text:
User eric zaslow - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:46:26Z http://mathoverflow.net/feeds/user/1186 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94347/eigenforms-for-laplacian-on-a-non-flat-two-torus Eigenforms for Laplacian on a non-flat two-torus Eric Zaslow 2012-04-18T02:20:17Z 2012-04-20T11:40:44Z
Does anyone know an explicit, exact description of the eigenforms of the Laplacian on a non-flat two-torus?
 http://mathoverflow.net/questions/36444/definition-of-an-e-infinity-algebra Definition of an E-infinity algebra Eric Zaslow 2010-08-23T13:20:53Z 2011-05-26T21:20:19Z
Hello. Can anyone give me a plain-and-simple definition of an E-infinity algebra without using the words "operad," "ring spectrum," or "stable homotopy"?

Sorry, but I honestly couldn't find it using all on-line resources at my disposal.

Thanks!
 http://mathoverflow.net/questions/58169/why-do-a-infty-functors-form-an-a-infty-category/60014#60014 Answer by Eric Zaslow for Why do A_\infty functors form an A_\infty category? Eric Zaslow 2011-03-29T19:24:38Z 2011-03-30T01:59:07Z
I can explain the pictures I usually draw to think of $A_\infty$ functors, but I don't know if they're standard. Anyway, I'll describe what is just a rubric for ingesting the long formulas, nothing more.

Let's consider first the Yoneda embedding $Y$, which re-thinks an object $L$ in an $A_\infty$-category $A$ as an $A$-module, or functor from $A^{op}$ to chain complexes. So $Y_L(M) = hom_A(M,L).$

I confess that when I confront these formulas/concepts, I always think in terms of the Fukaya category, which is very amenable to pictures and for which the $A_\infty$ structures are geometric.

So I draw a curve on a piece of paper and label it $L$. (The curve is literally a Lagrangian submanifold of my ${\mathbb R}^2$ piece of paper.) When I want to think of $L$ in terms of its Yoneda image, I draw the SAME curve, but as a squiggly line.

So what is the data that the squiggly line gives us? For each object $M$ (a regular curve on my paper), we have the intersection points, which form a graded vector space $hom_A^*(M,L).$ This vector space has the structure of a chain complex (Floer), with differential given by football-shaped bi-gons with one regular side and one squiggly side. For a pair of other objects, $M_1, M_2,$ we get a map $$\mu^2: hom_A(M_2,L)\otimes hom_A(M_1,M_2) \rightarrow hom_A(M_1,L),$$ and so on for all the structure of a module (section 1j, p. 19).

For the Fukaya category, the equations 1.19 follow (for non-squiggly lines) from studying degenerations of 1-parameter families of holomorphic polygons. Now squigglifying those same pictures gives 1.19 for an arbitrary module, and the equations are similar for not just modules but arbitrary functor between two $A_\infty$-categories.

What data do we have if we have two squiggly lines $L_1$ and $L_2$? They should intersect at a morphism between functors (and it should have a degree). This morphism of functores gives more data, using the Fukaya perspective. If we added one normal line $M$, we'd have the spaces $Y_{L_1}(M)$ and $Y_{_2}(M)$, and have a triangle which is a map between them. Higher polygons and the relations between them (by considering one-parameter families) should give you all the equations and give you a hint as to verify them. (But no promises!)

Hope that lengthy and pretty vague description was worth our time. (Oh, geez, this was a March 11 question? Probably stale by now!)
 http://mathoverflow.net/questions/57520/examples-in-mirror-symmetry-that-can-be-understood/58199#58199 Answer by Eric Zaslow for Examples in mirror symmetry that can be understood. Eric Zaslow 2011-03-11T21:08:08Z 2011-03-11T21:08:08Z
Here is my biased view of a simple example: the two-torus. Everything I know about homological mirror symmetry stems from this example.

Because the example is one-dimensional, a symplectic form is just an area form, and Lagrangians are simply curves, and the holomorphic maps which are part of the Fukaya category are simply topological disks. (By uniformization of Riemann surfaces, there is one holomorphic map for each topological disk satisfying the appropriate boundary conditions.) Even better, you can go to the universal cover, which is $R^2,$ and just draw Lagrangians as straight lines with rational slope. The holomorphic disks which determine compositions in the category are simply triangles.

On the mirror side, we're talking about a complex two-torus, or elliptic curve. A typical object would be a line bundle on the elliptic curve, such as the theta line bundle, whose sections are theta functions, once we lift them up to the complex plane.

The two-torus is circle-fibered over a base circle, and the elliptic curve is circle-fibered by the dual circle (i.e., $U(1)$ local systems on the original circle). This is called T-duality, and it explains how to construct the mirror equivalence going from Lagrangians to line bundles, or vice versa. For example, the Lagrangian ${y=0}$ represents a family of trivial $U(1)$ local systems, corresponding to the trivial holomorphic line bundle whose sections are just holomorphic functions. The Lagrangian ${y=nx}$ corresponds to a line bundle of degree $n$. After making these definitions, one checks that compositions match up.
 http://mathoverflow.net/questions/51204/is-there-a-crash-course-book-on-abelian-varieties-e-g-an-introduction-for-ph/51212#51212 Answer by Eric Zaslow for Is there a "crash-course" book on Abelian varieties (e.g., an introduction for physicists)? Eric Zaslow 2011-01-05T16:08:53Z 2011-01-05T16:08:53Z
The book by Polishchuk should be very helpful, and close in spirit to the nonexistent crash course on Mumford's paper discussed by Charles Matthews in his reponse:

http://www.cambridge.org/gb/knowledge/isbn/item1169225/?site_locale=en_GB
 http://mathoverflow.net/questions/19065/when-your-paper-makes-a-borderline-case-for-a-top-journal/50309#50309 Answer by Eric Zaslow for When your paper makes a borderline case for a top journal Eric Zaslow 2010-12-24T22:04:28Z 2010-12-24T22:04:28Z
This requires a conditional response. First, who is submitting?

Tenured professor: Not much repurcussion either way. Worth the wait if you think it deserves publication.

Graduate student: Some downside to waiting a long time and not having the thing published come application time, but this can happen at a lesser journal, too (for grads, the paper is often submitted only shortly before they are graduating). And postdoc hiring is not as publications-based as tenure-track hiring. Letters of recommendation mean more.

Postdoc. Here, submitting with a high probability of a rejection after a long wait can be a major gambit. Mitigating factor: does the postdoc have other worthy publications? If this will be the flagship result, it's hard not to think that a slightly lesser journal would have a higher expected yield (in terms of jobs). Note that it is true that some journals may reject your paper quickly -- then you can turn around and submit somewhere else -- but those papers are not really the marginally-great ones being asked about.

In the end, you are left with a hard decision. I don't think there's a formula which can help. In this case, you either go for broke or you don't.

Collaboration: decide based on the member with the most to lose.

 http://mathoverflow.net/questions/49024/mirror-symmetry-with-algebraic-geometry/49488#49488 Answer by Eric Zaslow for mirror symmetry with algebraic geometry? Eric Zaslow 2010-12-15T06:23:15Z 2010-12-15T06:23:15Z
Here are a few scattered observations:

Our ability to construct examples (e.g. of CY manifolds) is limited, and the tools of algebraic geometry are perfectly suited to doing so (as has been noted).

Toric varieties are a source of many examples -- Batyrev-Borisov pairs -- and they are even "more" than algebraic, they're combinatorial. In fact, the whole business is really about integers in the end, so combinatorics reigns supreme.

The fuzziness of $A_\infty$ structures is more suited to algebraic topology rather than geometry.

Continuity of certain structures (which are created from counting problems) across walls, scores some points for analysis over combinatorics and algebra.

Elliptic curves are only "kinda" algebraic, and the mirror phenomenon there is certainly transcendental.

Physics indeed does not care too much about how the spaces are constructed, but (as has been noted) even the non-topological version of mirror symmetry is an equivalence of a very algebraic structure (which includes representations of superconformal algebras).

I was hoping to unify these idle thoughts into a coherent response, but I don't think I can. Maybe the algebraic geometric aspects just grew faster because the mathematics is "easier" (or at least better understood by more mathematicians): witness the slow uptake of BCOV and its antiholomorphicity within mathematics.

To respond personally: these days, I try to transfer the algebraic and symplectic structures to combinatorics so that I can hold them in my hand and try to understand them better.
 http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/48704#48704 Answer by Eric Zaslow for Examples of non-rigorous but efficient mathematical methods in physics Eric Zaslow 2010-12-09T03:28:18Z 2010-12-09T03:28:18Z
Finally, a Math Overflow question that addresses my specialty: non-rigor!

Here are a few examples of non-rigor as applied to evidence for dualities:

Heterotic-Type II. In earlier times, the best evidence for heterotic-Type-II duality was a) counting the number of supersymmetries of the theory, and (b) comparing the moduli spaces.

AdS-CFT. For AdS-CFT the earliest and best comparisons were counting the so-called anomalous dimensions of various operators. To date, I think the tests are far from rigorized (and yes, this would be a great problem to make mathematically precise).

Mirror Symmetry, early days. Recall that mirror symmetry in CY moduli space came from constructing a chart of the Euler characteristics of CY complete intersections and noticing the symmetry of the chart about zero. Other non-rigorous arguments involve counting the dimensions (just the dimensions) of the moduli of purportedly mirror objects. Then there's the old compute-on-flat-space-and-let-supersymmetry-take-care-of-the-rest trick.

Low energy effective field theory. The "fact" that string theory reduces to an oft-identifiable QFT in a low energy limit is a huge source of argumentation/inspiration in string theory. Accounting for (effective) black holes helped lead to M-theory in one context, and to the microscopic description of black-hole entropy in another. One can also argue for dualities by identifying equivalent field contents in two different models. This brings up another point.

Invariance of BPS states under perturbation. It is great to take a quantity that does not vary and evaluate it in a limit where it is easy to compute. This argument appears again and again in physics -- and also in math, of course (e.g. in the heat-kernel proof of the index theorem). BPS numbers are just that. (Of course, they do vary, and the continuity of the relevant physical parameters [numbers are not necessarily physical quantities] is what underlies interesting explanations of wall-crossing.)

I'm probably including too many that don't fit and excluding a lot that do. Very non-rigorous of me!
 http://mathoverflow.net/questions/47852/examples-of-mixed-hodge-structures Examples of Mixed Hodge Structures Eric Zaslow 2010-12-01T01:59:00Z 2010-12-03T14:54:26Z
Does anyone know a user-friendly, example-laden introduction to mixed Hodge structures? I get from Wikipedia how to calculate for a punctured and pinched curve (http://en.wikipedia.org/wiki/Hodge_structure#Mixed_Hodge_structures), but I want more! I want tables and numbers and everything explicit and spoonfed. Thanks!
 http://mathoverflow.net/questions/47770/string-theory-computation-for-math-undergrad-audience/47790#47790 Answer by Eric Zaslow for String theory "computation" for math undergrad audience Eric Zaslow 2010-11-30T13:17:29Z 2010-11-30T13:17:29Z
Two counting problems -- from my own very biased and personal viewpoint -- that can perhaps be motivated:

Counting triangles on the torus = theta function relation for elliptic curve. (I tried to squeeze this into a public lecture one time.)

Counting symmetric polynomials of degree k in 24 variables = partition function of chiral bosonic string => counting curves on K3 by heterotic duality: 24, 24 + 24*25/2 = 324, etc.

But these can't beat calculating an actual partition function (as in Richard Eager's answer), unless you're trying to emphasize mathiness.
 http://mathoverflow.net/questions/47466/how-do-i-compute-the-compact-cohomology-of-a-hypersurface How do I compute the compact cohomology of a hypersurface? Eric Zaslow 2010-11-26T21:48:20Z 2010-11-27T23:18:52Z
How do I compute the compact cohomology of a hypersurface? For example, let $f$ be a Newton polynomial of a polytope in $\mathbb{R}^n$ and let $X = (f=0)$ inside $(\mathbb{C}^*)^n$ (maybe there is some dependency on the coefficients of $f\;$?). Can you tell me anything about $H^*_c(X)$? Perhaps I should know better, but I don't. Thanks!
 http://mathoverflow.net/questions/47082/exterior-derivative-on-almost-complex-manifolds/47092#47092 Answer by Eric Zaslow for Exterior derivative on almost complex manifolds Eric Zaslow 2010-11-23T13:27:20Z 2010-11-23T13:27:20Z
In writing $\omega$ you used a symbol $dz$ which doesn't make sense unless there is a holomorphic coordinate. Your $dz$ should really be an element of a frame of (1,0) 1-forms, which need not be closed (as you have assumed).
 http://mathoverflow.net/questions/46524/non-kahler-calabi-yau/46543#46543 Answer by Eric Zaslow for Non-Kahler "Calabi-Yau"? Eric Zaslow 2010-11-18T21:54:54Z 2010-11-18T21:54:54Z
This is covered in Andrei Halanay's answer, but it's worth mentioning the simplest examples, which are primary Kodaira surfaces. For the simplest of these:

Take C^2 and quotient by the group generated by these a_k:

a_1 : z -> z + 1

a_2 : z -> z + i

a_3 : w -> w + z + 1

a_4 : w -> w - iz + i

(I think this is it.)

The quotient group is nonabelian. Here z is the fiber and w the base.
 http://mathoverflow.net/questions/46418/relation-between-toric-geometry-and-log-geometry/46454#46454 Answer by Eric Zaslow for relation between toric geometry and log geometry Eric Zaslow 2010-11-18T03:43:08Z 2010-11-18T03:43:08Z
Sloppy stab at a reformulation:

I think we are assuming P is commutative? (\sigma^\vee \cap M certainly is.) If so, then isn't P spanned by all linear combinations of the generators? But then there are relations, too, etc. So the question might be, does P embed as a submonoid of a free abelian group over Z? (I don't know.) If so, convexity of the corresponding domain is clear, and finite generatedness means it's cut out by a finite number of conditions. Your lattice is the Z-span of P and your cone is then the (dual of the) R-convex hull.
 http://mathoverflow.net/questions/45802/undergraduate-math-research/46040#46040 Answer by Eric Zaslow for Undergraduate math research Eric Zaslow 2010-11-14T13:39:36Z 2010-11-14T13:39:36Z
Most major topics have been covered in discussion, so just two remarks/experiences:

While director of graduate studies at Northwestern (2007-2010), I led a committee which valued undergraduate preparedness over research experience. So at least as far as Northwestern was concerned during that time frame, research (especially research for which an undergrad may not be fully prepared) did not help as much as you might have thought.

In trying to use RTG funds toward undergraduates, rather than try to simulate a research environment, I and my co-PI's created an "undergraduate conference" to try to offer a supplement to standard undergraduate curricula, without yet getting on toward research. Here is the link: http://www.math.northwestern.edu/summerconference/ Maybe we'll do it again next year?

 http://mathoverflow.net/questions/17325/why-is-cotangent-more-canonical-than-tangent Why is cotangent more canonical than tangent? Eric Zaslow 2010-03-06T22:40:59Z 2010-10-24T19:28:48Z
You don't need a metric to define the differential of a function, and the cotangent bundle carries a canonical one-form.

But you do need a metric to define the gradient, and the tangent bundle does not have a canonical vector field.

These are not difficult truths, but still... why the preference toward "co"?
 http://mathoverflow.net/questions/38651/compute-the-kahler-moduli-of-an-elliptic-curve/38723#38723 Answer by Eric Zaslow for compute the Kähler moduli of an elliptic curve Eric Zaslow 2010-09-14T18:34:07Z 2010-09-14T18:43:13Z
The curve you wrote in equations lies in C^2, while the "elliptic curve" of your text is presumably a compact projective variety -- meaning you imagine making your equations homogeneous (or even quasi-homogeneous) and considering the closure of the set of points described by your equation in a (quasi-)projective plane. Not every "homogenization" will lead to an elliptic curve (Calabi-Yau) upon compactification, so you have to do this correctly (as noted by Kevin Buzzard above).

Having said that, the answer is that every projective variety is also Kahler: just restrict e.g. the Fubini-Study(-like) Kahler form. In plain English, since a Kahler form on a complex curve is just a volume form, the volume of the compact curve inside projective space gives you your answer.
 http://mathoverflow.net/questions/36596/refereeing-a-paper/38069#38069 Answer by Eric Zaslow for Refereeing a Paper Eric Zaslow 2010-09-08T16:07:00Z 2010-09-08T16:07:00Z
I've had long discussions about whether the referee report should be author-dependent. If mathematician X and mathematician Y submit the same paper, should the report be the same?
 http://mathoverflow.net/questions/37384/mirror-of-flop/37403#37403 Answer by Eric Zaslow for Mirror of Flop? Eric Zaslow 2010-09-01T17:07:26Z 2010-09-01T17:07:26Z
I assume the question regards the coherent sheaves on these two CY's. These CY's should be regarded as the "same" complex manifold with two different choices of complexified symplectic forms ("Kahler form," in physics terminology).

The mirrors are a "single" symplectic manifold with two different complex structures on it. There is a curve of complex structures relating the two.

That's about it. The tricky part is to "parallel transport" the category of coherent sheaves along this curve, using a "flat family of categories" defined by stability conditions. Doing so should provide a preferred isomorphism of the categories. Examples have been studied, but general statements (like the ones I have glibly been making) are not proven.
 http://mathoverflow.net/questions/30000/do-you-understand-syz-conjecture/32197#32197 Answer by Eric Zaslow for Do you understand SYZ conjecture Eric Zaslow 2010-07-16T16:00:36Z 2010-07-16T16:00:36Z
Hi-

Just saw this thread. Maybe I should comment. The conjecture can be viewed from the perspective of various categories: geometric, symplectic, topological. Since the argument is physical, it was written in the most structured (geometric) context -- but it has realizations in the other categories too.

Geometric: this is the most difficult and vague, mathematically, since the geometric counterpart of even a conformal field theory is approximate in nature. For example, a SUSY sigma model with target a compact complex manifold X is believed to lie in the universality class of a conformal field theory when X is CY, but the CY metric does not give a conformal field theory on the nose -- only to one loop. Likewise, the arguments about creating a boundary conformal field theory using minimal (CFT) + Lagrangian (SUSY) are only valid to one loop, as well. To understand how the corrections are organized, we should compare to (closed) GW theory, where "corrections" to the classical cohomology ring come from worldsheet instantons -- holomorphic maps contributing to the computation by a weighting equal to the exponentiated action (symplectic area). The "count" of such maps is equivalent by supersymmetry to an algebraic problem. No known quantity (either spacetime metric or Kahler potential or aspect of the complex structure) is so protected in the open case, with boundary. That's why the precise form of the instanton corrections is unknown, and why traction in the geometric lines has been made in cases "without corrections" (see the work of Leung, e.g.). Nevertheless, the corrections should take the form of some instanton sum, with known weights. The sums seem to correspond to flow trees of Kontsevich-Soibelman/ Moore-Nietzke-Gaiotto/Gross-Siebert, but I'm already running out of time.

Topological: Mark Gross has proven that the dual torus fibration compactifies to produce the mirror manifold.

Symplectic: Wei Dong Ruan has several preprints which address dual Lagrangian torus fibrations, which come to the same conclusion as Gross (above). I don't know much more than that.

Also-

Auroux's treatment discusses the dual Lagrangian torus fibration (even dual slag, properly understood) for toric Fano manifolds, and produces the mirror Landau-Ginzburg theory (with superpotential) from this.

With Fang-Liu-Treumann, we have used T-dual fibrations for the same fibration to map holomorphic sheaves to Lagrangian submanifolds, proving an equivariant version of homological mirror symmetry for toric varieties. (There are many other papers with similar results by Seidel, Abouzaid, Ueda, Yamazaki, Bondal, Auroux, Katzarkov, Orlov -- sorry for the biased view!)

Reversing the roles of A- and B-models, Chan-Leung relate quantum cohomology of a toric Fano to the Jacobian ring of the mirror superpotential via T-duality.

Help or hindrance?
 http://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners-list/22446#22446 Answer by Eric Zaslow for Mathematicians who were late learners?-list Eric Zaslow 2010-04-24T18:03:15Z 2010-04-24T18:03:15Z
Unfortunately, all these exceptions appear to be reaches, thus proving the rule.
 http://mathoverflow.net/questions/18631/mirror-of-local-calabi-yau/19320#19320 Answer by Eric Zaslow for Mirror of local Calabi-Yau Eric Zaslow 2010-03-25T15:45:29Z 2010-03-25T15:45:29Z
I think this is a stubborn case which does not fit into the general picture. For example, if you use the standard toric procedure to try to construct a differential equation for the log periods of the mirror, then try to find the GW invariants from the solutions (in this case, to recover the famous 1/d^3 formula), then it won't quite work. There are various adjustments you can make, based on knowing the answer ahead of time, and I once saw a paper (sorry, I forget where or by whom) which tried to make sense of all this, but I don't think there is a good general picture of this case. I would check Klemm's written record for some guidance.
 http://mathoverflow.net/questions/8692/higher-genus-closed-string-b-model/17319#17319 Answer by Eric Zaslow for Higher genus closed string B-model Eric Zaslow 2010-03-06T21:42:29Z 2010-03-06T21:42:29Z
One thing missing from this discussion is the even-more-mysterious holomorphic ambiguity (not "anomaly"). BCOV is not deterministic, and should probably be thought of as part of a general schema for a B-model-type topological theory. What I mean is that there are other solutions to the BCOV equation which do not yield GW invariants. The reason is that BCOV determines the partition function only up to a holomorphic function at each genus. The choice of this function -- analogous to initial conditions -- is usually set by matching to GW invariants or by imposing some structure at the singularities of CY moduli space.

I don't think there is a comprehensive understanding of how this ambiguity is resolved, though there are cases where it is fixed by other symmetries or properties of the full partition function.
 http://mathoverflow.net/questions/17238/which-part-of-physical-b-model-is-not-rigorous/17283#17283 Answer by Eric Zaslow for Which part of physical B model is not rigorous? Eric Zaslow 2010-03-06T13:55:16Z 2010-03-06T13:55:16Z
To define (as Kevin Lin does above) the B-model purely as the derived category of coherent sheaves is fine and rigorous, but it ignores the higher-genus aspects of mirror symmetry -- which was the original question. As I wrote above, Kevin Costello gives a rigorous description of the higher-genus amplitudes, but it is still conjectural whether this agrees with the physics. The issue is that higher-genus string amplitudes depend on an integration over the moduli space of Riemann surfaces (or a space of maps from them, depending on the model), and this demands compactification. The full, non-topological theory is of course an ordinary two-dimensional quantum field theory, with all the usual difficulties in making the path integral rigorous.
 http://mathoverflow.net/questions/17238/which-part-of-physical-b-model-is-not-rigorous/17245#17245 Answer by Eric Zaslow for Which part of physical B model is not rigorous? Eric Zaslow 2010-03-05T23:47:56Z 2010-03-05T23:47:56Z
Kevin Costello's mathematical definition of the B-model (math/0509264) is rigorous. It's an open problem whether this definition agrees with the BCOV construction, as far as I know.
 http://mathoverflow.net/questions/7823/negative-gromov-witten-invariants/7873#7873 Answer by Eric Zaslow for Negative Gromov-Witten invariants Eric Zaslow 2009-12-05T16:14:30Z 2009-12-05T16:14:30Z
Let me use an example. If I recall correctly (I am too lazy to look), the GW invariant for "local P^2" (i.e. the canonical bundle O(-3) over P^2) in genus zero and degree 1 is 3, and in degree 2 is -45/8. Now -45/8 breaks down, after trying to account for multiple covers, to 3/2^3 - 6, giving an effective number of degree 2 curves as -6. What up? This number is how many degree 2 curves the P^2 should "account for," when it pops into existence within a family of Calabi-Yau's. (Here of course we have a family of maps into the P^2.) Then you should ask, "Does this mean that for a compact CY to have just rigid genus zero curves and an embedded P^2 (and no other surface) and lie in a family that includes a CY with only rigid genus zero curves and no other surface, that it must have at least 6 degree 2 rational curves somewhere else?" Presumably.
 http://mathoverflow.net/questions/101314/are-donaldson-thomas-invariants-a-model-or-b-model/103701#103701 Comment by Eric Zaslow Eric Zaslow 2012-08-15T12:31:43Z 2012-08-15T12:31:43Z Fun! DT is B-model and GW is A-model. Nothing spooky there (yet). [Richard: GW numbers are locally invariant under symplectic deformations, just like your DT numbers. (I say "locally" because we could cross Kahler cone walls.) The "dependence" is in the parameter governing the generating function.] The DT partition function encodes nonperturbative (in the string coupling, g) contributions from the B-branes, while the GW partition function encodes perturbative (in g) contributions from A-branes. The relation between the perturb/nonperturb expansions in Type IIB is called S-duality. http://mathoverflow.net/questions/103120/does-the-moduli-space-of-smooth-curves-of-genus-g-contain-an-elliptic-curve/103140#103140 Comment by Eric Zaslow Eric Zaslow 2012-07-26T05:42:09Z 2012-07-26T05:42:09Z Jim, if I understood their talks correctly, your old result was just used by Donagi and Witten to show that the moduli space of super-Riemann surfaces does not split -- meaning superstring perturbation theory itself needs to be "revisited." http://mathoverflow.net/questions/99352/important-open-questions-in-the-field-of-tropical-geometry Comment by Eric Zaslow Eric Zaslow 2012-06-13T14:27:48Z 2012-06-13T14:27:48Z Perhaps the best way to demonstrate why I voted for this question is to reveal my own ignorance and confusion about this subject. Many younger researchers may carry the same uncertainties. How much of the package'' of algebraic geometry carries over to the tropical geometry? When working with the tropical semigroup ring, what topological spaces can you construct from tropical varieties and how exactly do they compare to varieties over C? What general results have been proven in tropical Gromov-Witten theory? If these questions are not open, there must be related ones that are, right? http://mathoverflow.net/questions/31789/statistical-physics-of-string-theory/31791#31791 Comment by Eric Zaslow Eric Zaslow 2012-05-20T01:46:29Z 2012-05-20T01:46:29Z I'm confused by this dispute. If I understand, Clay Cordova says "gravitons are always there," so gravity does not emerge through some averaging process, while jeremy says that the thermodynamics of classical gravitational systems is described by microstates, so represents an aggregate notion. It seems both parties are conscientious and knowledgeable and both are correct by their own definition of what it means to be "statistical" (a weighted word). The problem lies in the vagueness of the question, not in the answers. http://mathoverflow.net/questions/94347/eigenforms-for-laplacian-on-a-non-flat-two-torus/94627#94627 Comment by Eric Zaslow Eric Zaslow 2012-04-20T15:44:42Z 2012-04-20T15:44:42Z Thanks, Robert! That's exactly where I had arrived, having considered a torus with $S^1$ isometry. For particular $F$ you can make it look a lot like, but not precisely equal to, the "quantum pendulum." That system is described by the well-studied Matthieu equation, but again it's not precisely the correct equation (or at least I can't make it be). http://mathoverflow.net/questions/94347/eigenforms-for-laplacian-on-a-non-flat-two-torus Comment by Eric Zaslow Eric Zaslow 2012-04-18T11:43:59Z 2012-04-18T11:43:59Z Ooops! Sorry, Robert et al! Yes, the question I wrote is trivial. I meant to say the eigenforms of the Hodge Laplacian! Stupid me. Will update. http://mathoverflow.net/questions/85856/places-to-learn-about-landau-ginzburg-models Comment by Eric Zaslow Eric Zaslow 2012-02-18T06:58:06Z 2012-02-18T06:58:06Z If you know that in physics the energy is a sum of kinetic energy and potential energy, then the appearance of a potential in a physical theory should not be too surprising. The potential here is constructed from a holomorphic function, of which there is none for a compact manifold, but possibly many if the target space is noncompact. http://mathoverflow.net/questions/84954/can-you-make-the-cotangent-bundle-to-a-complex-manifold Comment by Eric Zaslow Eric Zaslow 2012-01-06T05:14:07Z 2012-01-06T05:14:07Z The Nijenhuis tensor lives upstairs and doesn't have so many symmetries, while the curvature tensor lives downstairs (with half the number of vectors) with more symmetries -- so the relation is not direct, and involves the splitting of T(TM) into TM+TM. Maybe the exact relation is in Kobayashi-Nomizu? http://mathoverflow.net/questions/84954/can-you-make-the-cotangent-bundle-to-a-complex-manifold Comment by Eric Zaslow Eric Zaslow 2012-01-05T15:06:38Z 2012-01-05T15:06:38Z The metric-induced almost complex structure is not integrable unless M is flat. (The Nijenhuis tensor is essentially the curvature.) Now interpreting Tim Perutz's excellent response to the question you cite: if M is compact, then there is an integrable complex structure on T*M by Eliashberg's result. It is not Kahler, in general. A Kahler metric exists on T*M = TM in a neighborhood of the zero section ("Grauert tube"). http://mathoverflow.net/questions/57520/examples-in-mirror-symmetry-that-can-be-understood/58199#58199 Comment by Eric Zaslow Eric Zaslow 2011-03-16T12:27:11Z 2011-03-16T12:27:11Z The paper with Polishchuk works through the example in detail, as opposed to just the sketch above. You seem to be still somewhat dissatisfied. Why don't you say precisely which aspect of mirror symmetry you are looking to uncover in your example? (Three references for torus fibrations are Arinkin-Polishchuk, Leung-Yau-Z, and Mark Gross's "Topological Mirror Symmetry.") http://mathoverflow.net/questions/57520/examples-in-mirror-symmetry-that-can-be-understood/58199#58199 Comment by Eric Zaslow Eric Zaslow 2011-03-14T17:31:08Z 2011-03-14T17:31:08Z Well, I can offer you my own article with Polishchuk: arxiv.org/abs/math/9801119 Below, AByer mentions that HMS implies that we get an isomorphism of our category for each loop in moduli space. These isomorphisms ("autoequivalences") alone can lead to the mirror map, as demonstracted by a calculation for this example in arxiv.org/abs/math/0506359. The T-duality aspect can be pushed in many other directions and examples, too. (Sorry for the lazy self-promotion! There are, of course, many other articles by other authors, some of which are included in the answers below.) http://mathoverflow.net/questions/51656/mathematics-of-gun-control Comment by Eric Zaslow Eric Zaslow 2011-01-11T14:13:24Z 2011-01-11T14:13:24Z @Charles: Thanks, I'll try again. @Steven Gubkin: That was a Ph.D. question in econonmics, as far as I understand, and a scholarly paper (with the opposite conclusion), so it would probably be a question under the tag "reference request." Here is a link: en.wikipedia.org/wiki/… http://mathoverflow.net/questions/51656/mathematics-of-gun-control Comment by Eric Zaslow Eric Zaslow 2011-01-11T09:43:58Z 2011-01-11T09:43:58Z @Andy Putman: Thanks for the reference! I don't think I am permitted to contribute to meta, so I cannot broach the issue of whether any question with math content (not nec. mine) but also with political ramifications is ever permissible on MO. Admittedly unnerving, this meta issue should (maybe already has been) be addressed, as MO is the only major free forum for math-related exchange. I "get" slippery-slope reluctance (lots of down votes and negative remarks!), but not all slopes are equally slippery. Up-voters may feel the issue compelling enough for the math community to weigh in on. http://mathoverflow.net/questions/51656/mathematics-of-gun-control Comment by Eric Zaslow Eric Zaslow 2011-01-11T03:04:06Z 2011-01-11T03:04:06Z @Pete L. Clark: Following your suggestion, I asked a professor of statistics. This is actually a problem of the application of statistics to law, a legitimate and specialized branch of the discipline. I doubt that a non-statistician could do this well, even given the data (but this is conjectural; certainly I cannot) -- and which data would be the best to collect? Now there is no Statistics Overflow (as far as I know), but there is a statistics tag on MO. The academic bona fides of the question are no worse than many soft questions, at the least. (AFAIK, issue's not in mission of UCS.) http://mathoverflow.net/questions/51656/mathematics-of-gun-control Comment by Eric Zaslow Eric Zaslow 2011-01-10T16:44:36Z 2011-01-10T16:44:36Z @JSE: Thank you for the explanation. I was hoping for a statistically meaningful and rigorous way of analyzing data. I don't statistics well enough to attach probabilities/error-bounds to any conclusions. What can one say after collecting lots of pieces of evidence and crossing against local/national laws? (I won't post any actual data, so as to keep this meta-conversation on-topic.) Is this settled matter? We have data, but not a controlled experiment. Obviously, the topic is incendiary. We could replace "gun-control" with "math-homework-enforcement" but we wouldn't have good data.


text:
1
At a room temperature of $300K$, calculate the thermal noise generated by two resistors of $10K\Omega$ and $20K\Omega$ when the bandwidth is $10KHz$. $1.2868\times10^{-6}V, 1.819\times10^{-6}V$ $6.08\times10^{-6}V, 15.77\times10^{-6}V$ $16.66\times10^{-6}V, 2.356\times10^{-6}V$ $1.66\times10^{-6}V, 0.23\times10^{-6}V$
2
In testing phase, the effort distribution is upto 10% 20% 40% 50%
3
HTML (HyperText Markup Language) has language element which permit certain actions other than describing the structure of the web document. Which one of the following actions is NOT supported by pure HTML (without any server or client side scripting) pages? (A) Embed ... a specified interval (C) Automatically redirect to another page upon download (D) Display the client time as part of the page
4
Which of the following statements is/are true? P : An XML document with correct syntax as specified by W3C is called Well Formed . Q : An XML documented validated against a DTD is both Well formed and valid . R : <xml version= 1.0 encoding = UTF-8 > is syntactly ... document. Select the correct answer from the options given below: P and Q only P and R only Q and R only All of P, Q and R
5
Consider the Boolean function $z(a,b,c)$. Which one of the following minterm lists represents the circuit given above? $z=\sum (0,1,3,7)$ $z=\sum (1,4,5,6,7)$ $z=\sum (2,4,5,6,7)$ $z=\sum (2,3,5)$
6
Suppose $\begin{pmatrix} 0&1 &0&0&0&1 \\ 1&0&1&0&0&0 \\ 0&1&0&1&0&1 \\ 0&0&1&0&1&0 \\ 0&0&0&1&0&1 \\ 1&0&1&0&1&0 \end{pmatrix}$ is the adjacency matrix of an undirected graph with six vertices: that is, the rows and columns are indexed by vertices of the graph, and an entry is $1$ if the ... has the above adjacency matrix? Only $(i)$ Only $(ii)$ Only $(iii)$ Only $(iv)$ $(i)$ and $(ii)$
7
An organization requires a range of IP address to assign one to each of its $1500$ computers. The organization has approached an Internet Service Provider (ISP) for this task. The ISP uses CIDR and serves the requests from the available IP address space $202.61.0.0/17$. The ISP wants to assign an ... $202.61.64.0/21$ $202.61.144.0/21$ I and II only II and III only III and IV only I and IV only
8
The figure below shows an annular ring with outer and inner as $b$ and $a$, respectively. The annular space has been painted in the form of blue colour circles touching the outer and inner periphery of annular space. If maximum $n$ ... $\pi [(b^{2}-a^{2})+n(b-a)^{2}]$
9
Two transactions $T_1$ and $T_2$ are given as $T_1:r_1(X)w_1(X)r_1(Y)w_1(Y)$ $T_2:r_2(Y)w_2(Y)r_2(Z)w_2(Z)$ where $r_i(V)$ denotes a $\textit{read}$ operation by transaction $T_i$ on a variable $V$ and $w_i(V)$ denotes a $\textit{write}$ operation by transaction $T_i$ on a variable $V$. The total number of conflict serializable schedules that can be formed by $T_1$ and $T_2$ is ______
10
Consider the recurrence function $T(n) = \begin{cases} 2T(\sqrt{n})+1, & n>2 \\ 2, & 0 < n \leq 2 \end{cases}$ Then $T(n)$ in terms of $\Theta$ notation is $\Theta(\log \log n)$ $\Theta( \log n)$ $\Theta (\sqrt{n})$ $\Theta(n)$
11
If the characteristic polynomial of a 3 $\times$ 3 matrix $M$ over $\mathbb{R}$ (the set of real numbers) is $\lambda^3 – 4 \lambda^2 + a \lambda +30, \quad a \in \mathbb{R}$, and one eigenvalue of $M$ is 2, then the largest among the absolute values of the eigenvalues of $M$ is _______
12
Consider the following languages. $L_1 = \{a^p \mid p \text{ is a prime number} \}$ $L_2 = \{ a^nb^mc^{2m} \mid n \geq 0, m \geq 0 \}$ $L_3 = \{a^n b^n c^{2n} \mid n \geq 0 \}$ $L_4 = \{ a^n b^n \mid n \geq 1\}$ Which of the ... $L_2$ is not context free $L_3$ is not context free but recursive $L_4$ is deterministic context free I, II and IV only II and III only I and IV only III and IV only
13
Consider a machine with a byte addressable main memory of $2^{32}$ bytes divided into blocks of size 32 bytes. Assume that a direct mapped cache having 512 cache lines is used with this machine. The size of the tag field in bits is _______
14
Let $L(R)$ be the language represented by regular expression $R$. Let $L(G)$ be the language generated by a context free grammar $G$. Let $L(M)$ be the language accepted by a Turing machine $M$. Which of the following decision problems are undecidable? Given a regular expression $R$ and a ... $w$, is $w \in L(M)$? I and IV only II and III only II, III and IV only III and IV only
15
The read access times and the hit ratios for different caches in a memory hierarchy are as given below: $\begin{array}{|l|c|c|} \hline \text {Cache} & \text{Read access time (in nanoseconds)}& \text{Hit ratio} \\\hline \text{$ ... are for instruction fetch and $40$% are for memory operand fetch. The average read access time in nanoseconds (up to $2$ decimal places) is _________
16
$G$ is an undirected graph with $n$ vertices and $25$ edges such that each vertex of $G$ has degree at least $3$. Then the maximum possible value of $n$ is _________ .
17
Match the following according to input (from the left column) to the compiler phase (in the right column) that processes it: ... $\text{P-iii; Q-iv; R-i; S-ii}$ $\text{P-i; Q-iv; R-ii; S-iii}$
18
In a B+ Tree , if the search-key value is $8$ bytes long , the block size is $512$ bytes and the pointer size is $2$ B , then the maximum order of the B+ Tree is ____
19
Consider the set of process with arrival time (in milliseonds), CPU burst time (in millisecods) and priority ($0$ ... The average waiting time (in milli seconds) of all the process using premtive priority scheduling algorithm is ______
20
Consider two hosts $X$ and $Y$, connected by a single direct link of rate $10^6 \hspace{0.1cm} bits/sec$. The distance between the two hosts is $10,000 \hspace{0.1cm} km$ and the propagation speed along the link is $2 \times 10^8 \hspace{0.1cm} m/sec$. Host $X$ ... value of $p$ and $q$ are $p=50$ and $q=100$ $p=50$ and $q=400$ $p=100$ and $q=50$ $p=400$ and $q=50$
21
$X$ is a $30$ digit number starting with the digit $4$ followed by the digit $7$. Then the number $X^3$ will have $90$ digits $91$ digits $92$ digits $93$ digits
22
A test has twenty questions worth $100$ marks in total. There are two types of questions. Multiple choice questions are worth $3$ marks each and essay questions are worth $11$ marks each. How many multiple choice questions does the exam have? $12$ $15$ $18$ $19$
23
A system shares $9$ ... of the following best describes current state of the system? Safe, Deadlocked Safe, Not Deadlocked Not Safe, Deadlocked Not Safe, Not Deadlocked
24
The representation of the value of a $16-bit$ unsigned integer $X$ in hexadecimal number system is $BCA9$. The representation of the value of $X$ in octal number system is $571244$ $736251$ $571247$ $136251$
25
Consider the following C program. #include<stdio.h> #include<string.h> int main() { char* c=”GATECSIT2017”; char* p=c; printf(“%d”, (int)strlen(c+2[p]-6[p]-1)); return 0; } The output of the program is _______
26
Consider the following C function int fun(int n) { int i, j; for(i=1; i<=n; i++) { for (j=1; j<n; j+=i) { printf("%d %d", i, j); } } } Time complexity of $fun$ in terms of $\Theta$ notation is $\Theta(n \sqrt{n})$ $\Theta(n^2)$ $\Theta(n \: \log n)$ $\Theta(n^2 \log n)$
If $f(x) = R \: \sin ( \frac{\pi x}{2}) + S, f’\left(\frac{1}{2}\right) = \sqrt{2}$ and $\int_0^1 f(x) dx = \frac{2R}{\pi}$, then the constants $R$ and $S$ are $\frac{2}{\pi}$ and $\frac{16}{\pi}$ $\frac{2}{\pi}$ and 0 $\frac{4}{\pi}$ and 0 $\frac{4}{\pi}$ and $\frac{16}{\pi}$
Consider the following function implemented in C: void printxy(int x, int y) { int *ptr; x=0; ptr=&x; y=*ptr; *ptr=1; printf(“%d, %d”, x, y); } The output of invoking $printxy(1,1)$ is: $0, 0$ $0, 1$ $1, 0$ $1, 1$

format: plain


text:
# Math Help - Mixing problem

1. ## Mixing problem

A country has 5 billion in paper currency. Each day, 30 million is deposited into banks and the same amount loaned out. There is a regime change and each day the 30 million is replaced by a new currency when loaned out. How long will it take for 90% of the currency to be new?

I know the answer, but I'm having trouble arriving at it.

2. If the new currencies of one day are not changed with the new currencies of the following days, then you need to solve $30n=5000(90/100)$. The solution is $n=150$

3. The solution is 384. Not all of the money coming in is old after day 1. Thanks though, I solved it last night.


text:
## Solutions of nonlinear Dirac equations.(English)Zbl 1126.49003

Existence and multiplicity results for stationary solutions of the Dirac equation $$-i\partial _{t}\psi =ic\hslash \sum_{k=1}^{3}a_{k}\partial _{k}\psi -mc^{2}\beta \psi +\nabla _{\psi }G(x,\psi )$$ are established via variational method. The associated Lagrangian functional is strongly indefinite and the Palais-Smale (PS) condition is not satisfied. Some recent saddle point theorems with Cerami type (PS) condition are applied to the considered functional.

### MSC:

 49J35 Existence of solutions for minimax problems 35Q40 PDEs in connection with quantum mechanics
Full Text:

### References:

 [1] Balabane, M.; Cazenave, T.; Douady, A.; Merle, F., Existence of excited states for a nonlinear Dirac field, Comm. math. phys., 119, 153-176, (1988) · Zbl 0696.35158 [2] Balabane, M.; Cazenave, T.; Vazquez, L., Existence of standing waves for Dirac fields with singular nonlinearities, Comm. math. phys., 133, 53-74, (1990) · Zbl 0721.35065 [3] Bartsch, T.; Ding, Y.H., On a nonlinear Schrödinger equation with periodic potential, Math. ann., 313, 15-37, (1999) · Zbl 0927.35103 [4] Bartsch, T.; Ding, Y.H., Homoclinic solutions of an infinite-dimensional Hamiltonian system, Math. Z., 240, 289-310, (2002) · Zbl 1008.37040 [5] T. Bartsch, Y.H. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, preprint · Zbl 1117.58007 [6] Bartsch, T.; Pankov, A.; Wang, Z.Q., Nonlinear Schrödinger equations with steep potential well, Commun. contemp. math., 3, 549-569, (2001) · Zbl 1076.35037 [7] Bjorken, J.D.; Drell, S.D., Relativistic quantum fields, (1965), McGraw-Hill · Zbl 0184.54201 [8] Cazenave, T.; Vazquez, L., Existence of local solutions for a classical nonlinear Dirac field, Comm. math. phys., 105, 35-47, (1986) · Zbl 0596.35117 [9] Coti-Zelati, V.; Ekeland, I.; Séré, E., A variational approach to homoclinic orbits in Hamiltonian systems, Math. ann., 288, 133-160, (1990) · Zbl 0731.34050 [10] Coti-Zelati, V.; Rabinowitz, P., Homoclinic type solutions for a semilinear elliptic PDE on $$\mathbb{R}^N$$, Comm. pure appl. math., 45, 1217-1269, (1992) · Zbl 0785.35029 [11] Dautray, R.; Lions, J.L., Mathematical analysis and numerical methods for science and technology, vol. 3, (1990), Springer-Verlag Berlin [12] Esteban, M.J.; Séré, E., Stationary states of the nonlinear Dirac equation: A variational approach, Comm. math. phys., 171, 323-350, (1995) · Zbl 0843.35114 [13] Esteban, M.J.; Séré, E., An overview on linear and nonlinear Dirac equations, Discrete contin. dyn. syst., 8, 281-397, (2002) · Zbl 1162.49307 [14] Finkelstein, R.; Lelevier, R.; Ruderman, M., Non-linear spinor fields, Phys. rev., 83, 326-332, (1951) · Zbl 0043.21603 [15] Kryszewki, W.; Szulkin, A., Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. differential equations, 3, 441-472, (1998) · Zbl 0947.35061 [16] Lions, P.L., The concentration-compactness principle in the calculus of variations. the locally compact cases, part II, Ann. inst. H. Poincaré anal. non linéaire, 1, 223-283, (1984) · Zbl 0704.49004 [17] Merle, F., Existence of stationary states for nonlinear Dirac equations, J. differential equations, 74, 50-68, (1988) · Zbl 0696.35154 [18] Ranada, A.F., Classical nonlinear Dirac field models of extended particles, () [19] Reed, M.; Simon, B., Methods of mathematical physics, vols. I-IV, (1978), Academic Press [20] Séré, E., Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209, 27-42, (1992) · Zbl 0725.58017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
View more editions
Solutions
Fundamentals of Momentum, Heat and Mass Transfer

# TEXTBOOK SOLUTIONS FOR Fundamentals of Momentum, Heat and Mass Transfer 5th Edition

• 837 step-by-step solutions
• Solved by publishers, professors & experts
• iOS, Android, & web
Over 90% of students who use Chegg Study report better grades.
May 2015 Survey of Chegg Study Users
Chapter: Problem:
SAMPLE SOLUTION
Chapter: Problem:
• Step 1 of 5

Figure depicts steady state heat conduction through the asbestos pad in x direction.

• Step 2 of 5

Consider a cross section at a distance of from the small end of the asbestos pad and the side of that cross section is .

Use the following relation to determine the value of :

Here, is the side of square at small end of pad, is the side of square at large end of pad and L is the length of the asbestos pad.

Substitute for , for and for L.

• Step 3 of 5

Now, the area of the section at a distance is calculated as follows:

Substitute for .

• Step 4 of 5

Consider a small element of length at where the temperature difference on the two surfaces is .

Therefore, according to Fourier’s law, the heat conduction rate in the x-direction is written as follows:

Here, k is the thermal conductivity of asbestos.

Substitute for .

…… (1)

• Step 5 of 5

Integrate both sides of equation (1), apply limits 0 to for x and to for T and substitute for k.

Therefore, the heat flow rate in -direction is .

Corresponding Textbook

Fundamentals of Momentum, Heat and Mass Transfer | 5th Edition
9780470128688ISBN-13: 0470128682ISBN:


text:
# Poincaré-Hopf theorem and its applications

I'm reading the basics of differential topology to try to understand the Poincaré-Hopf theorem, its proof and its applications. My plan is as follows:

1) Study transversality: its homotopy stability + genericity (any application is homotopic (isotopic) to a transverse application to a given submanifold).

2) Consider the intersection number modulo 2 + oriented intersection number.

3) Study Lefschetz maps (and their fixed points).

So far, this is technical but not too complicated...I do not know how to show that:

1) the formula $\sum_i\mathrm{ind}(X,x_i)$ gives the Euler-Poincaré characteristic (that I can define from de Rham cohomology I know and not from the singular homology, I do not know yet).

2) Show that the formula in the Poincaré-Hopf theorem does not depend on the choice of the vector field. In particular how to link this issue to the Lefschetz fixed point?

On the other hand, I have a problem to calculate the index of a vector field in an isolated singular point. If the point is non-degenerate, the calculation is simple. Otherwise, I do not know whether to go through the calculation of topological index of the Gauss map (where calculations are not always easy, because I must start from a regular value $y$, determine the fiber $f^{-1}(y)=\{x_1,...,x_k\}$ and calculate the sum $\sum\mathrm{sign}\det T_{x_i}f$).

I am afraid the following reasoning I made is wrong: the Euler characteristic of any Lie group is zero. For this, I start with a compact Lie group, as it is parallelizable, it admits a vector field without singularities and from the Poincaré-Hopf theroem, its Euler characteristic is zero. By the rigisity theorem of Mostow, any Lie group has the homotopy type of a compact Lie group and therefore also of Euler characteristic zero.

Thank you to enlighten me on these points and suggest me the necessary literature to treat this questions.

• Your argument that the Euler characteristic of (connected !) compact lie groups is zero works perfectly fine. I don't know the rigidity theorem, so I don't know if your conclusion is ok. I hope you considered having a look at Milnor, Topology from the differentiable viewpoint. He explains (or sketches) two points you're looking for: using the Gauss map to get independence of the vector field, and using Morse theory to build the bridge to the Euler characteristic. I don't exactly know how to avoid Morse theory at that point, maybe you have to have a look at that too. – Ben Jul 2 '13 at 21:22
• Many thanks, Ben! Now, I am sure, the Euler Characteristic of any Lie group is zero. The book by Milnor is great, but he only sketches the proof of the Poincaré-Hopf theorem... – amine Jul 11 '13 at 17:18


text:
# truc9

Depiction of a large-scale random hexagonal tiling and a key to indicate which value of $p$ in the one-parameter family of models is being depicted.

Eleanor is a maths PhD student at UCL. She works on biomechanical models of nerves and is interested in the applications of mathematical modelling to biological and medical problems.

• ### In conversation with Trachette Jackson

We chat with Trachette about her work in mathematical oncology, her role models, and boosting diversity in mathematics
• ### Page 3 model: Cooking spaghetti

Some saucy modelling
• ### Page 3 model: Bees

You won't bee-lieve it
• ### Black mathematician month 2019

A month celebrating the contributions of black mathematicians
• ### Page 3 model: Game of Thrones

Using graph theory to predict who will sit the iron throne
• ### Black Mathematician Month 2018

We look back at last year's Black Mathematician Month, and give a preview of what to expect this October.


text:
# Is this transfinite Cheryl's birthday problem correct?

I found Joel David Hamkins' riddle quite fun and wanted to construct one for a larger ordinal. I removed the fraction aspect since I think that would add too much difficulty to this version. However, I'm not 100% sure that the puzzle is unambiguous/correct as stated so I'm hoping people can point out if there's an error. I'll spoiler the answer in case anyone wants to think about it independently first.

Cheryl is throwing a birthday party, and as an excellent host decides to give each guest a gift. As a not so great host, her gift to each of them consists of one of the two following items:

• A month of the year
• A pair consisting of a month and a finite length, non-empty sequence of non-negative integers

Two gifts may be compared for "size" as follows:

• If both gifts are just months, the later month is larger
• If one gift is a month and the other is a pair, the month is larger
• If both are pairs and one has a later month, the pair with the later month is larger
• If both months are equal, sequences are compared first by length (longer sequences are bigger), then lexicographically

Albert and Bernard arrive and receive distinct gifts, which they each keep hidden.

Cheryl: Which of you do you think has the larger gift?

Albert: I don't know

Bernard: Neither do I

Albert: I still don't know

Bernard: And still either do I

Cheryl: Well, no matter how long you two go back and forth like that you'll never know who has the bigger gift

Albert: Despite that hint, I still don't know

Bernard: Neither do I

Albert: I still don't know

Bernard: And still neither do I

Cheryl: Let me call that last statement $$S_0$$, and say that no matter how many times we went back and forth with you two saying you don't know, and me saying $$S_0$$, you still wouldn't know who has the bigger gift

Albert: Despite that hint I still don't know, and also I find that subscript a bit ominous

Bernard: I also don't know

Cheryl: Call my last statement $$S_1$$, and now I will tell you that no matter how many times we alternate with you two saying you don't know, and myself saying $$S_0$$ or $$S_1$$, you still will not know who has the larger number

Albert: And yet I still don't know

Bernard: Me neither

Cheryl: Let me now define $$S_n$$ inductively to be the statement that no matter how many times we alternate with you two saying you don't know, and myself saying any $$S_k$$ for $$k < n$$, you two will still not know who has the larger gift. I now tell you that all statements $$S_n$$ are true.

Bernard: Me neither

Albert: Ah I now know who has the bigger gift!

Bernard: Oh! Now I know as well, and also I know what gift you got

Albert: Then I know what gift you got as well

I believe that with this set of utterances, the solution is that

Albert received a pair of February, and a one element sequence: $$[1]$$, and Bernard received a pair of February and $$[2]$$

My intention is that the gifts are ordered with order type

$$\omega^\omega \cdot 12 + 12$$. The $$+ 12$$ is added as a confounder, since I think the obvious wrong answer would be that all pairs are ruled out so they both must each have only a month.

Edit: Let me also add my reasoning so that someone can comment directly on whether it is correct:

I believe that each "I don't know" rules out the lowest available final number of a finite-sequence for the speaker, so the initial I don't knows are stepping through (January, $$[1]$$), (January, $$[2]$$) etc. They also eliminate the month-only gifts in descending order.

Cheryl asserting the truth of $$S_n$$ increments the lowest possible $$n$$-th element of the sequence in a (January, sequence) pair by 1 (and $$S_0$$ rules out all month-only gifts). Therefore, since all $$S_n$$ are true, exactly the pairs with January as their month are ruled out, so we're left with the just the version on the natural numbers (the 1 element sequences)

• Can the integers in a sequence be negative? Feb 6 at 0:28
• Ah great catch, they should be natural numbers let me edit that. Feb 6 at 0:30


text:
# What percent of 80 is 56?

Aug 21, 2016

70%

#### Explanation:

We can also ask this as "56 is what percent of 80?

Dividing $\frac{56}{80}$ will give decimal answer of $0.70$

Change it into a percentage by xx 100%

56/80 xx 100% = 0.70 xx 100% = 70%

We could also have worked with equivalent fractions where one fraction has a denominator of 100.

56/80 xx 1.25/1.25 = 70/100 = 70%#


text:
# American Institute of Mathematical Sciences

• Previous Article
A parallel water flow algorithm with local search for solving the quadratic assignment problem
• JIMO Home
• This Issue
• Next Article
Closed-loop supply chain network equilibrium model with retailer-collection under legislation
January  2019, 15(1): 221-233. doi: 10.3934/jimo.2018040

## A comparative study on three graph-based constructive algorithms for multi-stage scheduling with blocking

 1 School of Economics and Management, University of Electronic Science and Technology of China, Chengdu 611731, China 2 School of Economics and Management, Fuzhou University, Fujian Province 350108, China 3 School of Mathematical Sciences, Queensland University of Technology, 2 George St, Brisbane Queensland 4001, Australia

* Corresponding author: Shi Qiang Liu

Received  May 2017 Revised  November 2017 Published  January 2019 Early access  April 2018

Fund Project: This work was partially supported by the National Natural Science Foundation of China under Grant Nos. 71571032, 71531003, 71572156 and 71572030.

In this paper, the blocking conditions are investigated in permutation flow shop, general flow shop and job shop environments, in which there are no buffer storages between any pair of machines. Based on an alternative graph that is an extension of classical disjunctive graph, a new and generic polynomial-time algorithm is proposed to construct a feasible schedule with a given job processing sequence, especially for satisfying complex blocking constraints in multi-stage scheduling environments. To highlight the state-of-the-art of the proposed algorithm, a comparative analysis is conducted in comparison to two other constructive algorithms in the literature. The comparison shows that the proposed algorithm has the following advantages: $i$) it is more adaptive because it can be applied to three different types of scheduling problems (i.e., permutation flow-shop, general flow-shop and job-shop) without any modifications; $ii$) it is able to quickly evaluate whether a schedule is feasible (acyclic) or infeasible (cyclic) through checking the availability of the topological order in a directed alternative graph model; $iii$) it is able to determine the critical path which is useful to design the neighborhood moves in the development of metaheuristics.

Citation: Pengyu Yan, Shi Qiang Liu, Cheng-Hu Yang, Mahmoud Masoud. A comparative study on three graph-based constructive algorithms for multi-stage scheduling with blocking. Journal of Industrial & Management Optimization, 2019, 15 (1) : 221-233. doi: 10.3934/jimo.2018040
##### References:

show all references

##### References:
Blocking conditions on a pair of operations processed on the same machine
The result of a three-machine four-job BPFSS instance, obtained by the proposed alternative-graph-based constructive algorithm
The result of a three-machine four-job BPFSS instance, obtained by the directed-graph-based constructive algorithm
A directed alternative graph for a feasible BJSS schedule
A cyclic directed alternative graph for an infeasible BGFSS schedule
A cyclic directed alternative graph for an infeasible BGFSS schedule
The processing times of four jobs in a numerical example
 M1 M2 M3 J1 p1=1 p5=3 p9=3 J2 p2=1 p6=2 p10=2 J3 p3=4 p7=1 p11=4 J4 p4=2 p8=2 p12=2
 M1 M2 M3 J1 p1=1 p5=3 p9=3 J2 p2=1 p6=2 p10=2 J3 p3=4 p7=1 p11=4 J4 p4=2 p8=2 p12=2
 [1] Ming-Jong Yao, Tien-Cheng Hsu. An efficient search algorithm for obtaining the optimal replenishment strategies in multi-stage just-in-time supply chain systems. Journal of Industrial & Management Optimization, 2009, 5 (1) : 11-32. doi: 10.3934/jimo.2009.5.11 [2] Sun-Ho Choi, Hyowon Seo, Minha Yoo. A multi-stage SIR model for rumor spreading. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2351-2372. doi: 10.3934/dcdsb.2020124 [3] Li Deng, Wenjie Bi, Haiying Liu, Kok Lay Teo. A multi-stage method for joint pricing and inventory model with promotion constrains. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1653-1682. doi: 10.3934/dcdss.2020097 [4] Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109 [5] Jinghuan Li, Shuhua Zhang, Yu Li. Modelling and computation of optimal multiple investment timing in multi-stage capacity expansion infrastructure projects. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020154 [6] Jinghuan Li, Yu Li, Shuhua Zhang. Optimal expansion timing decisions in multi-stage PPP projects involving dedicated asset and government subsidies. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2065-2086. doi: 10.3934/jimo.2019043 [7] Jingwen Zhang, Wanjun Liu, Wanlin Liu. An efficient genetic algorithm for decentralized multi-project scheduling with resource transfers. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020140 [8] Binghai Zhou, Yuanrui Lei, Shi Zong. Lagrangian relaxation algorithm for the truck scheduling problem with products time window constraint in multi-door cross-dock. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021151 [9] Ruijun Zhao, Jemal Mohammed-Awel. A mathematical model studying mosquito-stage transmission-blocking vaccines. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1229-1245. doi: 10.3934/mbe.2014.11.1229 [10] Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781 [11] Adel Dabah, Ahcene Bendjoudi, Abdelhakim AitZai. An efficient Tabu Search neighborhood based on reconstruction strategy to solve the blocking job shop scheduling problem. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2015-2031. doi: 10.3934/jimo.2017029 [12] Tugba Sarac, Aydin Sipahioglu, Emine Akyol Ozer. A two-stage solution approach for plastic injection machines scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1289-1314. doi: 10.3934/jimo.2020022 [13] Jiping Tao, Ronghuan Huang, Tundong Liu. A $2.28$-competitive algorithm for online scheduling on identical machines. Journal of Industrial & Management Optimization, 2015, 11 (1) : 185-198. doi: 10.3934/jimo.2015.11.185 [14] Chunmei Zhang, Wenxue Li, Ke Wang. Graph-theoretic approach to stability of multi-group models with dispersal. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 259-280. doi: 10.3934/dcdsb.2015.20.259 [15] Didem Cinar, José António Oliveira, Y. Ilker Topcu, Panos M. Pardalos. A priority-based genetic algorithm for a flexible job shop scheduling problem. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1391-1415. doi: 10.3934/jimo.2016.12.1391 [16] Huan Su, Pengfei Wang, Xiaohua Ding. Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 253-269. doi: 10.3934/dcdsb.2016.21.253 [17] Lalida Deeratanasrikul, Shinji Mizuno. Multiple-stage multiple-machine capacitated lot-sizing and scheduling with sequence-dependent setup: A case study in the wheel industry. Journal of Industrial & Management Optimization, 2017, 13 (1) : 413-428. doi: 10.3934/jimo.2016024 [18] Guo Zhou, Yongquan Zhou, Ruxin Zhao. Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 533-548. doi: 10.3934/jimo.2019122 [19] Jiping Tao, Zhijun Chao, Yugeng Xi. A semi-online algorithm and its competitive analysis for a single machine scheduling problem with bounded processing times. Journal of Industrial & Management Optimization, 2010, 6 (2) : 269-282. doi: 10.3934/jimo.2010.6.269 [20] Xuewen Huang, Xiaotong Zhang, Sardar M. N. Islam, Carlos A. Vega-Mejía. An enhanced Genetic Algorithm with an innovative encoding strategy for flexible job-shop scheduling with operation and processing flexibility. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2943-2969. doi: 10.3934/jimo.2019088

2020 Impact Factor: 1.801


text:
# Nuclear Science and Techniques

《核技术》(英文版) ISSN 1001-8042 CN 31-1559/TL     2019 Impact factor 1.556

Nuclear Science and Techniques ›› 2018, Vol. 29 ›› Issue (11): 161

• NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH •

### Influence of the shell effects on evaporation residue cross section of superheavy nuclei

D. Naderi 1 • S. A. Alavi 2

1. 1 Department of Physics, Faculty of Basic Sciences, Razi University, Kermanshah 67149, Iran
2 Department of Physics, University of Sistan and Baluchestan, Zahedan 98135, Iran
• Contact: S. A. Alavi E-mail:s.a.alavi@phys.usb.ac.ir
PDF ShareIt Export Citation
D. Naderi, S. A. Alavi. Influence of the shell effects on evaporation residue cross section of superheavy nuclei.Nuclear Science and Techniques, 2018, 29(11): 161
Citations
Altmetrics

Abstract:

In order to study the influence of the shell effects on the formation and fission of superheavy elements, we applied multidimensional Langevin equations. The evaporation residue cross sections have been calculated for 3n, 4n, and 5n evaporation channels using three (K =0)- and four (K ≠0)-dimensional Langevin equations. Calculations were done for 48Ca + 238U and 48Ca + 244Pu hot fusion reactions with 3n, 4n evaporation channels and 70Zn +208Pb, and 54Cr +209Bi cold fusion reactions with 1n and 2n evaporation channels. The calculations were performed for 4n and 5n evaporation channels of the 26Mg + 238U reaction, as well. Our results show that with increasing dimension of Langevin equations the residue cross section increases, whereas the fission cross section decreases. The obtained results with four-dimensional Langevin and considering shell effects are in better agreement with experimental data in comparison with three- and four-dimensional Langevin equations without shell effects.


text:
# Gender Ratio of NYC Citibike Riders Above the Age of 45

Abstract
The Citi Bike project in New York City was launched in 2013 and has since seen growth in usage throughout the city.  In this experiment, we want to explore the age distribution of male and female bikers to examine how the Citi Bike delivery system can be better designed to serve user needs and attract more customers.  Data cleaning and manipulation were implemented in Python.  A null hypothesis significance test was conducted with a one tail z-score test. By designing the experiment with rigorous scientific theories and reproducible mechanism, the result shows middle-aged men are less likely to ride a bike than middle-aged women.

Data
Trip histories of Citi Bike riders were obtained through the NYC Citi Bike System Data portal.  Data from March 2015 was used having the appropriate size for this exploratory analysis.  The task of processing the data required identifying relevant variables, filtering for the appropriate records and then calculating the correct gender and age groups.  Python was used to run the analysis trimming the dataset to just the following fields: tripduration, usertype, birth year and gender.  The data set was then filtered for the “Subscriber” user type to remove data from one-time users identified as “Customer”.  This was a remedial step to ensure that the analysis focus on more frequent users.  In order to calculate the ratio of riders by age and gender, male and female groupings were each further grouped by birth year.  The age of 45 was selected for this analysis which placed the birth year cutoff at 1971.  Those born after 1971 were counted and labeled as above 45 for both genders.  Below are the Python scripts:

Remove fields not required.
df.drop(['starttime', 'stoptime', 'start station id', 'start station name', 'start station latitude', 'start station longitude', 'end station id', 'end station name', 'end station latitude', 'end station longitude', 'bikeid'], axis=1, inplace=True)
Filter data set to remove one-time users.
df1 = df[df.usertype != 'Customer']
Identify and count number of male riders above 45
df_m_above45 = (df1['birth year'][df1['gender'] == 1]).groupby(df1['birth year'] < 1971.0).count()
Identify and count number of female riders above 45
df_w_above45 = (df1['birth year'][df1['gender'] == 2]).groupby(df1['birth year'] < 1971.0).count()

Analysis
The null hypothesis was set as “the ratio of man above age 45 to man aged 45 or below riding a bike is the same or greater than the ratio of woman above age 45 to woman age 45 or below riding a bike.” The alternative hypothesis is that “the ratio of man above age 45 to man aged 45 or below riding a bike is the smaller than the ratio of woman above age 45 to man aged 45 or below riding a bike”. Furthermore, the significance level was set at alpha=0.05.
Replace this text with your caption
Since we used a large sample with more than 30, we decide to use a z-test. With alpha = 0.05, the critical values of z are -1.65 and +1.65. The null hypothesis is rejected if z < -1.65 or z > +1.65. In our z-test, z=18.61 so we can reject our null hypothesis.


text:
Opto-Electronic Advances  2019, Vol. 2 Issue (10): 190011      DOI: 10.29026/oea.2019.190011
A review on control methodologies of disturbance rejections in optical telescope
Tao Tang1,2, Shuaixu Niu1,2,3, Jiaguang Ma1,2, Bo Qi1,2, Ge Ren1,2, Yongmei Huang1,2
1. Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China;
2. Key Laboratory of Optical Engineering, Chinese Academy of Sciences, Chengdu 610209, China;
3. University of Chinese Academy of Sciences, Beijing 100049, China

Introduction

Disturbance rejection is one of the hottest topics in the high-performance servoing control areas1-5. Optical telescopes also face this issue, because the telescopes' imaging quality and their ability to observe faint stars at the diffraction limit is limited by the disturbances induced by wind shaking, equipment vibration and platform vibration6-10. High control bandwidth is the simplest method to reject vibrations. However, long integration time of the image sensor is always required for a high signal to noise, which restricts the closed-loop control bandwidth. Therefore, the vibrations induced by telescope structures could not be fully compensated by classical control loops11, 12. If disturbances are measurable, it is obvious that a disturbance feedforward control strategy is the most effective way to attenuate them13-15. Although the structural vibrations can be directly measured by extra inertial sensors, it is difficult to extract useful information to control the Tip-Tilt mirror. Furthermore, the estimation accuracy of disturbance using inertial sensors is affected by random drifting. To overcome these difficulties, one intuitive idea (called based-model control) for estimating disturbances through known variables to reject vibration is proposed, such that the disturbances can be compensated by the estimations rather than other sensors. These based-model control techniques are attractive in many practical scenarios, such as spacecraft, airplanes, vehicles and other platforms, due to its simplification and effectiveness. Related works are organized as follows. Frequency-based design of Modal-Control16 is presented to reduce the effects of distortions in ground-based telescopes. Simulations show that the proposed approach is a promising technique to improve the performance of the closed-loop control system. An adaptive control algorithm17, 18 to reject vibrations is developed and adapted to the complex control architectures. This adaptive vibration cancellation algorithm was integrated into a telescope currently operating at the European Southern Observatory in Chile and verified experimentally. Recently, some popular controllers focusing on Linear Quadratic Gaussian (LQG) control laws or using H/H2 synthesis methods19-22 have been successfully implemented. Existing experiments demonstrate that LQG control can achieve better on-sky performance than a feedback integrator controller in the condition of the optimally identified process model. As far as these techniques are concerned, the control model and the disturbance model play a crucial role in the Tip-Tilt control system. Large model errors usually deteriorate the closed-loop performance, and evenly cause the instability of the control system. Therefore, model identification is the priority when these methods are deployed. Although the control model of the Tip-Tilt mirror exhibits good linearity in the low-frequency domain, the middle-frequency or high-frequency nonlinearities and dynamics cause difficulties in model identification, resulting in hindering to reject the middle-frequency or high-frequency disturbance.

To relax these conditions, the disturbance observer control23-27 is reviewed and discussed for Tip-Tilt mirror control system in this paper. Furthermore, the error-based DOBC controller28-31 of Tip-Tilt mirror only based on CCD is proposed here. It can be plugged into an existing feedback loop that leads to a generalized version of 1-Q (Q is the designed filter) brought in the numerator of the original sensitivity function, resulting in the overall sensitivity function equal to zero in theory at the expected frequencies. To suppress different kinds of vibrations, relevant new Q-filters are also optimized to reject low-frequency and high-frequency disturbances. In frequency domain, the error-based DOBC with the improved Q-filters are analyzed in details, and its design procedure and implementation are also introduced. The remainder of this paper is arranged as follows: firstly we present a detailed introduction to conventional proportional integral (PI) control of the Tip-Tilt mirror and give a brief introduction to LQG controller as well as disturbance feedforward (DFF) controller. Secondly, the new error-based DOBC is proposed, which analyzes its characteristics, gives design process, and also sets up simulations and experiments to testify the error-based DOBC of Tip-Tilt mirror. And then, a new repetitive control32-34 based DOBC is proposed. Eventually, we give the conclusions.

Conventional control of Tip-Tilt mirror

The optical configuration of the Tip-Tilt mirror system28, 29 in the astronomical telescope is illustrated in Fig. 1(a), mainly including image sensor, splitter mirror, control unit, Tip-Tilt mirror, adaptive mirror, and image processing unit. The image sensor can detect the Tip-Tilt error caused by the telescope's structural vibrations. The classical control structure of the Tip-Tilt mirror is briefly demonstrated in Fig. 1(b).

 Fig. 1 Schematic view of the Tip-Tilt mirror system. (a) Typical configuration of Tip-Tilt mirror system. (b) A typical control structure.

G(s) is the controlled plant, and C(s) is the PI controller. The time delay ${{\rm{e}}^{ - \tau s}}$ represents the time delay of the control system. D(s) represents the disturbances. R(s) is the target trajectory, and Y(s) is the output. E(s) is the Tip-Tilt error provided by the image sensor. The close-loop error attenuation function ${S'_R}(s)$ and disturbance attenuation function ${S'_D}(s)$ are expressed below respectively.

 ${S'_R}(s) = \frac{{E(s)}}{{R(s)}} = \frac{1}{{1 + C(s)G(s){{\rm{e}}^{ - \tau s}}}},$ (1)
 ${S'_D}(s) = \frac{{Y(s)}}{{D(s)}} = \frac{1}{{1 + C(s)G(s){{\rm{e}}^{ - \tau s}}}},$ (2)

The Tip-Tilt mirror open-loop nominal response can be expressed as follows

 $G(s) = \frac{1}{{\frac{{{s^2}}}{{\omega _n^2}} + 2\frac{\xi }{{{\omega _n}}}s + 1}}\frac{1}{{{T_{\rm{e}}}s + 1}}.$ (3)

The natural frequency of Tip-Tilt mirror is usually up to kHz, and the damping factor ξ is much smaller than 1, leading to $G(s) \approx {({T_{\rm{e}}}s + 1)^{ - 1}}$. If the resonance frequency is low, a control strategy based on zero-pole cancellation35 is proposed to extend control bandwidth due to the stability of mechanical characteristic. Defining wc as the crossover frequency and wg as the gain frequency of the open-loop transfer function ${G_{{\rm{open}}}}(s) = C(s)\;G(s)$, Eq. (4) can be obtained according to the definition of phase margin and gain margin

 $\left\{ \begin{array}{l} \arg [P({\rm{j}}{w_{\rm{c}}})] \ge \frac{{\rm{ \mathsf{ π} }}}{4}\;, \;\;\;\;\;\;\;\;\;\;\;\;\left| {P({\rm{j}}{w_{\rm{c}}})} \right| = 1\\ - 20\lg \left| {P({\rm{j}}{w_{\rm{g}}})} \right| \ge 6\;, \;\;\;\arg [P({\rm{j}}{w_{\rm{g}}})] = - {\rm{ \mathsf{ π} }} \end{array} \right..$ (4)

Based on Eq. (4), we can derive that ${w_{\rm{c}}} = {\rm{ \mathsf{ π} }}/4\tau$ and ${w_{\rm{g}}} = {\rm{ \mathsf{ π} }}/2\tau$. As a result, the controller can be expressed as follows:

 $C(s) = \frac{{\rm{ \mathsf{ π} }}}{{4\tau }}\frac{{{T_{\rm{e}}}s + 1}}{s}.$ (5)

This controller can stabilize the plant with phase margin no less than 45 degrees and magnitude margin more than 6 dB. Substituting Eq. (5) into Eq. (1), we can have

 $\hat S \approx \frac{1}{{1 + \frac{{\rm{ \mathsf{ π} }}}{{4\tau }}\frac{1}{s}{{\rm{e}}^{ - \tau s}}}}.$ (6)

From Eq. (6), ${\left| {\hat S({\rm{j}}w)} \right|_{w = {w_c}}} = \sqrt 2 /2$ results in ${w_{\rm{c}}} \approx {\rm{ \mathsf{ π} }}\sqrt 2 /(4\tau )$, which is the bandwidth of sensitivity function. Thus, the control performance of the Tip-Tilt mirror is restricted by the time delay of the image sensor.

From above analysis, the closed-loop performance is restricted by linear PI controller due to a low bandwidth. For a linear control system driven by additive white Gaussian noise, the linear-quadratic-Gaussian (LQG) control problem36, 37 is to determine an optimal control law in the sense of minimizing the expected errors. The basic LQG control structure of Tip-Tilt mirror is depicted in Fig. 2.

 Fig. 2 LQG control structure of Tip-Tilt mirror.

The closed-loop performance of LQG control is dependent on the optimal controller, which is determined by the estimator. The key process in design LQG control is to choose an optimal metric to evaluate the performance. The accuracy of vibrations model plays an important role in the metric. Therefore, many works about LQG controller concentrate on model identifications in the Tip-Tilt control system. Although LQG controller is implemented in discrete-time domain, the closed-loop transfer function of Tip-Tilt mirror with the LQG controller can be obtained easily through bilinear transformation when the optimal controller is determined by the estimator.

Disturbance feed forward (DFF) controller

It is unavoidable for image sensors to induce time delay, resulting in low control bandwidth to limit vibration rejection, so other sensors independent of image sensors are added to measure vibrations directly so that a disturbance feed forward controller is implemented to eliminate vibrations37. Firstly, the vibrations are measured by additional inertial sensors such as accelerometers at the telescope structure and the vibrations are reconstructed by established filter techniques. The reconstructed signals are used feedforward for controlling the Tip-Tilt mirror loop. The basic control structure is shown in Fig. 3.

 Fig. 3 DFF control structure of Tip-Tilt mirror.

The transfer function of the Fig. 3 can be derived as follows:

 $Y(s) = \frac{{C(s)G(s){{\rm{e}}^{ - \tau s}}}}{{1 + C(s)G(s){{\rm{e}}^{ - \tau s}}}}R(s) + \frac{{1 - DFF(s)G(s)}}{{1 + C(s)G(s){{\rm{e}}^{ - \tau s}}}}D(s).$ (7)

The vibrations can be totally cancelled under the following condition:

 $DFF(s) = \frac{1}{{G(s)}}.$ (8)

Obviously, this perfect condition cannot be satisfied due to model errors existed in the Tip-Tilt control system. Although the partial compensation of Eq. (8) is effective to low-frequency vibrations, the final performance is directly affected by the reconstructed accuracy of vibrations. In fact, DFF is effective only with vibrations generated by the optical platform. However, these disturbances can occur at any point along the optical link due to structural flexibility, and therefore could not be reconstructed precisely to feedforward control of the Tip-Tilt mirror.

Disturbance observer control (DOBC)

To avoid complexities of extra load on the Tip-Tilt control system, the DOBC is proposed to improve the closed-loop system. In comparisons with LQG controller, this new method can enhance the original PI control system without deteriorating stability. Furthermore, the control system in the DOBC mode can obviously exhibit the open-loop and closed-loop characteristics in frequency domain. Fig. 4 shows the conventional DOBC structure25, 26. Q(s) is the designed filter. The inverse of the control plant G(s) is described by $G_m^{ - 1}(s)$. In the image sensor-based control system, the position sensor Y(s) cannot measure the disturbance D(s) directly. Thus, a new control block diagram29, 30 in the presence of the disturbance is proposed in Fig. 5.

 Fig. 4 Conventional DOBC structure of Tip-Tilt mirror.

 Fig. 5 Error-based DOBC structure of Tip-Tilt mirror.

The new closed-loop sensitivity transfer functions illustrated in Fig. 5 can be expressed as follows:

 ${S_R}(s) = \frac{{1 - Q(s)}}{{1 + C(s)G(s){{\rm{e}}^{ - \tau s}} + ({{\rm{e}}^{ - \tau s}}G(s)G_m^{ - 1}(s) - 1)Q(s)}}$ (9)
 ${S_D}(s) = \frac{{1 - Q(s)}}{{1 + C(s)G(s){{\rm{e}}^{ - \tau s}} + ({{\rm{e}}^{ - \tau s}}G(s)G_m^{ - 1}(s) - 1)Q(s)}}.$ (10)

${S_R}(s) = [1 - Q(s)]{S'_R}(s)$ and ${S_D}(s) = [1 - Q(s)]{S'_D}(s)$ can be obtained if $\left[ {{{\rm{e}}^{ - \tau s}}G(s)G_m^{ - 1}(s) - 1} \right]Q(s) \approx 0$. In this control mode, the main objective is to make $1 - Q(s)$ close to zero at the disturbance frequencies, and meanwhile, not to deteriorate the closed-loop performance of the original control system. The characteristic polynomial $W(s)$ of the closed-loop system can be expressed as

 $\begin{array}{l} W(s) = 1 + {{\rm{e}}^{ - \eta s}}C(s)\;G(s) + \left( {G_m^{ - 1}(s)\;G(s) - 1} \right)Q(s)\\ \;\;\;\;\;\;\;\; = \left( {1 + {{\rm{e}}^{ - \eta s}}C(s)\;G(s)} \right)\left( {1 + \frac{{\left( {G_m^{ - 1}(s)\;G(s) - 1} \right)Q(s)}}{{1 + {{\rm{e}}^{ - \eta s}}C(s)\;G(s)}}} \right), \end{array}$ (11)

Because the original feedback system is stable, it implies that the stability condition of the closed-loop control system has to satisfy the following condition according to Small Gain Theorem:

 ${\left\| {\frac{{\left( {G_m^{ - 1}(s)\;G(s) - 1} \right)\;Q(s)}}{{1 + {{\rm{e}}^{ - \eta s}}C(s)\;G(s)}}} \right\|_\infty } < 1.$ (12)
Frequency based design of Q-filter Low-pass filter

Without doubt, the Q-filter could be designed as a low-pass filter26, because the disturbance to be rejected is usually with low or medium frequency, whereas the sensor noise is with medium or high frequency. Therefore, the error-based DOBC is able to estimate the disturbance and uncertainty in a low and medium frequency range but filter out the high-frequency measurement noise. The general form of low-pass filters can be described as follows:

 ${Q_{\rm{L}}}(s) = \frac{{\sum\limits_{k = 2}^{m - 2} {C_m^k(} \tau s{)^k} + 1}}{{{{(\tau s + 1)}^m}}}.$ (13)

The general filters of Eq. (13) are subject to the orders of the filter and the time delay. Q31-filter39, 40 is considered as an optimal low-pass filter for the closed-loop performance in terms of bandwidth and robustness. $\tau = 15\eta$ is an optimal parameter of Q31-filter to make the phase margin and magnitude margin of the open-loop transfer function being robust.

There is an example of the Tip-Tilt mirror that compares Bode responses of the sensitivity functions in the modes of the PI and the DOBC controller in Fig. 6. The sampling frequency is 2000 Hz, and the time delay approximates with 0.0015. Obviously, Fig. 6 shows that the DOBC controller can achieve an extra improvement below the frequencies of 10 Hz.

 Fig. 6 Bode response of the sensitivity function with I and QL(s).

From Fig. 7, the DOBC with the low-pass filter is verified effectively to mitigate the low-frequency disturbances.

 Fig. 7 The closed-loop Tip-Tilt errors.
Band-pass filter

High rejection bandwidth in the low-pass filter control mode could be limited due to the stability condition of Eq. (8), leading to no mitigation of high-frequency disturbances. To overcome this problem, another way of canceling disturbances29 is expressed as follows

 $1 - Q(s)]D(s) = 0.$ (14)

Suppose that the disturbance D(s) can be defined as:

 $D(s) = {A_i}\sum\limits_{i = 0}^k {\varphi (s, {w_i})} ,$ (15)

where wi is the i-th disturbance with the maximum amplitude of Ai, and $\left| {\varphi (jw, {w_i})} \right| \le 1$. When $1 - Q(s)] =$$\prod\limits_{i = 1}^k {(\frac{{{s^2}}}{{w_i^2}} + 1)}$, we can derive that

 $1 - Q(s)]D(s) = \prod\limits_{i = 1}^k {(\frac{{{s^2}}}{{w_i^2}} + 1)} D(s) = 0.$ (16)

The above condition is impractical, because $\prod\limits_{i = 1}^k {(\frac{{{s^2}}}{{w_i^2}} + 1)}$ is non-causal, and cannot be implemented in a digital control system. A proposed filter to reconstruct Eq. (16) can be designed, and its extra sensitivity functions ESF(s) can be expressed as follows:

 $ESF(s) = 1 - Q(s) = \prod\limits_{i = 1}^k {\frac{{\frac{{{s^2}}}{{w_i^2}} + {\zeta _i}\frac{s}{{{w_i}}} + 1}}{{\frac{{{s^2}}}{{w_i^2}} + {\alpha _i}{\zeta _i}\frac{s}{{{w_i}}} + {\beta _i}}}} ,$ (17)

here, the three parameters follow that ${\alpha _i} > 0$, $1 > {\zeta _i} > 0$, ${\beta _i} > 0$. With the filter expressed in Eq. (17), disturbances can be reduced approximately to $\sum\nolimits_{i = 0}^k {{A_i}/{\alpha _i}}$. Q(s) characterizes a band-pass filter, which can be expressed below.

 $\begin{array}{l} Q(s) = 1 - \prod\limits_{i = 1}^k {\frac{{\frac{{{s^2}}}{{w_i^2}} + {\zeta _i}\frac{s}{{{w_i}}} + 1}}{{\frac{{{s^2}}}{{w_i^2}} + {\alpha _i}{\zeta _i}\frac{s}{{{w_i}}} + 1}}} \\ \;\;\;\;\;\;\; = \frac{{\prod\limits_{i = 1}^k {\left( {\frac{{{s^2}}}{{w_i^2}} + {\alpha _i}{\zeta _i}\frac{s}{{{w_i}}} + 1} \right)} - \prod\limits_{i = 1}^k {\left( {\frac{{{s^2}}}{{w_i^2}} + {\zeta _i}\frac{s}{{{w_i}}} + 1} \right)} }}{{\prod\limits_{i = 1}^k {\left( {\frac{{{s^2}}}{{w_i^2}} + {\alpha _i}{\zeta _i}\frac{s}{{{w_i}}} + 1} \right)} }}. \end{array}$ (18)

From Eq. (18), there is one differentiator in the numerator of Q(s), and meanwhile Q(s) features a low-pass characteristic since its relative degree is one. Therefore, the Q(s) is a band-pass filter. In a below example, the open-loop vibrations are shown in Fig. 8 that include multiple peak areas.

 Fig. 8 Spectra of Tip-Tilt vibrations.

Due to three areas of energetic vibrations existing in the high frequency, ESF(s) can be designed as follows

 $ESF(s) = ES{F_1}(s) \times ES{F_2}(s) \times ES{F_3}(s),$ (19)
 ${\rm{Where,}}\;ES{F_1}(s) = \frac{{0.000659{s^2} + 2.567{\rm{e}} - 5s + 1}}{{0.000659{s^2} + 0.0154s + 1}},$ (20)
 $ES{F_2}(s) = \frac{{2.093{\rm{e}} - 4{s^2} + 1.447{\rm{e}} - 5s + 1}}{{2.093{\rm{e}} - 4{s^2} + 0.002894s + 1}},$ (21)
 $ES{F_3}(s) = \frac{{5.234{\mathop{\rm e}\nolimits} - 5{s^2} + 7.234{\rm{e}} - 6s + 1}}{{5.234{\rm{e}} - 5{s^2} + 7.234{\rm{e}} - 4s + 1}}.$ (22)

As can be seen in Fig. 9, the 6-order notch filter shown in Eq. (19) is designed at the centered frequencies of 6 Hz, 11 Hz, and 21 Hz respectively. As a result, Q(s) is a band-pass filter.

 Fig. 9 Bode response of Q(s) and ESF(s).

The correction results are given in Fig. 10. The closed-loop errors are 2.32 μrad RMS and 0.63 μrad RMS respectively. In the DOBC mode, the RMS errors are less than 30% of that in the original loop. Peak disturbances still appear at 6 Hz, 11Hz and 21 Hz in the PI control mode, while they disappear in the DOBC mode.

 Fig. 10 Spectra of Tip-Tilt vibrations in DOBC mode.
Repetitive control of Tip-Tilt mirror

In this chapter, a learning-type control strategy called repetitive control is applied to a new Q-filter for coping with unknown disturbances. An improved Q-filter41 based on moving average filter is proposed to reduce the waterbed effect, which implies that additional gain amplifications can be mitigated between both periodic frequencies in comparison with conventional repetitive controller.

Q-filter design and performance analysis

The classical repetitive controller (CRC) is expressed as follows:

 ${Q_{{\rm{CRC}}}}({{\rm{e}}^{ - sT}}) = {{\rm{e}}^{ - sNT}}q({{\rm{e}}^{ - sT}}, l).$ (23)

A low-pass filter $q({{\rm{e}}^{ - sT}}, l)$ is used to block high-frequency noise and imperfect dynamics. The low-pass filter is usually designed as a zero phase filter as follows:

 $\begin{array}{l} q({{\rm{e}}^{ - sT}}, l) = {a_l}{{\rm{e}}^{slT}} + {a_{l - 1}}{{\rm{e}}^{s(l - 1)T}} + \cdots {a_0} + \cdots \\ \;\;\;\;\;\;\;\;\;\;\;\; + {a_{l - 1}}{e^{ - s(l - 1)T}} + {a_l}{e^{ - slT}}, \end{array}$ (24)

here l is a positive integer, and the coefficients meet $2({a_l} + {a_{l - 1}} + \cdots {a_1}) + {a_0} = 1$. In the CRC mode, the extra sensitivity function is given

 ${E_{{\rm{CRC}}}}({{\rm{e}}^{ - sT}}) = 1 - {{\rm{e}}^{ - sNT}}q({{\rm{e}}^{ - sT}}, l).$ (25)

The maximum value of Eq. (25) is equal to 2. The magnitude response of Eq. (25) is shown in Fig. 11, showing waterbed effect at nonperiodic frequencies.

 Fig. 11 Bode response of the Eq. (25).

For relaxing disturbance amplification at the nonrepetitive frequencies, we proposed an improved repetitive controller (IRC), and the new extra sensitivity function is given below

 ${E_{{\rm{IRC}}}}({{\rm{e}}^{ - sT}}) = \frac{{1 - {{\rm{e}}^{ - sNT}}q({{\rm{e}}^{ - sT}}, l)}}{{1 - \alpha {{\rm{e}}^{ - sNT}}q({{\rm{e}}^{ - sT}}, l)}}\;, \;\;\;\alpha \in [0, 1].$ (26)

Due to $q({{\rm{e}}^{ - sT}}, l) = q({{\rm{e}}^{sT}}, l) = 1$ below the frequency of ${w_{\rm{c}}}$, the ${E_{{\rm{IRC}}}}({{\rm{e}}^{ - sT}}){|^2}$ is simplified as

 ${\left| {{E_{{\rm{IRC}}}}({{\rm{e}}^{ - sT}})} \right|^2} = \frac{{2 - 2\cos (w/N)}}{{1 + {\alpha ^2} - 2\alpha \cos (w/N)}}.$ (27)

The maximum value of Eq. (27) at the frequencies of $w = {\rm{ \mathsf{ π} }} + 2k{\rm{ \mathsf{ π} }}\;(k = 0, {\kern 1pt} {\kern 1pt} 1, {\kern 1pt} {\kern 1pt} 2, \cdots )$ is 2, which implies that the worst amplification is amplified by 2. Reversely, if α is assigned a large value close to 1, the magnitude of the Eq. (27) is equal to 1. According to Bode's integral theorem, the amplification of nonperiodic disturbance can be reduced with the increase of α, while the disturbance reduction is also decreased. This phenomenon can be exhibited in Fig. 12.

 Fig. 12 Bode response of the Eq. (26).

The Q-filter of the IRC is derived below

 $\begin{array}{l} {Q_{{\rm{IRC}}}}({{\rm{e}}^{ - sT}}) = 1 - {E_{{\rm{IRC}}}}({{\rm{e}}^{ - sT}})\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{{(1 - \alpha ){{\rm{e}}^{ - sNT}}q({{\rm{e}}^{ - sT}}, l)}}{{1 - \alpha {{\rm{e}}^{ - sNT}}q({{\rm{e}}^{ - sT}}, l)}}. \end{array}$ (28)

In this section, a design example of the Tip-Tilt mirror under the condition of structural vibrations is exhibited. The sensitivity transfer functions of the Tip-Tilt mirror with IRC, CRC and I (integrator) are shown in Fig. 13. The benchmark line in blue is the closed-loop rejection with I controller, of which the rejection bandwidth is about 80 Hz, and no more than -20 dB suppression at the basic frequency of repetitive disturbance. IRC rejects the vibration peak with slightly magnifications of other frequencies, leading to an efficient improvement.

 Fig. 13 Sensitivity function responses in frequency domain.

Figure 14 shows the correction results in three different kinds of controllers. The resulting Tip-Tilt errors are 1.28, 0.91 and 0.66 μrad RMS respectively with I, CRC and IRC. The CRC has reduced Tip-Tilt error at the steady state, less than 72% of the original error, while in IRC mode the error is about 52% of the original error. The spectra are shown in Fig. 15 when the repetitive controllers are employed. The peak vibrations at the basic frequency of 6.6 Hz (such as 6.6 Hz, 13.2 Hz, 19.8 Hz and 26.4 Hz and so on) are significantly reduced.

 Fig. 14 Tip-Tilt errors in different control modes.

 Fig. 15 Spectra of Tip-Tilt vibrations with CRC and IRC.
Conclusion

In this paper, the control methodologies of Tip-Tilt mirror to reject structural vibrations in optical telescopes are reviewed and discussed. We focus on the survey of an error-based disturbance observer controller of the Tip-Tilt mirror. In this mode, a generalized version of 1-Q is cascaded into the original sensitivity function, and therefore the design of the Q-filter plays a vital role in the closed-loop control system of the Tip-Tilt mirror. The low-pass filter, band-pass filter and repetitive filter of the Q-filter are proposed to cope with vibrations in different frequency modes. The implementation of the proposed DOBC structure, the optimization of the control parameters and the analysis of the close-loop characteristics from the viewpoint of its practical implementation are provided in this paper. The key problem of designing the improved Q-filter is to determinate disturbance frequencies. A scanning method42 can make sure the accuracy in detection of the interesting vibrations. The control bandwidth of the Tip-Tilt mirror is not improved with this proposed controller, but disturbance attenuation is greatly enhanced at the disturbance frequencies. Furthermore, this improved control mode cannot cause big magnifications at non-disturbance frequencies, and therefore an effective improvement of the closed-loop performance is obtained. In comparison with existed methods, this improved technique can be carried out in real time if disturbance frequencies are given. Disturbance suppression, especially beyond the Nyquist frequency may be a continuing and attractive research in the mechanical control area. Nonlinear techniques43-46 such as sliding mode control, repetitive control, backstepping control and active disturbance rejection control are promising for further rejection performance in the high precision control system.

Acknowledgements

This work was in part supported by Youth Innovation Promotion Association, Chinese Academy of Sciences

Competing interests

The authors declare no competing financial interests.

References


text:
# BUS 352 Module 7 DQ 2 Your manager, who just read an abridged version of a statistics..

A
0 points

A+ Solutions

BUS-352 Module 7 DQ 2

Business Statistics - Hypothesis Testing With Two Samples

Your manager, who just read an abridged version of a statistics book, wants you to test hypotheses for the difference in two population means. The sample sizes for each are 23. He is adamant that you perform a z-test. What would you tell him? What specific explanation would you give?

BUS 352
a+tutor

A
0 points

#### Oh Snap! This Answer is Locked

Thumbnail of first page

Excerpt from file: BasedonwhatIhavelearnedsofarIwouldhavetofirstcomparetheTtest,againsttheZtest withthenumbersthathegavemeandIwouldmakesurewhichonewouldbethebestformula forthenumbersthathewants,andbasedonthisinformationIthinkthatthebestmethodwould

Filesize: < 2 MB

Print Length: 1 Pages/Slides

Words: NA

Surround your text in *italics* or **bold**, to write a math equation use, for example, $x^2+2x+1=0$ or $$\beta^2-1=0$$

Use LaTeX to type formulas and markdown to format text. See example.