User sandeepj - MathOverflow

Answer by SandeepJ for Strategies for digging through literature

2010-11-17T23:17:08Z

I use the MSC codes on MathSciNet to narrow down the search to relevant papers.

See http://www.ams.org/mathscinet/msc/msc.html for the PDF file

You have to use with the old codes as well for older papers.

Answer by SandeepJ for Conformal Mappings for hyperbolic polygon

2010-11-16T18:32:06Z

See Harmer and Martin's work on Conformal Mappings from the Upper Half Plane to Fundamental Domains on the Hyperbolic Plane.

Some of the ideas developed by Christopher Bishop in the context of computational geometry may also be of interest. See his talks and papers on conformal maps.

Answer by SandeepJ for Most memorable titles

2010-10-31T20:20:28Z

Given the atmosphere of terror and fear in recent years, I did a double take when I first glanced at Bruce Berndt's paper "Ramanujan's association with radicals in India".

Answer by SandeepJ for Laplace's summation formula

2010-10-25T14:04:23Z

However, the usual suspects (the arXiv, Wikipedia, MathWorld, Google) aren't turning up much.

You forgot google books!

There are references to the Laplace summation formula in two books.

Page 248 in The rise and development of the theory of series up to the early 1820s by Giovanni Ferraro http://books.google.com/books?id=vLBJSmA9zgAC
Page 192 in A history of numerical analysis from the 16th through the 19th century by Herman Goldstine http://books.google.com/books?id=20csAQAAIAAJ

Answer by SandeepJ for Looking for a collection of entry level proofs

2010-10-22T01:26:31Z

You can try Aigner and Ziegler's book Proofs from the book

Answer by SandeepJ for Values of the j-function

2010-08-21T00:48:37Z

In case the Gross-Zagier paper doesn't meet your needs, you can also refer to the following

Harold Baier Efficient computation of singular moduli
Noriko Yui On The Singular Values Of Weber Modular Functions

and of course David Cox's book Primes of the form $x^2 + Ny^2$, Section 3.12

The bad thing about the j-function is that it is a level 1 modular function so the coefficients of its defining polynomial are going to explode with increasing degree. Its easier to compute the singular moduli using a modular function of some higher level (e.g. Weber func has level 48) as demonstrated in the papers mentioned above as well in Cox's book.

Answer by SandeepJ for Bounds on remainder term of power series of elementary functions

2010-08-17T01:39:46Z

finding bounds for the error or remainder term in partial power series expansions

I think you want the Euler-Maclaurin Summation formula. That bounds the remainder terms, although it would require knowing the closed form of the integral representation of the function you are calculating.

$ \sum_{n=a}^b f(n) \sim \int_a^b f(x)\,dx + \frac{f(a)+f(b)}{2} + \sum_{k=1}^\infty \,\frac{B_{2k}}{(2k)!}\left(f^{(2k-1)}(b)-f^{(2k-1)}(a)\right) $

The paper by Apostol "Elementary view of Euler-Maclaurin" AMM vol 106 (1999) pp. 409-418 is very accessible.

The following papers/books may also be helpful

R.P. Boas "Estimating Remainders." Math. Mag. 51, pp 83-89, (1978)
http://www.tricki.org/article/Estimating_sums
Bridger and Frampton Bounding Power Series Remainders Math. Mag. 71 (1998), pp. 204-207
Sofo. Computational Techniques for the Summation of Series
Ross. Methods of Summation
Davis. Summation of Series.

Answer by SandeepJ for Finding all cycles of a certain length in a graph

2010-08-13T19:28:26Z

Is your graph topologically planar or non-planar, weighted or unweighted, directed or undirected? Do you want an algorithm and/or a formula/bound?

For bounds on planar graphs, see Alt et al. On the number of simple cycles in planar graphs

For an algorithm, see the following paper. It incrementally builds k-cycles from (k-1)-cycles and (k-1)-paths without going through the rigourous task of computing the cycle space for the entire graph. It also handles duplicate avoidance.

Hongbo Liu; Jiaxin Wang; , "A new way to enumerate cycles in graph," Telecommunications, 2006. AICT-ICIW '06. International Conference on Internet and Web Applications and Services/Advanced International Conference on , vol., no., pp. 57- 57, 19-25 Feb. 2006 doi: 10.1109/AICT-ICIW.2006.22 URL: http://www.ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1602189&isnumber=33674

Answer by SandeepJ for Can Fuchsian functions solve the general equation of degree n?

2010-08-10T02:23:56Z

Fuchsian functions are described on wikipedia

(1) Can anyone state, in a form suitable for reduction to a computer algorithm, a solution or family of solutions of the general equation of degree n (whether based on Fuchsian functions or not)?

I had prepared notes on this topic sometime ago so here goes! There are three principal ways to solve an algebraic equation of degree n:

Algebraically by using radicals in a number field. This method doesn't work if the Galois group of the equation is not solvable, which happens for general equations beyond degree 4. In terms of differential equations, the method is analogous to integrating by quadrature if the Automorphism group(Lie group) is solvable. For more on this, see Cox's book on Galois theory and Gaal's book on Galois theory Sec 4.5
Transcendental - by reducing equation to some familiar "modular equation" : The genus of the function which is used as the solution generally depends on the degree of the equation. (See Hilbert's 13th problem) For degree {2, 3, 4}, you use trigonometric functions. For degree 5, (use genus 1) elliptic functions. For degree 6, you need genus 2 theta functions. For degree 7, depending on the Galois group, requires hyperelliptic or theta functions of genus 3. Beyond degree 7, you need to see the Umemura paper mentioned by Charles Seigel in his answer. I have omitted some information here but for general computational references, see Bruce King's book Beyond Quartic. and McKean Moll's book Elliptic Curves Chapter 5
Solution by complex dynamics : You have to find an iterative map working in a function field such that attractor(map)=root(equation). The Newton-Raphson method which we apply for degree 2 and 3 equations is actually a primitive version of this method. You can see the Julia set for Newton's method on wolfram. For more of this method, see the papers of Scott Crass and Doyle McMullen. A general reference for the method of complex dynamics is Shurman's book Geometry of the quintic. He illustrates the similarities between this method and the method of adjoining radicals in a number field. In both cases, you are building an extension over a number/function field. The number field criterion says that says that splitting field can be constructed if Galois group is solvable while the function field criterion says the that Galois group must be nearly solvable (i.e. nearly solvable implies factor groups must be Mobius groups - rotation groups of the sphere)

Answer by SandeepJ for Simultaneously computing a complete elliptic integral and its complement

2010-08-04T00:06:25Z

There are two possible ways to attack this problem

Both K and K' can be expressed in terms of the Theta function as described here http://mathworld.wolfram.com/EllipticModulus.html. If you compute $\Theta_3$, you can get both at the same time.
The other way is to observe that both K and K' are expressible in terms of the hypergeometric function $_2F_1(\frac{1}{2}, \frac{1}{2} ; 1; m)$. They are solutions of the same self-adjoint Gauss hypergeometric differential equation (since the equation is invariant under the transformation (m $\rightarrow$ 1-m))

$(k^3 - k)\frac{d^2y}{dk^2} + (3k^2 -1)\frac{dy}{dk} + ky = 0 $

By virtue of this fact, both K and K' are connected. You will find the following series expansion for K'(k) derived in Borwein's book Pi and AGM Section 1.3

$ K'(k) = \frac{2}{\pi} log \frac {4}{k} K(k) - 2 [(\frac{1}{2})^2(\frac{1}{1.2}k^2 + (\frac{1.3}{2.4})^2(\frac{1}{1.2} + \frac{1}{3.4})k^4 + (\frac{1.3.5}{2.4.6})^2(\frac{1}{1.2} + \frac{1}{3.4} + \frac{1}{5.6})k^6 $.....(infinite series) + ]

You may also find Chapter 5 of Armitage and Eberlein's book on Elliptic Functions useful.

EDIT1: I put in the complete series expansion for K'(k).

Regarding the computation of E(k), E(k) and K(k) are connected by the differential equation $ \frac{dK}{dk} = \frac{E - (1-k^2)K}{k(1-k^2)} $ which is how the Legendre relation you mention above comes about.

Again Borwein has the solution for this problem(buy the book!). Exercise 3 in Sec 1.4 has the formula based on the quartic AGM iteration $ E(k) = K(k)[1 - \sum_{n=0}^{\infty} 4^n [\alpha_n^4 - (\frac{\alpha_n^2+\beta_n^2}{2})^2 ] $ where

$\alpha_n = (a_{2n})^{\frac{1}{2}} and \beta_n = (b_{2n})^{\frac{1}{2}}$

and $ a_n, b_n$ and $c_n $ satisfy the AGM relation

Answer by SandeepJ for Number of conjugacy classes in generic finite group?

2010-08-03T17:01:09Z

In elementary terms, you have to analyze the following class equation $ n = 1 + h_2 + ... + h_r $ where

n is the order of the group G
$h_k$ denotes the number of elements in the k-th conjugacy class, and $ n = c_k.h_k$.

Dividing by n, you get $1 = \frac{1}{n} + \frac{1}{c_2} + ... + \frac{1}{c_r} $ which has a finite number of solutions.

Christine Ayoub in her paper On the number of conjugate classes in a group (Proc. Internat. Conf Theory of Groups Canberra 1967) has worked out this analysis for p-groups and there are probably more recent papers on this aspect, which Scott and others allude to in the comments. See for example

MR2557143 Keller, Thomas Michael . Lower bounds for the number of conjugacy classes of finite groups. Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 3, 567--577.

Another way of looking at your question is to see that the number of conjugacy classes is the same as the number of irreducible representations. The character table is always square. Therefore, one could ask "what are the number of irreducible characters Irr(G) in a finite group of order n?". The number of linear characters are [G:G'] where G'=commutator subgroup but the nonlinear ones are tougher and there are papers establishing various bounds for these.

MR2526321 (2010d:20010) Aziziheris, Kamal ; Lewis, Mark L. Counting the number of nonlinear irreducible characters of a finite group. Comm. Algebra 37 (2009), no. 5, 1572--1578.
MR0689258 (84d:20014) Wada, Tomoyuki . On the number of irreducible characters in a finite group. Hokkaido Math. J. 12 (1983), no. 1, 74--82.
MR0798751 (87a:20006) Wada, Tomoyuki . On the number of irreducible characters in a finite group. II. Hokkaido Math. J. 14 (1985), no. 2, 149--154.

Answer by SandeepJ for Lehmer's conjecture for Ramanujan's tau function

2010-07-22T00:17:09Z

Lehmer's conjecture has an equivalent result in the theory of Harmonic Maass forms. The non-vanishing of the tau function is equivalent to the irrationality of the coefficients of Harmonic Maass forms.

Specifically there is a correspondence between the spaces $ \zeta_{2-k} : H_{2-k}(N, \chi) \rightarrow S_k(N, \chi) $. where

$ \zeta_{2-k}$ is a differential operator
H = Harmonic Maass forms
S = cusp forms (referred to as the shadow of the Maass form)

The discriminant function $\Delta(z)$ is the shadow of the Harmonic Maass form $\frac{1}{11!} Q^+(-1, 12, 1; z) $

See Theorem 12.5 in the paper Unearthing the visions of a master: harmonic Maass forms and number theory by Ken Ono. Also see Algebraicity of Harmonic Maass forms

Answer by SandeepJ for Recovering representation from its character

2010-07-21T22:50:11Z

This complements David's response which I can't hope to match :-)

I found Rao's book on Linear Algebra and Group theory for Physicists very useful while pondering on this problem sometime ago. It lists steps somewhat similar to those given by David (don't have the book right now). Given a group G = (X | R), it proceeds to find the center, the central idempotents, the basis of the 2-sided ideals, and finally the irrep. Detailed proofs are given as to how the idempotent leads to the irrep. Every irrep leaves a positive definite Hermitian form invariant as was noted by Moore in Math. Ann. 50 p 213(1898). Rao's book then constructs the Dirac algebra representation by way of example.

Gerhard Hiss's paper documents recent work in computational representation theory. It is quite possible that this problem has already been addressed in GAP

Answer by SandeepJ for distributed incremental SVD

2010-07-16T14:20:23Z

The people aspiring for the Netflix prize like incremental SVDs. See

https://issues.apache.org/jira/browse/MAHOUT-371
B.M. Sarwar, G.Karypis, J.A. Konstan, and J. Reidl. Incremental singular value deocmposition algorithms for highly scalable recommender systems. In Proceedings of the Fifth International Conference on Computer and Information Technology (ICCIT), 2002. (ONLINE)
M. Brand. Fast online svd revisions for lightweight recommender systems. In Proceedings of the 3rd SIAM International Conference on Data Mining, 2003. and his tech report

Have you tried searching the ACM digital library for parallel SVD or singular value decomposition?

EDIT1 based on new input : See the following two papers by Hall, Marshall and Martin

Answer by SandeepJ for Are there other nice math books close to the style of Tristan Needham?

2010-07-15T02:14:38Z

if you are interested in dynamical systems/oscillators/differential equations, Pikovsky's Synchronization: A Universal Concept in Nonlinear Sciences is very well-written.

Answer by SandeepJ for Intuitions/connections/examples for "eigen-*"

2010-07-14T14:25:03Z

When you see the word eigen, replace it with the term spectrum of an operator (see spectral theory) View the matrix as a continuous or discrete linear transform acting on a vector. Similar matrices ($B = MAM^{-1}$) represent the same transform with respect to a different base.

When you diagonalize the matrix, you are actually trying to obtain an orthogonal decomposition of the transform as a linearly independent eigensystem.

If there are n independent eigenvectors, you will obtain a full diagonalization of your matrix.
If less than n, you have two choices. If all eigenvalues are in the ground field, you will get a Jordan decomposition. Otherwise, you have to settle with a rational canonical form.

In addition to Gilbert Strang's excellent book and lectures on Linear Algebra, I recommend browsing through Castillo's Orthogonal sets and polar methods in linear algebra. Throughout the book, the matrix is seen as a transform rather than something which must be numerically manipulated.

Answer by SandeepJ for When is a surface in a threefold contractible to a curve?

2010-07-11T21:42:05Z

You want a divisorial contraction. This paper may be the answer.

MR2041612 (2005c:14019) Tziolas, Nikolaos . Terminal 3-fold divisorial contractions of a surface to a curve. I. Compositio Math. 139 (2003), no. 3, 239--261.

from the paper "This paper studies divisorial contractions of a surface to a curve, i.e. when dim $\Gamma = 1$ and X has only index 1 terminal singularities along $\Gamma$. It is not always true that given $\Gamma \subset X$, there is a terminal contraction of a surface to $\Gamma$. We investigate when there is one, give criteria for existence or not and in the case that there is a terminal contraction we also describe the singularities of Y."

Answer by SandeepJ for CM field to Torus to Abelian Variety?

2010-07-09T22:39:33Z

This is not the answer. I am adding some relevant papers (some possessing good examples) which won't fit in the comments section.

I have been wanting to know the answer to this question as well. It seems one has to find an ample line bundle, and then calculate the Riemann theta relations which define the projective embedding. The equation(s) of the embedding is decided by relation between the theta functions. The wikipedia page on equations for abelian varieties was written Charles Matthews. Perhaps he might have more to say on this question...

MR0946234 (89i:14038) Barth, Wolf . Abelian surfaces with $(1,2)$-polarization. Algebraic geometry, Sendai, 1985, 41--84, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987.

MR1257320 (95e:14033) Barth, W. ; Nieto, I. Abelian surfaces of type $(1,3)$ and quartic surfaces with $16$ skew lines. J. Algebraic Geom. 3 (1994), no. 2, 173--222.

MR1336597 (96h:14064) Barth, W. Quadratic equations for level-$3$ abelian surfaces. Abelian varieties (Egloffstein, 1993), 1--18, de Gruyter, Berlin, 1995.

MR1602020 (99d:14046) Gross, Mark ; Popescu, Sorin . Equations of $(1,d)$-polarized abelian surfaces. Math. Ann. 310 (1998), no. 2, 333--377.

MR2194379 (2006m:14054) Gunji, Keiichi . Defining equations of the universal abelian surfaces with level three structure. Manuscripta Math. 119 (2006), no. 1, 61--96.

MR0611469 (82h:14028) Sasaki, Ryuji . Some remarks on the equations defining abelian varieties. Math. Z. 177 (1981), no. 1, 49--60.

MR1048533 (91e:14043) Birkenhake, Ch. ; Lange, H. Cubic theta relations. J. Reine Angew. Math. 407 (1990), 167--177.

Answer by SandeepJ for Relation between indefinite quadratic forms and continued fractions

2010-07-09T17:05:53Z

Dror is right about the cycles.

For definite forms($\Delta < 0$), there is only one reduced form per class but for indefinite forms($\Delta > 0$), there is actually a cycle of reduced forms. Consider for example, the cycle of indefinite reduced forms

(a, b, c) $\rightarrow$ (-a, b, -c)$\rightarrow$ (c, b, a) $\rightarrow$ (-c, -b, -a).

The cycle of the equivalent reduced indefinite forms corresponds to the convergents of the periodic continued fraction. The quadratic forms also correspond to the ideal classes of the quadratic field

Mollin's book on Quadratics that Dror mentioned has worked examples. Also see the paper "Computing in Quadratic Orders" on John Robertson's site

Answer by SandeepJ for The Vanishing of Ramanujan's Function tau(n)

2010-07-08T16:18:11Z

The key to your question is lacunarity in modular functions.

The tau function, as we know, occurs as the coefficient of the Discriminant function, which in turn is the 24th power of the Eta function. The Eta function was known to be lacunary (having gaps or zero coefficients). Therefore it was natural for Lehmer in 1947 to wonder if coefficients of powers of eta are also zero. See the opening passage of the following paper

MR0021027 (9,12b) Lehmer, D. H. The vanishing of Ramanujan's function $\tau(n)$. Duke Math. J. 14, (1947). 429--433. http://projecteuclid.org/euclid.dmj/1077474140

Answer by SandeepJ for Ways to prove an inequality

2010-07-07T16:44:44Z

Steele in his book Cauchy-Schwarz Master Class identifies three pillars on which all inequalities rest

Monotonicity
Positivity
Convexity, which he says is a second-order effect (Chap 6)

These three principles apply to inequalities whether they be

discrete or integral or differential
additive or multiplicative
in simple or multi-dimensional spaces (matrix inequalities).

In Chap 13 of the book, he shows how majorization and Schur's convexity unify the understanding of multifarious inequalities.

I am still not done reading the book but it also mentions a normalization method which can convert an additive inequality to a multiplicative one.

Answer by SandeepJ for Demystifying complex numbers

2010-07-01T12:52:31Z

Several motivating physical applications are listed on wikipedia

Why do we need to study numbers which do not belong to the real world?

You may want to stoke the students' imagination by disseminating the deeper truth - that the world is neither real, complex nor p-adic (these are just completions of Q). Here is a nice quote by Yuri Manin picked from here

On the fundamental level our world is neither real nor p-adic; it is adelic. For some reasons, reflecting the physical nature of our kind of living matter (e.g. the fact that we are built of massive particles), we tend to project the adelic picture onto its real side. We can equally well spiritually project it upon its non-Archimediean side and calculate most important things arithmetically. The relations between "real" and "arithmetical" pictures of the world is that of complementarity, like the relation between conjugate observables in quantum mechanics. (Y. Manin, in Conformal Invariance and String Theory, (Academic Press, 1989) 293-303 )

Answer by SandeepJ for roots of sum of two polynomials

2010-06-30T18:13:32Z

If they are complex polynomials or can be treated as such, then you could apply Rouche's theorem, where the location of the zeros is determined by the dominant polynomial within the sum. ("Walk the dog on the leash")

Possibly related: you could use the Wronskian to determine the values of A that make $P_n(x)$ and $Q_n(x)$ linearly independent.

Your question is related to Mason's theorem. There are a few papers which explore this specifically

MR1923392 (2003j:30012) Kim, Seon-Hong . Factorization of sums of polynomials. Acta Appl. Math. 73 (2002), no. 3, 275--284.
MR2103113 (2005h:30011) Kim, Seon-Hong . On zeros of certain sums of polynomials. Bull. Korean Math. Soc. 41 (2004), no. 4, 641--646
MR1911767 (2003d:11036) Pintér, Á. Zeros of the sum of polynomials. J. Math. Anal. Appl. 270 (2002), no. 1, 303--305.

Answer by SandeepJ for List of centers of finite groups

2010-06-27T18:20:55Z

In Humphrey's Group theory, Chapter 21 gives the construction of Central Extensions. [A group G is a central extension of N by H with Z(G)=N.] You could use that construction to build appropriate examples.

Wreath products are also semidirect products. In Weinstein's Examples of Groups, Sec 4.4 on wreath products, Theorem 4.4.7 constructs and proves the restricted wreath product W=(G rwr A) such that Z(W) = Z (diag($G^A$) x {1}) where diag is the diagonal subgroup (result applies only if A is finite)

Meldrum's book on Wreath Products Chap 1 might provide more constructions.

Answer by SandeepJ for What does the expression count?

2010-06-26T20:23:02Z

This is a partial answer...perhaps someone will improve on it!

Historically, most of the $q$-analog formulae (beginning from Euler) were derived based on the assumption that $|q|< 1$ (to ensure series convergence) or $q=p^k$ for a prime $p$. John Baez in one of his weekly finds (week184) discusses the geometric interpretation of $q=1$ (counting over $\mathbb CP^n$), $q=-1$ (counting over $\mathbb RP^n$) and $q=$a prime power (counting over PG($\mathbb F_q$)). There is no discussion for other values of $q$.

However, in Gasper and Rahman's Basic Hypergeometric Series, there is an inversion identity on page 4 which can be used when $|q| > 1$:

$(a; q)_n = (a^{-1}; p)_n (-a)^n p^{-n(n-1)/2} $ where $p=1/q$.

This returns a new expression in base $|1/q| < 1$. You can see some examples of the identity being applied in Gasper's Lecture Notes on q-series (Exercise 1.1, Exercise 2.3, pg 14)

I have no idea how to interpret the result geometrically. Could it be relevant to Buildings, Buekenhout geometry or $p$-adic geometry?

Answer by SandeepJ for Characterizing visual proofs

2010-06-24T22:42:33Z

Question: What would be a nice way to characterize which assertions have such visual proofs? What definitions would one need?

I am not sure if this helps but you might want to check out Visualizing Inequalities by Alsina and Nelsen. Roger Nelsen is also the author of two other books (Proofs without Words) which is a collection of many of those columns in the Mathematics magazine.

Coming back to the book, it gives no definite characterization of visual proofs but it does list various methods by which inequalities can be represented through geometric figures. It provides the beginnings of a "Representation theory for inequalities " (!) if you may call it that.

It discusses how the circumcircle and the in-circle (Chapter 4) can be used to represent an inequality between the radius of circle and the length, perimeter or area of n-gons. This doesn't always work for n-gons with n > 3 since they may not have both the circumcircle and the in-circle.

To visually prove geometric inequalities, one can use isometric transformations such as reflection or rotation (Chap 5 and 6). One can also use non-isometric transformations (Chap 7) such as

Similarity of figures which preserve shape but change measure
Measure-preserving transformations which change shape
Projections which change shape.

I particularly liked Figures 1.4, 1.7, 1.14, 1.17, 2.6, 2.17, 4.4, 7.8, 7.9, 8.17, 8.18, 8.20.

one will have to define a computational model for the ``visual verifier''

Regarding this part of the question, there is prior work on Geometric Theorem Proving. (Is that what you meant?) A couple of books which come to mind are Mechanical Geometry Theorem Proving by Chou and Wu's Mathematics Mechanization. Also see the GEO-Prover

Some relevant papers are

Kapur, D. 1986. Using Grobner bases to reason about geometry problems. J. Symb. Comput. 2, 4 (Dec. 1986), 399-408. DOI= http://dx.doi.org/10.1006/jsco.1995.1056
Kutzler, B. and Stifter, S. 1986. On the application of Buchberger's algorithm to automated geometry theorem proving. J. Symb. Comput. 2, 4 (Dec. 1986), 389-397.’
Kutzler Stifter Automated geometry theorem proving using Buchberger's algorithm http://doi.acm.org/10.1145/32439.32480

Answer by SandeepJ for Relation between Tate's thesis and Class Field Theory

2010-06-23T01:28:49Z

Boyarsky has already answered the question in the comments section. These are a couple of expositions which put John Tate's contribution in perspective.

Stephen Kudla's chapter on Tate's thesis
Stephen Gelbart's article on Elementary introduction to Langlands' program. See the section beginning page 194

Answer by SandeepJ for What is the relationship between modular forms and the Rogers-Ramanujan identities?

2010-06-23T00:11:15Z

At the root, these identities arise because there exist theta function identities (e.g. Jacobi triple product) which connect infinite series to infinite products. The infinite products have partition-theoretic interpretation as number of partitions of certain type mod k - etc while the q-series (generating functions) are also modular functions, which satisfy modular equations between moduli. By virtue of this, two types of partitions get connected into a partition identity. The Bailey lemma also comes up in this context.

David Bressoud in his book Analytic and combinatorial generalizations of the Rogers-Ramanujan identities explains that Rogers-Ramanujan identities can be stated combinatorially (set bijection) or analytically (using the function theory of Riemann surfaces) and each approach has generalizations. The analytic statement was discovered by Rogers, Ramanujan and Schur and the combinatorial statement was discovered by MacMahon and Schur.

Generalizations have been proved - see Gordon-Gollnitz identities and Andrews-Gordon identity

In addition to the links given by Will Jagy, a couple of papers listed below by Bruce Berndt discuss how modular equations of various degree are linked to certain types of partitions.

http://www.math.uiuc.edu/~berndt/publications.html

Partition identities and Ramanujan's modular equations (with N. D. Baruah), J. Comb. Thy. (A) 114 (2007), 1024-1045 (pdf).
Partition identities arising from theta function identities (with N. D. Baruah), Acta Math. Sinica 24 (2008), 955-970 (pdf).

Answer by SandeepJ for Is there any literature on multivariable theta functions?

2010-06-22T23:13:13Z

There are three ways to view theta functions

as classical homomorphic functions in vector z and/or period matrix T
as matrix coefficients of a representation of the Heisenberg and/or Metaplectic grp
as sections of Line bundles on the Abelian variety and/or moduli space of the abelian variety

Ram Murty's Theta functions - from the classical to the modern discusses Weil's representation-theoretic interpretation of theta functions. See chapter 3 by Hoffstein on Eisenstein series and theta functions on the metaplectic group. It is the connection to the metaplectic group which gives rise to the functional equation of the multivariable theta function, which you will also find in the chapter on the Metaplectic group in vol 3 of Mumford's Tata Lectures

Bellman's Brief introduction to Theta functions Section 61 alludes to theta functions in several complex variables.

You may also want to search for material in books on Abelian varieties. For example, Baker's Abelian functions Chapter X develops the theory based on the period matrix. Also Murty's book on Abelian varieties and Polishchuk's Abelian Varieties, Theta Functions and the Fourier Transform

Tyurin's Quantization, Classical and Quantum Field Theory, and Theta Functions might also be a useful reference, which I haven't browsed.

Answer by SandeepJ for Good combinatorics textbooks for teaching undergraduates?

2010-06-22T22:02:24Z

Mazur's recently published Combinatorics guided tour is quite well-organized. This is the MAA page for the book

Comment by SandeepJ

2013-02-05T02:23:43Z

@L Spice, there are specific MSC codes which reveal articles which are expositions; then you have to narrow it down from there For example code 14-02 is "Publications (1973-now) Research exposition (monographs, survey articles) " to find all MSC codes for exposition, search for "exposition" in this list ams.org/mathscinet/msc/msc.html

Comment by SandeepJ

2010-10-25T17:34:31Z

I searched google and got Allen's thesis Multiplicites of Linear Recurrence Sequences. math.ucla.edu/~pballen/PAllen_MMath_Thesis.pdf

Comment by SandeepJ

2010-08-10T02:26:01Z

I just discovered a previous related question mathoverflow.net/questions/23094/…

Comment by SandeepJ

2010-08-04T13:24:14Z

@ Mangaldan, I don't know how your algo is structured but you can continue to use AGM for computing K(k) and then use the second answer I gave (series expansion) to calculate K' in terms of K. Doesnt' that work? Buy Borwein's Pi and AGM even otherwise :-) Its a great book! Another good and free book is King's 1924 "On the direct numerical calculation of elliptic functions and integrals" online at archive.org/details/ondirectnumerica00kinguoft

Comment by SandeepJ

2010-07-25T17:31:09Z

Qiaochu already mentioned Gannon. See books.google.com/…

Comment by SandeepJ

2010-07-21T16:10:02Z

@Ofer, here is some background since you are a computer vision student. What engineers call Fourier analysis is called Harmonic Analysis in Pure Mathematics. Now for different types of spaces (euclidean or not, discrete or continuous, etc), the basis functions are going to be change (you can't decompose with sin and cos everywhere!). The basis functions of a space are determined by the Laplacian on that space. For spaces possesing spherical symmetry, the basis functions are spherical harmonics. Think, for example celestial mechanics...that's where this could come into play.

Comment by SandeepJ

2010-07-18T17:47:43Z

Thanks Hans. The series of pictures are hilarious...and exactly how it feels.

Comment by SandeepJ

2010-07-15T14:11:26Z

Possibly related mathoverflow.net/questions/9220/…

Comment by SandeepJ

2010-07-14T21:25:57Z

Comment by SandeepJ

2010-07-14T14:43:09Z

@Amitesh, See amazon.com/… and math.unicaen.fr/~reyssat/dico/dicofa.html and french.about.com/library/vocab/bl-math.htm. If you search, you might find more.

Comment by SandeepJ

2010-07-12T13:11:21Z

@JME, I can't access these two papers right now but see alpha.science.unitn.it/~andreatt/bravo/rev.html (MR1620110, MR1638131). "...They achieve in dimension four what Mori completed for threefolds. In this they constitute the first significant step of the last decade in the four-dimensional minimal model program..." If not, ask Donu Arapura !

Comment by SandeepJ

2010-07-11T18:07:05Z

See page 68 of Property testing and parameter testing for permutations by Hoppen et al. siam.org/proceedings/soda/2010/…

Comment by SandeepJ

2010-07-09T01:59:32Z

See en.wikipedia.org/wiki/…

Comment by SandeepJ

2010-07-02T12:41:16Z

To supplement Scott, see Gunning's Lectures on Modular Forms, Chapter 3. This "sum over cosets" technique is discussed there. books.google.com/…

Comment by SandeepJ

2010-07-01T19:27:32Z

You could change the problem to finding values of m for which [F(x) - m.G(x)]=0 can be factorized. The your question is related to mathoverflow.net/questions/30072