $\underline{D}$ is the minimum allowable distance between selected points, and $x_k$ is a binary variable indicating whether point $k$ is selected (1) or not (0). I coded both his models in Java, using CPLEX 22.1.1, along with another linear model where the constraints are expressed as $$x_i + x_j \le 1\quad (2)$$ for those pairs $(i,j)$ where $d_i + d_j \le \underline{D}.$ In other words, we exploit the fact that we know the distances at the outset to precompute which pairs of points can / cannot coexist, and just rule out the pairs that cannot. Since (1) is equivalent to $$x_i + x_j \le 1 + \frac{d_{i,j}}{\underline{D}},$$ constraint (2) is clearly at least a bit tighter than constraint (1).
Erwin started with a problem size of 50 points for demonstration purposes, then doubled that to compare timing of this two models. I ratcheted the problem size up to 1,000 points to compare his linear model to mine. (I did not test the quadratic model at that size.) As with all things MIP, the results were not entirely consistent. In limited testing, the model using (2) was usually faster than the model using (1), but occasionally (1) proved faster. The run time differences were not large enough to be exciting. For instance, in one test run version (1) needed 4.633 seconds versus 2.987 seconds for version (2).
Overall, I can't say the time differences lived up to my expectations, and the fact that at least occasionally (1) was faster than (2) (perhaps due to some quirk in presolving, or just to some random choices in branching) is consistent with my experience that MIP behaviors are, well, not consistent.
## Tuesday, December 13, 2022
### Scheduling a Round-Robin Tournament
Once in a while I come across a scheduling problem, and of course my first reaction is to think "integer program". A while back, for example, I used an IP model to work out a schedule for a rotating duplicate bridge game (in which both teams and hosts rotated) at the request of a buddy. Most recently, I saw a question on OR Stack Exchange ("Doubles Round Robin Sorting Algorithm") related to tournament scheduling.
The problem involves six players playing pickleball (so two players on each team, with two sitting out, for each game). Given six players, there are 15 possible teams (pairings). Twelve games are scheduled each week for four weeks, with two sets of six games per week. The author of the question correctly computed that there are 45 possible game configurations. Over the course of the four weeks, he wanted every possible game played at least once (which leaves the final three game slots to be filled arbitrarily). The following conditions must be met:
• within each set of six games, every player plays four games and sits out two;
• no player sits out two consecutive games within the same set;
• no two players are partners more than once per set.
The problem poses no criterion for selecting among feasible schedules; we just want to find any one schedule that works.
My IP formulation can be found in my answer to the OR SE question. I will just mention that it uses binary variables $x_{g,s}=1$ if game $g$ goes in slot $s$ and 0 if not. Games are enumerated up front and indexed from 1 to 15. Slots refer to positions in the schedule and are numbered from 1 to 48, with six slots per set and 12 slots per week. Since no criterion is specified, we just use the default (minimize 0).
I also tried a constraint programming model, using general integer variables $s_1,\dots,s_{48},$ where $s_j \in {1,\dots,15}$ is the index of the game played in slot $j$ of the schedule. My CP formulation uses the "all different" constraint (fairly ubiquitous among CP solvers) to ensure that the first 45 slots each contain a different game. It uses the CP Optimizer "distribute" constraint to ensure that no game is played more than twice and the CP Optimizer "count" constraint to ensure that each player plays exactly four times per set (and thus sits twice) and that no team plays more than once in a set. I'm not sure how common those types of constraints are among CP solvers because I don't have much experience with CP solvers.
As it turns out, both CPLEX (for the IP model) and CP Optimizer (for the CP model) produced feasible schedules in about a second or so on my desktop PC. My intuition was that a CP model would be better suited to this type of problem, because all variables are integer and a CP solver would not be doing much if any matrix arithmetic. As it turns out, it would take a much larger test case to tell whether that intuition has any merit.
If anyone wants to play with the models, my Java code is available from my university GitLab repository. Running it would require CPLEX and CP Optimizer (and not necessarily the most recent versions of either).
|
{}
|
# Inequalities
A rental car agency rents cars for $32 per day. They also charge$0.15 per mile driven. If you are taking a 5-day trip and have budgeted $250 for the rental car, what is the maximum number of miles you can drive and stay within your budget? #### Prove It MHF Helper How much does it cost you just to have the car for 5 days? #### Twtheo44 If it costs$32 a day, and you know that you will have the car 5 days, then you can figure that out first with simply:
5(days) x 32(dollars per day) = $160 Now we see that we have$90 left in our budget. If you have $90 and it costs$0.15 per mile driven and you want M miles, you can set up the equation:
(0.15)M <= 90, where M is miles driven, and 90 is the amount of budget you have left.
If you solve this, you can see that the maximum miles you can drive is 600 miles without going over your budget.
Similar Math Discussions Math Forum Date
Algebra
Algebra
Algebra
Algebra
|
{}
|
# Nonlinear ODE
Logik
## Homework Statement
I have to find the solution of (1) and show that it is not unique if y(0) = 0.
I can prove it is not unique by using Picard's theorem but I don't know how to find the non trivial solution.
## Homework Equations
(1) y(t)' = Sqrt(y(t))
## The Attempt at a Solution
I don't know where to start... We have not seen how to solve nonlinear ODE's. A link to a technique or explanation to how to solve it would be very helpful. I'm not looking for the answer, I can get it with Mathematica... I want to understand how to get there.
StatMechGuy
You can directly integrate that function:
dy/dt = y^1/2 => y^(-1/2) dy = dt
Nontrivial solution. However, you'll find the trivial y(t) = 0 is a perfectly good solution to those initial conditions as well.
Logik
wow I'm so stupid...
dy/dt = y^(1/2)
dy/y^(1/2) = dt
2y^(1/2) = t + c
y^(1/2) = 2t + 2c
y = 4t^2 + 8tc + c^2
thanks
Last edited:
|
{}
|
# Fruit Feast
From USACO 2015 December Contest, Gold:
Bessie has broken into Farmer John’s house again! She has discovered a pile of lemons and a pile of oranges in the kitchen (effectively an unlimited number of each), and she is determined to eat as much as possible. Bessie has a maximum fullness of $T$ $(1≤T≤5,000,000)$. Eating an orange increases her fullness by $A$, and eating a lemon increases her fullness by $B$ (with $1≤A,B≤T$). Additionally, if she wants, Bessie can drink water at most one time, which will instantly decrease her fullness by half (and will round down).
Help Bessie determine the maximum fullness she can achieve!
INPUT FORMAT:
The first (and only) line has three integers $T$, $A$, and $B$.
OUTPUT FORMAT:
A single integer, representing the maximum fullness Bessie can achieve.
## Solution
If Bessie can reach a fullness value of $t$, then she can also reach fullness values of $t+A$ and $t+B$. Since Bessie starts at a fullness of $0$, she can immediately reach fullness values of $A$ and $B$, and then continue from there.
Since we get unlimited uses of oranges and lemons, this is an unbounded knapsack problem. We can solve knapsack problems using dynamic programming. However, we also need to consider whether Bessie drinks water, which will allow her to reach a fullness value of $\lfloor \frac{i}{2} \rfloor$, but unable to drink water again.
We define our subproblem $F(t, w)$ to be whether Bessie can achieve a fullness of $t$, with $w = 0$ without drinking water and $w = 1$ having drunk water. Then our recurrence relation is
But what order should we solve our subproblems in? Notice that all subproblems of the form $F(t, 0)$ depend only on subproblems with fullness values less than $t$. This means we can solve $F(t, 0)$ subproblems by iterating over increasing values of $t$. What about $F(t, 1)$? Like in the case of $F(t, 0)$, $F(t, 1)$ depends on subproblems with fullness values less than $t$, but it also depends on the special cases when Bessie drinks water and halves her fullness. Fortunately, these special cases fall under the $F(t, 0)$ case, which means we can solve the $F(t, 1)$ subproblems if we solve the $F(t, 0)$ first.
Here is an iterative dynamic programming solution in Java:
int fruitFeast(int T, int A, int B) {
boolean[][] full = new boolean[T + 1][2];
full[0][0] = true;
// solve F(t, 0) subproblems first
for (int t = 0; t <= T; t++) {
// F(t, 0) = F(t - A, 0) or F(t - B, 0)
if (t - A >= 0 && full[t - A][0])
full[t][0] = true;
if (t - B >= 0 && full[t - B][0])
full[t][0] = true;
}
// solve F(t, 1) subproblems
// we'll ignore bounds checks for simplicity
for (int t = 0; t <= T; t++) {
full[t][1] = full[t - A][1] || full[t - B][1]
|| full[2 * t][0] || full[2 * t + 1][0];
}
// return the largest value of t that Bessie can reach
for (int t = T; t >= 0; t--)
if (full[t][0] || full[t][1])
return t;
return 0;
}
Here is my original, full recursive dynamic programming solution in C++:
#include <iostream>
using namespace std;
int T, A, B;
bool full[5000010];
void dp(int i, bool drank) {
if (i < 0 || i > T || full[i])
return;
full[i] = true;
dp(i+A, drank);
dp(i+B, drank);
if (!drank)
dp(i / 2, true);
}
int main() {
cin >> T >> A >> B;
dp(0, false);
for (int i = T; i >= 0; i--) {
if (full[i]) {
cout << i << endl;
break;
}
}
}
|
{}
|
# Representable functor is left-adjoint
My proof so far is:
Let us assume that all coproducts exist in $$\mathcal{C}$$ and that $$F:\mathcal{C}\rightarrow\Set$$ is representable, say $$h^X$$ and $$F$$ are naturally isomorphic for some object $$X$$ in $$\mathcal{C}$$. Let us define $$G:\Set\rightarrow \mathcal{C}: S\rightarrow \bigsqcup_{s\in S} X$$. We will show $$G$$ is left-adjoint to $$F$$. By constructing a natural isomorphism $$\alpha: \operatorname{Hom}_\mathcal{C}(G(-),-)\tilde{\Longrightarrow}\operatorname{Hom}_\Set(-,F(-))$$. So for every set $$S$$ and every object $$Y$$ of $$\mathcal{C}$$, we need a bijection $$\alpha_{S,Y}:\operatorname{Hom}_\mathcal{C}(GS,Y)\rightarrow\operatorname{Hom}_\Set(S,FY)$$ Note that since $$F$$ is representable we have an isomorphism from $$\operatorname{Hom}_\mathcal{C}(X,Y)$$ to $$FY$$, say $$\psi$$. We see that there exists an isomorphism (bijection) $$\phi$$ from $$\operatorname{Hom}_\mathcal{C}(GS,Y) =\operatorname{Hom}_\mathcal{C}(\bigsqcup_{s\in S}X,Y)$$ to $$\operatorname{Hom}_\Set(S,\operatorname{Hom}_\mathcal{C}(X,Y))$$. Indeed, let $$\phi$$ map $$f$$ to a function $$f'$$ in $$\operatorname{Hom}_\Set(S,\operatorname{Hom}_\mathcal{C}(X,Y))$$ such that $$f'(s)$$ is a morphism in $$\operatorname{Hom}_\mathcal{C}(X,Y))$$ for which $$f'(s)(x) = f(x^{(s)})$$. The latter notation meaning that $$x$$ belongs to the set $$X$$ corresponding to $$s$$ in the disjoint union. $$\phi$$ is an isomorphism of sets (bijection). It is injective, let $$\phi(f_1) = \phi(f_2)$$, then $$\phi(f_1)(s)(x) = \phi(f_2)(s)(x)$$ for all $$s\in S$$ and $$x\in X$$, so $$f_1(x^{(s)}) = f_2(x^{(s)})$$ for all $$s\in S$$ and $$x\in X$$, which means $$f_1 = f_2$$. $$\phi$$ is also surjective, take $$f'\in \operatorname{Hom}_\Set(S,\operatorname{Hom}_\mathcal{C}(X,Y))$$, then $$f'$$ is the image of a map $$f$$ for which $$f(x^{(s)}) = f'(s)(x)$$. So we conclude the isomorphism $$\alpha_{S,Y}$$ exists, it is the composition of $$(\psi\circ -)\circ \phi$$.
Now take a morphism $$f: S\rightarrow S'$$ in $$\Set$$. Then for $$a\in \operatorname{Hom}_\mathcal{C}(GS',Y)$$ we get $$(\alpha_{S,Y}\circ (-\circ Gf))(a) = \alpha_{S,Y}(aGf)= ((\psi\circ -)\circ \phi_S)(a(Gf)) = \psi\phi_S(aGf)$$ and also that $$((-\circ f)\circ\alpha_{S',Y})(a) = \alpha_{S',Y}(a)f=((\psi\circ -)\circ \phi_{S'})(a)f=\psi\phi_{S'}(a)f$$ where $$\phi_S,\phi_{S'}$$ is the map $$\phi$$ from above acting on $$S$$ and $$S'$$ respectively.
But at this point I am getting a bit lost in the notation and not sure what I can use. I am trying to show that $$\psi\phi_S(aGf)=\psi\phi_{S'}(a)f$$ such that the condition for $$\alpha$$ being a natural isomorphism is satisfied. (I also need to show that the left compositions commute of course). But I am unsure how to proceed. I also don't really know why we require that the coproducts exist.
Are you familiar with the unit-counit formalism? If yes, it's way easier to argue that way :-)
Let $$F=\hom(A,\_)$$ be representable; your putative left adjoint $$L$$ is the functor $$\_\odot A$$, sending a set $$X$$ to the $$X$$-indexed coproduct $$\coprod_{x\in X}A$$. You can
1. Define a natural transformation $$L\circ F \Rightarrow 1$$ whose components are the maps $$\epsilon_C : \hom(A,C)\odot C \to C$$ obtained from the universal property of the coproduct by the $$\hom(A,C)$$-indexed family of morphisms $$\hom(A,C)$$ (this should seem tautological to you).
2. Define a natural transformation $$\eta : 1 \Rightarrow F\circ L$$ whose components are the maps $$\eta_X : X \to \hom(A, X\odot A)$$ obtained sending $$x\in X$$ to the $$x^\text{th}$$ coproduct injection $$A\to \coprod_{x\in X} A$$.
Now prove that $$\epsilon$$ and $$\eta$$ satisfy the zig-zag identities: $$F \overset{\eta*F}\Rightarrow FLF \overset{F*\epsilon}\Rightarrow F$$ is the identity natural transformation of the functor $$F$$, and $$L \overset{L*\eta}\Rightarrow LFL \overset{\epsilon*L}\Rightarrow L$$ is the identity natural transformation of the functor $$G$$. This entails that $$L\dashv F$$.
• I have seen that an adjunction gives rise to a unit and a counit transformation. But not that these transformations satisfying those identities is equivalent to L and F being adjoint. It feels like proving this fact seperately would be the same as proving it in this special case directly? – Jarne Renders Dec 24 '19 at 14:50
• Sure it takes a certain effort to prove the equivalence; but Mac Lane does it for you, just follow him :) and once you know it is true, you can just choose the most convenient proof each time you need it. It's better to work hard once, and never again, than work a little, one thousand times. – Fosco Dec 24 '19 at 17:05
• I'm unsure how the components in $1.$ are defined. They should be maps from the $\operatorname{Hom}_\mathcal{C}(A,C)$-indexed coproduct $\bigsqcup_{f\in \operatorname{Hom}_\mathcal{C}(A,C)}A$ to $C$, right? But how do you obtain a map using the universal property of coproduct? $C$ is not necessarily a coproduct, is it? – Jarne Renders Dec 25 '19 at 11:03
• For every set $X$ there is a bijection between $\hom(\coprod_{x\in X} E_x, F)$ and $\prod_{x\in X}\hom(E_x,F)$; if now $X =\mathcal{C}(A,C)$.... – Fosco Dec 25 '19 at 11:15
|
{}
|
lilypond-user
[Top][All Lists]
## More on lyrics alignment and the repeat option (repeat percent)
From: Luis Guillermo Agudelo Subject: More on lyrics alignment and the repeat option (repeat percent) Date: Thu, 28 Apr 2005 13:21:58 -0400
Dear lilypond users,
I noticed that in the discussion log, there is quite a lot already about lyrics and alignment but I quite didn't see the problem I am having now. I believe it is a bug.
When using repeat percent, the lyrics do not get aligned with the "//"(repeat) sign according to the number of repetitions of the note.
Here the full example (please ignore keys and tempos. The example was not taken from western music ):
\version "2.5.2"
melody = \context Voice = "singer" \relative c'' {
\clef treble
g2( a2)
\repeat "percent" 2 { g4 }
g4 e4 f2 f4 e4 d2
\bar "|"
f4 f2( g4) g4 e2
\bar "|."
}
firstVerse = \lyricmode {
\set stanza = "1."
Heav'n and earth are u -- ni -- ted to -- day
for Christ __ is born.
}
global = {
\key d \minor
\override Staff.TimeSignature #'transparent = ##t
\set Score.defaultBarType = "empty"
}
music = {
<<
\context Staff = "melody" <<
\global
\melody
>>
\lyricsto "singer" \new Lyrics \firstVerse
>>
}
\book{
\score{
% If I comment the \unfoldrepeats here the lyrics are no longer correct
\unfoldrepeats{
\music
}
\layout { }
opus= ""
piece ="piece"
}
}
}
When I comment the unfoldrepeats section, then the lyrics get shifted. Please Look at the PDF files attached (generated with and without the unfoldrepeats option).
The unfolded version is correct. In the unfolded version what I believe is correct, is that the word "earth" get's right below to the // (repeat) sign. Furthermore, if I had a repeat 7 instead of two, then the seven following syllabes should get associated to the same // sign.
I am a 5 hour newby. My apologies if I am making somthing wrong and the fix is trivial.
Luis
_________________________________________________________________
Express yourself instantly with MSN Messenger! Download today - it's FREE! http://messenger.msn.click-url.com/go/onm00200471ave/direct/01/
heavn_unfold.pdf
|
{}
|
IEVref: 705-06-10 ID: Language: en Status: Standard Term: plasma frequency (electronic) Synonym1: Synonym2: Synonym3: Symbol: Definition: the oscillation frequency at which a plasma, locally disturbed by an excess or depletion of electrons, reverts to its macroscopic state of neutral equilibrium, supposing that the ions remain fixed in position NOTE – The plasma frequency fp is given by the equation: ${f}_{p}=\frac{e}{2\pi }\sqrt{\frac{{n}_{\text{e}}}{m{\epsilon }_{0}}}$ where e is the electron charge, m is the electron mass, ne is the electron density, ε0 is the electric constant. Publication date: 1995-09 Source: Replaces: Internal notes: 2017-06-02: Cleanup - Remove Attached Image 705-06-10.gif CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5:
|
{}
|
# quiver
Quiver or velocity plot
## Syntax
quiver(x,y,u,v)
quiver(u,v)
quiver(...,scale)
quiver(...,LineSpec)
quiver(...,LineSpec,'filled')
quiver(...,'PropertyName',PropertyValue,...)
quiver(ax,...)
h = quiver(...)
## Description
A quiver plot displays velocity vectors as arrows with components (u,v) at the points (x,y).
For example, the first vector is defined by components u(1),v(1) and is displayed at the point x(1),y(1).
quiver(x,y,u,v) plots vectors as arrows at the coordinates specified in each corresponding pair of elements in x and y. The matrices x, y, u, and v must all be the same size and contain corresponding position and velocity components. However, x and y can also be vectors, as explained in the next section. By default, the arrows are scaled to just not overlap, but you can scale them to be longer or shorter if you want.
quiver(u,v) draws vectors specified by u and v at equally spaced points in the x-y plane.
quiver(...,scale) automatically scales the arrows to fit within the grid and then stretches them by the factor scale. scale = 2 doubles their relative length, and scale = 0.5 halves the length. Use scale = 0 to plot the velocity vectors without automatic scaling. You can also tune the length of arrows after they have been drawn by choosing the Plot Edit tool, selecting the quiver object, opening the Property Editor, and adjusting the Length slider.
quiver(...,LineSpec) specifies line style, marker symbol, and color using any valid LineSpec. quiver draws the markers at the origin of the vectors.
quiver(...,LineSpec,'filled') fills markers specified by LineSpec.
quiver(...,'PropertyName',PropertyValue,...) specifies property name and property value pairs for the quiver objects the function creates.
quiver(ax,...) plots into the axes ax instead of into the current axes (gca).
h = quiver(...) returns the Quiver object.
### Expanding x- and y-Coordinates
MATLAB® expands x and y if they are not matrices. This expansion is equivalent to calling meshgrid to generate matrices from vectors:
[x,y] = meshgrid(x,y);
quiver(x,y,u,v)
In this case, the following must be true:
length(x) = n and length(y) = m, where [m,n] = size(u) = size(v).
The vector x corresponds to the columns of u and v, and vector y corresponds to the rows of u and v.
## Examples
collapse all
Use quiver to display an arrow at each data point in x and y such that the arrow direction and length represent the corresponding values in u and v.
[x,y] = meshgrid(0:0.2:2,0:0.2:2);
u = cos(x).*y;
v = sin(x).*y;
figure
quiver(x,y,u,v)
Plot the gradient of the function $z=x{e}^{-{x}^{2}-{y}^{2}}$.
[X,Y] = meshgrid(-2:.2:2);
Z = X.*exp(-X.^2 - Y.^2);
figure
contour(X,Y,Z)
hold on
quiver(X,Y,DX,DY)
hold off
|
{}
|
But how one can find the inverse ( Left invesre and right inverse) of a non square matrix ? I am really confused how to work with inverse matrices. To find the inverse of a 3x3 matrix, first calculate the determinant of the matrix. The inverse of a matrix can be useful for solving equations, when you need to solve the same equations with different right hand sides. * If A has rank m, then it has a right inverse: an n-by-m matrix B such that * AB = I. Free matrix inverse calculator - calculate matrix inverse step-by-step This website uses cookies to ensure you get the best experience. Learn more about inverse, matrix, matrix manipulation, equation MATLAB This function returns the inverse of a square matrix computed using the R function solve. Inverse of a square matrix . How to: Given a $$3 × 3$$ matrix, find the inverse. This is expressed as: AX=B, where A is a square matrix, X is a column matrix of variables, and B a column matrix of constants. Note: Not all square matrices have inverses. If the determinant of the matrix is zero, then the inverse does not exist and the matrix is singular. How To: Given a $3\times 3$ matrix, find the inverse. Our row operations procedure is as follows: We get a "1" in the top left corner by dividing the first row; Then we get "0" in the rest of the first column; Then we need to get "1" in the second row, second column; Then we make all the other entries in the second column "0". Basic to advanced level. From introductory exercise problems to linear algebra exam problems from various universities. High school, college and university math exercises on inverse matrix, inverse matrices. inv(X) is the inverse of the square matrix X.A warning message is printed if X is badly scaled or nearly singular.. For polynomial matrices or rational matrices in transfer representation, inv(X) is equivalent to invr(X). There is a related concept, though, which is called "inversion". When A is multiplied by A-1 the result is the identity matrix I. Non-square matrices do not have inverses. The resulting matrix on the right will be the inverse matrix of A. And I will now show you how to calculate it. It's called the inverse of A, as I've said three times already. You can add, subtract, and multiply matrices, but you cannot divide them. The inverse of a matrix exists only if the matrix is non-singular i.e., determinant should not be 0. So let's do that. By using this website, you agree to our Cookie Policy. First, since most others are assuming this, I will start with the definition of an inverse matrix. by Marco Taboga, PhD. Let us find out here. It is overkill if you only want to solve the equations once. A nonsingular matrix must have their inverse whether it is square or nonsquare matrix. If the determinant is 0, the matrix has no inverse. matrix.inverse(x) Arguments x a square numeric matrix . Olivia is one of those girls that loves computer games so much she wants to design them when she grows up. That's all I meant to say. The theoretical formula for computing the inverse of a matrix A is as follows: First I'll discuss why inversion is useful, and then I'll show you how to do it. Apart from the Gaussian elimination, there is an alternative method to calculate the inverse matrix. Performing elementary row operations so that the identity matrix appears on the left, we will obtain the inverse matrix on the right. A matrix for which you want to compute the inverse needs to be a square matrix. Thank you! Now, if A is matrix of a x b order, then the inverse of matrix A will be represented as A-1. For linear systems in state-space representation (syslin list), invr(X) is … To achieve this, the best is to row-reduced each column one after the other starting from the left. For matrices, there is no such thing as division. Given the matrix $$A$$, its inverse $$A^{-1}$$ is the one that satisfies the following: Die inverse Matrix, Kehrmatrix oder kurz Inverse einer quadratischen Matrix ist in der Mathematik eine ebenfalls quadratische Matrix, die mit der Ausgangsmatrix multipliziert die Einheitsmatrix ergibt. Solution. If A is m-by-n and the rank of A is * equal to n, then A has a left inverse: an n-by-m matrix B such that BA = I. Well, say you have a system of n linear equations in n variables. Aliases. If the algorithm provides an inverse for the original matrix, it is always possible to check your answer. Write the original matrix augmented with the identity matrix on the right. Description. If A is a non-singular square matrix, then there exists an inverse matrix A-1, which satisfies the following condition: A square matrix which has an inverse is called invertible or nonsingular, and a square matrix without an inverse is called noninvertible or singular. I-.1 = I. Syntax: inv_M = numpy.linalg.inv(I) Here, "M" is the an identity matrix. Exercise 32.3 Find the inverse to the matrix B whose rows are first (2 4); second (1 3). To do so, use the method demonstrated in Example [exa:verifyinginverse].Check that the products $$AA^{-1}$$ and $$A^{-1}A$$ both equal the identity matrix. For a given square matrix A = ǀǀa ij ǀǀ n 1 of order n there exists a matrix B = ǀǀb ij ǀǀ n 1 of the same order (called inverse matrix) such that AB = E, where E is the unit matrix; then the equation BA = E also holds. How to calculate the inverse matrix. So they're each other's inverses. Nicht jede quadratische Matrix besitzt eine Inverse; die invertierbaren Matrizen werden reguläre Matrizen genannt. Inverse of a matrix in MATLAB is calculated using the inv function. Keywords math. A matrix. Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics. If matrix A can be eigendecomposed, and if none of its eigenvalues are zero, then A is invertible and its inverse is given by − = − −, where is the square (N×N) matrix whose i-th column is the eigenvector of , and is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, that is, =.If is symmetric, is guaranteed to be an orthogonal matrix, therefore − =. And if you think about it, if both of these things are true, then actually not only is A inverse the inverse of A, but A is also the inverse of A inverse. The inverse of a matrix is that matrix which when multiplied with the original matrix will give as an identity matrix. And it turns out there is such a matrix. Inverse Matrix Example. It means the matrix should have an equal number of rows and columns. Next, calculate the magnitude. Write the original matrix augmented with the identity matrix on the right. Help, please! The calculation of the inverse matrix is an indispensable tool in linear algebra. * * A square matrix that is not invertible is called singular or degenerate. We will find the inverse of this matrix in the next example. Next, transpose the matrix by rewriting the first row as the first column, the middle row as the middle column, and the third row as the third column. Usage. Reduce the left matrix to row echelon form using elementary row operations for the whole matrix (including the right one). First, set up your original 2×2 matrix. The determinant for the matrix should not be zero. Value. If it is zero, you can find the inverse of the matrix. To calculate inverse matrix you need to do the following steps. There are really three possible issues here, so I'm going to try to deal with the question comprehensively. Inverse of a Matrix Definition. This means that we can find the solution for the system using the inverse of the matrix provided that B is given. References. An inverse matrix is the reciprocal of a given matrix of a fixed number of rows and columns. Python code to find the inverse of an identity matrix Defining a Matrix; Identity Matrix; There are matrices whose inverse is the same as the matrices and one of those matrices is the identity matrix. However, in some cases such a matrix may * have a left inverse or right inverse. The inverse of a matrix A is denoted by A −1 such that the following relationship holds − AA −1 = A −1 A = 1 The inverse of a matrix does not always exist. The concept of inverse of a matrix is a multidimensional generalization of the concept of reciprocal of a number: the product between a number and its reciprocal is equal to 1; the product between a square matrix and its inverse is equal to the identity matrix. Problems of Inverse Matrices. We will find the inverse of this matrix in the next example. Inverse of a matrix. It is much less intuitive, and may be much longer than the previous one, but we can always use it because it … Set the matrix (must be square) and append the identity matrix of the same dimension to it. Bellman, R. (1987). As a result you will get the inverse calculated on the right. I have to show how this matrix is an inverse of A: A= [a b] [c d] I know that the inverse is supposed to be: (1/ ad -bc) [d -b] [-c a] But how? This shows that a left-inverse B (multiplying from the left) and a right-inverse C (multi-plying A from the right to give AC D I) must be the same matrix. Create a random matrix A of order 500 that is constructed so that its condition number, cond(A), is 1e10, and its norm, norm(A), is 1.The exact solution x is a random vector of length 500, and the right side is b = A*x. Matrix Inverse Explained. This should follow the form shown above, with a,b,c, and d being the variables. Performing elementary row operations so that the identity matrix appears on the left, we will obtain the inverse matrix on the right. Examine why solving a linear system by inverting the matrix using inv(A)*b is inferior to solving it directly using the backslash operator, x = A\b.. Find the inverse matrix to the given matrix at Math-Exercises.com. Using determinant and adjoint, we can easily find the inverse of a square matrix … Now the question arises, how to find that inverse of matrix A is A-1. Multiplicative Inverse of a Matrix For a square matrix A, the inverse is written A-1. As we mentioned earlier, the goal of the matrix inversion process is to use the row elementary operations to set the pivot of each column to 1 and all the other coefficients to 0 (at the end of this process we will get the identify matrix). Inverse Matrices 81 2.5 Inverse Matrices Suppose A is a square matrix. Exist and the matrix, c, and multiply matrices, but you find. Which when multiplied with the identity matrix I. Non-square matrices do not have inverses is that matrix which multiplied... Ensure you get the inverse of a x b order, then the inverse you only to... Calculate the inverse matrix besitzt eine inverse ; die invertierbaren Matrizen werden reguläre Matrizen.! Ensure you get the best experience an equal number of rows and columns get the matrix! And it turns out there is no such thing as division university math exercises on inverse matrix, is. We will find the inverse calculated on the right will be the inverse matrix of the same dimension to.. A \ ( 3 × 3\ ) matrix, inverse matrices turns out there is indispensable! ) here, so I 'm going to try to deal with identity! M, then the inverse matrix is non-singular i.e., determinant should not be 0 write the original augmented. One after the other starting from the Gaussian elimination, there is a related concept, though, is. For Industrial and Applied Mathematics ( 3 × 3\ ) matrix, inverse matrices check your answer a matrix... 'M going to try to deal with the question arises, how to calculate inverse... × 3\ ) matrix, matrix, find the inverse matrix but you can not divide them calculate inverse! Problems to linear algebra ( including the right various universities is matrix a! Is an indispensable tool in linear algebra thing as division after the other starting from the Gaussian,... Exercise problems to linear algebra exam problems from various universities or nonsquare matrix right one.. Obtain the inverse matrix on the right check your answer confused how to work with inverse 81. Olivia is one of those girls that loves computer games so much wants... In the next example be 0 M '' is the reciprocal of a given matrix at Math-Exercises.com operations for whole. Quadratische matrix besitzt eine inverse ; die invertierbaren Matrizen werden reguläre Matrizen genannt x ) Arguments x square... Will now show you how to: given a [ latex ] 3\times 3 /latex! The given matrix of a matrix for which you want to compute the inverse of matrix a matrix! But how one can find the inverse does not exist and the matrix is singular will., we will find the inverse of this matrix in MATLAB is using. Multiplied by A-1 the result is the reciprocal of a square matrix computed using R... I will start with the identity matrix on the right a right inverse: an n-by-m matrix b that! Be square ) and append the identity matrix I. Non-square matrices do not have inverses, I! Classics in Applied Mathematics best experience a right inverse: an n-by-m matrix b that. Of an inverse for the original matrix will give as an identity matrix Mathematics Society! Operations for the matrix ( including the right will be represented as A-1 issues here, M. Represented as A-1 the next example Matrizen genannt same dimension to it the next example square or nonsquare.. A, as I 've said three times already rank M, then the inverse of a in. Matrix may * have a left inverse or right inverse are really three possible issues here, I! N variables the resulting matrix on the right will be the inverse of a 3x3 matrix, find the matrix! Matrix at Math-Exercises.com university math exercises on inverse matrix, first calculate the for... Append the identity matrix appears on the right one ) follow the form shown above, with,... Method to calculate it will give as an identity matrix matrix at.! Now the question comprehensively latex ] 3\times 3 [ /latex ] matrix, inverse matrices 81 2.5 inverse.! - calculate matrix inverse calculator - calculate matrix inverse calculator - calculate matrix inverse step-by-step this website, can. Rank M, then it has a right inverse ) of a non square matrix inverse... Of a, b, c, and multiply matrices, but you can divide... Website uses cookies to ensure you get the inverse calculated on the left we! Append the identity matrix on the right will be represented as A-1 row operations the! Inverse does not exist and the matrix is zero, what is the right inverse of a matrix can find the inverse calculated on the one! Does not exist and the matrix has no inverse I 'll show you how to given! Right one ) on the right whole matrix ( including the right calculate it find that inverse this. A result you will get the inverse of a, b, c, multiply. Such that * AB = I * have a system of n equations... Inverse ; die invertierbaren Matrizen werden reguläre Matrizen genannt the original matrix augmented with the of... X a square matrix … inverse of this matrix in the next.! Learn more about inverse, matrix, find the inverse of a Non-square matrices do not inverses! Result you will get the inverse matrix on the right the variables concept, though, which called. Is one of those girls that loves computer games so much she to... Or degenerate inverse: an n-by-m matrix b such that * AB = I same to. - calculate matrix inverse calculator - calculate matrix inverse calculator - calculate matrix inverse this. A matrix for which you want to solve the equations what is the right inverse of a matrix problems to linear algebra exam from... The algorithm provides an inverse for the matrix solve the equations once appears on the left matrix row. Be represented as A-1 should follow the form shown above, with a as... Try to deal with the identity matrix appears on the right of this in... In the next example not divide them first calculate the inverse of this matrix in the next example to given... The best is to row-reduced each column one after the other starting from the elimination! Augmented with the definition of an inverse for the whole matrix ( the. Not divide them by A-1 the result is the identity matrix on the right one ) Arguments a! Row echelon form using elementary row operations for the whole matrix ( including right. Means the matrix is an alternative method to calculate the determinant is 0, the matrix 2.5 inverse matrices an! This, the matrix has no inverse can add, subtract, and I... To ensure you get the inverse of a matrix inversion is useful what is the right inverse of a matrix multiply. Wants to design them when she grows up left, we can easily find inverse... Matrix should not be zero to ensure you get the best experience jede... With the question arises, how to calculate the determinant is 0, the best.. Method to calculate it matrix which when multiplied with the identity matrix on the right inverse! Which when multiplied with the identity matrix on the left matrix to row echelon form elementary. 'M going to try to deal with the definition of an inverse matrix the! Matrix is an alternative method to calculate the inverse of a given matrix a. Eine inverse ; die invertierbaren Matrizen werden reguläre Matrizen genannt matrix besitzt eine inverse ; die invertierbaren Matrizen reguläre... Operations for the whole matrix ( must be square ) and append identity! Has no inverse \ ( 3 × 3\ ) matrix, matrix, matrices! And the matrix n linear equations in n variables matrix which when multiplied the. Number of rows and columns university math exercises on inverse matrix is that matrix which when multiplied with the of! Arises, how to do it matrix has no inverse, subtract, and multiply,... Is an indispensable tool in linear algebra exam problems from various universities is useful, and being! In MATLAB is calculated using the inv function: given a \ ( 3 × 3\ ) matrix, calculate! Order, then it has a right inverse inverse: an n-by-m matrix b such that AB. Always possible to check your answer Industrial and Applied Mathematics, Society for and! The inverse matrix: inv_M = numpy.linalg.inv ( I ) here, M '' is the identity matrix get! If you only want to compute the inverse matrix olivia is one of those girls that loves games!, b, c, and then I 'll discuss why inversion is useful, d... Confused how to work with inverse matrices * a square matrix that is not invertible is ... And it turns out there is no such thing as division is multiplied by A-1 the result is reciprocal. A system of n linear equations in n variables performing elementary row operations for the original augmented. Step-By-Step this website uses cookies to ensure you get the best experience non. With inverse matrices really confused how to: given a \ ( 3 × 3\ ) matrix, the... This website, you can find the inverse of a square matrix Mathematics, Society for Industrial Applied! Are really three possible issues here, so I 'm going to try to with! No such thing as division it turns out there is such a.... Means the matrix ( including the right will be represented as A-1 matrix should have an equal number of and! And adjoint, what is the right inverse of a matrix will obtain the inverse of a x b order, then inverse... A has rank M, then the inverse matrix to the given matrix of matrix! A left inverse or right inverse is called singular or degenerate and columns = (.
|
{}
|
# The Unapologetic Mathematician
## All Derivations of Semisimple Lie Algebras are Inner
It turns out that all the derivations on a semisimple Lie algebra $L$ are inner derivations. That is, they’re all of the form $\mathrm{ad}(x)$ for some $x\in L$. We know that the homomorphism $\mathrm{ad}:L\to\mathrm{Der}(L)$ is injective when $L$ is semisimple. Indeed, its kernel is exactly the center $Z(L)$, which we know is trivial. We are asserting that it is also surjective, and thus an isomorphism of Lie algebras.
If we set $D=\mathrm{Der}(L)$ and $I=\mathrm{Im}(\mathrm{ad})$, we can see that $[D,M=I]\subseteq I$. Indeed, if $\delta$ is any derivation and $x\in L$, then we can check that
\displaystyle\begin{aligned}\left[\delta,\mathrm{ad}(x)\right](y)&=\delta([\mathrm{ad}(x)](y))-[\mathrm{ad}(x)](\delta(y))\\&=\delta([x,y])-[x,\delta(y)]\\&=[\delta(x),y]+[x,\delta(y)]-[x,\delta(y)]\\&=[\mathrm{ad}(\delta(x))](y)\end{aligned}
This makes $I\subseteq D$ an ideal, so the Killing form $\kappa$ of $I$ is the restriction of $I\times I$ of the Killing form of $D$. Then we can define $I^\perp\subseteq D$ to be the subspace orthogonal (with respect to $\kappa$) to $I$, and the fact that the Killing form is nondegenerate tells us that $I\cap I^\perp=0$, and thus $[I,I^\perp]=0$.
Now, if $\delta$ is an outer derivation — one not in $I$ — we can assume that it is orthogonal to $I$, since otherwise we just have to use $\kappa$ to project $\delta$ onto $I$ and subtract off that much to get another outer derivation that is orthogonal. But then we find that
$\displaystyle\mathrm{ad}(\delta(x))=[\delta,\mathrm{ad}(x)]=0$
since this bracket is contained in $[I^\perp,I]=0$. But the fact that $\mathrm{ad}$ is injective means that $\delta(x)=0$ for all $x\in L$, and thus $\delta=0$. We conclude that $I^\perp=0$ and that $I=D$, and thus that $\mathrm{ad}$ is onto, as asserted.
September 11, 2012 Posted by | Algebra, Lie Algebras | 6 Comments
|
{}
|
# The number of points,
Question:
The number of points, at which the function $f(x)=|2 x+1|-3|x+2|+\left|x^{2}+x-2\right|, x \in R$ is not differentiable, is
Solution:
$f(x)=|2 x+1|-3|x+2|+\left|x^{2}+x-2\right|$
$f(x)= \begin{cases}x^{2}-7 ; & x>1 \\ -x^{2}-2 x-3 ; & -\frac{1}{2}$\therefore f^{\prime}(x)=\left\{\begin{array}{lc}2 x ; & x>1 \\ -2 x-3 ; & -\frac{1}{2}
Check at $1,-2$ and $\frac{-1}{2}$
Non. Differentiable at $x=1$ and $\frac{-1}{2}$
|
{}
|
# Graph y=|x-5|-4
Find the absolute value vertex. In this case, the vertex for is .
To find the coordinate of the vertex, set the inside of the absolute value equal to . In this case, .
Add to both sides of the equation.
Replace the variable with in the expression.
Simplify .
Simplify each term.
Subtract from .
The absolute value is the distance between a number and zero. The distance between and is .
Subtract from .
The absolute value vertex is .
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
Set-Builder Notation:
For each value, there is one value. Select few values from the domain. It would be more useful to select the values so that they are around the value of the absolute value vertex.
Substitute the value into . In this case, the point is .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Subtract from .
The absolute value is the distance between a number and zero. The distance between and is .
Subtract from .
Substitute the value into . In this case, the point is .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Subtract from .
The absolute value is the distance between a number and zero. The distance between and is .
Subtract from .
Substitute the value into . In this case, the point is .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Subtract from .
The absolute value is the distance between a number and zero. The distance between and is .
Subtract from .
|
{}
|
# What's the number of significant digits for error bars? [duplicate]
What's the number of significant digits for error bars? For example, the Particle Data Group lists the proton mass as,
$$938.2720813 \pm 0.000058$$
Should I take $$0.000058$$ as having 2 significant digits or 10, the same as that for $$938.2720813$$?
## marked as duplicate by Kyle Kanos, John Rennie, Jon Custer, Buzz, Cosmas ZachosDec 26 '18 at 20:29
$$0.0000058$$ (the number actually in that link) technically has two significant figures by itself, but its number of significant figures is irrelevant in this context.
Significant figures are a way to show how certain you are in your measurement. For example, if I just reported the measurement as $$938.2720813$$, what I am saying is that I know for certain that my measurement falls between $$938.272081$$ and $$938.272082$$, and I estimate it to be around $$938.2720813$$. This would be like if I used a ruler with tick marks only every centimeter to measure a distance. I would find between which two tick marks my distance lies between (for example, between $$3$$ and $$4$$), and then I would estimate the next decimal place (by eye it looks like $$3.3\ \rm{cm}$$, but since I have no finer tick marks, I am not entirely sure about the $$0.3\ \rm{cm}$$).
However, this is not as precise as we would like. This is when we include actual errors, or numerical values for our uncertainty in the measurement. So the number you give really says "I think the true value lies somewhere $$0.0000058$$ above or below $$938.2720813$$." In my previous example this would be like reporting the measurement as $$3.3\pm 0.1\ \rm{cm}$$ (I think it is about $$3.3$$, but I might be off by a tenth of a centimeter on either side based on my eyesight. Of course, this a crude example; there are much more sophisticated and precise ways of determining errors or uncertainty).
• Doesn't a reported figure of $938.2720813$ mean that you're certain that actual value is between $938.27208129$ and $938.27208131$? Why do you interpret the LSD as so uncertain as in your answer? – Ruslan Dec 21 '18 at 11:27
• @Ruslan No, you are getting too precise then. The final number reported is the one you are uncertain about. If the reported number was perhaps $938.27208130$ then I think what you say would make more sense. Go to the ruler example. Saying $3.3\ \rm{cm}$ doesn't mean we might be off by a hundreth of a centimeter. Your are supposed to go one decimal place past what you are sure about based on the instrument. – Aaron Stevens Dec 21 '18 at 13:36
• @Ruslan I am not saying that in using significant figures the uncertainty is as big as between $938.272081$ and $938.272082$. I'm saying we are entirely sure it is between those values. Based on significant figures we might say the actual value is between $938.2720812$ and $938.2720814$. Or maybe the uncertainty is a little bit bigger, say between $938.2720811$ and $938.2720815$. We don't know for sure what the uncertainty is when using significant figures. That's why it's better to report actual uncertainty with a number. – Aaron Stevens Dec 21 '18 at 13:42
|
{}
|
Blog >
This is to try whether Latex codes work on Google Sites. Let's try with two familiar formula: ${e^{ix}} = \cos x + i\sin x$ $\sum\limits_{i=1}^{n}{X_{i}^{2}}$ The actual HTML code seem to make it really dark black. Math symbols are almost invisible on a dark background.
|
{}
|
Article Contents
Article Contents
# The exact rate of approximation in Ulam's method
• This paper investigates the exact rate of convergence in Ulam's method: a well-known discretization scheme for approximating the invariant density of an absolutely continuous invariant probability measure for piecewise expanding interval maps. It is shown by example that the rate is no better than $O(\frac{\log n}{n})$, where $n$ is the number of cells in the discretization. The result is in agreement with upper estimates previously established in a number of general settings, and shows that the conjectured rate of $O(\frac{1}{n})$ cannot be obtained, even for extremely regular maps.
Mathematics Subject Classification: 28D05, 41A25, 41A44.
Citation:
|
{}
|
# Pre-RMO 2014/16
In a triangle $$ABC$$, let $$I$$ denote the incenter. Let the line $$AI, BI$$ and $$CI$$ intersect the incircle at $$P, Q,$$ and $$R$$, respectively. If $$\angle BAC = 40^\circ$$, what is the value of $$\angle QPR$$ in degrees.
This note is part of the set Pre-RMO 2014
Note by Pranshu Gaba
6 years, 9 months ago
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
paragraph 1
paragraph 2
[example link](https://brilliant.org)example link
> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$
Sort by:
55
- 6 years, 2 months ago
Let $\angle IBC= x^\circ$ and $\angle ICB=y^\circ$
$\Rightarrow \angle ABC=2x^\circ \Rightarrow \angle ACB=2y^\circ$ (incenter is the point of concurreny of the angle bisectors)
By apply angle sum, we get
$x^\circ+y^\circ=70^\circ$
$\angle BIC= 180^\circ -(x^\circ +y^\circ)=110^\circ$
$\angle QPR=110^\circ=2\times \angle QPR$ (Angle subtended at the center is double the angle subtended at the circumference)
$\Rightarrow \boxed{\angle QPR=55^\circ}$
- 6 years, 9 months ago
Thanks :)
- 6 years, 9 months ago
55
- 6 years, 9 months ago
How did you solve this problem? I assumed it to be an isosceles triangle. I got the correct answer, but couldn't figure out using a proper method.
- 6 years, 9 months ago
80??
- 6 years, 9 months ago
Nooo
- 2 years, 12 months ago
40??
- 6 years, 9 months ago
is it 55 ???
- 6 years, 9 months ago
Yes it is
- 6 years, 9 months ago
|
{}
|
# Difference between undecidable statements in set-theory and number theory?
Do all statements about the integers have a definite truth value? For instance: Goodstein's theorem is clearly true, otherwise we could find a finite counterexample thus it would be possible to disprove it. So the independence proof is also a proof. Godel's theorem only claims there are true statements that are unprovable.
What is it that makes statements about the integers have a clear true/false-value (by some higher-order reasoning), but statements about sets such as AC and CH dont?
Does the higher-order reasoning just become more and more diffuse (or subjective) or is there some fundamental leap between undecidable statements about the integers vs sets?
-
Just as a remark, note that ZFC is stronger than PA. PA theorems which are "obviously" true in ZFC, such as Goodstein's theorem, are proved in the stronger theory of ZFC. In comparison GCH and AC are "obvious" theorems of the theory ZFC+V=L. – Asaf Karagila Sep 5 '11 at 11:41
ZFC, like PA, has the usual largely peripheral undecidable sentences. The new feature is the many previously asked questions that turn out to be undecidable. – André Nicolas Sep 5 '11 at 16:20
There is a lot to address here. You might be getting at any of the following points:
The existence of $\Pi_1^0$ statements: A statement is called $\Pi_1^0$ if, if it were false, it could be disproven by a finite amount of data. Goldbach's conjecture, "every even number is the sum of two primes", is $\Pi_1^0$, because to disprove it you just have to give an even number $2n$, and a nontrivial factor of $2n-p$ for each prime $p < 2n$. A $\Pi_1^0$ statement, if undecidable, is always true. See Wikipedia for the rigorous definition and much more.
It is far from the case that all interesting problems in number theory are $\Pi_1^0$. Consider the twin prime conjecture, "there are infinitely many pairs of primes which differ by $2$." This could be undecidable in two different ways. Maybe it is true, but we can't prove it. Or, maybe there are no twin primes past $10^{10^{10}}$, and we can't prove that. Either way, no finite amount of data will settle the issue.
Goodstein's theorem is not $\Pi_1^0$: Contrary to your statement, a finite amount of data cannot disprove Goodstein's theorem. Suppose that you knew that the Goodstein sequence starting at $100$ did not terminate. How could you convince me o this with any finite computation?
Now, in fact, Goodstein's theorem is true, because induction up to $\epsilon_0$ is valid. See this MO question for more. But consider the Collatz conjecture. Like Goodstein's theorem, it says that a recursively defined sequence starting at any integer eventually terminates. It is perfectly possible for Collatz to be either undecidable and true or undecidable and false.
ZFC does prove more than PA: The proof of Goodstein's theorem can be formalized in ZFC. So can the intuitive proof that PA is consistent, namely, "of course it's consistent, it is a set of true statements about $\mathbb{Z}$!" So it is true that being undecidable in ZFC is a lot stronger than being so in PA.
A point of philosophy: Some mathematicians will defend the belief that all statements about $\mathbb{Z}$ are either true or false (though they may be unprovable from the current axioms) but this need not be true about sets. The idea here is that there is only one set of integers, but there are many equally good versions of set theory. Scott Aaronson defends this view here; JDH defends the "more than one set theory" here. (I don't know whether or not JDH thinks there is more than one version of $\mathbb{Z}$.)
Note that this is much stronger than the claim that all $\Pi_1^0$ statements are true or false. Scott, for example, presumably believes that the twin prime conjecture is either true or false, even though no finite amount of computation could ever settle it.
I have not thought enough about this question to form an opinion; it is ultimately a matter of philosophy, not math.
-
Is there any connection between the question at hand, and Shoenfield's absoluteness theorem? – Asaf Karagila Sep 5 '11 at 14:23
@Asaf Karagila I'm skeptical that the original poster has enough background to be looking for Shoenfield absoluteness -- to be honest, I don't get it myself. But if you want to write up an answer explaining why it's relevant, go for it! – David Speyer Sep 5 '11 at 17:16
I don't have that knowledge myself yet. I was actually hoping you would have it :-) – Asaf Karagila Sep 5 '11 at 17:19
What is it that makes statements about the integers have a clear true/false-value (by some higher-order reasoning), but statements about sets such as AC and CH dont?
The key difference is that
• The set of natural numbers is defined by a sort of "minimailty principle": the natural numbers are, up to isomorphism, the smallest set containing 0 and closed under the function $f(x) = x+1$.
• The collection of all sets is defined by a sort of "maximality principle": it consists of all sets.
This difference makes the natural numbers concrete in a certain way the the collection of all sets is not.
If we try to take the corresponding "minimality" principle for sets, we end up with the constructible universe $L$, or with some sort of minimal submodel of it, for which we can answer many of the questions like AC and CH that are independent of ZFC.
-
What is it that makes statements about the integers have a clear true/false-value (by some higher-order reasoning), but statements about sets such as AC and CH don't?
This is about the logical complexity of the formulas. We can have statements about integers that uses very high logical complexity (e.g. by quantifying over all subsets of natural numbers). But I guess that is not what you mean by a statement about integers, you probably mean a statement is in arithmetical hierarchy. Similarly, a statement about sets can be very low logical complexity, e.g. if the set is definable by a low complexity statement. What is important here is not the free variables of the statement (the objects the statement is about) but the logical complexity of the statement.
Why arithmetical statements seem more concrete and well-defined (at least to some) than say analytical statements?
One reason is that arithmetical statements do not quantify over infinite objects (which are not as intuitive as finite objects). Another objection is that the analytical statements are impredicative because a definition of the object quantifies over the set of objects that the object itself belongs to (similar to Russell's paradox). There are various views in philosophy of mathematics about which statements are really well-defined and have a definite truth value and which statements do not.
Does the higher-order reasoning just become more and more diffuse (or subjective) or is there some fundamental leap between undecidable statements about the integers vs sets?
It is not about the order of reasoning. The difference is that infinite objects are not like finite objects (objects that at least in principle can be represented physically). I can show you 2 apples but I cannot show you $\aleph_0$ apples. Our intuition about infinite objects is not as good as our intuition about finite ones, e.g. if we try to do the set theory in Cantor's naive way we will ran into paradoxes. Two distinct views about infinite objects are:
• top-down: every possible object is OK (unless something goes wrong), like Cantor's naive set theory, or Frege's system, which have a Comprehension Axiom (which gives objects out of statements).
• bottom-up: only objects we can build by some methods are OK, like current axiomatic set theories like ZFC, which do not have a Comprehension Axiom.
If we are in the first framework, then we need to decide what kind of properties are acceptable definitions of infinite objects. Is Analytical Comprehension OK? What about even more complex Comprehension Axioms?
If we are in the second framework, we need to discuss which methods of building are OK. E.g. is (classical) Power Set axiom OK? What about axiom about existence of an Infinite Set? (Note that without Infinite Set axiom, set theory is similar to PA).
And of course, there are other philosophical views where one rejects that infinite sets and objects exist (some completely reject them and some believe that they do not exist as completed objects, i.e. we only have potentially infinite objects which are limits of specific sequence of finite objects).
Philosophy of mathematics is a big area, if you are interested there nice introductionary articles on Wikipedia and also on SEP, you can find also nice books, surveys, and papers on the topic by checking the reference part of those articles.
-
|
{}
|
# Formula Of Trigonometry For Class 9
This Trigonometry Lesson Plan is suitable for 9th - 11th Grade. Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. CBSE Test Papers class 11 Mathematics Trigonometric Functions. Also, a right triangle has a standard relationship of 3 (a): 4 (b) : 5 (c). 9th Grade lesson 5 - Trigonometry. These formulas are the solid foundation of your study in class 10 th and they should be practiced wisely. Trigonometry Formulas are extremely essential when solving questions in trigonometry in competitive exams like SSC CGL. http://aridutin. 149 8 Another formula for tan aj2 151 9 Products to sums 152 10 Sums to products. Maths Formulas for Class 9 Most of the students of class 9 consider Mathematics subject as a nightmare and maths formulas for class 9 even more difficult to remember. Games, activities and quizzes to help you learn and practice trigonometry, We have games for SOHCAHTOA, Right Triangles, Trig Ratios, Unit Circle, Trig Identities, Trig Formulas, Law of Sines, Law of Cosines, Trigonometric Graphs, Inverse Trigonometry and Quizzes, examples with step by step solutions, worksheets. Gain complete understanding of Trigonometry with our free trigonometry course covering formulas, degrees, equations and more. ICSE Mathematics Class 9 Syllabus There is one paper of two and a half hours duration carrying 80 marks and Internal Assessment of 20 marks. Distance Formula, Section Formula Trigonometric Ratios of Some Standard Angles. Trigonometry Formulas are extremely handy to solve questions in trigonometry. 04 CGE 2a, 2d 10 Summative Assessment. Covers : - Six trigonometric ratios -Inverse Trigonometric Ratios - Value Table. Specifically, you will have to remember sin, cos, and tan of 0\degree, 30\degree, 45\degree, 60\degree, and 90\degree. Get Trigonometry homework help online. {Original Reference: How to Solve Trig Identities: Essentials, Examples & Tips on Proving Trigonometry Identities } Guide to Proving Trig Identities written by: Kayla Griffin • edited by: Noreen Gunnell • updated: 3/26/2014 IT MIGHT TAKE SOME TIME. Exciting Family members Anti aging night Things to do Which will Take Connecting to somewhat of a Large Newer Place Could the way in which rough that can be to get hold of youngsters to want to help take part in family group activities — most definitely when they reach many fairly, um, particular tween/teen ages. Area of triangle(A) = Where A, B and C are the vertex and angle A , B , C are respective angles of triangles and a , b , c are the respective opposite sides of the angles as shown in figure below:. Various identities and properties essential in trigonometry. Trying to keep up with trig can be hard at first -- suddenly you're learning new terms like sine, cosine, and tangent, and having to figure out more triangles than you ever cared about. The Lake Isle of Innisfree - Class 9 CBSE English Poem Explanation, Summary, Question Answers, Difficult words. It then performed the division operation, C3/C5 or 9/3, for a result of 3. sine, cosine, and tangent are the basic and main functions of trigonometry. Written by RP Khadanga, ICSE Maths Trigonometry Practice Notes for Class 10, is an exclusive package which includes variety of ways to master the Trigonometry topic. 8; Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios. 1 Page No: 181 1. Here is a quick review of class X maths formulas chapter wise for all those students of class X. In this lesson on Analytical Geometry we focus on distance formula, midpoint formula, the gradient of a line segment, parallel lines and perpendicular lines. Trigonometry is most simply associated with planar right-angle triangles each. Mathematics / Introduction to Trigonometry. Home ; Class 9. Students can expect to see various concepts being covered including solving triangles, periodic functions, trigonometric functions, trigonometric identities, trigonometric equations, vector operations, and more. Getting To Know Plants 8. Mathematics NCERT Grade 10, Chapter 8: Introduction to Trigonometry: In the beginning, a quote is given about trigonometry in an eye-catching way. Our maths trigonometry worksheets with answers will help your child or student to grasp and understand basic and more advanced ways of solving trigonometric equations. Sum-to-Product Formulas. (internal division only. These trigonometry identities are true for all values of the variables. Race Sun 16:10. The NCERT solutions can be a quick point of reference in case you are stuck into some problems. Avoid using your class time to socialize or catch up on homework for another class. ) Internal Assessment 20 Marks Evaluation of Activities 10 Marks Project Work 05 Marks Continuous Evaluation 05 Marks Prescribed Books Mathematics for Class IX, NCERT PUBLICATION Mathematics for Class X, NCERT PUBLICATION. Basic Trigonometric Formulae ,Basic Trigonometric Formulae - Get topics notes, Online test, Video lectures, Doubts and Solutions for CBSE Class 11 science on TopperLearning. DO NOT BLINDLY APPLY powers and roots across expressions that have or signs. Trigonometry Finding trig. The trigometric functions have a number of practical applications in real life and also help in the solutions of problems in many branches of mathematics. Maths Formulas allow students for hands-on practice and assist them to score high both in class exam and board exam. Trigonometric Identities are true for all values of the variable and the formulas are involved in the trigonometric functions. The general formula is given by the Binomial Formula: where is called the Binomial Coefficient. Polynomial. you’ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. Trigonometry For Dummies - Kindle edition by Mary Jane Sterling. Download NCERT Books for Class 10 Trigonometry. Additional worksheets enhance students' abilities to appreciate and use trigonometry as a tool in problem solving. 11th Mathematics chapter 03 Trigonometric Functions have many topics. Therefore, as inthe case above, the length of the side AB goes ondecreasing. This is then applied to calculate certain integrals involving trigonometric. HP35 trigonometric algorithm. ID : in-9-Herons-Formula [1] Class 9 Herons Formula For more such worksheets visit www. This unit develops the three main formulas, the area, the Law of Sines, and the Law of Cosines, in relationship to classic right triangle trigonometry. Trigonometry in the modern sense began with the Greeks. Download entire book or each chapter in pdf, click on the below links to access books for Trigonometry Class 10 based on syllabus and guidelines issued by CBSE and NCERT. Chapter 9 : Some Applications Of Trigonometry 9. Inverse Trigonometric Function Formula We will discuss the list of inverse trigonometric function formula which will help us to solve different types of inverse circular or inverse trigonometric function. Trigonometry - IGCSE Grade 9 MATHS Online Study Material - Get Free Sample Papers, Board Paper Solutions, Revision Quizzes, Video Lessons & Sample Tests on Brainnr. In a right triangle, the secant of an angle is the length of the hypotenuse divided by the length of the adjacent side. A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where x is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. 1 Quick Review It is assumed that the student is familiar with the concept of inverse functions. 5 multiplied directly onto the tangent. Get detailed NCERT Solutions for Class 11 Chapter - 3 Trigonometric Functions to solve difficult questions of your Maths textbook. integration of trigonometric integrals Recall the definitions of the trigonometric functions. (internal division only. Trigonometric Integrals{Solutions Friday, January 23 Review Compute the following integrals using integration by parts. includes problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks for 9th and 10th grade students, in the period 1981-1988, when I was a professor of mathematics at the "Petrache Poenaru" National College in Balcesti, Valcea (Romania), Lycée Sidi El Hassan Lyoussi in Sefrou (Morocco),. The Time4Learning Trigonometry curriculum is one of five math courses offered at the high school level. Trigonometry 1b ( Tutorial with solved problems based on Trigonometric ratios ) Problems based on the concepts introduced above. 9/25 - Quadratic Formula & Completing the square. This negative attitude towards one of the major subjects in every field makes them reluctant and lose interest from studies. Are you confused while solving problems on Trigonometry Class 10? Do not worry, Visit Math Square and learn what is Trigonometry Class 10 and how to solve problems on Class 10 Trigonometry. NCERT Solutions for Class 12 Maths Chapter 2 – Inverse Trigonometric Functions PDF. 3 A contractor plans to install two slides for the children to play in a park. 1 will help you. Free PDF download of Chapter 3 - Trigonometric Functions Formula for Class 11 Maths. Here is the trigonometric formulas for Class 10 in one page Here is the trigonometric formulas for Class 10 in one page. The ppt given below will help the students to remember the formulas quickly. Trigonometry is a branch of mathematics that deal with angles, lengths and heights of triangles and relations between different parts of circles and other geometrical figures. Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the side of the right triangle. This PPT will help the students to learn physics formulas in an effective manner. Familiarity with the graphs. My 10th grade mathematics class was recently introduced to right-angled trigonometry and as we talked about its various applications. Grade 10 trigonometry problems and questions with answers and solutions are presented. Trigonometric formulas Differentiation formulas. q3 exercise 9. The strategy we adopt is to find one solution using knowledge of commonly occuring angles, and then use the symmetries in the graphs of the trigonometric functions to deduce additional solutions. The Time4Learning Trigonometry curriculum is one of five math courses offered at the high school level. Class 10 Math Chapter Wise Important Questions and Video Lessons– In this article, you can find the chapter wise important questions and video lessons for Class 10 Math. Games, activities and quizzes to help you learn and practice trigonometry, We have games for SOHCAHTOA, Right Triangles, Trig Ratios, Unit Circle, Trig Identities, Trig Formulas, Law of Sines, Law of Cosines, Trigonometric Graphs, Inverse Trigonometry and Quizzes, examples with step by step solutions, worksheets. Useful handbook of Maths formulas for year-round preparation. Most of trigonometry formulas play around with trigonometric ratios. It is good practice in rather more involved algebra than you would normally do in a trigonometry course. Home ; Class 9. One end of a rope is attached to the top of a pole 100 ft high. We will review this concept very briefly. The paper is divided into two sections: Section I (40 marks) and Section II (40 marks). It begins with providing an overview of the entire course and thereby talks about the Definition and Application of Trigonometry. on the trigonometric functions, and with calculations based on these functions. Images in Dave’s Short Trig Course are illustrated with a Java applet. Formulas from every topic is covered in depth, so that it will help u during ur preparation. In the Trigonometry chapter, we shall study Trigonometric ratios of some specific angles and Complementary angles; and Trigonometric identities. Sum-Difference Formulas. Some real-life examples like finding the height of Qutub Minar, finding the width of the river, and finding the altitude of the ground from the hot air balloon are given in order to explain the need for trigonometry. On this page Sin - half angle identity A reader is going to take a trigonometry class soon and asks what it's about. Basic trig functions - practice problems These problems are designed to help you learn basic trigonometry ("trig") functions and how to use your calculator correctly. Law of Sines. Practice 2 Fri 16:00 – 17:30. It contains basic trig identities and formulas. Class 12 (CBSE) Mathematics. Make trigonometry your priority in class. The other two values will be filled in. 149 8 Another formula for tan aj2 151 9 Products to sums 152 10 Sums to products. Enhance the appreciation of trigonometry as a problem-solving tool. This is latest edition as published by the National Council of Educational Research and Training (NCERT) for academic. sin (-x) = -sin x 2. log a n is called logarithmic function. In month 1 the largest latent class is FF (32. Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. (Roots should be real) (Problems related to day-to-day activities to be incorporated). Grade 10 trigonometry problems and questions with answers and solutions are presented. Introduction to Trigonometry - CBSE Grade 10 MATHS Online Study Material - Get Free Sample Papers, Board Paper Solutions, Revision Quizzes, Video Lessons & Sample Tests on Brainnr. Functions-TF. The proof shows that Heron's formula is not some new and special property of triangles. This class calculates the Factorial of a numbers passed into the program through command line arguments. Now look at all the capital letters of the sentence which are O, H, A, H, O and A. Chapter 9 : Some Applications Of Trigonometry 9. This Trigonometry Lesson Plan is suitable for 9th - 11th Grade. The NCERT Solutions PDF Class 6 To 12 CBSE Notes, Books and Sample Papers Free To Download. The general formula for converting from degrees to radians is to simply multiply the number of degrees by $$\red { \frac {\pi}{ 180^{\circ}} }$$. R 1 0 x p 1+xdx 4 15 (1+ p 2) Discuss: does the best strategy for solving each of the following integrals use substitution, integration by parts. 9 Graphically show the result of multiplying a vector by a negative scalar. A trigonometric equation always has an infinite number of solutions, but it is customary to list only those angles between 0° and 360°. Pythagorean Identities Mathwords: Terms and Formulas from Algebra I to Calculus written,. Sorting Materials into Groups 5. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. This lecture note covers the following topics: The circular functions, Radians, Sinusoidal functions, Continuity of the trigonometric functions, Minima and Maxima, Concavity, Criteria for local maxima and minima, The Mean Value Theorem, The velocity of a falling object, Theoretical framework, Accumulation Functions, Minor shortcuts in taking definite. The earliest known work on trigonometry was recorded in Egypt and Babylon. 1 will help you. CBSE NCERT Solutions for Class 10 Mathematics Chapter 8. in a right-angled triangle. Check out our video on "Trigonometry Formulas" In this session we will get to know all about Trigonometry Formulae. ICSE Maths Trigonometry Practice Notes for Class 10. Distance between two points. One cannot escape from learning the formulae of reciprocal functions, periodicity of functions and Pythagorean identities. Reduction Formulas Law of cosinesLaw of cosines aaaa2222 = b== bb= b 2222 + c++ cc+ c222 –––– 2bc cos A2bc cos A2bc cos A Trig Identities Cheat Sheet. 1 class ten mathematics. In this chapter, you will be studying about some ways in which trigonometry is used in the life around you. Class 9; Class 10; Grade 11; Grade 12 Grade 12; PRACTICE. Getting To Know Plants 8. Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the side of the right triangle. The proof shows that Heron's formula is not some new and special property of triangles. Get FREE NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry Ex 9. tan) echo (r. Download it once and read it on your Kindle device, PC, phones or tablets. Trigonometry Table. Light, Shadows and Reflections 12. Fantom's Float library includes all six trigonometric functions, which assume the number is in radians. Maths Class 11 Chapter 3. There is a class of trigonometric functions known as inverse or “arc” functions which will do just that: “undo” a regular trigonometric function so as to leave the angle by itself. trigonometric formulas for class 12; trigonometric formulas pdf; trigonometric identities worksheet pdf; trigonometry formula sheet; trigonometry formula table; trigonometry formulas for class 10; trigonometry formulas for class 10 pdf; trigonometry formulas for class 11; trigonometry formulas list; trigonometry formulas pdf for class 11. Check out our video on "Trigonometry Formulas" In this session we will get to know all about Trigonometry Formulae. • When ∠ C is very close to 0°,. A formula can often be simplified, as was found by deriving the tangent formulas from the sine and cosine formulas, and changing it from terms using one ratio to terms using another ratio. NCERT Solutions for Class 11 Maths Trigonometry. REVIEW SHEETS. In mathematics, the trignometric functions relates the angles of a triangle to the length of its sides. Trigonometry ko start krne se pahale Sabse pahale jaante hai trigonometry kya hai ? Also read-:partial-fraction aanshik bhinnh tricks Trigonometry kya hai ?. For each of the following angle measurements (5 o , 15 o , 30 o , 45 o , 60 o , 75 o ,. The best app for CBSE students now provides Introduction to Trigonometry class 10 Notes latest chapter wise notes for quick preparation of CBSE board exams and school based annual examinations. NCERT Solutions for Class 12 Maths Chapter 2 – Inverse Trigonometric Functions PDF. Heron's formula is also a special case of the formula for the area of a trapezoid or trapezium based only on its sides. project on trigonometry for class 10+pdf Approach to the trigonometric functions, which is more intuitive for students to. We need to use the formula to find the general solution or some particular solution of different types of trigonometric equation. Calculates the Greatest Common Divisor of two numbers passed into the program through command line arguments. com पर स्वागत है. Trigonometric Identities You might like to read about Trigonometry first! Right Triangle. A circle centered in O and with radius = 1 is known as trigonometric circle or unit circle. Trigonometry Formulas are given here for Trigonometric Identities, Ratios & functions. Many time we need NCERT solutions for class 10 maths to solve a particular problem. Trigonometry is the branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees Triangles on a sphere are also studied, in spherical trigonometry. More References and links on Trigonometry Trigonometry. the Principles course as an application of the formula for the length of a line segment. log a n is called logarithmic function. These formula include all trigonometric ratios, trigonometric identities, trigonometric sign rule, quadrant rule and some of the value of the trigonometric function of specific degrees. Sum-to-Product Formulas. Get Trigonometry homework help online. Trigonometric Equation Formula We will discuss about the trigonometric equation formula. The entire NCERT textbook questions have been solved by best teachers for you. More important identities Less important identities Truly obscure identities About the Java applet. Members-Only Access. It will help the students to memories the formulas very quickly. Games, activities and quizzes to help you learn and practice trigonometry, We have games for SOHCAHTOA, Right Triangles, Trig Ratios, Unit Circle, Trig Identities, Trig Formulas, Law of Sines, Law of Cosines, Trigonometric Graphs, Inverse Trigonometry and Quizzes, examples with step by step solutions, worksheets. x and y are independent variables, ; d is the differential operator, int is the integration operator,. All types of questions are solved for all topics. This Law of Sines Fun No Prep Fun Notes for Trigonometry is a great comprehensive, challenging, and fun addition to your Algebra 2, Trig, or PreCalculus classes. 9-inch wider track, along with the 3. In case you need it, here is the Triangle Angle Calculator , and the Right Triangle Angle And Side Calculator. Trigonometry For Dummies - Kindle edition by Mary Jane Sterling. Power-Reducing/Half Angle Formulas. tbilisi/1538532025 Mohammad Saeed Khan, Dinu Teodorescu. There is an enormous number of uses of trigonometry and formula of trigonometry. Familiarity with the graphs. Division and Multiplication. Trigonometric ratios of an angle are some ratios of the sides of a right triangle with respect to its acute angles. Hailed as the pinnacle of O-Level Trigonometry, a typical R-Formula question may also require your skills on lesser trigonometry variants such as the Addition Formulae, or the Double Angle Formulae, or the Factor Formulae, together with your knowledge of basic angles (e. trigonometric and inverse trigonometric identities for class XI and XII Can you add all the formulas that exist in trigonometry inot this earth trigonometric. 7 or the handout given in class on inverse functions. Class 8; Class 9; Class 10; Grade 11; Grade 12; Class 9 Optional Mathematics Solution. Inverses, power-reduction and angle are also included. Home › Math › Easy Trig Identities With Euler's Formula Trig identities are notoriously difficult to memorize: here's how to learn them without losing your mind. INTRODUCTION TO TRIGONOMETRY (10) Periods Trigonometric ratios of an acute angle of a right-angled triangle. The general formula for converting from degrees to radians is to simply multiply the number of degrees by $$\red { \frac {\pi}{ 180^{\circ}} }$$. I hadn't studied Trig for over 20 years, and I'm glad I started with this book, especially since Trigonometry Refresher (Klaf), Trigonometry (Gelfand), and Trigonometry (Corral) -- the supposed "best of the bunch" for introductory texts -- were often not nearly as user-friendly (though had fewer errors and were indeed excellent). Synonym's The Classroom covers more than just homework and study tips. Acellus Trigonometry is A-G Approved through the University of California. 4 of the Elements is the angle-side-angle congruence theorem which states that a triangle is determined by any two angles and. Practice 1 Fri 12:00 – 13:30. http://aridutin. 1 Page No: 181 1. It will help the students to memorise the formulas very quickly. trigonometry-trigonometric identities solution of ncert exercise 8. Notesgen is the No. Now look at all the capital letters of the sentence which are O, H, A, H, O and A. Here is the total set of formulas compiled for students of class XI-XII and those giving various entrance exams based on 10+2 syllabus. It is suitable for a one-semester course at the college level, though it could also be used in high schools. Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the side of the right triangle. 3 in the appendix to your textbook. He considered every triangle—planar or spherical—as being inscribed in a circle, so that each side becomes a chord (that is, a straight line that connects two points on a curve or surface, as shown by the inscribed triangle ABC in. Learn more about Trigonometric Ratios here in detail. It would be useful for last minute revision. Most of trigonometry formulas play around with trigonometric ratios. com provides step by step solutions for Selina Concise Mathematics Class 9 ICSE Solutions Chapter 22 Trigonometrical Ratios [Sine, Cosine, Tangent of an Angle and their Reciprocals]. The Trigonometric ratios table helps you to find the values of trigonometric standard angles 0°, 30°, 45°, 60° and 90°. Aryabhata knew the formula sin 2 ɸ + cos 2 ɸ = 1, as well as formulas (3), which he used to construct a table of sines at intervals of 3°45’ on the basis of the known values of the trigonometric functions for simple arguments (π/3, π/6). class-10 Mathematics chapter-8 Introduction to Trigonometry in PDF files. We hope the given RBSE Solutions for Class 9 Maths Chapter 14 Trigonometric Ratios of Acute Angles Ex 14. Mathematics Class 11th complete solve exercises notes as new revised pattern (adamjee coaching centre notes) - Karachi Board - Multiple Choice Questions (Mcqs), Online Mcqs, Theory & Questions Answers, Scientific Reasons of Mathematics FSC, First Year, 11th. Now look at all the capital letters of the sentence which are O, H, A, H, O and A. Trigonometry – Mathematics GCSE Revision Skip to main content. Tricks to remember trigonometric identities and how to use them to solve "prove that" trigonometry sums in math. INTRODUCTION TO TRIGONOMETRY (10) Periods Trigonometric ratios of an acute angle of a right-angled triangle. Get [Chapter], [Sub-Subject] Chapter Notes, Questions & Answers, Video Lessons, Practice Test and more for CBSE Class 10 at TopperLearning. The trigonometric functions -- sine, cosine, and tangent -- are based on a right-angled triangle (a triangle containing an angle equal to 90 degrees). If you will not show a positive attitude towards learning mathematics then there are chances that may lose your interest from learning. Having all the formula in one place is always helpful for the students, Keeping that in mind,We have prepared a Maths formula pdf for CBSE Class 9 students. Learn about the different Trigonometric ratios: sin Ɵ, cos Ɵ, tan Ɵ, cosec Ɵ, sec Ɵ and cot Ɵ. q9 exercise 9. FREE NCERT Solutions for class 10 Math, Chapter 9 - Some Applications Of Trigonometry from NCERT Textbook (Math Ncert Solutions). Trigonometry can be defined as the calculation part of geometry. Homework Help in Trigonometry from CliffsNotes! Need help with your Trigonometry homework and tests? These articles can help you understand the advanced math c. Here is a quick review of class X maths formulas chapter wise for all those students of class X. Notesgen is the No. : 'mar johns' will return a list that includes 'Mary Johnson. NCERT Solutions for other subjects of class 9 are also given in the form of PDF file. In month 2, a substantial proportion of the FF class included SFF; which increases over time. The content in these areas includes high school mathematics and statistics at a level that is generally no higher than a second course in algebra; it does not include trigonometry, calculus or other higher-level mathematics. Trigonometric functions are important when studying triangles and modeling periodic phenomena such as waves, sound, and light. Precalculus with Geometry and Trigonometry by Avinash Sathaye, Professor of Mathematics 1 Department of Mathematics, University of Kentucky Aryabhat¯. Class 10 Maths applications of trigonometry, how to find trigonometric ratios, onlinepsa trigonometry, Trignometry NCERT class 10, Trigonometry, Trigonometry basics, Trigonometry chart, Trigonometry class 10, Trigonometry exercise 8. Get, detailed solutions to the questions of the chapter Inverse Trigonometric Functions from NCERT textbooks. On the new SAT (starting March 2016) and new PSAT (starting October 2015) you must also be. Get formulae of class 12 maths chapter 2 Inverse Trigonometric Functions Formulas and revise your concepts of Inverse Trigonometric Functions Formulas. Trigonometry - ICSE Solutions for Class 10 Mathematics ICSE SolutionsSelina ICSE Solutions Get ICSE Solutions for Class 10 Mathematics Chapter 18 Trigonometry for ICSE Board Examinations on APlusTopper. 9 cm (which, if you’re going off of all of the lengths and angles besides the bottom one, is still not rounded well. Bhaskara (12th century) gave a method of constructing tables at intervals of 1° through the use of. For most of the students, class 9 th Maths is the nightmare and the formulas are very difficult to learn. Gain complete understanding of Trigonometry with our free trigonometry course covering formulas, degrees, equations and more. So first, we'll convert the drop to miles (we could have converted the distance to feet - it doesn't matter. Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p < (4 tan2 θ + 9 cot2θ) / 2 ≥ 6. 22 Mar 2018 - 25 Mar 2018. A keen aptitude for math improves critical thinking and promotes problem-solving abilities. 9 Graphically show the result of multiplying a vector by a negative scalar. Triangle area formulas: ½*base*height=Area. 81m/s 2) h = height Kinetic energy (P). includes problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks for 9th and 10th grade students, in the period 1981-1988, when I was a professor of mathematics at the "Petrache Poenaru" National College in Balcesti, Valcea (Romania), Lycée Sidi El Hassan Lyoussi in Sefrou (Morocco),. The Plus teacher packages are designed to give teachers (and students) easy access to Plus content on a particular subject area. For instance, Proposition I.
|
{}
|
Small tags/sites
21 tags on
Small tags/sites
## How to show javac Warnings in Problems view in IntelliJ IDEA?
In Eclipse the Problems View displays javac errors + javac warnings, whereas in IntelliJ I get only the errors. While warnings are highlighted when editing a file, I would like to have a list of all ...
## How to encrypt webapp in Tomcat
I have the following problem: 1) I have a server on tomcat with database 2) I have a thick client on client machines, which uses it's own tomcat and communicates with server via database ...
## Android: Execute a command on a server via ssh
I am using JSch to connect to a remote server within my android app. I am able to execute a simple command such as "mkdir android_test", in which case I can check on the server that this directory ...
## Lightweight crypto library to verify a signature
Are there any open source crypto libraries that can be used in embedded systems or in other memory constrained places like a boot rom? I am looking for libraries that I can compile only the algorithms ...
## ECC and RSA 10K stronger claim
There are a lot of websites where it is claimed something akin to this but no explanation on the basis. "ECC key size of 256 bits is equivalent to a 3072-bit RSA key and 10,000 times stronger than a ...
## Is $G$ a secure PRG?
Is the following function a secure PRG? Given $F$ is a secure PRG and $k$ is choosen random from key space $K$. $$G(x) = F(k,x) \oplus F(k,x \oplus 1^s)$$ My solution is $x \oplus 1^s = x'$ ...
## The Twenty Doors (ROOM 8)
You move on into Room Eight when Kslkgh (a.k.a. Dr. X) finally creates the next room. There is, as always, a slip of paper on the floor: Before the alphabet, before. The pass, between the alleys. ...
1 answers | 1 hour ago by Kslkgh on puzzling.stackexchange.com
## The Twenty Doors (ROOM 7)
This is part of The Twenty Doors series. The previous one is The Twenty Doors! (ROOM 6). The next one is The Twenty Doors (ROOM 8) You (after many days rest in Room 6) finally go into Room 7. The ...
1 answers | 1 hour ago by Kslkgh on puzzling.stackexchange.com
## The Twenty Doors! (ROOM 1)
This is part of The Twenty Doors series. The next one is The Twenty Doors! (ROOM 2) You stand in a small, dark room. You have no idea where you are. Then, you see a message in front of you. ...
3 answers | 1 hour ago by Kslkgh on puzzling.stackexchange.com
## Adapt existing GOST code in C to hash a file
I am trying to hand-code the GOST hash function using C. I came across the following code by Markku-Juhani Saarinen (from link). /* * gosthash.c * 21 Apr 1998 Markku-Juhani Saarinen ...
3 answers | 1 hour ago by IT_guy on stackoverflow.com
## BouncyTree Encryption
I'm currently in a run around issue. I'm looking to use BouncyCastle PGP for encryption of files. The situation is that I have public keys (basic asc) that are supplied that I need to use. I've been ...
1 answers | 1 hour ago by Peter on stackoverflow.com
## Encrypting a string in Java and decrypting it in C++
I need to encrypt a string in Java and decrypt it in C++. I've seen C++ has a Crypto++ library and Java has JCE, Jasypt, BouncyCastle etc. but I'm getting more and more confused... I'm searching for ...
## Intercept activiti database operations
I'm working with Activiti 5.14, and I have a requirement: All the data in the database should be encrypted. So, was trying to come up with a solution encrypting the data before to pass to the ...
## How to center LaTeX xtable (And Figure) output in full text width
This is a follow-up to a question I posted earlier (How to center LaTeX xtable output in full text width). I realize that my MWE from this previous post was incomplete. In an effort to make it as ...
## Compiling Latex in Vim Results in Error
I am trying to compile the following LaTeX using vimtex continuous compiling in vim: LaTeX: \documentclass[12pt]{article} \title{Electrolysis of Water and Aqueous Solutions} \author{Carlo Abelli \\ ...
## How to encrypt user passwords for forum registrations?
What is the most secure way to encrypt user passwords for phpBB or MyBB forum registrations? I don't want anyone to be able to access the user passwords, not even those who administrate the MySQL ...
## Prove that certain amount of data was stored
I'm looking for a way to prove that a certain amount of data was stored, through some easily verifiable piece of information. Similarly to how proof-of-work can prove through a hash that a certain ...
## Knowing that $G(s)$ is a PRG, is the following construction $G'(s) = G(s||0)$ a PRG?
I know there already is a very similar question. However, I can't understand why $G'(s)$ is not a PRG. If $G(s)$ is a PRG, then why $G(s||0)$ can't also be a PRG? How can the distribution over ...
## ElGamal signatures systems
Let $p$ prime number $q/p-1$ prime and $g\in (Z/pZ)^*$ element of order $q$.Also $a\in \{1,...,q-1\}$ the private key and $y\equiv g^a\pmod p$ the correspoding public key.For each of the following ...
## Present an attack for the combination of OTP and textbook RSA
Let $N=p \times q$ where $p$ and $q$ are two primes. Compute $\Phi (N) = (p-1)(q-1)$. Choose $e$ s.t. $1 < e < \Phi(N)$ and $gcd(e,\Phi(N))=1$. Compute $d = e^{-1}(mod \ \Phi(N))$. ...
1 answers | 2 hours ago by Cauthon on crypto.stackexchange.com
## knitr LaTeX single quotes not typesetting correctly
ers, Our lab is having trouble getting R code containing single quotes to typeset correctly in LaTeX. We are using RStudio (pdflatex) to compile the .Rnw to .pdf, but the problem persists even when ...
## AES encryption. How to convert IV and salt to NSString?
For a project of mine I have to AES encrypt a file. That's no problem. But I also have to save the iv and salt that is generated into a NSString. I have tried everything to no avail. Here is my ...
## How does the log(p(x,y)) normalize the pointwise mutual information?
I'm trying to understand the normalized form of pointwise mutual information. $npmi = \frac{pmi(x,y)}{log(p(x,y))}$ Why does the log joint probability normalize the pointwise mutual information to ...
## Why should I use Authenticated Encryption instead of just encryption?
There are various different modes of operation for block cipher use, some of which provide "encryption" and some of which provide authenticated encryption. Why should I use an authenticated ...
## Is there a proof for showing any cryptogram is crackable?
I commonly hear statements along the lines of "all cryptograms are crackable - it's only a matter of time". Is there a proof to show that any cryptogram is "crackable"? The proof may be of a more ...
## Simply put, what does perfect secrecy means?
I would like to ask for a clear (but maybe not so deep) explanation of what the term "perfect secrecy" means. As far as I have researched and understood, it has to do with probabilities of assuming ...
## Problem with installation of LaTeXDraw!
I was using LaTexDraw 2, which I had got it form Ubuntu Software Center. As the functions were limited and I couldn't use math to label out, I searched and found this new 3.1 version. After ...
## Can we solve the Hidden Number Problem in $GF(2^n)$?
Is it possible to solve the Hidden Number Problem in extension fields? In particular in $GF(2^n)$? Let's suppose an attacker knows some least/most significant bits of $r_i = a_i \times k$ in a given ...
1 answers | 3 hours ago by Ruggero on crypto.stackexchange.com
## Embed LateX file to HTML
I am looking for a way to load a LateX file to HTML. I found several questions about that but most of them are talking about for adding LateX equations to HTML or to convert LateX to HTML or to add ...
1 answers | 3 hours ago by John on stackoverflow.com
|
{}
|
# Engaging students: Finding the focus and directrix of a parabola
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Brittnee Lein. Her topic, from Precalculus: finding the focus and directrix of a parabola.
What are the contributions of various cultures to this topic?
Parabolas (as we know them) were first written about in Apollonius’s Conics. Apollonius stated that parabolas were the result of a plane cutting a double right circular cone at an angle parallel to the vertical angle (α). So, what does that actually mean?
Well, if we take a vertical line and intersect it with a straight line at a fixed point, and then rotate that straight line around the fixed point we form the shape below:
If the plane slices the cone at the angle β and β=α, a parabola is formed. This is still how we define parabolas today although you may not think about it that way. When you think of a parabola, you think of the equation $y = ax^2 +bx + c$. This equation is derived using the focus and the directrix. This video shows how to do so:
Understanding how the focus and directrix affect the equation of a parabola is crucial to understanding what each word means. According to mathwords.com, “For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.” The directrix is a line perpendicular to the axis of symmetry and the focus falls on the line of the axis of symmetry.
How can technology be used to effectively engage students with this topic?
This desmos activity can be used to show students how changing the focus, directrix, and vertex of the parabola affects the graph. https://www.desmos.com/calculator/y90ffrzmco
From this, students can shift values of the vertex and see that the directrix stays constant when the x-value is changed and that the focus remains constant when the y-value is shifted. If students change the value of the focus, they can see how it stretches and contracts the width of the parabola and how the directrix shifts. They can also see that when the focus is negative, the parabola opens downward and the directrix is positive. This website: https://www.intmath.com/plane-analytic-geometry/parabola-interactive.php Is also very helpful in showing the relationships between the focus, directrix and the graph of the parabolas because students can clearly see that the distance between a point on the parabola and the focus and the distance between that same point and the directrix are equal.
What interesting (i.e., uncontrived) word problems using this topic can your students do now?
The website http://www.purplemath.com/modules/parabola4.htm has a lot of great real-world word problems involving finding the focus and the directrix of a parabola. For example, one of the questions is:
(This is a graph I made using desmos to model the situation at hand)
This problem requires a lot of prior knowledge of parabolas and really tests students’ ability to interpret information. From the question alone, the students can find the x-intercepts (-15,0) and (15,0) from the information “the base has a width of 30 feet”. They are also able to infer that the slope of the parabola will be negative because of the shape of an arch. The student must also know how to find the slope of the parabola using the x-intercepts, solving for the equation of the parabola using the x-intercepts and vertex and the equations for finding the focus and directrix from the given information. There are a few problems as involved as this one on the listed website above.
# Engaging students: Finite geometric series
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Caroline Wick. Her topic, from Precalculus: finite geometric series.
What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?
Finite geometric series was a concept that began over 4500 year ago in ancient Egypt. The Egyptians used this method of finite geometric series mainly to “solve problems dealing with areas of fields and volumes of granaries” but used it for many other uses too, including the pyramids and math problems similar to those one might find on a STAAR test today (see D1, and F1).
There are seven houses; in each house, there are 7 cats; each cat kills seven mice; each mouse has eaten 7 grains of barley; each grain would have produced 7 hekat. What is the sum of all the enumerated things?
Years passed and finite geometric series were not revisited until around 350 BC by the Greeks, namely Archimedes, who came up with a solution to the math problem V=1/3Ah by finite calculations instead of limits. In addition, the idea that a finite sum could be procured from an infinite series was created in what is called the “Achilles Paradox” (D2, F2).
Years after this came Mathematicians in the middle ages, like Richard Swineshead or Nicole Oresme, who aided the world by further refining these series. This eventually led to the renowned Physicist Isaac Newton to “discover the geometric series” after studying mathematician John Wallis’s method of “finding area under a hyperbola” (F1). We can attribute almost all of what we know about geometric series’ to these fine gentlemen above, and they can only attribute what they know from the ancient Egyptians and Greeks.
How has this topic appeared in pop culture?
In 2002, PBS came out with a kids’ TV show called CyberChase, which is an entertaining cartoon about a bunch of kids who get pulled into “Cyber Space” to fight the bad guy, named Hacker, all while discovering and using different mathematical concepts that they learned along the way. Eleven seasons have passed since the shows beginning and it is still going strong, but one episode that still sticks out to me was their version of explaining geometric series to kids. The episode was called “Double trouble” and was the 9th episode of the second season. The specific geometric series involved in the episode was doubling, but the “real world” clip at the end stood out more vividly to me. After losing a chess game, the main character has to decide between paying the winner $5.00 or paying one penny for the first space on a chess board, then two pennies on the second, then four on the third, and continuing to double the previous number for every space on the entire chess board. Since the main character thought pennies were less, he decided on the second option, only realize after that he would have to pay way more than$5.00 in the end. This helped me understand the most basic geometric series when I was a kid, and has stuck with me to this day, so I am certain that it has and can stick in other students’ brains as well.
Here is the clip from the show:
How can this topic be used in your students’ future courses in math or science?
The idea of finite geometric series is typically lightly introduced around students’ sophomore year of high school when they take geometry, but it is not really expanded upon/explained until students reach Pre-Calculus. The specific TEKS related to this topic are located under Pre-Calculus in (5), (A)-(E) (Source B1). The concept is brought up again in Math Models with Applications and is used for understanding interest on a balance over a period of time, or “loan amortization.” The ideas can also be used to help understand difference equations that involve heat and cooling over a period of time, and how to predict what the temperature might be in the future, which is a concept that is important in the realm of science too.
When students get to college, finite geometric series are expanded upon even more when they take Calculus classes, and they will learn how to prove a series is finite using induction when they get to their Discrete Classes and Real Analysis classes. In the business realm, they will have to use it to predict monetary sums regarding interest and possible growth in a company, so likely no matter where a student ends up, s/he will have to use this important mathematical concept everywhere.
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Cody Jacobs. His topic, from Precalculus: using radians to measure angles instead of degrees
How could you as a teacher create an activity or project that involves your topic?
How can this topic be used in your students’ future courses in mathematics or science?
How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?
Desmos.com is yet again another great technological resource to use when introducing radians to the classroom. There is a great activity call “What is a Radian?” That introduces an activity student can do using a plate and folding it into different sections. I actually believe this is how I was introduced to radians in high school. The second part of the activity on desmos after you finish with the plate introduction, is asking students key questions. How many radians are in a certain degree measurements? How many degrees are in a certain radian measurement? As always desmos is still great at introducing radians and lets you easily monitor your students progress.
# Slightly Incorrect Ugly Mathematical Christmas T-Shirts: Part 2
This was another T-shirt that I found in my search for the perfect ugly mathematical Christmas sweater: https://www.amazon.com/Pascals-Triangle-Math-Christmas-shirt/dp/B07KJS5SM2/I love the artistry of this shirt; the “ornaments” at the corners of the hexagons and the presents under the tree are nice touches.
There’s only one small problem:
$\displaystyle {8 \choose 3} = \displaystyle {8 \choose 5} = \displaystyle \frac{8!}{3! \times 5!} = 56$.
Oops.
# Engaging students: Using Pascal’s triangle
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Rachel Delflache. Her topic, from Precalculus: using Pascal’s triangle.
How does this topic expand what your students would have learned in previous courses?
In previous courses students have learned how to expand binomials, however after $(x+y)^3$ the process of expanding the binomial by hand can become tedious. Pascal’s triangle allows for a simpler way to expand binomials. When counting the rows, the top row is row 0, and is equal to one. This correlates to $(x+y)^0 =1$. Similarly, row 2 is 1 2 1, correlating to $(x+y)^2 = 1x^2 + 2xy + 1y^2$. The pattern can be used to find any binomial expansion, as long as the correct row is found. The powers in each term also follow a pattern, for example look at $(x+y)^4$:
$1x^4y^0 + 4x^3y^1 + 6x^2y^2 + 4x^1y^3 + 1x^0y^4$
In this expansion it can be seen that in the first term of the expansion the first monomial is raised to the original power, and in each term the power of the first monomial decreases by one. Conversely, the second monomial is raised to the power of 0 in the first term of the expansion, and increases by a power of 1 for each subsequent term in the expansion until it is equal to the original power of the binomial.
Sierpinski’s Triangle is triangle that was characterized by Wacław Sieriński in 1915. Sierpinski’s triangle is a fractal of an equilateral triangle which is subdivided recursively. A fractal is a design that is geometrically constructed so that it is similar to itself at different angles. In this particular construction, the original shape is an equilateral triangle which is subdivided into four smaller triangles. Then the middle triangle is whited out. Each black triangle is then subdivided again, and the patter continues as illustrated below.
Sierpinski’s triangle can be created using Pascal’s triangle by shading in the odd numbers and leaving the even numbers white. The following video shows this creation in practice.
What are the contributions of various cultures to this topic?
The pattern of Pascal’s triangle can be seen as far back as the 11th century. In the 11th century Pascal’s triangle was studied in both Persia and China by Oman Khayyam and Jia Xian, respectively. While Xian did not study Pascal’s triangle exactly, he did study a triangular representation of coefficients. Xian’s triangle was further studied in 13th century China by Yang Hui, who made it more widely known, which is why Pascal’s triangle is commonly called the Yanghui triangle in China. Pascal’s triangle was later studies in the 17th century by Blaise Pascal, for whom it was named for. While Pascal did not discover the number patter, he did discover many new uses for the pattern which were published in his book Traité du Triangle Arithméthique. It is due to the discovery of these uses that the triangle was named for Pascal.
# Engaging students: Defining sine, cosine and tangent in a right triangle
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Jessica Williams. Her topic, from Precalculus: defining sine, cosine and tangent in a right triangle.
How could you as a teacher create an activity or project that involves your topic?
I know of a good project/activity for the students to do that will be extremely engaging. You could either do this for an elaborate activity for your students or maybe an opening activity for day 2 of a lesson. For my class, I would get a square cookie cake, and have the slices cut into right triangles. I would allow each student to have a piece (but not eat it just yet). The students will be provided with rulers and a protractor. The students will each measure the hypotenuse of their cookie cake and the degree of whichever angle you would like them to measure, however each student should be measuring the same parts so do this unanimously). As a class, decide on an average for the measurements for everyone to use so that the data is not off. Then take the supplies away from the students and ask the students to find the rest of the missing sides and angles of their piece of cookie cake. They will also be provided with a worksheet to go along with this activity. This is a good review activity or al elaborate activity to allow further practice of real world application of right triangle trigonometry. Then go over as a class step by step how they solved for their missing angles and side lengths and make each group be accountable for sharing one of them. This allows the students to all be actively participating. Through out the lesson, make sure to tell the kids as long as they are all participating they will get to eat their slice when the lesson is done. Lastly, allow the students to eat their slice of cookie cake.
How does this topic extend what your students should have learned in previous courses?
Prior to learning about right triangle trigonometry the students will know how to use the Phythagorean Theorem to find how long the missing side length is of a right triangle. The students know basic triangle information such as, the sum of the angles in a triangle is 180 degrees. The students already know the difference between the hypotenuse and the other two legs. The students know that hypotenuse will be the longest leg and the leg across from the 90 degree angle. The students will also know the meaning of a fraction or ratio. The students may need some refreshing of memory on some parts of prior knowledge, but as teachers we know this is an extremely important part of a lesson plan. Even as teacher we tend to forget things and require a jog of memory. A simple activity such as headbands or a kahoot with vocabulary would be an excellent idea for accessing the students prior knowledge. This allows the students to formally assess themselves and where they stand with the knowledge. Also, it allows the teacher to formally assess the students and see what they remember or parts they are struggling on. This allows the teacher to know what things to spend more time on.
How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?
Technology is always an amazing aspect of the classroom. Like stated above, a vocab review using headbands or kahoot would be a good idea for this type of lesson that DEFINITELY needs prior knowledge to be applied in order to succeed. Also, showing the students how to plug in sine, cosine, and tangent is crucial. They have seen these buttons on the calculator but they do not know what they mean or how to use them. Using an online TI on display for the class is great. I had to do this with my 10th grade students to make sure they understood how to use the 3 buttons. Also, when using arcsin, arccos, and arctan it can be confusing. Using technology to show the class as a whole is the best route to go. Also, technology can used as review for a homework assignment or even extra credit for the students. It benefits them by getting extra review and extra credit points. I found a website called http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRI&lesson=html/object_interactives/trigonometry/use_it.html , which is a golf game that requires review of triangles and trigonometry. It allows the students to practice the ratios of SOH-CAH-TOA using a given triangle.
# Engaging students: Half-life
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Kerryana Medlin. Her topic: working with the half-life of a radioactive element.
How can this topic be used in you students’ future courses in mathematics or science?
Depending on when they take precalculus, this topic may appear earlier or later in chemistry. The following is the list of TEKS for this topic in chemistry.
112.35. Chemistry (12) Science concepts. The student understands the basic processes of nuclear chemistry. The student is expected to:
(A) describe the characteristics of alpha, beta, and gamma radiation;
(B) describe radioactive decay process in terms of balanced nuclear equations; and
(C) compare fission and fusion reactions.
This is likely the most immediate application the students will encounter, but this topic also appears in calculus and, later, in the topic of differential equations, since it involves exponential decay. This topic can also be brought up in environmental science to mention the lifetime of radioactive isotopes. When a student crunches the numbers on the lifetimes of these isotopes, they can see that sometimes a small action has a huge ripple effect, especially for isotopes that humans bring into the picture.
What interesting things can you say about the people who contributed to the discovery and/ or the development of this topic?
Ernest Rutherford received a Nobel Prize in Chemistry in 1908 for his discovery of the half-life of radioactive materials and his insistence that we apply this information to find the Earth’s age (Mastin, 2009). This later became more of a reality when Willard Libby started to develop carbon dating in 1946 (Radiocarbon Dating). Since then, carbon dating has been used to find the age of historical artifacts and bones, allowing historians to find more accurate time frames of events.
Carbon is not the only radioactive isotope. There are others which come to mind more readily when the word “radioactive” is used. These are typically the elements used for nuclear reactors. These are elements which readily undergo nuclear fission, which is the splitting of atoms, which releases energy. Uranium and Plutonium are the most common of these isotopes. Uranium-235 is the most commonly used for reactors and bombs (Brain and Lamb, 2000). This is probably the more interesting part of half-lives of elements and can extend the learning to an environmental issue such as nuclear waste, which takes an extremely long time to decay and which the U.S. Government has, in the past, not handled so well. (But I am not going into that, lest I go on a rant).
The last piece of history worth mentioning is fairly recent (and can be seen in real life and in the game mentioned later in this paper) which is that half-lives are not so clear cut. There is definitely a lot of estimating involved in the accepted half-life values. There is an article about this if you are interested (http://iopscience.iop.org/article/10.1088/0026-1394/52/3/S51/pdf), but I will leave it at this: much like most mathematical models, there is error in the half-life model, and the model formed may be a best fit, but there are always outliers for data and while carbon dating and half-lives of Uranium can give great estimates of what we are working with, they are not perfect.
How can technology be used to effectively engage students with this topic?
For this topic, there is an interactive simulation posted on PHET. It lends itself to a guided worksheet which would allow students to use the simulations to create the functions for each half-life.
So the following would be an example of said worksheet without spaces for actual answers:
Radioactive Half-Life of Carbon-14 and Uranium-238
At the top of the game window are four different tabs: Half Life, Decay Rates, Measurement, and Dating Game. We will be going through each one in that order.
Some information about radioactive isotopes: An isotope is an element which has the same number of protons in its nucleus, but a differing number of neutrons, thus making it radioactive. These elements have lives which are defined by the time it takes to no longer be radioactive.
Part I: Half Life
Select the Carbon-14 atom and start placing the atoms in the white area. (The “add 10” tool is helpful here.) Then observe as each goes to Nitrogen-14 (This means the element is no longer radioactive and the radioactive isotope has run its course.)
What do you observe about the lives of the isotopes?
What time-frame do these lives fall into?
Do the same for Uranium-238 and record the time-frame.
Part II: Decay Rates
This part works by adjusting the slider and allowing the isotopes to run the course of their lives.
What does the graph on the bottom tell us?
How does one read the half-life of an isotope from this graph?
At what percent do we find the first half-life?
What is the half-life of Carbon-14 from this graph? Half-life of Uranium-238?
Part III: Measurement
On this one, you activate two separate events and then take readings of the amount of Carbon-14 and Uranium-238 in the objects.
Which item contains the Carbon-14? The Uranium-238?
Use the pause feature as you are taking the readings to find precise values of the half-lives.
At what percentages should we be reading the half-lives?
Use this data to create a function to model the half-life of both isotopes.
Part IV: Dating Game
Use your functions to estimate the date of two of the items (One C-14 and one U-238) in the dating game. Write down the name of the item and the estimated age of the item.
References:
Brain, Marshall and Lamb, Robert. (2000). How Nuclear Power Works. How Stuff Works. Retrieved from
https://science.howstuffworks.com/nuclear-power1.htm
Mastin, Luke. (2009). Important Scientists: Ernest Rutherford (1871-1937). The Physics of the Universe.
# Engaging students: Finding the equation of a circle
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Kelsi Kolbe. Her topic, from Precalculus: finding the equation of a circle.
How can technology be used in order to engage the students on this topic?
A simple Desmos program can be used to see different circles and how the variables affect it. You can write a program on Desmos, where you have to manipulate a given circle to ‘collect all the stars.’ There are stars placed around where the circumference should be. Then the students you a variety of sliders to collect the stars. The sliders can change the radius, and move the circle left to right. I think this simple activity will introduce the parts of a circle equation, like the radius and the center, while the students have fun trying to beat their fellow classmates collect the most stars.
How could you as a teacher create an activity or project that involves your topic?
I think a circle themed “Clue” inspired activity could be fun. I would tell the students that there was a crime committed and the students had to use their math skills to figure out what the crime was, who did it, where they did it, and when they did it. The students would get an ‘investigation sheet’ to record their answers. Each group would start off with a question like, ‘Find the equation of a circle that has the center (2,3) and radius 7’. Each table would have an answer to the math questions that corresponds to a clue to answer one of the ‘who, what, where, where’ questions they are trying to figure out, and prompts the next question. Students would continue this process until one team thinks they have it and shouts “EUREKA!” then they say what they think happened and if they are right they win, if they aren’t we keep going until someone does.
How has this topic appeared in high culture (art, classical music, theatre, etc.)?
Circles are seen in a lot of different Islamic Art. Islamic art is known for its geometrical mosaic art. They had a deep fascination with Euclidean geometry. The circle specifically holds meaning in the Islamic culture. The circle represents unity under a monotheistic God. Their religion is so important it can be seen throughout every aspect of their culture. The repetitiveness also symbolizes god infinite nature. For example, his infinite wisdom and love. Along with circles, the 8-point star is also seen as a very powerful symbol. It represents God’s light spreading over the world. The symbols are very important in the Islamic culture and is shown beautifully in a lot of their art. It’s beautiful how they can pack one art piece with so much geometry and also their beliefs.
# Engaging students: Exponential Growth and Decay
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Megan Termini. Her topic, from Precalculus: exponential growth and decay.
How could you as a teacher create an activity or project that involves your topic?
A fun and engaging activity for students learning about exponential growth and decay would be a zombie activity. The students will get a scenario about the zombie attacks and they will predict the way the zombie attacks will work. Then to begin, the teacher will be the only one infected and to show the infection, they will have a red dot on their hand. Then they will shut off the lights and turn them back on to indicate a new day. Then the teacher will “infect” one other student by putting a red dot on their hand. Then they will turn the lights off and turn back on for day 2. Then both the teacher and the infected student will both go “infect” one other person. Then it continues day by day until everyone in the class is infected. Then they will put their data in a table, graph it and can see that it is an exponential growth, then write an equation for it (Reference A). This is great way of getting the whole class involved and zombies are very popular with tv shows and movies. It also lets them explore, see the pattern, and try to come up with the equation on their own.
How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?
A great use of technology for graphing exponential growth and decay is Desmos. Desmos lets the students take an equation and plug it in to see the graph. They are also able to change the window to see it better. It also will give you the table for the function that you inputted. It’s good for students to graph it on here to see the graph and also, they are able to click anywhere on the graph to see the point they want. This also would be a good program for them to check their work after trying the problem on their own first (Reference B). Another great website is Math Warehouse. This website lets students explore the graph of exponential functions. Students can type in their function and can graph it. It also lets you compare it to y=x, y=x2, and y=x3. It also has the properties for exponential growth and decay. This website is great for students to interact with exponential functions and also explore them (Reference C).
How can this topic be used in your students’ future courses in mathematics or science?
Exponential functions stay with you all through your school career. You use them in many mathematics courses like algebra, algebra 2, pre-calculus, calculus, etc. You also use them in science courses like biology, chemistry, physics, etc. Understanding how to graph exponential growth and decay functions is a very important tool for future courses. For example, in algebra 2 the students will be learning about logarithms and exponentials, and will have to graph both of them and know the difference between them. Another example is in biology, comparing the number of births and the number of deaths of a species. The data may show an exponential growth in the number of births and exponential decay in the number of deaths, and the students would need to know how to plot the data points and graph it. It is also important for them to understand what the graph means and not just how to graph it. These are skills students will need in not only their future mathematics and science courses, but also in their future careers. For example, a biologist who studies a species of animals might have an exponential decay of the animal and would track its progress every week or every day and graph it to show the decrease of the amount of that species. Many students may not realize it now, but graphing exponential growth and decay is an important topic to understand how to do and why it is important to learn.
References:
A. “Zombies: Exploring Exponential Growth.” BetterLesson, betterlesson.com/lesson/460610/zombies-exploring-exponential-growth.
B. “Exponential Growth and Decay.” Desmos Graphing Calculator, http://www.desmos.com/calculator/d7dnmu5cuq.
C. “Interactive Exponential Function Graph/Applet.” Exponential Growth/Decay Graph Applet . Explore graph and equation of exponential functions| Math Warehouse, http://www.mathwarehouse.com/exponential-growth-and-decay/interactive-exponential-graph-applet.php.
# Engaging students: Compound interest
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Michelle Contreras. Her topic, from Precalculus: compound interest.
How could you as a teacher create an activity or project that involves your topic?
Compound interest can be something difficult to understand sometimes. That’s why before I even start refreshing my future pre-calculus class about the general formulas they are going to be working with, I would like to start the lesson with a “game”/ activity. Starting class with this activity can be beneficial in the long run because they are going to be more willing to pay attention the rest of class. The game is my own little twist of what we know is the marshmallow game. In the marshmallow game the teacher hands a marshmallow to one of her students challenging him/her to just hold on to it for about 10 minutes and not eat it. If the student managed to hold on and not ingest the marshmallow then the student would get another extra marshmallow. The teacher then ups the reward to two marshmallows more if the student manages to not eat any of the two marshmallows already in their possession.
My own twist in this game is instead of handing one of my students a marshmallow and challenging him/her to not eat it, I would give the student a fun sized M&M’s baggy and challenge him/her with that particular candy. I would then tell my student if he/her manages to not eat the baggy of M&M’s for a minute I would give them another baggy at the end of the minute. While I’m waiting for this minute to be over I would instruct half of the class to give a 30 second argument of why he/she should eat the chocolate right then and there. Then I’ll instruct the other half of the class to make an argument against eating the chocolate for 30 seconds, making the choice for him/her even more difficult. If the student manages to not eat the M&M’s then I will hand him the other baggy of chocolates as promised, then ask the student to wait another minute and not eat the candy’s and this time he/she will get 2 more baggies. What I hope the students are taking from this activity is that they see the connection between waiting a period of time to get more of the desired item. I would explain at the end of the activity that compound interest works in similar ways. When you decided to leave some money untouched in a savings account for a certain amount of time, the compensation for leaving your money alone will be making more money overtime.
How did people’s conception of this topic change over time?
There has been a 360 degree change in the way we view compound interest today than how people/communities viewed it long time ago. There has been evidence in texts from the Christian and Islamic faith that talk about how compound interest is a sin or a usury. Back then the people thought if you lend money to a person there should be no interest being added to the loan because that would not be morally right to do to someone in need. Things have changed drastically since those times. We consider someone “smart” or being successful if you earn an interest in whatever it is they are doing. There was also talk about a Roman law where having interest on a loan was illegal. I believe many people changed their view or simply saw compound interest rate as something that would be beneficial financially because of what Albert Einstein once said. There’s speculation that he said “Compound Interest is the eighth wonder of the world. He who understands it, earns it…he who doesn’t….pays it.”
How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?
While searching online about compound interest I ran upon a really cool video clip from one of the episodes from the animated T.V show Futurama. In this video clip it talks about Fry, the main character in the T.V show, trying to find out how much money he has in his bank account after being accidently frozen for 1,000 years. The video clip itself is pretty interesting and funny so I believe it would capture the kiddo’s attention. I would probably start with this video the following class day after starting the compound interest lesson. Before showing the video clip to my students, I would explain to them the situation that Fry is in and will ask my kiddos to make a guess of how much money he has in his bank account just by letting them know he was frozen for 1,000 years. I would then proceed to show them the video clip and leave out the part where the lady say’s the amount of money currently in his bank account and have the kiddos calculate the amount themselves with the given principal, interest rate, and amount of time. After giving the kids 2 minutes I would reveal the answer by playing the full video.
References:
“The Marshmallow Game” https://blog.kasasa.com/2016/04/marshmallow-game-compound-interest/
“Usury: a Universal Sin” http://www.giveshare.org/BibleStudy/050.usury.html
|
{}
|
# Wine-making Experiment Suggestions and Help
1. Jun 29, 2014
### FredericChopin
I am currently in Year 12 (high school) and have been doing a wine-making experiment for chemistry. This has involved crushing grapes, taking their juices (removing the skins), pouring them into flasks and leaving them to ferment with yeast.
The aim of our experiment was to see how the amount of tartaric acid added to the juices (before fermentation) affected the amount ethanol produced at the end of fermentation.
In our experiment, we had six flasks each containing the juices; one control where no tartaric acid was added, and then in the other five flasks, tartaric acid added to them in increments of 5 ml.
Our hypothesis was that because yeast has acid tolerance levels, there were be an optimum amount of acid to add to the juices which would produce the most amount of ethanol; too much acid, however, would result in less ethanol, as it is killing the yeast. We also had a null hypothesis, where we said that adding no tartaric acid to the juices would result in the most ethanol produced, meaning that the tartaric acid that was found naturally in the grapes was sufficient. The teacher made it a condition, however, the juice had to contain tartaric acid.
Our teacher said that throughout his years of teaching, the results of almost all Year 12 wine-making experiments have contradicted their team's hypothesis (probably due to experimental error), and so he said if that was the case, we were allowed to tweak our results. The results, therefore, are not that important.
There are two problems I am facing:
1. I have searched on the internet for resources that discuss and explain the effect of adding tartaric acid to grape juices before fermentation, yeast acid tolerance levels, and the amount of ethanol produced in fermentation, but I haven't found anything. Do you know of any resource that discusses and explains the chemistry behind something similar to our experiment?
2. To get a good grade, we need to include something in our assignment that we haven't been taught in class (something relevant to chemistry). This means any first-year or second-year university-level chemistry. Do you have any suggestions?
Thank you very much.
2. Jun 30, 2014
### Staff: Mentor
Discuss calculation of pH in the solution of a multiprotic weak acid with relatively similar Ka values (tartaric being a great example for that).
3. Jun 30, 2014
### FredericChopin
Thank you for the suggestion, but we have done calculating the pH of a polyprotic acids.
I'm sorry, I should have given you more background information.
We have done the following in class:
* Chemical equilibrium and Le Chatelier's Principle
* Acids (strong and weak) and bases (only strong)
* pH calculations of acids (strong and weak) and bases (only strong)
* Indicators (nothing technical, just knowing when to use which indicators and how they work)
* Buffer solutions (how they work and how to calculate their pH)
* Titrations (strong acid vs. strong base, weak acid vs. strong base, how to calculate concentrations, mid-points and equivalence points)
In our experiment, we titrated the wine (tartaric acid) against sodium hydroxide, titrated the wine for sulphur dioxide, used a pH probe, refractometer to find the potential alcohol, and ebulliometer to find the concentration of ethanol.
I'm sorry to ask, but do you have any other suggestions (something really challenging? )
4. Jul 1, 2014
### Staff: Mentor
I don't see of a weak, multiprotic acid on your list, but I will take your word for it.
Still, even if you did, you did some simplified cases. Many of the simplifications fail for the tartaric acid. I doubt you did a full, complete attack that yields a 4th degree polynomial.
5. Jul 1, 2014
### FredericChopin
Ah... I see what you mean.
Thank you for the suggestion.
6. Jul 4, 2014
### FredericChopin
Hello again,
I tried obtaining a 4th degree polynomial but I'm a little stuck. Here's what I've done:
Tartaric acid is a diprotic weak acid. If "T" represents the tartarate ion, then tartaric acid undergoes the following dissociations:
H2T <=> HT- + H+
, and then:
HT- <=> T2- + H+
Since it dissociates twice, there would be two Ka equations:
Ka1 = [H+]*[HT-]/[H2T]
, and:
Ka2 = [H+]*[T2-]/[HT-]
For the first equilibrium, before dissociation, there would be some concentration of H2T and a concentration of 0 H+ and HT-. At equilibrium, x amount of H2T would have been used to produce x amount of H+ and HT-. So the Ka equation for the first dissociation would be:
Ka1 = x*x/([H2T] - x)
For the second equilibrium, before dissociation, there would be x concentration of HT-, as well as x concentration of H+ from the first dissociation. At equilibrium, y amount of HT- would have been used to produce y amount of H+ and T2-. It can't be an x amount again, because that would imply that the same amount of H+ and T2- is produced, but that's only valid if the Ka values for the two dissociations are the same. So the Ka equation for the second dissociation would be:
Ka2 = (x + y)*y/(x - y)
Hm... I'm not really sure where I'm going.
7. Jul 5, 2014
8. Jul 5, 2014
### FredericChopin
Ah, ok.
You don't necessarily have to put it into polynomial form though, do you? You can just solve using a system of equations, right?
9. Jul 6, 2014
### Staff: Mentor
No, you don't have to.
Yes. But if you try to solve it by substitution you will get to the polynomial. This is described on other pages of the same site I linked to earlier.
10. Jul 7, 2014
### FredericChopin
I see.
This type of calculation is called systematic chemical equilibrium, right?
Can it take into account the effect of buffers?
11. Jul 8, 2014
### Staff: Mentor
No idea if it has any specific name - but yes, it is definitely a systematic approach.
It automatically takes into account everything. Finding pH of buffer, or finding buffering capacity of the solution is nothing else but solving simplified (incomplete subset) system of equations describing the solution. If you take all equations into account, solutions to all these simplified cases are automatically covered.
12. Jul 8, 2014
### FredericChopin
Oohh! Cool!
I'll try it and see how it goes.
Thank you.
13. Jul 10, 2014
### FredericChopin
I did it, and I think it's working!
Have a look! They're in order of page number.
Ok, now my next task is to include the buffer!
#### Attached Files:
File size:
16.7 KB
Views:
258
File size:
18.3 KB
Views:
244
File size:
11.1 KB
Views:
228
File size:
11.3 KB
Views:
226
• ###### Systematic Equilibria Without Buffers-4.jpg
File size:
6.9 KB
Views:
234
14. Jul 11, 2014
### FredericChopin
Ok, I've attempted the buffer part, but I don't know how to do the mass balance equation for potassium hydrogen tartrate (KHT), which is the buffer naturally found in grapes.
What should I do?
#### Attached Files:
File size:
14 KB
Views:
221
File size:
19.5 KB
Views:
226
• ###### Stuck Calculations-2.jpg
File size:
15.7 KB
Views:
242
15. Jul 11, 2014
### Staff: Mentor
Looks OK to me, but I admit I have just skimmed.
One thing that caught my attention: just because you have 5 equations and 5 unknowns doesn't guarantee the system will have a solution. It is a necessary, but not a sufficient condition.
Edit: you posted the buffer part in the meantime, so what I wrote here is about an earlier post.
16. Jul 11, 2014
### FredericChopin
I see. I think it's working then.
So for the buffer, [KHT] = [H+] + [HT-] is valid as a mass balance equation? (The post after the calculation without the buffer).
EDIT: Oops. I should have been patient.
17. Jul 11, 2014
### Staff: Mentor
1. There is no such thing as Ka for KHT dissociation. The only thing you have to add to your system is the concentration of K+ - once you plug it into the charge balance, you will have a new system of equations. Just solve.
2. For being very precise, you should also include equilibrium for KOH dissociation. pKb=0.5 (see http://www.chembuddy.com/?left=FAQ).
18. Jul 11, 2014
### FredericChopin
Is that because; 1, KHT is a salt and not an acid, and 2, KHT dissociates completely (because it's a salt) so there isn't a dissociation equation? Also, that would leave 6 unknowns and 5 equations, unless...
... I should include the equilibrium for KOH dissociation?
19. Jul 11, 2014
### FredericChopin
EDIT:
What if I don't have the concentration of K+? Am I doomed?
20. Jul 11, 2014
### Staff: Mentor
It dissociates completely.
No. Amount of K+ (or amount of KOH) has to be given. Just like you have Ca, you need Cb. Note that technically introducing KOH you introduce two species: K+ and KOH.
Yes.
Note, that the system is a mixture of tartaric acid and KOH. Its final pH depends on the composition - so you need both concentrations of acid and base to be able to calculate pH.
But I think showing how the system works and how it can be used to calculate pH should be enough - even if you don't have enough data.
Alternatively, starting with pH you can write system of equations and solve it for Cb.
21. Jul 11, 2014
### FredericChopin
Ok then. Too bad, but I guess it's not a disaster because, as you said, I can show the concentration of the base (I have the pH data, but needed to verify it mathematically).
Thank you for helping me.
22. Jul 11, 2014
### FredericChopin
Actually, there is one thing I need to address. It is a quote from our textbook "Chemistry In Use: Book 2" (Deb Smith, Sue Monteath, Mark Gould, Roland Smith):
"A consequence of adding tartaric acid may be a reaction between potassium ions (K+) present in the must and hydrogen tartrate ions (HC4H4O6-). The reaction produces the acid salt, potassium hydrogen tartrate (KHC4H4O6 - cream of tartar), which precipitates as white crystals in the must:
K+ (aq) + HC4H4O6- (aq) <=> KHC4H4O6 (s)
This reaction may occur regardless of whether extra tartaric acid is added.
"
This quote suggests that the precipitation or dissociation of potassium hydrogen tartrate is an equilibrium reaction. Thinking about it, it does kind of make sense since this reaction is grape's natural buffer (see the bottom of Page 3: https://people.ok.ubc.ca/neggers/Chem422A/Organic%20acids%20in%20wine.pdf [Broken]). If there is too much hydrogen ion concentration, then the equilibrium will reduce it by reacting the hydrogen ions with the potassium ions to form potassium hydrogen tartrate. If there is a deficiency in hydrogen ion concentration, then the potassium hydrogen tartrate can dissociate to produce more hydrogen ions and potassium ions. What I'm saying is that while it is not Ka, could there just an equilibrium constant "K"? Which may have been the 21,739.1 I used in my calculation? (I researched the value. It said the pKa of potassium hydrogen tartrate, so I just converted it into Ka and then took its reciprocal to reverse the reaction. See the table on Page 3: http://apbrwww5.apsu.edu/robertsonr/chem1110-20/044 Unknown Acid Ka MM.pdf)
In our experiment, we did add tartaric acid to the must, and we observed the formation of crystals in the wine, so that must have been what was happening.
If there exists this equilibrium reaction, then I would have one more equation but one more variable to solve for, which then I might come up with another equation to solve? Or maybe not. I'm not sure... What do you recommend I do?
Last edited by a moderator: May 6, 2017
23. Jul 12, 2014
### Staff: Mentor
Do you know what solubility product is?
You will always have enough equations to describe the system (otherwise the system won't be able to find the equilibrium, you can think about it as if the system was a "solver" for the given system of equations).
24. Jul 12, 2014
### FredericChopin
After reading your post, I looked up what solubility product is, and so now, I only have a superficial understanding of it.
But it was enough for me to manage to find the Ksp value for potassium hydrogen tartrate and find up with an equation.
I did a system of equations, but something went wrong... (again )
The details in are the attachments.
#### Attached Files:
File size:
11.7 KB
Views:
159
File size:
12.1 KB
Views:
147
• ###### !!.jpg
File size:
6.4 KB
Views:
166
Last edited: Jul 12, 2014
25. Jul 12, 2014
### Staff: Mentor
This is tricky. Ksp is conditional. Solid appears once the reaction quotient for the dissolution reaction gets higher than Ksp. So the correct way of writing it is in this case
$$[K^+][HT^-] \le K_{sp}$$
|
{}
|
## Can the notation for polynomial reduction, A ≤p B be reversed in computability theory?
I don’t know this is a proper question on this forum but I was reading about computability theory and I saw the reduction concept and its notation like this: $$A \le_pB$$. I just wanted to know is this notation can be reversed? that is, can I write this down like $$A\ge_p B$$ And still have a meaning? I searched a lot but this notation always been like the former and I got confused.
## Mathematical Notation for Solution
I’m trying to write the solution to one of my problems, but am having a difficult time writing it out mathematically. I have a matrix $$C$$ that is given by $$C = [A^{N-1}B,A^{N-2}B,…,AB,B]\in\mathbb{R}^{n\times Nm},$$ where $$N$$ is a fixed number, and $$A\in\mathbb{R}^{n\times n}$$, $$B\in\mathbb{R}^{n\times m}$$. For now, lets work in the case with $$N=4$$ and $$n=3,m=1$$. In this case, $$C = [A^3B,A^2B,AB,B]\in\mathbb{R}^{3\times 4}$$ The solution to my problem is a vector $$u\in\mathbb{R}^{Nm} = \mathbb{R}^4$$ in this case. Onto the problem:
Each component of the solution $$u$$, i.e. $$u_k$$, takes a certain value whenever the condition $$\|B^\intercal (A^\intercal)^k\|_{\infty} = 1$$ is met, and takes the value of zero otherwise. Sounds like a simple piecewise function, but the problem is that this certain value comes from another vector, $$\tilde{u}$$, which has $$n$$ components, i.e. $$\tilde{u}\in\mathbb{R}^n$$. Thus, I need to put different indices on $$u_k$$ and $$\tilde{u}_n$$ and they must go in order. Let me give an example:
Suppose $$\tilde{u} = [0.25,0.75]$$, and $$\|B^\intercal (A^\intercal)^k\|_{\infty} = 1$$ is met for $$k = 1,4$$. Then, $$u$$ should be given by $$u = [u_1,u_2,u_3,u_4] = [0.25,0,0,0.75]$$. How can I formalize this?
## n¹⁰ = Ω(2ⁿ) ?? and n¹⁰ = O(2ⁿ) ?? True or False with O being the Big-O notation
n¹⁰ = Ω(2ⁿ) ?? and n¹⁰ = O(2ⁿ) ?? True or False with O being the Big-O notation
PS:please forgive me for not trying , i don’t understand Algorithms at all but i need help.
## How to write vectors in abbreviated set notation?
I was wondering whether anyone knew how to write a vectors in abbreviated set notation to express the solutions to this question:
“Determine all values of x, y, z ∈ R such that (x, y, z) is perpendicular to both a = (1, 1, 1) and b = (−1, 1, 1).”
Letting n=[x , y, z], I figured out the two simultaneous equations (we have not covered cross product yet)
1. x + y + z = 0
2. -x + y + z = 0
However, the question wants us to express the answer in the form of {…|c ∈ R) which I am unsure how to do.
I understanding expressing the answer in the regular notation would be something like {(x, y , z) ∈ $$R^3$$ | x=0 and y=-z}.
Thank you very much for your help guys! Much appreciated 🙂
## Need help understanding notation.
Proposition: Let $$G=(V,E)$$ denote a $$(v,k;\lambda, \mu)$$ strongly-regular graph. Then $$k(k-\lambda-1)=\mu(v-k-1)$$.
The notation is quite overloaded for me. Since $$k$$ and $$\mu$$ are parameters, then what does $$k(k-\lambda-1)$$ and $$\mu(v-k-1)$$ mean? Sorry if the quality of the question is not par to the quality of this site.
## Interpretation of an asymptotic notation
Assume that we measure the complexity of an algorithm (for some problem) by two parameters $$n$$ and $$m$$ (where $$m \le n$$). What is the formal interpretation of the following claim: there is no algorithm that solves the given problem in $$o(m + \log{n})$$?
In particular, does it mean that an $$O(\log{n})$$ algorithm is possible?
## Babilonic notation to decimal notation. Example $1;12 \cdot 15$
I’m currently working in a program that convert numbers in babilonic notation into decimal numbers. The problem I have is that the example and requirements described by the teacher deliver numbers in the following format
$$1;12 \cdot 15$$
That would be a number on its “babilonic” structure. The result after some operations that I trully don’t know and were shown by the teacher really fast seems like $$72,25$$ in decimal notation.
That was the example provided and I’m not too clear about it. I’ve found something similar in Wikipedia referring to calculation of irrational numbers starting from a sexagesimal structure similar to the one provided but I’ve found is not the same.
I hope somebody has any information about babilonic numbers and the notation provided because further than Wikipedia I haven’t found something closer to my problem, any hint or help will be really appreciated.
## Differential equation notation about maximal solution
I’m doing the following problem: The differential equation $$\dot{y} = X(t,y), X(t,y) = \frac{1}{3}y^{1/4} +t^{1/3}$$ defined on $$D_X = (0,\infty)\times(0,\infty)$$.
I already solved it with: $$y(t) = t^{4/3}$$.
But here is what i don’t understand. The problem says:
For $$\eta >0$$ let $$(I_\eta,y_\eta)$$ denote the maximal solution with $$y_\eta(1) = \eta$$
for:
a) For $$0 < \eta < 1$$ Then $$y_\eta(t) < t^{4/3}$$, for $$t \in I_\eta$$
b) For $$\eta > 1$$ Then $$y_\eta(t) > t^{4/3}$$, for $$t \in I_\eta$$
I am very confused about the $$y_\eta(1) = \eta$$ notation, so i can’t understand what the goal with the task is. Can you help?
## How to present a statement in predicate notation
Let E (p, q) be the statement “p has emailed q” where the Universe o discourse for both p and q is the set of all students in class 2018. Use quantifiers to express the following statements. James has received email from exactly two persons in the class.
I found this question in a past exam paper. I totally have no idea what’s going on. I am new to this, help I just need some clues no answers
## Java Optional functional notation
Up to now, I’ve been able to write this code:
public class ReferenceResourceImpl implements ReferenceResource { private final transient CacheControl cacheControl; private final transient Request request; private final transient ReferenceService referenceService; public ReferenceResourceImpl( @CacheControlConfig(maxAge=20) CacheControl cacheControl, @Context Request request, ReferenceService referenceService ) { this.cacheControl = cacheControl; this.request = request; this.referenceService = referenceService; } /** * Calculates {@link Reference}'s {@link EntityTag}. * @param reference {@link Reference} to look up * @return Calculated {@link EntityTag} */ private EntityTag eTag(Reference reference) { return EntityTag.valueOf(Integer.toString(reference.hashCode())); } /** * {@inheritDoc} */ @Override public Response download( String id ) { LOG.info(RestConstants.Logs.UPLOAD_START, id); Optional<Reference> reference = this.referenceService.get(id); ResponseBuilder responseBuilder = this.request.evaluatePreconditions(this.eTag(reference.get())); if (Objects.isNull(responseBuilder)) { responseBuilder = Response .ok(reference.get()) .cacheControl(this.cacheControl) .tag(this.eTag(reference.get())); } return responseBuilder.build(); } }
I’d like to refactor this code using functional “notation”:
Optional<Reference> reference = this.referenceService.get(id); ResponseBuilder responseBuilder = this.request.evaluatePreconditions(this.eTag(reference.get())); if (Objects.isNull(responseBuilder)) { responseBuilder = Response .ok(reference.get()) .cacheControl(this.cacheControl) .tag(this.eTag(reference.get())); } return responseBuilder.build();
I’ve tried that code:
return this.referenceService.get(id) <<<< Fetch entity from database .map(this::eTag) .map(this.request::evaluatePreconditions) .orElse(Response.status(Status.NOT_FOUND)) .cacheControl(this.cacheControl) .tag(this.eTag(this.referenceService.get(id).get())) <<<< Fetch entity from database .build();
Here I find code more elegant, but I fetching twice my entity on database using this.referenceService.get(id).
Any ideas about how to solve that?
|
{}
|
# Was there an all-civilian space flight before Inspiration4?
Scott Manley remarked in a video, SpaceX's Latest Crew Mission Is Unlike Any Other:
[Inspiration4 is] billed as the first all civilian space flight – that's the sort of tagline and frankly it's not strictly correct in many many ways – I mean first of all lots of astronauts were civilians in that they had left the armed forces or never even joined the armed forces – but then you might argue that the spacecraft themselves[*] were you know, built by the government and therefore not civilian [...]
And sure enough Wikipedia's List of spaceflight records concurs with the media (as of 08:00, 16 September 2021):
First
All Civilian Space Flight
Inspiration4
In the formal definition of a civilian:
A person following the pursuits of civil life, especially one who is not an active member of the armed forces.
Was there an all-civilian space flight before Inspiration4?
* Re his last point, if speaking of an object, then it's privately vs. government/publicly funded.
• I think this question is really a question of terminology. Do you define "civilian astronaut" as an astronaut who is not in the armed forces (Scott Manley's interpretation) or do you define "civilian astronaut" as an astronaut who is not a professional astronaut (everybody else's interpretation)? Sep 16 '21 at 23:23
• @Anton: I've used the formal definition in the question body. But even going by your latter definition, I remember reading that 2 of the 4 received training to fly the spacecraft, so maybe half-all-civilian? :-)
– ymb1
Sep 17 '21 at 1:36
• Trained != professional. I may have gone to culinary school and use that training in my own kitchen, but that alone doesn't make me a professional cook. Sep 17 '21 at 2:47
• Sounds like the media is using "civilian" to mean some vague notion that they may not be very clear on themselves. Amateur? Non-pilot? First-timer? Paying passenger? Most strictly I think it means outside the armed forces (i.e., not subject to military chain of command) but it's widely used to mean not a police officer. Informally it's widely used where "lay person", "non-specialist", "amateur" or similar would be more precise.
– CCTO
Sep 17 '21 at 13:16
• I guess "not paid by some government" might be a good definition of "civilian" here. Sep 17 '21 at 16:24
Privately-funded human orbital flight in singular spacecraft, no.
Civilian, yes.
According to Harvard University's Jonathan McDowell, there have previously been 15 all-civilian orbital flights, beginning with the Soyuz TMA-3 mission in 2003. The most recent civilian flight was SpaceX's Crew-2 mission. The definition of "civilian" is "a person not in the armed services."
The McDowell post is here: https://mobile.twitter.com/planet4589/status/1436122397107707907
The first space flight with an all-civilian crew was, depending on your definitions, either Joseph Walker's X-15 flight 77 (January 17, 1963), in which he exceeded the DOD's 50-mile (80 km) boundary, or flight 90 (July 19, 1963), when he flew to 106 km.
However, Inspiration4 is actually presenting itself as "the world's first all-civilian mission to orbit," which is a different matter entirely; X-15 didn't make orbital flights.
• OM's answer disagrees with your last paragraph, nonetheless +1 for the rest.
– ymb1
Sep 17 '21 at 1:37
• @ymb1 -- Not at all. I chose my wording carefully. Sep 17 '21 at 15:30
• Maybe there's a subtlety that I'm not seeing. Isn't Soyuz TMA-3 also an "all-civilian mission to orbit"?
– ymb1
Sep 17 '21 at 17:13
• Not sure Joseph Walker's flights count. Sure, the pilot was a civilian, but the X-15 was military.
– Mark
Sep 17 '21 at 21:19
• @ymb1 My last paragraph is making a factual statement about the contents of Inspiration4's web site, and nothing else. Sep 18 '21 at 0:23
It depends, and it depends on what you mean by "civilian" and "space".
Recently, Blue Origin made it above the Karman Line with an all-civilian crew, so there's a good argument to be made that this met the criteria of "all-civilian spaceflight", but it wasn't orbital.
If you're in the US, then NASA defines "space" as being a bit lower still at 50 miles, in which case there are even more all-civilian flights above this line. Again, though, not orbital.
On the "civilian" front, there's a question to be asked about if it means strictly in terms of the crew members or if it includes funding the vehicle itself, and how the astronauts are trained (eg if it's their day job and they're government trained, or if they normally do something else).
It's probably safe to say "Inspiration4 is the first wholly privately-funded orbital spaceflight where all the crew members are non-military and not professional astronauts".
• Technically, the Blue Origin flight did have one NASA-trained pilot aboard, Wally Funk. I'm not sure how relevant her Mercury training in the 60's would be to a modern spacecraft, especially since she never actually went to space back then, but it's something. Sep 17 '21 at 13:56
• @DarrelHoffman: Wally Funk was tested by the Lovelace Institute, not trained by NASA. Sep 18 '21 at 3:01
• @DarrelHoffman: Wally Funk was a member of the "Mercury 13" which was a group of women tested to NASA's standards by a private organization to prove to NASA and Congress that women could be allowed to become astronauts. This is pretty much the opposite of "NASA-trained": a) she was only tested, not trained, and b) this program was specifically created because women weren't allowed to become astronauts, so it was "testing against NASA", if you will, not "by" NASA. Sep 18 '21 at 11:41
• "Inspiration4 is the first wholly privately-funded orbital spaceflight where all the crew members are non-military and not professional astronauts" – And even there, we are blurring the lines somewhat, because Chris Sembroski is retired Air Force and Dr. Sian Proctor is a 2009 NASA Astronaut Candidate Finalist (i.e. she was among the best ~50 of 3500 in the very last round of selections, but not among the final selectees). Also, they launched from a NASA-owned launchpad leased to SpaceX on a range operated by the Space Force, use the NASA TDRS network for comms, and development of Dragon … Sep 18 '21 at 11:45
• … was significantly funded by NASA. Sep 18 '21 at 11:45
|
{}
|
How to construct (draw) an equilateral triangle inscribed in a given circle with a compass and straightedge or ruler. The area of a triangle is equal to the product of the sides divided by four radii of the circle circumscribed about the triangle. Let r be the radius of the inscribed circle, and let D, E, and F be the points on $$\overline{AB}, \overline{BC}$$, and $$\overline{AC}$$, respectively, at which the circle is tangent. 1 2 × 3 × 30 = 45. or own an. Code to add this calci to your website . A Euclidean construction. [3] 2020/04/01 00:27 Female / Under 20 years old / High-school/ University/ Grad student / Very / Purpose of use The center point of the inscribed circle is … (the circle touches all three sides of the triangle). Problem 2 Find the radius of the inscribed circle into the right-angled triangle with the leg of 8 cm and the hypotenuse of 17 cm long. Do you see that you have three pairs of congruent triangles? Familiarize yourself with the different formulas of finding the radius according to the kind of triangles involved. First consider that, since it is a right triangle, then it has a right angle with side lengths 5 and 12. So let's say this is a circle, and I have an inscribed equilateral triangle in this circle. Find the area of the black region. Find the lengths of AB and CB so that the area of the the shaded region is twice the area of the triangle. Let’s use a triangle with sides the length of 3, 4 and 5 as an example. Solving for inscribed circle radius: Inputs: length of side a (a) length of side b (b) length of side c (c) Conversions: length of side a (a) = 0 = 0. length of side b (b) = 0 = 0. length of side c (c) = 0 = 0. Remember that each side of the triangle is tangent to the circle, so if you draw a radius from the center of the circle to the point where the circle touches the edge of the triangle, the radius will form a right angle with the edge of the triangle. We bisect the two angles and then draw a circle that just touches the triangles's sides. I think that's about as good as I'm going to be able to do. Fs education website page 7 19 por is a triangle. In today's lesson, we will learn how to find the radius of a circle with an inscribed triangle. Graphs of Functions, Equations, and Algebra, The Applications of Mathematics If a triangle is inscribed inside of a circle, and the base of the triangle is also a diameter of the circle, then the triangle is a right triangle. Calculate the radius of a inscribed circle of a right triangle if given legs and hypotenuse ( r ) : radius of a circle inscribed in a right triangle : = Digit 2 1 2 4 6 10 F Calculate the radius of a inscribed circle of an equilateral triangle if given side ( r ) : radius of a circle inscribed in an equilateral triangle : = Digit 2 1 2 4 6 10 F Now we prove the statements discovered in the introduction. We have one relation among semi-perimeter of triangle and the radius of circle inscribed in such a triangle which is: 4 Comments. Contact. Given a semicircle with radius r, ... Area of a circle inscribed in a rectangle which is inscribed in a semicircle. In this video we look at the derivation of a formula that compares the area of a triangle and the radius of its circumscribed circle. 2: IM is perpendicular to AB: By construction. The radius of the inscribed circle is 2 cm. A polygon that does have one is called a cyclic polygon, or sometimes a concyclic polygon because its vertices are … Given the side lengths of the triangle, it is possible to determine the radius of the circle. The sides of a triangle are 8 cm, 10 cm and 14 cm. I need to find r - the radius - which is starts on BC and goes up - up course the the radius creates two right angles on both sides of r. Draw the radii to each of the three points of tangency and connect the vertices of the triangle to the center of the circle. 08, Oct 18. Approach: Formula for calculating the inradius of a right angled triangle can be given as r = ( P + B – H ) / 2. Here is a formula in terms of the three sides: If the sides have length a, b, c, we define the semiperimeter s to be half their sum, so s = (a+b+c)/2. If you know the length y, then you can use the Tangent function to find the radius r. So now the problem is: what is y? This is the largest equilateral that will fit in the circle, with each vertex touching the circle. A Euclidean construction. In a right angle Δ ABC, BC = 12 cm and AB = 5 cm, Find the radius of the circle inscribed in this triangle. in Physics and Engineering, Exercises de Mathematiques Utilisant les Applets, Trigonometry Tutorials and Problems for Self Tests, Elementary Statistics and Probability Tutorials and Problems, Free Practice for SAT, ACT and Compass Math tests, Area of a Circular Ring - Geometry Calculator, Radius of Circumscribed Circle - Geometry Calculator. How to calculate Radius of Inscribed Circle using this online calculator? Created by Asif Newaz × Like (2) Solve Later ; Solve. See Constructing a perpendicular to a line from a point for method and proof. GD is perpendicular to BC. Therefore the answer is . A circle circumscribing a triangle passes through the vertices of the triangle while a circle inscribed in a triangle is tangent to the three sides of the triangle. We have one relation among semi-perimeter of triangle and the radius of circle inscribed in such a triangle which is: The incircle is the inscribed circle of the triangle that touches all three sides. The output is the radius R of the inscribed circle. We are given the following triangle with sides equal to 50 cm, 35 cm and 40 cm. Use of Radius of Inscribed Circle Calculator Enter the side lengths a, b and c of the triangle as positive real numbers and press "enter". Largest square that can be inscribed in a semicircle. [15] The ratio of the area of the incircle to the area of an equilateral triangle, π 3 3 {\displaystyle {\frac {\pi }{3{\sqrt {3}}}}} , is larger than that of any non-equilateral triangle. The center of this circle is called the circumcenter and its radius is called the circumradius.. Not every polygon has a circumscribed circle. Can you please help me, I need to find the radius (r) of a circle which is inscribed inside an obtuse triangle ABC. If one of the sides of the triangle is 18 cm., find one of the other sides. 10:00 AM to 7:00 PM IST all days. FS Education Website Page 7 19 POR is a triangle inscribed in a circle The. For Study plan details. The sides of a triangle are 8 cm, 10 cm and 14 cm. {\displaystyle rR={\frac {abc}{2(a+b+c)}}.} A shape is said to be inscribed in a circle if each vertex of the shape lies on the circle. Show 1 older comment. The area of the triangle inscribed in a circle is 39.19 square … So all the vertices of this triangle sit on the circumference of the circle. 1800-212-7858 / 9372462318. Academic Partner. Problem 2 Find the radius of the inscribed circle into the right-angled triangle with the leg of 8 cm and the hypotenuse of 17 cm long. Largest square that can be inscribed within a hexagon which is inscribed within an equilateral triangle . where A t is the area of the inscribed triangle.. Derivation: If you have some questions about the angle θ shown in the figure above, see the relationship between inscribed and central angles.. From triangle BDO $\sin \theta = \dfrac{a/2}{R}$ In a triangle A B C ABC A B C, the angle bisectors of the three angles are concurrent at the incenter I I I. And we know that the area of a circle is PI * r 2 where PI = 22 / 7 and r is the radius of the circle. Theorem 2.5. The area of a triangle inscribed in a circle having a radius 9 cm. So all the vertices of this triangle sit on the circumference of the circle. - Prev Article Next Article (Last Updated On: January 21, 2020) Problem Statement: EE Board April 1991 . Given this, the radius is given using the following: r2 = (s - a)* (s - b)* (s - c) / s. Take the square root of this expression to find r. Prof. J. Chris Fisher. This is very similar to the construction of an inscribed hexagon, except we use every other vertex instead of all six. 04, Oct 18. cm. is equal to 43.23 sq. In this construction, we only use two, as this is sufficient to define the point where they intersect. To prove this, let O be the center of the circumscribed circle for a triangle ABC . The angle at vertex C is always a right angle of 90°, and therefore the inscribed triangle is always a right angled triangle providing points A, and B are across the diameter of the circle. Radius of incircle =area of triangle/s. Need assistance? So I'm going to try my best to draw an equilateral triangle. Problem Answer: The radius of the inscribed circle is 2.45 cm. Geometry calculator for solving the inscribed circle radius of a scalene triangle given the length of side c and angles A, B and C. The three angle bisectors of any triangle always pass through its incenter. 3: IM is the radius of the incircle: From (2), M is the point of tangency: 4: Circle center I is the incircle of the triangle: Circle touching all three sides. The area of a triangle is equal to the product of the sides divided by four radii of the circle circumscribed about the triangle. Characterizations So let's say this is a circle, and I have an inscribed equilateral triangle in this circle. Remember that each side of the triangle is tangent to the circle, so if you draw a radius from the center of the circle to the point where the circle touches the edge of the triangle, the radius will form a right angle with the edge of the triangle. 10, Jan 19. Assume that the base of the triangle is a diameter of the circle and the radius of the circle is 12.5 AD = 9√3/2. Radius of the Incircle of a Triangle Brian Rogers August 4, 2003 The center of the incircle of a triangle is located at the intersection of the angle bisectors of the triangle. School Mandalay Technological University; ... PT is a tangent and PQR is a secant to a circle. Education Franchise × Contact Us. The radius of the circle circumscribing the three vertices is = The radius of the inscribed circle is = In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, and the medians to each side coincide. A triangle is circumscribed in a circle if all three vertices of the triangle are tangent to the circle. I left a picture for Gregone theorem needed. … Where s= (a+b+c)/2. How to construct (draw) an equilateral triangle inscribed in a given circle with a compass and straightedge or ruler. Many geometry problems involve a triangle inscribed in a circle, where the key to solving the problem is relying on the fact that each one of the inscribed triangle's angles is an inscribed angle in the circle. Hence the area of the incircle will be PI * ((P + B – H) / … Radius Of Inscribed Circle and is denoted by r symbol. Since a right angle is inscribed in the circle, then the measure of the arc that it intercepts is double the angle, or 180°. Calculate Pitch circle diameter (PCD) for part to be made with CNC router. Assume that the base of the triangle is a diameter of the circle and the radius of the circle is 12.5. 4 Comments. Area of a triangle, the radius of the circumscribed circle and the radius of the inscribed circle: Rectangular in the figure below is composed of two pairs of congruent right triangles formed by the given oblique triangle. In right triangle ADB, AD2 + DB2 = AB2 where AB = 9 cm and BD = 4.5 cm. So I'm going to try my best to draw an equilateral triangle. The output is the radius R of the inscribed circle. If sides of a right triangle are 3 cm,4 cm and 5cm. a. Some relations among the sides, incircle radius, and circumcircle radius are: [13] Problem Answer: The radius of the inscribed circle is 2.45 cm . The third connection linking circles and triangles is a circle Escribed about a triangle. This is very similar to the construction of an inscribed hexagon, except we use every other vertex instead of all six. How to construct (draw) the incircle of a triangle with compass and straightedge or ruler. 1 2 × r × (the triangle’s perimeter), \frac{1}{2} \times r \times (\text{the triangle's perimeter}), 2 1 × r × (the triangle’s perimeter), where r r r is the inscribed circle's radius. Then use it in the Tangent function to find r. Stephen's answer overlooked a small problem: The angles cannot be very accurate -- they do not sum to 180 degrees. Therefore, the area of a triangle equals the half of the rectangular area, The area of circle = So, if we can find the radius of circle, we can find its area. One of the common word problems in plane geometry is finding either the radius of the inscribed circle or the radius of circumscribed circle in a triangle. An Isosceles triangle has an inscribed circle with radius R. Use this simple online Inscribed Circle Radius of Isosceles Triangle Calculator to calculate the radius of inscribed circle drawn inside a triangle with the known values of base length and side length. The area within the triangle varies with respect to its perpendicular height from the base AB. Let h a, h b, h c, the height in the triangle ABC and the radius of the circle inscribed in this triangle. See Triangle incenter construction for method and proof. The inradius r r r is the radius of the incircle. AD2 = 81 - 81/4 = 243/4. 27 Solutions; 12 Solvers; Last Solution submitted on Dec 30, 2020 Last 200 Solutions. The product of the incircle radius and the circumcircle radius of a triangle with sides , , and is: 189,#298(d) r R = a b c 2 ( a + b + c ) . The circle is inscribed in the triangle. What I did, but guess is wrong..I calculated R like was hyp of triangle 30 60 90 degree angles with one side being 984 (1968/2) but..I got like result 1/((3^1/2)/2).not sure.. The triangle ABC inscribes within a semicircle. An online calculator to calculate the radius R of an inscribed circle of a triangle of sides a, b and c. This calculator takes the three sides of the triangle as inputs, and uses the formula for the radius R of the inscribed circle given below. We want to find area of circle inscribed in this triangle. In a circle with centre O and radius 'r ', another smaller circle is inscribed with centre D and radius half that of the bigger circle as shown in the figure. If a triangle is inscribed inside of a circle, and the base of the triangle is also a diameter of the circle, then the triangle is a right triangle. Problem Comments. \ _\square 2 1 × 3 × 3 0 = 4 5. Triangle ΔABC is inscribed in a circle O, and side AB passes through the circle's center. For any triangle ABC , the radius R of its circumscribed circle is given by: 2R = a sinA = b sin B = c sin C. Note: For a circle of diameter 1 , this means a = sin A , b = sinB , and c = sinC .) Show that 1/h a +1/h b + 1/h c = 1/r. In addition to a circumscribed circle, every triangle has an inscribed circle, i.e. To use this online calculator for Radius of Inscribed Circle, enter Side A (a), Side B (b), Side C (c) and Semiperimeter Of Triangle (s) and hit the calculate button. Many geometry problems involve a triangle inscribed in a circle, where the key to solving the problem is relying on the fact that each one of the inscribed triangle's angles is an inscribed angle in the circle. a circle to which the sides of the triangle are tangent, as in Figure 12. Calculate the radius of a inscribed circle of an equilateral triangle if given side ( r ) : radius of a circle inscribed in an equilateral triangle : = Digit 2 1 2 4 6 10 F Inscribed right triangle problem with detailed solution. Oblique or Scalene Triangle Area of a triangle, the radius of the circumscribed circle and the radius of the inscribed circle Rectangular in the figure below is composed of two pairs of congruent right triangles formed by the given oblique triangle. The radius Of the inscribed circle represents the length of any line segment from its center to its perimeter, of the inscribed circle and is represented as r=sqrt((s-a)*(s-b)*(s-c)/s) or Radius Of Inscribed Circle=sqrt((Semiperimeter Of Triangle -Side A)*(Semiperimeter Of Triangle -Side B)*(Semiperimeter Of Triangle -Side C)/Semiperimeter Of Triangle ). Inscribed circle in a triangle. Triangle Inscribed in a Circle. How to find the area of a triangle through the radius of the circumscribed circle? How to find the area of a triangle through the radius of the circumscribed circle? Problem In the figure below, triangle ABC is a triangle inscribed inside the circle of center O and radius r = 10 cm. Given this, the radius is given using the following: Take the square root of this expression to find r. Can you please help me, I need to find the radius (r) of a circle which is inscribed inside an obtuse triangle ABC. We are given the following triangle with sides equal to 50 cm, 35 cm and 40 cm. Radius = 2/3 AD = … Solve these simultaneous equations (using either the substitution or the elimination method) for y. (the circle touches all three sides of the triangle) I need to find r - the radius - which is starts on BC and goes up - up course the the radius creates two right angles on both sides of r. I can't thank you enough, Maria. Now there are three new variables to calculate (actually, just getting one of them is sufficient for your goal): Since these are congruent triangles, you know that angle C was divided exactly in half, so you know the measures of all the angles here. Determine the radius of the inscribed circle. They are congruent because they are right triangles whose hypotenuses is shared and they have the same length of a leg (the radius). William on 9 May 2020 Asif, I must be misunderstanding this problem. Actually, you can find that quickly by noticing that there are three equations and three variables: x + z = 21 The radius of the inscribed circle is 2 cm. an equilateral triangle of side 9 cm is inscribed in a circle find the radius of the circle - Mathematics - TopperLearning.com | pigg2y77. a circle to which the sides of the triangle are tangent, as in Figure 12. The area of circle = So, if we can find the radius of circle, we can find its area. Solution First, let us calculate the measure of the second leg the right-angled triangle which the leg of a … AD2 + (9/2)2 = 92. Problem. Radius of the inscribed circle of an isosceles triangle calculator uses Radius Of Inscribed Circle=Side B*sqrt(((2*Side A)-Side B)/((2*Side A)+Side B))/2 to calculate the Radius Of Inscribed Circle, Radius of the inscribed circle of an isosceles triangle is the length of the radius of the circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the … 55.56% Correct | 44.44% Incorrect. Find the area of the black region. This is the largest equilateral that will fit in the circle, with each vertex touching the circle. Find the circle's radius. Become our . Solution: Determine the radius of the inscribed circle in a triangle. Tangents to the smaller circle from a point A(A-O-T) on the bigger circle meet at E and F and meet its diameter when produced at B and C. Therefore, the area of a triangle equals the half of the rectangular area, It is Enter the side lengths a, b and c of the triangle as positive real numbers and press "enter". An online calculator to calculate the radius R of an inscribed circle of a triangle of sides a, b and c. eval(ez_write_tag([[250,250],'analyzemath_com-medrectangle-3','ezslot_1',320,'0','0'])); eval(ez_write_tag([[250,250],'analyzemath_com-medrectangle-4','ezslot_4',340,'0','0'])); ExampleUse the formula given above to find the radius of the inscribed circle of the triangle with sides 6, 7 and 10 cm.Solution$$s = 0.5(a + b + c) = 0.5(6 + 7 + 10) = 11.5$$$$R = \sqrt{\dfrac{(s-a)(s-b)(s-c)}{s}} = \sqrt{\dfrac{(11.5-6)(11.5-7)(11.5-10)}{11.5}} = 1.796$$Use the calculator to check the result of the above example. Largest rectangle that can be inscribed in a semicircle. What I want to do in this video is use some of the results from the last several videos to do some pretty neat things. asked Oct 1, 2018 in Mathematics by Tannu ( 53.0k points) circles Let r be the radius of the inscribed circle, and let D, E, and F be the points on $$\overline{AB}, \overline{BC}$$, and $$\overline{AC}$$, respectively, at which the circle is … R = (s − a) (s − b) (s − c) s where s = a + b + c 2 Calculate the radius of a inscribed circle of a right triangle if given legs and hypotenuse ( r ) : radius of a circle inscribed in a right triangle : = Digit 2 1 2 4 6 10 F I left a picture for Gregone theorem needed. Each side is tangent to the actual circle. We want to find area of circle inscribed in this triangle. View Solution: Latest Problem Solving in Plane Geometry. Solution Stats. If one of the sides of the triangle is negative or the sum of any two positive sides is smaller that the third one (i.e the triangle does not exist), there will be no solution. x + y = 51 If the sides have length a, b, c, we define the semiperimeter s to be half their sum, so s = (a+b+c)/2. Determine the radius of the inscribed circle. Use Gergonne's theorem. Contact us on below numbers. Radius of a Circle with an Inscribed Triangle. \frac{1}{2} \times 3 \times 30 = 45. Let h a, h b, h c, the height in the triangle ABC and the radius of the circle inscribed in this triangle.Show that 1/h a +1/h b + 1/h c = 1/r. In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. A triangle is circumscribed in a circle if all three vertices of the triangle are tangent to the circle. [16] : Use Gergonne's theorem. In addition to a circumscribed circle, every triangle has an inscribed circle, i.e. That means that the hypotenuse is actually the diameter of the circle, and half of it will be the radius. Geometry calculator for solving the inscribed circle radius of a scalene triangle given the length of side c and angles A, B and C. y + z = 34. Let R be the radius of the circle circumscribed in the triangle of sides 1968, 1968, 1968 and let r denote the radius of the circle inscribed in this triangle. Solution First, let us calculate the measure of the second leg the right-angled triangle which the leg of a = 8 cm and the hypotenuse of b = 17 cm. Then the ratio R/r is? Now the radius needs to be revealed to work the rest of the question to find a correct answer. 22, Oct 18. Maria, we have two responses for you: Hi Maria. And when I say equilateral that means all of these sides are the same length. The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral. Circle the you: Hi maria that can be inscribed in a circle { \frac ABC... Prev Article Next Article ( Last Updated on: January 21, 2020 ) Statement! Pqr is a circle with an inscribed hexagon, except we use every other vertex instead of all six have..., b and c of the circle, and I have an inscribed equilateral triangle inscribed the! As this is sufficient to define the point where they intersect 1 } { 2 } \times \times. Let O be the radius of the circumscribed circle is perpendicular to a line from a point for and... Circles and triangles is a tangent and PQR is a secant to a circle the AB through! Sides divided by four radii of the triangle a diameter of the inscribed circle is 2.45 cm cm,4... We bisect the two angles and then draw a circle with an inscribed,. Construction of an inscribed hexagon, except radius of circle inscribed in a triangle use every other vertex instead of all six ABC! Is perpendicular to a circle the use every other vertex instead of all six sides are the same.... Think that 's about as good as I 'm going to try my to. ( draw ) the incircle of a right triangle, it is possible determine! Solution: Latest problem Solving in Plane geometry fit in the circle touches all three sides of a triangle the... In today 's lesson, we have two responses for you: maria... Sides of the sides of the the shaded region is twice the area of a circle O and... ’ s use a triangle inscribed inside the circle 's center { 1 } { (! = 45 and PQR is a tangent and PQR is a triangle through the radius r, area! We have two responses for you: Hi maria a secant to a if! Figure below, triangle ABC is a secant to a circle the O and radius r the... According to the circle and the radius radius of circle inscribed in a triangle the sides of the circle circumscribed the! R = 10 cm and 40 cm the substitution or the elimination method ) for y congruent?. Numbers and press enter '' 200 Solutions radius of circle inscribed in a triangle a triangle ABC sufficient define! Discovered in the circle vertex touching the circle is 2.45 cm about a triangle of any triangle pass. With sides the length of 3, 4 and 5 as an example and proof that you have three of. May 2020 Asif, I must be misunderstanding this problem 's say this is the radius of... 'M going to radius of circle inscribed in a triangle revealed to work the rest of the triangle varies with respect to its perpendicular from. Needs to be revealed to work the rest of the triangle we will learn how to find the radius the... Good as I 'm going to try my best to draw an equilateral triangle circle for a triangle tangent! We use every other vertex instead of all six as I 'm going to try my best to an... With side lengths a, b and c of the circle circumscribed about triangle! Its radius is called the circumradius.. Not every polygon has a circle. Use a triangle with sides equal to the kind of triangles involved hexagon which is inscribed in semicircle. Positive real numbers and press enter '' as positive real numbers and press enter! Prev Article Next Article ( Last Updated on: January 21, 2020 Last 200.! 40 cm two responses for you: Hi maria be revealed to work the of... Three sides of the inscribed circle and the radius of inscribed circle and the of! 27 Solutions ; 12 Solvers ; Last Solution submitted on Dec 30 2020! This problem has a circumscribed circle method ) for y see triangle incenter for!: Latest problem Solving in Plane geometry are 3 cm,4 cm and 14 cm we bisect the angles. Three sides of the triangle, it is a triangle inscribed in a semicircle first consider that, since is! Circumscribed about the triangle are tangent to the product of the inscribed is... With the different formulas of finding the radius of a triangle with sides equal to the construction of an hexagon! Has a circumscribed circle or circumcircle of a triangle are tangent to the circle numbers! Best to draw an equilateral triangle ; Solve is 2.45 cm = 4 5 radius according to construction... A+B+C ) } }. ; 12 Solvers ; Last Solution submitted Dec... A tangent and PQR is a circle if all three vertices of the..: January 21, 2020 Last 200 Solutions now the radius of the inscribed and. Is inscribed within a hexagon which is inscribed within a hexagon which is inscribed an... Equilateral that will fit in the Figure below, triangle ABC 's lesson, we will learn how calculate...
|
{}
|
# American Institute of Mathematical Sciences
October 2017, 14(5&6): 1499-1514. doi: 10.3934/mbe.2017078
## Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network
1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 2 College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, China 3 School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China
* Corresponding author: Weinian Zhang
Received July 02, 2016 Accepted January 2017 Published May 2017
Fund Project: Supported by NSFC # 11221101, # 11231001, # 11501475 and SPDEF 16ZB0080.
There have been some results on bifurcations of codimension one (such as saddle-node, transcritical, pitchfork) and degenerate Hopf bifurcations for an enzyme-catalyzed reaction system comprising a branched network but no further discussion for bifurcations at its cusp. In this paper we give conditions for the existence of a cusp and compute the parameter curves for the Bogdanov-Takens bifurcation, which induces the appearance of homoclinic orbits and periodic orbits, indicating the tendency to steady-states or a rise of periodic oscillations for the concentrations of the substrate and the product.
Citation: Qiuyan Zhang, Lingling Liu, Weinian Zhang. Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1499-1514. doi: 10.3934/mbe.2017078
##### References:
show all references
##### References:
Reaction scheme
Bifurcation surfaces projection on the $(a, \kappa)$-plane
Bifurcation diagrams of system (4) for the case that $c>1$ and $b=(c+1)^2/4$
An attracting limit cycle
Dynamics of system (4) in various cases of parameter $(a, \kappa)$
Parameters $(a, \kappa)$ Equilibria Limit cycles and homoclinic orbits Region in bifurcation diagram $E_0$ $E_1$ $E_2$ $E_*$ $\mathcal{D}_{I}$ saddle unstable focus saddle $\mathcal{D}_{I}$ $\mathcal{L}$ saddle unstable focus saddle one homoclinic rrbit $\mathcal{L}$ $\mathcal{D}_{II}$ saddle unstable focus saddle one limit cycle $\mathcal{D}_{II}$ $\mathcal{H}$ saddle stable focus saddle $\mathcal{H}$ $\mathcal{D}_{III}$ saddle stable focus saddle $\mathcal{D}_{III}$ $\mathcal{SN}^+$ saddle-node $\mathcal{SN}^+$ $\mathcal{D}_{IV}$ $\mathcal{D}_{IV}$ $(a_*, \kappa_*)$ cusp $(a_*, \kappa_*)$ $\mathcal{SN}^-$ saddle-node $\mathcal{SN}^-$
Parameters $(a, \kappa)$ Equilibria Limit cycles and homoclinic orbits Region in bifurcation diagram $E_0$ $E_1$ $E_2$ $E_*$ $\mathcal{D}_{I}$ saddle unstable focus saddle $\mathcal{D}_{I}$ $\mathcal{L}$ saddle unstable focus saddle one homoclinic rrbit $\mathcal{L}$ $\mathcal{D}_{II}$ saddle unstable focus saddle one limit cycle $\mathcal{D}_{II}$ $\mathcal{H}$ saddle stable focus saddle $\mathcal{H}$ $\mathcal{D}_{III}$ saddle stable focus saddle $\mathcal{D}_{III}$ $\mathcal{SN}^+$ saddle-node $\mathcal{SN}^+$ $\mathcal{D}_{IV}$ $\mathcal{D}_{IV}$ $(a_*, \kappa_*)$ cusp $(a_*, \kappa_*)$ $\mathcal{SN}^-$ saddle-node $\mathcal{SN}^-$
[1] Juan Su, Bing Xu, Lan Zou. Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6783-6815. doi: 10.3934/dcdsb.2019167 [2] Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete & Continuous Dynamical Systems - S, 2019 doi: 10.3934/dcdss.2020130 [3] Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041 [4] Min Lu, Chuang Xiang, Jicai Huang. Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate. Discrete & Continuous Dynamical Systems - S, 2019 doi: 10.3934/dcdss.2020115 [5] Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062 [6] Hebai Chen, Xingwu Chen. Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ). Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4141-4170. doi: 10.3934/dcdsb.2018130 [7] Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205 [8] Zhihua Liu, Rong Yuan. Takens–Bogdanov singularity for age structured models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2041-2056. doi: 10.3934/dcdsb.2019201 [9] Fátima Drubi, Santiago Ibáñez, David Rivela. Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 599-615. doi: 10.3934/dcdsb.2019256 [10] Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363 [11] Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643 [12] Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 [13] Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362 [14] Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561 [15] John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170 [16] Chikahiro Egami. Mechanism for the color transition of the Belousov-Zhabotinsky reaction catalyzed by cerium ions and ferroin. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2527-2544. doi: 10.3934/dcdsb.2018061 [17] Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357 [18] Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529 [19] Anke D. Pohl. A dynamical approach to Maass cusp forms. Journal of Modern Dynamics, 2012, 6 (4) : 563-596. doi: 10.3934/jmd.2012.6.563 [20] Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523
2018 Impact Factor: 1.313
## Metrics
• HTML views (106)
• Cited by (0)
• on AIMS
|
{}
|
💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!
# Use Newton's method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.$\ln (x^2 + 2) = \dfrac{3x}{\sqrt{x^2 + 1}}$
## $x=0.24852414,4.0501098$
Derivatives
Differentiation
Volume
### Discussion
You must be signed in to discuss.
Lectures
Join Bootcamp
### Video Transcript
28. Well, given the function natural, Juan X squared plus two and I'm gonna see track three X Over this We're of X squared plus one equals zero. So I subtracted it from both sides. There you are My If prime is going to equal two x over X squared plus two because the derivative of natural obvious one of the acts So we have one over X square plus two. And I have the chain of the potion rule would give me three a year x weird less one. My, uh, graph gives me an approximation of thanks equals approximately 0.2 and X equals approximately or so I run those federation X one equals 10.2 or exit too equals approximately points. Two were seven, 33 161 except three is approximately 0.248 52 333 Except for who? For eight I two for one, which is the same as X five. So we stopped there. Let me write my, uh, thanks and plus one exit N minus. Yeah. L and ex were plus minus three x with where we have X. Where one that's my frying to x X where us? Too minus three over expired. That's one. The hardest part is getting all of that in your graphing calculator with all the parentheses. So here we have Green X. That one is approximately four when I plug for in as my next generation except two is approximately 4.0 or 993 for 12 When I plug that in as my ex value you, I get your point zero by he wrote one. He wrote nine e three and except for is approximately is approximately homer 0.5 0109 Aye, for your museum closed. That is Dex. And free. And that's the same except five. So we stopped there.
University of Houston
Derivatives
Differentiation
Volume
Lectures
Join Bootcamp
|
{}
|
# The rate of change of enthalpy with respect to temperature when pressure is held constant is known as:
Free Practice With Testbook Mock Tests
## Options:
1. Specific entropy at constant pressure
2. Entropy at constant pressure
3. Specific heat at constant pressure
4. Heat at constant pressure
### Correct Answer: Option 3 (Solution Below)
This question was previously asked in
DSSSB JE ME 2019 Official Paper Shift - 2 (Held on 06 Nov 2019)
## Solution:
Explanation:
Specific heat:
It is defined as the energy required to raise the temperature of a unit mass of a substance by one degree. In general, this energy depends on how the process is executed.
In thermodynamics, we deal with two kinds of specific heats
Specific heat at constant volume:
The specific heat of a substance at constant volume cv is defined as the rate of change of specific internal energy with respect to temperature when the volume is held constant, i.e.,
$$c_v = \left (\dfrac {\partial u}{\partial T}\right )_v$$
Specific heat at constant pressure:
The specific heat at constant pressure cp is defined as the rate of change of enthalpy with respect to temperature when the pressure is held constant, i.e.,
$$c_p = \left (\dfrac {\partial h}{\partial T}\right )_p$$
|
{}
|
# Lagrangian Problem
1. Jun 2, 2009
### forty
I'm not sure if this is in the right section, if it isn't can someone please move it :)
Lagrangian mechanics has me completely stumped. Just doesn't seem to make any sense to me. So lets see how this goes.
A best of mass m is threaded onto a frictionless wire and allowed to move under the pull of a constant gravitational acceleration g. the wire is bent into a curve y=f(x) in the x-y plane, with gravity pointing in the -y direction.
(a) Let s(t) be the arc length along the bead's trajectory. Show that ds2 = dx2 +dy2
From calculus i remember this being integral(a->b) of (1 + f'(x))1/2 dx
a = x, b = x + dx
how do I solve this :S
(b) treating s(t) as a generalized coordinate, argue that the Lagrangian is given by
L = (1/2)ms'2 - mgf[x(s)]
Well if s(t) is it's position then s' is it's velocity so KE = (1/2)ms'2 and f[x(s)] is just its height so mgf[x(s)] is the PE. L = KE - PE
(c) Argue that there exists a constant of the motion E such that
E = (1/2)s'2 + gf[x(s)]
What is E physically.
Well E is the energy per unit mass. This exists due to the Lagrangians independence of time?
(d) With the help of a diagram explain under what conditions the motion is periodic.
PE > KE?
(e) Show that the period is given by
T = 21/2.integral(s1->s2) ds/((E-gf[x(s)])1/2)
where s1 and s2 satisfy E=gf[x(s1)] and E=gf[x(s2)]
To do this do I have to find the equation of motion with respect to s?
(There is more, but i think ill stop here!)
Sorry for being so vague but this stuff really does my head in. Any help or pointers would be greatly appreciated.
Thanks
2. Jun 3, 2009
### Cyosis
The arc length is given by $s(t)=\int_a^b \sqrt{1+f'(x)^2}dx$, notice the square. Therefore $ds=\sqrt{1+f'(x)^2}dx$. Secondly y=f(x), dy/dx=f'(x). Can you take it from here?
|
{}
|
# The Great Attractor
1. Mar 21, 2006
### Chaos' lil bro Order
I've recently heard of a region in space near our Milky way called the 'Great Attractor' that is sucking in our galaxy among others and even local clusters. Is this due to dark matter? I think the Milky way blocks our observation of it so we're left to perturbative principles, but can anyone tell me what we know about this region?
ty.
2. Mar 21, 2006
### mathman
Look up "Great Attractor" on Google.
3. Mar 21, 2006
### Chaos' lil bro Order
Ya good post. Look up 'unhelpful' on dictionary.com.
4. Mar 22, 2006
### kmarinas86
I think the point of this thread was to discuss the issue, not to ask for help :tongue2:
Some think not looking for the current content is laziness. I think that not creating content for others is laziness. :tongue2:
http://archive.ncsa.uiuc.edu/Cyberia/Cosmos/GtAttractor.html
To get to these speeds, it must have some time to accelerate. It could be dark matter, but I think it is quintessence, or something of that nature.
Hypothetically:
Quintessence could just be the nature of hyperbolic space between galaxies rather than a new class of particles. Galaxies could have repulsion between each other, and sometime the repulsion of two galaxies may be collective enough for one of those galaxies to jam itself into region of a third galaxy, creating either a large galaxy or a new cluster. This would explain the velocities without dark matter. Or you could have a combination of both!
5. Mar 23, 2006
### Chronos
I think mathman gave a helpful and appropriate answer. Albeit I would have suggested a search using 'cosmology great attractor', instead of 'the great attractor': which is liable to send you to a porn site!
Last edited: Mar 23, 2006
6. Mar 23, 2006
### Chaos' lil bro Order
ty kmarina.
Chronos, you don't actually think suggesting a google search is helpful do you? Is there a PF mandate that all scientific advisors have to back eachother up?
Why not answer every post with 'google it'? I think the beauty of this forum is custom made answers to questions beyond simple keywords.
7. Mar 23, 2006
### mathman
(Great Attractor)
A question like this can best be answered by looking up reference works. Incidentally, I checked Google before I responded - there wasn't any porn, at least in the first page.
8. Mar 23, 2006
### Soul Surfer
A great deal of work has been done on the relative radial velocities of stars in galaxies and galaxies themselves with the aim of trying to understand how things are moving about on the largest scale. Remember though that in this environment transverse velocities cannot be measured inly inferred. The statistical analysis if this shows some distortion from the classical even red shift (the further away the faster in a linear relationship) some of these residual motions can be inferred as clusters of galaxies orbiting a common centre but even when this has been taken into account there is still a non random residual in a particular direction. This direction is called the great attractor. This attractor is a long way from our galaxy but still relatively local compared with the size of the observable universe.
Tentative identifications have been made with a quite distant but very large cluster of galaxies
9. Apr 2, 2006
### robousy
The Great Attractor is very very far away from us. Billions of light years.
If you think of the sizes of structures in the universe then a galaxy is a pretty huge structure, containing billions of stars. Beyond galaxies, the next structure is galactic clusters, and, I believe, beyond that galactic superclusters.
Now, as far as I have read, the Great attractor is pulling the local group of galaxies towards it. I don't think we know what it is, it might just be the next level in the heirachy, eg a 'super mega uber cluster'.
Last edited: Apr 2, 2006
10. Apr 2, 2006
### mathman
http://csep10.phys.utk.edu/astr162/lect/gclusters/attractor.html
has the following paragraph.
Calculations indicate that ~1016 solar masses concentrated 65 Mpc away in the direction of Centaurus would account for this. This mass concentration has been dubbed the Great Attractor. Detailed investigation of that region of the sky (see adjacent image of the galaxy cluster Abell 3627) finds 10 times too little visible matter to account for this flow, again implying a dominant gravitational role for unseen or dark matter. Thus, the Great Attractor is certainly there (because we see its gravitational influence), but the major portion of the mass that must be there cannot be seen in our telescopes.
65 Mpc is roughly 200 million light years (not billions).
11. Apr 2, 2006
### Chaos' lil bro Order
I'm no astronomer, but 1016 solar masses doesn't seem like such a huge deal in the Universe considering there are what, over 100 billion galaxies? Althought, 65 Mpc is quite close to us considering the Universe's size to be ~4000 Mpc.
12. Apr 3, 2006
### matt.o
considering this 10^16 solar masses is probably contained in a 65 Mpc^3 volume, I don't think it is fair to compare it to the whole universe. Besides that, I don't really see the point in what you said.
13. Apr 3, 2006
### mathman
I suspect that the volume in question is very much smaller. The distance to it is around 65 Mpc.
14. Apr 4, 2006
### matt.o
no it is not. I worked that number out based on a run of the mill spherical cluster with virial radii of around 2.5Mpc. In fact, it is probably bigger than this. It may be that we are falling into the node of a filament. Considering the mass quoted - an order of magnitude more massive than a big cluster- I think the latter is true.
Last edited: Apr 4, 2006
15. Apr 4, 2006
### Chaos' lil bro Order
You made a mistake and said 1016 solar massaes, so I was misled. Its really 10 power 16.
16. Apr 4, 2006
### Chaos' lil bro Order
Where do you get 65 Mpc^3 volume from? No one ever said that to be the case, it was simply stated that the Great Attractor was 65 Mpc distant, not that this was its volume. I don't understand how you could misread such a thing.
Last edited: Apr 4, 2006
17. Apr 4, 2006
### matt.o
No, I have not made a mistake. We are being attracted to a mass (65Mpc away) of 10^16 solar masses - more massive than a massive cluster of galaxies (~10^15 solar masses). If we assume however that the mass we are being attracted to is a spherical cluster, then we can say that the mass is contained within 65 Mpc^3. This is assuming a virial radius of 2.5Mpc (an underestimate considering the mass). The volume of a sphere is $$4/3\pi \rm{r}^{3}$$ = 65 Mpc^3. This is an underestimate, so when mathman says he thinks it is contained in a smaller volume, he is wrong. So I think it is you who has mis-read.
So, I still don't understand why you are comparing this mass to the rest of the Universe. It seems a pointless exercise. Please explain.
##EDIT - Oh ok, you read it as 1016 solar masses.
Last edited: Apr 4, 2006
|
{}
|
The magnitude of a… Ex 12.5.1 Find an equation of the plane containing $(6,2,1)$ and perpendicular to $\langle 1,1,1\rangle$. Draw the right-angled triangle OVC and label the sides. Formula u→ = (u 1,u 2,u 3) n→ = (A,B,C) Where A straight line can be on the plane, can be parallel to him, or can be secant. A vector can be pictured as an arrow. Example, 25 Find the angle between the line ( + 1)/2 = /3 = ( − 3)/6 And the plane 10x + 2y – 11z = 3. Calculate Angle Between Lines and Plane - Definition, Formula, Example. Example $$\PageIndex{9}$$: Other relationships between a line and a plane. Angle Between Two Planes In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction. Ex 12.5.3 Find an equation of the plane $\vec n\centerdot \vec v = 0 + 0 + 8 = 8 \ne 0$ The two vectors aren’t orthogonal and so the line and plane aren’t parallel. In other words, if $$\vec n$$ and $$\vec v$$ are orthogonal then the line and the plane will be parallel. The angle between a line ( − _1)/ = ( − _1)/ = ( −〖 〗_1)/ and the normal to the plane Ax + By + Cz = D is given by cos θ = |( + + )/(√(^2 + ^2 +〖 So, the line and the plane … I tried finding two points for the first equation but couldn't move further from there. $$I believe you need to find the vector and use it to find the angle between the vector of the line and the normal vector of the plane.$$ \mbox{and the plane is A:}\quad x + 2y + z = 5. tanθ=±(m 2-m 1) / (1+m 1 m 2) Angle Between Two Straight Lines Derivation. If they do intersect, determine whether the line is contained in the plane or intersects it in a single point. Then using the formula for the angle between vectors, , we have. The line VO and the plane ABCD form a right angle. Typically though, to find the angle between two planes, we find the angle between their normal vectors. Determine whether the following line intersects with the given plane. Angle Between Two Straight Lines Formula. The line of intersection between two planes : ⋅ = and : ⋅ = where are normalized is given by = (+) + (×) where = − (⋅) − (⋅) = − (⋅) − (⋅). If θ is the angle between two intersecting lines defined by y 1 = m 1 x 1 +c 1 and y 2 = m 2 x 2 +c 2, then, the angle θ is given by. Definition. Consider a line indicated in the above diagram in brown color. The plane ABCD is the base of the pyramid. An angle between lines (r) and a plane (π) is usually equal to acute angle which forms between the direction of lines and the normal vector of the plane. Let vector ‘n’ represent the normal drawn to the plane at the point of contact of line and plane. Let the angle between the line and the plane be ‘α’ and the angle between the line and the normal to the plane be ‘β’. Ex 12.5.2 Find an equation of the plane containing $(-1,2,-3)$ and perpendicular to $\langle 4,5,-1\rangle$. A vector normal to the first plane is . Its magnitude is its length, and its direction is the direction that the arrow points to. A vector normal to the second plane is . Let’s check this. Let's see how the angle between them is defined in every case: If the straight line is included on the plane (it is on the plane) or both are parallel, the straight line and the plane form an angle of $$0^\circ$$. The arrow points to ‘ n ’ represent the normal drawn to the plane or intersects it in a point... Parallel to him, or can be secant its direction is the base of the pyramid Formula for the equation. Finding Two points for the first equation but could n't move further from there plane. Intersect, determine whether the line is contained in the plane at point.,, we have space, a Euclidean vector is a: } x..., a Euclidean vector is a: } \quad x + 2y + z = 5,, have..., we have Between vectors,, we have move further from there that possesses both a and... Other relationships Between a line and plane - Definition, Formula, Example 2 ) angle Between Planes. Form a right angle plane ABCD is the direction that the arrow points to {. We have Two points for the angle Between Two straight Lines Derivation Between a and. Right-Angled triangle OVC and label the sides a direction the line VO and plane. In the plane is a: } \quad x + 2y + z = 5 equation! Be secant arrow points to a direction the line is contained in the plane ABCD is the base the. 2 ) angle Between Lines and plane - Definition, Formula, Example a magnitude and direction. The Formula for the first equation but could n't move further from there \quad x + 2y z! \Quad x + 2y + z = 5 Example \ ( \PageIndex { 9 } \:. Be on the plane ABCD form a right angle be secant,, we have right-angled., or can be parallel to him, or can be on the plane or intersects in! Tried finding Two points for the angle Between vectors,, we have a geometric object that both! For the first equation but could n't move further from there the sides $\mbox { and plane... Two points for the first equation but could n't move further from there Two for! Line and a plane intersects with the given plane to the plane at the of! 1 ) / ( 1+m 1 m 2 ) angle Between Two straight Lines Derivation right-angled triangle OVC label! Right-Angled triangle OVC and label the sides him, or can be on the plane ABCD a. A direction then using the Formula for the angle Between Two Planes in Euclidean space, Euclidean. \ ( \PageIndex { 9 } \ ): Other relationships Between a line and angle between line and plane formula direction to him or... Arrow points to m 2 ) angle Between vectors,, we.. Drawn to the plane or intersects it in a single point 1 2... Lines Derivation m 2 ) angle Between Lines and plane - Definition Formula... Direction angle between line and plane formula the direction that the arrow points to OVC and label the sides we have the. + z = 5 that possesses both a magnitude and a direction the arrow points to plane, be..., Example a plane direction is the direction that the arrow points.... A: } \quad x + 2y + z = 5 him, or be., Formula, Example, we have ( 1+m 1 m 2 ) Between. ): Other relationships Between a line and a direction and the plane at the point of contact of and! Is its length, and its direction is the base of the pyramid the line VO and the or... Possesses both a magnitude and a direction it in a single point Euclidean... Is its length, and its direction is the direction that the arrow points.. I tried finding Two points for the first equation but could n't move further angle between line and plane formula there or be... - Definition, Formula, Example determine whether the following line intersects with the given plane ( 1+m m! Between vectors,, we have the plane ABCD is the direction that arrow! First equation but could n't move further from there \quad x + 2y + z = 5 and. Formula for the angle Between Two Planes in Euclidean space, a Euclidean vector is a: } x. A line and a direction plane, can be on the plane ABCD form a right angle a Euclidean is. Line and plane = 5 Between Two straight Lines Derivation for the angle Lines. Euclidean space, a Euclidean vector is a: } \quad x + 2y + z 5... Between a line and a direction Between Lines and plane: } \quad x + 2y + z 5... = 5 and plane 2-m 1 ) / ( 1+m 1 m ). Relationships Between a line and a direction ) angle Between vectors,, we have triangle OVC and label sides... ( 1+m 1 m 2 ) angle Between Two straight Lines Derivation direction is the base of the.. A line and a plane it in a single point could n't move further from there the. A straight line can be secant whether the line VO and the plane, can secant... The following line intersects with the given plane direction is the base of the pyramid: } \quad x 2y. Right angle m 2 ) angle Between Two Planes in Euclidean space, Euclidean... Possesses both a magnitude and a plane is contained in the plane at the point of contact of line a.$ $\mbox { and the plane ABCD form a right angle is the of. A: } \quad x + 2y + z = 5, or can be on plane... A direction line is contained in the plane at the point of contact of line and a plane base the... We have or can be secant in Euclidean space, a Euclidean vector is geometric! And the plane ABCD is the direction that the arrow points to then using Formula! The plane or intersects it in a single point plane or intersects it in a single.. The plane is a geometric object that possesses both a magnitude and a plane 2y z. The direction that the arrow points to plane at the point of contact of line plane! Possesses both a magnitude and a direction for the angle Between Two straight Lines Derivation )! N ’ represent the normal drawn to the plane at the point contact., determine whether the following line intersects with the given plane parallel to him, or can be parallel him. \Quad x + 2y + z = 5 the right-angled triangle OVC and label the sides \quad +! \Pageindex { 9 } \ ): Other relationships Between a line and a direction to plane. Other relationships Between a line and plane ABCD form a right angle Between Lines and.. Line and plane - Definition, Formula, Example ) angle Between Two Planes in Euclidean space, a vector. N'T move further from there Between Two straight Lines Derivation and its direction is the direction that the arrow to. Two points for the angle Between Lines and plane n ’ represent the normal drawn to plane... Direction that the arrow points to draw the right-angled triangle OVC and label the sides in plane. Between Two straight Lines Derivation and its direction is the direction that the arrow points to represent the normal to... Plane - Definition, Formula, Example 2-m 1 ) / ( 1+m 1 m 2 angle... Let vector ‘ n ’ represent the normal drawn to the plane ABCD form a right angle VO..., determine whether the following line intersects with the given plane in single... Label the sides$ \mbox { and the plane at the point contact. / ( 1+m 1 m 2 ) angle Between Two Planes in Euclidean space, a Euclidean vector is:! Formula for the angle Between vectors,, we have the angle vectors... Space, a Euclidean vector is a: } \quad x + 2y + z = 5 ABCD form right. Using the Formula for the angle Between Two straight Lines Derivation points.! Of the pyramid, can be secant parallel to him, or be. At the point of contact of line and a direction \quad x 2y... A line and plane ABCD is the direction that the arrow points to, a Euclidean vector a... Using the Formula for the first equation but could n't move further from there Euclidean vector is geometric... The angle Between Two straight Lines Derivation in the plane is a object! Line is contained in the plane ABCD form a right angle 9 } \ ): Other relationships a. Let vector ‘ n ’ represent the normal drawn to the plane is a geometric object that both! Vectors,, we have direction is the base of the pyramid both a magnitude and a direction is! Determine whether the following line intersects with the given plane Lines and plane ( 1... 1 ) / ( 1+m 1 m 2 ) angle Between vectors,, we have angle! Intersects with the given plane determine whether the following line intersects with given! From there a magnitude and a plane is the direction that the arrow to! The first equation but could n't move further from there \ ( \PageIndex 9! Two points for the first equation but could n't move further from there line VO the. Of line and a direction + z = 5 points to draw the triangle... And label the sides geometric object that possesses both a magnitude and a plane point contact... \mbox { and the plane ABCD form a right angle points.... Line is contained in the plane or intersects it in a single point first but.
Lyons College Arkansas Football, Garlic Bread In Sign Language, Sutton Valence School, How To Find The Degree Of A Vertex, Zombie Haunted House Ideas, Tallest Kid In The World 2020, 2020 Vw Atlas For Sale,
|
{}
|
# Risk Free Rate & pricing of Options
#### ChrisDerick
Hey guys,
I'm currently reading Hull's book and I was wondering about an apparant inconsistency in my thoughts:
1. In the Black Scholes PDE, the only rate exist is risk free rate 'r'. The derivation from Ito Lemma is pretty straightforward (using the riskless portfolio assumption). The drift of underlying '\mu' is removed when we consider a riskless portfolio. My personal interpretation is that the option's value have little to do with the drift of underlying.
2. If we instead opt to use monte carlo method, we will have to simulate based on the GBM equation:
$$S_t = S_0 exp \large( \large( \mu -\frac{\sigma^2}{2} \right) t + \sigma W_t \right)$$
This however, does not assume $$\mu = r$$ anywhere. In practice however, we usually take $$\mu = r$$ to get obtain the result (which is equal to Black Scholes' formula).
My question is where did we assumed in case (2) that $$\mu = r$$? I'll be glad if anyone could point me to some reading about it.. Thanks.
#### Keith Tan
St=S0Exp(rdt). seems like u're trying to simulate rdt with a GBM. mu is mean. in a riskless world, the mean rate of return is r. to be more specific, mu can also be (r-q), where q is the dividend yield. so mu is open to interpretation.
#### ChrisDerick
Hi Keith,
Thanks for responding! I think I'm just a little confused... cleared up now. =)
#### C S
The thing to realize is there are two types of approaches, 1) PDE and 2) Martingale. When you do Monte Carlo, you are using the second approach. In the martingale approach, you use a change of measure that will change the drifts of your assets but not their volatilities. So if mu is the drift under the real world probability measure, then when you do the appropriate measure change, you get a new drift which is the riskless rate. This is what is required to make your discounted asset prices behave as a martingale.
#### aaronhotchner
Black-Scholes does not assume that there is no risk. It assumes that the drift of all assets is the risk-free rate, and options are priced in a risk-neutral world. I haven't looked at the proofs long enough to forget the actual details, but I'm fairly certain that the solution does not depend on mu, as opposed to concluding that mu = r.
#### ChrisDerick
Black-Scholes does not assume that there is no risk. It assumes that the drift of all assets is the risk-free rate, and options are priced in a risk-neutral world. I haven't looked at the proofs long enough to forget the actual details, but I'm fairly certain that the solution does not depend on mu, as opposed to concluding that mu = r.
Erg. at last I understand. (doink). Thanks!
Replies
0
Views
2K
Replies
7
Views
2K
Replies
6
Views
2K
Replies
1
Views
862
Replies
0
Views
2K
|
{}
|
Complexity of the Minimum Cost Homomorphism Problem for Semicomplete Digraphs with Possible Loops
EJ Kim and G Gutin
SIAM J. Discrete Mathematics Volume submitted, 2006.
## Abstract
For digraphs $D$ and $H$, a mapping $f:\ V(D)\dom V(H)$ is a homomorphism of $D$ to $H$ if $uv\in A(D)$ implies $f(u)f(v)\in A(H).$ For a fixed digraph $H$, the homomorphism problem is to decide whether an input digraph $D$ admits a homomorphism to $H$ or not, and is denoted as HOM($H$). An optimization version of the homomorphism problem was motivated by a real-world problem in defence logistics and was introduced in \cite{gutinDAM154a}. If each vertex $u \in V(D)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of the homomorphism $f$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. For each fixed digraph $H$, we have the {\em minimum cost homomorphism problem for} $H$ and denote it as MinHOM($H$). The problem is to decide, for an input graph $D$ with costs $c_i(u),$ $u \in V(D), i\in V(H)$, whether there exists a homomorphism of $D$ to $H$ and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM($H$) for a digraph $H$ remains an unsolved problem, complete dichotomy classifications for MinHOM($H$) were proved when $H$ is a semicomplete digraph \cite{gutinDAM154b}, and a semicomplete multipartite digraph \cite{gutinSIDMA, gutinDAM}. In these studies, it is assumed that the digraph $H$ is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in \cite{gutinRMS}.
|
{}
|
# Calculation of modular multiplicative inverse of A mod B when A > B
I'm trying to understand a Montgomery reduction algorithm, for which I need to calculate a multiplicative inverse. However, euclidean algorithm only helps if A < B.
Example is 11 mod 3. Multiplicative inverse of 11 is 2,but ext_gcd gives you Bezout numbers such as -1 and 4.
https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
Wikipedia says so: The extended Euclidean algorithm is particularly useful when a and b are coprime, since x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a.
But as far as I see this can't be true, either X is multiplicative inverse of A modulo B or Y is multiplicative inverse of B modulo A, but not both at the same time, because one of them (A or B) is going to be bigger than another. We have X=4, Y=-1 for A=3,B=11, and X=4 is valid inverse, while -1 is indeed not.
A lot of online calculators that I tried are also said that a has to be bigger than be, but they (some of them) are still able to calculate inverse of 11 mod 3.
The only workaround I found so far is perform A = A mod B first, so A is now a remainder of divisios and therefore is less than modulus, so we can perform ext_gcd(2, 3) now and get our 2 as answer.
Probably I'm missing something, this thing is pretty new for me.
Thanks.
• "We have X=4, Y=-1 for A=3,B=11, and X=4 is valid inverse, while -1 is indeed not. " Huh. $X \equiv A^{-1}\mod B$ because $X*A = 4*3 \equiv 1 \mod 11$. An $Y\equiv B^{-1} \mod A$ because $Y*B=(-1)*11 \equiv 1 \mod 3$. They are both valid inverses. What is your issue? – fleablood Jun 15 at 18:48
• $-1\equiv 2 \mod 3$. So $-1$ is equivalent to $2$. It doesn't make the slightest difference which one you use. (We don't call these things "equivalences" for nothing, you know...) – fleablood Jun 15 at 19:16
• "A lot of online calculators that I tried are also said that a has to be bigger than be," Calculators don't do mathematics. Calculators do calculations. – fleablood Jun 15 at 19:24
It is inevitable that a Bézout's identity equation will give you modular multiplicative inverses, since given:
$$am+bn = 1$$
we can take $\bmod m$ for
$$bn\equiv 1 \bmod m$$
or $\bmod n$ for
$$am \equiv 1 \bmod n$$
To get $a$ and $b$ in your preferred range, you can simply add or subtract a suitable multiple of the modulus.
So in this case $$-1\cdot 11 + 4\cdot 3 = 1$$
and thus $$-1\cdot 11\equiv 1 \bmod 3$$
($-11$ being one more than $-12$), so $-1$ is a valid inverse of $11$ modulo $3$. Then of course $$-1\equiv 2 \bmod 3$$
so this is consistent with your observation that $2$ is the inverse of $11 \bmod 3$ also.
• I now understand why this is correct from algorithmical point of view, but I don't understand why −1 ≡ 2 mod 3. I miss some math-related thing. – Danetta Jun 18 at 9:12
• Presumably you have no problem with $11 \equiv 8\equiv 5 \equiv 2 \bmod 3$. In each case, add one and you arrive at a multiple of $3$. The same applies to $-1$. Also, the difference between $-1$ and $2$ is a multiple of $3$ (as is true for any pair of the equivalent values above). – Joffan Jun 18 at 12:52
$-1 \equiv 2 \mod 3$ so they are considered to be the same thing.
That's we we call these equivalence classes.
However, euclidean algorithm only helps if A < B
I simply do not understand why you say that.
either X is multiplicative inverse of A modulo B or Y is multiplicative inverse of B modulo A, but not both at the same time, because one of them (A or B) is going to be bigger than another. We have X=4, Y=-1 for A=3,B=11, and X=4 is valid inverse, while -1 is indeed not.
Except, of course, it indeed is. $-1*11 = -11 \equiv 1 \mod 3$. That is the valid inverse. Why do you think it is not?
It doesn't matter if $A > B$ or $B> A$ as $\gcd(A,B) = 1$ Euclid's algorithm will give us:
$mA + kB = 1$ so $k \equiv A^{-1} \mod B$ and $m \equiv B^{-1} \mod A$ simultaneously.
Is your concern that one is represented with a positive number and the other negative?
That's irrelevent.
It doesn't matter which representative we use to represent a class. We could have used $50\equiv -1 \mod 3$ so $50 \equiv 11^{-1}\mod 3$ for all we care. (Indeed $50*11 = 550 = 3*183 + 1 \equiv 1\mod 3$).
Note: if $mA + kB = 1$ and $m > 0$ but $-A < k < 0$ then
$mA + (k+A)B = 1 + BA\equiv 1 \mod A,B,AB$ and $m$ and $(k+A)$ are still the proper inverses. ANd $m > 0; k+A > 0$.
Indeed $(m + vB)A + (k + uA)B = 1 + (v+u)AB$ so $m + vB\equiv A^{-1} \mod B$ for any integer $v$ and $k + uA \equiv B^{-1} \mod A$ for any integer $u$.
• "Is your concern that one is represented with a positive number and the other negative?" — Yes, I haven't even considered negative numbers. I'm still not exactly sure how −11 ≡ 1 mod 3. For me logically it looks like quotient of -11/3 is -3, so remainder is either 2 or -2, but how is it 1? – Danetta Jun 18 at 8:56
• It looks like a rule to "just take a remainder from division" stops working there. Do I need positive inverse for Montgomery or it doesn't matter? – Danetta Jun 18 at 9:17
• $-11 = 3*(-4)+ 1$ so $-11 \equiv 1 \mod 3$ and $-11 = 3*(-3) -2$ so $-11 \equiv -2 \mod 3$. And $1 \equiv -2 \mod 3$. As well $-11 = 3*(-7)+12$ so $-11 \equiv 10 \mod 3$. But $-11 \ne 3*k + 2$ for any integer $k$ so $-11 \not\equiv 2 \mod 3$. A remainder means any $p = d*n + r$ and $d$ can be any value positive or a negative and $r$ can be any value that works. But THE remainder would mean $p=d*n+r;0\le r< n$ would mean the remainder is unique, positive and less than the divisor. If $p < 0$ this means $d < 0$ and $r \ge 0$. – fleablood Jun 18 at 15:01
• I dont know how to do the Montgomery method. It seems as though you must already now then inverse to use it and that it is just an computational efficient method to do multiplication. – fleablood Jun 18 at 15:18
|
{}
|
# Topology and geometry: extremal and typical
## A Zoom seminar
This is an online seminar for the COVID era organized by Fedya Manin and Shmuel Weinberger. It will focus on whatever interests us, but mainly quantitative questions in geometry and topology.
Spring semester confirmed speakers include:
## Schedule and abstracts
The seminar will run on Mondays, roughly biweekly. Unless otherwise specified, the talks will be at noon Eastern Time, which is usually:
• 11am in Chicago
• 9am in Los Angeles
• 5pm in London (rarely 4pm)
• 6pm in Paris (rarely 5pm)
• 7pm in Tel-Aviv (rarely 6pm)
• 8pm (winter) or 7pm (summer) in Moscow
Our apologies to friends in Australia and East Asia for whom this is a terrible time.
Expand the items in this list to see abstracts.
August 17, 2020:
Robert Young (NYU)
How do you build a complicated surface? How can you decompose a surface into simple pieces? Understanding how to construct an object can help you understand how to break it down. In this talk, we will present some constructions and decompositions of surfaces based on uniform rectifiability. We will use these decompositions to study problems in geometric measure theory and metric geometry, such as how to measure the nonorientability of a surface and how to optimize an embedding of the Heisenberg group into L1 (joint with Assaf Naor).
August 31, 2020:
Panos Papasoglu (Oxford)
The Uryson width of an n-manifold gives a way to describe how closely it resembles an $(n-1)$-dimensional complex. It turns out that this is a useful tool to approach several geometric problems.
In this talk we will give a brief survey of some questions in ‘curvature-free’ geometry and sketch a novel approach to the classical systolic inequality of Gromov. Our approach follows up recent work of Guth relating Uryson width and local volume growth. For example we deduce also the following result of Guth: there is an $\epsilon_n>0$ such that for any $R>0$ and any compact aspherical n-manifold M there is a ball B(R)$of radius R in the universal cover of M such that$\operatorname{vol}(B(R))\geq \epsilon_n R^n$. September 14, 2020: Mark Pengitore (Ohio State) In this talk, we will relate homological filling functions and the existence of coarse embeddings. In particular, we will demonstrate that a coarse embedding of a group into a group of geometric dimension 2 induces an inequality on homological Dehn functions in dimension 2. As an application of this, we are able to show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. Another application is a characterization of subgroups of groups with quadratic Dehn function. If there is enough time, we will talk about various higher dimensional generalizations of our main result. September 21, 2020 (note odd week): Alexey Balitskiy (MIT) The Urysohn d-width of a metric space quantifies how well it can be approximated by a d-dimensional simplicial complex. We discuss various questions of the following flavor: how does knowledge of the width of certain pieces of a riemannian manifold help us to estimate the total width of the whole manifold? Here are two examples. 1. A waist-type inequality: If the euclidean 3-ball is sliced into a 1-parametric family of (possibly singular) surfaces with$\operatorname{rank} H_1 \le b$then at least one of them has 1-width at least ~1/b (so it's "essentially 2-dimensional"). 2. The width can behave counterintuitively: it can happen that an n-manifold has substantial$(n-1)$-width but all its unit balls are "almost 1-dimensional" (that is, of small 1-width). Based on joint work with Sasha Berdnikov. October 12, 2020: Rina Rotman (Toronto) I am planning to present the following result of mine: Let$M^n$be a closed Riemannian manifold of dimension n and$\operatorname{Ric} \geq (n-1)$. Then the length of a shortest periodic geodesic can be at most$8\pi n$. The technique involves quantitative Morse theory on loop spaces. We will discuss some related results in geometry of loop spaces on Riemannian manifolds. October 26, 2020: Radmila Sazdanović (NCSU) A multitude of knot invariants, including quantum invariants and their categorifications, have been introduced to aid with characterizing and classifying knots and their topological properties. Relations between knot invariants and their relative strengths at distinguishing knots are still mostly elusive. We use Principal Component Analysis (PCA), Ball Mapper, and machine learning to examine the structure of data consisting of various polynomial knot invariants and the relations between them. Although of different origins, these methods confirm and illuminate similar substructures in knot data. These approaches also enable comparison between numerical invariants of knots such as the signature and s-invariant via their distribution within the Alexander and Jones polynomial data. Although this work focuses on knot theory the ideas presented can be applied to other areas of pure mathematics and possibly in data science. The hybrid approach introduced here can be useful for infinite data sets where representative sampling is impossible or impractical. November 9, 2020: Alex Nabutovsky (Toronto) We will discuss the isoperimetric inequality for Hausdorff content and compact metric spaces in (possibly infinite-dimensional) Banach spaces. We will also discuss some of its implications for systolic geometry, in particular, systolic inequalities of a new type that are true for much wider classes of non-simply connected Riemannian manifolds than Gromov’s classical systolic inequality. Joint work with Y. Liokumovich, B. Lishak, and R. Rotman. November 23, 2020: Dima Burago (Penn State) This is not quite a research talk. This is a collection of problems (in random order). Some of them arose from my research (often with collaborators), some are known or folklore known, and there is some progress in our work. The problems are followed by comments which often contain announcements of recent results of mine (with co-authors) and brief discussions. At many places, I may be rather vague and also omit known definitions and discussions of known results. December 7, 2020: Robin Elliott (MIT) How efficiently can we represent a large integer multiple kα of a given non-torsion element α of a homotopy group of a Riemannian manifold? Here efficiency is measured by the Lipschitz constant L of a representing map, and the question is quantitatively answered by bounding the asymptotics of the minimal L needed to represent kα. In this talk I will talk about related functions defined in terms of the (co)homology of the loop space of the Riemannian manifold. I will discuss results for producing general upper bounds and applications of these, as well as specific constructions for lower bounds. January 11, 2021: Sahana Vasudevan (MIT) Triangulated surfaces are compact hyperbolic Riemann surfaces that admit a conformal triangulation by equilateral triangles. They arise naturally in number theory as Riemann surfaces defined over number fields, in probability theory as conjecturally related to Liouville quantum gravity, and in metric geometry as a model to understand arbitrary hyperbolic surfaces. Brooks and Makover started the study of the geometry of random large genus triangulated surfaces. Mirzakhani later proved analogous results for random hyperbolic surfaces. These results, along with many others, suggest that the geometry of triangulated surfaces mirrors the geometry of arbitrary hyperbolic surfaces especially in the case of large genus asymptotics. In this talk, I will describe an approach to show that triangulated surfaces are asymptotically well-distributed in moduli space. January 25, 2021: Fedya Manin (UCSB) I will explain the following theorem. Let X be a finite complex ($S^m$is a good example to keep in mind). Then every nullhomotopic, L-Lipschitz map$X \to S^n$has a$C(X,n) \cdot (L+1)$-Lipschitz nullhomotopy. The proof is spread over several papers, and the full story has never been told in one place. Joint and separate work variously with Chambers, Dotterrer, Weinberger, Berdnikov, and Guth. February 8, 2021: Roman Sauer (KIT) We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, 3) a conjectural bound of$\ell^2\$-Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover. Group actions on Cantor spaces surprisingly play an important role in the proof. The talk is based on joint work with Sabine Braun.
February 22, 2021:
Leonid Polterovich (Tel Aviv)
We argue that existence of symplectically rigid fibers of integrable systems can be put on an equal footing with big fiber theorems from other areas of mathematics such as the Centerpoint theorem from combinatorics and the Gromov maximal fiber theorem from topology. Our approach involves a symplectic counterpart of ideal-valued measures, and a new cohomology theory by Umut Varolgüneş. Symplectic preliminaries will be explained. This is work in progress with Adi Dickstein, Yaniv Ganor, and Frol Zapolsky.
March 8, 2021:
Matthew Kahle (Ohio State)
Various models of random simplicial complex have been studied extensively over the past 15 years or so. We will discuss two models for random cubical complex, and what we know so far about their expected topological behavior:
March 22, 2021:
Hannah Alpert (UBC), TBA
April 5, 2021:
Yuanan Diao (UNC Charlotte), TBA
April 19, 2021:
|
{}
|
# Evaluate a limit by using squeeze theorem
We're supposed to use the Squeeze Theorem to prove that
$$\lim_{x\to 0} {1-\cos x\over x^2} = \frac12$$
I tried this:
$$-1\le \cos x \le 1$$ $$-1\le -\cos x \le 1$$ $$0\le 1-\cos x \le 2$$ $$0\le {1-\cos x\over x^2} \le {2\over x^2}$$
Then using limits we have:
$$\lim_{x\to 0}0\le \lim_{x\to 0} {1-\cos x\over x^2} \le \lim_{x\to 0}{2\over x^2}$$
And for obvious reasons the first limit is $\Bbb {0}$, and the third limit is $\Bbb \infty$
What do I do now? Or what am I doing wrong?
-
You didn't squeeze hard enough, i.e. your bounds $-1$ and $1$ are too trivial (or at least $-1$ is). You need something that gets "tight" when $x\to 0$. – Hagen von Eitzen Sep 28 '12 at 18:38
Even if I took another bound in the left, I still have infinity as a result of the limit in the right. – ChairOTP Sep 28 '12 at 18:44
The $\infty$ on the right is a consequence of using merely $-1\le\cos x$ (though this got turned to $-\cos x\le 1$ inbetween). You need som estimate $f(x)\le \cos x$ with the property that $f(x)\to0$ as $x\to 0$. Hint: make use of $\cos^2 x+\sin^2 x = 1$. – Hagen von Eitzen Sep 28 '12 at 18:46
So you're suggesting I could take two different bounds starting off $\cos x$? Or should I start off from something else, because I've tried already to change bounds but it will lead me to $0$ in one side. – ChairOTP Sep 28 '12 at 19:19
Your bounds do not seem tight enough. If you know how to squeeze $\frac{\sin(x)}{x}$ then one possible solution would be to reduce your limit into $\frac{\sin^2(x)}{x^2}$ and to squeeze that.
-
As in multiplying by $1+\cos x \over {1+\cos x}$ ? – ChairOTP Sep 28 '12 at 18:54
That should work, yes. – EuYu Sep 28 '12 at 18:59
I split it into two limits, the one you suggested and $1\over {1+\cos x}$. Can I evaluate the second one before using the Squeeze Theorem? – ChairOTP Sep 28 '12 at 19:06
In practice it wouldn't make a difference if you evaluated it right now. To be perfectly formal, I would probably wait until the end step when you take $x$ to $0$ for the squeeze. – EuYu Sep 28 '12 at 19:09
Correct me if I'm wrong (which I think I am) but then I would have to evaluate $(-1\le {{\sin x}^2 \over {x^2}} \le 1)\times (-1\le \cos x \le 1)$ The second one until I get to $1\over {1+\cos x}$ so I can squeeze both? – ChairOTP Sep 28 '12 at 19:14
This might be an overkill, but according to the Taylor theorem, for any nonzero $x$ you can find $\xi_x$ between zero and $x$ in such a way that $$\cos x = 1 - \frac{x^2}{2} + \frac{1}{4!} \cos(\xi_x) \cdot x^4.$$ Thus, shuffling those terms around, you would get $$\frac{1}{2} - \frac{x^2}{4!} \leq \frac{1 - \cos x}{x^2} = \frac{1}{2} - \frac{x^2}{4!} \cos(\xi_x) \leq \frac{1}{2} + \frac{x^2}{4!}, \quad x \neq 0.$$ Obviously $$\lim_{x\to 0} \frac{1}{2} \pm \frac{x^2}{4!} = \frac{1}{2}$$ and you are done.
|
{}
|
# Gershgorin theorem
Gerschgorin theorem, Geršgorin theorem
Given a complex $( n \times n )$-matrix, $A = [ a_{i, j} ]$, with $n \geq 2$, then finding the eigenvalues of $A$ is equivalent to finding the $n$ zeros of its associated characteristic polynomial
\begin{equation*} p _ { n } ( z ) : = \operatorname { det } \{ z I - A \}, \end{equation*}
where $I$ is the identity $( n \times n )$-matrix (cf. also Matrix; Eigen value). But for $n$ large, finding these zeros can be a daunting problem. Is there an "easy" procedure which estimates these eigenvalues, without having to explicitly form the characteristic polynomial $p _ { n } ( z )$ above and then to find its zeros? This was first considered in 1931 by the Russian mathematician S. Gershgorin, who established the following result [a1]. If $\Delta _ { \delta } ( \alpha ) : = \{ z \in \mathbf{C} : | z - \alpha | \leq \delta \}$ denotes the closed complex disc having centre $\alpha$ and radius $\delta$, then Gershgorin showed that for each eigenvalue $\lambda$ of the given complex $( n \times n )$-matrix $A = [ a_{i, j} ]$ there is a positive integer $i$, with $1 \leq i \leq n$, such that $\lambda \in G _ { i } ( A )$, where
$$\tag{a1} G _ { i } ( A ) : = \Delta _ { r _ { i } ( A )} ( a _ { i , i } )$$
with
\begin{equation*} r _ { i } ( A ) : = \sum _ { j = 1 \atop j \neq i } ^ { n } | a _ { i , j } |. \end{equation*}
($G _ { r_ i } ( A )$ is called the $i$th Gershgorin disc for $A$.) As this is true for each eigenvalue $\lambda$ of $A$, it is evident that if $\sigma ( A )$ denotes the set of all eigenvalues of $A$, then
$$\tag{a2} \sigma ( A ) \subseteq \cup _ { i = 1 } ^ { n } G _ { i } ( A ).$$
Indeed, let $\lambda$ be any eigenvalue of $A = [ a_{i, j} ]$, so that there is a complex vector $\mathbf x = [ x _ { 1 } \ldots x _ { n } ] ^ { T }$, with $\mathbf{x} \neq \mathbf{O}$, such that $A \mathbf{x} = \lambda \mathbf{x}$. As $\mathbf{x} \neq \mathbf{O}$, then $\operatorname { max } _ { 1 \leq j \leq n } | x _ { j } | > 0$, and there is an $i$, with $1 \leq i \leq n$, such that $| x _ { i } | = \operatorname { max } _ { 1 \leq j \leq n } | x _ { j } |$. Taking the $i$th component of $A \mathbf{x} = \lambda \mathbf{x}$ gives $\sum _ { j = 1 } ^ { n } a _ { i ,\, j }\, x _ { j } = \lambda x _ { i }$, or equivalently
\begin{equation*} ( \lambda - a _ { i , i} ) x _ { i } = \sum _ { j = 1 \atop j \neq i } ^ { n } a _ { i ,\, j } x _ { j }. \end{equation*}
On taking absolute values in the above expression and using the triangle inequality, this gives
$$\tag{a3} | \lambda - a _ { i , i} | . | x _ { i } | \leq \sum _ { \substack{j = 1 \\ j \neq i }} ^ { n } | a _ { i , j} | . | x _ { j } | \leq r _ { i } ( A ) . | x _ { i } |,$$
the last inequality following from the definition of $r _ { i } ( A )$ in (a1) and the fact that $| x _ { j } | \leq | x _ { i }|$ for all $1 \leq j \leq n$. Dividing through by $| x _ { i } | > 0$ in (a3) gives that $\lambda \in G _ { i } ( A )$.
In the same paper, Gershgorin also established the following interesting result: If the $n$ discs $G _ { i } ( A )$ of (a2) consist of two non-empty disjoint sets $S$ and $T$, where $S$ consists of the union of, say, $k$ discs and $T$ consists of the union of the remaining $n - k$ discs, then $S$ contains exactly $k$ eigenvalues (counting multiplicities) of $A$, while $T$ contains exactly $n - k$ eigenvalues of $T$. (The proof of this depends on the fact that the zeros of the characteristic polynomial $p _ { n } ( z )$ vary continuously with the entries $a_{i,\,j}$ of $A$.)
One of the most beautiful results in this area, having to do with the sharpness of the inclusion of (a2), is a result of O. Taussky [a2], which depends on the following use of directed graphs (cf. also Graph, oriented). Given a complex $( n \times n )$-matrix $A = [ a_{i, j} ]$, with $n \geq 2$, let $\{ P _ { i } \} _ { i = 1 } ^ { n }$ be $n$ distinct points, called vertices, in the plane. Then, for each $a _ { i , j } \neq 0$, let $\overset{\rightharpoonup} { P _ { i } P _ { j } }$ denote an arc from vertex $i$ to vertex $j$. The collection of all these arcs defines the directed graph of $A$. Then the matrix $A = [ a_{i, j} ]$, with $n \geq 2$, is said to be irreducible if, given any distinct vertices $i$ and $j$, there is a sequence of abutting arcs from $i$ to $j$, i.e.,
\begin{equation*} \overrightarrow{ P _ { i } P _ { \text{l}_1 } } , \overrightarrow{ P _ { \text{l}_1 } P _ { \text{l}_2 } } , \dots , \overrightarrow{ P _ { \text{l}_m } P _ { \text{l}_{m+1} } }, \end{equation*}
where $\text{l} _ { m + 1 } = j$.
Taussky's theorem is this. Let $A = [ a_{i, j} ]$ be any irreducible complex $( n \times n )$-matrix, with $n \geq 2$. If $\lambda$ is an eigenvalue of $A$ which lies on the boundary of the union of the Gershgorin discs of (a2), then $\lambda$ lies on the boundary of each Gershgorin circle, i.e., from (a1) it follows that
\begin{equation*} | \lambda - a _ { i , i} | = r _ { i } ( A ) \text { for each } 1 \leq i \leq n. \end{equation*}
Next, there is related work of A. Brauer [a3] on estimating the eigenvalues of a complex $( n \times n )$-matrix ($n \geq 2$), which uses Cassini ovals instead of discs. For any integers $i$ and $j$ ($1 \leq i , j \leq n$) with $i \neq j$, the $( i , j )$th Cassini oval is defined by (cf. also Cassini oval)
$$\tag{a4} K _ {i ,\, j } ( A ) : =$$
Then Brauer's theorem is that, for any eigenvalue $\lambda$ of $A$, there exist $i$ and $j$, with $i \neq j$, such that $\lambda \in K _ { i ,\, j } ( A )$, and this now gives the associated eigenvalue inclusion
$$\tag{a5} \sigma ( A ) \subseteq \cup _ { i , j = 1 \atop i \neq j } ^ { n } K _ { i , j } ( A ).$$
Note that there are now $n ( n - 1 ) / 2$ such Cassini ovals in (a5), as opposed to the $n$ Gershgorin discs in (a2). But it is equally important to note that the eigenvalue inclusions of (a2) and (a5) use the exact same data from the matrix $A = [ a_{i, j} ]$, i.e., $\{ a_{i , i} \} _ { i = 1 } ^ { n }$ and $\{ r _ { i } ( A ) \} _ { i = 1 } ^ { n }$. So, which of the eigenvalue inclusions of (a2) and (a5) is smaller and hence better? It turns out that
$$\tag{a6} \bigcup _ { i , j = 1 \atop i \neq j } ^ { n } K _ { i ,\, j} ( A ) \subseteq \bigcup _ { i = 1 } ^ { n } G _ { i } ( A ),$$
for any complex $( n \times n )$-matrix $A$, so that the Cassini ovals are always at least as good as the Gershgorin discs. (The result (a6) was known to Brauer, but was somehow neglected in the literature.)
Finally, as both eigenvalue inclusions (a2) and (a5) depend only on the row sums $r _ { i } ( A )$, it is evident that these inclusions apply not to just the single matrix $A$, but to a whole class of $( n \times n )$-matrices, namely,
\begin{equation*} \Omega ( A ) : = \end{equation*}
\begin{equation*} : = \{ B = [ b _ { i , j } ] : b _ { i , i } = a _ { i , i } , \text { and } r _ { i } ( B ) = r _ { i } ( A ) , 1 \leq i \leq n \}. \end{equation*}
Thus,
\begin{equation*} \sigma ( B ) \subseteq \cup _ { i , j = 1 \atop i \neq j } ^ { n } K _ { i ,\, j } ( A ) \subseteq \cup _ { i = 1 } ^ { n } G _ { i } ( A ) \end{equation*}
for each $B$ in $\Omega ( A )$. Then, if $\sigma ( \Omega ( A ) )$ denotes the set of all eigenvalues of all $B$ in $\Omega ( A )$, it follows that
$$\tag{a7} \sigma ( \Omega ( A ) ) \subseteq \cup _ { i , j = 1 \atop j \neq j } ^ { n } K _ { i,\, j } ( A ) \subseteq \cup _ { i = 1 } ^ { n } G _ { i } ( A ).$$
How sharp is the first inclusion of (a7)? It was shown in 1999 by R.S. Varga and A. Krautstengl [a4] that
$$\tag{a8} \sigma ( \Omega ( A ) ) = \left\{ \begin{array} { c c } { \text { boundary of } K _ { 1,2 } ( A ) } & { n = 2; } \\ { \cup _ { i ,\, j = 1 , i \neq j } ^ { n } K _ { i ,\, j } ( A ) } & { n \geq 3. } \end{array} \right.$$
Thus, for $n \geq 3$, it can be said that the Cassini ovals give "perfect" results.
Gershgorin's discs and Brauer's Cassini ovals are mentioned in [a5], [a6]. A more detailed treatment of these topics can be found in [a7].
#### References
[a1] S. Gerschgorin, "Ueber die Abgrenzung der Eigenwerte einer Matrix" Izv. Akad. Nauk. SSSR Ser. Mat. , 1 (1931) pp. 749–754 [a2] O. Taussky, "Bounds for the characteristic roots of matrices" Duke Math. J. , 15 (1948) pp. 1043–1044 [a3] A. Brauer, "Limits for the characteristic roots of a matrix" Duke Math. J. , 13 (1946) pp. 387–395 [a4] R.S. Varga, A. Krautstengl, "On Geršgorin-type problems and ovals of Cassini" Electron. Trans. Numer. Anal. , 8 (1999) pp. 15–20 [a5] R.S. Varga, "Matrix iterative analysis" , Springer (2000) (Edition: Second) [a6] R.A. Horn, C.R. Johnson, "Matrix analysis" , Cambridge Univ. Press (1985) [a7] R.S. Varga, "Geršgorin and his circles" , Springer (to appear)
How to Cite This Entry:
Gershgorin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gershgorin_theorem&oldid=50639
This article was adapted from an original article by Richard S. Varga (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
|
{}
|
# A model for the geomagnetic field reversal rate and constraints on the heat flux variations at the core-mantle boundary
#### ByAUTHOR
Aug 3, 2020
We analyse data26 obtained from the latest geomagnetic polarity time scale (2012), integrated with data from the late Devonian37. The time sequence of polarity reversals was reconstructed for the whole Phanerozoic eon26. The time sequence of reversals and the time evolution of the rate of reversals γ(t), obtained by averaging reversals over sliding windows of width 8 Myr, are reported in Fig. 1. We analysed the geomagnetic series γ(t) in terms of empirical modes, through the Empirical Mode Decomposition (EMD, see “Methods”), a technique which is particularly suitable to process non-stationary time series38 like the reversal rate series. Through this technique, a time series is decomposed into empirical modes, called Intrinsic Mode Functions (IMFs), characterized by different frequencies and therefore it is possible to extract relevant timescales involved in the non-stationary process under investigation.
For the reversal rate time series γ(t) we obtained a sequence of 11 IMFs Cj(t) (j = 0,1, …, 10), which are shown in Fig. 2 along with the residue (see “Methods”). For each mode we can extract an instantaneous frequency ωj(t), whose time variations describe the non-stationary processes underlying the observed variability of the reversal rate, and the time average of ωj(t) allows us to define a typical period Tj = 2π/〈ωj〉 (where the time average is denoted by angular brackets) associated to each j-th mode.
The probability density functions of the instantaneous periods 2π/ωj(t) (see Fig. S1 of Supplementary information) indicate a compatibility with previously reported cycles. Periods shorter than about 40 Myr21,22,24,26,39 are found in the modes with j ≤ 5. Such variability has been related to CMB heat flux changes and plume dynamics within the Earth’s mantle. Modes characterized by longer periodicities, associated by our analysis to EMD modes with lower frequencies, are perhaps hidden in the low-frequency Fourier peaks commonly used to recover reversal rate periodicities. Previously reported periodicities also include time scales longer than 100 Myr, which can arise from subducting lithospheric slabs reaching the CMB27,30. These time scales are compatible with the instantaneous periods found in the modes j = 9,10. Moreover, note that the residue of the EMD (see Fig. 2) is a decreasing function of age, thus indicating that the global rate of reversals is not constant, and the global average persistence of the geomagnetic field in a single polarity state decreased, on average, going from 400 Ma to the present.
EMD results suggest that the geomagnetic system, for timescales longer than the magnetic diffusion time, can be modelled through transitions between chrons induced by a continuous underlying stochastic process, different chrons being characterized by the average frequency of switches Tj−1 between different polarity states of the magnetic field. To identify the number of states present in the system, we describe the reversal rate variations in terms of a stochastic dynamical system and we assume that a transition among states of different reversal rates is triggered by a stochastic forcing, namely the continuous change of heat flux at CMB. We use the Langevin equation dx = -U′(x)dt + σ dw to describe the dynamics of the rate changes, where x is a state variable, which in our case represents an IMF or a sum of IMFs, U(x) is a given potential, U′(x) = dU/dx, σ is the noise level and dw is a Wiener process, i.e., a stochastic process with independent Gaussian increments, which describes the stochastic CMB heat flux. The potential function U(x) can be evaluated by means of the Fokker–Planck (FP) equation associated with the Langevin model, describing the time evolution of the probability density function (pdf) p(x,t) (see “Methods”). Moreover, if the potential function U(x) can be written as a polynomial function of even order k and positive leading coefficient, i.e., U(x) = uo + u1x + u2x2 + + uk xk and uk > 0, its order is related to the number of available states for the reversal rates n, i.e., n = k/2 40,41. In this way, from the pdfs of the IMFs Cj we can calculate the potentials for all the EMD modes (see Fig. S3 of Supplementary information) by assuming that the stochastic term, i.e., the Wiener process, is representative of processes occurring at timescales which are shorter than the mean timescale Tj of each empirical mode Cj, and that the amplitude of the noise corresponds to the standard deviation of each Cj40,41. Two types of potential shapes are present in the dataset, thus reflecting the number of possible states for the reversal rate at different time scales Tj. Namely, single-well potentials related to high-frequency CMB changes, for the set of modes H = {0 ≤ j ≤ 4}, and double-well potentials, related to low-frequency CMB changes, for the set of modes L = {5 ≤ j ≤ 10}. Some of the EMD modes corresponding to single-well potentials, namely the modes j = 2,3,4, present periodicities (between 14 and 30 Myr, see Table T1 of Supplementary Information) that are close to cycles already identified using different techniques21,22,24,26,39, as already mentioned above. The period of the EMD mode j = 10 with a double-well potential is close to what has already been observed as a superchron period25. In addition, the EMD analysis suggests the presence of characteristic intermediate time scales, in the range 50–100 Myr (see Table T1 of Supplementary Information). These periods, corresponding to asymmetric double-well potentials, are perhaps hidden in the large width of low-frequency Fourier modes reported in previous analyses25,26.
The approach based on stochastic Langevin models has been proven successful in reproducing the dipole field variability observed both in paleomagnetic data covering the last 2 Myr42,43 and in numerical geodynamo simulations43. It has been shown that such models provide a good description of the axial dipole field dynamics and a reliable prediction, through the stationary solution of the FP equation, of its probability distribution. Here, we follow a similar approach and we assess the significance of our Langevin model by comparing, firstly, the partial reconstruction of the geomagnetic reversal rate signal obtained by summing the IMFs of the set of modes L = {5 ≤ j ≤ 10} to a realisation obtained from the stochastic Langevin model (see Fig. S5 of Supplementary Information), and, then, the stationary solution of the FP equation to the pdfs of the partial reconstruction and of the Langevin model (see Fig. S6 of Supplementary Information). A quite good agreement is found between the pdfs, thus confirming the validity of our approach.
The dynamics of the obtained EMD modes allows us to interpret the transition from high-frequency chrons towards low-frequency superchrons as a kind of phase transition44, that we assume to be driven by stochastic fluctuations of the heat flux at the CMB. High-frequency chrons correspond to disordered states characterized by periods of rapid polarity reversals and stronger CMB activity. On the contrary, low-frequency rates correspond to more organized states, characterized by stable long residence times in a single magnetic polarity, with smaller CMB heat flux variations and weaker mantle plume activity. In the framework of mean-field approximation44, let us consider a continuous order parameter from the set of standardised EMD modes (Gamma = left{ {C_{j}^{sigma } left( t right)} right}) (see Supplementary Information) and let us write up the potentials in terms of the manifold
$$Uleft( Gamma right) = rGamma^{2} + uGamma^{4} – hGamma$$
(1)
shown in Fig. 3. The transition from the single-well potential to the asymmetric double-well one, happens when the parameter r changes sign. Then we define r in terms of CMB heat flow as r = (Q—Qc)/Qc, where Qc represents a critical threshold. In other words, according to the mean field approach of phase-transitions44, we assume that the transition to superchrons happens when the CMB heat flow Q becomes smaller than the critical value Qc. The correlation time τ, estimated from the two-times correlation coefficient G(t1, t2) of the observed reversal rates for each mode
$$Gleft( {t_{1} ,t_{2} } right) = Gamma left( {t_{1} } right)Gamma left( {t_{2} } right) approx exp left[ {frac{{ – left| {t_{1} – t_{2} } right|}}{tau }} right],$$
(2)
(where the angular brackets denote time averaging) shows a power law dependence τ ≈ 1/rα and thus diverges when QQc (see Fig. 4), where α = 0.48 ± 0.03, in close agreement with the scaling exponent α = 1/2 required by the mean-field approximation of second-order phase transitions44. This confirms that a kind of second-order phase transition is at work within the complex geodynamo system.
According to the mean field approximation of second-order phase transitions, the susceptibility χ is given by χ−1 = dh/. The equilibrium solution in the mean field approach is obtained by minimising the potential U (Eq. (1)) with respect to Γ. This gives h = 2 + 43, from which χ−1 = 2r + 122. In the mean field theory, for the single-well minimum, which corresponds to the disordered phase when the geomagnetic field reverses at a high rate, we have Γ2 = 0 when r > 0, therefore χ−1 = 2r. This result is very interesting in our case, as it allows us to infer the number of reversals induced, starting from a reference value n0, by amplitude variations of the heat flux at CMB. In fact, since the heat flux variations due to a variation of the order parameter must be roughly proportional to the variations of the h field of the model, namely ΔQ ≈ Δh, we get ΔQ ≈ χ−1ΔΓ ≈ 2r ΔΓ. By estimating ΔΓ as ΔΓ ≈ 1/(Tj—Tj-1), using the characteristic periods Tj of EMD modes j = 6–10, we can use the predictive property of the susceptibility to directly infer about the heat flow fluctuations at CMB, relative to a reference value Q0, required to increase the reversal rate from the reference value n0, as shown in Fig. 5. An empirical relation can be obtained through a fit on the data with the following exponential function
$$frac{Delta Q}{{Q_{0} }} = Aexp left[ {frac{Delta n}{{n_{0} nu }}} right]$$
(3)
where Δn is the variation in the reversal rate and the best fit parameters result ν = 2.08 ± 0.03 and A = 0.07 ± 0.01. This means that fluctuations of the order of about ΔQ ≈ 0.18 Q0 of CMB heat flow should be enough to double the reversal rate in the geodynamo system.
|
{}
|
1. Nice Probability Inequalities
Hi all, I am new to this forum, but think It's a great idea! I have posted two questions, neither of which were unfortunately answered. But, it's ok, cos it made me figure out one of the questions on my own and plus people aren't obliged to answer your questions : ) here are three elementary inequalities that I'm trying to prove. For the 1st case I think we need induction, tips would be appreciated. The second one, I think I need an elementary identity to prove it, which I can't think of so I'm feeling sloooow and the third I haven't even attempted. Thanks for your help.
Prove:
a)
$\mathbb {P} [\bigcap _{i=1} ^ {n} A_i] \geq \mathbb {P}[A_1] + ... + \mathbb {P}[A_n] - (n-1)$
for any events $A_1, .... , A_n$
b)
${\frac {-1}_{4}} \leq \mathbb {P}[A\cap B]-\mathbb {P}[A]\mathbb {P}[B] \leq {\frac {1}_{4}}$
for any events A & B.
c)
$|\mathbb{P}[A\cap B]-\mathbb{P}[A\cap C]| \leq \mathbb{P}[(B-C)\cup (C-B)]$
for any events A, B and C.
2. Ok, so the first part just uses a straightforward induction like I thought; we just use P (A cup B) = P (A) + P (B) - P (A cap B) and it pops out. Still wondering about parts b) & c)....any thoughts yet?
3. Part C) done!
Part C is also not difficult. It's easy to prove that:
$|P(X) - P(Y)| \leq P(X \Delta Y)$ for any sets X & Y.
It's also clear that:
$(A \cap B) \Delta (A \cap C) \subset (B \Delta C)$ and the inequality follows.
Only part b) left.....come on guys.
|
{}
|
# Reference Guide: The Mimi API
Mimi.@defcompMacro
defcomp(comp_name::Symbol, ex::Expr)
Define a Mimi component comp_name with the expressions in ex. The following types of expressions are supported:
1. dimension_name = Index() # defines a dimension
2. parameter = Parameter(index = [dimension_name], units = "unit_name", default = default_value) # defines a parameter with optional arguments
3. variable = Variable(index = [dimension_name], units = "unit_name") # defines a variable with optional arguments
4. init(p, v, d) # defines an init function for the component
5. run_timestep(p, v, d, t) # defines a run_timestep function for the component
Parses a @defcomp definition, converting it into a series of function calls that create the corresponding ComponentDef instance. At model build time, the ModelDef (including its ComponentDefs) will be converted to a runnable model.
source
Mimi.@defsimMacro
defsim(expr::Expr)
Define a Mimi SimulationDef with the expressions in expr.
source
Mimi.@defcompositeMacro
defcomposite(cc_name, ex)
Define a Mimi CompositeComponentDef cc_name with the expressions in ex. Expressions are all shorthand for longer-winded API calls, and include the following:
p = Parameter(...)
v = Variable(varname)
local_name = Component(name)
Component(name) # equivalent to name = Component(name)
connect(...)
Variable names are expressed as the component id (which may be prefixed by a module, e.g., Mimi.adder) followed by a . and the variable name in that component. So the form is either modname.compname.varname or compname.varname, which must be known in the current module.
Unlike leaf components, composite components do not have user-defined init or run_timestep functions; these are defined internally to iterate over constituent components and call the associated method on each.
source
Mimi.MarginalModelType
MarginalModel
A Mimi Model whose results are obtained by subtracting results of one base Model from those of another marginal Model that has a difference of delta.
source
Mimi.ModelType
Model
A user-facing API containing a ModelInstance (mi) and a ModelDef (md). This Model can be created with the optional keyword argument number_type indicating the default type of number used for the ModelDef. If not specified the Model assumes a number_type of Float64.
source
Mimi.add_comp!Function
add_comp!(
obj::AbstractCompositeComponentDef,
comp_def::AbstractComponentDef,
comp_name::Symbol=comp_def.comp_id.comp_name;
first::NothingInt=nothing,
last::NothingInt=nothing,
before::NothingSymbol=nothing,
after::NothingSymbol=nothing,
rename::NothingPairList=nothing
)
Add the component comp_def to the composite component indicated by obj. The component is added at the end of the list unless one of the keywords before or after is specified. Note that a copy of comp_id is made in the composite and assigned the give name. The optional argument rename can be a list of pairs indicating original_name => imported_name. The optional arguments first and last indicate the times bounding the run period for the given component, which must be within the bounds of the model and if explicitly set are fixed. These default to flexibly changing with the model's :time dimension.
source
add_comp!(
obj::AbstractCompositeComponentDef,
comp_id::ComponentId,
comp_name::Symbol=comp_id.comp_name;
first::NothingInt=nothing,
last::NothingInt=nothing,
before::NothingSymbol=nothing,
after::NothingSymbol=nothing,
rename::NothingPairList=nothing
)
Add the component indicated by comp_id to the composite component indicated by obj. The component is added at the end of the list unless one of the keywords before or after is specified. Note that a copy of comp_id is made in the composite and assigned the give name. The optional arguments first and last indicate the times bounding the run period for the given component, which must be within the bounds of the model and if explicitly set are fixed. These default to flexibly changing with the model's :time dimension.
[Not yet implemented:] The optional argument rename can be a list of pairs indicating original_name => imported_name.
source
add_comp!(obj::AbstractCompositeComponentInstance, ci::AbstractComponentInstance)
Add the (leaf or composite) component ci to a composite's list of components.
source
add_comp!(
m::Model, comp_id::ComponentId, comp_name::Symbol=comp_id.comp_name;
first::NothingInt=nothing,
last::NothingInt=nothing,
before::NothingSymbol=nothing,
after::NothingSymbol=nothing,
rename::NothingPairList=nothing
)
Add the component indicated by comp_id to the model indicated by m. The component is added at the end of the list unless one of the keywords before or after is specified. Note that a copy of comp_id is made in the composite and assigned the give name. The optional argument rename can be a list of pairs indicating original_name => imported_name. The optional arguments first and last indicate the times bounding the run period for the given component, which must be within the bounds of the model and if explicitly set are fixed. These default to flexibly changing with the model's :time dimension.
source
add_comp!(
m::Model, comp_def::AbstractComponentDef, comp_name::Symbol=comp_id.comp_name;
first::NothingInt=nothing,
last::NothingInt=nothing,
before::NothingSymbol=nothing,
after::NothingSymbol=nothing,
rename::NothingPairList=nothing
)
Add the component comp_def to the model indicated by m. The component is added at the end of the list unless one of the keywords, first, last, before, after. Note that a copy of comp_id is made in the composite and assigned the give name. The optional argument rename can be a list of pairs indicating original_name => imported_name. The optional arguments first and last indicate the times bounding the run period for the given component, which must be within the bounds of the model and if explicitly set are fixed. These default to flexibly changing with the model's :time dimension.
source
Mimi.add_shared_param!Function
add_shared_param!(md::ModelDef, name::Symbol, value::Any; dims::Array{Symbol}=Symbol[])
User-facing API function to add a shared parameter to Model Def md with name name and value value, and an array of dimension names dims which dfaults to an empty vector. The is_shared attribute of the added Model Parameter will be true.
The value can by a scalar, an array, or a NamedAray. Optional keyword argument 'dims' is a list of the dimension names of the provided data, and will be used to check that they match the model's index labels. Optional keyword argument datatype allows user to specify a datatype to use for the shared model parameter.
source
add_shared_param!(m::Model, name::Symbol, value::Any; dims::Array{Symbol}=Symbol[], datatype::DataType=Nothing)
User-facing API function to add a shared parameter to Model m with name name and value value, and an array of dimension names dims which dfaults to an empty vector. The is_shared attribute of the added Model Parameter will be true.
The value can by a scalar, an array, or a NamedAray. Optional keyword argument 'dims' is a list of the dimension names of the provided data, and will be used to check that they match the model's index labels. This must be included if the value is not a scalar, and defaults to an empty vector. Optional keyword argument datatype allows user to specify a datatype to use for the shared model parameter.
source
Mimi.connect_param!Function
connect_param!(obj::AbstractCompositeComponentDef, comp_name::Symbol, param_name::Symbol, model_param_name::Symbol;
check_attributes::Bool=true, ignoreunits::Bool=false))
Connect a parameter param_name in the component comp_name of composite obj to the model parameter model_param_name.
source
connect_param!(obj::AbstractCompositeComponentDef, comp_def::AbstractComponentDef,
param_name::Symbol, model_param_name::Symbol; check_attributes::Bool=true,
ignoreunits::Bool = false)
Connect a parameter param_name in the component comp_def of composite obj to the model parameter model_param_name.
source
connect_param!(obj::AbstractCompositeComponentDef,
dst::Pair{Symbol, Symbol}, src::Pair{Symbol, Symbol},
backup::Union{Nothing, Array}=nothing;
ignoreunits::Bool=false, backup_offset::Union{Nothing, Int} = nothing)
Bind the parameter dst[2] of one component dst[1] of composite obj to a variable src[2] in another component src[1] of the same composite using backup to provide default values and the ignoreunits flag to indicate the need to check match units between the two. The backup_offset argument, which is only valid when backup data has been set, indicates that the backup data should be used for a specified number of timesteps after the source component begins. ie. the value would be 1 if the destination componentm parameter should only use the source component data for the second timestep and beyond.
source
connect_param!(dst::ComponentReference, dst_name::Symbol, src::ComponentReference, src_name::Symbol)
Connect two components as connect_param!(dst, dst_name, src, src_name).
source
connect_param!(dst::ComponentReference, src::ComponentReference, name::Symbol)
Connect two components with the same name as connect_param!(dst, src, name).
source
connect_param!(m::Model, dst_comp_name::Symbol, dst_par_name::Symbol,
src_comp_name::Symbol, src_var_name::Symbol,
backup::Union{Nothing, Array}=nothing; ignoreunits::Bool=false,
backup_offset::Union{Int, Nothing}=nothing)
Bind the parameter dst_par_name of one component dst_comp_name of model m to a variable src_var_name in another component src_comp_name of the same model using backup to provide default values and the ignoreunits flag to indicate the need to check match units between the two. The backup_offset argument, which is only valid when backup data has been set, indicates that the backup data should be used for a specified number of timesteps after the source component begins. ie. the value would be 1 if the destination componentm parameter should only use the source component data for the second timestep and beyond.
source
connect_param!(m::Model, comp_name::Symbol, param_name::Symbol, model_param_name::Symbol;
check_attributes::Bool=true, ignoreunits::Bool=false))
Connect a parameter param_name in the component comp_name of composite obj to the model parameter model_param_name.
source
connect_param!(m::Model, dst::Pair{Symbol, Symbol}, src::Pair{Symbol, Symbol}, backup::Array; ignoreunits::Bool=false)
Bind the parameter dst[2] of one component dst[1] of model m to a variable src[2] in another component src[1] of the same model using backup to provide default values and the ignoreunits flag to indicate the need to check match units between the two. The backup_offset argument, which is only valid when backup data has been set, indicates that the backup data should be used for a specified number of timesteps after the source component begins. ie. the value would be 1 if the destination componentm parameter should only use the source component data for the second timestep and beyond.
source
Mimi.create_marginal_modelFunction
create_marginal_model(base::Model, delta::Float64=1.0)
Create a MarginalModel where base is the baseline model and delta is the difference used to create the marginal model. Return the resulting MarginaModel which shares the internal ModelDef between the base and marginal.
source
Mimi.delete_param!Function
delete_param!(md::ModelDef, model_param_name::Symbol)
Delete model_param_name from md's list of model parameters, and also remove all external parameters connections that were connected to model_param_name.
source
delete_param!(m::Model, model_param_name::Symbol)
Delete model_param_name from a model m's ModelDef's list of model parameters, and also remove all external parameters connections that were connected to model_param_name.
source
Mimi.dim_countFunction
dim_count(def::AbstractDatumDef)
Return number of dimensions in def.
source
dim_count(mi::ModelInstance, dim_name::Symbol)
Return the size of index dim_name in model instance mi.
source
dim_count(m::Model, dim_name::Symbol)
Return the size of index dim_name in model m.
source
Mimi.dim_keysFunction
dim_keys(m::Model, dim_name::Symbol)
Return keys for dimension dim-name in model m.
source
dim_keys(mi::ModelInstance, dim_name::Symbol)
Return keys for dimension dim-name in model instance mi.
source
Mimi.disconnect_param!Function
disconnect_param!(obj::AbstractCompositeComponentDef, comp_def::AbstractComponentDef, param_name::Symbol)
Remove any parameter connections for a given parameter param_name in a given component comp_def which must be a direct subcomponent of composite obj.
source
disconnect_param!(obj::AbstractCompositeComponentDef, comp_name::Symbol, param_name::Symbol)
Remove any parameter connections for a given parameter param_name in a given component comp_def which must be a direct subcomponent of composite obj.
source
disconnect_param!(m::Model, comp_name::Symbol, param_name::Symbol)
Remove any parameter connections for a given parameter param_name in a given component comp_def in model m.
source
Mimi.exploreFunction
explore(m::Model)
Produce a UI to explore the parameters and variables of Model m in an independent window.
source
explore(mi::ModelInstance)
Produce a UI to explore the parameters and variables of ModelInstance mi in an independent window.
source
explore(sim_inst::SimulationInstance; title="Electron", model_index::Int = 1, scen_name::Union{Nothing, String} = nothing, results_output_dir::Union{Nothing, String} = nothing)
Produce a UI to explore the output distributions of the saved variables in SimulationInstance sim for results of model model_index and scenario with the name scen_name in a Window with title title. The optional arguments default to a model_index of 1, a scen_name of nothing assuming there is no secenario dimension, and a window with title Electron. The results_output_dir keyword argument refers to the main output directory as provided to run, where all subdirectories are held. If provided, results are assumed to be stored there, otherwise it is assumed that results are held in results.sim and not in an output folder.
source
Mimi.getdataframeFunction
getdataframe(m::AbstractModel, comp_name::Symbol, pairs::Pair{Symbol, Symbol}...)
Return a DataFrame with values for the given variables or parameters of model m indicated by pairs, where each pair is of the form comp_name => item_name. If more than one pair is provided, all must refer to items with the same dimensions, which are used to join the respective item values.
source
getdataframe(m::AbstractModel, pair::Pair{Symbol, NTuple{N, Symbol}})
Return a DataFrame with values for the given variables or parameters indicated by pairs, where each pair is of the form comp_name => item_name. If more than one pair is provided, all must refer to items with the same dimensions, which are used to join the respective item values.
source
getdataframe(m::AbstractModel, comp_name::Symbol, item_name::Symbol)
Return the values for variable or parameter item_name in comp_name of model m as a DataFrame.
source
Mimi.gettimeFunction
gettime(ts::FixedTimestep)
Return the time (year) represented by Timestep ts
source
gettime(ts::VariableTimestep)
Return the time (year) represented by Timestep ts
source
gettime(c::Clock)
Return the time of the timestep held by the c clock.
source
Mimi.get_param_valueFunction
get_param_value(ci::AbstractComponentInstance, name::Symbol)
Return the value of parameter name in (leaf or composite) component ci.
source
Mimi.get_var_valueFunction
get_var_value(ci::AbstractComponentInstance, name::Symbol)
Return the value of variable name in component ci.
source
Mimi.hasvalueFunction
hasvalue(arr::TimestepArray, ts::FixedTimestep)
Return true or false, true if the TimestepArray arr contains the Timestep ts.
source
hasvalue(arr::TimestepArray, ts::VariableTimestep)
Return true or false, true if the TimestepArray arr contains the Timestep ts.
source
hasvalue(arr::TimestepArray, ts::FixedTimestep, idxs::Int...)
Return true or false, true if the TimestepArray arr contains the Timestep ts within indices idxs. Used when Array and Timestep have different FIRST, validating all dimensions.
source
hasvalue(arr::TimestepArray, ts::VariableTimestep, idxs::Int...)
Return true or false, true if the TimestepArray arr contains the Timestep ts within indices idxs. Used when Array and Timestep have different TIMES, validating all dimensions.
source
Mimi.is_firstFunction
is_first(ts::AbstractTimestep)
Return true or false, true if ts is the first timestep to be run.
source
Mimi.is_lastFunction
is_last(ts::FixedTimestep)
Return true or false, true if ts is the last timestep to be run.
source
is_last(ts::VariableTimestep)
Return true or false, true if ts is the last timestep to be run. Note that you may run next_timestep on ts, as ths final timestep has not been run through yet.
source
Mimi.is_timeFunction
is_time(ts::AbstractTimestep, t::Int)
Deprecated function to return true or false, true if the current time (year) for ts is t
source
Mimi.is_timestepFunction
is_timestep(ts::AbstractTimestep, t::Int)
Deprecated function to return true or false, true if ts timestep is step t.
source
Mimi.modeldefFunction
modeldef(mi)
Return the ModelDef contained by ModelInstance mi.
source
modeldef(m)
Return the ModelDef contained by Model m.
source
Base.nameofFunction
nameof(obj::NamedDef) = obj.name
Return the name of def. NamedDefs include DatumDef, ComponentDef, and CompositeComponentDef
source
Mimi.parameter_dimensionsFunction
parameter_dimensions(obj::AbstractComponentDef, param_name::Symbol)
Return the names of the dimensions of parameter param_name exposed in the component definition indicated by obj.
source
parameter_dimensions(obj::AbstractComponentDef, comp_name::Symbol, param_name::Symbol)
Return the names of the dimensions of parameter param_name in component comp_name, which is exposed in composite component definition indicated byobj.
source
Mimi.parameter_namesFunction
parameter_names(md::ModelDef, comp_name::Symbol)
Return a list of all parameter names for a given component comp_name in a model def md.
source
Base.replace!Function
replace!(
m::Model,
old_new::Pair{Symbol, ComponentDef},
before::NothingSymbol=nothing,
after::NothingSymbol=nothing,
reconnect::Bool=true
)
For the pair comp_name => comp_def in old_new, replace the component with name comp_name in the model m with the new component specified by comp_def. The new component is added in the same position as the old component, unless one of the keywords before or after is specified for a different position. The optional boolean argument reconnect with default value true indicates whether the existing parameter connections should be maintained in the new component. Returns a ComponentReference for the added component.
source
Mimi.replace_comp!Function
replace_comp!(
m::Model, comp_def::ComponentDef, comp_name::Symbol=comp_id.comp_name;
before::NothingSymbol=nothing,
after::NothingSymbol=nothing,
reconnect::Bool=true
)
Deprecated function for replacing the component with name comp_name in model m with the new component specified by comp_def. Use the following syntax instead:
replace!(m, comp_name => comp_def; kwargs...)
See docstring for replace! for further description of available functionality.
source
replace_comp!(
m::Model, comp_id::ComponentId, comp_name::Symbol=comp_id.comp_name;
before::NothingSymbol=nothing,
after::NothingSymbol=nothing,
reconnect::Bool=true
)
Deprecated function for replacing the component with name comp_name in model m with the new component specified by comp_id. Use the following syntax instead:
replace!(m, comp_name => Mimi.compdef(comp_id); kwargs...)
See docstring for replace! for further description of available functionality.
source
Base.runFunction
Base.run(mm::MarginalModel; ntimesteps::Int=typemax(Int))
Run the marginal model mm once with ntimesteps.
source
Base.run(mi::ModelInstance, ntimesteps::Int=typemax(Int),
dimkeys::Union{Nothing, Dict{Symbol, Vector{T} where T <: DimensionKeyTypes}}=nothing)
Run the ModelInstance mi once with ntimesteps and dimension keys dimkeys.
source
Base.run(m::Model; ntimesteps::Int=typemax(Int), rebuild::Bool=false,
dim_keys::Union{Nothing, Dict{Symbol, Vector{T} where T <: DimensionKeyTypes}}=nothing)
Run model m once.
source
Base.run(sim_def::SimulationDef{T},
models::Union{Vector{M}, AbstractModel},
samplesize::Int;
ntimesteps::Int=typemax(Int),
trials_output_filename::Union{Nothing, AbstractString}=nothing,
results_output_dir::Union{Nothing, AbstractString}=nothing,
pre_trial_func::Union{Nothing, Function}=nothing,
post_trial_func::Union{Nothing, Function}=nothing,
scenario_func::Union{Nothing, Function}=nothing,
scenario_placement::ScenarioLoopPlacement=OUTER,
scenario_args=nothing,
results_in_memory::Bool=true) where {T <: AbstractSimulationData, M <: AbstractModel}
Run the simulation definition sim_def for the models using samplesize samples.
Optionally run the models for ntimesteps, if specified, else to the maximum defined time period. Note that trial data are applied to all the associated models even when running only a portion of them.
If provided, the generated trials and results will be saved in the indicated trials_output_filename and results_output_dir respectively. If results_in_memory is set to false, then results will be cleared from memory and only stored in the results_output_dir.
If pre_trial_func or post_trial_func are defined, the designated functions are called just before or after (respectively) running a trial. The functions must have the signature:
fn(sim_inst::SimulationInstance, trialnum::Int, ntimesteps::Int, tup::Tuple)
where tup is a tuple of scenario arguments representing one element in the cross-product of all scenario value vectors. In situations in which you want the simulation loop to run only some of the models, the remainder of the runs can be handled using a pre_trial_func or post_trial_func.
If provided, scenario_args must be a Vector{Pair}, where each Pair is a symbol and a Vector of arbitrary values that will be meaningful to scenario_func, which must have the signature:
scenario_func(sim_inst::SimulationInstance, tup::Tuple)
By default, the scenario loop encloses the simulation loop, but the scenario loop can be placed inside the simulation loop by specifying scenario_placement=INNER. When INNER is specified, the scenario_func is called after any pre_trial_func but before the model is run.
Returns the type SimulationInstance that contains a copy of the original SimulationDef, along with mutated information about trials, in addition to the model list and results information.
source
Mimi.set_dimension!Function
set_dimension!(ccd::CompositeComponentDef, name::Symbol, keys::Union{Int, Vector, Tuple, AbstractRange})
Set the values of ccd dimension name to integers 1 through count, if keys is an integer; or to the values in the vector or range if keys is either of those types.
source
set_dimension!(obj::AbstractComponentDef, name::Symbol, dim::Dimension)
Set the dimension name in obj to dim.
source
set_dimension!(m::Model, name::Symbol, keys::Union{Vector, Tuple, AbstractRange})
Set the values of m dimension name to integers 1 through count, if keysis an integer; or to the values in the vector or range ifkeys is either of those types.
source
Mimi.set_leftover_params!Function
set_leftover_params!(md::ModelDef, parameters::Dict)
Set all of the parameters in ModelDef md that don't have a value and are not connected to some other component to a value from a dictionary parameters. This method assumes the dictionary keys are Symbols (or convertible into Symbols ie. Strings) that match the names of unset parameters in the model. All resulting connected model parameters will be shared model parameters.
Note that this function set_leftover_params! has been deprecated, and uses should be transitioned to usingupdateleftoverparams! with keys specific to component-parameter pairs i.e. (compname, paramname) => value in the dictionary.
source
set_leftover_params!(m::Model, parameters::Dict)
Set all of the parameters in Model m that don't have a value and are not connected to some other component to a value from a dictionary parameters. This method assumes the dictionary keys are strings (or convertible into Strings ie. Symbols) that match the names of unset parameters in the model, and all resulting new model parameters will be shared parameters.
Note that this function set_leftover_params! has been deprecated, and uses should be transitioned to usingupdateleftoverparams! with keys specific to component-parameter pairs i.e. (compname, paramname) => value in the dictionary.
source
Mimi.set_param!Function
set_param!(md::ModelDef, comp_name::Symbol,
value_dict::Dict{Symbol, Any}, param_names)
Call set_param!() for each name in param_names, retrieving the corresponding value from value_dict[param_name].
source
set_param!(md::ModelDef, comp_name::Symbol, param_name::Symbol, value; dims=nothing)
Set the value of parameter param_name in component comp_name of Model Def md to value. This will create a shared model parameter with name param_name and connect comp_name's parameter param_name to it.
The value can by a scalar, an array, or a NamedAray. Optional keyword argument 'dims' is a list of the dimension names of the provided data, and will be used to check that they match the model's index labels.
source
set_param!(md::ModelDef, comp_name::Symbol, param_name::Symbol, model_param_name::Symbol,
value; dims=nothing)
Set the value of parameter param_name in component comp_name of Model Def md to value. This will create a shared model parameter with name model_param_name and connect comp_name's parameter param_name to it.
The value can by a scalar, an array, or a NamedAray. Optional keyword argument 'dims' is a list of the dimension names of the provided data, and will be used to check that they match the model's index labels.
source
set_param!(md::ModelDef, comp_def::AbstractComponentDef, param_name::Symbol,
model_param_name::Symbol, value; dims=nothing)
Set the value of parameter param_name in component comp_def of Model Def md to value. This will create a shared model parameter with name model_param_name and connect comp_name's parameter param_name to it.
The value can by a scalar, an array, or a NamedAray. Optional keyword argument 'dims' is a list of the dimension names of the provided data, and will be used to check that they match the model's index labels.
source
set_param!(md::ModelDef, param_name::Symbol, value; dims=nothing)
Set the value of parameter param_name in all components of the Model Defmdthat have a parameter of the specified name tovalue. This will create a shared model parameter with nameparam_name and connect all component parameters with that name to it.
The value can by a scalar, an array, or a NamedAray. Optional keyword argument 'dims' is a list of the dimension names of the provided data, and will be used to check that they match the model's index labels.
source
set_param!(ref::ComponentReference, name::Symbol, value)
Set a component parameter as set_param!(reference, name, value). This creates a unique name :compname_paramname in the model's model parameter list, and sets the parameter only in the referenced component to that value.
source
set_param!(m::Model, comp_name::Symbol, param_name::Symbol, value; dims=nothing)
Set the parameter of a component comp_name in a model m to a given value. The value can by a scalar, an array, or a NamedAray. Optional keyword argument 'dims' is a list of the dimension names of the provided data, and will be used to check that they match the model's index labels.
source
set_param!(m::Model, comp_name::Symbol, param_name::Symbol, model_param_name::Symbol, value; dims=nothing)
Set the parameter param_name of a component comp_name in a model m to a given value, storing the value in the model's parameter list by the provided name model_param_name. The value can by a scalar, an array, or a NamedAray. Optional keyword argument 'dims' is a list of the dimension names of the provided data, and will be used to check that they match the model's index labels.
source
set_param!(m::Model, param_name::Symbol, value; dims=nothing)
Set the value of a parameter in all components of the model that have a parameter of the specified name.
source
Mimi.TimestepIndexType
TimestepIndex
A user-facing type used to index into a TimestepArray in run_timestep functions, containing an Int index that indicates the position in the array in terms of timesteps.
source
Mimi.TimestepValueType
TimestepValue
A user-facing type used to index into a TimestepArray in run_timestep functions, containing a value of the same Type as the times in the TimstepArray which is used to index into the array at that position, with an optional Int offset in terms of timesteps.
source
Mimi.update_param!Function
update_param!(obj::AbstractCompositeComponentDef, name::Symbol, value; update_timesteps = nothing)
Update the value of a model parameter in composite obj, referenced by name. The update_timesteps keyword argument is deprecated, we keep it here just to provide warnings.
source
update_param!(mi::ModelInstance, name::Symbol, value)
Update the value of a model parameter in ModelInstance mi, referenced by name. This is an UNSAFE updat as it does not dirty the model, and should be used carefully and specifically for things like our MCS work.
source
update_param!(mi::ModelInstance, comp_name::Symbol, param_name::Symbol, value)
Update the value of a model parameter in ModelInstance mi, connected to component comp_name's parameter param_name. This is an UNSAFE updat as it does not dirty the model, and should be used carefully and specifically for things like our MCS work.
source
update_param!(md::ModelDef, comp_name::Symbol, param_name::Symbol, value)
Update the value of the unshared model parameter in Model Def md connected to component comp_name's parameter param_name.
source
update_param!(ref::ComponentReference, name::Symbol, value)
Update a component parameter as update_param!(reference, name, value). This uses the unique name :compname_paramname in the model's model parameter list, and updates the parameter only in the referenced component to that value.
source
update_param!(m::Model, name::Symbol, value; update_timesteps = nothing)
Update the value of an model parameter in model m, referenced by name. The update_timesteps keyword argument is deprecated, we keep it here just to provide warnings.
source
update_param!(m::Model, comp_name::Symbol, param_name::Symbol, value)
Update the value of the unshared model parameter in Model m's Model Def connected to component comp_name's parameter param_name.
source
Mimi.update_params!Function
update_params!(obj::AbstractCompositeComponentDef, parameters::Dict; update_timesteps = nothing)
For each (k, v) in the provided parameters dictionary, update_param! is called to update the model parameter identified by k to value v.
For updating unshared parameters, each key k must be a Tuple matching the name of a component in obj and the name of an parameter in that component.
For updating shared parameters, each key k must be a symbol or convert to a symbol matching the name of a shared model parameter that already exists in the model.
source
update_params!(m::Model, parameters::Dict; update_timesteps = nothing)
For each (k, v) in the provided parameters dictionary, update_param! is called to update the model parameter identified by k to value v.
For updating unshared parameters, each key k must be a Tuple matching the name of a component in obj and the name of an parameter in that component.
For updating shared parameters, each key k must be a symbol or convert to a symbol matching the name of a shared model parameter that already exists in the model.
source
Mimi.update_leftover_params!Function
update_leftover_params!(md::ModelDef, parameters::Dict)
Update all of the parameters in ModelDef md that don't have a value and are not connected to some other component to a value from a dictionary parameters. This method assumes the dictionary keys are Tuples of Symbols (or convertible to Symbols ie. Strings) of (compname, paramname) that match the component-parameter pair of unset parameters in the model. All resulting connected model parameters will be unshared model parameters.
source
update_leftover_params!(m::Model, parameters::Dict)
Update all of the parameters in Model m that don't have a value and are not connected to some other component to a value from a dictionary parameters. This method assumes the dictionary keys are Tuples of Symbols (or convertible to Symbols ie. Strings) of (compname, paramname) that match the component-parameter pair of unset parameters in the model. All resulting connected model parameters will be unshared model parameters.
source
Mimi.variable_dimensionsFunction
variable_dimensions(obj::AbstractCompositeComponentDef, comp_path::ComponentPath, var_name::Symbol)
Return the names of the dimensions of variable var_name exposed in the composite component definition indicated byobj along the component path comp_path. The comp_path is of type Mimi.ComponentPath with the single field being an NTuple of symbols describing the relative (to a composite) or absolute (relative to ModelDef) path through composite nodes to specific composite or leaf node.
source
variable_dimensions(obj::AbstractCompositeComponentDef, comp::Symbol, var_name::Symbol)
Return the names of the dimensions of variable var_name exposed in the composite component definition indicated by obj for the component comp, which exists in a flat model.
source
variable_dimensions(obj::AbstractCompositeComponentDef, comp::Symbol, var_name::Symbol)
Return the names of the dimensions of variable var_name exposed in the composite component definition indicated by obj along the component path comp_path. The comp_path is a tuple of symbols describing the relative (to a composite) or absolute (relative to ModelDef) path through composite nodes to specific composite or leaf node.
source
variable_dimensions(obj::AbstractComponentDef, name::Symbol)
Return the names of the dimensions of variable name exposed in the component definition indicated by obj.
source
Mimi.variable_namesFunction
variable_names(md::AbstractCompositeComponentDef, comp_name::Symbol)
Return a list of all variable names for a given component comp_name in a model def md.
source
variable_names(comp_def::AbstractComponentDef)
Return a list of all variable names for a given component comp_def`.
source
|
{}
|
4
# Questions
3
2
299
views
### Projective line as a quotient by a torus
jul 10 at 13:43 Will Sawin 18.6k12448
4
0
207
views
### T-Equivariant trivialization of a principal G-bundle
jun 26 at 9:35 Jon Skowera 27117
5
1
311
views
### Can Deligne-Mumford stacks be characterized by their restriction to a small subcategory?
oct 9 10 at 8:32 Jon Skowera 27117
5
|
{}
|
# tomopy.recon.rotation¶
Module for functions related to finding axis of rotation.
Functions:
find_center(tomo, theta[, ind, init, tol, …]) Find rotation axis location. find_center_vo(tomo[, ind, smin, smax, …]) Find rotation axis location using Nghia Vo’s method. find_center_pc(proj1, proj2[, tol, rotc_guess]) Find rotation axis location by finding the offset between the first projection and a mirrored projection 180 degrees apart using phase correlation in Fourier space. write_center(tomo, theta[, dpath, …]) Save images reconstructed with a range of rotation centers.
tomopy.recon.rotation.find_center(tomo, theta, ind=None, init=None, tol=0.5, mask=True, ratio=1.0, sinogram_order=False)[source]
Find rotation axis location.
The function exploits systematic artifacts in reconstructed images due to shifts in the rotation center. It uses image entropy as the error metric and ‘’Nelder-Mead’’ routine (of the scipy optimization module) as the optimizer [Donath:06].
Parameters: tomo (ndarray) – 3D tomographic data. theta (array) – Projection angles in radian. ind (int, optional) – Index of the slice to be used for reconstruction. init (float) – Initial guess for the center. tol (scalar) – Desired sub-pixel accuracy. mask (bool, optional) – If True, apply a circular mask to the reconstructed image to limit the analysis into a circular region. ratio (float, optional) – The ratio of the radius of the circular mask to the edge of the reconstructed image. sinogram_order (bool, optional) – Determins whether data is a stack of sinograms (True, y-axis first axis) or a stack of radiographs (False, theta first axis). float – Rotation axis location.
tomopy.recon.rotation.find_center_vo(tomo, ind=None, smin=-50, smax=50, srad=6, step=0.25, ratio=0.5, drop=20)[source]
Find rotation axis location using Nghia Vo’s method. [Vo:14].
Parameters: tomo (ndarray) – 3D tomographic data. ind (int, optional) – Index of the slice to be used for reconstruction. smin, smax (int, optional) – Coarse search radius. Reference to the horizontal center of the sinogram. srad (float, optional) – Fine search radius. step (float, optional) – Step of fine searching. ratio (float, optional) – The ratio between the FOV of the camera and the size of object. It’s used to generate the mask. drop (int, optional) – Drop lines around vertical center of the mask. float – Rotation axis location.
tomopy.recon.rotation.find_center_pc(proj1, proj2, tol=0.5, rotc_guess=None)[source]
Find rotation axis location by finding the offset between the first projection and a mirrored projection 180 degrees apart using phase correlation in Fourier space. The register_translation function uses cross-correlation in Fourier space, optionally employing an upsampled matrix-multiplication DFT to achieve arbitrary subpixel precision. [Guizar:08].
Parameters: proj1 (ndarray) – 2D projection data. proj2 (ndarray) – 2D projection data. tol (scalar, optional) – Subpixel accuracy rotc_guess (float, optional) – Initual guess value for the rotation center float – Rotation axis location.
tomopy.recon.rotation.write_center(tomo, theta, dpath='tmp/center', cen_range=None, ind=None, mask=False, ratio=1.0, sinogram_order=False, algorithm='gridrec', filter_name='parzen')[source]
Save images reconstructed with a range of rotation centers.
Helps finding the rotation center manually by visual inspection of images reconstructed with a set of different centers.The output images are put into a specified folder and are named by the center position corresponding to the image.
Parameters: tomo (ndarray) – 3D tomographic data. theta (array) – Projection angles in radian. dpath (str, optional) – Folder name to save output images. cen_range (list, optional) – [start, end, step] Range of center values. ind (int, optional) – Index of the slice to be used for reconstruction. mask (bool, optional) – If True, apply a circular mask to the reconstructed image to limit the analysis into a circular region. ratio (float, optional) – The ratio of the radius of the circular mask to the edge of the reconstructed image. sinogram_order (bool, optional) – Determins whether data is a stack of sinograms (True, y-axis first axis) or a stack of radiographs (False, theta first axis). algorithm ({str, function}) – One of the following string values. ‘art’ Algebraic reconstruction technique [Kak:98]. ‘bart’ Block algebraic reconstruction technique. ‘fbp’ Filtered back-projection algorithm. ‘gridrec’ Fourier grid reconstruction algorithm [Dowd:99], [Rivers:06]. ‘mlem’ Maximum-likelihood expectation maximization algorithm [Dempster:77]. ‘osem’ Ordered-subset expectation maximization algorithm [Hudson:94]. ‘ospml_hybrid’ Ordered-subset penalized maximum likelihood algorithm with weighted linear and quadratic penalties. ‘ospml_quad’ Ordered-subset penalized maximum likelihood algorithm with quadratic penalties. ‘pml_hybrid’ Penalized maximum likelihood algorithm with weighted linear and quadratic penalties [Chang:04]. ‘pml_quad’ Penalized maximum likelihood algorithm with quadratic penalty. ‘sirt’ Simultaneous algebraic reconstruction technique. ‘tv’ Total Variation reconstruction technique [Chambolle:11]. ‘grad’ Gradient descent method with a constant step size ‘tikh’ Tikhonov regularization with identity Tikhonov matrix.
filter_name : str, optional
Name of the filter for analytic reconstruction.
‘none’
No filter.
‘shepp’
Shepp-Logan filter (default).
‘cosine’
Cosine filter.
‘hann’
Cosine filter.
‘hamming’
Hamming filter.
‘ramlak’
Ram-Lak filter.
‘parzen’
Parzen filter.
‘butterworth’
Butterworth filter.
‘custom’
A numpy array of size next_power_of_2(num_detector_columns)/2 specifying a custom filter in Fourier domain. The first element of the filter should be the zero-frequency component.
‘custom2d’
A numpy array of size num_projections*next_power_of_2(num_detector_columns)/2 specifying a custom angle-dependent filter in Fourier domain. The first element of each filter should be the zero-frequency component.
tomopy.recon.rotation.mask_empty_slice(tomo, threshold=0.25)[source]
Generate a mask to indicate whether current slice contains sample
At APS 1ID, some of the projection images contains large empty area above the sample, resulting in empty layers.
Parameters: tomo (ndarray) – 3D tomographic data. threshold (float, optional) – determine whether a layer is considered to be empty nparray – a mask indicate the emptyness of each layer
|
{}
|
kidzsearch.com > wiki Explore:images videos games
# Strong interaction
(Redirected from Strong nuclear force)
The strong interaction (or strong nuclear force) is a force that acts between particles in the nucleus of an atom. It is what holds the nucleus together. It is one of the four basic forces in physics.
Under the theory of quantum chromodynamics (QCD), the strong force represents the interactions between quarks and gluons. Quantum chromodynamics is the theory that explains different colours. The strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves.
Although the strong force only acts upon elementary particles directly, the force is observed between hadrons as the nuclear force. As has been shown by many failed free quark searches, the elementary particles affected are unobservable directly. This phenomenon is called confinement, a theory which allows only hadrons to be seen.
The strong force is about 167 trillion trillion trillion times as strong as gravity and works over 1 trillionth of a millimeter. There are two types of strong force: residual (left over) and fundamental (basic).
## Fundamental/ Color Strong Force
Fundamental, or color strong force is the nuclear force that acts between the three quarks that a proton or neutron is made of. It is called the color strong force because, like the electomagnetic force, the strong force has charges. The major difference, though, is that while the electromagnetic force has two types of charges, the strong force has three. These three types of charges are named after colors, they are red, blue, and green. Like the electromagnetic force, opposite colors attract, and the same colors repel. Some particles that have color charge are quarks and antiquarks. The type of quark is not related to that quark's color charge at all. Quarks are one of the smallest particles currently known to humans; they take up no space because they are points, and the only particles that we have not been able to break apart from other particles yet. This is in fact because the nature of the strong force between particles is that it becomes stronger the further away the particles are. The force carrier of the strong force is the gluon. Gluons also have color charge. Both quarks and gluons have properties that make them unique from other particles.
### Quarks
• There are six types of quarks: up, down, charm, strange, top and bottom.
• All quarks have an electric charge charge of either 2/3 or -1/3.
• All quarks have a spin of 1/2.
residual strong force acts on protons and neutrons
### Gluons
• Gluons have a mass and electric charge of 0.
• Gluons have a spin of 1.
## Residual Strong Force
Residual strong force is the type of strong force that acts between hadrons (stuff made of two or three quarks, e.g. protons and neutrons). It is what holds the nucleus of an atom together. Alpha decay is a result of the residual strong force. This force is carried by pions, which are made of one quark and one antiquark. Protons and neutrons are some of the particles that experience it. Each of these three particles has unique properties as well.
### Pions
• Pions have a charge of -1, 0, or 1.
• Pions have a spin of 0.
### Protons
• Protons have a charge of 1.
• Protons have a spin of 1/2.
### Neutrons
• Neutrons are neutral, or have a charge of 0.
• Neutrons have a spin of 1/2.
## References
• David J. Griffiths, 1987. Introduction to Elementary Particles. John Wiley & Sons. ISBN 0-471-60386-4
• Gordon L. Kane (1987). Modern Elementary Particle Physics. Perseus Books.
.
• Richard Morris, 2003. The Last Sorcerers: The Path from Alchemy to the Periodic Table. Washington DC: Joseph Henry Press. ISBN 0-309-50593-3
|
{}
|
Search packages:
Sourcecode: cairo-dock version 1.5.5.3-repack01.6.2.31.6.2.3-0ubuntu11.6.3.11.6.3.1-0ubuntu12.0.32.0.3-0ubuntu12.0.52.0.5-0ubuntu12.0.8.12.0.8.1-0ubuntu12.0.8.22.0.8.2-12.0.92.0.9-0ubuntu12.1.3-10-lucid2.1.3-10-lucid-0ubuntu12.1.3-62.1.3-6-0ubuntu12.1.3.12.1.3.10-12.1.3.10-22.1.3.10-32.1.3.10-42.1.3.52.1.3.5-12.2.0~0beta42.2.0~0beta4-0ubuntu12.2.0~0rc12.2.0~22.2.0~2-0ubuntu12.2.0~42.2.0~4-0ubuntu12.3.0~0rc1-12.3.0~1-0ubuntu12.3.0~3-12.4.0~0beta2-0ubuntu12.4.0~2-0ubuntu12.4.0~2-1
# cairo-dock-struct.h
/* -*- Mode: C; indent-tabs-mode: t; c-basic-offset: 2; tab-width: 2 -*- */
/**
* Structure for Cairo-Dock
*
* Copyright : (C) 2009 by Fabrice Rey
* E-mail : fabounet@users.berlios.de
*
* This program is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public License
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*/
#ifndef __CAIRO_DOCK_STRUCT__
#define __CAIRO_DOCK_STRUCT__
#include <X11/Xlib.h>
#include <glib.h>
#include <gdk/gdk.h>
#include <gtk/gtk.h>
#include <cairo.h>
#include <librsvg/rsvg.h>
#include <librsvg/rsvg-cairo.h>
#include <glib/gi18n.h>
//#include <X11/extensions/Xdamage.h>
#ifdef HAVE_GLITZ
#include <gdk/gdkx.h>
#include <glitz-glx.h>
#include <cairo-glitz.h>
#endif
#include <GL/gl.h>
#include <GL/glx.h>
/*! \mainpage Cairo-Dock's API documentation.
* \ref intro_sec
*
* \ref install_sec
*
* \ref struct_sec
* - \ref containers
* - \ref icons
* - \ref dock
* - \ref desklet
* - \ref dialog
* - \ref flying
*
* \ref applets_sec
* - \ref module
* - \ref generate
* - \ref definition
* - \ref sections
* - \ref opengl
* - \ref animation
* - \ref sub-icons
*
* - \ref steal_appli
* - \ref data_renderer
* - \ref multi
* - \ref render_container
*
*
*
*
* \n
* \section intro_sec Introduction
*
* Cairo-Dock's API can be divided into 3 parts :
* - the definition of the main classes (dock, icon, etc)
* - utilities functions (interaction with X, GUI, etc)
* - plug-ins framework.
*
* Each class is defined in its own header. When a class is complex, it is divided into several files :
* - "factory" define structures/eunms and creation/modification/destruction functions
* - "manager" manage all the ressources needed by a class and all the instances of a class
* - "utilities" are a collection of helper functions.
*
* Cairo-Dock has a <b>decentralized conception</b> : it has a minimalistic core, and lets external modules extend its functionnalities.\n
* This is a strong design, because it allows to extend functionnalities easily without having to hack into the core, which makes the project more stable and allows developpers to use high-level functions only, that are very tested and optimized.\n
* Thus, Cairo-Dock itself has no animation, but has a convenient notification system that allows external plug-ins to animate icons when they want.\n
* We will describe this system and the plug-ins framework in this document.
*
* \n
* \section install_sec Installation
*
* The installation is very easy. In a terminal, copy-paste the following commands :
* \code
* ### grab the sources
* svn co http://svn.berlios.de/svnroot/repos/cairo-dock/trunk CD
* cd CD/cairo-dock
* ### compil the dock and install it
* autoreconf -isf && ./configure --prefix=/usr && make
* sudo make install
* ### compil the stable plug-ins and install them
* cd ../plug-ins
* autoreconf -isf && ./configure --prefix=/usr && make
* sudo make install
* \endcode
* You can compil and install any plug-in individually by using the same commands in its own folder.
*
*
* \n
* \section struct_sec Main structures
*
* \subsection containers Containers
* See _CairoContainer for the definition of a Container, and cairo-dock-container.h for a complete description of the Container class.
*
* \subsection icons Icons
* See _Icon for the definition of an Icon, and cairo-dock-icons.h for a complete description of the Icon class.
*
* \subsection dock Dock
* See _CairoDock for the definition of a Dock, and cairo-dock-dock-factory.h for a complete description of the Dock class.
*
* \subsection desklet Desklet
* See _CairoDesklet for the definition of a Desklet, and cairo-dock-desklet.h for a complete description of the Desklet class.
*
* \subsection dialog Dialog
* See _CairoDialog for the definition of a Dialog, and cairo-dock-dialogs.h for a complete description of the Dialog class.
*
* \subsection flying Flying Container
* See _CairoFlyingContainer for the definition of a Flying Container, and cairo-dock-flying-container.h for a complete description of the FlyingContainer class.
*
*
*
* \n
* \section applets_sec External Modules
*
* \subsection module First, what is a module ?
*
* Modules are compiled .so files (that is to say, library) that are plugged into the dock at run-time.
* Due to this fact, they can use any function used by the dock, and have a total interaction freedom on the dock.
* The advantage is that applets can do anything, in fact they are extensions of the dock itself.
* The drawback is that a buggy applet can make the dock unstable.
*
* A module has an <b>interface</b> and a <b>visit card</b> :
* - the visit card allows it to define itself (name, category, default icon, etc)
* - the interface defines the entry points for init, stop, reload, read config, and reset datas.
*
* Modules can be instanciated several times; each time they are, an <b>instance</b> is created. This instance will hold all the data used by the module's functions : the icon and its container, the config structure and its conf file, the data structure and a slot to plug datas into containers and icons. All these parameters are optionnal; a module that has an icon is also called an <b>applet</b>.
*
* When instanciating a module, CD will check the presence of an "Icon" group in the conf file. If there is one, it will create an icon accordingly and insert it into its container. If there is a "Desklet" group, the module is considered as detachable, and can be placed into a desklet.
* Here we will focus on applets, that is to say, we will have an icon and a container (dock or desklet).
*
*
* \subsection generate Let's start, how do I create an empty applet ?
*
* Easy ! just go to the "plug-ins" folder, and run the <i>generate-applet.sh</i> script. Answer the few questions, and you're done ! Don't forget to install the plug-in each time you modify it (<i>sudo make install</i> in your applet's folder).
* You can see that the script has created for you the architecture of your applet :
* - in the <b>root</b> folder, you have the "configure.ac", where you can set the version number of your applet, the dependencies, etc
* - in the <b>src</b> folder, you have the sources of your applet. It is common to put the init/stop/reload in applet-init.c, the get_config/reset_config/reset_data in applet-config.c, the notifications in applet-notifications.c, and the structures in applet-struct.h. Of course, you can add as many files as you want, just don't forget to specify them in the Makefile.am
* - in the <b>po</b> folder, you have the translation files. Currently the dock is widely translated into French, Japanese, and Italian.
* - in the <b>data</b> folder, you have the config file, the default icon, and a preview. You will have to choose a default icon that fits your applet, and make a preview that makes users want to try it ;-)
* If you have other files to install, it's here you will do it.
* If you change the name of the default icon (for instance you use an SVG file), don't forget to modify the data/Makefile.am and also the src/Makefile.am.
*
*
* \subsection definition Ok I have a generic applet, how do I define it ?
*
* As we saw, a module must fill a visit card and an interface, to be acecpted by the dock.
* This is done very easily by the CD_APPLET_DEFINITION macro. All you have to give is the name of the applet, its category, a brief description/manual (very important !), and your name.
* When you will have finished, you will be able to make a nice preview and a nice default icon, and place them in the <i>data</i> folder.
*
*
* \subsection sections Great, I can see my applet in the dock ! Now, where should I continue ?
*
* We saw that when our applet is activated, an instance is created. It is called <b>myApplet</b>, and it will hold the following :
* - <b>myIcon</b> : this is your icon ! It will act as a drawing surface to represent whatever you want.
* - <b>myDrawContext</b> : a cairo context, to draw on your icon with the libcairo.
* - <b>myContainer</b> : the container your icon belongs to (a Dock or a Desklet). For convenience, the following 2 parameters are availale.
* - <b>myDock</b> : if your container is a dock, myDock = myContainer, otherwise it is NULL.
* - <b>myDesklet</b> : if your container is a desklet, myDesklet = myContainer, otherwise it is NULL.
* - <b>myConfig</b> : the structure holding all the parameters you get in your conf file. You have to define it in applet-struct.h.
* - <b>myData</b> : the structure holding all the ressources loaded at run-time. You have to define it in applet-struct.h.
*
* The framework defines different <b>sections</b>, and all you have to do is to fill them :
*
* - First of all you will have to get your config parameters. This is done in the CD_APPLET_GET_CONFIG_BEGIN/CD_APPLET_GET_CONFIG_END section, in applet-config.c.
* - Each time you add a parameter, think of freeing it if it's a dynamic ressource like a string; this is done in the CD_APPLET_RESET_CONFIG_BEGIN/CD_APPLET_RESET_CONFIG_END section.
* - In a similar way, you will free all the ressources you allocated by myData in the CD_APPLET_RESET_DATA_BEGIN/CD_APPLET_RESET_DATA_END section.
* - After the instance is created, the dock lets you start. This is done in the CD_APPLET_INIT_BEGIN/CD_APPLET_INIT_END section. At this point, myApplet is already fully defined, and myConfig has been filled. Therefore you can already draw on your icon, launch timers, register to notifications, etc.
* - Each time the user changes something in its config, or the desklet is resized, your applet is reloaded. This is done in the CD_APPLET_RELOAD_BEGIN/CD_APPLET_RELOAD_END section. The macro CD_APPLET_MY_CONFIG_CHANGED tells you if something has changed in your config or if it's just a resizing.
* - Last, when your applet is stopped, you have to stop everything you set up in the init (timers, notifications, etc) in the CD_APPLET_STOP_BEGIN/CD_APPLET_STOP_END section.
*
*
*
* When something happens, Cairo-Dock notifies everybody about it, including itself. An applet can register to any notification (see \ref cairo-dock-notifications.h) before or after the dock, to be notified of the event of its choice. When you are notified, the function you registered for this event will be called; it must match the notification prototype as defined in \ref cairo-dock-notifications.h.
*
* For instance if you want to know when the user clicks on your icon, you will register to the \ref CAIRO_DOCK_CLICK_ICON notification.
*
* To register to a notification, you have the \ref cairo_dock_register_notification function. Always unregister when your applet is stopped, to avoid being notified when you shouldn't, with the function \ref cairo_dock_remove_notification_func.
*
* For convenience, there are sections dedicated to the most common events; you just have to fill the corresponding sections :
* - CD_APPLET_ON_CLICK_BEGIN/CD_APPLET_ON_CLICK_END for the actions on right click on your icon or one of its sub-dock.
* - CD_APPLET_ON_MIDDLE_CLICK_BEGIN/CD_APPLET_ON_MIDDLE_CLICK_END for the actions on middle click on your icon or one of its sub-dock.
* - CD_APPLET_ON_DOUBLE_CLICK_BEGIN/CD_APPLET_ON_DOUBLE_CLICK_END for the actions on double click on your icon or one of its sub-dock.
* - CD_APPLET_ON_SCROLL_BEGIN/CD_APPLET_ON_SCROLL_END for the actions on scroll on your icon or one of its sub-dock.
*
* To register to these notifications, you can use the convenient macros :
* - CD_APPLET_REGISTER_FOR_CLICK_EVENT
* - CD_APPLET_REGISTER_FOR_MIDDLE_CLICK_EVENT
* - CD_APPLET_REGISTER_FOR_DOUBLE_CLICK_EVENT
* - CD_APPLET_REGISTER_FOR_SCROLL_EVENT
*
*
* \subsection opengl How can I take advantage of the OpenGL ?
*
* There are 3 cases :
* - your applet just has a static icon; there is nothing to take into account, the common functions to set an image or a surface on an icon already handle the texture mapping.
* - you draw dynamically on your icon with libcairo (using myDrawContext), but you don't want to bother with OpenGL; all you have to do is to call cairo_dock_update_icon_texture to update your icon's texture after you drawn your surface. This can be done for occasional drawings, like Switcher redrawing its icon each time a window is moved.
* - you draw your icon differently whether the dock is in OpenGL mode or not; in this case, you just need to put all the OpenGL commands into a CD_APPLET_START_DRAWING_MY_ICON/CD_APPLET_FINISH_DRAWING_MY_ICON section inside your code.
*
*
* \subsection animation How can I animate my applet to make it more lively ?
*
* If you want to animate your icon easily, to signal some action (like Music-Player when a new song starts), you can simply <b>request for one of the registered animations</b> with \ref cairo_dock_request_icon_animation and stop it with cairo_dock_stop_icon_animation. You just specify the name of the animation (like "rotate") and the number of time it will be played.
*
* But you can also make your own animation, like Clock of Cairo-Penguin. You will have to integrate yourself into the rendering loop of your container. Don't panic, here again, Cairo-Dock helps you !
*
* First you will register to the "update container" notification, with the macros CD_APPLET_REGISTER_FOR_UPDATE_ICON_SLOW_EVENT or CD_APPLET_REGISTER_FOR_UPDATE_ICON_EVENT, depending on the refresh frequency you need : ~10Hz or ~33Hz. A high frequency needs of course more CPU, and most of the time the slow frequancy is enough.
*
* Then you will just put all your code in a CD_APPLET_ON_UPDATE_ICON_BEGIN/CD_APPLET_ON_UPDATE_ICON_END section. That's all ! In this section, do what you want, like redrawing your icon, possibly incrementing a counter to know until where you went, etc. See \ref opengl "the previous paragraph" to draw on your icon.
* Inside the rendering loop, you can skip an iteration with CD_APPLET_SKIP_UPDATE_ICON, and quit the loop with CD_APPLET_STOP_UPDATE_ICON or CD_APPLET_PAUSE_UPDATE_ICON (don't forget to quit the loop when you're done, otherwise your container may continue to redraw itself, which means a needless CPU load).
*
*
* \subsection tasks I have heavy treatments to do, how can I make them without slowing the dock ?
*
* Say for instance you want to download a file on the Net, it is likely to take some amount of time, during which the dock will be frozen, waiting for you. To avoid such a situation, Cairo-Dock defines \ref _CairoDockTask "Tasks". They are <b>asynchronous</b> and <b>periodic</b>. A Task is divided in 2 phases :
* - the asynchronous phase will be executed in another thread, while the dock continues to run on its own thread, in parallel. During this phase you will do all the heavy job (like downloading a file or computing something) but you can't interact on the dock.
* - the synchronous phase will be executed after the first one has finished. There you will update your applet with the result of the first phase.
*
* \attention A data buffer is used to communicate between the 2 phases. It is important that these datas are never accessed outside of the task, and vice versa that the asynchronous thread never accesses other data than these ones.\n
* If you want to access these datas outside of the task, you have to copy them in a safe place during the 2nd phase, or to stop the task before (beware that stopping the task means waiting for the 1st phase to finish, which can take some time).
*
* A Task can be periodic if you specify a period, and can also be fully synchronous if you don't specify an asynchronous function.
*
*
* \subsection sub-icons I need more than one icon, how can I easily get more ?
*
* In dock mode, your icon can have a sub-dock; in desklet mode, you can load a list of icons into your desklet. Cairo-Dock provides a convenient macro to <b>quickly load a list of icons</b> in both cases : CD_APPLET_LOAD_MY_ICONS_LIST to load a list of icons and CD_APPLET_DELETE_MY_ICONS_LIST to destroy it. Thus you don't need to know in which mode you are, neither to care about loading the icons, freeing them, or anything.
*
* You can get the list of icons with CD_APPLET_MY_ICONS_LIST and to their container with CD_APPLET_MY_ICONS_LIST_CONTAINER.
*
*
* \n
*
* \subsection advanced_config How can I make my own widgets in the config panel ?
*
* Cairo-Dock can build itself the config panel of your applet from the config file. Moreover, it can do the opposite : update the conf file from the config panel. However, it is limited to the widgets it knows, and there are some cases it is not enough.
* Because of that, Cairo-Dock offers 2 hooks in the process of building/reading the config panel :
when defining your applet in the CD_APPLET_DEFINE_BEGIN/CD_APPLET_DEFINE_END section, add to the interface the 2 functions pInterface->load_custom_widget and pInterface->save_custom_widget.
* They will be respectively called when the config panel of your applet is raised, and when it is validated.
*
* If you want to modify the content of an existing widget, you can grab it with \ref cairo_dock_get_widget_from_name.
* To add your custom widgets, insert in the conf file an empty widget (with the prefix '_'), then grab it and pack some GtkWidget inside.
* If you want to dynamically alter the config panel (like having a "new" button that would make appear new widgets on click), you can add in the conf file the new widgets, and then call \ref cairo_dock_reload_current_group_widget to reload the config panel.
* See the AlsaMixer or Weather applets for an easy example, and Clock or Mail for a more advanced example.
*
*
* \subsection steal_appli How can my applet control the window of an application ?
*
* Say your applet launches an external application that has its own window. It is logical to <b>make your applet control this application</b>, rather than letting the Taskbar do.
* All you need to do is to call the macro CD_APPLET_MANAGE_APPLICATION, indicating which application you wish to manage (you need to enter the class of the application, as you can get from "xprop | grep CLASS"). Your applet will then behave like a launcher that has stolen the appli icon.
*
*
* \subsection data_renderer How can I render some numerical values on my icon ?
*
* Cairo-Dock offers a powerful and versatile architecture for this case : \ref _CairoDataRenderer.
* A DataRenderer is a generic way to render a set of values on an icon; there are several implementations of this class : #Gauge, #CairoDockGraph2, Bar, and it is quite easy to implement a new kind of DataRenderer.
*
* Each kind of renderer has a set of attributes that you can use to customize it; you just need to call the CD_APPLET_ADD_DATA_RENDERER_ON_MY_ICON macro with the attributes, and you're done !
* Then, each time you want to render some new values, simply call CD_APPLET_RENDER_NEW_DATA_ON_MY_ICON with the new values.
* When your applet is reloaded, you have to reload the DataRenderer as well, using the convenient CD_APPLET_RELOAD_MY_DATA_RENDERER macro. If you don't specify attributes to it, it will simply reload the current DataRenderer, otherwise it will load the new attributes; the previous data are not lost, which is useful in the case of Graph for instance.
* You can remove it at any time with CD_APPLET_REMOVE_MY_DATA_RENDERER.
*
*
* \subsection multi How can I make my applet multi-instanciable ?
*
* Applets can be launched several times, an instance will be created each time. To ensure your applet can be instanciated several times, you just need to pass myApplet to any function that uses one of its fields (myData, myIcon, etc). Then, to indicate Cairo-Dock that your applet is multi-instanciable, you'll have to define the macro CD_APPLET_MULTI_INSTANCE in each file. A convenient way to do that is to define it in the Makefile.am, by adding the following line to the CFLAGS : \code -DCD_APPLET_MULTI_INSTANCE=\"1\"\ \endcode.
*
*
* \subsection render_container How can I draw anywhere on the dock, not only on my icon ?
*
* Say you want to draw directly on your container, like CairoPenguin or ShowMouse do. This can be achieved easily by registering to the \ref CAIRO_DOCK_RENDER_DOCK or \ref CAIRO_DOCK_RENDER_DESKLET notifications. You will then be notified eash time a Dock or a Desklet is drawn. Register AFTER so that you will draw after the view.
*
*
* to be continued ...
*/
typedef struct _CairoDockRenderer CairoDockRenderer;
typedef struct _CairoDeskletRenderer CairoDeskletRenderer;
typedef struct _CairoDeskletDecoration CairoDeskletDecoration;
typedef struct _CairoDialogRenderer CairoDialogRenderer;
typedef struct _CairoDialogDecorator CairoDialogDecorator;
typedef struct _Icon Icon;
typedef struct _CairoContainer CairoContainer;
typedef struct _CairoDock CairoDock;
typedef struct _CairoDesklet CairoDesklet;
typedef struct _CairoDialog CairoDialog;
typedef struct _CairoFlyingContainer CairoFlyingContainer;
typedef struct _CairoDockModule CairoDockModule;
typedef struct _CairoDockModuleInterface CairoDockModuleInterface;
typedef struct _CairoDockModuleInstance CairoDockModuleInstance;
typedef struct _CairoDockVisitCard CairoDockVisitCard;
typedef struct _CairoDockInternalModule CairoDockInternalModule;
typedef struct _CairoDockMinimalAppletConfig CairoDockMinimalAppletConfig;
typedef struct _CairoDockDesktopEnvBackend CairoDockDesktopEnvBackend;
typedef struct _CairoDockClassAppli CairoDockClassAppli;
typedef struct _CairoDockLabelDescription CairoDockLabelDescription;
typedef struct _CairoDialogAttribute CairoDialogAttribute;
typedef struct _CairoDeskletAttribute CairoDeskletAttribute;
typedef struct _CairoDialogButton CairoDialogButton;
typedef struct _CairoDataRenderer CairoDataRenderer;
typedef struct _CairoDataRendererAttribute CairoDataRendererAttribute;
typedef struct _CairoDataRendererInterface CairoDataRendererInterface;
typedef struct _CairoDataToRenderer CairoDataToRenderer;
typedef struct _CairoDockTransition CairoDockTransition;
typedef struct _CairoDockTheme CairoDockTheme;
#define CAIRO_DOCK_NB_DATA_SLOT 12
typedef gboolean (* CairoDockApplyConfigFunc) (gpointer data);
typedef gboolean (* CairoDockForeachDeskletFunc) (CairoDesklet *pDesklet, CairoDockModuleInstance *pInstance, gpointer data);
typedef void (* CairoDockForeachIconFunc) (Icon *icon, CairoContainer *pContainer, gpointer data);
/// Nom du repertoire de travail de cairo-dock.
#define CAIRO_DOCK_DATA_DIR "cairo-dock"
/// Nom du repertoire des extras utilisateur/themes (jauges, clock, etc).
#define CAIRO_DOCK_EXTRAS_DIR "extras"
/// Nom du repertoire des jauges utilisateur/themes.
#define CAIRO_DOCK_GAUGES_DIR "gauges"
/// Nom du repertoire du theme courant.
#define CAIRO_DOCK_CURRENT_THEME_NAME "current_theme"
/// Nom du repertoire des lanceurs.
#define CAIRO_DOCK_LAUNCHERS_DIR "launchers"
/// Nom du repertoire des icones locales.
#define CAIRO_DOCK_LOCAL_ICONS_DIR "icons"
/// Mot cle representant le repertoire local des icones.
#define CAIRO_DOCK_LOCAL_THEME_KEYWORD "_LocalTheme_"
/// Nom du dock principal (le 1er cree).
#define CAIRO_DOCK_MAIN_DOCK_NAME "_MainDock_"
/// Nom de la vue par defaut.
#define CAIRO_DOCK_DEFAULT_RENDERER_NAME N_("default")
#define CAIRO_DOCK_LAST_ORDER -1e9
#define CAIRO_DOCK_NB_MAX_ITERATIONS 1000
#define CAIRO_DOCK_UPDATE_DOCK_SIZE TRUE
#define CAIRO_DOCK_ANIMATE_ICON TRUE
#define CAIRO_DOCK_INSERT_SEPARATOR TRUE
typedef enum {
CAIRO_DOCK_MAX_SIZE,
CAIRO_DOCK_NORMAL_SIZE,
CAIRO_DOCK_MIN_SIZE
} CairoDockSizeType;
typedef enum {
CAIRO_DOCK_UNKNOWN_ENV=0,
CAIRO_DOCK_GNOME,
CAIRO_DOCK_KDE,
CAIRO_DOCK_XFCE,
CAIRO_DOCK_NB_DESKTOPS
} CairoDockDesktopEnv;
typedef enum {
CAIRO_DOCK_BOTTOM = 0,
CAIRO_DOCK_TOP,
CAIRO_DOCK_RIGHT,
CAIRO_DOCK_LEFT,
CAIRO_DOCK_INSIDE_SCREEN,
CAIRO_DOCK_NB_POSITIONS
} CairoDockPositionType;
typedef enum {
CAIRO_DOCK_LAUNCHER_FROM_DESKTOP_FILE = 0,
CAIRO_DOCK_LAUNCHER_FOR_CONTAINER,
CAIRO_DOCK_LAUNCHER_FOR_SEPARATOR,
CAIRO_DOCK_NB_NEW_LAUNCHER_TYPE
} CairoDockNewLauncherType;
#endif
Generated by Doxygen 1.6.0 Back to index
|
{}
|
Feeds:
Posts
## Banach algebras, the Gelfand representation, and the commutative Gelfand-Naimark theorem
Banach algebras abstract the properties of closed algebras of operators on Banach spaces. Many basic properties of such operators have elegant proofs in the framework of Banach algebras, and Banach algebras also naturally appear in areas of mathematics like harmonic analysis, where one writes down Banach algebras generalizing the group algebra to study topological groups.
Today we will develop some of the basic theory of Banach algebras, our goal being to discuss the Gelfand representation of a commutative Banach algebra and the fact that, for commutative C*-algebras, this representation is an isometric isomorphism. This implies in particular a spectral theorem for self-adjoint operators on a Hilbert space.
This material can be found in many sources; I am working from Dales, Aiena, Eschmeier, Laursen and Willis’ Introduction to Banach Algebras, Operators, and Harmonic Analysis.
Below all vector spaces are over $\mathbb{C}$, all algebras are unital, and all algebra homomorphisms preserve units unless otherwise stated. In the context of Banach algebras, the last two assumptions are not standard, but in practice non-unital Banach algebras are studied by adjoining units first, so we do not lose much generality.
Read Full Post »
## Hilbert spaces (and dagger categories)
Hilbert spaces are a particularly nice class of Banach spaces. They axiomatize ideas from Euclidean geometry such as orthogonality, projection, and the Pythagorean theorem, but the ideas apply to many infinite-dimensional spaces of functions of interest to various branches of mathematics. Hilbert spaces are also fundamental to quantum mechanics, as vectors in Hilbert spaces (up to phase) describe (pure) states of quantum systems.
Today we’ll develop and discuss some of the basic theory of Hilbert spaces. As with the theory of Banach spaces, there are (at least) two types of morphisms we might want to talk about (unitary operators and bounded operators), and we will discuss an elegant formalism that allows us to talk about both. Things written by John Baez will be cited excessively.
Read Full Post »
|
{}
|
# Pearce Wiki
### Site Tools
notes:rebuilding_kernel_modules
# Differences
This shows you the differences between two versions of the page.
notes:rebuilding_kernel_modules [2012/10/17 10:22]andy created notes:rebuilding_kernel_modules [2012/10/17 10:32] (current)andy 2012/10/17 10:32 andy 2012/10/17 10:26 andy 2012/10/17 10:23 andy 2012/10/17 10:22 andy created Next revision Previous revision 2012/10/17 10:32 andy 2012/10/17 10:26 andy 2012/10/17 10:23 andy 2012/10/17 10:22 andy created Line 1: Line 1: ====== Rebuilding Linux Kernel Modules ====== ====== Rebuilding Linux Kernel Modules ====== - This page specifically concerns building kernel modules to work with an existing running kernel without having to rebuild the entire kernel or reboot the system. + This page specifically concerns building kernel modules to work with an existing running kernel without having to rebuild the entire kernel or reboot the system. More detailed information on this topic can be found in the [[http://git.kernel.org/?p=linux/kernel/git/torvalds/linux-2.6.git;a=blob_plain;f=Documentation/kbuild/modules.txt;h=3fb39e0116b4c8e42d40009357ed5cf13c1f2888;hb=HEAD|Documentation/kbuild/modules.txt]] file within the kernel source package. There are two prerequisites for building modules to work with an existing kernel: There are two prerequisites for building modules to work with an existing kernel: Line 13: Line 13: * ''/usr/src/linux-headers-$(uname -r)/Module.symvers'' * ''/usr/src/linux-headers-$(uname -r)/Module.symvers'' - <note>The latter file is located within the ''linux-headers-$(uname -r)'' package. + <note>On Ubuntu, the latter file is located within the ''linux-headers-$(uname -r)'' package. Unzip a copy of the Linux kernel sources and change into the root directory of it. Then execute the following commands: Unzip a copy of the Linux kernel sources and change into the root directory of it. Then execute the following commands: Line 34: Line 34: <code> <code> - make M=/sound/usb + make M=sound/usb + + Once the build is complete you should find the ''.ko'' files in their respective subdirectories. + + If there are no ''.ko'' files or there are other inexplicable build failures, double check that your ''GREP_OPTIONS'' environment variable isn't set, and unset it if it is. After unsetting you may need to repeat everything from ''make clean'' onwards.
notes/rebuilding_kernel_modules.1350469371.txt.gz · Last modified: 2012/10/17 10:22 by andy
|
{}
|
# Is this possible to integrate?
1. Aug 23, 2006
### jason17349
$$\int_0^x \frac{1}{\sqrt{(A+Bx^2+Cx^3+Dx^4)}} \,dx$$
I have no idea where to start or if this is even possible.
2. Aug 23, 2006
### StatusX
I could be obnoxious, and say yes, just pull that big constant outside the integral, but I won't (or maybe I just did). But if you mean for those x's to be replaced by v's, then this integral is generally only possible if you can factor the denominator. Then you would proceed by the method of partial fractions, which you can look up in google to find an easy explanation of.
3. Aug 23, 2006
### mepcotterell
maybe it's....
$$\frac{x}{\sqrt{(A+Bx^2+Cx^3+Dx^4)}}$$
4. Aug 23, 2006
### jason17349
You are wrong on both counts, the dv is supposed to be dx
5. Aug 23, 2006
### StatusX
I thought of that, but then you lose v altogether, which I figured you might want, and plus you'll have to change the limits of integration. It doesn't really matter.
6. Aug 23, 2006
### Hurkyl
Staff Emeritus
Of course it's integrable.
I'm 99.9% sure its not expressible in terms of "elementary" functions, though.
7. Aug 23, 2006
### StatusX
Oh, was that square root always there? I must have missed it. Then, yes, no, you can't integrate it in general without resorting to elliptic itegrals and other messy functions.
|
{}
|
# Why can't alcohols form hydrogen-bonded dimers like carboxylic acids?
Carboxylic acids such as acetic acid are capable of forming dimers:
I'm wondering why alcohols like ethanol don't generally form dimers. In the diagram below, the oxygen atom on the left ethanol molecule should be capable of accepting a hydrogen bond from the partially positive $\ce{H}$ atom on the right ethanol molecule. Does it have something to do with the $\ce{-OH}$ bond angle?
• Because the ring would be too strained. – DHMO Oct 10 '16 at 12:01
• That I don't believe in, hydroxyl groups are just not acidic/basic enough to engage in that type of hydrogen bonding, no matter the ring size. If you would add in water molecules the rings would get extended and still you wouldn't get dimers with incorporated water. – logical x 2 Oct 18 '16 at 11:58
The strength of a hydrogen bond somewhat depends on the $\ce{X-H\bond{...}X}$ angle that the hydrogen-bonding hydrogen forms with the two electronegative elements $\ce{X}$. In our case, carboxylic acids or alcohols, $\ce{X} = \ce{O}$ so the angle is $\ce{O-H\bond{...}O}$. The ideal angle for this fragment is $180^\circ$.
As you have drawn for carboxylic acids, it is very easy to allow this linear arrangements. If you wish, you could describe the entire $\ce{(R-COOH)2}$ feature as a benzene-like ring streched in a single direction. Most importantly, the acceptors and donors line up nicely, the carboxyl function features an angle of $120^\circ$ and the $\ce{C-O-H}$ angle (and the $\ce{C=O\bond{...}H}$ one!) is also close to $120^\circ$. These arae the theoretically predicted, unstrained angles.
For two alcohol (e.g. ethanol) molecules to attempt a similar arrangement, we would end up with only four atoms that have to form a rectangle with $\ce{O, H, O, H}$ at the four corners. Thus, the $\ce{O-H\bond{...}O}$ hydrogen bond angle would be much closer to $90^\circ$ — a very bad arrangement. Furthermore, this would put the two electronegative oxygens (which thus feature a negative partial charge) closer together with nothing in-between which would cause destabilisation due to like charges approaching. The hydrogen-bonding hydrogens cannot alleviate this unfavourable interaction since they are at the corners of the square.
The situation for alcohols is much better if they create a network of a number of molecules that allow for much more favourable angles. With four molecules, you could already create a cyclic structure of approximately $100^\circ$ $\ce{H\bond{...}O-H}$ angles and $170^\circ$ $\ce{O-H\bond{...}O}$ angles — much more favourable. Of course, in the actual solution this number will fluctuate strongly as hydrogen bonds are broken and reformed constantly and different ring sizes happen all the time.
• "As you have drawn for carboxylic acids, it is very easy to allow this linear arrangements. If you wish, you could describe the entire (R−COOH)2 feature as a benzene-like ring streched in a single direction." Well except that in benzene, no bond is linear as they are all tilted by 60° (so that the bond angle is 120°)." Also the two alcohols dont necessarily have to form a rectangle, as the h bonds might just as well be longer than the covalent O-H bonds. Im still not convinced... – logical x 2 Nov 20 '16 at 23:45
• @ketbra Re angle: The first sentence is about a general $\ce{X-H\bond{...}X}$ angle. The second sentence tells you which atoms we are dealing with. And since $\ce{X}$ is $\ce{O}$, it also specifies that the angle is $\ce{O-H\bond{...}O}$ in this case. An angle will always be made up by three points in space; in the context of molecules/chemistry, typically three atoms. – Jan Nov 20 '16 at 23:52
• Re streched benzene ring: Yes, benzene consists of six atoms while a carboxylic acid dimer consists of eight. Hoever, imagine two bonds of benzene which are transformed onto one another by a mirror image parallel to both extended in a way such that an additional atom fits between the two carbons. These eight atoms are now a ‘benzene-like ring structure’; six $120^\circ$ angles and two $180^\circ$. – Jan Nov 20 '16 at 23:55
• Re rectangle: A rectangle is a shape with four corners, two sets of parallel sides and four $90^\circ$ angles. I don’t see how that should be unable to accomodate H bonds ‘longer than the covalent $\ce{O-H}$ bonds’? I explicitly said rectangle, not square. – Jan Nov 20 '16 at 23:57
• Four 90 ° angles would be the definition of a square though... But then I dont get the next statement that the "hydrogen bond angle would be much closer to 90° — a very bad arrangement". How do you come up with that? I would say that there is considerable stabilization between hydroxyl groups (i.e. alcohols) by hydrogen bonding e.g. in water and there also exists the water dimer (or methanol) so that more or less refutes this statement. – logical x 2 Nov 21 '16 at 7:36
The carboxylic acid dimer is more likely than the alcohol dimer because the carboxylic acid dimer has two attachment points, whereas the alcohol dimer only has one. While the hydrogen bonds may be of comparable strength in the two cases, the carboxylic acid case is also favored entropically.
Consider, by analogy, the chelate effect. Chelating (multidentate - more than one binding site) ligands bind more strongly to metals than monodentate ligands.
Here are some data for the comparable systems of copper/ethylenediamine $\ce{Cu(en)_2^2+}\ (\ce{en}=\ce{H2NCH2CH2NH2})$ and copper/methylamine $\ce{Cu(CH3NH2)_4^2+}$. Here, I am presenting the overall formation constants $\beta$ (whcih are the products of the equilibrium/formation/binding constants of each individual association step). The formation constant for $\ce{Cu(en)_2^2+}$ is higher than for $\ce{Cu(CH3NH2)_4^2+}$ by a factor of $\sim 10^4$, which corresponds to the chelated complex having a more negative free energy of binding by 23 kJ/mol. The major component in this lower energy state is entropy. Data from here.
$$\begin{array}{|c|c|c|c|c|}\hline \mathrm{Equilibrium} & \beta & \Delta G^\circ& \Delta H^\circ& -T \Delta S^\circ\\ \hline \ce{Cu^2+ + 2en <=> Cu(en)_2^2+} & 4.17 \times 10^{10} &-60.67 & -56.48 & -4.19 \\ \hline \ce{Cu^2+ + 4CH3NH2 <=> Cu(CH3NH2)_4^2+} & 3.55 \times 10^6 & -37.4 & -57.3 & 19.9 \\ \hline \end{array}$$ (data of last three columns in $\mathrm{kJ\ mol^{-1}}$).
It's all a matter of relative energies. Carboxylic acids form two aligned hydrogen bonds, so the $\Delta E$ associated with this process is fairly negative. There's also some extra electron density on the carbonyl O that likely contributes to a better hydrogen bond. Now, alcohols can form one hydrogen bond. You almost certainly make the hydrogen bond in solution, but any collision from a solvent molecule could break the hydrogen bond.
Edit: Apparently, my answer was not clear. No, the bond angle is not the problem.
The main reason for the higher extend of H bonding in carboxylic acids is due the higher acidity of the carboxyl group (e.g. ethanol: $~\mathrm pK_\mathrm a = 16$, acetic acid: $~\mathrm pK_\mathrm a=4.5$) making it a good hydrogen bond donor, and the ability of the carbonyl oxygen to act as a good hydrogen bond acceptor. Thus the extend of H bonding can be explained solely based on acidity/basicity arguments. Compared to that a hydroxide is just too basic to be formed out of the hydroxyl group, so hydrogen bonded dimers of alcohols make no sense. What is also important to point out, is that the resulting dimer will be completely symmetric (due to resonance) making the situation energetically even more favourable.
• I have a feeling you were rather confused here... – Mithoron Nov 19 '16 at 22:14
• @Mithoron Thanks for getting personal : ) Maybe try finding arguments for your position next time, if you have any. But I really don't believe in the points mentioned in the other answers. It just has to do with the strength of the O-H bond, i.e. its lower aciditiy, compared to carboxylic acids and not with the alignment. – logical x 2 Nov 20 '16 at 8:52
• Well we could talk about it in chat (chat.stackexchange.com/rooms/3229/the-periodic-table) or maybe separate room. Or you could just have faith in Jan's answer :D – Mithoron Nov 20 '16 at 21:07
|
{}
|
The amplifier would be embedded within a 2-way speaker. I used this notation because we connect the COM and VSS port of the IR2110 to the "-30V" rail. In this paper a new high-speed, low-current levelshifter and a robust deadtime control arrangement are presented that are essential for a high quality switching power stage. It is better to use a converter IC directly like the LTC6992 https://ibb.co/zm1s04H . Joined Oct 15, 2017 45. The values of the resistor and the capacitor set a frequency of approximately 200kHz. Can i use CD4504 level shifter instead of this 2 trasistor 2N5401 ? The 7805 voltage regulator will have no effect on stabilising the voltage. A class D amplifier comprises three basic sections: the Pulse Width Modulator (PWM), ... entail combining the two functions in a single integrated circuit. Hi, all looks really great! It is good post & good job thanks admin i will made it. please let me know and thanks a bunch for your help, Sorry for replying to you so late. System ÎGate Drive ÎMOSFET ÎDesign Example An electronic project where you not only see the results but also hear them? In theory, an op-amp can be used as a comparator, but in reality op-amps are designed for other types of work, so make sure you use an actual comparator. Finally the filter. Thanks for the great article Cezar, I have a question about a bipolar power supply here in the US I can only find a dual +- 20vdc, is there a project you can recommend for building a bipolar +- 30vdc power supply. 1001++ Electronic Circuit Schematic. The high side MOSFET needs to be driven by a gate voltage that is about 12V above the switching node, VS. Add to Wish List Add to Compare. Just understand how the circuit works. Again, thank you for taking your time to give your feedback, I truly appreciate it. This is a 200W power amplifier circuit project. This High Power Amplifier circuit is a class D power amplifier, which has a high enough power to generate 3000W of power at 4 Ohm impedance - and also more power up to 4500W at 2 Ohm impedance. When you will finally hear that crisp sound coming from your speaker, it will all be worth it. To be honest I do not know much about Ultrasound amplifier, but I will start with a schematic of that and see the requirements. $185.00. Unlike the simple design of Class AB amplifiers, Class D amps are much more complex. There are also unexpensive devices with just 5 mOhm channel resistance in order to improve the overall efficiency. The same way the regulators in this circuit are being used to produce "non 5V" power rails. The IR2010 or IR2011 as well as the comparator are relatively slowly devices. You can buy one already made but I would suggest that you wind your own—this is a DIY project after all. Hey Cezar, I had a few questions about this project: I am trying to build it at home. However, I do not think that having -25V going out of a 5V regulator is right. We will work it out. I checked and the guys at Infineon (IRAUDAMP1 reference design) use the same type of notation for their power supply. Theory is one aspect and practice is another. For the triangle generator, I used an LMC555, which is the CMOS variant of the famous 555 chip. SDS-450C 4 Channel Class D Amplifier Kit . 170W Audio Power Amplifier. The voltage mode Class D amplifier is defined as a switching circuit that results in the generation of a half-sinusoidal current waveform and a square voltage waveform. Let's start with that first sentence. Input and output waveforms of the comparator are shown in the figure below. The complete Bill of Materials can be found in the files below, where you can also find the PCB files both in PDF format and as KiCAD files. The frequency of the sawtooth waveform is usually selected 10 times the maximum frequency of interest in the input audio signal. It is not necessary but highly recommended that you use a heatsink for BD241C as it gets quite hot. Can it be good, also to ampliy ultrasound? On the PCB, The ground connectors are near the -30V connector. Class D amplifier uses MOSFETs that are either ON or OFF. In a class D amplifier, power field-effect transistors (FETs) are driven to produce an output square-wave that switches between a high and low level at … Conversely, the class AB amplifier will always have some current passing through and some voltage remaining across the switching element. The rectangular signal is amplified, and then a low-pass filter results in a higher-power version of the original analog signal. Somewhere between 120-140W. One of the critical aspects of having excellent audio performance is reducing interference. The audio escapes … One question what is the power out one can expect from this circuit? This requires a voltage that is higher than the positive supply; the IR2110 provides this drive voltage with the help of our bootstrap capacitor, C10. please let me know and thanks a bunch for your help. How do you get +12v from a -30v supply ???? Or could the IC be bad? Hi, SET 2A3 Tube Amplifier Schematic (EF86 input) - [3.5 Watts, SET, class-A] SET 2A3 Tube Amp Schematic by Loftin-White (6SL7 SRPP input) - [3.5 Watts, SET, class-A] 300B - … Add to Cart. Class D power amplifier is a type of audio amplifier were the power handling devices are operated as binary switches. Class D power amplifiers are much power efficient when compared to its predecessors like Class A, Class B and Class AB. Here is the output voltage seen in a typical Class-D amplifier with a slightly underdamped output filter. Although there are a number of different design variations, Class D amplifiers are essentially switching amplifiers or Pulse Width Modulator (PWM) designs. Hello Guys , i was also building same class D amplifier ,i have built the circuit ,when i am powering the circuit ,555 Timer gives a nice triangular wave and Lm 393 gives a output of PWM wave , but when i am providing this signal to the hex inverter , the output is not coming,zero volts is coming .Anyone can help me about ? Hey Cezar, I had few more questions, I substituted 2n5401 with MPS751 bc it is unavailable in the market now. I hope that the information in this article is sufficient for you to build your own audio power amplifier. You can easily connect with the computer and have a marvelous audio experience with either headphones or passive speakers. Create one now. Almost all power drawn is supplied to the load. There are two of them: One as an input, one as an output for the speaker, although it is better to connect the speakers direct to the power supply ground, to remove some humming noise. does that mean we need 2 power supplies? This class D amplifier design consist of three ic are TL071, CD4049, and IR2110. TAS5611A TL494 CLASS D AMPLIFIER COMPLETED. The high frequency response is dependent on the loudspeaker impedance. For some reason I keep getting the notification emails in my spam inbox. They have already been shared as KiCAD files. The switching circuit is generally designed around MOSFETs. Amplifier modules. All connections are 100% correct even though my PCB layout differs slightly. Hi, thanks for the post! A PWM of 200 kHz is allready very high to operate with these devices. Traditional amplifiers, like the class AB, operate as linear devices. This circuit can be used in different amp circuits on a separate PCB. Compare this to switching amplifiers, so called because the power transistors (the MOSFETs) are acting like switches, changing their st… I have a Sony subwoofer speaker unit with 2 ohms and 180 watts RMS (Aluminium tube -dual speaker -internally wired parallel) .Can I use your circuit for this speaker unit?.If so what changes in this circuit I need to do ? Since Class D amplifiers are highly power efficient, they require a smaller heatsink and a smaller power supply. I designed this amplifier for an output power of about 100-150W. These devices may offer somewhat improved performance, but they could also be more expensive. Audio frequencies range from about 20 Hz to 20 kHz, so the amplifier must have good frequency response over this range (less when driving a band-limited speaker, such as a woofer or a tweeter). You can go higher than this, but for voltages of about ±40V you need to make sure that you change the values of the resistors R4 and R5 to 2K2. eval(ez_write_tag([[300,250],'circuitstoday_com-medrectangle-4','ezslot_4',109,'0','0'])); The task of the low pass filter is to filter out useful low frequency components from the output of the switching circuit. Tags: amplifier … I am wondering if the power supply ground is isolated from the circuit ground. Click on the circuit diagram to view in high resolution It is that simple. If you have any trouble with your build, comment here or post on the forum using as much information as possible. High performing Class-D amplifiers for a range of audio applications Address the needs of any audio application with the industry’s broadest portfolio of speaker amplifiers (including Class-D, Class-D boosted, Class-AB and smart amps) ranging from 5 W to more than 50 W of output power and a range of topologies, performance and features. I would suggest to operate with the LM5104 for the half bridge driver together with the STP16NF06 NMOS transistors to operate at 500 kHz. Add to Cart. Class D amps are switching amplifiers, meaning the output transistors act as a switch; either on or off. This High Power Amplifier circuit is a class D power amplifier, which has a high enough power to generate 3000W of power at 4 Ohm impedance - and also more power up to 4500W at 2 Ohm impedance. The same is true in rectification, filter power stage .. If you're looking for an all-in-one solution to your home stereo needs, it's hard to top the … You will find lots of information by googling the "synchronous buck converter" and "half bridge circuit". Hey cezar thanks for the quick reply. No harm in doing it this way, though. BUT, if we put the black probe on the 0V ground, on the multimeter we would have -18V. This 200W power amplifier using the complementary transistors of 2SC5200 and 2SA1943 as the main parts. My email is johnjol399@aol.com if you'd rather email me. Can this amplifier deliver 60-120 watts into 16 ohms? High performing Class-D amplifiers for a range of audio applications Address the needs of any audio application with the industry’s broadest portfolio of speaker amplifiers (including Class-D, Class-D boosted, Class-AB and smart amps) ranging from 5 W to more than 50 W of output power and a range of topologies, performance and features. The typical class d amplifier consists of a sawtooth waveform generator circuit. This is done using PNP transistor and 1N4148 diodes. Input and output waveforms of the switching circuit are shown in the figure below. The actual frequency of the triangle signal is much higher, on the order of hundreds of kHz, so that we can later extract our original signal. This is two comparators in one package, and we just swap the inputs for the second comparator. It does not matter if it does not work on the first try. does that mean we need 2 power supplies? The latest class D audio amplifier is switch type amplifier by using Pulse Width Modulator (PWM) amplifier. To remove the hum noise (50/60 Hz, from the mains frequency), I used a star-ground configuration; this means connecting all grounds (amplifier ground, signal ground, and speaker ground) at the same point, preferably on the power supply PCB, after the rectifier circuit. Good circuit and a good DIY project from you….... Hello, I have been working on this circuit for quite some time. By the argument you put forward, any variable power supply that uses, for example, a LM317 regulator would have a 1.25V output all the time as it is a 1.25V voltage regulator. Press Esc to cancel. To ensure maximum system robustness, an advanced protection strategy has been implemented to provide overvoltage, overtemperature and overcurrent protection. So I removed the MOSFETS and measure pulses at pins 1 and 7. A real filter, not an ideal one, does not have a perfect "brick-wall" transition from passband to stopband, so we want the triangle signal to have a frequency at least 10 times higher than 20KHz, which is the upper human hearing limit. The cut-off frequency is 40kHz, and the load resistance is 4 ohms because we have a 4-ohm speaker (the values used here will also work with an 8-ohm speaker, but it is best to adjust the filter according to the speaker you choose). •PWM technique is used to express analog audio signals with ON or OFF states in output devices. As you have them it is rather confusing, even though you mention they are referenced to the -30V line. (google sound card buffer) if possible could you show me how to add it? Is it before the input of the amplifier? 170w Class D Amplifier Schematic Diagram 3000 Watts Power Amplifier Class D Mosfet Irfp260 Irfp4227 Pau Mosfet Class D Amplifier Offset Voltage Electrical Engineering Just fire up that soldering iron, etch your PCB, and start working. Great article! Hey cezar thanks for the quick reply. The LM317 is just a part of a circuit that happens to be a 1.25V regulator, but produces a variable output. HI IS IT NECESSARY TO HAVE 220N AT +TO G AND -TO G ??? But it is also possible to make a good converter by use of 555 timer ICs https://ibb.co/cDqcYyT . PWM technique is used to express analog audio signals with ON or OFF states in output devices. The bass sure is tight, deep and fast but lacks some meat on the bones. Negative feedback loops are often included in between the low pass filter output and the comparators audio input in order to fight the errors. I dont mind the power output, it could be 50W-100W. Please add more info about proper supply rates. What’s more, the lower power dissipation accounts for the minimal heat generated. The amplifier module is based on the TPA3116 circuit comprising two bridged power amplifier channels with common switching, muting … Because the VSS pin of the IC is tied to the negative power supply, we need to level shift the signals from the comparator. One more thing, I am familiar with Eagle, is there anyway to convert KiCad files to eagle or do I have to manually rebuild the whole thing on eagle? I wanted to give a look at the KiCad files but the link gives me “Error 404 HiFi Class-D Discrete Power Amplifier Class-d power amplifier circuit using discrete components (transistors, resistors, capacitors) witho... High Power Amplifier Crown Power amplifier Crown XLS can be supplied with a voltage of at least 45V DC to 90V DC, and to get m... 2.1 Power Amplifier SOCL504 Power Amplifier SOCL 504 ; You might like. Two issues are the rise and fall time of the devices in the power stage and the fact that we are using an NMOS transistor for the high-side driver. The quality factor $Q = \frac{1}{\sqrt{2}}$. With LM4651 & LM4652. When a transistor is off, the current through it is zero. The heat sink barely gets warm! The typical class d amplifier consists of a sawtooth waveform generator circuit. The circuit is a class-D 6W inductor free audio amplifier for BTL (bridged-Tied Load) stereo speaker at up to 6Ω to 8Ω per channel. Aside from being light on energy use, they're generally quite easy on the wallet, and … I am trying to build one myself and I have a few issues. Thanks for your knowledge and have a great day. Class D Audio Amplifier- Schematic Diagram. If you are using a transformer (I recommend toroidal because of their size), and not a SMPS, you will need a bridge rectifier and some beefy filtering caps (I personally used 2x10 000uF per branch - positive/negative). The main difference is that instead of ΔΣ modulation, mine uses PWM. The answer could be just a sentence long: It is a switching amplifier. Class D Amplifier Operation Class D amplifiers consist mainly of 3 stages: the input switching stage, the power amplification stage, and the output filter stage. Also could you link me the heatsink you used?, I opened the files in kicad and it said that a lot of the libraries you used are missing, is it possible to send me your kicad libraries in a zip? When it is OFF the entire voltage remains across it and no current will flow through it. This seems a pretty simple circuit containing all the main parts of a Class D amplifier. That's it. If its possible how much power i can get? But all rails are measured W.R.T. at the Class D amp input. Ncore® is the first Class-D amplifier not just to nudge the best linear amplifiers, but to surpass them in every aspect relevant to sound quality. A class-D amplifier or switching amplifier is an electronic amplifier in which the amplifying devices operate as electronic switches, and not as linear gain devices as in other amplifiers. Hello, I have looked at the symmetric diagram, but I am not too sure why there is 2 set of inductors at the end of Vb and Vs port of IR2110, can anyone give me a helping hand, please:), I built this amplifier but it burns out the mosfets as soon as power is applied. These capacitors need to be polypropylene or polyester—in general it's not a great idea to use ceramic capacitors with audio signals. And for output power that can be issued this IC can reach up to 340W x 1 @ 8 Ohm; 170W x 2 @ 4 Ohm. And you need to make sure that the capacitors that you are using for filtering are rated for high voltage, at least 100VAC (more doesn't hurt). There is only a pulse at pin 7 but nothing at pin 1. A typical 5 V linear regulator (such as the LM7805) will make the output voltage 5 V higher than the "ground" voltage. We measure the electric potential difference, V2-V1. eval(ez_write_tag([[468,60],'circuitstoday_com-medrectangle-3','ezslot_2',122,'0','0']));Higher efficiency means low thermal dissipation and it means it dissipates less power when compared to the predecessors (The Class A, Class B, Class AB and Class D). Furthermore, these ICs provide the boosted gate voltage needed for the high-side NMOS. Imagine that we use a multimeter and we put the black probe (ground) to the -30V rail. This amplifier is designed very hard to the limits of its devices. Another approach is to use a comparator that has two outputs, such as the LT1016 from Linear Technology. I have modified the diagrams and I now hope it is less confusing to others. On my previous test, the same as this, I didn't had any problems. In this instructable I will layout the basic components of a Class D amplifier built from common ICs. It has smaller heat dissipation, so small heatsink is needed. The basic Class-D block diagram is shown in Figure 2. As you can see from the main image, we have made the circuit on a piece of perfboard. Depends on the frequency, but keep in mind that this is an amplifier designed to go well with frequencies under 20kHz. The transistors used can function between cutoff and saturation. On the reguatlor you have got 18 volt on the input and 25 volt on the output. CDA-120 2 CHANNEL AMPLIFIER KIT . I'm guessing there is a whole lot of distortion, since there isn't any negative feedback to compensate for all the imperfections in the triangle wave and comparators and output stage; it should be fine for signals with no dynamic range like a siren, or a compressed speech over a bullhorn. Class D is the only option for combining all these requirements together. i am feeding in a PWM input to the class D amplifier and then converting it back to an analog signal by a low pass filter. 0 volts. I thought u wouldnt even see my post. Is it possible (because I would like to build my own studio speakers) ? Design of audio amplifiers: selection guide for Class-D audio amplifier and circuit ideas. Something similar happens when you try to add negative feedback to improve stability, bandwidth and THD, I need to determine the feedback gain (and with that the total gain) by design so that I can then choose the value of certain components. Below is the schematic circuit of TDA8950TH. BECAUSE WE HAVE ALREADY 100N PARALLEL TO POWER CAPS. You mentioned increasing input voltage to 40V but what is the lowest value? Just wondering, why exactly are you shifting the comparators outputs to -25V-30V with the PNP before going into the IR2110 instead of the standard 0V? TPA3116 Class D Amplifier Circuit A small class-D mid-range stereo headset that will be used in car headphones, active speakers, or PC-audio. For the filtering stage, one of the best ways to do this is to use a Butterworth filter. Class-D PAs use two or more transistors as switches to generate a square drain-voltage waveform. how about a nice power supply project along with a simple enclosure to complete the package?! Shouldn’t it be the other way round? Any higher than this and we will run into trouble because the comparator and the MOSFET driver are not the fastest devices. For class D amplifiers to operate in switch mode, pulse-width modulation (PWM) can be used. The combination of the LM4651 driver IC and the LM4652 power MOSFET Class D power amplifier IC provides a high efficiency amplifier solution, suitable for self-powered speakers, subwoofers and quality car boosters.. Previous 12V 19V DC TO DC CONVERTER FOR LAPTOP UC3843D SCHEMATIC CIRCUIT DIAGRAM. It varies greatly (100-150W) depending on your power supply. This class of amplifier uses a technique similar to PWM to control the output. 1000++ Circuit. Class D or Class T According to Wikipedia, “A Class T amplifier is an audio amplifier IC design.$445.00. Cezar, They operate by rapidly switching back and forth between the supply rails, being fed by a modulator using pulse width, pulse density, or related techniques to encode the audio input into a pulse train. Us I heard about digikey, mouser and farnell/newark but I also see +-5V going to the class amplifier. Ic directly like the LTC6992 https: //ibb.co/zm1s04H neither of the transistors low, which to... But it is unavailable in the previous section what have I to modify to have in output devices signals... Mode amplifier similar in operation to the load as possible, VS,... Power efficient when compared to its predecessors like class a and class B class! Project along with a simple enclosure to complete the package? audio … schematic of critical... As load, the low-voltage audio input in order to get a power supply project along with a slightly output. In output devices share the PCB, Im having trouble finding the common ground you talked about be 50W-100W a., Sorry for replying to you so late a great idea to use same! No voltage across it when it is simply not as high as what ’ amplifier! Its transistor on and OFF this amplifier for an amplifier with a simple enclosure to complete package. An audio amplifier IC design in mind that this is to digitize the input signals with on or states. Also provide schematic and PCB some design choices and how to make one yourself change the output act. The ground connectors are near the -30V rail which has low on resistance and smaller power supply reduces the and! So that it was best to use a triangle converter solution Forums ; new Posts ; thread starter Hassan Malik! Brüel & Kjær or sound Technology these days Holiday Season analog Alarm, pulse-width ). By Bernhard Dwersteg, TRINAMIC Motion control, meaning the output can class d amplifier schematic! Converter IC directly like the class D amplifier or switching amplifier which uses pulse Width,... By a low-pass filter to PWM to control the output of the best ways to do is. On it and no current will flow through it with the click of a regulator., mouser and farnell/newark but I also see +-5V going to the circuit on perfboard that... Aac which seems to be iron powder ; ferrite can work on loudspeaker... Be stable as the main advantage of a class D amplifier project features the LM3615 vu circuit! I heard about digikey, mouser and farnell/newark but I would suggest that you wind your own—this is switching... Two or more transistors as switches to generate a train of fixed-amplitude square.! % ( in class d amplifier schematic ) that is regulated from the +-30V power supply project along with slightly. Some meat on the PCB layout of this website to help understand these amps difference is instead... Change master value and Rightmark audio Analyzer can measure noise and distortion and frequency response is dependent on the time! ) depending on your power supply using the IRF540N or the IRFB41N15D can transform. Amplifier module circuit shown below: building the circuit is not as gutsy and has less slam regular. But it does the job pulse-width modulation ) analog signal you, by Bernhard,... What have I to modify to have 220N at +TO G and -TO G??????... Mosfets connected infereon, that is regulated from the main job of gain... 5V regulator is right maximum efficiency of 78.5 % dependent on the multimeter digit we would have -18V power... Generally you would need to ensure that the MOSFET has an adequate maximum (! Please tell me what I need to ensure maximum system robustness, an advanced protection strategy has been to... ) to the limits of its devices and has less dead time that I want to will! & uses, RFID Reader and Tag – ultimate guide on RFID module rail... Uses a technique similar to PWM to control the output Q = \frac { 1 } { \sqrt 2! ) analog signal by mixing it with no voltage across it and no current will flow through.! The STP16NF06 NMOS transistors to operate with these devices may offer somewhat improved performance, but keep in mind this. Eminent feature in class D amplifier is shown in the figure below main advantage of a 5V regulator and.! I find the time, maybe only achieve a maximum theoretical efficiency of basic. Add it the figure below power MOSFETs go, I bought them from tme, eu, which can to! Have all the resistors, unless noted ( R4, class d amplifier schematic ), on bones. 50 % Where did you connect your ground on the PCB, Im having trouble finding the ground. Ab amplifiers can drop as low as 50 % ground is isolated from the +-30V power reduces... Features high power, good definition and very low noise design also to!, these ICs provide the boosted gate voltage that is something you might wan na use are necessary means the... Up the IR2110 get an expresion of the list the most efficient class AB amplifiers can as. “ -25V ” relatively slowly devices times the maximum frequency of the sawtooth waveform is usually 10... Are compared with the computer and have a marvelous audio experience with either class d amplifier schematic or passive.! Entire voltage remains across it when it is a switching amplifier s V going the! 30V to 100V DC asymmetric power supply, but keep in mind that this is an project! They require a smaller heatsink and smaller power supply with ±30V rails just a component, the... Being a separate PCB circuitstoday.com is an audio amplifier IC design asymmetric power supply reduces the and!: //ibb.co/zm1s04H 50 % modulation ) analog signal very flat response in the below. To teach you all its nooks and crannies see +-5V going to the op amps power. Analyzer can measure noise and distortion and frequency response with the chopping sawtooth waveform for sampling input... For class D amplifier audio players or controlling the motors as well increasing! Slam than regular big transistor amps highly power efficient when compared to its predecessors class. Ab only have a maximum efficiency of class AB, operate as linear.. Forum using as much information as possible what ’ s amplifier – a class D circuit! Where did you connect your ground on the 0V ground, on the multimeter we would -18V! Amplifier, I used this notation because we connect the COM and VSS port the! Smaller power supply using the two Zenner diodes D1 and D2 even though you mention they are used! Is better to use a bipolar power supply project along with a enclosure... In different amp circuits on a design for school and have been looking for insight from various projects ... Depends on the 0V ground, on the reguatlor you have all the resistors, unless noted ( R4 R5... But the RDS ( on ) is not working and some voltage remaining across the switching node,.! '' rail an 25 ohm impedance that you wind your own—this is table! This as a direct replacement for IR2011 AAC which seems to be fast be polypropylene or general! And has less dead time for through shoot protection there a reputable parts supplier that you your. Ab amplifiers, meaning the output voltage seen in a higher-power version of parts. The voltages will be a little complicated by using pulse Width Modulator ( PWM ) be... Switching mode amplifier similar in operation to the circuit in order to allow me to have 220N at +TO and! Have 220N at +TO G and -TO G????????. Voltage controlled PWM converter I would like higher perfomance devices, faster MOSFET,! These Types of filters have a look at the Class-B schematic shown in figure.... As power MOSFETs go, I suggest using the IRF540N or the IRFB41N15D some of... And less gate capacity and gate charge for faster switching and low side MOSFETs would read +30V even you. Quality sound reproduction ( IC ) as the minus 30V rail ( line ) is necessary. Amplifier employs MOSFETs which has low on resistance ), high impedance?... By using pulse Width modulated, rectangular signal is converted into a pulse at pin 7 but at! Non-Linear switching mode amplifier similar in operation to the load it delivers about 200 watts in 4 ohms:... Talked about power CAPS with frequencies under 20kHz and gate charge for faster switching and low side MOSFETs be a... A DSP for your knowledge and have a marvelous audio experience with either headphones or passive speakers suggest. Used, maybe I will now tell you some design choices and how to my... With a Manual volume control switch, 3: th etc ) Class-D 2-channel audio amplifier design, let s... Together with the LM5104 for the minimal heat generated, are 1/4W need! The module IC PAM8402 amplifier chief installed, offers low THD+N, allowing it to high. - 30 watts range, what changes should be referenced to the circuit ground thing is instead... The class D amplifier or switching amplifier is a non-linear switching mode amplifier similar in operation the..., are 1/4W of transfer function in order to allow me to have 220N at +TO and! Will produce high-frequency sawtooth waveform generator class d amplifier schematic -30V rail the PWM freqency to be its possible how much power can. Project after all ) is not as high as what ’ s needed for a Class-A amplifier to something! Pass all current through it with no voltage across it is not stable next audio … schematic the... Can help me out drive the high and low RDS ( on ) is not final. Some meat on the PCB layout of this mixing will be stable as the comparator is to use converter... If something goes wrong, we need +-30V, but the RDS ( on is!
Weather In Yerevan 15 Days, Mitchell Johnson Spell Ashes, Cardiology Research Reddit, The Witcher: Monster Slayer Reddit, Mario Kart 7 Custom Karts, Naman Ojha In Ipl 2020, Best Dp For Instagram,
|
{}
|
# Proof-verification: Existence of an explicit formality morphism from the Barratt-Eccles Koszul dual cooperad
I know asking for proof-verification on MO is a tricky thing. On one hand interesting research level proofs are usually subject of articles and can not be discussed here in detail. On the other hand most simple proof which can be written on a forum are not "high-level" enough for MO and other places are more appropriate. After all MO does not have a "proof-verification" tag, like math.stackexchange.
Anyway, for my personal taste at least, the following is research level. So lets see where this goes:
I want to proof the following statement:
Consider everything over the field $\mathbb{Q}$. For a fixed, given $n\geq 2$, let $\mathcal{E}_{n}$ be the $E_{n}$-suboperad of the Barratt-Eccles operad $\mathcal{E}$, $\mathcal{E}_{n}^{i}$ its Koszul dual cooperad in the sense described in the paper "Koszul duality of En-operads" by Benoit Fresse and let $e_{n}$ be the operad of $(n-1)$-Gerstenhaber algebras. Then there exist a solution to the Maurer Cartan equation in the convolution dg Lie algebra
$\Pi_{k\in\mathbb{N}}Hom_{\Sigma_{k}}(\mathcal{E}_{n}^{i}(k),\Omega e_{n}^{i}(k))$
where $\Omega e_{n}^{i}$ is the minimal model of $e_{n}$.
Proof:
Since $\mathcal{E}_{n}$ is an $E_{n}$-operad, by the definition of $E_{n}$-operads there is a zig-zag of quasi-isomorphisms of dg-operads
$\mathcal{E}_{n}\overset{\simeq}{\longleftarrow}\bullet\overset{\simeq}{\longrightarrow}\cdots\overset{\simeq}{\longleftarrow}\bullet\overset{\simeq}{\longrightarrow}e_{n}$
where we consider $e_{n}$ as a differential graded operad with trivial differential in each arity. Now since in both cases ($\mathcal{E}_{n}$ as well as $e_{n}$), the appropriate Koszul dual cooperads $\mathcal{E}_{n}^{i}$ and $e_{n}^{i}$ are the linear duals "up to tensoring with appropriate shifting cooperads", this implies the existence of the following diagram of dg-cooperad quasi-isomorphisms:
$\mathcal{E}_{n}^{i}\overset{\simeq}{\longrightarrow}\bullet\overset{\simeq}{\longleftarrow}\cdots\overset{\simeq}{\longrightarrow}\bullet\overset{\simeq}{\longleftarrow}e_{n}^{i}$
since the linear dual of a quasi-isomorphism is a quasi-isomorphism. Now if we change the category of differential graded cooperads with morphisms of differential graded cooperads into the category of dg cooperads with infinity morphisms of dg cooperads (Such an infinity morphism $F_{\infty}:\mathcal{C}_{1}\rightsquigarrow\mathcal{C}_{2}$ is defined as (or equivalent to ) a morphism of dg operads $\Omega F_{\infty}:\Omega\mathcal{C}_{1}\to\Omega\mathcal{C}_{2}$ ) then any quasi-isomorphism has an actual inverse in terms of these infinity morphisms (To emphasis these different kind of maps, I write $\rightsquigarrow$ for them). Therefore in this other category, there exist the following diagram of dg-cooperad infinity-isomorphisms
$\mathcal{E}_{n}^{i}\overset{\simeq}{\rightsquigarrow}\bullet\overset{\simeq}{\rightsquigarrow}\cdots\overset{\simeq}{\rightsquigarrow}\bullet\overset{\simeq}{\rightsquigarrow}e_{n}^{i}$
and by composition, we get a single infinity isomorphism of dg-cooperads $\mathcal{E}_{n}^{i}\rightsquigarrow e_{n}^{i}$. By definition of these infinity morphisms, this is equivalent to the existence of an ordinary isomorphism of dg-operads
$\Omega\mathcal{E}_{n}^{i}\to\Omega e_{n}^{i}$
which in turn is equivalent to the existence of a solution to the Maurer Cartan equation in $\Pi_{k\in\mathbb{N}}Hom_{\Sigma_{k}}(\mathcal{E}_{n}^{i}(k),\Omega e_{n}^{i}(k))$.
q.e.d.
Second question: The proof relays on the transition from ordinary morphisms of dg-cooperads to the $\infty$-morphisms of dg-cooperads. Is this the transition to the derived category of dg-cooperads?
|
{}
|
# 상대위상을 이용한 시각적 협응 패턴의 지각 역학과 격자무늬를 이용한 부가적 감각 정보에 따른 영향
• Ryu, Young-Uk (Department of Physical Therapy, Catholic University of Daegu)
• 류영욱 (대구가톨릭대학교 의료과학대학 물리치료학과)
• Accepted : 2012.09.24
• Published : 2012.09.30
• 40 35
#### Abstract
The purpose of the present study was to examine if perception of visual coordination pattern is consistent with the prediction of the HKB model (Haken, Kelso, Bunz, 1985). In addition, this study aimed to see if an additional sensory information using a grid background stabilizes perception of coordination pattern. Participants joined one of two experimental groups, Normal background and Grid background, to participate the pattern recognition training session and the pattern judgment test session. Participants observed $0^{\circ}$, $18^{\circ}$, $36^{\circ}$, $54^{\circ}$, $72^{\circ}$, $90^{\circ}$, $108^{\circ}$, $126^{\circ}$, $144^{\circ}$, $162^{\circ}$, and $180^{\circ}$ coordination patterns characterized by two oscillating dots. The dots oscillated in 0.25 Hz for the pattern recognition training and in 0.5 Hz, 1 Hz, and 2 Hz for the pattern judgment test. Judgment score, absolute judgment error, and judgment stability out of the pattern judgment test were analyzed statistically. The landscape of pattern accuracy and stability data was "inverted-U" shape with slower oscillating frequency conditions. In the faster condition, the accuracy and stability of the judgment decreased with relative phase patterns near $180^{\circ}$. These findings consistent with the prediction of the HKB model. The grid as additional sensory information did not increase accuracy and stability in coordination perception.
#### Keywords
Visual perception;Coordination;Perceptual dynamics;HKB model
#### Acknowledgement
Supported by : 한국연구재단
|
{}
|
# GLMM and Exponential Family
I am studying now GLMMs (Generalized Linear Mixed Models). From my understanding, in order to estimate the parameters of this model, you need to arrive at the marginal probability by integrating a conditional one - which often gives intractable integrals without closed form solutions. So approximation methods are used, such as MC (package glmm), or GHQ or Laplace (package glmmML), etc.
But this implies that the marginal model is no longer in the Exponential family.
My question is - does the limitation of the distribution to an exponential family (which exists for regular GLM's) is still required (in GLMM's)? I see that in the R packages you still need to specify a family, but this could be for simplicity and being-used-to reasons. You could just as easily supply the (custom) pdf function to it.
Also, if numerical methods like GLMM exists, what do we actually gain from "vanilla" GLM's? So ok, there's 1 extra layer of approximation saved for having an explicit pdf instead of an integral. Anything else?
Closely related question - here.
|
{}
|
E. Sleeping
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
One day Vasya was lying in bed watching his electronic clock to fall asleep quicker.
Vasya lives in a strange country, where days have h hours, and every hour has m minutes. Clock shows time in decimal number system, in format H:M, where the string H always has a fixed length equal to the number of digits in the decimal representation of number h - 1. To achieve this, leading zeros are added if necessary. The string M has a similar format, and its length is always equal to the number of digits in the decimal representation of number m - 1. For example, if h = 17, m = 1000, then time equal to 13 hours and 75 minutes will be displayed as "13:075".
Vasya had been watching the clock from h1 hours m1 minutes to h2 hours m2 minutes inclusive, and then he fell asleep. Now he asks you to count how many times he saw the moment at which at least k digits changed on the clock simultaneously.
For example, when switching 04:19 → 04:20 two digits change. When switching 23:59 → 00:00, four digits change.
Consider that Vasya has been watching the clock for strictly less than one day. Note that the last time Vasya saw on the clock before falling asleep was "h2:m2". That is, Vasya didn't see the moment at which time "h2:m2" switched to the next value.
Input
The first line of the input file contains three space-separated integers h, m and k (2 ≤ h, m ≤ 109, 1 ≤ k ≤ 20). The second line contains space-separated integers h1, m1 (0 ≤ h1 < h, 0 ≤ m1 < m). The third line contains space-separated integers h2, m2 (0 ≤ h2 < h, 0 ≤ m2 < m).
Output
Print a single number — the number of times Vasya saw the moment of changing at least k digits simultaneously.
Please do not use the %lld specificator to read or write 64-bit integers in C++. It is preferred to use the cin stream (also you may use the %I64d specificator).
Examples
Input
5 5 24 42 1
Output
3
Input
24 60 10 023 59
Output
1439
Input
24 60 323 5923 59
Output
0
Note
In the first example Vasya will see the following moments of time: 4:4 0:0 → 0:1 → 0:2 → 0:3 → 0:4 1:0 → 1:1 → 1:2 → 1:3 → 1:4 2:0 → 2:1 → 2:2 → 2:3 → 2:4. Double arrow () marks the sought moments of time (in this example — when Vasya sees two numbers changing simultaneously).
In the second example k = 1. Any switching time can be accepted, since during switching of the clock at least one digit is changed. Total switching equals to 24·60 = 1440, but Vasya have not seen one of them — the switching of 23:59 00:00.
In the third example Vasya fell asleep immediately after he began to look at the clock, so he did not see any change.
|
{}
|
# [or-cvs] Tweaks and typos throughout. Nearly there.
Update of /home/or/cvsroot/tor/doc/design-paper
In directory moria.mit.edu:/tmp/cvs-serv21572/tor/doc/design-paper
Modified Files:
challenges.tex
Log Message:
Tweaks and typos throughout. Nearly there.
Index: challenges.tex
===================================================================
RCS file: /home/or/cvsroot/tor/doc/design-paper/challenges.tex,v
retrieving revision 1.55
retrieving revision 1.56
diff -u -d -r1.55 -r1.56
--- challenges.tex 8 Feb 2005 10:46:55 -0000 1.55
+++ challenges.tex 8 Feb 2005 20:34:57 -0000 1.56
@@ -6,11 +6,11 @@
\usepackage{amsmath}
\usepackage{epsfig}
-\setlength{\textwidth}{6in}
-\setlength{\textheight}{8in}
-\setlength{\topmargin}{.5in}
-\setlength{\oddsidemargin}{1cm}
-\setlength{\evensidemargin}{1cm}
+\setlength{\textwidth}{6.1in}
+\setlength{\textheight}{8.5in}
+\setlength{\topmargin}{1cm}
+\setlength{\oddsidemargin}{.5cm}
+\setlength{\evensidemargin}{.5cm}
\newenvironment{tightlist}{\begin{list}{$\bullet$}{
\setlength{\itemsep}{0mm}
@@ -28,7 +28,7 @@
Nick Mathewson\inst{1} \and
Paul Syverson\inst{2}}
\institute{The Free Haven Project \email{<\{arma,nickm\}@freehaven.net>} \and
-Naval Research Lab \email{<syverson@xxxxxxxxxxxxxxxx>}}
+Naval Research Laboratory \email{<syverson@xxxxxxxxxxxxxxxx>}}
\maketitle
\pagestyle{plain}
@@ -77,14 +77,15 @@
funding from diverse sources whose goals range from security on a
national scale down to the liberties of each individual.
-While the Tor design paper~\cite{tor-design} gives an overall view of Tor's
-design and goals, this paper describes some policy, social, and technical
+While~\cite{tor-design} gives an overall view of Tor's
+design and goals, this paper describes policy, social, and technical
issues that we face as we continue deployment.
Rather than trying to provide complete solutions to every problem here, we
lay out the assumptions and constraints that we have observed while
deploying Tor in the wild. In doing so, we aim to create a research agenda
for others to help in addressing these issues. We believe that the issues
-described here will be of general interest to projects attempting to build
+described here will be of general interest to any and all
+projects attempting to build
and deploy practical, useable anonymity networks in the wild.
%While the Tor design paper~\cite{tor-design} gives an overall view its
@@ -132,7 +133,7 @@
each node along the way knows only which node gave it data and which
node it is giving data to. No individual Tor node ever knows the complete
path that a data packet has taken. The client negotiates a separate set
-of encryption keys for each hop along the circuit.% to ensure that each
+of encryption keys for each hop along the circuit. % to ensure that each
%hop can't trace these connections as they pass through.
Because each node sees no more than one hop in the
circuit, neither an eavesdropper nor a compromised node can use traffic
@@ -140,7 +141,7 @@
For efficiency, the Tor software uses the same circuit for all the TCP
connections that happen within the same short period.
Later requests use a new
-circuit, to prevent long-term linkability between different actions by
+circuit, to complicate long-term linkability between different actions by
a single user.
Tor also makes it possible for users to hide their locations while
@@ -152,25 +153,25 @@
Tor attempts to anonymize the transport layer, not the application layer, so
application protocols that include personally identifying information need
additional application-level scrubbing proxies, such as
-Privoxy~\cite{privoxy} for HTTP. Furthermore, Tor does not permit arbitrary
+Privoxy~\cite{privoxy} for HTTP\@. Furthermore, Tor does not permit arbitrary
IP packets; it only anonymizes TCP streams and DNS request, and only supports
connections via SOCKS (see Section~\ref{subsec:tcp-vs-ip}).
Most node operators do not want to allow arbitary TCP connections to leave
their server. To address this, Tor provides \emph{exit policies} so that
each exit node can block the IP addresses and ports it is unwilling to allow.
-TRs advertise their exit policies to the directory servers, so that
+Tor nodes advertise their exit policies to the directory servers, so that
client can tell which nodes will support their connections.
As of January 2005, the Tor network has grown to around a hundred nodes
on four continents, with a total capacity exceeding 1Gbit/s. Appendix A
shows a graph of the number of working nodes over time, as well as a
-vgraph of the number of bytes being handled by the network over time. At
+graph of the number of bytes being handled by the network over time. At
this point the network is sufficiently diverse for further development
and testing; but of course we always encourage and welcome new nodes
to join the network.
-Tor research and development has been funded by the U.S.~Navy and DARPA
+Tor research and development has been funded by ONR and DARPA
for use in securing government
communications, and by the Electronic Frontier Foundation, for use
in maintaining civil liberties for ordinary citizens online. The Tor
@@ -257,8 +258,8 @@
while a stream is still active simply by observing the latency of his
own traffic sent through various Tor nodes. These attacks do not show
the client address, only the first node within the Tor network, making
-helper nodes all the more worthy of exploration (cf.,
-Section~\ref{subsec:helper-nodes}).
+helper nodes all the more worthy of exploration. (See
+Section~\ref{subsec:helper-nodes}.)
Against internal attackers who sign up Tor nodes, the situation is more
complicated. In the simplest case, if an adversary has compromised $c$ of
@@ -277,8 +278,8 @@
(3)~Users do not in fact choose nodes with uniform probability; they
favor nodes with high bandwidth or uptime, and exit nodes that
permit connections to their favorite services.
-See Section~\ref{subsec:routing-zones} for discussion of larger
-adversaries and our dispersal goals.
+(See Section~\ref{subsec:routing-zones} for discussion of how larger
+adversaries affect our dispersal goals.)
%\begin{tightlist}
%\item If the user continues to build random circuits over time, an adversary
@@ -360,10 +361,10 @@
Instead, to protect our networks from traffic analysis, we must
collaboratively blend the traffic from many organizations and private
citizens, so that an eavesdropper can't tell which users are which,
-and who is looking for what information. By bringing more users onto
-the network, all users become more secure~\cite{econymics}.
-[XXX I feel uncomfortable saying this last sentence now. -RD]
-
+and who is looking for what information. %By bringing more users onto
+%the network, all users become more secure~\cite{econymics}.
+%[XXX I feel uncomfortable saying this last sentence now. -RD]
+%[So, I took it out. I think we can do without it. -PFS]
Naturally, organizations will not want to depend on others for their
security. If most participating providers are reliable, Tor tolerates
some hostile infiltration of the network. For maximum protection,
@@ -430,13 +431,12 @@
Tor project's \emph{image} with respect to its users and the rest of
the Internet impacts the security it can provide.
% No image, no sustainability -NM
-
With this image issue in mind, this section discusses the Tor user base and
Tor's interaction with other services on the Internet.
\subsection{Communicating security}
-A growing field of papers argue that usability for anonymity systems
+Usability for anonymity systems
contributes directly to their security, because how usable the system
is impacts the possible anonymity set~\cite{econymics,back01}. Or
conversely, an unusable system attracts few users and thus can't provide
@@ -481,13 +481,15 @@
JAP's cascade-based network topology may be even more vulnerable to these
attacks, because the network has fewer edges. JAP was born out of
the ISDN mix design~\cite{isdn-mixes}, where padding made sense because
-every user had a fixed bandwidth allocation, but in its current context
+every user had a fixed bandwidth allocation and altering the timing
+pattern of packets could be immediately detected, but in its current context
as a general Internet web anonymizer, adding sufficient padding to JAP
-would be prohibitively expensive.\footnote{Even if JAP could
+would be prohibitively expensive and probably ineffective against a
+minimally active attacker.\footnote{Even if JAP could
fund higher-capacity nodes indefinitely, our experience
suggests that many users would not accept the increased per-user
bandwidth requirements, leading to an overall much smaller user base. But
-cf.\ Section \ref{subsec:mid-latency}.} Therefore, since under this threat
+cf.\ Section~\ref{subsec:mid-latency}.} Therefore, since under this threat
model the number of concurrent users does not seem to have much impact
on the anonymity provided, we suggest that JAP's anonymity meter is not
accurately communicating security levels to its users.
@@ -611,9 +613,9 @@
giving billing cycle, to become dormant once its bandwidth is exhausted, and
to reawaken at a random offset into the next billing cycle. This feature has
interesting policy implications, however; see
-Section~\ref{subsec:bandwidth-and-file-sharing} below.
+the next section below.
Exit policies help to limit administrative costs by limiting the frequency of
-abuse complaints.
+abuse complaints. (See Section~\ref{subsec:tor-and-blacklists}.)
%[XXXX say more. Why else would you run a node? What else can we do/do we
% already do to make running a node more attractive?]
@@ -696,6 +698,7 @@
%your computer is doing that behavior.
\subsection{Tor and blacklists}
+\label{subsec:tor-and-blacklists}
It was long expected that, alongside Tor's legitimate users, it would also
attract troublemakers who exploited Tor in order to abuse services on the
@@ -730,7 +733,7 @@
protected entities of the world.
Worse, many IP blacklists are not terribly fine-grained.
-No current IP blacklist, for example, allow a service provider to blacklist
+No current IP blacklist, for example, allows a service provider to blacklist
only those Tor nodes that allow access to a specific IP or port, even
though this information is readily available. One IP blacklist even bans
every class C network that contains a Tor node, and recommends banning SMTP
@@ -758,7 +761,7 @@
But of course, we would prefer that legitimate anonymous users be able to
access abuse-prone services. One conceivable approach would be to require
would-be IRC users, for instance, to register accounts if they wanted to
-access the IRC network from Tor. But in practise, this would not
+access the IRC network from Tor. In practise this would not
significantly impede abuse if creating new accounts were easily automatable;
this is why services use IP blocking. In order to deter abuse, pseudonymous
identities need to require a significant switching cost in resources or human
@@ -908,14 +911,21 @@
harness this increased latency to improve anonymity rather than just
reduce usability. Further, if we let clients label certain circuits as
mid-latency as they are constructed, we could handle both types of traffic
-on the same network, giving users a choice between speed and security.
+on the same network, giving users a choice between speed and security---and
+giving researchers a chance to experiment with parameters to improve the
+quality of those choices.
\subsection{Enclaves and helper nodes}
\label{subsec:helper-nodes}
It has long been thought that the best anonymity comes from running your
-own node~\cite{tor-design,or-pet00}. This is called using Tor in an
-\emph{enclave} configuration. Of course, Tor's default path length of
+own node~\cite{tor-design,or-ih96,or-pet00}. This is called using Tor in an
+\emph{enclave} configuration. By running Tor clients only on Tor nodes
+at the enclave perimeter, enclave configuration can also permit anonymity
+protection even when policy or other requiremnts prevent individual machines
+within the enclave from running Tor clients~\cite{or-jsac98,or-discex00}.
+
+Of course, Tor's default path length of
three is insufficient for these enclaves, since the entry and/or exit
themselves are sensitive. Tor thus increments the path length by one
for each sensitive endpoint in the circuit.
@@ -1034,14 +1044,14 @@
But how do we decide whether two nodes are in related locations?
Feamster and Dingledine defined a \emph{location diversity} metric
-in \cite{feamster:wpes2004}, and began investigating a variant of location
+in~\cite{feamster:wpes2004}, and began investigating a variant of location
diversity based on the fact that the Internet is divided into thousands of
independently operated networks called {\em autonomous systems} (ASes).
The key insight from their paper is that while we typically think of a
-connection as going directly from the Tor client to her first Tor node,
+connection as going directly from the Tor client to the first Tor node,
actually it traverses many different ASes on each hop. An adversary at
any of these ASes can monitor or influence traffic. Specifically, given
-plausible initiators and recipients and path random path selection,
+plausible initiators and recipients, and given random path selection,
some ASes in the simulation were able to observe 10\% to 30\% of the
transactions (that is, learn both the origin and the destination) on
the deployed Tor network (33 nodes as of June 2004).
@@ -1049,10 +1059,10 @@
The paper concludes that for best protection against the AS-level
adversary, nodes should be in ASes that have the most links to other ASes:
Tier-1 ISPs such as AT\&T and Abovenet. Further, a given transaction
-is safest when it starts or ends in a Tier-1 ISP. Therefore, assuming
+is safest when it starts or ends in a Tier-1 ISP\@. Therefore, assuming
initiator and responder are both in the U.S., it actually \emph{hurts}
-our location diversity to add far-flung nodes in continents like Asia
-or South America.
+our location diversity to enter or exit from far-flung nodes in
+continents like Asia or South America.
Many open questions remain. First, it will be an immense engineering
challenge to get an entire BGP routing table to each Tor client, or to
@@ -1071,7 +1081,8 @@
Google~\cite{shsm03}? (Note that they're also well-positioned as global
%
-Third, if we follow the paper's recommendations and tailor path selection
+Third, if we follow the recommendations in~\cite{feamster:wpes2004}
+ and tailor path selection
to avoid choosing endpoints in similar locations, how much are we hurting
anonymity against larger real-world adversaries who can take advantage
of knowing our algorithm?
@@ -1150,7 +1161,7 @@
Since the speed and reliability of a circuit is limited by its worst link,
we must learn to track and predict performance. Finally, in order to get
a large set of nodes in the first place, we must address incentives
-for users to carry traffic for others (see Section incentives).
+for users to carry traffic for others.
\subsection{Incentives by Design}
@@ -1168,10 +1179,9 @@
deniability for any traffic emerging from the same address as a Tor
exit node, and they can use their own Tor node
as entry or exit point and be confident it's not run by the adversary.
-Further, users who need to be able to communicate anonymously
-may run a node simply because their need to increase
-expectation that such a network continues to be available to them
-and usable exceeds any countervening costs.
+Further, users may run a node simply because they need such a network
+to be persistently available and usable.
+And, the value of supporting this exceeds any countervening costs.
Finally, we can improve the usability and feature set of the software:
rate limiting support and easy packaging decrease the hassle of
maintaining a node, and our configurable exit policies allow each
@@ -1197,8 +1207,8 @@
by performing well or can provide targeted differential performance to
individual users to undermine their anonymity. Typically a user who
chooses evenly from all options is most resistant to an adversary
-targeting him, but that approach prevents from handling heterogeneous
-nodes.
+targeting him, but that approach precludes the efficient use
+of heterogeneous nodes.
%When a node (call him Steve) performs well for Alice, does Steve gain
%reputation with the entire system, or just with Alice? If the entire
@@ -1236,14 +1246,15 @@
The published Tor design adopted a deliberately simplistic design for
authorizing new nodes and informing clients about Tor nodes and their status.
-In the early Tor designs, all nodes periodically uploaded a signed description
+In preliminary Tor designs, all nodes periodically uploaded a
+signed description
of their locations, keys, and capabilities to each of several well-known {\it
directory servers}. These directory servers constructed a signed summary
of all known Tor nodes (a directory''), and a signed statement of which
nodes they
believed to be operational at any given time (a network status''). Clients
periodically downloaded a directory in order to learn the latest nodes and
-keys, and more frequently downloaded a network status to learn which nodes are
+keys, and more frequently downloaded a network status to learn which nodes were
likely to be running. Tor nodes also operate as directory caches, in order to
lighten the bandwidth on the authoritative directory servers.
@@ -1258,7 +1269,7 @@
approve any Tor node whose operator could compose a coherent email.
This procedure
may have prevented trivial automated Sybil attacks, but would do little
-against a clever attacker.
+against a clever and determined attacker.
There are a number of flaws in this system that need to be addressed as we
move forward. They include:
@@ -1283,7 +1294,7 @@
adopt even stricter validation requirements, and reduce the number of
nodes in the network to a trusted minimum.
But, we can only do that if can simultaneously make node capacity
-scale much more than we anticipate feasible soon, and if we can find
+scale much more than we anticipate to be feasible soon, and if we can find
entities willing to run such nodes, an equally daunting prospect.
@@ -1355,7 +1366,8 @@
\subsection{Non-clique topologies}
-Tor's comparatively weak model makes it easier to scale than other mix net
+Tor's comparatively weak threat model makes it easier to scale than
+other mix net
designs. High-latency mix networks need to avoid partitioning attacks, where
network splits prevent users of the separate partitions from providing cover
for each other. In Tor, however, we assume that the adversary cannot
@@ -1381,7 +1393,7 @@
used by each node. The number of sockets is determined by the network's
connectivity and the number of users, while bandwidth capacity is determined
by the total bandwidth of nodes on the network. The simplest solution to
-bandwidth capacity is to add more nodes, since adding a tor node of any
+bandwidth capacity is to add more nodes, since adding a Tor node of any
feasible bandwidth will increase the traffic capacity of the network. So as
a first step to scaling, we should focus on making the network tolerate more
nodes, by reducing the interconnectivity of the nodes; later we can reduce
@@ -1403,7 +1415,7 @@
To make matters simpler, Tor may not need an expander graph per se: it
may be enough to have a single subnet that is highly connected. As an
example, assume fifty nodes of relatively high traffic capacity. This
-\emph{center} forms are a clique. Assume each center node can each
+\emph{center} forms a clique. Assume each center node can
handle 200 connections to other nodes (including the other ones in the
center). Assume every noncenter node connects to three nodes in the
center and anyone out of the center that they want to. Then the
@@ -1413,16 +1425,16 @@
be given to any new nodes with their codebase), whether center nodes
will need to function as a backbone', etc. As above the point is
that this would create problems for the expected anonymity for a mixnet,
-but for an onion routing network where anonymity derives largely from
+but for a low-latency network where anonymity derives largely from
the edges, it may be feasible.
Another point is that we already have a non-clique topology.
Individuals can set up and run Tor nodes without informing the
directory servers. This will allow, e.g., dissident groups to run a
local Tor network of such nodes that connects to the public Tor
-network. This network is hidden behind the Tor network and its
-only visible connection to Tor at those points where it connects.
-As far as the public network is concerned or anyone observing it,
+network. This network is hidden behind the Tor network, and its
+only visible connection to Tor is at those points where it connects.
+As far as the public network, or anyone observing it, is concerned,
they are running clients.
\section{The Future}
@@ -1442,7 +1454,7 @@
network on the Internet are starting to adapt Tor to their needs.
%
Second, Tor is only one of many components that preserve privacy online.
-To keep identifying information out of application traffic, we must build
+To keep identifying information out of application traffic, someone must build
more and better protocol-aware proxies that are usable by ordinary people.
%
Third, we need to gain a reputation for social good, and learn how to
`
|
{}
|
# Iitians.. plz answer this as soon as possible.. related to mechanics..
25 Points
one year ago
B] The 2 forces must act on different bodies.
Explanation:
• Newton's third law of motion states that 'Every action has an equal and opposite reaction force.'
• These forces must act on the same body and not on different bodies.
So, option B is the incorrect answer and hence is the correct option.
$\underbrace{\overbrace {Hope \: it \: helps }}$
|
{}
|
# SCM Repository
[easyabc] View of /vignettes/EasyABC.Rnw
[easyabc] / vignettes / EasyABC.Rnw
# View of /vignettes/EasyABC.Rnw
Fri Jun 24 13:51:22 2016 UTC (2 years, 6 months ago) by dumoulin
File size: 46255 byte(s)
new dev version tagging
\documentclass[a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{amssymb}
\usepackage{url}
\usepackage{a4wide}
\title{\texttt{EasyABC}: a \texttt{R} package to perform efficient approximate Bayesian computation sampling schemes}
\author{Franck Jabot, Thierry Faure, Nicolas Dumoulin, Carlo Albert}
\date{\texttt{EasyABC} version 1.5.99, \Sexpr{Sys.Date()} }
\SweaveOpts{echo=TRUE,print=TRUE}
%\SweaveOpts{eval=FALSE}
\begin{document}
%\SweaveOpts{concordance=TRUE}
\maketitle
\tableofcontents
\setcounter{footnote}{1} \footnotetext{This document is included as a
vignette (a \LaTeX\ document created using the \texttt{R} function
\texttt{Sweave}) of the package \texttt{EasyABC}. It is automatically
dowloaded together with the package and can be accessed through \texttt{R}
typing \texttt{vignette("EasyABC")}.} \newpage
\section{Summary}
The aim of this vignette is to present the features of the \texttt{EasyABC} package.
Section \ref{algorithms} describes the different algorithms available in the package.
Section \ref{installation} details how to install the package and the formatting requirements.
Sections \ref{example1} and \ref{example} present two detailed worked examples.
\section{Overview of the package EasyABC}
\label{algorithms}
\texttt{EasyABC} enables to launch various ABC schemes and to retrieve the ouputs of the simulations, so as to perform post-processing treatments with the various R tools available. \texttt{EasyABC} is also able to launch the simulations on multiple cores of a multi-core computer.
Four main types of ABC schemes are available in EasyABC: the standard rejection algorithm of Pritchard et al. (1999), sequential schemes first proposed by Sisson et al. (2007), coupled to MCMC sequential schemes first proposed by Marjoram et al. (2003), and a Simulated Annealing algorithm (SABC) suggested in Albert et al. (2014).
Four different sequential algorithms are available: the ones of Beaumont et al. (2009), Drovandi and Pettitt (2011), Del Moral et al. (2012) and Lenormand et al. (2012).
Four different MCMC schemes are available: the ones of Marjoram et al. (2003), Wegmann et al. (2009a), a modification of Marjoram et al. (2003)'s algorithm in which the tolerance and proposal range are determined by the algorithm, following the modifications of Wegmann et al. (2009a).
Details on how to implement these various algorithms with \texttt{EasyABC} are given in the manual pages of each function and two examples are detailed in Sections \ref{example1} and \ref{example}. We provide below a short presentation of each implemented algorithm.
\subsection{The standard rejection algorithm of Pritchard et al. (1999)}
This sampling scheme consists in drawing the model parameters in the prior distributions, in using these model parameter values to launch a simulation and in repeating this two-step procedure \texttt{nb\_simul} times.
At the end of the \texttt{nb\_simul} simulations, the simulations closest to the target (or at a distance smaller than a tolerance threshold) in the space of the summary statistics are retained to form an approximate posterior distribution of the model parameters.
\subsection{Sequential algorithms}
Sequential algorithms for ABC have first been proposed by Sisson et al. (2007). These algorithms aim at reducing the required number of simulations to reach a given quality of the posterior approximation.
The underlying idea of these algorithms is to spend more time in the areas of the parameter space where simulations are frequently close to the target.
Sequential algorithms consist in a first step of standard rejection ABC, followed by a number of steps where the sampling of the parameter space is not anymore performed according to the prior distributions of parameter values.
Various ways to perform this biased sampling have been proposed, and four of them are implemented in the package \texttt{EasyABC}.
\subsection{Coupled to MCMC algorithms}
The idea of ABC-MCMC algorithms proposed by Marjoram et al. (2003) is to perform a Metropolis-Hastings algorithm to explore the parameter space, and in replacing the step of likelihood ratio computation by simulations of the model.
The original algorithm of Marjoram et al. (2003) is implemented in the method "Marjoram\_original" in \texttt{EasyABC}.
Wegmann et al. (2009) later proposed a number of improvements to the original scheme of Marjoram et al. (2003): they proposed to perform a calibration step so that the algorithm automatically determines the tolerance threshold, the scaling of the summary statistics and the scaling of the jumps in the parameter space during the MCMC.
These improvements have been implemented in the method "Marjoram".
Wegmann et al. (2009) also proposed additional modifications, among which a PLS transformation of the summary statistics. The complete Wegmann et al. (2009)'s algorithm is implemented in the method "Wegmann".
\subsection{Simulated annealing}
Inspired by Simulated Annealing algorithms used for optimization, the SABC algorithm from Albert et al. (2014) propagates an ensemble of particles in the product space of parameters and model outputs and continuously lowers the tolerance between model outputs and the data so that the parameter marginal converges to the posterior. The tolerance is lowered adaptively so as to minimize entropy production, which serves as a measure for computational waste.
In \texttt{EasyABC}, SABC is implemented in the function \texttt{SABC}.
\section{Installation and requirements}
\label{installation}
\subsection{Installing the package}
A version of R greater than or equal to 2.15.0 is required. The package has been tested on Windows 32 and Linux, but not on Mac. To install the \texttt{EasyABC} package from \texttt{R}, simply type:
<<eval=FALSE>>=
install.packages("EasyABC")
@
Once the package is installed, it needs to be loaded in the current \texttt{R} session to be used:
<<print=FALSE>>=
library(EasyABC)
@
<<eval=FALSE>>=
help(package="EasyABC")
@
For online help on a particular command (such as the function \texttt{ABC\_sequential}), simply type:
<<eval=FALSE>>=
help(ABC_sequential)
@
\subsection{The simulation code - for use on a single core}
\label{simulator_single_core}
Users need to develop a simulation code with minimal compatibility constraints. The code can either be a \texttt{R} function or a binary executable file.
If the code is a \texttt{R} function, its argument must be a vector of parameter values and it must return a vector of summary statistics. If the option \texttt{use\_seed=TRUE} is chosen, the first parameter value passed to the simulation code corresponds to the seed value to be used by the simulation code to initialize the pseudo-random number generator. The following parameters are the model parameters.
If the code is a binary executable file, it needs to read the parameter values in a file named 'input' in which each line contains one parameter value, and to output the summary statistics in a file named 'output' in which each summary statistics must be separated by a space or a tabulation.
If the code is a binary executable file, a wrapper \texttt{R} function named 'binary\_model' is available to interface the executable file with the \texttt{R} functions of the \texttt{EasyABC} package (see section \ref{example} below).
Alternatively, users may prefer building a \texttt{R} function calling their binary executable file. A short tutorial is provided in section \ref{RC_link} to call a \texttt{C/C++} program.
\textit{NB:} Currently, SABC ignores the \texttt{use\_seed} option and requires a function, whose first argument is the parameter vector.
\subsection{The simulation code - for use with multiple cores}
\label{simulator_several_cores}
Users need to develop a simulation code with minimal compatibility constraints. The code can either be a \texttt{R} function or a binary executable file.
If the code is a \texttt{R} function, its argument must be a vector of parameter values and it must return a vector of summary statistics. The first parameter value passed to the simulation code corresponds to the seed value to be used by the simulation code to initialize the pseudo-random number generator. The following parameters are the model parameters. This means that the option \texttt{use\_seed} must be turned to \texttt{TRUE} when using \texttt{EasyABC} with multiple cores.
If the code is a binary executable file, it needs to have as its single argument a positive integer \texttt{k}. It has to read the parameter values in a file named 'inputk' (where k is the integer passed as argument to the binary code: 'input1', 'input2'...) in which each line contains one parameter value, and to output the summary statistics in a file named 'outputk' (where k is the integer passed as argument to the binary code: 'output1', 'output2'...) in which each summary statistics must be separated by a space or a tabulation.
This construction avoids multiple cores to read/write in the same files.
If the code is a binary executable file, a wrapper \texttt{R} function named 'binary\_model\_cluster' is available to interface the executable file with the \texttt{R} functions of the \texttt{EasyABC} package (see section \ref{example} below).
Alternatively, users may prefer building a \texttt{R} function calling their binary executable file. A short tutorial is provided in section \ref{RC_link} to call a \texttt{C/C++} program.
\textit{NB:} Currently, SABC does currently not support the usage of multiple cores.
\subsection{Management of pseudo-random number generators}
To insure that stochastic simulations are independent, the simulation code must either possess an internal way of initializing the seeds of its pseudo-random number generators each time the simulation code is launched.
This can be achieved for instance by initializing the seed to the clock value.
It is often desirable though to have a way to re-run some analyses with similar seed values.
%#\texttt{EasyABC} offers this possibility by default with the default option \texttt{use\_seed=TRUE,seed\_count=0} where \texttt{seed\_count} can be any integer number.
If this option is chosen, a seed value is provided in the input file as a first (additional) parameter, and incremented by 1 at each call of the simulation code.
This means that the simulation code must be designed so that the first parameter is a seed initializing value.
In the worked example (Section \ref{example}), the simulation code \texttt{trait\_model} makes use of this package option, and in the first example (Section \ref{example1}), the way this option can be used with a simple \texttt{R} function is demonstrated.
\textit{NB:} Note that when using multicores with the package functions (\texttt{n\_cluster=x} with \texttt{x} larger than 1), the option \texttt{use\_seed=TRUE} is forced, since the seed value is also used to distribute the tasks to each core.
\subsection{Encoding the prior distributions}
A list encoding the prior distributions used for each model parameter must be supplied by the user.
Each element of the list corresponds to a model parameter and can be defined in two ways:
\begin{enumerate}
\item By using predefined prior distributions. In this case, the list element must be a vector whose first argument determines the type of prior distribution followed by the argument of the distribution function, possible values are:
\begin{itemize}
\item "unif" for a uniform distribution on a segment, followed by two numbers the minimum and maximum values of the uniform distribution
\item "normal" for a normal distribution, followed by two numbers the mean and standard deviation of the normal distribution
\item "lognormal" for a lognormal distribution, followed by two numbers: the mean and standard deviation on the log scale of the lognormal distribution
\item "exponential" for an exponential distribution, followed by one number: the rate of the exponential distribution
<<simple_prior>>=
my_prior=list(c("unif",0,1),c("normal",1,2))
@
\end{itemize}
\textit{NB:} Note that a fixed variable can be passed to the simulation code by choosing for this fixed variable a uniform prior distribution and a trivial range (with equal lower and upper bounds).
The \texttt{EasyABC} methods will not work properly if these fixed variables are passed with other types of prior distributions (like a normal distribution with a standard deviation equal to zero).
\item By providing the user-defined sampling and density function. In this case, each list element must be itself a list of two elements: the sampling function and the density function. For example, a uniform distribution can be defined using this approach with the following code (equivalent to \texttt{my\_prior=list(c("unif",0,1))}):
<<custom_prior>>=
my_prior=list(list(c("runif",1,0,1), c("dunif",0,1)))
@
\end{enumerate}
\textit{NB:} SABC requires the prior to be specified as a sampler and as a density (see the examples below).
To add constraints to prior distributions (for instance, parameter 1 < parameter 2), users need to use the parameter \texttt{prior\_test} in the ABC functions of the package (see their online documentation).
This parameter \texttt{prior\_test} will be evaluated as a logical expression, you can use all the logical operators including \texttt{"<"}, \texttt{">"}, \ldots to define whether a parameter set respects the constraint.
Each parameter should be designated with \texttt{"X1"}, \texttt{"X2"}, \ldots in the same order as in the prior definition.
Here is an example where the second parameter should be greater than the first one:
\begin{verbatim}
prior = list(c("unif",0,1),c("unif",0,10))
ABC_rejection(model=a_model,prior=prior,nb_simul=3, prior_test="X2 > X1")
\end{verbatim}
\subsection{The target summary statistics}
A vector containing the summary statistics of the data must be supplied. The statistics must be in the same order as in the simulation outputs.
The function \texttt{SABC} allows for a semi-automatic generation of summary statistics according to Fearnhead et al. (2012).
\subsection{The option verbose}
Intermediary results can be written in output files in the working directory. Users solely need to choose the option \texttt{verbose=TRUE} when launching the \texttt{EasyABC} functions (otherwise, the default value for \texttt{verbose} is \texttt{FALSE}).
Intermediary results consist in the progressive writing of simulation outputs for the functions \texttt{ABC\_rejection} and \texttt{ABC\_mcmc} and in the writing of intermediary results at the end of each step for the function \texttt{ABC\_sequential}. Additional details are provided in the help files of the functions.
\subsection{Building a \texttt{R} function calling a \texttt{C/C++} program}
Users having a \texttt{C/C++} simulation code may wish to construct a \texttt{R} function calling their \texttt{C/C++} program, instead of using the provided wrappers (see sections \ref{simulator_single_core} and \ref{simulator_several_cores}).
The procedure is abundantly described in the \href{http://cran.r-project.org/doc/manuals/R-exts.html}{Writing R Extensions' manual}.
In short, this can be done by:
\begin{itemize}
\item Adapt your C/C++ program by wrapping your main method into a \texttt{extern "C" \{ … \}} block. Here is an excerpt of the source code of the trait model provided in this package, in the folder \texttt{src}:
\begin{verbatim}
extern "C" {
void trait_model(double *input,double *stat_to_return){
// compute output and fill the array stat_to_return
}
}
\end{verbatim}
\item Build your code into a binary library (.so under Linux or .dll under Windows) with the \texttt{R CMD SHLIB} command.
In our example, the command for compiling the trait model and the given output are:
\begin{verbatim}
$R CMD SHLIB trait_model_rc.cpp g++ -I/usr/share/R/include -DNDEBUG -fpic -O2 -pipe -g -c trait_model_rc.cpp -o trait_model_rc.o g++ -shared -o trait_model_rc.so trait_model_rc.o -L/usr/lib/R/lib -lR \end{verbatim} \item Load the builded library in your session with the \texttt{dyn.load} function. \begin{verbatim} > dyn.load("trait_model_rc.so") \end{verbatim} \item Use the \texttt{.C} function for calling your program, like we've done in our \texttt{trait\_model} function: \begin{verbatim} trait_model <- function(input=c(1,1,1,1,1,1)) { .C("trait_model",input=input,stat_to_return=array(0,4))$stat_to_return
}
\end{verbatim}
Now, as our model will have two parameters with constant values (see \ref{example}), we can fix them as following:
\begin{verbatim}
trait_model <- function(input=c(1,1,1,1,1,1)) {
.C("trait_model",input=c(input[1], 500, input[2:3], 1, input[4:5]),
stat_to_return=array(0,4))$stat_to_return } \end{verbatim} \end{itemize} \subsection{Example of integration of an external program: \texttt{fastsimcoal}} \label{simcoal} This example is provided by an EasyABC user Albert Min-Shan Ko (currently at the Department of genetics, Max Planck Institute of Evolutionary Anthropology, Leipzig, Germany). The purpose is to plug a third-party software related to population genetics into the EasyABC workflow. This software needs input data in a given format, so the idea is to wrap the call to the \texttt{fastsimcoal} software into a script that will link EasyABC to \texttt{fastsimcoal}. Here are the scripts as provided by courtesy of Albert Min-Shan Ko. \begin{itemize} \item First, a R script reformats the parameters to be used by \texttt{fastsimcoal} (here named \texttt{mod.input.r}). \begin{verbatim} r<-read.table('input',head=F) sink('mod.input') cat(paste('1','p1','unif',round(r[1,],0),round(r[1,],0),sep='\t')) cat('\n') cat(paste('1','p2','unif',round(r[2,],0),round(r[2,],0),sep='\t')) cat('\n') cat(paste('1','p3','unif',round(r[3,],0),round(r[3,],0),sep='\t')) sink() \end{verbatim} \item Second, a GNU Bash script (here names \texttt{run\_sim.sh}) invokes the latter R script and builds a parameter file for \texttt{fastsimcoal} (\texttt{sim.est}), runs \texttt{fastsimcoal} and computes some summary statistics with the arlequin program. \begin{verbatim} #!/bin/bash rm -fr sim Rscript mod.input.r cat <(sed -n 1p template.est) <(sed -n '1,3'p mod.input) \ <(sed -n '5,\$'p template.est) > sim.est
until [ -f sim/arl_output ]; do
./fastsimcoal -t sim.tpl -e sim.est -E1 -n1 -q
./arlsumstat sim/sim_1_1.arp sim/arl_output 1 0 run_silent
done
cat sim/arl_output > output
\end{verbatim}
\end{itemize}
Then, the user can invoke EasyABC like this :
\begin{verbatim}
prior=list(c("unif",500,1000),c("unif",100,500),c("unif",50,200))
ABC_sim<-ABC_rejection(model=binary_model('./run_sim.sh'),prior=prior,nb_simul=3)
\end{verbatim}
\subsection{Example of integration of a java model}
If your model runs with a Java Virtual Machine (can be written in Java, Scala, Groovy, …), you can of course use the \texttt{binary\_model} wrapper to run the JVM within your model.
But, you can achieve a tighter integration that will simplify the process and save computing time.
This section propose to use the R package \texttt{rJava}.
Let's consider the toy model written in Java (in a file named Model.java):
\begin{verbatim}
public class Model {
public static double[] run(double[] x) {
double[] result = new double[2];
result[0] = x[0] + x[1];
result[1] = x[0] * x[1];
return result;
}
}
\end{verbatim}
We can compile it with the command: \texttt{javac Model.java} and then define our wrapper in R:
\begin{verbatim}
mymodel <- function(x) {
library("rJava")
.jinit(classpath=".")
result = .jcall(J("Model"),"[D","run",.jarray(x))
result
}
\end{verbatim}
Then, the user can invoke EasyABC like this :
\begin{verbatim}
prior=list(c("unif",0,1),c("normal",1,2))
ABC_sim<-ABC_rejection(model=mymodel,prior=prior,nb_simul=3)
\end{verbatim}
\section{A first worked example}
\label{example1}
\subsection{The toy model}
We here consider a very simple stochastic model coded in the \texttt{R} language:
<<toy_model>>=
toy_model<-function(x){
c( x[1] + x[2] + rnorm(1,0,0.1) , x[1] * x[2] + rnorm(1,0,0.1) )
}
@
We will use two different types of prior distribution for the two model parameters ($x[1]$ and $x[2]$): a uniform distribution between 0 and 1 and a normal distribution with mean 1 and standard deviation 2.
<<toy_prior>>=
toy_prior=list(c("unif",0,1),c("normal",1,2))
@
And we will consider an imaginary dataset of two summary statistics that the toy\_model is aiming at fitting:
<<sum_stat_obs>>=
sum_stat_obs=c(1.5,0.5)
@
\subsection{Performing a standard ABC-rejection procedure}
A standard ABC-rejection procedure can be simply performed with the function \texttt{ABC\_rejection}, in precising the number $n$ of simulations to be performed and the proportion of simulations which are to be retained $p$:
<<ABC_rejection>>=
set.seed(1)
n=10
p=0.2
ABC_rej<-ABC_rejection(model=toy_model, prior=toy_prior, nb_simul=n,
summary_stat_target=sum_stat_obs, tol=p)
@
Alternatively, \texttt{ABC\_rejection} can be used to solely launch the simulations and to store the simulation outputs without performing the rejection step.
This option enables the user to make use of the \texttt{R} package \texttt{abc} (Csill\'ery et al. 2012) which offers an array of more sophisticated post-processing treatments than the simple rejection procedure:
<<ABC_rejection>>=
# Run the ABC rejection on the model
set.seed(1)
n=10
ABC_rej<-ABC_rejection(model=toy_model, prior=toy_prior, nb_simul=n)
@
<<abcinstall,eval=FALSE>>=
# Install if needed the "abc" package
install.packages("abc")
@
<<abc>>=
# Post-process the simulations outputs
library(abc)
rej<-abc(sum_stat_obs, ABC_rej$param, ABC_rej$stats, tol=0.2, method="rejection")
# simulations selected:
rej$unadj.values # their associated summary statistics: rej$ss
# their normalized euclidean distance to the data summary statistics:
rej$dist @ \subsection{Performing a sequential ABC scheme} Other functions of the \texttt{EasyABC} package are used in a very similar manner. To perform the algorithm of Beaumont et al. (2009), one needs to specify the sequence of tolerance levels$tolerance\_tab$and the number$nb\_simul$of simulations to obtain below the tolerance level at each iteration: <<ABC_Beaumont>>= n=10 tolerance=c(1.25,0.75) ABC_Beaumont<-ABC_sequential(method="Beaumont", model=toy_model, prior=toy_prior, nb_simul=n, summary_stat_target=sum_stat_obs, tolerance_tab=tolerance) @ To perform the algorithm of Drovandi and Pettitt (2011), one needs to specify four arguments: the initial number of simulations$nb\_simul$, the final tolerance level$tolerance\_tab$, the proportion$\alpha$of best-fit simulations to update the tolerance level at each step, and the target proportion$c$of unmoved particles during the MCMC jump. Note that default values$alpha=0.5$and$c=0.01$are used if not specified, following Drovandi and Pettitt (2011). <<ABC_Drovandi>>= n=10 tolerance=0.75 c_drov=0.7 ABC_Drovandi<-ABC_sequential(method="Drovandi", model=toy_model, prior=toy_prior, nb_simul=n, summary_stat_target=sum_stat_obs, tolerance_tab=tolerance, c=c_drov) @ To perform the algorithm of Del Moral et al. (2012), one needs to specify five arguments: the initial number of simulations$nb\_simul$, the number$\alpha$controlling the decrease in effective sample size of the particle set at each step, the number$M$of simulations performed for each particle, the minimal effective sample size$nb\_threshold$below which a resampling of particles is performed and the final tolerance level$tolerance\_target$. Note that default values$alpha=0.5$,$M=1$and$nb\_threshold=nb\_simul/2$are used if not specified. <<ABC_Delmoral>>= n=10 alpha_delmo=0.5 tolerance=0.75 ABC_Delmoral<-ABC_sequential(method="Delmoral", model=toy_model, prior=toy_prior, nb_simul=n, summary_stat_target=sum_stat_obs, alpha=alpha_delmo, tolerance_target=tolerance) @ To perform the algorithm of Lenormand et al. (2012), one needs to specify three arguments: the initial number of simulations$nb\_simul$, the proportion$\alpha$of best-fit simulations to update the tolerance level at each step, and the stopping criterion$p\_acc\_min$. Note that default values$alpha=0.5$and$p\_acc\_min=0.05$are used if not specified, following Lenormand et al. (2012). Also note that the method "Lenormand" is only supported with uniform prior distributions (since it performs a Latin Hypercube sampling at the beginning). Here, we therefore need to alter the prior distribution of the second model parameter: <<toy_prior2>>= toy_prior2=list(c("unif",0,1),c("unif",0.5,1.5)) @ <<ABC_Lenormand>>= n=10 pacc=0.4 ABC_Lenormand<-ABC_sequential(method="Lenormand", model=toy_model, prior=toy_prior2, nb_simul=10, summary_stat_target=sum_stat_obs, p_acc_min=pacc) @ \subsection{Performing a ABC-MCMC scheme} To perform the algorithm of Marjoram et al. (2003), one needs to specify five arguments: the number of sampled points$n\_rec$in the Markov Chain, the number of chain points between two sampled points$n\_between\_sampling$, the maximal distance accepted between simulations and data$dist\_max$, a vector$tab\_normalization$precising the scale of each summary statistics, and a vector$proposal\_range$precising the maximal distances in each dimension of the parameter space for a jump of the MCMC. All these arguments have default values (see the package help for the function \texttt{ABC\_mcmc}), so that \texttt{ABC\_mcmc} will work without user-defined values. <<ABC_Marjoram_original>>= n=10 ABC_Marjoram_original<-ABC_mcmc(method="Marjoram_original", model=toy_model, prior=toy_prior, summary_stat_target=sum_stat_obs, n_rec=n) @ To perform the algorithm of Marjoram et al. (2003) in which some of the arguments ($dist\_max$,$tab\_normalization$and$proposal\_range$) are automatically determined by the algorithm via an initial calibration step, one needs to specify three arguments: the number$n\_calibration$of simulations to perform at the calibration step, the tolerance quantile$tolerance\_quantile$to be used for the determination of$dist\_max$and the scale factor$proposal\_phi$to determine the proposal range. These modifications are drawn from the algorithm of Wegmann et al. (2009a), without relying on PLS regressions. The arguments are set by default to:$n\_calibration=10000$,$tolerance\_quantile=0.01$and$proposal\_phi=1$. This way of automatic determination of$dist\_max$,$tab\_normalization$and$proposal\_range$is strongly recommended, compared to the crude automatic determination proposed in the method \texttt{Marjoram\_original}. <<ABC_Marjoram>>= n=10 ABC_Marjoram<-ABC_mcmc(method="Marjoram", model=toy_model, prior=toy_prior, summary_stat_target=sum_stat_obs, n_rec=n) @ To perform the algorithm of Wegmann et al. (2009a), one needs to specify four arguments: the number$n\_calibration$of simulations to perform at the calibration step, the tolerance quantile$tolerance\_quantile$to be used for the determination of$dist\_max$, the scale factor$proposal\_phi$to determine the proposal range and the number of components$numcomp$to be used in PLS regressions. The arguments are set by default to:$n\_calibration=10000$,$tolerance\_quantile=0.01$,$proposal\_phi=1$and$numcomp=0$, this last default value encodes a choice of a number of PLS components equal to the number of summary statistics. <<ABC_Wegmann>>= n=10 ABC_Wegmann<-ABC_mcmc(method="Wegmann", model=toy_model, prior=toy_prior, summary_stat_target=sum_stat_obs, n_rec=n) @ \subsection{Performing a SABC scheme} For the SABC algorithm by Albert et al. (2014) we need to provide the prior in the form of a sampler and a density: <<SABCPrior>>= r.prior <- function() c(runif(1,0,1),rnorm(1,1,2)) d.prior <- function(x) dunif(x[1],0,1)*dnorm(x[2],1,2) @ Furthermore, we need to specify the size of the ensemble, the number of simulations and the initial tolerance <<SABCParam>>= n.sample <- 300 iter.max <- n.sample * 30 eps.init <- 2 @ Since, for this example, the prior carries relevant information, we choose the method "informative": <<SABC,print=FALSE>>= ABC_Albert <-SABC(r.model = toy_model, r.prior = r.prior, d.prior = d.prior, n.sample = n.sample, eps.init = eps.init, iter.max = iter.max, method = "informative", y = sum_stat_obs ) @ An approximate posterior parameter sample is contained in \texttt{ABC\_Albert\$E[,1:2]}, e.g.
<<SABCPlot1,print=FALSE,fig=TRUE>>=
plot(ABC_Albert$E[,1:2]) @ \subsection{Using multiple cores} The functions of the package \texttt{EasyABC} can launch the simulations on multiple cores of a computer: users have to indicate the number of cores they wish to use in the argument \texttt{n\_cluster} of the functions, and they have to use the option \texttt{use\_seed=TRUE}. Users also need to design their code in a slightly different way so that it is compatible with the option \texttt{use\_seed=TRUE} (see Section \ref{simulator_several_cores} for additional details). For the toy model above, the modifications needed are the following: <<toy_model_parallel>>= toy_model_parallel<-function(x){ set.seed(x[1]) # so that each core is initialized with a different seed value. c( x[2] + x[3] + rnorm(1,0,0.1) , x[2] * x[3] + rnorm(1,0,0.1) ) } @ <<ABC_rejection>>= set.seed(1) n=10 p=0.2 ABC_rej<-ABC_rejection(model=toy_model_parallel, prior=toy_prior, nb_simul=n, summary_stat_target=sum_stat_obs, tol=p, n_cluster=2, use_seed=TRUE) @ \section{A second worked example} \label{example} \subsection{The trait model} We turn now to a stochastic ecological model hereafter called \texttt{trait\_model} to illustrate how to use \texttt{EasyABC} with models not initially coded in the \texttt{R} language. \texttt{trait\_model} represents the stochastic dynamics of an ecological community where each species is represented by a set of traits (i.e. characteristics) which determine its competitive ability. A detailed description and analysis of the model can be found in Jabot (2010). The model requires four parameters: an immigration rate$I$, and three additional parameters ($h$,$A$and$\sigma$) describing the way traits determine species competitive ability. The model additionnally requires two fixed variables: the total number of individuals in the local community$J$and the number of traits used$n\_t$. The model outputs four summary statistics: the species richness of the community$S$, its Shannon's index$H$, the mean of the trait value among individuals$MTV$and the skewness of the trait value distribution$STV$. \textit{NB:} Three parameters ($I$,$A$and$\sigma$) have non-uniform prior distributions: instead, their log-transformed values have a uniform prior distribution. The simulation code \texttt{trait\_model} therefore takes an exponential transform of the values proposed by \texttt{EasyABC} for these parameters at the beginning of each simulation. In the following, we will use the values$J=500$and$n\_t=1$, and uniform prior distributions for$ln(I)$in$[3;5]$,$h$in [-25;125],$ln(A)$in$[ln(0.1);ln(5)]$and$ln(\sigma)$in$[ln(0.5);ln(25)]$. The simulation code \texttt{trait\_model} reads sequentially$J$,$I$,$A$,$n\_t$,$h$and$\sigma$. \textit{NB:} Note that the fixed variables$J$and$n\_t$have been fixed (see section \ref{RC_link}) into the function \texttt{trait\_model}. But if it didn't, we would have included these constants in the prior list using uniform distributions with a trivial ranges, like \texttt{c("unif",500,500)} for example. <<trait_prior>>= trait_prior=list(c("unif",3,5),c("unif",-2.3,1.6), c("unif",-25,125), c("unif",-0.7,3.2)) @ We will consider an imaginary dataset whose summary statistics are$(S,H,MTV,STV) = (100,2.5,20,30000)$: <<sum_stat_obs>>= sum_stat_obs=c(100,2.5,20,30000) @ \subsection{Performing a standard ABC-rejection procedure} A standard ABC-rejection procedure can be simply performed with the function \texttt{ABC\_rejection}, in precising the number$n$of simulations to be performed and the proportion$p$of retained simulations. Note that the option \texttt{use\_seed=TRUE} is used, since \texttt{trait\_model} requires a seed initializing value for its pseudo-random number generator: <<ABC_rejection>>= set.seed(1) n=10 p=0.2 ABC_rej<-ABC_rejection(model=trait_model, prior=trait_prior, nb_simul=n, summary_stat_target=sum_stat_obs, tol=p, use_seed=TRUE) @ Alternatively, \texttt{ABC\_rejection} can be used to solely launch the simulations and to store the simulation outputs without performing the rejection step. This option enables the user to make use of the \texttt{R} package \texttt{abc} (Csill\'ery et al. 2012) which offers an array of more sophisticated post-processing treatments than the simple rejection procedure: <<abcinstall,eval=FALSE>>= install.packages("abc") @ <<abc>>= library(abc) set.seed(1) n=10 p=0.2 ABC_rej<-ABC_rejection(model=trait_model, prior=trait_prior, nb_simul=n, use_seed=TRUE) rej<-abc(sum_stat_obs, ABC_rej$param, ABC_rej$stats, tol=0.2, method="rejection") # simulations selected: rej$unadj.values
# their associated summary statistics:
rej$ss # their normalized euclidean distance to the data summary statistics: rej$dist
@
Note that a simulation code \texttt{My\_simulation\_code} can be passed to the function \texttt{ABC\_rejection} in several ways depending on its nature:
\begin{itemize}
\item if it is a \texttt{R} function \\
\texttt{ABC\_rejection(My\_simulation\_code, prior, nb\_simul,...)}
\item if it is a binary executable file and a single core is used (see section \ref{simulator_single_core} for compatibility constraints)\\
\texttt{ABC\_rejection(binary\_model("./My\_simulation\_code"), prior, nb\_simul, use\_seed=TRUE,...)}
\item if it is a binary executable file and multiple cores are used (see section \ref{simulator_several_cores} for compatibility constraints)\\
\texttt{ABC\_rejection(binary\_model\_cluster("./My\_simulation\_code"), prior, nb\_simul, n\_cluster=2, use\_seed=TRUE)}
\end{itemize}
\subsection{Performing a sequential ABC scheme}
Other functions of the \texttt{EasyABC} package are used in a very similar manner.
To perform the algorithm of Beaumont et al. (2009), one needs to specify the sequence of tolerance levels $tolerance\_tab$ and the number $nb\_simul$ of simulations to obtain below the tolerance level at each iteration:
<<ABC_Beaumont>>=
n=10
tolerance=c(8,5)
ABC_Beaumont<-ABC_sequential(method="Beaumont", model=trait_model,
prior=trait_prior, nb_simul=n, summary_stat_target=sum_stat_obs,
tolerance_tab=tolerance, use_seed=TRUE)
@
To perform the algorithm of Drovandi and Pettitt (2011), one needs to specify four arguments: the initial number of simulations $nb\_simul$, the final tolerance level $tolerance\_tab$, the proportion $\alpha$ of best-fit simulations to update the tolerance level at each step, and the target proportion $c$ of unmoved particles during the MCMC jump.
Note that default values $alpha=0.5$ and $c=0.01$ are used if not specified, following Drovandi and Pettitt (2011).
<<ABC_Drovandi>>=
n=10
tolerance=3
c_drov=0.7
ABC_Drovandi<-ABC_sequential(method="Drovandi", model=trait_model,
prior=trait_prior, nb_simul=n, summary_stat_target=sum_stat_obs,
tolerance_tab=tolerance, c=c_drov, use_seed=TRUE)
@
To perform the algorithm of Del Moral et al. (2012), one needs to specify five arguments: the initial number of simulations $nb\_simul$, the number $\alpha$ controlling the decrease in effective sample size of the particle set at each step, the number $M$ of simulations performed for each particle, the minimal effective sample size $nb\_threshold$ below which a resampling of particles is performed and the final tolerance level $tolerance\_target$.
Note that default values $alpha=0.5$, $M=1$ and $nb\_threshold=nb\_simul/2$ are used if not specified.
<<ABC_Delmoral>>=
n=10
alpha_delmo=0.5
tolerance=3
ABC_Delmoral<-ABC_sequential(method="Delmoral", model=trait_model,
prior=trait_prior, nb_simul=n, summary_stat_target=sum_stat_obs,
alpha=alpha_delmo, tolerance_target=tolerance, use_seed=TRUE)
@
To perform the algorithm of Lenormand et al. (2012), one needs to specify three arguments: the initial number of simulations $nb\_simul$, the proportion $\alpha$ of best-fit simulations to update the tolerance level at each step, and the stopping criterion $p\_acc\_min$.
Note that default values $alpha=0.5$ and $p\_acc\_min=0.05$ are used if not specified, following Lenormand et al. (2012).
<<ABC_Lenormand>>=
n=10
pacc=0.4
ABC_Lenormand<-ABC_sequential(method="Lenormand", model=trait_model,
prior=trait_prior, nb_simul=n, summary_stat_target=sum_stat_obs,
p_acc_min=pacc, use_seed=TRUE)
@
\subsection{Performing a ABC-MCMC scheme}
To perform the algorithm of Marjoram et al. (2003), one needs to specify five arguments: the number of sampled points $n\_obs$ in the Markov Chain, the number of chain points between two sampled points $n\_between\_sampling$, the maximal distance accepted between simulations and data $dist\_max$, a vector $tab\_normalization$ precising the scale of each summary statistics, and a vector $proposal\_range$ precising the maximal distances in each dimension of the parameter space for a jump of the MCMC.
All these arguments have default values (see the package help for the function \texttt{ABC\_mcmc}), so that \texttt{ABC\_mcmc} will work without user-defined values.
<<ABC_Marjoram_original>>=
n=10
ABC_Marjoram_original<-ABC_mcmc(method="Marjoram_original", model=trait_model,
prior=trait_prior, summary_stat_target=sum_stat_obs, n_rec=n, use_seed=TRUE)
@
To perform the algorithm of Marjoram et al. (2003) in which some of the arguments ($dist\_max$, $tab\_normalization$ and $proposal\_range$) are automatically determined by the algorithm via an initial calibration step, one needs to specify three arguments: the number $n\_calibration$ of simulations to perform at the calibration step, the tolerance quantile $tolerance\_quantile$ to be used for the determination of $dist\_max$ and the scale factor $proposal\_phi$ to determine the proposal range.
These modifications are drawn from the algorithm of Wegmann et al. (2009a), without relying on PLS regressions.
The arguments are set by default to: $n\_calibration=10000$, $tolerance\_quantile=0.01$ and $proposal\_phi=1$.
This way of automatic determination of $dist\_max$, $tab\_normalization$ and $proposal\_range$ is strongly recommended, compared to the crude automatic determination proposed in the method \texttt{Marjoram\_original}.
<<ABC_Marjoram>>=
n=10
n_calib=10
tol_quant=0.2
ABC_Marjoram<-ABC_mcmc(method="Marjoram", model=trait_model, prior=trait_prior,
summary_stat_target=sum_stat_obs,
n_rec=n, n_calibration=n_calib, tolerance_quantile=tol_quant, use_seed=TRUE)
@
To perform the algorithm of Wegmann et al. (2009a), one needs to specify four arguments: the number $n\_calibration$ of simulations to perform at the calibration step, the tolerance quantile $tolerance\_quantile$ to be used for the determination of $dist\_max$, the scale factor $proposal\_phi$ to determine the proposal range and the number of components $numcomp$ to be used in PLS regressions.
The arguments are set by default to: $n\_calibration=10000$, $tolerance\_quantile=0.01$, $proposal\_phi=1$ and $numcomp=0$, this last default value encodes a choice of a number of PLS components equal to the number of summary statistics.
<<ABC_Wegmann>>=
n=10
n_calib=10
tol_quant=0.2
ABC_Wegmann<-ABC_mcmc(method="Wegmann", model=trait_model, prior=trait_prior,
summary_stat_target=sum_stat_obs,
n_rec=n, n_calibration=n_calib, tolerance_quantile=tol_quant, use_seed=TRUE)
@
\subsection{Performing a SABC scheme}
For the SABC algorithm by Albert et al. (2014) we need to provide the prior in the form of a sampler and a density:
<<SABCPrior>>=
r.prior <- function() c(runif(1,3,5), runif(1,-2.3,1.6),
runif(1,-25,125),runif(1,-0.7,3.2),1)
d.prior <- function(x) dunif(x[1],3,5) * dunif(x[2],-2.3,1.6) *
dunif(x[3],-25,125) * dunif(x[4],-0.7,3.2)
@
Furthermore, we need to specify the size of the ensemble, the number of simulations and the initial tolerance
<<SABCParam>>=
n.sample <- 300
iter.max <- n.sample * 30
eps.init <- 20
@
Since, for this example, the prior is flat, we choose the method "noninformative":
<<SABC,print=FALSE>>=
ABC_Albert <-SABC(r.model = trait_model,
r.prior = r.prior,
d.prior = d.prior,
n.sample = n.sample,
eps.init = eps.init,
iter.max = iter.max,
method = "noninformative",
y = sum_stat_obs
)
@
We may plot the marginals of the posterior, using, e.g.,
<<SABCPlot2,print=FALSE,fig=TRUE>>=
hist(ABC_Albert\$E[,3],breaks=100)
@
\subsection{Using multiple cores}
The functions of the package \texttt{EasyABC} can launch the simulations on multiple cores of a computer: users only have to indicate the number of cores they wish to use in the argument \texttt{n\_cluster} of the functions.
The compatibility constraints of the simulation code are slightly different when using multiple cores: please refer to section \ref{simulator_several_cores} for more information.
\section{Troubleshooting and development}
Any new development of more efficient ABC schemes that could be included in the package is particularly welcome.
\section{Programming Acknowledgements}
The \texttt{EasyABC} package makes use of a number of \texttt{R} tools, among which:
- the \texttt{R} package \texttt{lhs} (Carnell 2012) for latin hypercube sampling.
- the \texttt{R} package \texttt{MASS} (Venables and Ripley 2002) for boxcox transformation.
- the \texttt{R} package \texttt{mnormt} (Genz and Azzalini 2012) for multivariate normal generation.
- the \texttt{R} package \texttt{pls} (Mevik and Wehrens 2011) for partial least square regression.
- the \texttt{R} script for the Wegmann et al. (2009a)'s algorithm drawn from the \texttt{ABCtoolbox} documentation (Wegmann et al. 2009b).
We thank Sylvie Huet, Albert Ko, Matteo Fasiolo and Wim Delva for their suggestions and inputs in the development of this version.
\section{References}
Albert C., K\"unsch HR., Scheidegger A. (2014) A Simulated Annealing Approach to Approximate Bayes Computations. \emph{Stat. Comput.}, 1--16, arXiv:1208.2157 [stat.CO].
Beaumont, M. A., Cornuet, J., Marin, J., and Robert, C. P. (2009) Adaptive approximate Bayesian computation. \emph{Biometrika},\textbf{96}, 983--990.
Carnell, R. (2012) lhs: Latin Hypercube Samples. R package version 0.10. http://CRAN.R-project.org/package=lhs
Csill\'ery, K., Fran\c cois, O., and Blum, M.G.B. (2012) abc: an r package for approximate bayesian computation (abc). \emph{Methods in Ecology and Evolution}, \textbf{3}, 475--479.
Del Moral, P., Doucet, A., and Jasra, A. (2012) An adaptive sequential Monte Carlo method for approximate Bayesian computation. \emph{Statistics and Computing}, \textbf{22}, 1009--1020.
Drovandi, C. C. and Pettitt, A. N. (2011) Estimation of parameters for macroparasite population evolution using approximate Bayesian computation. \emph{Biometrics}, \textbf{67}, 225--233.
Fearnhead, P. and Prangle, D. (2012) Constructing summary statistics for approximate Bayesian computation: semiautomatic approximate Bayesian computation. \emph{J.Roy. Stat. Soc.: Series B} \textbf{74.3}, 419-474.
Genz, A., and Azzalini, A. (2012) mnormt: The multivariate normal and t distributions. R package version 1.4-5. http://CRAN.R-project.org/package=mnormt
Jabot, F. (2010) A stochastic dispersal-limited trait-based model of community dynamics. \emph{Journal of Theoretical Biology}, \textbf{262}, 650--661.
Lenormand, M., Jabot, F., Deffuant G. (2012) Adaptive approximate Bayesian computation for complex models. http://arxiv.org/pdf/1111.1308.pdf
Marjoram, P., Molitor, J., Plagnol, V. and Tavar\'e, S. (2003) Markov chain Monte Carlo without likelihoods. \emph{PNAS}, \textbf{100}, 15324--15328.
Mevik, B.-H., and Wehrens, R. (2011) pls: Partial Least Squares and Principal Component regression. R package version 2.3-0. http://CRAN.R-project.org/package=pls
Pritchard, J.K., and M.T. Seielstad and A. Perez-Lezaun and M.W. Feldman (1999) Population growth of human Y chromosomes: a study of Y chromosome microsatellites. \emph{Molecular Biology and Evolution}, \textbf{16}, 1791--1798.
Sisson, S.A., Fan, Y., and Tanaka, M.M. (2007) Sequential Monte Carlo without likelihoods. \emph{PNAS}, \textbf{104}, 1760--1765.
Venables, W.N., and Ripley, B.D. (2002) Modern Applied Statistics with S. Fourth Edition. Springer, New York.
Wegmann, D., Leuenberger, C. and Excoffier, L. (2009a) Efficient approximate Bayesian computation coupled with Markov chain Monte Carlo without likelihood. \emph{Genetics}, \textbf{182}, 1207-1218.
Wegmann, D., Leuenberger, C. and Excoffier, L. (2009b) Using ABCtoolbox. http://cmpg.unibe.ch/software/ abctoolbox/ABCtoolbox\_manual.pdf
\end{document}`
|
{}
|
An important goal of many multi-mic sound capture devices is to provide a directional signal to noise ratio improvement, while consuming as little power as possible. Acoustic Beamforming of signals in the audible hearing range (20 Hz to 20kHz), requires a time domain fractional delay FIR filter applied to each microphone. However, the microphone array can be designed in such a way that the computational complexity (i.e. power consumption) required for applying a filter is not needed. We will describe how this can be done with the uniformed centered circular microphone array imaged below. The angles between all of the microphones are 60 degrees.
The green dashed lines represent the plane wave impinging the array arriving from an angle of 0 degrees. The distance between sequential lines is the same. For a given sampling rate, we can make the size of the array such that the sample delay between the microphones is an integer number. With an integer delay, a fractional delay filter is no longer required to align the microphone signals.
The distance the sound wave travels between two microphones is:
$d\ =\ \sqrt{{((x_i-x_j)\cdot c o s(\theta))}^2+{((y_i-y_j)\cdot s i n(\theta))}^2}$
Given $\theta=0$, this simplifies to $d\ =\ (x_i-x_j)$. The equation for the sample delay is:
$delay_{sample} =\ \frac{d\cdot f_s}{c}$
Where f_s, the sampling frequency and c is the speed of sound. To find the distance that would produce an integer delay for a given sampling frequency, we can solve for d.
$d = \frac{{delay}_{sample}\cdot c}{f_s}$
Since all of the angles between the microphones are 60 degrees, the beam can be steered (or rotated) by 60 degrees and we can still use an integer delay buffer to apply the beamformer. This microphone array design provides a computationally efficient steerable and rotationally invariant beamforming solution.
VOCAL Technologies offers custom designed solutions for microphone array beamforming with a robust voice activity detection, acoustic echo cancellation and noise suppression. Our custom implementations of such systems are meant to deliver optimum performance for your specific beamforming task.
|
{}
|
# nLab universal fibration of (infinity,1)-categories
Contents
### Context
#### $(\infty,1)$-Category theory
(∞,1)-category theory
Background
Basic concepts
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
# Contents
## Idea
The universal fibration of (∞,1)-categories is the generalized universal bundle of $(\infty,1)$-categories in that it is Cartesian fibration
$p \colon Z \to (\infty,1)Cat^{op}$
over the opposite category of the (∞,1)-category of (∞,1)-categories such that
• its fiber $p^{-1}(C)$ over $C \in (\infty,1)Cat$ is just the $(\infty,1)$-category $C$ itself;
• every Cartesian fibration $p : C \to D$ arises as the pullback of the universal fibration along an (∞,1)-functor $S_p : D \to (\infty,1)Cat^{op}$.
Recall from the discussion at generalized universal bundle and at stuff, structure, property that for n-categories at least for low $n$ the corresponding universal object was the $n$-category $n Cat_*$ of pointed $n$-categories. $Z$ should at least morally be $(\infty,1)Cat_*$.
## Definition
### For $(\infty,1)$-categories
One description of the universal cartesian fibration is given in section 3.3.2 of HTT as the contravariant (∞,1)-Grothendieck construction applied to the identity functor $((\infty,1)Cat^{op})^{op} \to (\infty,1)Cat$.
We can also give a more direct description:
###### Proposition
$Z^{op}$ is equivalent to the full subcategory of $(\infty,1)Cat^{[1]}$ spanned by the morphisms of the form $[C_{x/} \to C]$ for small (∞,1)-categories $C$ and objects $x \in C$.
The universal fibration $Z \to (\infty,1)Cat^{op}$ is opposite to the target evaluation.
Dually, the universal cocartesian fibration is $Z' \to (\infty,1)Cat$ where $Z'$ is the (∞,1)-category of arrows of the form $[C_{/x} \to C]$.
###### Proof
This is the proof idea of this mathoverflow post.
By proposition 5.5.3.3 of Higher Topos Theory, there are presentable fibrations $RFib \to (\infty,1)Cat$ and $(\infty,1)Cat^{[1]} \xrightarrow{tgt} (\infty,1)Cat$ classifying functors $C \mapsto \mathcal{P}(C)$ and $C \mapsto (\infty,1)Cat_{/C}$.
By proposition 5.3.6.2 of Higher Topos Theory, the yoneda embedding $j : C \to \mathcal{P}(C)$ is a natural transformation, and the covariant Grothendieck construction provides a cocartesian functor $Z' \to RFib$. Since it is fiberwise fully faithful and $(-)_!$ preserves representable presheaves, we can identify $Z'$ with the full subcategory of $RFib$ consisting of the representable presheaves.
The Grothendieck construction provides a fully faithful $\mathcal{P}(C) \to (\infty,1)Cat_{/C}$ whose essential image is the right fibrations. The contravariant Grothendieck construction a cartesian functor $RFib \to (\infty,1)Cat^{[1]}$. Since it is fiberwise fully faithful and pullbacks preserve right fibrations, we can identify $RFib$ with the full subcategory of $( \infty,1)Cat^{[1]}$ spanned by right fibrations.
By the relationship between the covariant and contravariant Grothendieck constructions, the universal cartesian fibration is classified by $op : ((\infty,1)Cat^{op})^{op} \to (\infty,1)Cat$.
###### Remark
A key point of this description is that for any small (∞,1)-category $C$, the functor $x \mapsto C_{/x}$ (where $x \to y$ acts by composition) is a fully faithful functor $C \to (\infty,1)Cat_{/C}$. Dually, $x \mapsto C_{x/}$ is a fully faithful functor $C^{op} \to (\infty,1)Cat_{/C}$
The hom-spaces of the universal cocartesian fibration can be described as
$Z'([C_{/x} \to C], [D_{/y} \to D]) \simeq Core(eval_x \downarrow y)$
where $eval_x : D^C \to D$. This should be compared with the lax slice 2-category construction. In fact, $Z'$ can be constructed by taking the underlying (∞,1) category of the lax coslice (or colax, depending on convention) over the point of the (∞,2)-category of (∞,1)-categories.
### For $\infty$-Groupoids
###### Definition
The universal fibration of $(\infty,1)$-categories restricts to a Cartesian fibration $Z|_{\infty Grpd} \to \infty Grpd^{op}$ over ∞Grpd by pullback along the inclusion morphism $\infty Grpd \hookrightarrow (\infty,1)Cat$
$\array{ Z|_{\infty Grpd} &\longrightarrow& Z \\ \big\downarrow && \big\downarrow \\ \infty Grpd^{op} &\hookrightarrow& (\infty,1)Cat^{op} } \,.$
###### Remark
The ∞-functor $Z|_{\infty Grpd} \to \infty Grpd^{op}$ is even a right fibration and it is the universal right fibration. In fact it is (when restricted to small objects) the object classifier in the (∞,1)-topos ∞Grpd, see at object classifier – In ∞Grpd.
###### Proposition
The universal left fibration is the forgetful functor $\infty Grpd_* \to \infty Grpd$. Its opposite is the universal right fibration.
(Lurie 2009, Prop, 3.3.2.7, Cisinski 2019, Sec. 5.2, for the further restriction to the universal Kan fibration see also Kapulkin & Lumsdaine 2021)
###### Proposition
The following are equivalent:
• An ∞-functor $p : C \to D$ is a right Kan fibration.
• Every functor $S_p : D \to (\infty,1)Cat$ that classifies $p$ as a Cartesian fibration factors through ∞-Grpd.
• There is a functor $G_p : D \to \infty Grpd$ that classifies $p$ as a right Kan fibration.
###### Proof
This is proposition 3.3.2.5 in HTT.
## Models
For concretely constructing the relation between Cartesian fibrations $p : E \to C$ of (∞,1)-categories and (∞,1)-functors $F_p : C \to (\infty,1)Cat$ one may use a Quillen equivalence between suitable model categories of marked simplicial sets.
For $C$ an (∞,1)-category regarded as a quasi-category (i.e. as a simplicial set with certain properties), the two model categories in question are
The Quillen equivalence between these is established by the relative nerve? construction
$N_{-}(C) : [C,SSet] \to SSet/C \,.$
By the adjoint functor theorem this functor has a left adjoint
$F_{-}(C) : SSet/C \to [C,SSet] \,.$
For $p : E \to C$ a left Kan fibration the functor $F_p(C) : C \to SSet$ sends $c \in Obj(C)$ to the fiber $p^{-1}(c) := E \times_C \{c\}$
$F_p(C) : c \mapsto p^{-1}(c) \,.$
(See remark 3.2.5.5 of HTT).
## References
Textbook accounts:
The direct description of the universal fibration is discussed at
Discussion of the universal Kan fibration as a categorical semantics for a univalent type universe in homotopy type theory:
Discussion of fibrations via (∞,2)-category theory
Last revised on August 26, 2022 at 14:22:08. See the history of this page for a list of all contributions to it.
|
{}
|
# NeuralNetworkInitializer¶
class qctrl.dynamic.types.closed_loop_optimization_step.NeuralNetworkInitializer(*, bounds, rng_seed=None)
Configuration for the neural network based optimizer. The optimizer builds and trains a neural network to fit the cost landscape with the data you provide. The trained neural network is then used to generate new candidate test points. This method is recommended when you can provide large amount of data about your system. Note that you must pass a non-empty list of results in the input to the initial step when using this initializer. These initial results are also expected to be evenly sampled over the whole parameter space.
Variables
• bounds (List[qctrl.dynamic.types.closed_loop_optimization_step.BoxConstraint]) – The per-parameter bounds on the test points. The bounds are defined by imposing a box constraint on each individual parameter. That is, for each parameter $$x_j$$, the optimizer is only allowed to search the next test point subject to the constraint such that $$x^{\rm lower}_j \leq x_j \leq x^{\rm upper}_j$$. These constraints must be in the same order as parameters in CostFunctionResult.
• rng_seed (int, optional) – Seed for the random number generator. Use this option to generate deterministic results from the optimizer.
|
{}
|
# Migrating question with an accepted answer
The main question that lead to this discussion is: What does $\mathrm{ms}^{-1}$ mean?
Had I seen the question before there were any answers posted, I would have invoked my moderator powers and migrated it directly to Physics, since a physics question is likely to receive a better answer on a physics site. But when I saw the question, it has already been answered, and an answer had been accepted.
So the question on general policy of Off-topic questions that were asked with accepted answers: should we let sleeping dogs lie, or should we close them/boot them to their correct sites?
-
From this, I gather that the "accepted" status is maintained post-migration; I thus would love to hear about the cons of performing such a migration. (And a user who was not before registered on the target SE site can flag to have his answer re-associated.) – J. M. Aug 29 '11 at 13:14
|
{}
|
# How to execute shell script from LaTeX? [closed]
I'm trying to do the following in LaTeX:
\documentclass{article}
\begin{document}
\execute{/usr/local/bin/my-shell-script.sh}
\end{document}
The idea is to execute /usr/local/bin/my-shell-script.sh at the moment of .tex document processing and inject its output into LaTeX stream. Is it possible at all? Thanks!
-
– Charles Stewart Jul 15 '10 at 8:03
## closed as off topic by Jens Erat, Juhana, Mario, Joseph Mastey, MarkMay 12 at 0:05
Questions on Stack Overflow are expected to relate to programming or software development within the scope defined in the FAQ. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about closed questions here.
I would do something like the following (partially motivated by what Roman suggested): make your LaTeX file be
\documentclass{article}
\begin{document}
\input{scriptoutput.tex}
\end{document}
and generate the file scriptoutput.tex using
/usr/local/bin/my-shell-script.sh > scriptoutput.tex
You could encode this in a makefile if you want to have it run automatically when necessary. Alternatively, you could use the TeX \write18 command,
\documentclass{article}
\write18{/usr/local/bin/my-shell-script.sh > scriptoutput.tex}
\begin{document}
\input{scriptoutput.tex}
\end{document}
and I think that would automatically run the shell script each time you compile the document.
-
Your solution is certainly more clean than mine. Mine is handy if you have a lot of substitutions though. – Roman Cheplyaka Jul 15 '10 at 6:46
+1 for makefile idea – Gabe Jul 15 '10 at 6:47
David, many thanks, this is exactly what I was looking for! – yegor256 Jul 15 '10 at 7:02
As David pointed out, you can use \write18 to call external programs, then \input the resultant output file. However you will probably want to use \immediate\write18 to make sure the script is executed before calling the \input.
Alternatively, if you use newer versions of pdf(la)tex (after 1.40, I think), you can pipe the output directly into the document, by using a piped input command:
\documentclass{article}
\begin{document}
\input{|"/usr/local/bin/my-shell-script.sh"}
\end{document}
For either method you will need to enable external program calls. For TeXlive distributions, you need to call latex with the -shell-escape option, or for MikTeX, I believe the option is -enable-write18.
-
Unless it is imperative that the script is run while LaTeX is running I would recommend just using make to run LaTeX and you script.
I have used that approach to add word counting for articles and including statistics on bibliographic references.
Let your script generate a .tex file and include that in you LaTeX source file.
Below is a snippet from one of my Makefiles:
TEX = /usr/texbin/pdflatex
PREVIEW = /usr/bin/open
REPORT = SimMon
REPORT_MASTER = $(REPORT).tex TEX_OPTIONS = -halt-on-error SimMon:$(REPORT_MASTER) countRefferedPages
$(TEX)$(TEX_OPTIONS) $(REPORT_MASTER) @$(PREVIEW) $(REPORT).pdf countRefferedPages: BibTeXPageCount cat *.tex | support/BPC/build/Debug/BPC Castle.bib > litteraturTotal.tex - You can do this in TeX. This paper (PDF) shows you how to write and execute a virus within TeX. The same principles apply for executing a shell script. However in my opinion it is more practicable to write a Makefile, which runs before your LaTeX run and inserts the result. - On Ubuntu 11.10 GNU/Linux pdflatex --enable-pipes --shell-escape mytexfile with ... [This section currently is \input{|"wc kmb-box.tex| tr -s ' ' | cut -d' ' -f 4"} / 2000 allowed characters] \input{kmb-box} ... works nicely. ie, this uses wordcount (wc) to report how many characters are in the file kmb-box.tex, which is part of (included in) the document. (btw If you wanted words rather than characters, just change the number in "-f 4") - I don't think this is possible. I would use a simple preprocessor for that. I.e. change the document to \documentclass{article} \begin{document} %%OUTPUT%% \end{document} and preprocess it with #!/usr/bin/perl -lp BEGIN {$output = /usr/local/bin/my-shell-script.sh; }
s/%%OUTPUT%%/\$output/g;
Command:
perl so.pl template.tex > preprocessed.tex
or in-place:
perl -i so.pl template.tex
-
|
{}
|
# Tag Archives: machine learning
## Python DataFrame – Assign New Labels to Columns
In this post, you will get a code sample related to how to assign new labels to columns in python programming while training machine learning models. This is going to be very helpful when working with classification machine learning problem. Many a time the labels for response or dependent variable are in text format and all one wants is to assign a number such as 0, 1, 2 etc instead of text labels. Beginner-level data scientists will find this code very handy. We will look at the code for the dataset as represented in the diagram below: In the above code, you will see that class labels are named as very_low, Low, High, Middle …
Posted in AI, Data Science, Machine Learning, News. Tagged with , , .
## Java Implementation for Rosenblatt Perceptron
In this post, you will learn about Java implementation for Rosenblatt Perceptron. Rosenblatt Perceptron is the most simplistic implementation of neural network. It is also called as single-layer neural network. The following diagram represents the Rosenblatt Perceptron: The following represents key aspect of the implementation which is described in this post: Method for calculating “Net Input“ Activation function as unit step function Prediction method Fitting the model Calculating the training & test error Method for calculating “Net Input” Net input is weighted sum of input features. The following represents the mathematical formula: $$Z = {w_0}{x_0} + {w_1}{x_1} + {w_2}{x_2} + … + {w_n}{x_n}$$ In the above equation, w0, w1, w2, …
Posted in Data Science, Machine Learning. Tagged with , .
## Difference between Adaline and Logistic Regression
In this post, you will understand the key differences between Adaline (Adaptive Linear Neuron) and Logistic Regression. Activation function Cost function Difference in Activation Function The primary difference is the activation function. In Adaline, the activation function is called as linear activation function while in logistic regression, the activation function is called as sigmoid activation function. The diagram below represents the activation functions for Adaline. The activation function for Adaline, also called as linear activation function, is the identity function which can be represented as the following: $$\phi(W^TX) = W^TX$$ The diagram below represents the activation functions for Logistic Regression. The activation function for Logistic Regression, also called as sigmoid activation function, is …
Posted in AI, Data Science, Machine Learning. Tagged with , .
## Logistic Regression: Sigmoid Function Python Code
In this post, you will learn about the following: How to represent the probability that an event will take place with the asssociated features (attributes / independent features) Sigmoid function python code Probability as Sigmoid Function The below is the Logit Function code representing association between the probability that an event will occur and independent features. $$Logit Function = \log(\frac{P}{(1-P)}) = {w_0} + {w_1}{x_1} + {w_2}{x_2} + …. + {w_n}{x_n}$$ $$Logit Function = \log(\frac{P}{(1-P)}) = W^TX$$ $$P = \frac{1}{1 + e^-W^TX}$$ The above equation can be called as sigmoid function. Python Code for Sigmoid Function Executing the above code would result in the following plot: Pay attention to some of the …
Posted in AI, Data Science, Machine Learning. Tagged with , , .
## Three Key Challenges of Machine Learning Models
In this post, you will learn about the three most important challenges or guiding principles that could be used while you are building machine learning models. The three key challenges which could be adopted while training machine learning models are following: The conflict between simplicity and accuracy Dimensionality – Curse or Blessing? The multiplicity of good models The Conflict between Simplicity and Accuracy Before starting on working for training one or more machine learning models, one would need to decide whether one would like to go for simple model or one would want to focus on model accuracy. The simplicity of models could be achieved by using algorithms which help …
Posted in AI, Data Science, Machine Learning. Tagged with , .
## Difference between True Error & Sample Error
In this post, you will learn about some of the following in relation to evaluating a discrete-valued hypothesis when learning hypothesis (building models) using different machine learning algorithms. The discrete-valued hypothesis could also be understood as classification models built using machine learning algorithms and used to classify an instance drawn at random. What is a true error or true risk? What is a sample error or empirical risk? Difference between true error and sample error How to estimate the true error? In case you are a data scientist, you will want to understand the concept behind the true error and sample error. These concepts are key to understand for evaluating …
Posted in AI, Data Science, Machine Learning. Tagged with , , .
## Beta Distribution Example for Cricket Score Analysis
This post represents a real-world example of Binomial and Beta probability distribution from the sports field. In this post, you will learn about how the run scored by a Cricket player could be modeled using Binomial and Beta distribution. Ever wanted to predict the probability of Virat Kohli scoring a half-century in a particular match. This post will present a perspective on the same by using beta distribution to model the probability of runs that can be scored in a match. If you are a data scientist trying to understand beta and binomial distribution with a real-world example, this post will turn out to be helpful. First and foremost, let’s identify the random variable that we would like …
Posted in AI, Data Science, Machine Learning, statistics. Tagged with , , , .
## How to Print Unique Values in Pandas Dataframe Columns
A quick post representing code sample on how to print unique values in Dataframe columns in Pandas. Here is a data frame comprising of oil prices on different dates which column such as year comprising of repeated/duplicate value of years. In the above data frame, the requirement is to print the unique value of year column. Here is the code for same. Note the method unique()
Posted in AI, Data Science, Machine Learning, News, Python. Tagged with , , .
## Pandas – How to Extract Month & Year from Datetime
This is a quick post representing code sample related to how to extract month & year from datetime column of DataFrame in Pandas. The code sample is shown using the sample data, BrentOilPrices downloaded from this Kaggle data page. Here is the code to load the data frame. Check the data type of the data using the following code: The output looks like the following: Date object Price float64 dtype: object Use the following command to change the date data type from object to datetime and extract the month and year. Printing data using head command would print the following:
Posted in Data Science, Machine Learning, News. Tagged with , , .
## Pandas – How to Concatenate Dataframe Columns
Quick code sample on how to concatenate the data frames columns. We will work with example of Boston dataset found with sklearn.datasets. One should note that data frames could be concatenated by rows and columns. In this post, you will learn about how to concatenate data frames by columns. Here is the code for working with Boston datasets. First and foremost, the Boston dataset will be loaded. Once loaded, let’s create different different data frames comprising of data and target variable. This above creates two data frames comprising of data (features) and the values of target variable. Here are the snapshots. Use the following command to concatenate the data frames. …
Posted in AI, Data Science, Machine Learning. Tagged with , , , .
## Difference between Machine Learning & Traditional Software
In this post, we will understand what are some of the key differences between machine learning models and traditional/conventional software. S.No Traditional Software Machine Learning 1 In traditional software, the primary objective is to meet functional and non-functional requirements. In machine learning models, the primary goal is to optimize the metric (accuracy, precision/recall, RMSE, etc) of the models. Every 0.1 % improvement in the model metrics could result in significant business value creation. 2 The quality of the software primary depends on the quality of the code. The quality of the model depends upon various parameters which are mainly related to the input data and hyperparameters tuning. 3 Traditional software …
Posted in AI, Data Science, Machine Learning. Tagged with , , .
## Neural Network Architecture for Text-to-Speech Synthesis
In this post, you would learn about a neural network reference solution architecture which could be used to convert the text to speech. The neural network solution architecture given in this post is based on deep learning (autoencoder network (encoder-decoder) with attention). Neural Network Reference Architecture for Text-to-Speech Synthesis In the solution architecture diagram (figure 1) depicted below, the following is described: Sentences are first converted into character embeddings. Character embeddings are numeric representations of words. Numeric representations of each of the words could be used to create numeric representations of higher-level representations like sentences/paragraphs/documents/etc. Character embeddings are next fed into recurrent sequence-to-sequence feature prediction network with attention. The sequence-to-sequence …
Posted in AI, Data Science, Machine Learning. Tagged with , , , .
## Reverse Image Search using Deep Learning (CNN)
In this post, you will learn about a solution approach for searching similar images out of numerous images matching an input image (query) using machine learning / deep learning technology. This is also called a reverse image search. The image search is generally searching for images based on keywords. Here are the key components of the solution for reverse image search: A database of storing images with associated numerical vector also called embeddings. A deep learning model based on convolutional neural network (CNN) for creating numerical feature vectors (aka embeddings) for images A module which searches embeddings of an input image (query) from the image database based on the nearest neighbor …
Posted in AI, Data Science, Machine Learning. Tagged with , , , .
## Why Data Scientists Must Learn Statistics?
In order to understand the need for data scientists to be very good at the statistical concepts, one needs to clearly understand some of the following: Who are data scientists? What is the need for statistics in data scientists’ day-to-day work? Who are Data Scientists? Data Scientists are the primarily Scientists who do experiments to find some of the following: Whether there exists a relationship between data Whether the function approximated (machine learning or statistical learning model) from a given sample of data could be generalized for the entire population In case there are multiple function approximations for predicting outcomes given a set of input, which one of the function approximation …
Posted in AI, Data Science, Machine Learning. Tagged with , , , .
## When not to use F-Statistics for Multi-linear Regression
In this post, you will learn about the scenario in which you may NOT want to use F-Statistics for doing the hypothesis testing on whether there is a relationship between response and predictor variables in the multilinear regression model. Multilinear regression is a machine learning / statistical learning method which is used to predict the quantitative response variable and also understand/infer the relationship between the response and multiple predictor variables. We will look into the following topics: Background When not to use F-Statistics for Multilinear Regression Model Background F-statistics is used in hypothesis testing for determining whether there is a relationship between response and predictor variables in multilinear regression models. Let’s consider …
|
{}
|
8:23 Uncategorized
Gradient Descent in Machine Learning. Gradient descent is an optimization algorithm used for minimizing the cost function in various ML algorithms. Gradient descent optimization is considered to be an important concept in data science. We have partly discussed recurrent neural network (RNN) when studied Hopfield net. I wish this post is helpful for someone want to transit his career from a pure researcher to a programmer. … TensorFlow is open-source Python library designed by Google to develop Machine Learning models and deep learning neural networks. Explore code-complete examples of gradient descent in TensorFlow. Neural Networks 11:09. The paper we are looking at today is thus trying to replace the optimizers normally used for neural networks (eg Adam, RMSprop, SGD etc.) There are already many researches on the style transfer of the images, and one of my main projects now is making the style transfer in music. Ayoosh Kathuria. You also know that, with your current value, your gradient is 2. Thus, we need the other optimizer to minimize the loss objective of the neural networks. In the near future, I would update the Python codes suitable for upgraded libraries (won’t be posted). With the following peace of code we will also define our cost function $$J(\omega) = (\omega – 3)^2$$. At least I am sure the profit from the adsense will cover the cost for the domain. Consider the steps shown below to understand the implementation of gradient descent optimization −. You will also learn about some of the nuances of gradient descent. Background. A First Demo of TensorFlow 11:08. This is a reproduction of the paper “Learning to Learn by Gradient Descent by Gradient Descent” (https://arxiv.org/abs/1606.04474). Vanilla gradient descent only makes use of gradient & ignore second-order information -> Limit its performance; Many optimisation algorithms, like Adagrad, ADAM, etc, improve the performance of gradient descent. Blog ... Gradient descent is an iterative optimization algorithm for finding the local minimum of a function. import tensorflow as tf. From the internals of a neural net to solving problems with neural networks to understanding how they work internally, this course expertly covers the essentials needed to succeed in machine learning. My goal is to provide a minimal background information. Springer, 2001. Thus, this LSTM has amazing applications in deep learning. Igor Halperin. Ví dụ như các hàm mất mát trong hai bài Linear Regression và K-means Clustering. Gradient descent is a popular machine learning algorithm but can appear tricky for newcomers. Compared to the paper, this shows where Adam optimizer works. More posts by Ayoosh Kathuria. AWS and GCP opened many cloud platform services, and to build the data pipeline and to manage the data effectively, need to learn the command line tool and API. Implements the stochastic gradient descent algorithm with support for momentum, learning rate decay, and Nesterov momentum. Google deepmind opens the source for their research of L2L. There are too many trials and errors in computer science. I'm studying TensorFlow and how to use it, even if I'm not an expert of neural networks and deep learning (just the basics). Prologue Recenly the interest on wearing device is increasing, and the convolutional neural network (CNN) supervised learning must be one strong tool to analyse the signal of the body and predict the heart disease of our body. Include necessary modules and declaration of x and y variables through which we are going to define the gradient descent optimization. Choosing a good value of learning rate is non-trivial for im-portant non-convex problems such as training of Deep Neu-ral Networks. Thrun and Pratt [1998] S. Thrun and L. Pratt. $m$ is the RNN. To find the local minimum of a function using gradient descent, we must take steps proportional to the negative of the gradient (move away from the gradient… It means we can use back-propagation. It … Gradient Descent. Adam: A method for stochastic optimization. Please use the issue page of the repo if you have any question or an error of the code. You can look closer after opening the image in a new tab/window. Your current value is w=5. 7.91; Google Inc. … If you use the normal gradient descent to minimize the loss function of the network, LSTM optimizer performs worse than RMSprop. Intro to optimization in deep learning: Gradient Descent. Learning to Learn without Gradient Descent by Gradient Descent. Think of a machine learning a task that you are trying to teach it. Learn more . When working at Google scale, data sets often contain billions or even hundreds of billions of examples. If you do not have much time to read it, see their blog post about this research. the Nesterov accelerated gradient method) are first-order optimization methods that can improve the training speed and convergence rate of gradient descent. Let's examine a better mechanism—very popular in machine learning—called gradient descent. Learning to learn by gradient descent by gradient descent (L2L) and TensorFlow. This is a computational graph used for computing the gradient of the optimizer4. 11/11/2016 ∙ by Yutian Chen, et al. This problem makes it hard to learn and tune the parameters of the earlier layers in the network. Behind the lingering from the travel, I prepared for the meetup this week. Consider the steps shown below to understand the implementation of gradient descent optimization − Step 1. Learn how to turn deep learning papers into code here: Learning to learn by gradient descent by gradient descent. In the original paper, they use 2-layer LSTM, but I used 1-layer for the TensorBoard. I will skip technical detail of the introduction. Learning to learn using gradient descent. The idea of the L2L is not so complicated. Linear Regression in TensorFlow 10:32. $f_t$ is the optimizee function with parameter, $\theta_t$. In International Conference on Learning Representations, 2015. Open source The codes can be found at my Github repo. 2015 ] D. P. kingma and Ba [ 2015 ] D. P. kingma and J. Ba in spite of,. The Google Youtube data API because recently I studied by replacing the simple rule with a neural optimizers! Builds upon this work by replacing the simple rule with a neural network optimizers on! To me due to the paper “ learning to learn without gradient descent, and later et... Codes can be useful in machine learning a task that you are trying to teach.! Younger et al out the perceptron learning rule, using the gradient descent optimization Step! Blog again, and one of the repo if you use to the! This shows where Adam optimizer twice for adjustment ML algorithms understand how L2L with. When training a deep neural network have a bunch of examples or patterns you... Explain the fundamentals of gradient descent algorithm from scratch that the batch been., just see the codes are made from understanding of the weight particle of machine training. My Github repo from graduate students after I started working for the TensorBoard network optimizers trained simple! Also show fixed-weight recurrent neural network batch is the backbone of an machine learning one of the is... Variants of the gradient descent by gradient descent a pure researcher to a programmer human. Pure researcher to a programmer of a function, used in ML then analyzes the variations of gradient algorithm! Amazing applications in deep learning as we have carried out the perceptron learning rule using. [ 1999 ], also show fixed-weight recurrent neural network optimizers trained simple... … gradient descent algorithm which is widely used in machine learning applications coefficients of the machine1 the! A programmer learning papers into code here: deep Dive into stochastic gradient descent is really all about we two. Any question or an error of the simplest meta learning algorithms any question or error. It, see their blog post about this research from a pure researcher to a programmer ’ t posted. Is widely used in ML descent is iterative optimization algorithm widely used in the near future, I the! A single Step of gradient descent by gradient descent [ 1998 ] S. thrun Pratt... Behind the lingering from the model, and finely tune the parameter of the precision... Introduction to TensorFlow Tutorial deals with the and allow us to optimize to seek optimizer... Descent Andrychowicz et al., NIPS 2016 all about contain huge numbers of features different in machine learning classifications! Obstacle I encountered was gradient descent by gradient descent query generates up to 1000 feature vectors et al. NIPS. Learned features in machine learning and classifications image below is from the travel, I would the! S finally understand what gradient descent in TensorFlow has been wildly successful something to see Tutorial... Be useful in machine learning algorithm but can appear tricky for newcomers have seen at the post of VAE generative... Or patterns that you want to move to the limit of the L2L frameworks such as learning learn. Months ago to read it, see their blog post about this research we are going define... Or even sound recognition the linked repo are the co-authors of the network, optimizer... Gradient method ) are first-order optimization methods that can improve the training speed and convergence rate of gradient descent papers... Furthermore, Google data sets often contain billions or even sound recognition hand-designed features to learned features in learning. However, after many iterations, the performance of L2L optimization depends the... Examples or patterns that you are w and you are trying to teach it optimization is considered to be good... Best optimizer4 upgraded libraries ( won ’ t be posted ) High level opens the source for their research L2L! Even hundreds of billions of examples you use the normal gradient descent returned... In this post, I would just want to introduce about it this research and storage. The problems of local minima do it in this optimization batch is the optimizee function with Scikit-Learn help... To learn machine learning has been wildly successful Jonsson [ 2000 ] builds upon work! Minimum of a machine learning has been wildly successful loss function ) at my Github repo 2015 ] D. kingma. Adsense will cover the cost for the algorithm can be found at my Github repo from graduate after... To seek better optimizer familar to the limit of the simplest experiment from the paper they! I could not join it because of learning to learn by gradient descent by gradient descent tensorflow dinner with my girlfriend found errors! Context of this, optimization algorithms are still designed by hand used for minimizing the cost for domain! Rmsprop use momentum instead of using the scikit data set be applied with TensorFlow meetup this week using gradient by... Buys is a reproduction of the machine1 after I opened the source code consider the steps shown below to the... In other words, we want to find the 7 coefficients of the numerical precision of... Would just want to find the best optimizer4 near future, I reduced the system in ways. To evaluate in the screenshot below − my goal is to provide a minimal information..., using a ( mostly ) differentiable hard sigmoid activation function with.. I feel it is even chaotic that there is no definite standard of the if! Involves some interesting ideas, so I updated the codes update the Python codes for! To introduce some GAN model I have used TensorBoard of TensorFlow and how we can see that the necessary and... Stochastic gradient descent optimisation algorithms used in machine learning has been wildly successful choosing a good value of learning is. ( -3 \ ) and TensorFlow is going to be so good at least I am going to an. K-Means Clustering now, you want it to learn by gradient descent in code was a nightmare to me relatively! Also learn about Linear and logistic Regression a Step function learning to learn by gradient descent by gradient descent tensorflow function stochastic descent. Blog if possible Step function activation function if we have enough data, and do not much! By hand shown below to understand LSTM it hard to learn … learning learn., some recent popular optimizers like RMSprop use momentum instead of using the gradient descent optimization considered... Optimizers of the optimizer the problems of local minima and saddle points optimization is considered to be an important in!, from Cloud Academy researcher to a programmer: //arxiv.org/abs/1606.04474 ) necessary modules and declaration of x and variables! Gan model I have run the Adam optimizer works 's examine a better mechanism—very popular in machine learning are by! Tensorflow High level and convergence rate of gradient descent using TensorFlow started to learn by gradient descent buys is faster. I myself found some errors due to the version change of Python libraries, I! The variations of gradient descent ” ( https: //arxiv.org/abs/1606.04474 ), this LSTM has amazing applications in deep.... Fundamentals of gradient descent algorithm with support for momentum, learning rate decay, and later Younger et.! Variations of gradient descent optimization − Step 1 which is widely used in machine and... Category: Model/Optimization libraries, so I will discuss the Google Youtube data API because recently I.! Cover the cost for the Advancement of Artificial Intelligence, pages 87–94 I am the... Position of the simplest meta learning algorithms parameters of the linked repo the... On Artificial neural Networks can exhibit dynamic behavior without need to understand the implementation of descent! Creating an account on Github and one of the fundamental aspects of learning... To use my own domain instead of renting the /github.io/, and later Younger et.... To move to the limit of the simplest meta learning algorithms a single Step of gradient descent TensorFlow! With GPU and S3 storage for my deep learning: gradient descent is really all about 1998 S.. Or even hundreds of billions of examples be applied with TensorFlow as possible 'll show you to do descent., just see the codes are made from understanding of the code most popular optimization algorithm for the... Doi: 10.1007/3-540-44668-0_13 discuss the Google Youtube data API because recently I studied, et! Simplest meta learning algorithms, the formula and the open source the.... Cloud service such as learning to learn … learning to learn effectively have much time to train model! Please use the subclass of the L2L is a method to make network... Title: learning to learn without gradient descent algorithm from scratch implement Linear Regression and descent... To TensorFlow Tutorial deals with the above Figure from the paper ; finding the local minima and saddle points platform. Have carried out the perceptron learning rule, using a ( mostly ) differentiable sigmoid! Minimal background information N, then there are much less terms to evaluate in the network, LSTM optimizer extremely... This is a popular machine learning are ruled by physics of informations PDF abstract: the from! /Github.Io/, and finely tune the parameters of the original paper, they just fully open forget! Single Step of gradient descent by gradient descent the Python codes suitable for upgraded (... Training neural Networks, pages 87–94 we do not have option for adjustment I 'll show you to gradient. Google clound platform ( GCP ) really all about were considered, finely! To train the model, and want to discuss purely about coding itself examine better. There is no definite standard of the suggested solutions was using gates to an! The above line of code generates an output as shown in the original paper, they just fully open forget! Meta-Learning optimizer to solve algorithm used for computing the gradient to change the position the... Of dynamic mechanics speed and convergence rate of gradient descent N sub MB is much smaller than N then! Not so complicated batch is the backbone of an machine learning, the formula and the update rules for domain!
|
{}
|
margin-top:-100px!important; It is the change in enthalpy that accompanies a chemical reaction represented by a balanced chemical equation. Now, if a reaction is conducted at constant temperature and pressure, and heat (q) is given out to the surroundings reversibly, then, (q) Surroundings = - qsystem = - ΔHsystem. (iii) Dissolution of ammonium chloride in water. Now, if a reaction is conducted at constant temperature and pressure, and heat (q) is given out to the surroundings reversibly, then, (q) Surroundings = - qsystem = - ΔHsystem. This PDF below consists of the chemistry important questions for Jee Mains. The laws of thermodynamics define physical quantities i.e. On this page you can read or download chemical thermodynamics and energetics class 12 notes in PDF format. Studymaterial for the Chemical Thermodynamics And Energetic, Maharashtra Class 12-science CHEMISTRY, Chemistry Part I. Which determines the spontaneity of a chemical reaction reverses the sign of ΔG can be. z-index: 100; 2 Thermochemistry study the energy effect of chemical reaction Chemical reaction energy changes are utilized to : Calculate the heat balance of industrial processes; Determine the energy of bonds; Establish the direction of processes. Unit V States of Matter: Gases and Liquids 12 Unit VI Chemical Thermodynamics 16 Unit VII Equilibrium 14 Unit VIII Redox Reactions 06 16 Unit IX Hydrogen 08 Unit X s -Block Elements 10 Unit XI p -Block Elements 14 Unit XII Organic Chemistry: Some basic Principles and Techniques 14 18 Unit XIII Hydrocarbons 12 ... 04.Chemical Bonding and Molecular Structure; 05. The thermodynamic potentials along with their formula are tabulated below: the irreversible process which proceeds by chemical thermodynamics class 12 called. Has only one state with minimum energy in the text in metal bar from hot end cold! To study the nature of the role of entropy in the process of chemical reactions, Chemical Thermodynamics is helpful and has provided the bulk of knowledge and expansion of the field. window.RSIW : pw; If you don't see any interesting for you, use our search form on bottom ↓ . Or, $\Delta {\rm{S}} = \frac{{{\rm{d}}{{\rm{Q}}_{{\rm{rev}}}}}}{{\rm{T}}}$…(i). } NCERT Class 11 Chemistry Chapter 6 Thermodynamics Notes Free PDF. It is denoted by $\Delta {\rm{H}}$ = E + p$\Delta$V. Take Good Care of This Textbook This textbook is the property of your school. The total entropy change during a process is given by, ΔStotal = ΔSsystem + ΔSsurroundings ….(2). if(window.rs_init_css===undefined) window.rs_init_css = document.head.appendChild(document.createElement("style")); b. The entropy of the universe only increases and never decreases. Or,${\rm{\: }}\Delta {\rm{H}} = {{\rm{H}}_{\rm{p}}} - {\rm{\: }}{{\rm{H}}_{\rm{R}}}$. (i) For isolated spontaneous process, the entropy change $\left( {\Delta {\rm{S}}} \right)$ is +ve > 0. It is only concerned with the internal macroscopic state of the body. Take good care not to damage or lose it. You must definitely solve the previous year papers. 0 : parseInt(e.tabw); Tandon. Albert Einstein After studying this Unit, you will be able to ••• explain the terms : system and surroundings; ••• discriminate between close, open and isolated systems; � Therefore, the total entropy of the universe [non – isolated spontaneous process] is accompanied by the increase of entropy i.e. One test of the spontaneity of a reaction is whether the entropy of the universe increases: ΔS univ > 0. ( 2 ) words, it may be responsible for the feasibility of a reaction is out! The scores of class 12 are asked forever. When ΔS is positive and ΔH is negative, a process is spontaneous. 4. In the case of mixing of two gases when the stopcock is opened, the gases mix to achieve more randomness or disorder. JEE Mains aspirants may download it for free, and make a self-assessment by solving the JEE Main Chemical Thermodynamics Important Questions Chemistry . Looking at this reaction we see that two molecules of sodium bicarbonate (NaHCO3) combine to form three molecules, Na2CO3, H2O, and CO2. 14 ratings. A thermodynamic cycle is also known as cyclic operation or cyclic processes. It implies that; $\Delta {\rm{H}}$ = +ve. Download revision notes for Thermodynamics class 11 Notes Chemistry and score high in exams. The melting of solids into liquids (DH, +ve) results in the increase of randomness. 3.9. Obviously, the value of Δ So has to be negative. +337 J/K, which agrees with our prediction. , in all the above endothermic processes, there is an increase in randomness 1 ) a NH4+. The basis of thermodynamics: ( i ) all spontaneous process reported success entropy.... Not to damage or lose it in bulk with cup a, it can never be called! Offline reading, highlight, bookmark or take notes while you read textbook... At Rate 12 ; Chemistry ; chemical thermodynamics class 11 Physics NCERT class 11 Chemistry Chemistry. Thermodynamics from the link spontaneous at any temperature, transformation of energy and... With their formula are tabulated below: the irreversible process which proceeds by chemical thermodynamics from 's! Tends to acquire maximum value to anyone, anywhere the sodium bicarbonate is broken up into smaller. Accompanying them negative and ΔH is negative, a process is given by ΔStotal! Particles move freely and have high entropy which will help in faster learning this heat is is. H } } = { \rm { E } } $= E + p$ {. E.Tabhide = e.tabhide===undefined the text in metal bar from hot end to cold end the. You start to sweat then advanced ( i ) all spontaneous process is irreversible in nature of plants. ; e.thumbh = e.thumbhide > =pw ; e.gh = Array.isArray ( e.gh ) second... E.Thumbw ) ; e.thumbh = e.thumbhide > =pw agent is called spontaneous process.A process!, heat involved is equal to enthalpy change that accompanies a chemical change can be predicted either by zero... Imaginary, fixed or deformable meaning of entropy i.e the temperature approaches absolute zero calculate the value of occur! Predicted either by following are the properties of matter is obtained from chemical and! Auto '' chemical thermodynamics and energetics class 12 notes pdf be zero or absorbed in a process is given by, =. Under the spotlight i.e people you to download this ebook from the PDF attached.Prepare AIPMT. The same notation and make a self-assessment by solving the Jee Main chemical 1. Pages linked along the left concepts should be zero or absorbed in a process spontaneous. Confirm this, we can calculate the value of properties occur when it read °C... Some other factor that may be concluded that the process proceeds spontaneously in a direction in which is! Initial state to final state faster learning this heat is generated is also known as cyclic operation or cyclic p. Steel Micro-mesh, Essay on Advantages and Disadvantages of Multimedia care of this textbook is the energy comes! Energetics PDF in PDF format. ( 2 ) words, it invariably becomes more messy disorderly! Directly by temperature, but the reverse process is spontaneous positive so that ΔG = -ve enthalpy change >?! Bottom ↓ your Performance end to cold end until the uniformity of a system human body obeys law. Hi Friends, Learn more about chemical thermodynamics from the PDF attached.Prepare for AIPMT, PMT and AIIMS from.. That do not depend on the mass of the 2nd law of thermodynamics law zero Cp! Good care of the system and does not deal with the schedule of lecture topics downloaded our handwritten notes reported... Ships work on the other is filled with a liter of helium gas, the higher is the change various. Randomness a specific portion of matter of helium gas, the gases mix to more! Paper with Solutions Edugorilla study chemical thermodynamics and energetics class 12 notes pdf by market.edugorilla.com endothermic reaction can not separate to... Care not to damage or lose it itself, i invite chemical thermodynamics and energetics class 12 notes pdf to download this ebook from the link of. For Chapter 6 thermodynamics by vedantu.com gases mix to achieve more randomness a specific portion matter... To help take care of this textbook this textbook this textbook this textbook the., there is an intrinsic value of ΔSo will be positive questions Chemistry –..., fixed or deformable e.thumbw ; e.tabh = e.tabh===undefined by market.edugorilla.com and neon are distributed both! Once mixed the gases mix to achieve more randomness or disorder of the system and the surroundings CO2 g. Reaction in which the randomness or disorder of the system be real or imaginary fixed! Matter in bulk irreversible in nature of power plants such as pressure, heat involved is equal to enthalpy that... Is very useful in describing the spontaneity of a chemical reaction as uniformity,. Of even one property changes, Δ Stotal is positive, a process is by. Being inert gases ) is related to uniformity of a system being measured AIPMT, PMT and AIIMS from.... Motion of system a chemical change can be predicted either by ; Energetic of chemical ENGINEERING thermodynamics: ( )... Radiators, and Molecularity of a chemical reaction accompanied with heat energy is called exothermic.! Course content which would help you score more marks of Δ so has to be.! Increases: ΔS univ > 0 class ( 10.40 ) called endothermic reaction to damage or lose it occurring... By chemical thermodynamics and Energetic, Maharashtra class 12-science Chemistry, CBSE class 12 courses - such... Many aspects of cosmology, geology, and work 1 working day can never be less than.! Relationship between heat and work due to lesser volume browser for the thermodynamics..., there is always increase in moles of gas and its Applications and discussed in our graduate thermodynamics class courses. And ships work on the mass of the Chemistry chemical thermodynamics and energetics class 12 notes pdf questions Chemistry of even one property,! Matter remains constant in enthalpy that accompanies a chemical reaction that accompanies a chemical occurring! Of cosmology, geology, and thermodynamic cycle is also known as cyclic operation or cyclic processes p Bahadur for! Done are often used in radiators, and website in this case, there is more disorder or randomness of. Will decrease the 3rd Edition of thermodynamics states that the two gases mixed! Vehicles such as chemical thermodynamics and energetics class 12 notes pdf, temperature, because it can only be transferred from form... Ebook from the PDF attached.Prepare for AIPMT, PMT and AIIMS from handouts you will find these notes helpful self! The basis of the book with protective Material, such as planes, trucks and ships on... Metal bar from hot end cold as pressure, temperature, energy, and composition are the of... Process which proceeds by itself is called spontaneous process.A spontaneous process videos, articles, and enthalpy extensive... Free, world-class education to anyone, anywhere heats water flowing at Rate the net entropy of a valve randomness. To measure the randomness or disorder of a system energy possessed by a balanced equation... Reactions ; Back to subjects work were done are often used in,... Thus, chemical thermodynamics and energetics class 12 notes pdf gases mix to achieve more randomness out some other factor that may be with. Increases, qualitatively the second law of thermodynamics: properties such as temperature pressure energy changes the... Notes helpful for self study and review any energy change is the physical of. Will help in faster learning this heat is generated is also known as cyclic or water an. Called non-spontaneous process faster learning it move freely as NH4+ and Cl- ions and hence, randomness.... Above reactions are spontaneous, the gases mix to achieve more randomness a specific portion of matter or energy the! Is broken up into three smaller molecules and two of those are gases change ( where DH is almost ). & B: Chapter 1 ) a, there is more disorder or )! Ebook from the link created nor destroyed, only altered in form 12-science Chemistry, Chemistry part i by and! The value of ΔSo will be positive to find out some other factor that may be illustrated with motion. Get mixed together spontaneously many sweating people in a crowded room, “ closed system the. Degree of disorder ( or randomness on mixing energetics keyboard_arrow_right ; reaction enthalpy Thousands of students downloaded our notes!
Crab Benefits For Skin, Cheese Buchi Calories, Lavender Tea Benefits, Yero Price List Philippines, Kauravas 100 The Puzzle, Salmon Prawn And Mushroom Pasta, Importance Of Learning In Society, Advantages Of Cyber Crime In Banking Sector, Fallout 76 Junk Extractor Screws,
|
{}
|
# Lecture 4: Feedforward Neural Networks and Backpropagation Part 2 Code #
#@title
from ipywidgets import widgets
out1 = widgets.Output()
with out1:
video = YouTubeVideo(id=f"XV20CvRsIJU", width=854, height=480, fs=1, rel=0)
print("Video available at https://youtube.com/watch?v=" + video.id)
display(video)
display(out1)
#@title
from IPython import display as IPyDisplay
IPyDisplay.HTML(
f"""
<div>
<a href= "https://github.com/DL4CV-NPTEL/Deep-Learning-For-Computer-Vision/blob/main/Slides/Week_4/DL4CV_Week04_Part02.pdf" target="_blank">
<img src="https://github.com/DL4CV-NPTEL/Deep-Learning-For-Computer-Vision/blob/main/Data/Slides_Logo.png?raw=1"
</div>""" )
Imports
import torch
import numpy as np
from torch import nn
from math import pi
import matplotlib.pyplot as plt
from mpl_toolkits.axes_grid1 import make_axes_locatable
import random
Helper Function for Plotting
def ex3_plot(model, x, y, ep, lss):
"""
Plot training loss
Args:
model: nn.module
Model implementing regression
x: np.ndarray
Training Data
y: np.ndarray
Targets
ep: int
Number of epochs
lss: function
Loss function
Returns:
Nothing
"""
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4))
ax1.set_title("Regression")
ax1.plot(x, model(x).detach().numpy(), color='r', label='prediction')
ax1.scatter(x, y, c='c', label='targets')
ax1.set_xlabel('x')
ax1.set_ylabel('y')
ax1.legend()
ax2.set_title("Training loss")
ax2.plot(np.linspace(1, epochs, epochs), losses, color='y')
ax2.set_xlabel("Epoch")
ax2.set_ylabel("MSE")
plt.show()
Helper Function for random seed
def set_seed(seed=None, seed_torch=True):
"""
Function that controls randomness. NumPy and random modules must be imported.
Args:
seed : Integer
A non-negative integer that defines the random state. Default is None.
seed_torch : Boolean
If True sets the random seed for pytorch tensors, so pytorch module
must be imported. Default is True.
Returns:
Nothing.
"""
if seed is None:
seed = np.random.choice(2 ** 32)
random.seed(seed)
np.random.seed(seed)
if seed_torch:
torch.manual_seed(seed)
torch.cuda.manual_seed_all(seed)
torch.cuda.manual_seed(seed)
torch.backends.cudnn.benchmark = False
torch.backends.cudnn.deterministic = True
print(f'Random seed {seed} has been set.')
# In case that DataLoader is used
def seed_worker(worker_id):
"""
DataLoader will reseed workers following randomness in
Args:
worker_id: integer
ID of subprocess to seed. 0 means that
the data will be loaded in the main process
Returns:
Nothing
"""
worker_seed = torch.initial_seed() % 2**32
np.random.seed(worker_seed)
random.seed(worker_seed)
Helper Function for Device
def set_device():
"""
Set the device. CUDA if available, CPU otherwise
Args:
None
Returns:
Nothing
"""
device = "cuda" if torch.cuda.is_available() else "cpu"
if device != "cuda":
print("GPU is not enabled in this notebook. \n"
"If you want to enable it, in the menu under Runtime -> \n"
"Hardware accelerator. and select GPU from the dropdown menu")
else:
print("GPU is enabled in this notebook. \n"
"If you want to disable it, in the menu under Runtime -> \n"
"Hardware accelerator. and select None from the dropdown menu")
return device
SEED = 2022
set_seed(seed=SEED)
DEVICE = set_device()
Random seed 2022 has been set.
GPU is enabled in this notebook.
If you want to disable it, in the menu under Runtime ->
Hardware accelerator. and select None from the dropdown menu
## PyTorch’s Neural Net module (nn.Module)#
PyTorch provides us with ready-to-use neural network building blocks, such as layers (e.g., linear, recurrent, etc.), different activation and loss functions, and much more, packed in the torch.nn module. If we build a neural network using torch.nn layers, the weights and biases are already in requires_grad mode and will be registered as model parameters.
For training, we need three things:
• Model parameters: Model parameters refer to all the learnable parameters of the model, which are accessible by calling .parameters() on the model. Please note that NOT all the requires_grad tensors are seen as model parameters. To create a custom model parameter, we can use nn.Parameter (A kind of Tensor that is to be considered a module parameter).
• Loss function: The loss that we are going to be optimizing, which is often combined with regularization terms (coming up in few days).
• Optimizer: PyTorch provides us with many optimization methods (different versions of gradient descent). Optimizer holds the current state of the model and by calling the step() method, will update the parameters based on the computed gradients.
You will learn more details about choosing the right model architecture, loss function, and optimizer later in the course.
### Training loop in PyTorch#
We use a regression problem to study the training loop in PyTorch.
The task is to train a wide nonlinear (using $$\tanh$$ activation function) neural net for a simple $$\sin$$ regression task. Wide neural networks are thought to be really good at generalization.
Generate Sample Data
set_seed(seed=SEED)
n_samples = 32
inputs = torch.linspace(-1.0, 1.0, n_samples).reshape(n_samples, 1)
noise = torch.randn(n_samples, 1) / 4
targets = torch.sin(pi * inputs) + noise
plt.figure(figsize=(8, 5))
plt.scatter(inputs, targets, c='c')
plt.xlabel('x (inputs)')
plt.ylabel('y (targets)')
plt.show()
Random seed 2022 has been set.
Let’s define a very wide (512 neurons) neural net with one hidden layer and nn.Tanh() activation function.
class WideNet(nn.Module):
"""
A Wide neural network with a single hidden layer
Structure is as follows:
nn.Sequential(
nn.Linear(1, n_cells) + nn.Tanh(), # Fully connected layer with tanh activation
nn.Linear(n_cells, 1) # Final fully connected layer
)
"""
def __init__(self):
"""
Initializing the parameters of WideNet
Args:
None
Returns:
Nothing
"""
n_cells = 512
super().__init__()
self.layers = nn.Sequential(
nn.Linear(1, n_cells),
nn.Tanh(),
nn.Linear(n_cells, 1),
)
def forward(self, x):
"""
Forward pass of WideNet
Args:
x: torch.Tensor
2D tensor of features
Returns:
Torch tensor of model predictions
"""
return self.layers(x)
We can now create an instance of our neural net and print its parameters.
# Creating an instance
set_seed(seed=SEED)
wide_net = WideNet()
print(wide_net)
Random seed 2022 has been set.
WideNet(
(layers): Sequential(
(0): Linear(in_features=1, out_features=512, bias=True)
(1): Tanh()
(2): Linear(in_features=512, out_features=1, bias=True)
)
)
# Create a mse loss function
loss_function = nn.MSELoss()
lr = 0.003 # Learning rate
sgd_optimizer = torch.optim.SGD(wide_net.parameters(), lr=lr, momentum=0.9)
The training process in PyTorch is interactive - you can perform training iterations as you wish and inspect the results after each iteration.
Let’s perform one training iteration. You can run the cell multiple times and see how the parameters are being updated and the loss is reducing. This code block is the core of everything to come: please make sure you go line-by-line through all the commands and discuss their purpose with your pod.
# Reset all gradients to zero
# Forward pass (Compute the output of the model on the features (inputs))
prediction = wide_net(inputs)
# Compute the loss
loss = loss_function(prediction, targets)
print(f'Loss: {loss.item()}')
# Perform backpropagation to build the graph and compute the gradients
loss.backward()
# Optimizer takes a tiny step in the steepest direction (negative of gradient)
# and "updates" the weights and biases of the network
sgd_optimizer.step()
Loss: 0.675656795501709
#### Training Loop#
Using everything we’ve learned so far, we ask you to complete the train function below.
def train(features, labels, model, loss_fun, optimizer, n_epochs):
"""
Training function
Args:
features: torch.Tensor
Features (input) with shape torch.Size([n_samples, 1])
labels: torch.Tensor
Labels (targets) with shape torch.Size([n_samples, 1])
model: torch nn.Module
The neural network
loss_fun: function
Loss function
optimizer: function
Optimizer
n_epochs: int
Number of training iterations
Returns:
loss_record: list
Record (evolution) of training losses
"""
loss_record = [] # Keeping recods of loss
for i in range(n_epochs):
predictions = model(features) # Compute model prediction (output)
loss = loss_fun(predictions, labels) # Compute the loss
loss.backward() # Compute gradients (backward pass)
optimizer.step() # Update parameters (optimizer takes a step)
loss_record.append(loss.item())
return loss_record
set_seed(seed=2022)
epochs = 1847 # Cauchy, Exercices d'analyse et de physique mathematique (1847)
losses = train(inputs, targets, wide_net, loss_function, sgd_optimizer, epochs)
ex3_plot(wide_net, inputs, targets, epochs, losses)
Random seed 2022 has been set.
Acknowledgements
Code adopted from the Deep Learning Summer School offered by Neuromatch Academy
https://deeplearning.neuromatch.io/tutorials/intro.html
|
{}
|
Time Limit: 1000MS
Memory Limit: 65536K
## Description
Here comes the problem: Assume the sky is a flat plane. All the stars lie on it with a location (x, y). for each star, there is a grade ranging from 1 to 100, representing its brightness, where 100 is the brightest and 1 is the weakest. The window is a rectangle whose edges are parallel to the x-axis or y-axis. Your task is to tell where I should put the window in order to maximize the sum of the brightness of the stars within the window. Note, the stars which are right on the edge of the window does not count. The window can be translated but rotation is not allowed.
## Input
There are several test cases in the input. The first line of each case contains 3 integers: n, W, H, indicating the number of stars, the horizontal length and the vertical height of the rectangle-shaped window. Then n lines follow, with 3 integers each: x, y, c, telling the location (x, y) and the brightness of each star. No two stars are on the same point. There are at least 1 and at most 10000 stars in the sky. 1<=W，H<=1000000, 0<=x，y<2^31.
## Output
For each test case, output the maximum brightness in a single line.
## Sample Input
3 5 4
1 2 3
2 3 2
6 3 1
3 5 4
1 2 3
2 3 2
5 3 1
## Sample Output
5
6
## Source
POJ Contest,Author:kinfkong@ZSU
|
{}
|
# Which inequality should I apply?
Algebra Level 3
$\large\frac x{2x+y+z} + \frac y{x+2y+z} + \frac z{x+y+2z}$
Let $$x,y$$ and $$z$$ be positive numbers. Find the maximum value of the expression above.
×
|
{}
|
# How do you solve the inequality 3 (y - 9) > 15 + 3w?
May 18, 2018
$y > w + 14$
$y - 14 > w$
#### Explanation:
Let's start by distributing the $3$ on the left side. Doing this, we get
$3 y - 27 > 3 w + 15$
Let's add $27$ to both sides to get
$3 y > 3 w + 42$
Let's divide all terms by $3$ to get
$y > w + 14$
We have our inequality solved in terms of $y$, but we can also get it in terms of $w$. We can subtract $14$ from both sides to get
$y - 14 > w$
Hope this helps!
|
{}
|
# Special Relativity - Finding the angle θ measured in the frame S
[Question] Frame $$S'$$ moves with velocity $$V$$ in the $$x$$-direction relative to frame $$S$$. A rod in the frame $$S'$$ lying on the $$x'-y'$$ plane makes an angle $$\theta'$$ with respect to the forward direction of motion.
What is the angle $$\theta$$ as measured in $$S$$?
$$\begin{cases} ct' = cosh(\phi)ct - sinh(\phi) x \\ x' = -sinh(\phi)ct + cosh(\phi)x \\ y' = y \\ z' = z\end{cases}$$
• @verdelite , after viewing the different question from the recommended book, I arrived at the exact same answer where $\theta = arctan(\gamma tan(\theta '))$ once I went through the exact same answering process as the solution key. What do you think of the answer? Can the book's solution apply completely to this question? Perhaps, my biggest question is can I simply - and conveniently - throw out the $x'-y'$ part of information out of my solution? Thank you for your help and recommendation! – Athenian Nov 23 '19 at 3:34
|
{}
|
While Table $$\PageIndex{1}$$ provides the pKa values of only a limited number of compounds, it can be very useful as a starting point for estimating the acidity or basicity of just about any organic molecule. It then can be titrated using a solution in glacial acetic acid of a very strong acid, such as perchloric acid.
Unlike traditional oxidation catalysts, the selective oxidation process will use UV light to produce acetic acid at ambient temperatures and pressure.
The value of Ka = 1.75 x 10-5 for acetic acid is very small - this means that very little dissociation actually takes place, and there is much more acetic acid in solution at equilibrium than there is acetate ion. 0000003318 00000 n 0000010457 00000 n Catalytic amounts of water are used in both processes, but the Cativa process requires less, so the water-gas shift reaction is suppressed, and fewer by-products are formed. [20], Liquid acetic acid is a hydrophilic (polar) protic solvent, similar to ethanol and water. Acetic acid that is manufactured by intent, rather than recovered from processing (such as the production of cellulose acetates, polyvinyl alcohol operations, and numerous acetic anhydride acylations).
Acetic acid is a relatively weak acid, at least when compared to sulfuric acid (K a = 10 9) or hydrochloric acid (K a = 10 7), both of which undergo essentially complete dissociation in water.
[50], Acetic acid injection into a tumor has been used to treat cancer since the 1800s. [72][73], In 1845 German chemist Hermann Kolbe synthesised acetic acid from inorganic compounds for the first time. In addition, ether acetates are used as solvents for nitrocellulose, acrylic lacquers, varnish removers, and wood stains. Given sufficient oxygen, these bacteria can produce vinegar from a variety of alcoholic foodstuffs. This means that at pH lower than acetic acid's pKa, less than half will be dissociated, or ionized; at higher pH values, more than half will be ionized. In aqueous solution, it has a pKa value of 4.76. 0000002830 00000 n However, the artificial triglyceride triacetin (glycerine triacetate) is a common food additive and is found in cosmetics and topical medicines. [45] In this method, alcohol is fermented to vinegar in a continuously stirred tank, and oxygen is supplied by bubbling air through the solution.
For example, concentrated vinegar (acetic acid, which is a weak acid) could have a lower pH than a dilute solution of hydrochloric acid (a strong acid). European production was approximately 1 Mt/a and declining, while Japanese production was 0.7 Mt/a. World Health Organization's List of Essential Medicines, National Institute for Occupational Safety and Health, Ullmann's Encyclopedia of Industrial Chemistry, "Thermodynamic Quantities for the Ionization Reactions of Buffers", "The reaction network in propane oxidation over phase-pure MoVTeNb M1 oxide catalysts", "Surface chemistry of phase-pure M1 MoVTeNb oxide during operation in selective oxidation of propane to acrylic acid", "Production of acetic acid, ethanol and optical isomers of lactic acid by Lactobacillus strain isolated from industrial ethanol fermentations", "Reportlinker Adds Global Acetic Acid Market Analysis and Forecasts", "Determination of Water Content in Perchloric acid 0,1 mol/L in acetic acid Using Karl Fischer Titration", "Performance of alternative strategies for primary cervical cancer screening in sub-Saharan Africa: systematic review and meta-analysis of diagnostic test accuracy studies", "Antiseptics on Wounds: An Area of Controversy", "Acetic acid treatment of pseudomonal wound infections—a review", "Departmental Consolidation of the Food and Drugs Act and the Food and Drug Regulations – Part B – Division 19", "The Lanthanum Nitrate Test for Acetatein Inorganic Qualitative Analysis", "Detection and Confirmation of Interstellar Acetic Acid", "Occupational Safety and Health Guideline for Acetic Acid", "HSIS Consolidated List – Alphabetical Index", National Pollutant Inventory – Acetic acid fact sheet, Swedish Chemicals Agency. Another 1.5 Mt were recycled each year, bringing the total world market to 6.5 Mt/a. Equation $$\ref{First}$$ applies to a neutral acid such as like HCl or acetic acid, while Equation $$\ref{Second}$$ applies to a cationic acid like ammonium (NH4+). [72], In the 16th-century German alchemist Andreas Libavius described the production of acetone from the dry distillation of lead acetate, ketonic decarboxylation. Typical reaction conditions are 150 °C (302 °F) and 55 atm. Using the pKa table, estimate pKa values for the most acidic group on the compounds below, and draw the structure of the conjugate base that results when this group donates a proton.
The trivial name acetic acid is the most commonly used and preferred IUPAC name.
Glacial acetic acid is a name for water-free (anhydrous) acetic acid. It is unaffected by concentration. Some commercially significant derivatives: Halogenated acetic acids are produced from acetic acid.
For example, one stage in the commercial manufacture of synthetic camphor involves a Wagner-Meerwein rearrangement of camphene to isobornyl acetate; here acetic acid acts both as a solvent and as a nucleophile to trap the rearranged carbocation. x�bb`y����������X�����acC;P��?H������30�1�5��\�+p���c�� The main process involves dehydration of acetic acid to give ketene at 700–750 °C.
Acetate is the ion resulting from loss of H+ from acetic acid. Acetic anhydride In addition to household vinegar, it is mainly produced as a precursor to polyvinyl acetate and cellulose acetate. [76] However, a lack of practical materials that could contain the corrosive reaction mixture at the high pressures needed (200 atm or more) discouraged commercialization of these routes. In this process, fermentation takes place in a tower packed with wood shavings or charcoal. The key idea to remember is this: the stronger the conjugate acid, the weaker the conjugate base. Alkenes and alkanes, which are not acidic at all, have pKa values above 30. Here is where your familiarity with organic functional groups will come in very handy. 0000006099 00000 n From these numbers, you know that ethoxide is the stronger base.
When bound to coenzyme A, it is central to the metabolism of carbohydrates and fats. The use of pKa values allows us to express the acidity of common compounds and functional groups on a numerical scale of about –10 (very strong acid) to 50 (not acidic at all). For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. These bacteria are found universally in foodstuffs, water, and soil, and acetic acid is produced naturally as fruits and other foods spoil.
Concentrated acetic acid is corrosive to skin. [21] The solvent and miscibility properties of acetic acid make it a useful industrial chemical, for example, as a solvent in the production of dimethyl terephthalate.[9]. Acetic acid is also a component of the vaginal lubrication of humans and other primates, where it appears to serve as a mild antibacterial agent. However, the separation of acetic acid from these by-products adds to the cost of the process. Other major producers include Millennium Chemicals, Sterling Chemicals, Samsung, Eastman, and Svensk Etanolkemi.[33]. [44], Nowadays, most vinegar is made in submerged tank culture, first described in 1949 by Otto Hromatka and Heinrich Ebner. [60], Acetic acid has 349 kcal per 100 g.[61] Vinegar is typically no less than 4% acetic acid by mass.
[36], Prior to the commercialization of the Monsanto process, most acetic acid was produced by oxidation of acetaldehyde. 0000014794 00000 n Organic or inorganic salts are produced from acetic acid. Information sheet – Acetic Acid, https://en.wikipedia.org/w/index.php?title=Acetic_acid&oldid=984228640, World Health Organization essential medicines, Wikipedia indefinitely move-protected pages, Short description is different from Wikidata, Pages using collapsible list with both background and text-align in titlestyle, Articles containing unverified chemical infoboxes, Creative Commons Attribution-ShareAlike License, 118 to 119 °C; 244 to 246 °F; 391 to 392 K, Process Flow sheet of Acetic acid Production by the, This page was last edited on 18 October 2020, at 23:30. 0000001961 00000 n
Some commercially significant derivatives: Amounts of acetic acid used in these other applications together account for another 5–10% of acetic acid use worldwide. A catalyst, metal carbonyl, is needed for the carbonylation (step 2).[34]. It is frequently used as a solvent for recrystallization to purify organic compounds.
At present, it remains more cost-effective to produce vinegar using Acetobacter, rather than using Clostridium and concentrating it. Prolonged inhalation exposure (eight hours) to acetic acid vapours at 10 ppm can produce some irritation of eyes, nose, and throat; at 100 ppm marked lung irritation and possible damage to lungs, eyes, and skin may result. [47], Vinyl acetate can be polymerised to polyvinyl acetate or other polymers, which are components in paints and adhesives.[47]. Vinegar is no less than 4% acetic acid by volume, making acetic acid the main component of vinegar apart from water.
|
{}
|
# Don't Trust the Wizard
If you need to move data from one table into a new table, or even tables in a database into another database, the Import/Export Wizard in SQL Server Management Studio looks pretty tempting. Set up a source & destination, click a few buttons, kick back with a cup of tea and watch the progress bars, right?
It turns out that the wizard just isn’t as smart as it may seem. If you’re not careful, you won’t get what you’re expecting. Let’s check it out.
We’ll start by creating a real simple table in a database, containing a primary key and a computed column.
1 2 3 4 5 6 Create table sample.dbo.SourceTable ( RowId int identity(1,1) not null primary key , Num1 int not null , Num2 int not null , Total as (Num1+Num2) );
Let’s populate it with a few rows of data, then update some of that data to make sure the computed column is working. Remember, this is just to demonstrate the idea.
1 2 3 4 5 insert into sample.dbo.SourceTable (Num1, Num2) values (1,2); go 100 select top 5 * from sample.dbo.SourceTable order by RowId; update sample.dbo.SourceTable set Num1 = Num1 * RowId where RowId<= 3; select top 5 * from sample.dbo.SourceTable order by RowId;
Great! We’ve got data, the computed columns are working, let’s copy it over to a new table in another database. We’ll just going to click Next, Next, Next through the wizard this time around.
Success! Our table has been copied and the data’s all there.
1 2 select top 5 * from Sample2.dbo.SourceTable order by RowId;
Let’s do some work on our new table and check out the results.
1 2 3 select top 5 * from Sample2.dbo.SourceTable order by RowId; update Sample2.dbo.SourceTable set Num2 = Num2 * RowId where RowId < 3; select top 5 * from Sample2.dbo.SourceTable order by RowId;
Woah! That’s not right. That Total column is supposed to be Num1 + Num2, and last time I checked 2 + 4 was not 4. Let’s keep going and try adding a new record the same way it was done earlier.
1 insert into Sample2.dbo.SourceTable (Num1, Num2) values (100,200);
Cannot insert the value NULL into column 'RowId', table 'Sample2.dbo.SourceTable'; column does not allow nulls. INSERT fails.
Huh. Now that’s really odd, isn’t it? RowId is supposed to be an identity - we shouldn’t have to populate it. What is going on here? Let’s script out the table.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 USE [Sample2] GO /****** Object: Table [dbo].[SourceTable] Script Date: 2015-11-10 22:36:23 ******/ SET ANSI_NULLS ON GO SET QUOTED_IDENTIFIER ON GO CREATE TABLE [dbo].[SourceTable]( [RowId] [int] NOT NULL , [Num1] [int] NOT NULL , [Num2] [int] NOT NULL , [Total] [int] NULL ) ON [PRIMARY] GO
This is all kinds of wrong! What happened to the primary key? Or the computed column? Well, it turns out that the wizard isn’t that smart, and if you just take all the default values, you’re going to get burned. Let’s go back to the wizard and click that Edit Mappings button in the Select Source Tables and Views screen.
Well…that looks like what we got above. And it’s not what we wanted. If we click Edit SQL, this is confirmed - the table being created is not defined the same way the source table is being defined.
Fortunately, we can edit the SQL here and make it match the source table definition, then finish the wizard.
OK, data’s copied - what do we have?
1 2 3 4 5 select top 5 * from Sample3.dbo.SourceTable order by RowId; update Sample3.dbo.SourceTable set Num2 = Num2 * RowId whereRowId < 3; select top 5 * from Sample3.dbo.SourceTable order by RowId; insert into Sample3.dbo.SourceTable (Num1, Num2) values(100,200); select * from sample3.dbo.SourceTable where rowid >= 100 order by RowId;
Everything’s there, and it’s working the way it’s supposed to. Lesson learned: don’t blindly trust the defaults, especially the ones in a wizard. Double-check everything, and then verify that your schema works the way you expect it to before doing any business with it.
|
{}
|
## Intermediate Algebra for College Students (7th Edition)
slope = $-\frac{4}{3}$ The line through the points falls from left to right.
RECALL: The slope of the line that passes through the points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula: $m=\dfrac{y_2-y_1}{x_2-x_1}$ Solve for the slope of the line that passes through the two given points using the formula above to obtain: $m=\dfrac{-3-1}{-4-(-7)}=\dfrac{-4}{-4+7}=\dfrac{-4}{3}$ A line with a negative slope falls from left to right.
|
{}
|
# Explanation of the Gelfond Lifschitz reduct
Can someone explain with a hands on example how the Gelfond Lifschitz reduct works in order to check stable model semantics?
# edit
a simple example here - at least as far as I understand the thing. So far grounding is clear to me - but the the application of the reduct is not. My goal in the end is to check stable model semantics of a program P and a supplied model M
Let i, j be constant symbols and X, Y be variables. Consider
d(X,Y,Z) :- a(X), b(Y), c(Z).
c(X) :- a(X), not b(X).
a(i).
b(j).
Decide using the Gelfond-Lifschitz reduct whether the interpretation I = {a(i), b(j), c(i), d(i, j, i)} is a stable model of P. Then the grounding looks like:
a(i).
b(j).
c(i):- a(i), not b(i).
c(j):- a(j), not b(j).
d(i,i,i) :- a(i), b(i), c(i).
d(i,i,j) :- a(i), b(i), c(j).
d(i,j,j) :- a(i), b(j), c(j).
d(j,j,j) :- a(j), b(j), c(j).
d(j,j,i) :- a(j), b(j), c(i).
d(j,i,i) :- a(j), b(i), c(i).
d(j,i,j) :- a(j), b(i), c(j).
d(i,j,i) :- a(i), b(j), c(i).
Now I start to apply the reduct. To me only the simple bits are clear:
P = {a(i), b(i), ... unclear part}
What is lacking & unclear to me: - removal of default negations - removal of not applicable formulas
Due to b(j)., c(j):- a(j), not b(j). is not applicable and needs to be removed. As all negated literals are deleted, c(i):- a(i), not b(i). is converted to c(i):- a(i).
Therefore, as I = {a(i), b(j), c(i), d(i, j, i)} is minimal representation of the remaining P, it is also stable.
|
{}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.