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[–]Graph Theory 21 points22 points  (39 children)

This guy has a classic Russian view of mathematics. It's all analysis, all the time, and if it's not analysis (or linear algebra to a lesser extent), it's not math.

I'm obviously oversimpling but this sort of thinking is common among the Russian faculty members I've met at US universities. They tend to discount the importance of fields like combinatorics and group theory. For example, in the proposed curriculum the group theory looks like it's geared toward getting the student to advanced topological ideas and not deeper in to group theory.

I personally don't like his proposal. No graph theory or combinatorics? No thanks.

The bottom line is that people tend to overvalue the things they're interested in and undervalue other fields. I think that's what's happening here.

[–][S] 8 points9 points  (0 children)

I just noticed that this was written for students at ITEP, which seems to have been a research group interested primarily in mathematical physics and string theory. So that would explain most of the bias.

[–] 24 points25 points  (9 children)

Totally agree.

The fact that he says

Relevance to physics is the only criterion that we have left

Just shows me how out of touch he is. Biology, Economics, and Cog Sci/Neuroscience, are relying increasingly on math, and the math they rely on is far from Analysis. This doesn't even touch on the elephant in the room, Comp Sci, which is arguably the fastest growing mathematical discipline, and founded on discrete maths.

Big Data is here, and analysis is NOT the tool to deal with it. Just the pure absurdity of his position in an age with Google, Watson, etc, boggles my mind. Frankly I would go the other direction. Apart from theoretical physics, the world is build increasingly on combinatorics, graph theory and other discrete maths. If anything Analysis is loosing relevance.

I wouldn't submit to this curriculum with a gun to my head.

[–] 2 points3 points  (1 child)

This is indeed the case. Though I should add that with Big Data is actually more of trend in theoretical computer science towards linear algebra in high dimensional space and probability (in addition to the previously successful reliance on discrete math).

[–] 0 points1 point  (0 children)

Meh, probability may seem very continuous, but the fundamentals are rooted in discrete maths. I would say that Linear Algebra/Probability is the flip side of combinatorics. At the very least I would say the two are much close to discrete math than analysis, (which is kinda off by itself with topology). Also, although it will earn me lots of flack I consider TCS to be a branch of math. I see no reason why it is not.

[–] 2 points3 points  (6 children)

That's a curriculum for theoretical mathematics, so relevance to Big Data and economics does not serve as a criterion.

Also the list doesn't have that much analysis and almost no hard analysis.

[–] 2 points3 points  (5 children)

Why is relevance to physics admissible, but relevance to economics not? Granted one is much younger and less developed field, but that should only make us more eager to apply the rigor of mathematics to it, not discount it entirely.

Stuffy mathematicians like these are imho the biggest threat to the growth of the discipline, as they are the ones who will oppose true and meaningful curriculum reform at every level.

[–] 1 point2 points  (4 children)

By definition: it's a syllabus for someone who would be interested in physics-inspired theoretical math. Someone who wants something else (be it quant finance, graph theory or playing violin) should pick a different one.

[–] 0 points1 point  (3 children)

I'm not sure that's true. He seems to be indicating that this is superior to any math syllabus.

I do not think that all of mathematics is equally valuable, and I am sure that the intrinsic value of mathematics itself has not.

[–] -1 points0 points  (2 children)

Those who wish to follow the math (in the sense in which that word is understood above) are invited...

[–] 1 point2 points  (1 child)

Even that quote is arrogance. He is claiming the entirety of the field of math for himself (with his own definition), and everything he doesn't care for isn't math.

[–] -1 points0 points  (0 children)

At this point you are putting words in his mouth.

I'm not really interested in discussing personalities, tone and so on.

[–] 1 point2 points  (0 children)

I agree. I know that analysis is extremely important, but I have little interest in it. My true love is logic, but I am also drawn to graph theory, combinatorics, discrete optimization. I have no interest in physics.

This proposed curriculum has very little to offer people like me. I suspect that under such a system we would end up majoring in compsci and try to avoid math as much as possible...

[–] 3 points4 points  (25 children)

A very small part of it is analysis, it's mostly algebraic and differential geometry actually.

As for graph theory, combinatorics and other "second culture" topics they are not included because there is no coherent graph theory, just a bunch of definitions and results (relatively speaking, compare it to e.g. algebraic topology) that are accessible to any competent person. So unless you are excited about it for some reason, you'd be better off studying the topics of the "first culture". That's one of the ideas behind the list.

[–]Graph Theory 7 points8 points  (24 children)

This is exactly the kind of short sighted, dismissive attitude I'm talking about. It's offensive of you to completely discount entire fields of mathematics because you personally don't care for them.

And trying to tell me what I'd be "better off" studying is asinine. You have no idea what I should or should not study and the level of arrogance it took to assume that you do is astounding.

[–] -1 points0 points  (23 children)

That's not my position, it's the position of the person who wrote the list (I happen to be familiar with it). The point was to give a roadmap for a student of modern physics-inspired math (where modern applies to both "physics" and "math"). The author gives arguments in favor of its relative importance (unlike you). If you are not interested in it, just move along.

Here's what I think: if you like graph theory or combinatorics, that's fine. I like them too. But if you like them so much as to be unable to tell analysis from algebraic geometry, that may be a sign of poor understanding of the bigger picture.

Also I wonder if you have some observation other than "things I like are not included fffuuuuu".

[–] 1 point2 points  (1 child)

Here's what I think: if you like graph theory or combinatorics, that's fine. I like them too. But if you like them so much as to be unable to tell analysis from algebraic geometry, that may be a sign of poor understanding of the bigger picture.

You've confused me. I like graph theory quite a bit. Not once has my interest in graph theory made it difficult for me to distinguish between analysis and algebraic geometry. Are you talking about the possible situation where someone studies graph theory so single-mindedly that when confronted with topics in analysis and AG they are ill-equipped to grapple with them?

[–] 1 point2 points  (0 children)

Pretty much.

[–]Graph Theory 0 points1 point  (15 children)

Also I wonder if you have some observation other than "things I like are not included fffuuuuu".

No part of my post was like this. My "fffuuuuu" as you describe it was at your arrogance, not the original post. You claim that the position was not your own, but you wrote everything as if it were, and you continue to defend the position so I doubt that you were sincere in your claim.

In particular I have a problem with this statement (bolding added for emphasis):

As for graph theory, combinatorics and other "second culture" topics they are not included because there is no coherent graph theory, just a bunch of definitions and results (relatively speaking, compare it to e.g. algebraic topology) that are accessible to any competent person.

That's complete and utter bullshit, and is a clear demonstration of your own bias. If anyone here is guilty of "things I like" vs "things I don't like" it's you. I'd love to hear you defend this statement. What makes any field of math more than just a bunch of definitions and results? Have you ever actually studied graph theory or are you just a talking out of your ass?

I'd also like to know what your background is. What type of math did you study and where? I'm sure the answer to that question will make clear why you have such a skewed view of mathematics.

[–] 1 point2 points  (0 children)

Tim Gowers (who largely studies problems on graphs, although I wouldn't call him a graph theorist) basically wrote the same thing in this essay. Graph theory does not have a monolithic body of essential theory that people need to comprehend before they can approach problems in the field (in the same way that, say, algebraic geometry does).

Another point which most people seem to have missed is that Verbitsky's teaching experience is mostly with very capable students, many of whom probably come from a competitive math background. Such students are generally quite strong in combinatorics and discrete math upon entering university due to the topics tested in math olympiads, and generally have a much stronger grasp of the topic before even entering university than a typical US undergraduate student would after having completed an undergraduate course or two in discrete math.

[–] 0 points1 point  (13 children)

No part of my post was like this.

I think this part is pretty close:

I personally don't like his proposal. No graph theory or combinatorics? No thanks.

...

You claim that the position was not your own, but you wrote everything as if it were

That's what I've said:

That's one of the ideas behind the list.

Also I might add I'm not the author of the list, so those are not my ideas. If you are not going to take my word on it, I don't see the point in persuading you.

What makes any field of math more than just a bunch of definitions and results?

That's not what I've said, you ignored the second part. A bunch of definitions and results ... that are accessible to any competent person. In some fields you have to study for years just to understand the problems, they build layers upon layers of abstraction. Graph theory is not like that.

Also it doesn't have a solid, unavoidable core. E.g. if you study analysis, you'll use derivatives and integrals for sure, in topology you'll use homology without a doubt and so on.

If anyone here is guilty of "things I like" vs "things I don't like" it's you.

I've explicitly stated I like graph theory.

Have you ever actually studied graph theory...

To some extent.

I'd also like to know what your background is.

I'm not interested in discussing myself and I don't think it's relevant anyway.

[–]Graph Theory 0 points1 point  (1 child)

This whole discussion is silly. You have your opinion, I have mine, and we aren't going to convince one another of anything meaningful. Why don't we just agree to respectfully walk away from this and call it a day?

[–] 0 points1 point  (0 children)

I'm sure I can be convinced if I'm wrong.

[–]Graph Theory 0 points1 point  (10 children)

That's not what I've said, you ignored the second part. A bunch of definitions and results ... that are accessible to any competent person. In some fields you have to study for years just to understand the problems, they build layers upon layers of abstraction. Graph theory is not like that.

You've completely dodged the question.

What field of math is not just a bunch of definitions and results that are accessible to any competent person? You later admit that you've studied graph theory "to some extent" and yet you feel qualified to make the claim that the entire field is completely approachable to any competent person with out years of study? Again, your arrogance is astounding.

I directly challenge you to support the following statement:

In some fields you have to study for years just to understand the problems, they build layers upon layers of abstraction. Graph theory is not like that.

I admit that many of the problems are deceptively simple, but that doesn't mean that you can dismiss the entire field of study. Take the Four Color Theorem as an example. Sure it meets your criterion of being easy to understand but does that mean that it's somehow not a problem worth studying? Or consider the Graceful Tree Conjecture. I could certainly explain the problem in simple terms that just about anyone could understand but that doesn't make it an easy problem.

Perhaps your only exposure to graph theory has been with this kind of easy to understand question. I would expect that of someone who has only studied graph theory "to some extent". But it's not fair to assume that your limited exposure represents all of graph theory. If you believe that you can attack a problem involving, for example, Turan Hypergraphs without taking considerable time to understand fundamental graph theoretical concepts than I'd love to work with you as soon as possible because you could be the next Paul Erdős. And that doesn't even begin to touch on things like spectral graph theory as explored by people like Fan Chung. Do you really think that a reasonably competent person can understand the importance of the eigenstructure of directed graphs without years of education?

[–] 1 point2 points  (9 children)

You've completely dodged the question.

Sure, because I think you wouldn't ask it if you understood what I'm saying.

What field of math is not just a bunch of definitions and results that are accessible to any competent person?

The "competent person" I was talking about is not a expert in graph theory, just a mathematician or maybe a physicist or computer scientist. Example of a field that is not that accessible: modern homology theory, you pretty much have to be a topologist or an algebraic geometer to even understand the definitions.

That is not a statement about the difficulty of the field. It's about the dependency graphs: problems and definitions in graph theory have relatively few dependencies, its dependency graph has relatively small diameter. Relatively, that is compared to the things in the syllabus. For many topics of its fifth year you'd have to read dozens of books (in a particular order), not counting the usual undergraduate stuff. Graph theory has smaller diameter, so to speak (think diameter of the dependency graph).

...that doesn't mean that you can dismiss the entire field of study.

Never had the intention.

...does that mean that it's somehow not a problem worth studying?

I've never said it. By the way, the author of the syllabus doesn't say that either.

...that doesn't make it an easy problem.

Also never said it.

By the way, hard or easy is not a criterion. Building houses with matches is hard, but it's not worth doing unless you enjoy it.

If you believe that you can attack a problem involving...

See above.

...And that doesn't even begin to touch on things like spectral graph theory as explored by people like Fan Chung. Do you really think that a reasonably competent person can understand the importance of the eigenstructure of directed graphs without years of education?

Well, yeah. It should be obvious to anyone who has studied Riemannian geometry or physics, anything with Laplace operators really. As in "spectra of important operators are important".

[–]Graph Theory 0 points1 point  (8 children)

Ah, I see. So you're saying that any individual part of graph theory would be obvious to any who's studied the material required to understand that part of graph theory. Sure, that's vacuously true and a stupid thing to say.

[–] 0 points1 point  (7 children)

No, that's not what I'm saying. "Riemannian geometry or physics" are not at all required for spectral graph theory (nothing except elementary linear algebra is), but they have common idioms so a person who has studied them wouldn't be lost. And I don't think it works the other way around.

Why do you always choose the most idiotic reading?

[–] -1 points0 points  (4 children)

If it is in fact a syllabus intended for students who have already expressed an interest in physics inspired math then I think peoples main problem with this is his tone.

He expressed his belief in this curriculum with extreme arrogance, and in the process insults quite a few other perfectly valid branches of math.

Also, there is quite a bit of analysis in this curriculum, regardless of what you say. Moreover, every other topic introduced is introduced primarily with the results that make it relevant to analysis.

[–] -1 points0 points  (3 children)

I think peoples main problem with this is his tone.

Sure.

Also, there is quite a bit of analysis in this curriculum, regardless of what you say.

The analysis parts are easy (relatively) and there is much more algebra, geometry and topology than analysis.

Moreover, every other topic introduced is introduced primarily with the results that make it relevant to analysis.

That's simply not true at all. It's actually the other way around: the analysis in the syllabus is there because of the geometric applications (index theorems, Hodge decompositions, complex transcendental techniques and so on).

[–] 1 point2 points  (2 children)

That's simply not true at all. It's actually the other way around: the analysis in the syllabus is there because of the geometric applications (index theorems, Hodge decompositions, complex transcendental techniques and so on).

Perhaps I miss-phrased. The only included topics are topics with relevance to analysis/physics based math.

I think we have come to understand the main misunderstanding here:

This is a curriculum for people who have already decided to specialize in this particular type of math. In my opinion, he made a COLOSSAL misstep in presenting this as the overall best curriculum for undergrads in mathematics, (regardless of breadth), and in insulting maths which are not included in his curriculum. That said, we should recognize that the fact that he is an asshole has no bearing on the quality of this curriculum as it was intended to be judge (namely a physics-inspired math curriculum), if this was indeed the case.

TL;DR, This may be a very fine physics inspired math-curriculum, AND this guy may be an asshole. The two are not mutually exclusive.

[–] -2 points-1 points  (1 child)

It's a matter of context, "presenting this as the overall best curriculum for undergrads in mathematics, (regardless of breadth)" is not really what it was supposed to be. At least that's how I understand it.

The only included topics are topics with relevance to analysis/physics based math.

If you replace "analysis/physics based math" with "physics based math" that's what he is saying in the explanation, so that's really not that surprising.

[–] -1 points0 points  (0 children)

Yes, possibly. I'm not really interested in intent, but his explanation was clearly interpreted by many as a assertion of superiority on the part of physics based math. The near universality of this interpretation strike me as a failure of communication on his part.

Also, machine translation may have aggravated the problem.

[–] 0 points1 point  (0 children)

Well, he's fine with group theory as long as it's geometry group theory a la Gromov.

[–][S] 8 points9 points  (3 children)

### Mathematics program should be designed so

School program (exam Matshkolnik)

• Euclidean geometry, complex numbers, scalar multiplication, the Cauchy-Schwarz inequality. Principles of Quantum Mechanics (Kostrikin-Manin). Group of transformations of the plane and space. Conclusion of trigonometric identities. The geometry of the upper half plane (hyperbolic). Properties of inversion. Effect of linear fractional transformations.
• Rings and fields. Linear algebra, finite groups, Galois theory. Proof of Abel. Basis, rank, determinant, classical Lie group. Dedekind. Determination of the real numbers. The definition of the tensor product of vector spaces.
• Set theory. Zorn's lemma. A well-ordered set. Cauchy-Hamel basis. Cantor-Bernstein theorem. Uncountable set of real numbers.
• Metric spaces. The set-theoretic topology (defined continuous maps, compact, custom maps). Countable base. The definition of compactness in terms of convergent sequences for spaces with a countable base.Homotopy, the fundamental group, homotopy equivalence.
• p-adic numbers, Ostrowski theorem, multiplication, and division of the p-adic numbers in a column
• Differentiation, integration, the Newton-Leibniz. Delta epsilon formalism lemma policeman.

The first course

• Analysis on \$R ^ n \$. Differential display. lemma contracting maps. Implicit Function Theorem. Riemann and Lebesgue. ("Analysis" of Laurent Schwartz, "Analysis" Zorich, "Problems and theorems of functions. Analysis" Kirillov-Gvishiani)
• Hilbert spaces, Banach spaces (definition). The existence of a basis in the Hilbert space. Continuous and discontinuous linear operators. Criteria for continuity. Examples of compact operators. ("Analysis" of Laurent Schwartz, "Analysis" Zorich, "Problems and theorems of functions. Analysis" Kirillov-Gvishiani)
• Smooth manifolds, submersion immersion Sard's theorem. Partition of unity. Differential Topology (Milnor-Wallace). Transversality. Degree of the map as a topological invariant.
• Differential forms, de Rham operator, Stokes theorem, Maxwell's equations of the electromagnetic field. Gauss's theorem as a particular example.
• Comprehensive analysis of a single variable (based on the book by Henri Cartan, or the first volume of Shabbat). Contour integrals, Cauchy, Riemann mapping theorem of any simply connected subset of \$C \$ in a circle theorem on the extension of borders, Picard's theorem on the achievement of an entire function of all values except three. Multivalent functions (for example, the logarithm).
• Category theory, the definition of the functors, equivalence, adjoint functors (McLane, Categories for working mathematician, Gelfand-Manin, the first chapter.)
• Lie groups and algebras. Lie groups. Lie algebra as linearization. Universal enveloping algebra, the Poincare-Birkhoff-Witt. Free Lie algebra. Campbell-Hausdorff series and the construction of a Lie group by its algebra (yellow Serre, first half).

The second course

• Algebraic Topology (Fuchs-Fomenko). Cohomology (simplicial, singular, de Rham), their equivalence, Poincaré duality, the homotopy groups. Dimension. Bundle (Serre), the spectral sequence (Mishchenko, "Vector bundles ..."). Computation of the cohomology of classical Lie groups and projective space.
• Vector bundles, connection, the Gauss-Bonnet, the Euler class, Black, Pontryagin, Stiefel-Whitney test. Multiplicativity of the Chern character. Classifying spaces ("Characteristic Classes", Milnor and Stashef).
• Differential Geometry. Levi-Civita connection, the curvature of the algebraic and differential Bianchi identity. Killing Fields. Gaussian curvature of two-dimensional Riemannian manifold. Cell decomposition of the loop space in terms of geodesics. Morse theory on the loop space (based on the book of Milnor, "Morse theory" and Arthur Besse "Einstein manifolds"). Principal bundles and connections in them.
• Commutative algebra (Atiyah-Macdonald). Noetherian, Krull dimension, Lemma Nakayama adic completion, integrally closed, discrete valuation ring. Flat modules, local criterion plane.
• Beginning of algebraic geometry. (Chapter Hartshorne either green or Shafarevich Mumford). Affine variety, projective variety, projective morphism, the image of a projective variety is projective (via resultants).Beams. Zariski topology. Algebraic variety as a ringed space. Hilbert's theorem on zeros. The spectrum of the ring.
• Beginning of homological algebra. Group Ext, Tor for modules over a ring, resolutions, projective and injective modules (Atiyah-Macdonald). Construction of injective modules. Grothendieck duality (in the book Springer Lecture Notes in Math, Grothendieck Duality, number about 21 and 40).
• Number theory, local and global field discriminant rule the ideal class group (blue book Cassels and Frohlich).
• Reductive groups, root systems, representations of semisimple groups, the weight, the Killing form. Groups generated by reflections, their classification. Lie algebra cohomology. Computation of the cohomology in terms of invariant forms. Singular cohomology of a compact Lie group and its cohomology algebra. Invariants of classical Lie groups. (Yellow Serre, the second half, Hermann Weyl, "Invariants of classical groups"). Design of special Lie groups. Hopf algebra. Quantum groups (definition).

The third course

• K-theory as a cohomological functor, Bott periodicity, Clifford algebra. Spinors (Atiyah's book "K-theory" or Mishchenko "the vector bundles and their application"). Spectra. Eilenberg-Mac Lane. Infinite loop space (the book Switzer or yellow book or Adams Adams "Lectures on generalized cohmology", 1972).
• Differential operators, pseudodifferential operators, symbol, elliptic operators. Properties of the Laplace operator. Self-adjoint operators with discrete spectrum. Green operator and applications to Hodge theory on Riemannian manifolds. Quantum mechanics. (Book R.Uellsa analysis or Mishchenko "vectors of the bundle and its application").
• The index formula (Atiyah-Bott-Patodi, Mishchenko), the Riemann-Roch. Zeta function of the operator with discrete spectrum and its asymptotic behavior.
• Homological algebra (Gelfand-Manin, all in the last intermediate). Cohomology of Sheaves, derived category, triangulated category, derived functor, the spectral sequence of a double. The composition of triangulated functors and the corresponding spectral sequence. Verdier duality. The formalism of the six functors and perverse sheaves.
• Scheme algebraic geometry, the scheme over the ring, the projective spectra, derivatives, Serre duality, coherent beams, replacement base. Own and separated scheme, valyuativny property and separability criterion (Hartshorne). Functors representability, the moduli space. Direct and inverse images of sheaves, higher direct images. With proper mapping higher direct images are coherent.
• Cohomological methods in algebraic geometry, semicontinuity cohomology theorem of Zariski connectedness theorem Stein decomposition.
• Kahler manifolds, Lefschetz, Hodge theory, the ratio of Kodaira, properties of the Laplace operator (zero head head Griffiths-Harris, clearly stated in the book André Weil "Kahler manifolds"). Hermitian bundle.Line bundles and their curvature. Line bundles with positive curvature. Kodaira-Nakano Vanishing on (Griffiths-Harris).
• Holonomy, the Ambrose-Singer theorem, special holonomy holonomy classification, Calabi-Yau, hyper, theorem of Calabi-Yau manifolds.
• Spinors on the manifold, the Dirac operator, the Ricci curvature, the formula Weitzenbock-Lichnerowicz, Bochner's theorem. Bogomolov's theorem on the decomposition of manifolds with zero canonical class (Arthur Besse, "Einstein manifolds").
• Tate cohomology class field theory (Cassels-Frohlich, blue book). The calculation of the factor Galois number field commutator. Brauer and its applications.
• Ergodic theory. Ergodicity of billiards.
• Complex curves, pseudo-conformal mappings, Teichmüller space, the theory of Ahlfors-Bers (the book Ahlfors thin).

[–][S] 3 points4 points  (1 child)

Fourth year.

• Rational homotopy type and profinite étale cover nerve cell complex homotopy equivalent to its profinite type. Topological definition of étale cohomology. Galois action on profinite homotopy type (Sullivan, "Geometric Topology").
• Étale cohomology in algebraic geometry, the comparison functor, Henselian, geometric point. Base change. Any smooth variety over a field locally in the étale topology is isomorphic to \$A ^ n \$. Étale fundamental group (Milne, review of Daniel VINITI and SGA 4 1/2, the first article of Deligne).
• Elliptic curves, j-invariant, automorphic forms, Taniyama-Weil conjecture and its applications to number theory (Fermat's theorem).
• Rational homotopy (in the last chapter of the book or the Gelfand-Manin article Griffiths-Morgan-Long-Sullivan). Massey operations and rational homotopy type. Vanishing Massey operations on a Kahler manifold.
• Chevalley group of generators and relations (the book Steinberg). K_2 computation of the field (Milnor, Algebraic K-theory).
• Algebraic K-theory of Quillen, \$BGL ^ + \$ and \$Q \$-structure (review Suslin in the 25th volume of VINITI, lectures Quillen - Lecture Notes in Math. 341).
• Complex analytic manifold, coherent beams, the coherence theorem of Oka, Hilbert's Nullstellensatz for ideals in the sheaf of holomorphic functions. Noetherian ring of germs of functions, Weierstrass theorem on division, Weierstrass preparation theorem. Theorem on a branched covering. Grauert-Remmert (image of a compact analytic space under a holomorphic analytic morphism). Hartogs' theorem on the continuation of an analytic function. Multidimensional Cauchy formula and its applications (uniform limit of holomorphic functions is holomorphic).

Fifth Year

• Kodaira-Spencer theory. Deformation of a manifold and solutions Maurer-Cartan equations. Solvability of the Maurer-Cartan and Massey operations on the DG-Lie algebra cohomology of vector fields. Moduli space and finite (see lecture Kontsevich, or the collected works of Kodaira). Theorem Bogomolov-Tian-Todorov on deformations of Calabi-Yau manifolds.
• Symplectic reduction. Moment map. Kempf-Ness theorem.
• Deformations of coherent bundles and bundles in algebraic geometry. Geometric invariant theory. Moduli space of bundles on a curve. Stability. Uhlenbeck compactification, Gieseker and Maruyama. Geometric invariant theory is the symplectic reduction (third edition of Mumford's geometric invariant theory, application, Frances Kirwan).
• Instantons in the four-dimensional geometry. Donaldson theory. Donaldson invariants. Instantons on Kahler surfaces.
• The geometry of complex surfaces. Kodaira classification, Kahler and nekelerovy surface of the Hilbert scheme of points on the surface. Castelnuovo-Enriques criterion, the Riemann-Roch inequality Bogomolov-Miyaoka-Yau. The ratio of invariants of the surface. Elliptic surfaces, the surface Kummer K3 surfaces and Enriques.
• Elements of the program Mori: Kawamata-Viehweg zero freedom theorem of base points, the theorem of Mori cone (Clemens-Kollar-Mori, "Multidimensional complex geometry", plus not translated Kollar-Mori and Kawamata-Matsuki-Massoud) .
• Stable bundles as instantons. Yang-Mills equation on a Kahler manifold. Theorem Donaldson-Uhlenbeck-Yau metrics on the Yang-Mills theory on a stable bundle. Its interpretation in terms of the symplectic reduction. Stable bundles and instantons on hyperkähler manifolds, an explicit solution of the Maurer-Cartan equation in terms of the Green's operator.
• Pseudoholomorphic curves in symplectic manifold. Gromov-Witten. Quantum cohomology. Mirror conjecture and interpretation. The structure of the group of symplectomorphisms (under the Kontsevich-Manin book Polterovich, "Symplectic geometry" green book of pseudoholomorphic curves and lecture notes McDuff and Salamon).
• Complex spinors, Seiberg-Witten, Seiberg-Witten. Why Seiberg-Witten invariants are Gromov-Witten.
• Hyperkähler reduction. Flat bundles, and the equation of the Yang-Mills theory. Hyperkähler structure on the moduli space of flat bundles (Hitchin-Simpson).
• Mixed Hodge structure. Mixed Hodge structure on the cohomology of algebraic varieties. Mixed Hodge structures on Malcev completion of the fundamental group. Variations of mixed Hodge structures. Theorem on a nilpotent orbit. Theorem on \$SL (2)-orbit. Close and vanishing cycles. The exact sequence of the Clemens-Schmid (the red book Griffiths "Transcendental methods in algebraic geometry"). • Nonabelian Hodge theory. Variations of Hodge structures as fixed points of \$ C ^ * \\$-action on the moduli space of Higgs bundles (dissertation Simpson).
• Weil conjecture and proof. L-adic sheaves, perverse sheaves, the Frobenius automorphism, its weight, purity theorem (Beilinson, Bernstein, Deligne, plus Deligne, Weil Conjectures II).
• Quantitative algebraic topology Gromov (the book Gromov "Metric structures for Riemannian and non-Riemannian spaces"). Gromov-Hausdorff metric, prekompaktnost set of metric spaces, hyperbolic manifolds and hyperbolic groups, harmonic maps into hyperbolic space, the proof of Mostow rigidity (two compact Kahler manifolds, covered by one and the same symmetric space X of negative curvature, isometric if their fundamental groups isomorphic, and dim X> 1).
• Variety of general type, Kobayashi and Bergman metrics, analytical stiffness (Sioux).

[–] 0 points1 point  (0 children)

I'm impressed by his list in Year 4 and 5. I'm sure many PhD in the US only manage to master 2-3 topics listed here. It is not even realistic to try to find so much people to teach such courses.

[–] 7 points8 points  (0 children)

Be wary of anyone selling you a new curriculum as the answer to education woes.

[–] 3 points4 points  (0 children)

What's a "program"? This is a just huge list of mathematical topics, many of them extremely specialized. I don't think most working mathematicians know most of this stuff beyond the "second year" unless it is their field of interest, and not more than a handful would be familiar with a majority of it.

[–] 10 points11 points  (0 children)

Good stuff, but very few people would find keeping up to it realistic. It is heavily influenced by Misha's interests.

[–]Mathematical Physics 6 points7 points  (5 children)

Sorry to be that guy... but I'm not really sure what you're asking. In what sort of setting would this curriculum be implemented? Is this a suggested program of study for undergraduate students?

[–] 7 points8 points  (2 children)

Came here to check what my 16 year-old son should study next. I'm off to ask ELI5 Delta Epsilon Formalism Lemma Policeman!?

[–] 8 points9 points  (0 children)

Lemma policeman is probably ''Théorème des gendarmes" aka "Squeeze Theorem" or "Sandwich Theorem"

Delta-Epsilon formalism is the delta-epsilon definition of convergance, limit, etc.

[–][S] 6 points7 points  (0 children)

Probably an abbreviation in Russian that google mistranslated.

[–][S] 4 points5 points  (1 child)

I'm not really asking anything specifically. Just thought it might spark an interesting discussion. Might be useful to some students looking for direction as well.

[–] 2 points3 points  (0 children)

Honestly, after participating in the discussion, then reading this comment I feel trolled.

You posted an extremely specialized curriculum, presented as a generalized one by an arrogant asshole. Of course people are going to have strong emotional reactions to it. But I don't think we really learned anything from the discussion, except that this guy is an asshole, and Russians like analysis.

[–] 3 points4 points  (0 children)

...And this is all supposed to be covered in what period of time? I feel like learning all that would take me a decade.

[–] 2 points3 points  (0 children)

Based on the conversations below, I'm going to TL;DR this for the OP:

It seems that this is a specialized curriculum for students interested in physics-inspired math, and it does not represent a balanced study of mathematics. That said, he frames it as the only curriculum one should study, and in the process manages to insult many branches of math.

It may well be a quality curriculum in this area, (IANAE), but it is for specialists.

Finally, this guy seems like an asshole.

[–] 0 points1 point  (0 children)

This post and comment discussions are fascinating. I feel like I'm seeing the distinction between purists and application dictating where research goes.

I must be a romantic because I feel a bias towards... both sides :(


text:
# Motion of a body given acceleration as a function of velocity

I'm having trouble figuring this out. I'm trying to determine the final time and final velocity for the motion of a vehicle over a given distance along a straight line.

$t$ is time
$v$ is velocity
$a$ is acceleration
$r$ is displacement
$m$ and $c$ are constant

Known:
$t_0$, $v_0$, $r_0$, $\Delta r$

Acceleration of the vehicle is described by:
$a=\frac{f(v)-c}{m}$

I want to derive expressions for:
$\Delta t$, and $v$

$f(v)$ is a function which returns the force which the engine can apply at a given velocity. This is essentially a lookup against a table of values from the vehicle manufacturer. I can perform an accumulation over this if required.

I'm currently estimating a result to this problem by iteratively calculating using small intervals of time.

Any help would be greatly appreciated.

-
Are these vectors or scalars? For example, is displacement the same as the distance travelled? Is the motion in a straight line? –  Paresh Feb 12 '13 at 14:59
Scalars, the motion is indeed in a straight line. –  Jack Feb 12 '13 at 15:11
@Jack What do you mean by final time? Do you mean the time at which velocity reaches some value? –  oks Feb 12 '13 at 15:53
@oks The time at which the displacement has reached some value; i.e. the travel time –  Jack Feb 12 '13 at 16:13
How are you interpolating $f(v)$ between the points in the lookup table? Is it just linear interpolation? –  Rahul Feb 12 '13 at 17:33

I think you can get much farther with explicit methods than the other answers suggest. First, given $a$ as a function of $v$, we can find $t$ as a function of $v$ by integration: $$\begin{gather} \frac{\mathrm dv}{\mathrm dt}=a=\frac{f(v)-c}m.\\ \frac m{f(v)-c}\,\mathrm dv=\mathrm dt.\\ \int_{v_0}^{v_1}\frac{m\,\mathrm dv}{f(v)-c}=\int_{t_0}^{t_1}\mathrm dt=t_1-t_0. \end{gather}$$ Since $f(v)-c$ is piecewise linear, $\int_{v_0}^{v_1}m\,\mathrm dv/(f(v)-c)$ is piecewise logarithmic and you can compute it explicitly. This gives us $t$ as an increasing function of $v$, which I'll call $\tau$: $$t_1=\tau(v_1):=t_0+\int_{v_0}^{v_1}\frac{m\,\mathrm dv}{f(v)-c}.$$ Now we get $r$ by integrating again: $$\begin{gather} \tau(v)=t.\\ \frac{\mathrm dr}{\mathrm dt}=v=\tau^{-1}(t).\\ r_1-r_0=\int_{t_0}^{t_1}\tau^{-1}(t)\,\mathrm dt. \end{gather}$$ It appears we have to integrate $\tau^{-1}$, which seems complicated. But not too complicated: let $t_0=\tau(v_0)$ and $t_1=\tau(v_1)$, and then we have $$\int_{t_0}^{t_1}\tau^{-1}(t)\,\mathrm dt=(t_1v_1-t_0v_0)-\int_{v_0}^{v_1}\tau(v)\,\mathrm dv,$$ and integrating the piecewise logarithmic $\tau$ can also be done explicitly. So now we have $r$ as another explicitly known, increasing function of $v$. Your problem amounts to finding the $v$ for the given $r$, which you will have to do via a numerical procedure, but something really simple like bisection search should work just fine.

-
This is brilliant, I'll implement this and let you know how it goes. I also have some termination conditions (like braking distance requirements), so a bisection search would have been required regardless... I think. –  Jack Feb 13 '13 at 7:46

Since $a = \dot{v}$, your equation of motion is the ODE $$m\dot{v} = f(v) - c$$

This is an autonomous first-order ODE; separating the variables gives $$m\int_{v(0)}^{v(t)} \frac{1}{f(w)-c}dw = t,$$ and if you had a formula for $f$ you could try to perform the integration on the LHS to get $mg(v) = t$, and then solve for $v$.

Since you only have numerical data, though, your best bet is numerical quadrature (which it sounds like you're already doing). For instance, pick a time step $h$ a good bit smaller than your estimated $\Delta t$, and iterate

\begin{align*} r_{i+1} &= r_i + h v_i\\ v_{i+1} &= v_i + h \frac{f(v_i) -c}{m}\\ t_{i+1} &= t_i + h \end{align*}

until $r_i-r_0 > \Delta r$.

-

You are trying to calculate $\frac 1m \int f(v)-c \; dt$ and unless you have a nice formula for $f(v)$ you will be doing it numerically, which basically amounts to what you are doing-add up accelerations times small timesteps. If $f(v)$ is reasonably smooth, you can reduce the calculation time and make it more robust by using an adaptive integration routine. Any numerical analysis text will have a discussion. I like Numerical Recipes, which is free online and has routines in C.

-

As user7530 says, you could integrate numerically by using a first order Euler method to integrate numerically

\begin{align*} r_{i+1} &= r_i + h v_i\\ v_{i+1} &= v_i + h a_{i+1} \\ a_{i+1} & = \frac{f(v_i) -c}{m}\\ t_{i+1} &= t_i + h \end{align*}

until $r_i-r_0 > \Delta r$.

But you could use different methods to integrate numerically. One way to test whether it's worth using a more complicated method would be to replace your looked up acceleration function with something that has a nice formmula for f(v), and then compare the numerical result with the exact result.

-

format: plain


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## Monday, October 13, 2008

### CS: Fast and Efficient Dimensionality Reduction Using Structurally Random Matrices, SAMP code, Nuclear Norm Minimization using SRMs

Following last week's entry that mentioned the connection between Random Projections used in CS and the ones used in Dimensionality Reduction based on the Johnson-Lindenstrauss Lemma, Thong Do pointed out to me his recently submitted paper entitled:

Fast and Efficient Dimensionality Reduction Using Structurally Random Matrices by Thong Do, Lu Gan , Yi Chen, Nam Nguyen and Trac Tran. The abstract reads:

Structurally Random Matrices (SRM) are first proposed in [1] as fast and highly efficient measurement operators for large scale compressed sensing applications. Motivated by the bridge between compressed sensing and the Johnson-Lindenstrauss lemma [2] , this paper introduces a related application of SRMs regarding to realizing a fast and highly efficient embedding. In particular, it shows that a SRM is also a promising dimensionality reduction transform that preserves all pairwise distances of high dimensional vectors within an arbitrarily small factor ², provided that the projection dimension is on the order of O(\epsilon^(-2) log^3 N), where N denotes the number of d-dimensional vectors. In other words, SRM can be viewed as the suboptimal Johnson-Lindenstrauss embedding that, however, owns very low computational complexity O(d log d) and highly efficient implementation that uses only O(d) random bits, making it a promising candidate for practical, large scale applications where efficiency and speed of computation are highly critical.

He is also hosting a webpage on the subject of Structurally Random Matrices (SRMs) I am now changing the attendant link of the Big Picture to that page.

Sariel Har-Peled shows in ex 3.1 of this manuscript on the Johnson-Lindenstrauss lemma that one can also conserve angles as well as distance. Angle preservation was also the subject of interest of Jarvis Haupt and Robert Nowak in A generalized restricted isometry property. One wonders if the SRMs can do the same.

We mentioned SAMP before, Thong Do has released the attendant matlab code of the SAMP algorithm. It is listed in the reconstruction section of the Big Picture.

In a different area, Thong Do, Yi Chen, Nam Nguyen, Lu Gan and Trac Tran have also released A Fast and Efficient Heuristic Nuclear Norm Algorithm for Rank Minimization. The abstract reads:

The problem of affine rank minimization seeks to find the minimum rank matrix that satisfies a set of linear equality constraints. Generally, since affine rank minimization is NP-hard, a popular heuristic method is to minimize the nuclear norm that is a sum of singular values of the matrix variable [1]. A recent intriguing paper [2] shows that if the linear transform that defines the set of equality constraints is nearly isometrically distributed and the number of constraints is at least O(r(m + n) logmn), where r and m £ n are the rank and size of the minimum rank matrix, minimizing the nuclear norm yields exactly the minimum rank matrix solution. Unfortunately, it takes a large amount of computational complexity and memory buffering to solve the nuclear norm minimization problem with known nearly isometric transforms. This paper presents a fast and efficient algorithm for nuclear norm minimization that employs structurally random matrices [3] for its linear transform and a projected subgradient method that exploits the unique features of structurally random matrices to substantially speed up the optimization process. Theoretically, we show that nuclear norm minimization using structurally random linear constraints guarantees the minimum rank matrix solution if the number of linear constraints is at least O(r(m+n) log3 mn). Extensive simulations verify that structurally random transforms still retain optimal performance while their implementation complexity is just a fraction of that of completely random transforms, making them promising candidates for large scale applications.

Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie Institution of Washington, The First Image After Closest Approach of Fly-By 2 of Messenger at 200 kilometers above the surface, Release Date: October 8, 2008


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# Anti matter elements? [closed]

Quite a self-explanatory question I guess. I wonder whether there is any information or theory on this. I seem to like remember there is such an element with positron in center and electron rotating around but nothing else.

Can there be any use if we somehow get these? Don't think like immediate destruction, the electrostatic force between "normal" elements electron and the other types can still repel each other and prevent immediate annihilation, in my view.

Can there be any spot in universe with mostly antimatter?

## closed as too broad by Mithoron, airhuff, pentavalentcarbon, Tyberius, Todd MinehardtMay 10 '18 at 2:17

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Whether/where/how stabke antimatter is, is more of a physics questions. But assuming sufficient short time scales or stable antimatter, you can apply the very same quantum chemical methods to describe the chemistry of such elements. The one you are refering to is conveniently called positronium, and can be considered an excotic, super light weight hydrogen isotop – Feodoran May 8 '18 at 19:54
• anti-hydrogen consists of an anti-proton with a positron in orbit. Not a positron and en electron. A positron is the anti-particle of the electron not the proton. – matt_black May 9 '18 at 12:34

Antihydrogen has been made at CERN in small amounts.

This is made by the alpha experiment, from poistrons and antiprotons. So far they have not been able to do any chemistry experiments.

• But also they have not observed any notable differences between the properties and the properties of normal matter. So it looks like chemistry would be the same. – matt_black May 9 '18 at 12:36

There is no theory, nor is there a need for one. To put in in other words: yes, there is a theory, and that's the one you are already familiar with. Antimatter elements are precisely the same as ordinary matter elements, only the other way around. There "are" (read: "could be", though some were actually detected) antihydrogen, antihelium (all with the same energy levels as their real counterparts), also anticarbon, antiwater, anti-$\ce{NaCl}$, antiglucose, and pretty much everything anty-else.

To compose a substance of particles and antiparticles together is an entirely different question. Yes, there is a compound of electron and positron; it is called positronium and has the lifetime of a few microseconds or something. Electrostatic forces attract the opposite particles, rather than repel them (otherwise it would not be a compound, BTW), so it annihilates pretty quickly. There are other similar compounds, equally or more short-lived. Matter and antimatter do not sit well with each other; there is no going around it.

As for the spots of the Universe composed of antimatter, there was once a mainstream view that there must be many of those, about as many as those made of matter. Then the global symmetry broke, and things went... complicated. I'd rather leave it at that. Go ask at Phys.SE for a more comprehensive take.

So it goes.

• Thanks for the answer, I'll try to ask this in Phys.SE as well. – Güray Hatipoğlu May 9 '18 at 4:17


text:
## Monthly Archives: August 2016

### Time has stopped !!!

My clock has stopped !! 😦 😦 😦

My only time piece is a wall clock. One day I forgot to wind it and it stopped. I went to visit a friend whose watch is always correct, stayed awhile, and returned home. There I made a simple calculation and set the clock right.

How did I do this when I had no watch on me to tell how long it took me to return from my friend’s home?

-Nalin Pithwa

🙂 🙂 🙂

Mathematics is a fantastic escape from the humdrum and mundane monotonous affairs of life !!

Express 100 with five 1’s. Express 100 three ways with five 5’s. You can use brackets, parentheses, and these signs: +, -, x and $\div$.

Have fun !

Nalin Pithwa

### A Russian Duel !!

(From a translated Russian literature)

The Mathematics Circle in our school had this custom: Each applicant was given a simple problem to solve — a little mathematical nut to crack, so to speak. You became a full member only if you solved the problem.

An applicant named Vitia was given this array:

$1 \hspace{0.1in} 1 \hspace{0.1in} 1 \hspace{0.1in}$

$3 \hspace{0.1in} 3 \hspace{0.1in} 3 \hspace{0.1in}$

$5 \hspace{0.1in} 5 \hspace{0.1in } 5 \hspace{0.1in}$

$7 \hspace{0.1in} 7 \hspace{0.1in} 7 \hspace{0.1in}$

$9 \hspace{0.1in} 9 \hspace{0.1in} 9 \hspace{0.1in}$

He was asked to replace 12 digits with zeros so that the sum would be 20. Vitia thought a little, then wrote rapidly:

$0 \hspace{0.1in} 1 \hspace{0.1in} 1 \hspace{0.1in}$

$0 \hspace{0.1in} 0 \hspace{0.1in} 0 \hspace{0.1in}$

$0 \hspace{0.1in} 0 \hspace{0.1in} 0 \hspace{0.1in}$

$0 \hspace{0.1in} 0 \hspace{0.1in} 0 \hspace{0.1in}$

$0 \hspace{0.1in} 0 \hspace{0.1in} 9 \hspace{0.1in}$

$----$

$X \hspace{0.1in} 2 \hspace{0.1in} 0 \hspace{0.1in}$

and

$0 \hspace{0.1in} 1 \hspace{0.1in} 0 \hspace{0.1in}$

$0 \hspace{0.1in} 0 \hspace{0.1in} 3 \hspace{0.1in}$

$0 \hspace{0.1in} 0 \hspace{0.1in} 0 \hspace{0.1in}$

$0 \hspace{0.1in} 0 \hspace{0.1in} 7 \hspace{0.1in}$

$0 \hspace{0.1in} 0 \hspace{0.1in} 0 \hspace{0.1in}$

$----$

$X \hspace{0.1in} 2 \hspace{0.1in} 0 \hspace{0.1in}$.

He smiled and said, “If you substitute just ten zeros for digits, the sum will be 1111. Try it!”

The Circle’s president was taken aback briefly. But, he not only solved Vitia’s problem, but improved upon it:

“Why not replace only nine digits with 0’s —- and still get 1111?”

As the debate continued, ways of getting 1111 by replacing 8, 7, 6, and 5 digits with zeros were found.

Solve the six forms of this problem.

More duels for you to prove your mettle are coming !!

Nalin Pithwa

### From Russia with love of numbers !!

Problem:

There are three ways to add four odd numbers and get 10:

$1+1+3+5=10$

$1+1+1+7=10$

$1+3+3+3=10$

Changes in the order of numbers do not count as new solutions.

Now add eight odd numbers to get 20. To find all eleven solutions, you will need to be systematic.

-Nalin Pithwa

PS: I found this gem in a Russian book (translated).

### Applications of mathematics in sports — careers in mathematics

In India, as is well-known, the most popular out-door sport is cricket. There are various statistics associated with cricket, as with any other sport. Especially, in cricket, we find the following parameters discussed about in every match:

• General statistics (matches, catches, stumpings)
• Batting statistics (innings, Not Outs, runs, highest score, batting average, centuries, half-centuries, balls faced, strike rate, run rate, Net run rate).
• Bowling Statistics: Overs. Bowls, Maiden Overs, Runs, Bowling Analysis, Wickets, No balls, Wides, Bowling Averages, Strike Rate, Economy Rate, Best Bowling In Innings, Best Bowling in Match, Five Wickets in an Inning, Ten Wickets in a match,
• Dynamic and graphical statistics: The advent of saturation television coverage of professional cricket has provided an impetus to develop new and interesting forms of presenting statistical data to viewers. Television networks have thus invented several new ways of presenting statistics.
• The T-20’s and ODI’s might have different statistics because their purposes are different. The county matches have a yet different purpose.

There are numerous books that deal with the applications of mathematics in sports. Three very good ones are:

• de Mestre, N (1990): The Mathematics of Projectiles in Sport. Cambridge, UK: Cambridge University Press.
• Hart, D. and T. Croft (1988). Modelling with Projectiles, Chicester, UK: Ellis Horwood.
• Townend, M. S (1984), Mathematics in Sport, Chicester, UK: Ellis Horwood.

More later,

Nalin Pithwa

### Apples from Moscow !!

Here is a puzzle, which I found in a Russian book of puzzles!!

Five Apples

Five apples are in a basket. How do you divide them among five girls so that each girl gets an apple, but one apple remains in the basket?

Well, an apple a day keeps the doctor away; a puzzle a day keeps the brain fresh !!

-Nalin Pithwa

### Applications of sinc function

The occurrence of the function $\frac{\sin {x}}{x}$ in calculus is an isolated event. The function arises in diverse fields such as quantum physics (where it appears in solutions of the wave equation) and electrical engineering(in signal analysis and DSP filter design) as well as in the mathematical fields of differential equations and probability theory.

-Nalin Pithwa.

### Sunday Coffee Time Puzzles !

1. A train, an hour after starting, meets with an accident which detains it an hour, after which it proceeds at three-fifths of its former rate and arrives 3 hours after time; but, had the accident happened 50 miles further on the line, it would have arrived 1.5 hours sooner: find the length of the journey.
2. A body of men were formed into a hollow square, three deep, when it was observed, that with the addition of 25 to  their number a solid square might be formed, of which the number of men in each side would be greater by 22 than the square root of the number of men in each aide of the hollow square: required the number of men.
3. A set out to walk at the rate of 4 miles an hour; after he had been walking $2\frac{3}{4}$, B set out to overtake him and went $4\frac{1}{2}$ miles the first hour, $4\frac{3}{4}$ miles the second, 5 the third, and so gaining a quarter of a mile every hour. In how many hours would he overtake A?
4. A man wishing his two daughters to receive equal portions when they came of age bequeathed to the elder the accumulated interest of a certain sum of money invested at the time of his death in 4 per cent stock at 88; and to  the younger he bequeathed the accumulated interest of a sum less than the former by $\pounds 3500$ invested at the same time in the 3 per cents, at 63. Supposing their ages at the time of their father’s death to have been 17 and 14, what was the sum invested in each case, and what was each daughter’s fortune?
5. A and B travelled on the same road and at the same rate from Huntington to London. At the $50^{th}$ milestone from London, A overtook a drove of geese which were proceeding at the rate of 3 miles in 2 hours; and two hours afterwards met a waggon, which was moving at the $46^{th}$ milestone, and met the waggon exactly 40 minutes before he came to the $31^{st}$ milestone. Where was B when A reached London?
6. To complete a certain work, a workman A alone would take m times as many days as B and C working together; B alone would take a times as many days as A and C together; C alone would take p times as many days as A and B together; show that the numbers of days in which each would do it alone are as $m+1:n+1:p+1$.
7. A traveller set out from a certain place, and went 1 mile the first day, 3 the second, 5 the next, and so on, going every day 2 miles more than he had gone the preceding day. After he had been gone three days, a second sets out, and travels 12 miles the first day, 13 the second and so on. In how  many days, will the second overtake the first? Explain the double answer.
8. A number of persons were engaged to do a piece of work which would have occupied them 24 hours if they had commenced at the same time; but instead of doing so, they commenced at equal intervals and then continued to work till the whole was finished, the payment being proportional to the work done by each: the first comer received eleven times as much as the last; find the time occupied.
9. There are three towns A, B and C; a person by walking from A to B, driving from B to C, and riding from C to A makes the journey in 15.5 hours; by driving from A to B, riding from B to C, and walking from C to A, he could make the journey in 12 hours. On foot he could make the journey in 22 hours, oh horseback in 8.25 hours, and driving in 11 hours. To walk a mile, ride a mile, and drive a mile he takes altogether half an hour; find the rates at which he travels, and the distance between the towns.
10. In a mixed company consisting of Poles, Turks, Greeks, Germans and Italians, the Poles are one less than the one-third of the number of Germans, and three less than half the number of Italians. The Turks and Germans outnumber the Greeks and Italians by 3; the Greeks and Germans form one less than the half the company; while the  Italians and Greeks form seven-sixteenths of the company; determine the number of each nation.

More fun later,

Nalin Pithwa

### Some Applications of Derivatives — Part II

Derivatives in Economics.

Engineers use the terms velocity and acceleration to refer to the derivatives of functions describing motion. Economists, too, have a specialized vocabulary for rates of change and derivatives. They call them marginals.

In a manufacturing operation, the cost of production c(x) is a function of x, the number of units produced. The marginal cost of production is the rate of change of cost (c) with respect to a level of production (x), so it is $dc/dx$.

For example, let c(x) represent the dollars needed needed to produce x tons of steel in one week. It costs more to produce x+h units, and the cost difference, divided by h, is the average increase in cost per ton per week:

$\frac{c(x+h)-c(x)}{h}=$ average increase in cost/ton/wk to produce the next h tons of steel

The limit of this ratio as $h \rightarrow 0$ is the marginal cost of producing more steel when the current production level is x tons.

$\frac{dc}{dx}=\lim_{h \rightarrow 0} \frac{c(x+h)-c(x)}{h}=$ marginal cost of production

Sometimes, the marginal cost of production is loosely defined to be the extra cost of producing one unit:

$\frac{\triangle {c}}{\triangle {x}}=\frac{c(x+1)-c(x)}{1}$

which is approximately the value of $dc/dx$ at x. To see why this is an acceptable approximation, observe that if the slope  of c does not change quickly near x, then the difference quotient will be close to its limit, the derivative $dc/dx$, even if $\triangle {x}=1$. In practice, the approximation works best for large values of x.

Example: Marginal Cost

Suppose it costs $c(x)=x^{3}-6x^{2}+15x$  dollars to produce x radiators when 8 to 30 radiators are produced. Your shop currently produces 10 radiators a day. About how much extra cost will it cost to produce one more radiator a day?

Example : Marginal tax rate

To get some feel for the language of marginal rates, consider marginal tax rates. If your marginal income tax rate is 28% and your income increases by USD 1000, you can expect to have to pay an extra USD 280 in income taxes. This does not mean that you pay $28$ percent of your entire income in taxes. It just means that at your current income level I, the rate of increase of taxes I with respect to income is $dT/dI = 0.28$. You will pay USD 0.28 out of every extra dollar you earn in taxes. Of course, if you earn a lot more, you may land in a higher tax bracket and your marginal rate will increase.

Example: Marginal revenue:

If $r(x) = x^{3}-3x^{2}+12x$ gives the dollar revenue from selling x thousand candy bars, $5<= x<=20$, the marginal revenue when x thousand are sold is

$r^{'}(x) = \frac{d}{dx}(x^{3}-3x^{2}+12x)=3x^{2}-6x+12$.

As with marginal cost, the marginal revenue function estimates the increase in revenue that will result from selling one additional unit. If you currently sell 10 thousand candy bars a week, you can expect your revenue to increase by about $r^{'}(10) = 3(100) -6(10) +12=252$ USD, if you increase sales to 11 thousand bars a week.

Choosing functions to illustrate economics.

In case, you are wondering why economists use polynomials of low degree to illustrate complicated phenomena like cost and revenue, here is the rationale: while formulae for real phenomena are rarely available in any given instance, the theory of  economics can still provide valuable guidance. the functions about which theory speaks can often be illustrated with low degree polynomials on relevant intervals. Cubic polynomials provide a good balance between being easy to work with and being complicated enough to illustrate important points.

Ref: Calculus and Analytic Geometry by G B Thomas.

More later,

Nalin Pithwa

### Some Applications of Derivatives — Part I

Sensitivity to Change.

When a small change in x produces a large change in the value of a function $f(x)$, we say that the function is relatively sensitive to changes in x. The derivative $f^{'}(x)$ is a measure of the senstivity to change at x.

Example:

The Austrian monk Gregor Johann Mendel (1822-1884), working with garden peas and other plants, provided the first scientific explanation of hybridization. His careful records show that if p (a number between 0 and 1) is the frequency of the gene for smooth skin in peas (dominant) and $(1-p)$ is the frequency of the gene for wrinkled skin in peas, then the proportion of smooth-skinned peas in the population at large is

$y = 2p(1-p)+p^{2}=2p-p^{2}$

The graph of y versus p is an inverted parabola. If you plot and see, it suggests that the value of y is more sensitive to a change in p, when p is small than when p is large. Instead, this is borne out by the derivative graph which shows that ($\frac{dy}{dp}$) is close to 2, when p is  near zero, and close to zero, when p is near 1.

More later,

Nalin Pithwa

PS: why peas wrinkle

British geneticists had discovered that the wrinkling trail comes from an extra piece of DNA that prevents the  gene that directs starch synthesis from functioning properly. With the plant’s starch conversion impaired, sucrose and water build up in the young seeds. As the seeds mature, they lose much of this water, and the shrinkage leaves them wrinkled.


text:
### Home > CCA2 > Chapter 7 > Lesson 7.1.5 > Problem7-81

7-81.

Greg was working on his homework. He completed the square to change $y=2x^2−6x+2$ to graphing form and identify the vertex of the parabola. He did the work below and identified the vertex to be $\left(\frac{3}{2},-\frac{1}{4}\right)$. When he got back to class and checked his answers, he discovered that his vertex was wrong, but he cannot find his mistake. Examine Greg's work and explain to him where the mistake occurred. Then show him how to correct the mistake and state the vertex.

$y=2x^2−6x+2\\y = 2(x^2 − 3x) + 2\\y = 2(x^2 − 3x + \frac { 9 } { 4 }) + 2 − \frac { 9 } { 4 }\\y = 2(x − \frac { 3 } { 2 })^2 − \frac { 1 } { 4 }$

Use the Distributive Property to rewrite the third equation. Is it equivalent to the first equation?
It should be.


text:
# Determination of chloride content in aluminium chloride by titration

TLDR: Is there an as simple and direct as possible method to measure chloride content by titration?

I want to measure the content (mass fraction) of chloride in $$\mathrm{AlCl_3}$$ powder by titration. Purpose is to measure the purity of the aluminium chloride. For example assume the measured chloride content is 75 % then that means the $$\mathrm{AlCl_3}$$ is 94.0 % pure.

When I search for how to determine chloride by titration I get the following three methods, which appear to be more complex than needed: Mohr method, Volhard method, Fajans method. All of which use an indicator and require specific conditions and treatments or reagents. The Volhard method is also a back titration and seems to be the default method for some reason which I could not find.

I found the following procedure, which is basically the Volhard method but with an additional reagents (of which I do not know the purpose) and an additional dilution step.

1.) Prepare an flask with stopper with 50 ml water and chill in the freezer to form ice.

2.) Weigh about 1 g of sample and transfer it in to the flask along with the paper and put the stopper.

3.) Allow it to come to room temperature. Add a few drops of phenolphthalein indicator and titrate with 1N NaOH solution to the end point.

4.) Dilute the solution to 250 mL. Pipette out 20 mL in to another flask.

5.) Add 3-4 mL of conc. nitric acid , 50 mL of 0.1 N silver nitrate solution, 2 mL of nitro benzene and a few drops of ferric ammonium sulfate indicator.

6.) Titrate with 0.1 N ammonium thiocyanate solution to reddish brown end point.

No idea what the nitro benzene is for. And the additional dilution step is certainly unnecessary and lowers precision; could just weight less sample and not dilute. The NaOH step is probably to react HCl (which is formed when $$\mathrm{AlCl_3}$$ reacts with water and may escape and thus lower the chlorine content) to NaCl; but also not sure about that.

What speaks against just directly titrating the chloride containing sample solution with the 0.1N $$\mathrm{AgNO_3}$$ solution and detect the end point potentiometric? Am I missing something? Why would people do a back titration with two solutions needed, instead of just a direct titration?

• You can consult Vogel's Quantitative Analysis. There you will find all titrimetric details for chloride and explanations. Web search will only lead to confusion. Feb 11 at 16:15
• All your answers to the question are available in Vogel. Feb 11 at 16:16
• @M.Farooq - What, no love for Skoog and West? Feb 11 at 19:33
• @ToddMinehardt, I like classical books when it comes to titrimetric and gravimetric methods. Vogel's classical Quantitative Analysis from the 60s was a massive book with >1200 pages, all extensive details such as holding volumetric pipette. Feb 11 at 22:02
• Here is the textbook for the OP on Internet Archive. archive.org/details/textbookofquanti0006voge/page/n7/mode/… Feb 11 at 22:03

This problem is not as easy as it looks. It starts from the hypothesis that $$\ce{AlCl3}$$ is not pure in the original flask- The impurity will be supposed not to interfere later on. Only $$\ce{H+}$$ and $$\ce{Cl-}$$ will be determined by titration.

Let's suppose that the original sample of substance weighs $$m_o$$ and contains $$n\ce{_o mol AlCl3}$$. When dissolved into water, this substance reacts according to : $$\ce{AlCl3 + 3 H2O <=> 3 H+ + 3 Cl^- + Al(OH)3} \tag{1}$$ So, if the hydrolysis is complete (and it will be during titrating by $$\ce{NaOH}$$), the total amount of $$\ce{H+}$$ and $$\ce{Cl^-}$$ in the solution is $$3n_o$$. The first titration, with $$\ce{NaOH}$$, determines the amount $$n_1$$ = $$n\ce{(NaOH)}$$ = $$n(\ce{H+})$$ = $$\ce{n({Cl^-})}$$ = $$3n_o$$ produced by the reaction $$(1)$$. But, apparently, the result of this titration is not reliable enough. The author of the problem is afraid of some unwanted acidic contamination. So the actual amount $$n_2$$ of $$\ce{H+}$$ and of $$\ce{Cl-}$$ due to ($$1$$) will be obtained by another way, as will be described now.

The second titration is made to determine the amount $$n_2$$ of $$\ce{Cl-}$$ ion in solution. But it cannot be done directly, as, after the first titration by $$\ce{NaOH}$$, the solution contains $$\ce{NaCl}$$ and maybe some $$\ce{NaOH}$$. It is first acidified by adding some drops of $$\ce{HNO3}$$ to destroy this unwanted $$\ce{NaOH}$$ . Then a known excess of $$n_3$$ mol silver nitrate is added. All the $$n_2$$ mol $$\ce{Cl-}$$ ions are precipitated as insoluble $$\ce{AgCl}$$. $$\ce{Ag^+ + Cl- -> AgCl(s)} \tag{2}$$ The solution contains now an excess of $$n_3 - n_2$$ mol $$\ce{Ag+}$$ ions. Some drops of ferric ions are added (plus nitrobenzene - I don't know why). The excess of $$\ce{Ag+}$$ ions is titrated by gradually adding a thiocyanate solution. In the beginning, it produces the reaction : $$\ce{Ag+ + SCN^- -> AgSCN(s)}\tag{3}$$ $$\ce{AgSCN}$$ is a white insoluble precipitate, which looks like $$\ce{AgCl}$$.

The end point can be determined with a high degree of precision. Because, when the precipitation reaction (3) is finished, the next drop of thiocyanate reacts with the ferric ions to produce a dark red color, made of the ion $$\ce{Fe(SCN)^{2+}}$$ which is dark red. So as soon as the last amount of $$\ce{AgSCN}$$ is precipitated, the solution gets red. And this point can be determined with a high precision.

If $$n_4$$ mol thiocyanate have been added to get to the final red color, the amount $$n_2$$ of $$\ce{Cl-}$$ ions can be determined : $$n_2 = n_4 - n_3$$. And the original mass of $$\ce{AlCl3}$$ is equal to $$m = \frac{(n_4 - n_3) M(\ce{AlCl3})}{3}$$, if $$M(\ce{AlCl3})$$ is the molar mass of $$\ce{AlCl3}$$. One should find that $$m < m_o$$

• Maurice, I think HAlCl4 does not even exist. Feb 11 at 22:46
• @M. Farooq. You may be right, after all. I know that the chloraluminate ion $\ce{AlCl_4^-}$ does exist. But the existence of $\ce{HAlCl4}$ is not proven. So I will edit and modify my answer. Feb 12 at 20:32
• @Maurice Thank you for the explanation. The $\ce{AlCl3}$ is supposed to be pure or almost pure (> 99.5 %) as it is purchased from a supplier, but we have to test the assay of it before using it in a production process. Now, what I really do not understand is: Why is it not common to directly titrate the chloride (be it in the form of $\ce{[Al(H2O)6Cl3]}$ or $\ce{HCl}$ or $\ce{NaCl}$) with the $\ce{AgNO3}$ solution? Then the used volume of $\ce{AgNO3}$ solution directly corresponds to the amount of chloride. Why the detour via a back titration with thiocyanate? Feb 14 at 10:18
• @Wolfram. Even if the product is sold $99.5$% by the supplier, you cannot avoid that some part of $\ce{AlCl3}$ has been hydrolyzed in the flask during its storage. Feb 14 at 11:16


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# Local construction of a map on a manifold

Assume $(M,J)$ is a smooth manifold equipped with an almost complex structure. Let $p$ be a point of $M$, and define the linear map $\Theta_p \colon T_pM \to T_pM$ so that $$\Theta_p = -\frac{1}{2}\mathrm{id}_{T_pM} + \frac{\sqrt{3}}{2}J_p.$$ One has that $\Theta_p^3 = \mathrm{id}_{T_pM}$, so $\Theta_p$ is a linear isomorphism of $T_pM$. The goal is to show there exists a neighbourhood of $p$, $U(p)$, and a map $\theta_p \colon U(p) \to U(p)$ such that $\theta_{p*} = \Theta_p$.

I have tried to solve this exercise finding a candidate for $\theta$: locally around $p$ one can assume to have a metric and then a connection, so we can construct an exponential map $\exp_p \colon T_pM \to U(p)$, where $U(p)$ is some neighbourhood of $p$. If $t$ is a real parameter in a sufficiently small open interval containing $0$, then $\exp_p tX_p = \gamma(t,p)$, where $\gamma$ is a curve on $M$ such that $\gamma(0,p) = p$ and $\gamma'(0,p) = X_p$. Hereafter, $Y_p$ is a tangent vector at $p$ and $\alpha(t,p)$ its integral geodesic. Defining $\theta_p := \exp_p \circ \thinspace \Theta_p \circ\exp_p^{-1}\colon U(p) \to U(p)$, I get \begin{align} \theta_{p*}(Y_p) & = \frac{\mathrm{d}}{\mathrm{dt}}\left(\exp_p \circ \thinspace \Theta_p \circ \exp_p^{-1}(\alpha(t,p)) \right)\Bigr\lvert_{t=0} \\ & = \frac{\mathrm{d}}{\mathrm{dt}}\left(\exp_p \circ \thinspace \Theta_p(tY_p) \right)\Bigr\lvert_{t=0} \\ & = \frac{\mathrm{d}}{\mathrm{dt}}\left(\exp_p\left(t\left( -\frac{1}{2}Y_p+\frac{\sqrt{3}}{2}J_pY_p\right)\right)\right)\Biggr\lvert_{t=0} \\ & = -\frac{1}{2}Y_p+\frac{\sqrt{3}}{2}J_pY_p = \Theta_p(Y_p). \end{align} What I am not convinced of is that I am fixing a curve $\alpha$ which is the geodesic associated to some vector $Y_p$. I do not know if this can be an arbitrary curve, and then I am not quite sure I solved the problem. Probably the point is the definition of the exponential map. Other solutions are also welcome, of course.


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Shaded Area – Right Triangle Inscribed in a Circle

A right triangle is inscribed in a circle such that its longest side is the diameter of the circle. If the shorter sides of the triangle measure 6cm and 8cm, find the area of the shaded region. Use $\pi = 3.14$

Solution

A triangle inscribed in a circle with its longest side as the diameter of the circle is always a right triangle (by Thales’ Theorem). So, we can find the area $A_T$ of the triangle.

Finding the area of the triangle,

$A_T = \frac{6 \times 8}{2} = 24$Continue reading “Shaded Area – Right Triangle Inscribed in a Circle”

Grade 3 MTAP Reviewer Set 8

Below is the eight set of the MTAP Reviewer for Grade 3.

1.) ¾ is how many eighths?

2.) Write “six thousand fifty six” in numeric form.

3.) How many years are there in 7 ½ decades?

4.) Two numbers are in the ratio 2:3. Their sum is 45. What are the numbers?

5.) Using the digits 2, 5, 8 and 3. Write the largest number greater than 2000 but less than 8000?

6.) Sheila has 250 pesos. She spent 1/5 of it buying notebooks. Then she spent half of the remaining for her meal. How much money did she have left?  Continue reading “Grade 3 MTAP Reviewer Set 8”

Grade 2 MTAP Reviewer Set 8

Below is the eighth set of the MTAP Reviewer for Grade 2.

1.) How many months are there in 3 ½ years?

2.) What is the value of N? 7, 11, 15, 19, N, 27?

3.) What is LXV in Hindu Arabic numerals?

4.) How many much more is 9 × 6 than 12 + 15?

5.) If May 12 is a Monday, what day is May 27?

6.) A store has 120 skateboards. On a Monday, 90 were rented. What fraction was not rented?

7.) A rectangular flower garden has length 12m and width 9m. What is its perimeter? Continue reading “Grade 2 MTAP Reviewer Set 8”

Shaded Region – Circle Inscribed in a Square

A circle with radius 3 cm is inscribed in a square.

Find the area of the shaded region. Let pi = 3.14 and round your answer to the nearest tenths.

Solution

We know that

area of shaded region = area of square – area of circle  Continue reading “Shaded Region – Circle Inscribed in a Square”


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The current GATK version is 3.7-0
Examples: Monday, today, last week, Mar 26, 3/26/04

#### Howdy, Stranger!

It looks like you're new here. If you want to get involved, click one of these buttons!

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3. Include tool and Java versions.
4. Tell us whether you are following GATK Best Practices.
5. Include relevant details, e.g. platform, DNA- or RNA-Seq, WES (+capture kit) or WGS (PCR-free or PCR+), paired- or single-end, read length, expected average coverage, somatic data, etc.
6. For tool errors, include the error stacktrace as well as the exact command.
7. For format issues, include the result of running ValidateSamFile for BAMs or ValidateVariants for VCFs.
8. For weird results, include an illustrative example, e.g. attach IGV screenshots according to Article#5484.
9. For a seeming variant that is uncalled, include results of following Article#1235.

#### ☞ Formatting tip!

Wrap blocks of code, error messages and BAM/VCF snippets--especially content with hashes (#)--with lines with three backticks ( ` ) each to make a code block as demonstrated here.

Picard 2.10.4 has MAJOR CHANGES that impact throughput of pipelines. Default compression is now 1 instead of 5, and Picard now handles compressed data with the Intel Deflator/Inflator instead of JDK.
GATK version 4.beta.2 (i.e. the second beta release) is out. See the GATK4 BETA page for download and details.

# Joint calling in trios using UnifiedGenotyper

Member

Hi,

I have exome data for a few sets of parent-offspring trios, in which offspring have phenotypically related but probably genetically different diseases. Their parents are unaffected so I am particularly interested in identifying de novo mutations. My plan was to analyse each individual separately up to the variant calling phase and then to input three (mother, father, child) analysis-ready BAMs into the UnifiedGenotyper along with a ped file. My questions are:

1) Can you tell me whether the UnifiedGenotyper uses pedigree information in the ped file to call genotypes more accurately? In other words, is this better than calling variants jointly without supplying the ped file? If not, does GATK recommend any external tools for doing this step?

2) It is better to call variants jointly using all the trios (even though they are not related and probably don't share the same disease-causing mutations)?

Best wishes,

Kath

Tagged:


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• ### A United Model for the cosmic ray energy spectra and anisotropy in the energy range 100 - 100 000 GeV(1203.6458)

Nov. 10, 2012 astro-ph.HE
We propose a model where a supernova explodes in some vicinity of our solar system (some tens of parsecs) in the recent past (some tens of thousands years) with the energy release in cosmic rays of order of $10 ^ {51}$ erg. The flux from this supernova is added to an isotropic flux from other sources. We consider the case where the Sun's location is not in some typical for Our Galaxy average environment, but in the Local Superbubble about 100 pc across, in which the diffusion coefficient $D (E) = D_0 \times E ^ {0.6}$, with the value of $D_0 \sim 10 ^ {25} cm^ 2 s^ {-1}$. We describe the energy dependence of the anisotropy of cosmic rays in the TeV region, together with the observed features of the energy spectrum of protons found in direct measurements. Our model provides a natural explanation to the hardening of the proton spectrum at 200 GeV, together with the observed steepening of the spectrum above 50 TeV.
• ### Possible structure in the cosmic ray electron spectrum measured by the ATIC-2 and ATIC-4 experiments(1104.3452)

April 18, 2011 astro-ph.HE
A strong excess in a form of a wide peak in the energy range of 300-800 GeV was discovered in the first measurements of the electron spectrum in the energy range from 20 GeV to 3 TeV by the balloon-borne experiment ATIC (J. Chang et al. Nature, 2008). The experimental data processing and analysis of the electron spectrum with different criteria for selection of electrons, completely independent of the results reported in (J. Chang et al. Nature, 2008) is employed in the present paper. The new independent analysis generally confirms the results of (J. Chang et al. Nature, 2008), but shows that the spectrum in the region of the excess is represented by a number of narrow peaks. The measured spectrum is compared to the spectrum of (J. Chang et al. Nature, 2008) and to the spectrum of the Fermi/LAT experiment.
• ### Energy Spectra of Abundant Nuclei of Primary Cosmic Rays from the Data of ATIC-2 Experiment: Final Results(1101.3246)

Jan. 17, 2011 astro-ph.HE
The final results of processing the data from the balloon-born experiment ATIC-2 (Antarctica, 2002-2003) for the energy spectra of protons and He, C, O, Ne, Mg, Si, and Fe nuclei, the spectrum of all particles, and the mean logarithm of atomic weight of primary cosmic rays as a function of energy are presented. The final results are based on improvement of the methods used earlier, in particular, considerably increased resolution of the charge spectrum. The preliminary conclusions on the significant difference in the spectra of protons and helium nuclei (the proton spectrum is steeper) and the non-power character of the spectra of protons and heavier nuclei (flattening of carbon spectrum at energies above 10 TeV) are confirmed. A complex structure of the energy dependence of the mean logarithm of atomic weight is found.
• ### Energy dependence of Ti/Fe ratio in the Galactic cosmic rays measured by the ATIC-2 experiment(0905.0049)

May 1, 2009 astro-ph.HE
Titanium is a rare, secondary nucleus among Galactic cosmic rays. Using the Silicon matrix in the ATIC experiment, Titanium has been separated. The energy dependence of the Ti to Fe flux ratio in the energy region from 5 GeV per nucleon to about 500 GeV per nucleon is presented.
• ### The results of ATIC-2 experiment for elemental spectra of cosmic rays(astro-ph/0612377)

Dec. 14, 2006 astro-ph
The results of ATIC-2 baloon experiment (2002-2003) for energy spectra of protons, He, C, O, Ne, Mg, Si, Fe, some groups of nuclei, and all-particle spectrum in primary cosmic rays are presented in energy region 50GeV--200TeV. The conclusion is that the spectra of protons and helium nuclei are essentially different (the spectrum of protons is steeper) and the spectra of protons and heavy nuclei have no-power form.
• ### Three component model of cosmic ray spectra from 10 GeV to 100 PeV(astro-ph/0601475)

Aug. 24, 2006 astro-ph
A model to describe cosmic ray spectra in the energy region from 10 GeV to 100 PeV is suggested based on the assumption that Galactic cosmic ray flux is a mixture of fluxes accelerated by shocks from nova and supernova of different types. We analyze recent experimental data on cosmic ray spectra obtained in direct measurements above the atmosphere and data obtained with ground Extensive Air Shower arrays. The model of the three classes of cosmic ray sources is consistent with direct experimental data on cosmic ray elemental spectra and gives a smooth transition from the all particle spectrum measured in the direct experiments to the all particle spectrum measured with EAS.


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MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

One of the questions I asked above was if the addition of different size spheres changes maximum chromatic number. I can prove it does not for dimension equals 2. We have the set of graphs of tagent tangent circles is the same as the set of planar graphs. Now the maximum chromatic number of a planar graph is 4 by the four color theorem. What is needed is a graph of circles of the same radius which has chromatic number four. Assume that every such graph can be colored with three colors Now look any four circles of unit radius $a$, $b$, $c$ and $d$ with $a$, $b$ and $c$ mutually tangent and $b$, $c$ and mutually tangent. The only way this can happen is if $a$, $b$ and $c$ have their centers forming an equilateral triangle of side 2 as do $b$, $c$ and $d$. If there is a three coloring we have $a$ and $d$ forced to be the same color. We can arrange a series of these graphs in a cycle such that $a$, $d$,... $x$ have the same color and $x$ is tangent to $a$. This will give a contradiction. So this graph has chromatic number four and we are done.
One of the questions I asked above was if the addition of different size spheres changes maximum chromatic number. I can prove it does not for dimension equals 2. We have the set of graphs of tagent circles is the same as the set of planar graphs. Now the maximum chromatic number of a planar graph is 4 by the four color theorem. What is needed is a graph of circles of the same radius which has chromatic number four. Assume that every such graph can be colored with three colors Now look any four circles of unit radius $a$, $b$, $c$ and $d$ with $a$, $b$ and $c$ mutually tangent and $b$, $c$ and mutually tangent. The only way this can happen is if $a$, $b$ and $c$ have their centers forming an equilateral triangle of side 2 as do $b$, $c$ and $d$. If there is a three coloring we have $a$ and $d$ forced to be the same color. We can arrange a series of these graphs in a cycle such that $a$, $d$,... $x$ have the same color and $x$ is tangent to $a$. This will give a contradiction. So this graph has chromatic number four and we are done.


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# Integrating $\sec^{3}\theta$ . [duplicate]

How do I integrate the trigonometric term ?

$$\sec^{3} \theta$$

My assumption

I tried to write it in cosine form and apply formula of $\cos^{3}\theta$ , but to no purpose. What should I do ?

## marked as duplicate by Eric Towers, Lord Shark the Unknown, user228113, RRL, Tom-TomMar 11 '18 at 21:56

• Do you realize how many duplicate questions there are to this? – Paul Mar 11 '18 at 18:43
• Nope ... please give me some links...the rules might be same whereas the forms might be unseen by me . It would be helpful... – A.H.M. Mar 11 '18 at 18:44
• @Paul Entering "\int\sec^3\theta d\theta" in the search box turned up literally dozens of pages; all I checked had this integral. – user5713492 Mar 11 '18 at 18:52

You have\begin{align}\int\sec^3\theta\,\mathrm d\theta&=\int\sec^2\theta\sec\theta\,\mathrm d\theta\\&=\tan\theta\sec\theta-\int\tan\theta\sec'\theta\,\mathrm d\theta\\&=\tan\theta\sec\theta-\int\tan^2\theta\sec\theta\,\mathrm d\theta\\&=\tan\theta\sec\theta-\int\frac{\sin^2\theta}{\cos^3\theta}\,\mathrm d\theta\\&=\tan\theta\sec\theta-\int\frac{\cos\theta\sin^2\theta}{(1-\sin^2\theta)^2}\,\mathrm d\theta.\end{align}Now, you can compute this last primitive using the substitution $\sin\theta=x$ and $\cos\theta\,\mathrm d\theta=\mathrm dx$
use that $$\int\sec^m(x)dx=\frac{\sin(x)\sec^{m-1}(x)}{m-1}+\frac{m-2}{m-1}\int\sec^{m-2}(x)dx$$ and set $m=3$!
Hint: \begin{align} \int\sec^3(\theta)\,\mathrm{d}\theta &=\int\sec^4(\theta)\,\mathrm{d}\sin(\theta)\\ &=\int\frac1{\left(1-\sin^2(\theta)\right)^2}\,\mathrm{d}\sin(\theta)\\ \end{align} and use Partial Fractions.
$$\int \sec^2 x \sec x \ \mathrm dx =\int (1+\tan^2 x) \sec^2 x \ \mathrm dx$$ Set $u=\tan x$ $$\int 1+u^2 \ \mathrm du= u+\frac{u^3}{3}+C = \tan x+\frac{\tan^3 x}{3}+C$$
• The integrand on the left is equal to $\sec^3$ and the integrand on the right is equal to $\sec^4$. – George Coote Mar 11 '18 at 19:04


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Viṣṇu Purāṇa | Book 4 - Chapter 21

Chapter XXI

I WILL now enumerate the kings who, will reign in future periods:

The present monarch, Parīkṣit, will have four sons, Janamejaya, Śrutasena, Ugrasena, and Bhīmasena.

The son of Janamejaya will be Śatānīka, who will study the Vedas under Yājnyawalkya, and military science with Kripa; but becoming dissatisfied with sensual enjoyments, he will acquire spiritual knowledge from the instructions of Śaunaka, and ultimately obtain salvation.

His son will be Aśvamedha-datta (a son given by the gods in reward for the sacrifice of a horse); his son will be Asīma-kṛṣṇa; his son will be Nichakra, who will remove the capital to Kauśāmbī, in consequence of Hastināpura being washed away by the Ganges;

his son will be Uṣṇa; his son will be Chitraratha; his son will be Vṛṣṇimat; his son will be Susheṇa; his son will be Sunītha; his son will be Richa; his son will be Nrichakṣu;

his son will be Sukhīhala; his son will be Pariplava; his son will be Sunaya; his son will be Medhāvin; his son will be Nripanjaya; his son will be Mridu; his son will be Tigma;

his son will be Vrihadratha; his son will be Vasudāna; and his son will be another Śatānīka; his son will be Udayana; his son will be Ahīnara; his son will be Khaṇḍapāni; his son will be Niramitra; his son will be Kṣemaka: of him this verse is recited:

"The race which gave origin to Brahmans and Kṣatriyas, and which was purified by regal sages, terminated with Kṣemaka; in the Kali age."


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# Slides of Talk on Sage & Algebraic Techniques

The slides of my Icebreak talk on Sage and algebraic techniques for (lazy) cryptographers are now available (LaTeX sources here).

# BKW: Update

We have updated our pre-print titled “On the Complexity of the BKW Algorithm on LWE” on ePrint.

There are two main changes and the reasons why I am mentioning this update here.

1. We included a more thorough comparison with other approaches, in particular, with lattice reduction (reducing LWE to SIS). To our surprise, BKW is quite competitive even in relatively modest dimensions. For Regev’s and Lindner-Peikert’s parameter sets (as interpreted here) we get that BKW is at least as fast as BKZ starting in dimension $n \approx 250$, which I find very low (see Table 4 on page 19).
2. We also provide an alternative approximating for the running time of BKZ. The standard estimate due to Lindner-Peikert is $\log_2 T_{sec} = \log_2 1.8/\delta_0 - 110$ where $\delta_0$ is the targeted root hermit factor. Interpolating estimates from the BKZ 2.0 simulator and reflecting on the doubly exponential running time of BKZ in the blocksize $\beta$ we found: $\log_2 T_{sec} = \log_2 0.009/\delta^2_0 - 27$. However, since this might be controversial, we include estimates for both models.

# Faugère-Lachartre implementation for linear algebra for Gröbner bases

Fayssal’s code which implements the Faugère-Lachartre approach to linear algebra for Gröbner bases is available on Github now. Fayssal did a Master’s project on linear algebra for Gröbner bases in the team of Jean-Charles Faugère.

# A Generator for LWE and Ring-LWE Instances

We’re ready to announce our LWE/Ring-LWE generators for Sage:

We introduce software for the generation of instances of the LWE and Ring-LWE problems, allowing both the generation of generic instances and also particular instances closely-related to those arising from cryptomania proposals in the literature. Our goal is to allow researchers to attack different instances in order to assess the practical hardness of LWE and Ring-LWE. This will in turn give insight to the practical security of cryptographic systems based on both problems.

Both the Sage and the lmonade project were successful in applying to Google Summer of Code 2013. If you are a student head over to their respective GSOC pages and get in touch. If you want to do a project related the stuff I write about on this blog, i.e., with me as a mentor, get in touch as well.

# LPN and SVP

I am currently attending ESC 2013 in Mondorf, Luxembourg. Over dinner someone mentioned that there is no known reduction from LPN to lattice reduction, i.e., it is not known that you can solve LPN with LLL and friends.  This seems rather strange to me, because the standard lattice attack on LWE seems to be carrying over as is:

n = 100 # number of variables
m = 400 # number of samples
A = random_matrix(GF(2), m, n)
s = random_vector(GF(2), n) # our secret
p = 0.25 # our error rate

v = A*s + vector(GF(2),[1 if random() < p else 0 for _ in range(m)])

# we are searching for a short vector in the dual lattice
B = A.kernel().matrix()
L = B.change_ring(ZZ).LLL()

# because a short vector there, means few additions which means a higher bias in the sum
Av = A.augment(v)
sum(map(lambda x: abs(x) % 2,L[0])), (L[0]*Av)[-1]

Of course, this means running lattice reduction many times, but still: what am I missing?

PS: Obligatory, Sage cell here.

# Matrix Multiplication over GF(p^e)

After my talk at Sage Days 35 in Warwick (that was in winter 2011) David Harvey had an idea on how to speed up matrix multiplication over $\mathbb{F}_{p^n}$. We spend some time on this in Warwick and developed this idea further (adding fun stuff like Mixed Integer Programming in the process) but did not get around to do much on this project in the mean time (I have explained the idea at the end of my talk in Mykonos, though).

Just now, in a conversation with Richard Parker I was reminded of this dormant project, i.e., the question of how many multiplications i $\mathbb{F}_p$ it takes to do a multiplication in $\mathbb{F}_{p^n}$. In particular, I recalled to have written some code for Sage which gives some upper bound to this answer which is better than Karatsuba.

Well, here’s an interactive demo … gosh, I love the Sage cell server.

# M4RI 20121224

I have just pushed the button to release M4RI 20121224. The main feature of this release is a considerable performance improvement. It all started with Fast matrix decomposition in F2 by Enrico Bertolazzi and Anna Rimoldi showing up on the arXiv. Here’s the abstract

In this work an efficient algorithm to perform a block decomposition (and so to compute the rank) of large dense rectangular matrices with entries in F2 is presented. Depending on the way the matrix is stored, the operations acting on rows or block of consecutive columns (stored as one integer) should be preferred. In this paper, an algorithm that completely avoids the column permutations is given. In particular, a block decomposition is presented and its running times are compared with the ones adopted into SAGE.

… and that comparison made M4RI (which realises this functionality in Sage) look pretty bad. I did’t (and still don’t) share the implicit assumption that avoiding column swaps was the key ingredient in making this code so much faster than ours. I assume the impressive timings are due to a very efficient base case implementation. Anyway, we sat down  and looked for performance bottlenecks the result of which is 20121224. I actually have no idea whether we caught up to the code described in Enrico’s and Anna’s pre-print as they did not publish their sources.

Still, the performance improvements over 20120613 were worth the trouble. Below two plots of the (normalised) leading constants giving the leading constants for multiplication and elimination respectively (more plots on imgur) That is, it plots the running time divided by $n^{2.807} \cdot 10^9$. In theory these plots should all have slope 0.

Multiplication on Intel Core i7

PLE on Intel Core i7

Finally, here’s the plot for Fast matrix decomposition in F2 which starts very small but has a rather large slope. That’s why I concluded that the performance stems from a very efficient base case. I should get in touch with Enrio and Anna about this.

# Linear Algebra for Gröbner Bases over GF(2): M4RI

Two days ago I wrote about LELA’s implementation of Gaussian elimination for Gröbner basis computations over $\mathbb{F}_2$. Yesterday, I implemented LELA’s algorithm (which is from Faugere & Lachartre paper) in M4RI. Continue reading


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# Can 2 charged black holes repel each other?

Say there are 2 stellar black holes (equal mass and like charged), can they repel each other? I believe when people say things like charged black hole it is not the same as magnetar where it generates extremely powerful magnetic field right?

A real black hole is unlikely to have any significant electric charge, because a distribution of matter that has enough charge to overcome the gravitational attraction would — for that very reason — not collapse to form a black hole.

(Magnetic fields are a different story. Spinning black holes with accretion disks are thought to generate very strong magnetic fields, which are presumably responsible for the astrophysical jets that have been observed in several systems [1].)

For a more interesting answer, suppose we ignore the obstacles to forming such a highly charged object. The Reissner-Nordström solution describing a non-rotating charged black hole with mass $$M$$ and charge $$Q$$ is $$ds^2=-A(r)dt^2+\frac{dr^2}{A(r)}+r^2d\Omega^2 \tag{1}$$ with $$A(r)\equiv 1-\frac{2GM}{r}+\frac{GQ^2}{r^2} \tag{2}$$ in units with $$c=1$$ and in which Coulomb's law would have the simple form $$F=Q^2/r^2$$. Equations (1)-(2) are from equation (11.15) in [2]. There are three cases:

• The usual case is $$Q, which describes a black hole in the usual sense. The special case $$Q=0$$ is the Schwarzschild black hole. Two objects of this type attract each other, because the repulsive electric interaction is not strong enough to overcome the attractive gravitational interaction.

• The opposite case is $$Q>M$$, which doesn't describe a black hole at all, because there is no horizon in this case. This solution has a naked singularity instead. Two objects of this type would repel each other, but they're not black holes, so this doesn't satisfy what the question is asking for.

• The extremal case is $$Q=M$$. This case is remarkable, because two objects of this type (with same-sign charges) don't attract each other or repel each other. The class of exact solutions called the Majumdar-Papapetrou solutions describes any finite number of these objects in static equilibrium with each other, as explained in section 1.2 in [3]. These are black holes (they do have event horizons), but they don't quite repel each other, so this also doesn't satisfy what the question is asking for.

Based on these examples, the answer to the question seems to be "no": if the objects are black holes, then they don't repel each other; and if they do repel each other, then they're not black holes. But these are admittedly only examples, not a general proof.

By the way, although the question asked about electrical repulsion, the paper [4] claims that two black holes can repel each other for a different reason: angular momentum. Here's an excerpt:

the remarkable feature of the newly found binary black-hole configurations is that the black holes in them can repel each other due to prevailing of the spin-spin repulsion over gravitational attraction, which prevents the two black holes from merging into a single black hole.

I don't know if this claim has been independently checked yet, but it's an intriguing claim.

References:

[2] Marolf, 2012. "Chapter 11: Black holes and branes in supergravity," pages 273-324 in Black Holes in Higher Dimensions, edited by Horowitz (Cambridge University Press)

[3] Horowitz, 2012. “Chapter 1: Black holes in four dimensions," pages 3-25 in the same book

[4] Manko, Ruiz, "'Black hole-naked singularity' dualism and the repulsion of two Kerr black holes due to spin-spin interaction," https://arxiv.org/abs/1803.03301

• I would like to clarify, what happens to the Q=M black hole if we keep firing charged into it with an electron canon? It seems like the singularity will not disintegrate, so prevents us from forming a Q > M black hole that way? Oct 18, 2019 at 13:56
• @VladimirAkopyan Two things to consider: first, an electron has $Q>M$ already. Second, an electron cannon would need to propel the electrons with enough energy to overcome the immense electrostatic repulsion, and we need to account for this energy when calculating the final total mass. Oct 18, 2019 at 15:13