User alexander chervov - MathOverflow

Minimizing |FT(X)|_{\infty} by permutation of X_i - question on Fourier transform related to engineering problem (peak factor of OFDM system)

2012-10-15T10:38:32Z

Consider vector X =( X_1 ... X_N), consider the discrete Fourier transform $Y=F(X)$.

I am interested to minimize $|Y|_{\infty}$, by permutation of numbers X_i, how to do it ?

Here $|Y|_{\infty}$ is infinity norm of the vector Y i.e. just the maximum of absolute values of components of Y.

More close to life problem is a little more complicated: my numbers X_i are splited at several subsequences such that |X|=const in each subsequence. And I am allowed to make "block" permutations of these subsequences. The goal is the same as to minimize $|Y|_{\infty}$

Background: roughly speaking the OFDM based ( = most advanced) radio telecommunication systems (LTE, WiMax, new WiFi) make the Fourier transform before transmitting data symbols to the space. Average power is fixed, but people care also about the maximal instant power, which they do not want to be big. Instant power is just the maximal component of the vector.

Applications of algebraic geometry/commutative algebra to biology/pharmacology ?

2012-04-25T07:30:46Z

Are there applications of algebraic geometry/commutative algebra to biology/pharmacology ?

It might be that some Groebner basis technique is used somewhere ? I know there are some applications to robotics - in solving some complicated non-linear equations, may be something similar can happen in biology...

Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3...

2012-11-17T20:19:12Z

Classification of simple finite-dim Lie algebras for char >=5 has been accomplished not so long time ago, and char p=2,3 is open problem.

I wonder what is known/expected for char p=2,3 ?

More vague and soft question is the following - look at some famous classification problems: simple finite-dim Lie algebras, simple finite groups, some other things classified by ADE... We see the following pattern: there are some series of objects and finite number of "sporadic" objects. I.e. it never happens that there is infinite number of examples which are not in "series". So classification of simple objects is simple (in some very informal sense).

The question: can we expect this in advance, without obtaining classification ? (What are other examples/counter examples of similar phenomenon ?).

For example can we expect/prove this for simple Lie algs for char =2,3 ? I.e. there will be some finite number of series and finite number of "sporadic" examples ?

Convergence speed of Jacobi eigenvalue algorithm for parallel ordering(Brent-Luk) ?

2011-10-07T08:05:27Z

Is there estimate for convergence of the Jacobi eigenvalue algorithm for Hermitian matrices for "parallel ordring" (Brent-Luk ordering (see comment below)) ?

For example for 4 4 matrices parallel ordering is the following 1a) 12 1b) 34 2a) 23 2b) 14 3a) 13 3b) 24

[EDIT] Moreover convergence itself is not known for such ordering even in 4x4 case. I have consulted with many experts in the field - it is not proved. Numerical simulations (checked more than 10^10 matrices of different form) shows that the convergence exists.

There is certain subtlety in the definition of method. Which lead some authors to claim that there is NO convergence. But actually counter-example is not for "reasonable" implementation of details.

The detail is the following consider 2x2 matrix such that diagonal elements are equal. Then the rotation can be either +45 either -45 - no unique choice. What the authors claim that if we have a freedom to choose +45 or -45 by our own wish, in each step where ambiguity occurs - then there will be counterexample ! However this counter-example does NOT work if we fix +45 (or -45) once and forever ! I.e. in the case of ambiguity we ALWAYS choose angle to be the same. Simulations shows - that than there is no problem.

I spent about 2 weeks trying to prove this just in the 4x4 example - but I was unable to prove it. The difficulty is that we need to analyse about 3-4 sweeps. It can be shown that there always exists a matrix that can be arbitrary "BAD" after 1-2 sweeps...

[END of EDIT on 21 Jan. 2012]

As far as I can expect that there should be ultimate quadratic convergence [EDIT] actually as works of Walter F. Mascarenhas suggests their will be cubic ultimate convergence[EDIT] but I am interested at the first iteration - they should be at most linear convergence, but it is not clear for is there uniform convergence speed or there can be some matrices where convergence can be arbitrary bad ? (From simulation we see that probably there is NO bad examples - convergence seems rather fast, but there are certain difficulties in proving this theoretically).

Actually even the convergence for arbirary ordering is not clear for me.

Paper by Walter Mascarenhas:

SIAM. J. Matrix Anal. & Appl. 16, pp. 1197-1209 (13 pages) On the Convergence of the Jacobi Method for Arbitrary Orderings Walter F. Mascarenhas States only convergence of the diagonal elements. Non-diagonal elements may not converge, for some sophisticated orderings. He constructed examples in his PhD at MIT unpublished (private communication from him)

Mathematical properties of financial prices

2013-04-28T12:21:10Z

Prices of financial assets (stock-market prices or currency exchange rates) obviously resemble trajectories of stochastic processes.

What is known about their mathematical properties ?

I know there is huge (too huge) literature (see e.g. MO-Financial Mathematics Books) around it, I am familiar with some ideas, like below, but would be grateful for any comments/suggestions.

1) The individual distributions are better modelled by heavy-tailed distributions, rather than by normal distribution, reflecting that sometimes prices change heavily in short time periods. (See e.g. MO: Is there any straightforward way to substitute for Gaussian/Brownian assumptions in financial mathematics?).

2) To some extent they are similar to Brownian (log-Brownian) motion, more precisely the increments are independent at least at some time scales (which means that you cannot win money), (however there are other claims that short time increments are correlated).

3) There are claims that fractal (Hausdorff ) dimension is near to 1.5 (the same as for Brownian motion).

Answer by Alexander Chervov for classification for coadjoint orbits of lower or upper triangular matrices

2013-04-10T09:14:56Z

Let me first say, that I am not an expert, but was interested in the same question recently, so I would also be happy if someone provides more info on the question. Let me collect some facts, which I know.

1) As far as I understand the general classification of orbits is in certain sense "wild" problem.

2) Classification up to n=7 can found in Coadjoint orbits of the group $UT(7,K)$ (2006) , further papers by A.Panov and his students give partial results on general $n$, e.g. these ones Involutions in $S_n$ and associated coadjoint orbits Diagram method in research on coadjoint orbits (2009)

3) There is a lots of recents studies which are "related" to the question. Especially in the case of ground field is finite. In such a case people are greatly interested in understanding representation theory of U(n,F_q) and in particular of the "orbit method" approach to it, and hence in coadjoint orbits. See some comments at mathoverflow question: Finite Unipotent Groups: References.

4) If you restrict to ground field to be finite, then it is worth to mention several facts: a) number of adjoint and coadjoint orbits is the same b) it is the same with the number of conjugacy classes in the group c) hence the same as number of irreps d) it is related to interesting combinatorics, see e.g. paper by A.A. Kirillov, A. Melnikov On a Remarkable Sequence of Polynomials and other papers by these and other authors.

For the finite field, these MO questions, related: Finite Unipotent Groups: References, Representation theory of p-groups in particular upper tringular matrices over F_p Irreducible representations of the unitriangular group.

Answer by Alexander Chervov for Finite Unipotent Groups: References

2013-04-09T14:03:06Z

I guess U(n,F_q) - upper triangular matrices over F_q. I am not an expert in the field, but was recently interested in similar question, so I'll put these remarks.

There is certain amount of quite recent works related to representations of U(n,F_q) and some "big names" involved. There are certain conjectures which are easy to state, but hard to prove, and some more conceptual problems and recent breakthroughs.

One of the origins of the modern interest are papers by A.A. Kirillov (1995-2005), e.g. Variation on a triangular theme, Two more variations on a triangular theme, where he considered a question whether the "orbit method" can be extended to finite Lie groups such as U(n,F_q). (Originally (60-ies) A.A. Kirillov proposed "orbit method" for Lie groups over R, and U(n,R) where the first groups where he demonstrated its work).

Kirillov's papers are always pleasure to read, he puts the accent not on technical details, but on new ideas, problems and insighting observations. The moral that in certain cases "orbit method" can be applied, but there some problems, which are deserved to be further studied to achieve further progress. See e.g. Vipul Naik's site, where one can find lots of interesting and understandable information, worked out examples and further references: orbit method for U(3,p), orbit method for finite Lazard groups.

One of the major breakthroughs in the subject is an idea of "supercharacters". Let me quote from E. Arias-Castro P. Diaconis R. Stanley

The character theory of U(n,F_q) is a well known nightmare. In recent work, Carlos Andre, Roger Carter and Ning Yan have developed a theory based on certain unions of conjugacy classes (here called super-classes) and sums of irreducible characters (here called super-characters).

The main point is that "classification of characters" is to certain extent "wild" problem, while "supercharacters" are quite manageable to classify.

See more comments in MO question- "Irreducible representations of the unitriangular group" .

Let me also quote from F. Ladisch answer on MO question - "Representation theory of p-groups in particular upper triangular matrices over F_p"

There are, by now, many papers about the character theory of the upper triangular group and related topics, which is in part motivated by Higman's conjecture that for every n, the number of conjugacy classes of Un(Fq) is a polynomial in q with integer coefficients.

Let me also mention "23 page article with 28 authors :)" which is devoted to the subject: Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

Answer by Alexander Chervov for A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

2013-03-28T18:38:17Z

http://arxiv.org/abs/0712.2448

Gluing of Surfaces with Polygonal Boundaries E. T. Akhmedov, Sh. Shakirov

By pairwise gluing of edges of a polygon, o produces two-dimensional surfaces with handles and boundaries. In this paper, we count the number ${\cal N}_{g,L}(n_1, n_2, n_L)$ of different ways to produce a surfac of given genus $g$ with $L$ polygonal boundaries with given numbers of edges $n_1, n_2, >..., n_L$. Using combinatorial relations between graphs on real two-dimensional surfaces, we derive recursive relations between $\cal N_{g,L}$. We show that Harer-Zagier numbers appear as a particular case of ${\cal N}_{g,L}$ and deriv a new explicit expression for them. Comments: 7 pages, 9 figures

It seems proposes quite elementary proof. The key idea that they found some generalization which is more easy to prove.

Product of conjugacy classes - is there an analog of Tanaka-Krein reconstruction ?

2012-10-27T20:31:16Z

Consider a finite group G. The product of conjugacy classes can be defined in natural way just by multiplying the representatives and counting multiplicities (see e.g. MO 62088). So we get ring with a basis and structure constants are natural numbers. Similar to what one has for product of irreps. There are many analogies between conjugacy classes and irreps in particular see this article.

Tanaka-Krein duality states that group can be reconstructed from the tensor category of its representations which is semisimple for finite groups, and hence carries the same information as ring + basis of irreps.

Question: Can one reconstruct a group having (ring + basis) made of conjugacy classes ?

If not - what partial information (e.g. character table) one can get ?

Question: Is there any relation between this ring and ring of irreps of the same group ? or may be some other group ?

(Remark. For abelian group they are isomorphic.)

Question: Are there any further analogies between ring of irreps and conjugacy classes except mentioned in the paper cited above ?

Role of applications in modern mathematics

2013-03-06T18:12:13Z

Older days scientists were universalists and philosophy, physics and mathematics were a part the same question - understanding the world. Nowadays one may get feeling that the role of applications in development of modern mathematics is negligible - of course it depends on the field. And aim of the question is to get different opinions from different points of view.

Question 1 What is the role of applications in modern mathematics ?

Question 2 Different countries have different mechanisms to stimulate interaction between mathematics and applications - what are these mechanisms and what are their advantages and disadvantages ?

Question 3 (for pure mathematicians) What is your personal stance on applications ? Is it out of your scope of interests or you are have (trying to have) some contact with applications ?

Answer by Alexander Chervov for Not especially famous, long-open problems which higher mathematics beginners can understand

2012-07-02T21:04:50Z

The Hot spot conjecture The conjecture seems quite amazing and simple to formulate, it can be even understood by persons "from the street" seems its prediction can be tested experimentally. It is a subject of "polymath project 7". Let me quote:

The hotspots conjecture can be expressed in simple English as:

Suppose a flat piece of metal, represented by a two-dimensional bounded connected domain, is given an initial heat distribution which then flows throughout the metal. Assuming the metal is insulated (i.e. no heat escapes from the piece of metal), then given enough time, the hottest point on the metal will lie on its boundary.

In mathematical terms, we consider a two-dimensional bounded connected domain D and let u(x,t) (the heat at point x at time t) satisfy the heat equation with Neumann boundary conditions. We then conjecture that

For sufficiently large t > 0, u(x,t) achieves its maximum on the boundary of D

This conjecture has been proven for some domains and proven to be false for others. In particular it has been proven to be true for obtuse and right triangles, but the case of an acute triangle remains open. The proposal is that we prove the Hot Spots conjecture for acute triangles! Note: strictly speaking, the conjecture is only believed to hold for generic solutions to the heat equation. As such, the conjecture is then equivalent to the assertion that the generic eigenvectors of the second eigenvalue of the Laplacian attain their maximum on the boundary. A stronger version of the conjecture asserts that

For all non-equilateral acute triangles, the second Neumann eigenvalue is simple; and The second Neumann eigenfunction attains its extrema only at the boundary of the triangle.

(In fact, it appears numerically that for acute triangles, the second eigenfunction only attains its maximum on the vertices of the longest side.)

==========================================================

May be this problem can be mentioned when teaching determinants and in particular:

$\det(AB)= \det(A)\det(B).$

There are so-called Capelli identities which generalize this formula for specific matrices with non-commutative entries. In the paper Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities by Sergio Caracciolo, Andrea Sportiello, Alan D. Sokal they formulate certain conjectures of the type $$\det(A)\det(B)=\det(AB+\text{correction})$$ on the page 36 (bottom), conjectures 5.1, 5.2.

I think these are quite non-trivial, but probably some smart young mathematician may solve them, given some amount of time (some months may be). I spent some amount of time thinking on them without success, and moreover let me mention that D. Zeileberger and D. Foata also failed to find a combinatorial proof of the Capelli identity of very similar type -- the one proved by Kostant-Sahi and Howe-Umeda -- see their comments in Combinatorial Proofs of Capelli's and Turnbull's Identities from Classical Invariant Theory page 9 bottom: "Although we are unable to prove the above identity combinatorially ... ". So words above are some idications of non-triviality of the conjectures.

Personally I am quite interested in a proof, probably it can give clue for further generalizations.

Algorithm to solve Sokoban-like game on graphs - move chips from one set of vertices to another

2013-02-15T07:03:37Z

The question is close to the Sokoban game (thanks to Dima Pasechnik !), but a little different in details.

Consider a directed graph (multi-graph). Consider some set of marked chips (chip1, chipe2,..., chipM). Put chips on some set of vertices 'Init1','Init2','Init3'... And consider some other set of vertices 'Final1','Final2',..., 'FinalM'.

Question Propose an "efficient" algorithm which will determine is it possible to "MOVE" chips from positions "InitNN" to positions 'FinalNN'.

Where we are allowed to "MOVE" chip from a vertex to an outgoing edge and from incoming edge to corresponding vertex. With the CONSTRAINT that two chips are NOT allowed to be at the same place. One move - moves only ONE chip. ChipK should go to position FinalK - same "K".

Question There can be many approaches to solve the problem, I am interested in analysis their complexity. Any ideas are welcome. For example if graph is "random" in certain sense what can be the algorithm the least average complexity ?

Where complexity is counted in number of operations (write a C-code (I actually wrote a Matlab code), compile to and calculate the number of cycles - this is well-defined complexity measure, different compilers and CPU will give approximately same result).

Example of algorithm It seems the simplest way to solve a problem is the following. Essentially it can be reduced to determining where two vertices are connected in some bigger graph, which in turn can be solved by "breadth-first search" ("wave algorithm" in Russian) (I mean let us enumerate all possible chip configurations - it will give vertices of the "new graph". Let us connect two vertices (configurations) if there is a "MOVE" which goes form one to another.) By "breadth-first search" ("wave algorithm" in Russian) I mean the following - take an initial vertex and find all connected to it; next step find all vertices connected to vertices found on the previous step; and so on....

Question What about efficiency of this algorithm ? Can one propose better ?

Fiction books about mathematicians?

2012-07-08T11:16:59Z

What are some fiction books about mathematicians?

It seems to me rather difficult for writers to create good books on this subject. Some years ago I thought there were no such books at all. There are many reasons: it is difficult to describe the process of discovery and describe it in the exciting way. The subject has narrow audience and not the way to make best-seller...

Comments on how authors try to avoid these problems are also welcome. The movie "A Beautiful Mind" is a (beautiful for me) example, where the story of mathematician was mixed with love and spy stories to make it interesting for general audience, well not so much preserved from mathematician's story, but nevertheless I am quite positive about it.

Here is a related MO question:

Movies about mathematics mathematicians

Answer by Alexander Chervov for Lie algebra embeddings and the center of their enveloping algrabras

2013-02-18T18:45:11Z

Take a look at "Shifted Schur Functions"

Andrei Okounkov, Grigori Olshanski http://arxiv.org/abs/q-alg/9605042

Section 10: "Coherence property of quantum immanants and shifted Schur polynomials"

In particular formulas 10.4, 10.5 - they discuss "averaging operators" Z(U(gl(n)) -> ZU(gl(N)) , n < N

and later prove certain "good" (coherence) property of special generators of the centers Z(U(gl(k)) which has been studied by the authors and M. Nazarov.

Hope this helps...

What is very interesting for me personally - is try to generalize such things to the case of loop algebras Z(U(\hat gl)). Here certain "good" elements of the centers has been constructed by Talalaev's formula, it is natural to expect that Okounkov-Olshanski-... story can be generalized to loop algebra case

Math behind databases management and SQL ?

2013-02-08T08:32:34Z

Are there some mathematical theories/theorems/... behind modern development of database management systems and in particular of SQL ?

I am refreshing my knowledge of these things which are quite down-to-earth "how to use" (create table ..., insert..., select * from ...), but I think some deeper understanding what is behind would be helpful.

In particular in Wikipedia one may find some relations with 3-valued logic:

Along with True and False, the Unknown resulting from direct comparisons with Null thus brings a fragment of three-valued logic to SQL. The truth tables SQL uses for AND, OR, and NOT correspond to a common fragment of the Kleene and Lukasiewicz three-valued logic (which differ in their definition of implication, however SQL defines no such operation).

But it is not very clear for me what it means and how deep it is ?

Construction of one graph from another - known ? ("Sokoban graph", chips configurations)

2013-02-17T15:40:20Z

Consider a [directed] graph and natural number "m". Let us construct new [directed] graph[s] from it as follows. There is one idea behind the construction, but one can play with details and get several constructions.

Vertices of new graph - all possible configurations of "m" ([non]-marked) chips on vertices (edges) of original graph [with/without] constraint that two chips should be in different positions.

Edge of new graph - two vertices are connected if one configuration of chips can be obtained from another in one "MOVE".

Where by "MOVE" I mean that we can move ONE chip from vertex of original graph to neighboring (edge)-vertex of original graph.

Question Is this construction(s) well-known ? what is name/refrence ? How properties of the original graph are related to the one of the new graph ?

Motivation: Such graph appears if one thinks on the question:

http://mathoverflow.net/questions/121871/algorithm-to-solve-sokoban-like-game-on-graphs-move-chips-from-one-set-of-verti

The solution to question above is related considering the construction above and searching if two vertices of the new graph are connected.

Roland Bacher in his answer describes similar idea which is directly related to sokoban game.

Path search algorithms on graphs

2013-02-17T12:46:40Z

Consider a directed graph and two vertices on it. I need to determine is there a path between them. There is a "breadth-first search" ("wave algorithm" in Russian) algorithm (see description below).

Question What are the alternatives and is it known what kind of algorithm has less complexity on some specific types of graphs, e.g. random graphs, "sokoban-graphs" (see below) ?

Roughly speaking breadth-first algorithm - look at ALL paths outgoing from "A" of length 1, next step of length 2, next step length 3, ...

It has clear intuitive disadvantage if graph is "very connected" - we are looking for too many "short paths" - it would be better to take one "long path" which goes from vertex "A" "somewhere near" to destination "B", and then find path from "somewhere nearby B" to "B" by "breadth-first" search. Of course, here we should somehow be able to explain what means "somewhere near" and propose a strategy to find a path from "A" to it. For some classes of graphs - "trellis graphs" this is clear what means, I do not know in general.

Motivation comes from the question

http://mathoverflow.net/questions/121871/algorithm-to-solve-sokoban-like-game-on-graphs-move-chips-from-one-set-of-verti

The problem about chip movement or sokaban like problem can be reduced to the question of existence of the path between the two vertexes. However the graph appearing here is quite specific - vertices of the "big-new-graph" are configurations of chips on the original graph and they are connected if there is a "MOVE" from one configuration to another.

So taking these specific properties of that graph what algorithm should one use to settle the path existence problem.

Answer by Alexander Chervov for Trichotomies in mathematics

2013-02-11T17:07:44Z

Let me point out that Vladimir Arnold was quite interested in similar question. He called subj. "mathematical trinities", see e.g. his paper "Symplectization, Complexification and Mathematical Trinities". As far as I remember from his lectures, his ideas were that many of these "trinities" are actually related to each other; and he also considered subj. as a tool to invent to theories: see question marks at already cited "Arnold's table": jpg.

Let me also mention some "trinities" which occur in my own research related to Capelli identities (which are some non-commutative analogs of det(AB)=det(A)det(B) ).

Matrix trinity - a) generic b) symmetric c) antisymmetric

Here how it goes in Capelli (and related Cayley) identities:

a) generic matrices - original Capelli identity has been discovered by Capelli in 19-th century - it is for "generic matrices" $A=x_{ij}$ $B = \partial_{ji}$

b) symmetric matrices - analog of the Capelli identity has been discovered by Turnbull around 1940-ies - here $A=(x_{ij}+x_{ji})$ $B= \partial_{ji} + \partial_{ij} $.

c) antisymmetric matrices - analog has been found by Howe-Umeda and Kostant-Sahi around 1990, here $A=(x_{ij}-x_{ji})$ $B= \partial_{ji} - \partial_{ij} $.

Similar generalization were found for Cayley identity respectively: a) attributed to Cayley b) Garding 1948 c) Shimura 1984 - see arXiv:1105.6270 for quite a complete information.

My question: is it really trinity ? Or you can propose some analogs of Cayley-Capelli for some other matrices, say "symplectic" ?

It is might be strange, but other trinities like R,C,H also appears in the Capelli story - and they give different identities. Moreover trinities can be combined and we might get trinity^trinity^trinity...

Actually H-analog of the Capelli identity is not fully known for the momemnt - only analog for 1x1 matrices has been discovered quite recently by student of R. Borcherds, An Huang. Looking at this example I proposed some C-analogs of Capelli identities. Actually all generic/symmetric/antisymmetric can be complexified, hopefully there should exist quaternionic analogs and thus we might have trinity^trinity. Some partial results of trinity^trinity spirit for Cayley identity contained in loc. cit.

There are certain analogs of Capelli identities for classical Lie algebras: this can seen as gl/so/su trinity, well probably it is not the trinity in some strict sense. I have no idea can we have something like trinity^trinity^trinity ...

What is matrix A such that Hamming weight of [x, Ax] is maximal ? (Min distance of 1/2 block code?)

2012-07-06T09:32:07Z

Everything over F_2.

For any matrix $A$ define the number $N(A) = min_{x}$ HammingWeight $( [x , Ax])$. Where $x$ is vector and [a,b] is just concatenation of vectors: (a_1,...a_n, b_1,...,b_m).

Question What is $max_{A \in Mat(n,m) } (N(A))$ ?

Particular case n=m.

Motivation.

The map $x \to [x, Ax] $ can be considered as error-correcting coding, $x$ - information bits, $Ax$ are redundancy bits.

The code is good if distance between codewords is small.

Reformulation of question: what is the "best possible" code of type above ? ("best possible" in the sense of minimal distance -- it is not always "best" from practical point of view nevertheless).

papers archives? (especially not indexed by google)

2012-01-30T18:18:57Z

http://www.digizeitschriften.de/index.php?id=239&L=2 has many papers with free access (e.g. Inventiones Mathematicae) but when you search with scholar.google.com it does not index this site!

Are there any other archives like this?

Just in case let me list other archives (they are indexed by google as far as I understand).

http://projecteuclid.org

http://www.numdam.org/?lang=fr

http://www.math.uiuc.edu/K-theory/

e.g. I cannot find:

Koszul, J (1981), "Les algebres de Lie graduées de type sl (n, 1) et l'opérateur de A. Capelli", C.R. Acad. Sci. Paris (292): 139-141

Does it mean search skills are poor or it is really not available electronically?

Answer by Alexander Chervov for Classical limit and Drinfelds realization of quantum groups

2013-02-07T06:09:53Z

It is more like comment, but seems too long.

I would say yes. I cannot give precise reference, but by all the idealogy it is yes. Or you see some "underwater stones" - problems ?

Ideas are like this: If you start with RLL=LLR description, then you need to make "Gauss" or "tringular" decomposition of L = LowTriangular*D*UpperTrianular to extract Drinfeld's currents.

In the classical limit "everything" takes the form A = A_{cl} + O(h). Now if you take L = L_{cl} , Triangular = Triangular_{cla} + O(h)

The simple fact that should hint that the "yes" answer is the following. So the decomponsition L = LowTriangular*D*UpperTrianular in classical limit corresponds to L_{cl} = LowTriangular_{cl} + D_{cl} + UpperTrianular_{cl}

You see multiplication in the first order corresponds to addition. And this means that corresponding classical currents are just currents to appropriate upper-lower triangular parts - which corresponds to x(z).

Is the quantum algebra unique (up to isomorphism) in deformation quantization ?

2013-01-27T05:54:09Z

Consider a Poisson algebra A (i.e. commutative algebra with Poisson bracket).

Let $\hat A$ be a deformation quantization of the algebra A. We know that construction of deformation quantization and more generally "formality isomorphism of Kontsevich" depends on choices of certain data (in Kontsevich approach we can change "propagator" and coordinates on manifold, in Tamarakin's approach we can choose arbitrary associator.)

Question Is it known/true/expected that for different choices of "that datum" we nevertheless obtain isomorphic quantum algebras $\hat A$?

May be one needs certain restrictions on setup (e.g. only smooth algebras A, only "generic" quantization morphism, whatever...) to guarantee uniqueness ?

Kontsevich also mentions that Grothendieck-Teichmuller group should act on the set of all deformation quantizations. Is it at least true that two quantizations living in the same orbit of that group give isomorphic quantum algebras ?

Question in formally precise form Consider a Poisson algebra. Choose two different formality isomorphisms (e.g. with different propagators or associators).

Define two star-products $\star$ and $\star'$ with the help of these two formality isomorphisms.

Question Are the algebras defined by these two star-products isomorphic ? More strongly - are these star products "equivalent" ? (See definition of equivalence in Stefan Waldmann's answer below or in Kontsevich paper).

Some comments.

If our manifold is R^2n we canonical Poisson bracket { p_i q_j } = delta_ij, then undoubtly the quantum algebra should be unique and isomorphic to Heisenberg algebra: $[ \hat p_i , \hat q_j ] = delta_{ij}$. But I am not sure that even in this case it is that much obvious - if we change coordinates we can make Poisson bracket arbitrary weird, it would be non-obvious that we get isomorphism with Heisenberg algebra. And moreover it might depend on category we are working with (polynomial or smooth functions).

More general example is Lie algebra $g$ - corresponding quantization should be isomorphic to universal enveloping algebra, but again it is not that much obvious. (In Kontsevich paper he devoted some special (small) arguments to prove that his quantum algebra is isomorphic to U(g)).

Concerning choices of different coordinates in classical algebra A - it already states in Kontsevich paper that obtained algebras will be isomorphic. More strongly star-products will be "equivalent".
See the last formula on the page 3 of his paper. However nothing is said about choices of different propagators

I had discussed this question with some experts some years ago, but there was no clear answer.

The motivation to ask partly comes from MO-discussions here: http://mathoverflow.net/questions/119849/quantization-of-a-classical-system-e-g-the-case-of-a-billard

Answer by Alexander Chervov for Quantization of a classical system (e.g. the case of a billard)

2013-01-26T10:30:34Z

Let me add some comments. I think the question has many faces: 1) general principles of correspondence classical to quantum world 2) quite a concrete question about boundary conditions for quantization of billiards.

About (1) I have written something in http://mathoverflow.net/questions/106721/quantum-mechanics-basics/106723#106723 I can add more, but not sure it is appropriate...

About (2), let me add some comments, it is not full answer, but may be still of some use.

So Joel asks " But I am not sure why the wave function should be defined on R^2 instead of just on B, and even while it should be continuous."

Yes, I think from physical point of view it should be defined on R^2 and should be continuous, let me explain some arguments which come to my mind.

How can you confine a particle to restricted billiard region "B" in practice ? What physical experiment you keep in mind ?

The answer is the following - let us create a potential barrier with very high energy U(x) = U_0 - outside "B" and U(x) = 0 inside "B". Well, actually I think such discontinuous potential barrier is not practical, but we can smooth as much as we want.

Classical particle with energy < U_0 cannot go outside the barrier, but quantum particle can make tunneling inside barrier with exponentially decaying wave function.

Now we just want to consider the limit U_0 -> infinity. That would correspond to confining quantum particle to the region "B", again in practice there are NO infinities, so always small probability for particle to be outside region B, but as mathematical abstraction it is Okay to take U_0 = inf.

So now we come to mathematically well-formulated questions :

Consider smooth potentials U_n(x) which approximate U(x), where U(x) =inf in R^2\B and U(x) = 0, inside B. Consider the wave functions Psi(x) which is solution of the corresponding problem (Laplace + U_n(x) ) \Psi_n(x) = \Lambda Psi_n(x)

0) Is it true the limit \Psi (x) does not depend on approximating sequence U_n(x) ?

1) Is it true that limit Psi_n (x) is continuouos ?

2) Is it true that Psi_n(x) = 0 outside B (including the boundary) ?

I hope the answer is YES, on both questions, but I am not sure I know the arguments.

It is better to start with these question on R^1 not R^2 - this is done in any quantum mechanics textbook, I am sorry I a little forget the details.

Answer by Alexander Chervov for Financial Mathematics Books

2013-01-24T19:02:12Z

Some expert (physicist, working partly in finance) recommended me the book:

Jean-Philippe Bouchaud, Marc Potters (2003). Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management Amazon

It is econophysics approach to analysis of financial markets. It uses quite advanced mathematics including random matrices, stable distributions and so on. One can also look for the papers by these authors in arxiv.

Another expert recommended me the following site: http://www.opentradingsystem.com/quantNotes/main.html about quantative finances,

and the book " High-Frequency Trading. A Practical Guide to Algorithmic Strategies and Trading Systems" IRENE ALDRIDGE

Why all irreducible representations of compact groups are finite-dimensional ? [EDIT: Subtleties: AC,etc]

2013-01-20T15:22:50Z

About 20 years ago I read in textbook that "all irreducible representations of compact groups are finite-dimensional", but me and the proof of this fact never met each other :)

May I ask dear MO colleagues, is there (simple?) argument to prove it ?

As far as I heard this result can be generalized in the realm of non-commutative geometry, Woronowicz compact quantum groups (?).

So the "bonus" question - what is appropriate "compactness" condition for some algebra (and/or Hopf algebra) such that it will guarantee the same property (i.e. all irreps are finite-dim.) ?

[EDIT] Thanks very much for excellent answers ! Let me ask about some more details, to finally clarify.

1) What is maximal possible relaxation of the requirement on vector space V ? Is it enough to require arbitrary linear topological space or we need to restrict to Hausdorff (?) Banach (?) Hilbert (?), whatever spaces ? (It seems restrictions on the space may come from the Schur lemma, it is not clear for me what is appropriate generality it holds).

2) Do we need axiom of choice here ? (Probably not, we need existence of Haar measure, but Wikipedia writes that "Henri Cartan furnished a proof of existence of Haar measure which avoided AC use.[4]"

3) Informally: what is the hardest tool one uses in the proof ? (May be existence of Haar measure ?)

[END EDIT].

[EDIT]

Let me add sketch of arguments by Aakumadula, as I understand it. It might be helpful to clarify new questions.

1) Tool: Continuous functions on the group can be mapped to operators on V. (Need measure here). (Group algebra acts on V).

2) Fact: Continuous function will be mapped to COMPACT operators. (In R^n I know how to prove it, in general no).

3) Observe: Conjugation invariant function are mapped to operators which commute with action of group.

4) Schur Lemma: operators commuting with group in irrep are Lamda*Id. (What do we need from the space V for this to be true ? )

5) Corollary: If we find invariant continuous function which is mapped in NON-zero in V, then we are done, because by (2) it is compact operator and by (4) it is Lambda*Id.

So we need to find invariant function which will be non-zero in V.

6) Take arbitrary "approximate identity" i.e. sequence of continuous (non-invariant) functions f_n which converge as functionals to delta-function in identity of the group. (It is local fact. But how to prove it ? Do we need Axiom of choice here ? )

7) Make averaging over the group of f_n - get sequence of INVARIANT continuous functions which again converge to detla(e), since delta(e) is invariant.

8) Operators T(f_n) converge to identity operator, hence for some N they are NON-ZERO. WE ARE DONE by (5) ! Because T(f_N) is compact and Lambda*Id and Lambda is NON-ZERO.

[End EDIT].

"Soft" Voronoi cells or statistical criterias

2013-01-19T11:16:00Z

It is probably some basic statistics question, but...

Informally 1: How to choose "criteria", such that it will guarantee that error decision probability is less than "epsilon", and maximize probability of correct decision ? ("Criteria", "probabilities" will be explained below).

Informally 2 consider some points A_i in R^n. Task - for given point "x", find nearest point A_i. It is exactly the same determine to what Voronoi cell point "x" belongs to. Now imagine that point "x" is exactly on the same distance between several points A_i, then the answer to the question is not unique. Now let us add some randomness in this setup (see details below) so we should expect that if point "x" is "near" the edge of the Voronoi cell, our decision to say that the closest point is say A_1 will not be reliable. So I need to "decrease" Voronoi cells such that I can guarantee reliability is bigger than (1-epsilon).

My question will be - what is the shape of these "epsilon-soften" Voronoi cells ? (Of course, I will need to specify "probabilities" - see below).

Setup for formal question Consider some points S_i in R^n. Consider some R^n-valued random vector "N" (say Gaussian) (N - is "noise"). Define R=S+N, where S is random variable which uniformly takes values S_i (S-"sent signal", "R"-"received signal").

Question How to define subsets D_i in R^n, such that:

1) Probability that "R" belongs to D_i, under condition that $S\ne A_i$ is less than epsilon (small probability of incorrect signal detection)

2) For all other choices of subsets D_i satisfying (1), the probability of "R" belongs to D_i, under condition that $S=A_i$ is maximal possible over all choices of subsets D_i satisfying (1).

D_i - are our "soft" Voronoi cells.

The informal sense is the following - D_i are some neigbourhouds of points A_i, which are small enough to guarantee (1) (incorrect decision has small probability), but the biggest among all subsets satisfying (1) (we want to maximize the probability of correct decision).

Remark In R^1 this is standard simple statistics knowledge, but the question seems to be non-trivial even in R^2.

SubQuestion 1 What are the references ?

SubQuestion 2 Is there some simple knowledge that everybody must understand before start thinking on the question ?

SubQuestion 3 Is the question difficult ? Or there is some simple solution ? (Yes/No)

SubQuestion 4 Is it true that D_i cutted by hyperplanes (similar to Voronoi cells) ? (Yes/No)

SubQuestion 5 If problem is difficult in general, then is there some approximate solution (algorithm), which satisfy (1), and gives non-bad maximization in (2) ?

The Unreasonable Effectiveness of Physics in Mathematics. Why ? What/how to catch?

2012-12-16T15:42:04Z

Starting from 80-ies the ideas either coming from physics, or by physicists themselves (e.g. Witten) are shaping many directions in mathematics. It is tempting to paraphrase E. Wigner, saying about "The Unreasonable Effectiveness of Physics in Mathematics".

What can be the reasons for it ?

Do physicists have some tools/ideas/techniques which allow them to make insights, which are not seen for mathematicians? Or it is just because Witten&K are very ... very smart ?

If, yes, what are these tools/ideas ? How to learn/absorb/(put into math. framework) them?

What can be the further applications of these ideas ?

Being a mathematician, but working in physicists surrounding for many years, I have thought on this questions for quite a while. The recent MO question http://mathoverflow.net/questions/116251/mathematician-trying-to-learn-string-theory/116402 prompts me to ask it here.

I would think, that yes, there are such "ideas". But from some outstanding mathematicians I've heard an opposite opinion.

My vague feeling is that quantum field theory and string theory it is something like an analysis/differential geometry on infinite-dimensional manifolds. But these manifolds are not abstract, say Banach modeled manifolds, which theory is not so rich, but kind of maps from one finite-dim. manifold to another, which has certain specific structures which are not fully revealed by mathematicians. E.g. vertex operator algebras, arise from maps of circle to manifold, if we map not circle but something higher dimensional there should be something more complicated. Another issue is about Feynman integral, which allows physicist to use integration techniques in geometric problems, it is not well-defined mathematically, but it might be it cannot be defined in very general form of infinite-dimensional integrals, but again physicist have an intuition where it can be defined, where cannot, and proper mathematical theory should first clarify the setup where it exists, rather than trying to build general theory which might not exist. These words are probably very vague, so might be answers help to me clarify.

I think everybody knows the influence of physics happened from 80-ies, but for completeness let me mention just a few.

Donaldson used instanton moduli spaces in his study of 4-folds.

Faddeev, Drinfeld et. al. created quantum groups

Representation theory of infinite-dimensional algebras have been large influenced by conformal field theory developments.

Witten's contributions are numerous his Fields Medal says more than I can say.

Mirror Symmetry, quantum cohomology etc...

The works of Fields Medalist Kontsevich and Okounkov are largely influenced by physics.

So on an so forth...

Seeing topological (geom.) properties of the space via corresponding C^*-algebra

2013-01-06T21:42:24Z

Compact Hausdorff spaces bijectively correspond to C^*-algebras with identity. One needs to consider the algebra of continuous functions C(X) to go in one direction and spectrum to go in the other. (See e.g. Wikipedia). The situation is similar to algebraic geometry - affine manifolds correspond to commutative algebras... Basic skill in alg.geom. is to recast algebraic properties in geometric and vice versa e.g. projective modules - vector bundles... (the dictionary is lengthy).

I wonder about similar correspondence in C^*-algebra setup. In particular:

Question 1: if space "X" is topological manifold (i.e. locally R^n), is there some "nice" way to recognize it via C^*-algebra of continuous function ? (... is there non-commutative version ? ... )

Question 2: if "X" is smooth manifold, is there nice way to recognize it and define sub-algebra of smooth functions entirely in terms of C^*-algebra ? (... is there non-commutative version ? ... )

Question 3 Is it possible to characterize the set of all measures on "X" in terms of C(X) ? (... is there non-commutative version ? ... )

If you have further comments how interesting algebraic properties can be recasted in topological or vice versa, you are welcome to post.

Answer by Alexander Chervov for New grand projects in contemporary math

2013-01-05T13:10:28Z

In information theory (error-correcting codes) the grand achievements in 90-ies are turbo-codes and LDPC codes. Recent 2009 discovery which became hottest topic is polar codes.

It is tempting to say that paradigm-shift coming with turbo and LDPC codes instead of earlier popular approaches: convolutional codes, Reed-Solomon codes, BCH codes et.al. is shift from algebra to probability, from order to chaos. I mean that earlier constructions were much dominated by algebra considerations e.g. non-recursive convolutional codes are just the ideals in the ring $F_2[x]\oplus ... \oplus F_2[x]$. While turbo and LDPC are actually constructed and decoded with methods which much influenced by probabilistic and randomized considerations: roughly speaking good LDPC codes can be constructed by sufficiently sparse and random matrix. The decoding methods used for LDPC - belief propagation naturally belong to probability or machine learning maths. rather than algebra.

Actually turbo code is almost the same as convolutional code, modula one "small" detail - interleaver. Interleaver is "radomizer" added to the algebra-tasted convolutional code, it is crucial thing which makes all work. That what concerns the encoder. The decoder of turbo-codes "resembles" turbine and hence the name "turbo"-code, it is crucially based on probabilistic techniques in coding theory. Well, the key technique - BCJR algorithm was developed much earlier, so, of course, all division into old-new paradigms is not very precise, but nevertheless seems there is something behind it.

These ideas found rich practical applications. If someone is reading this with the help of smartphone - say thank to "turbo-codes" - they are working there.

New discovery - polar codes - probably can be characterized as algebra's strike back - they seems to be quite algebraic nature, sorry I cannot say much for the moment.

Hot-topics in error correcting coding related to interesting math. ?

2012-03-25T19:21:31Z

What are topics in error-correcting coding which are related to interesting math. ? I am primarely interested in nowdays hot topics, but old days topics are also welcome.

Let me try to mention what I heard about.

1) Hot topic in error-correction is finding LDPC codes with very low "error-floor" for code lengths dozens thoursands bits, this might be useful for optic transmission. However it is not clear for me what kind of math playing role here ? ("Error-floor" is related with codewords with small Hamming weight. So the code might be quite good - means majority of codewords have big Hamming weight, so in most case code performs well, but very small number having small Hamming weight will cause small number of errors - it can be seen on the BER/SNR plot as a "floor".)

2) There is certain number of papers applying number theory (lattices in algebraic number fields) to consruct good codes. One may see papers by F. Oggier, G. Rekaya-Ben Othman, J.-C. Belfiore, E. Viterbo: e.g. this one : http://arxiv.org/abs/cs/0604093. I am not aware how "hot" is this topic and how far it is from practical applications...

3) Polar codes is a hot topic. What kind of math is playing role here ?

4) Probably most classical example is the Golay code (1948) and sporadic simple Mathieu groups. Let me quote Wikipedia: http://en.wikipedia.org/wiki/Binary_Golay_code : "The automorphism group of the binary Golay code is the Mathieu group . The automorphism group of the extended binary Golay code is the Mathieu group . The other Mathieu groups occur as stabilizers of one or several elements of W." By the way - is it occasional coincidence of there is something behind it ?

Comment by Alexander Chervov

2013-04-28T14:05:03Z

It does not depend on size n? Hmmm...

Comment by Alexander Chervov

2013-04-21T12:12:54Z

What is multiplication on stable homotopy groups of spheres? (To make them ring? )

Comment by Alexander Chervov

2013-04-10T09:16:56Z

And also mathoverflow.net/questions/127010/… classification for coadjoint orbits of lower or upper triangular matrices

Comment by Alexander Chervov

2013-04-09T13:27:42Z

And also mathoverflow.net/questions/68207/… irreducible-representations-of-the-unitriangular-group

Comment by Alexander Chervov

2013-04-09T12:45:59Z

Let me mention: mathoverflow.net/questions/108406/…

Comment by Alexander Chervov

2013-04-09T12:39:22Z

Related question: mathoverflow.net/questions/126932/…

Comment by Alexander Chervov

2013-04-09T12:37:56Z

Related question mathoverflow.net/questions/106521/… in comments under it I have collected some references, which might be of interest

Comment by Alexander Chervov

2013-04-01T20:23:18Z

What is nondegenerate? aa

Comment by Alexander Chervov

2013-04-01T10:27:34Z

mathoverflow.net/questions/81415/… here is nice application given by Vladimir Dotsenko

Comment by Alexander Chervov

2013-03-17T11:49:58Z

Second Cohomology of semisinple lie alg vanishes. So any deformation is trivial. So the two algs are isomorphic.

Comment by Alexander Chervov

2013-03-09T07:37:08Z

META discussion meta.mathoverflow.net/discussion/1551/…

Comment by Alexander Chervov

2013-03-07T11:33:48Z

Thank you for the answer. My question is about applications of math outside math. I do not see the sense of specifying the "application" very precisely - hope everybody understands vague meaning and that is enough. I "assume a good will" - if some one thinks it is worth to write an in the answer about what he thinks deserves to be shared with the community - go on...

Comment by Alexander Chervov

2013-03-01T19:46:15Z

look also at quantization of lagrangian submanifolds

Comment by Alexander Chervov

2013-02-26T11:30:35Z

Be aware of stats.stackexchange.com

Comment by Alexander Chervov

2013-02-23T09:22:33Z

What are the references ?