Inverting the Schur complement, and large-dimensional Gelfand-Tsetlin patterns

Suppose we have an matrix that is expressed in block-matrix form as

where is an matrix, is an matrix, is an matrix, and is a matrix for some . If is invertible, we can use the technique of Schur complementation to express the inverse of (if it exists) in terms of the inverse of , and the other components of course. Indeed, to solve the equation

where are column vectors and are column vectors, we can expand this out as a system

Using the invertibility of , we can write the first equation as

and substituting this into the second equation yields

and thus (assuming that is invertible)

and then inserting this back into (1) gives

Comparing this with

we have managed to express the inverse of as

One can consider the inverse problem: given the inverse of , does one have a nice formula for the inverse of the minor ? Trying to recover this directly from (2) looks somewhat messy. However, one can proceed as follows. Let denote the matrix

(with the identity matrix), and let be its transpose:

Then for any scalar (which we identify with times the identity matrix), one has

and hence by (2)

noting that the inverses here will exist for large enough. Taking limits as , we conclude that

On the other hand, by the Woodbury matrix identity (discussed in this previous blog post), we have

and hence on taking limits and comparing with the preceding identity, one has

This achieves the aim of expressing the inverse of the minor in terms of the inverse of the full matrix. Taking traces and rearranging, we conclude in particular that

In the case, this can be simplified to

where is the basis column vector.

We can apply this identity to understand how the spectrum of an random matrix relates to that of its top left minor . Subtracting any complex multiple of the identity from (and hence from ), we can relate the Stieltjes transform of with the Stieltjes transform of :

At this point we begin to proceed informally. Assume for sake of argument that the random matrix is Hermitian, with distribution that is invariant under conjugation by the unitary group ; for instance, could be drawn from the Gaussian Unitary Ensemble (GUE), or alternatively could be of the form for some real diagonal matrix and a unitary matrix drawn randomly from using Haar measure. To fix normalisations we will assume that the eigenvalues of are typically of size . Then is also Hermitian and -invariant. Furthermore, the law of will be the same as the law of , where is now drawn uniformly from the unit sphere (independently of ). Diagonalising into eigenvalues and eigenvectors , we have

One can think of as a random (complex) Gaussian vector, divided by the magnitude of that vector (which, by the Chernoff inequality, will concentrate to ). Thus the coefficients with respect to the orthonormal basis can be thought of as independent (complex) Gaussian vectors, divided by that magnitude. Using this and the Chernoff inequality again, we see (for distance away from the real axis at least) that one has the concentration of measure

and thus

(that is to say, the diagonal entries of are roughly constant). Similarly we have

Inserting this into (5) and discarding terms of size , we thus conclude the approximate relationship

This can be viewed as a difference equation for the Stieltjes transform of top left minors of . Iterating this equation, and formally replacing the difference equation by a differential equation in the large limit, we see that when is large and for some , one expects the top left minor of to have Stieltjes transform

where solves the Burgers-type equation

with initial data .

Example 1 If is a constant multiple of the identity, then . One checks that is a steady state solution to (7), which is unsurprising given that all minors of are also times the identity.

Example 2 If is GUE normalised so that each entry has variance , then by the semi-circular law (see previous notes) one has (using an appropriate branch of the square root). One can then verify the self-similar solution

to (7), which is consistent with the fact that a top minor of also has the law of GUE, with each entry having variance when .

One can justify the approximation (6) given a sufficiently good well-posedness theory for the equation (7). We will not do so here, but will note that (as with the classical inviscid Burgers equation) the equation can be solved exactly (formally, at least) by the method of characteristics. For any initial position , we consider the characteristic flow formed by solving the ODE

with initial data , ignoring for this discussion the problems of existence and uniqueness. Then from the chain rule, the equation (7) implies that

and thus . Inserting this back into (8) we see that

and thus (7) may be solved implicitly via the equation

for all and .

Remark 3 In practice, the equation (9) may stop working when crosses the real axis, as (7) does not necessarily hold in this region. It is a cute exercise (ultimately coming from the Cauchy-Schwarz inequality) to show that this crossing always happens, for instance if has positive imaginary part then necessarily has negative or zero imaginary part.

Example 4 Suppose we have as in Example 1. Then (9) becomes

for any , which after making the change of variables becomes

as in Example 1.

Example 5 Suppose we have

as in Example 2. Then (9) becomes

If we write

one can calculate that

and hence

which gives

One can recover the spectral measure from the Stieltjes transform as the weak limit of as ; we write this informally as

In this informal notation, we have for instance that

which can be interpreted as the fact that the Cauchy distributions converge weakly to the Dirac mass at as . Similarly, the spectral measure associated to (10) is the semicircular measure .

If we let be the spectral measure associated to , then the curve from to the space of measures is the high-dimensional limit of a Gelfand-Tsetlin pattern (discussed in this previous post), if the pattern is randomly generated amongst all matrices with spectrum asymptotic to as . For instance, if , then the curve is , corresponding to a pattern that is entirely filled with ‘s. If instead is a semicircular distribution, then the pattern is

thus at height from the top, the pattern is semicircular on the interval . The interlacing property of Gelfand-Tsetlin patterns translates to the claim that (resp. ) is non-decreasing (resp. non-increasing) in for any fixed . In principle one should be able to establish these monotonicity claims directly from the PDE (7) or from the implicit solution (9), but it was not clear to me how to do so.

An interesting example of such a limiting Gelfand-Tsetlin pattern occurs when , which corresponds to being , where is an orthogonal projection to a random -dimensional subspace of . Here we have

and so (9) in this case becomes

A tedious calculation then gives the solution

For , there are simple poles at , and the associated measure is

This reflects the interlacing property, which forces of the eigenvalues of the minor to be equal to (resp. ). For , the poles disappear and one just has

For , one has an inverse semicircle distribution

There is presumably a direct geometric explanation of this fact (basically describing the singular values of the product of two random orthogonal projections to half-dimensional subspaces of ), but I do not know of one off-hand.

The evolution of can also be understood using the -transform and -transform from free probability. Formally, letlet be the inverse of , thus

for all , and then define the -transform

The equation (9) may be rewritten as

and hence

or equivalently

See these previous notes for a discussion of free probability topics such as the -transform.

Example 6 If then the transform is .

Example 7 If is given by (10), then the transform is

Example 8 If is given by (11), then the transform is

This simple relationship (12) is essentially due to Nica and Speicher (thanks to Dima Shylakhtenko for this reference). It has the remarkable consequence that when is the reciprocal of a natural number , then is the free arithmetic mean of copies of , that is to say is the free convolution of copies of , pushed forward by the map . In terms of random matrices, this is asserting that the top minor of a random matrix has spectral measure approximately equal to that of an arithmetic mean of independent copies of , so that the process of taking top left minors is in some sense a continuous analogue of the process of taking freely independent arithmetic means. There ought to be a geometric proof of this assertion, but I do not know of one. In the limit (or ), the -transform becomes linear and the spectral measure becomes semicircular, which is of course consistent with the free central limit theorem.

In a similar vein, if one defines the function

and inverts it to obtain a function with

for all , then the -transform is defined by

Writing

for any , , we have

and so (9) becomes

which simplifies to

replacing by we obtain

and thus

and hence

One can compute to be the -transform of the measure ; from the link between -transforms and free products (see e.g. these notes of Guionnet), we conclude that is the free product of and . This is consistent with the random matrix theory interpretation, since is also the spectral measure of , where is the orthogonal projection to the span of the first basis elements, so in particular has spectral measure . If is unitarily invariant then (by a fundamental result of Voiculescu) it is asymptotically freely independent of , so the spectral measure of is asymptotically the free product of that of and of .

Szemeredi’s proof of Szemeredi’s theorem

Szemerédi’s theorem asserts that all subsets of the natural numbers of positive density contain arbitrarily long arithmetic progressions.  Roth’s theorem is the special case when one considers arithmetic progressions of length three.  Both theorems have many important proofs using tools from additive combinatorics, (higher order) Fourier analysis, (hyper) graph regularity theory, and ergodic theory.  However, the original proof by Endre Szemerédi, while extremely intricate, was purely combinatorial (and in particular “elementary”) and almost entirely self-contained, except for an invocation of the van der Waerden theorem.  It is also notable for introducing a prototype of what is now known as the Szemerédi regularity lemma.

Back in 2005, I rewrote Szemerédi’s original proof in order to understand it better, however my rewrite ended up being about the same length as the original argument and was probably only usable to myself.  In 2012, after Szemerédi was awarded the Abel prize, I revisited this argument with the intention to try to write up a more readable version of the proof, but ended up just presenting some ingredients of the argument in a blog post, rather than try to rewrite the whole thing.  In that post, I suspected that the cleanest way to write up the argument would be through the language of nonstandard analysis (perhaps in an iterated hyperextension that could handle various hierarchies of infinitesimals), but was unable to actually achieve any substantial simplifications by passing to the nonstandard world.

A few weeks ago, I participated in a week-long workshop at the American Institute of Mathematics on “Nonstandard methods in combinatorial number theory”, and spent some time in a working group with Shabnam Akhtari, Irfam Alam, Renling Jin, Steven Leth, Karl Mahlburg, Paul Potgieter, and Henry Towsner to try to obtain a manageable nonstandard version of Szemerédi’s original proof.  We didn’t end up being able to do so – in fact there are now signs that perhaps nonstandard analysis is not the optimal framework in which to place this argument – but we did at least clarify the existing standard argument, to the point that I was able to go back to my original rewrite of the proof and present it in a more civilised form, which I am now uploading here as an unpublished preprint.   There are now a number of simplifications to the proof.  Firstly, one no longer needs the full strength of the regularity lemma; only the simpler “weak” regularity lemma of Frieze and Kannan is required.  Secondly, the proof has been “factored” into a number of stand-alone propositions of independent interest, in particular involving just (families of) one-dimensional arithmetic progressions rather than the complicated-looking multidimensional arithmetic progressions that occur so frequently in the original argument of Szemerédi.  Finally, the delicate manipulations of densities and epsilons via double counting arguments in Szemerédi’s original paper have been abstracted into a certain key property of families of arithmetic progressions that I call the “double counting property”.

The factoring mentioned above is particularly simple in the case of proving Roth’s theorem, which is now presented separately in the above writeup.  Roth’s theorem seeks to locate a length three progression in which all three elements lie in a single set.  This will be deduced from an easier variant of the theorem in which one locates (a family of) length three progressions in which just the first two elements of the progression lie in a good set (and some other properties of the family are also required).  This is in turn derived from an even easier variant in which now just the first element of the progression is required to be in the good set.

More specifically, Roth’s theorem is now deduced from

Theorem 1.5.  Let be a natural number, and let be a set of integers of upper density at least .  Then, whenever is partitioned into finitely many colour classes, there exists a colour class and a family of 3-term arithmetic progressions with the following properties:

1. For each , and lie in .
2. For each , lie in .
3. The for are in arithmetic progression.

The situation in this theorem is depicted by the following diagram, in which elements of are in blue and elements of are in grey:

Theorem 1.5 is deduced in turn from the following easier variant:

Theorem 1.6.  Let be a natural number, and let be a set of integers of upper density at least .  Then, whenever is partitioned into finitely many colour classes, there exists a colour class and a family of 3-term arithmetic progressions with the following properties:

1. For each , lie in .
2. For each , and lie in .
3. The for are in arithmetic progression.

The situation here is described by the figure below.

Theorem 1.6 is easy to prove.  To derive Theorem 1.5 from Theorem 1.6, or to derive Roth’s theorem from Theorem 1.5, one uses double counting arguments, van der Waerden’s theorem, and the weak regularity lemma, largely as described in this previous blog post; see the writeup for the full details.  (I would be interested in seeing a shorter proof of Theorem 1.5 though that did not go through these arguments, and did not use the more powerful theorems of  Roth or Szemerédi.)

 

Continuous analogues of the Schur and skew Schur polynomials

Fix a non-negative integer . Define an (weak) integer partition of length to be a tuple of non-increasing non-negative integers . (Here our partitions are “weak” in the sense that we allow some parts of the partition to be zero. Henceforth we will omit the modifier “weak”, as we will not need to consider the more usual notion of “strong” partitions.) To each such partition , one can associate a Young diagram consisting of left-justified rows of boxes, with the row containing boxes. A semi-standard Young tableau (or Young tableau for short) of shape is a filling of these boxes by integers in that is weakly increasing along rows (moving rightwards) and strictly increasing along columns (moving downwards). The collection of such tableaux will be denoted . The weight of a tableau is the tuple , where is the number of occurrences of the integer in the tableau. For instance, if and , an example of a Young tableau of shape would be

The weight here would be .

To each partition one can associate the Schur polynomial on variables , which we will define as

using the multinomial convention

Thus for instance the Young tableau given above would contribute a term to the Schur polynomial . In the case of partitions of the form , the Schur polynomial is just the complete homogeneous symmetric polynomial of degree on variables:

thus for instance

Schur polyomials are ubiquitous in the algebraic combinatorics of “type objects” such as the symmetric group , the general linear group , or the unitary group . For instance, one can view as the character of an irreducible polynomial representation of associated with the partition . However, we will not focus on these interpretations of Schur polynomials in this post.

This definition of Schur polynomials allows for a way to describe the polynomials recursively. If and is a Young tableau of shape , taking values in , one can form a sub-tableau of some shape by removing all the appearances of (which, among other things, necessarily deletes the row). For instance, with as in the previous example, the sub-tableau would be

and the reduced partition in this case is . As Young tableaux are required to be strictly increasing down columns, we can see that the reduced partition must intersperse the original partition in the sense that

for all ; we denote this interspersion relation as (though we caution that this is not intended to be a partial ordering). In the converse direction, if and is a Young tableau with shape with entries in , one can form a Young tableau with shape and entries in by appending to an entry of in all the boxes that appear in the shape but not the shape. This one-to-one correspondence leads to the recursion

where , , and the size of a partition is defined as .

One can use this recursion (2) to prove some further standard identities for Schur polynomials, such as the determinant identity

for , where denotes the Vandermonde determinant

or the Jacobi-Trudi identity

with the convention that if is negative. Thus for instance

We review the (standard) derivation of these identities via (2) below the fold. Among other things, these identities show that the Schur polynomials are symmetric, which is not immediately obvious from their definition.

One can also iterate (2) to write

where the sum is over all tuples , where each is a partition of length that intersperses the next partition , with set equal to . We will call such a tuple an integral Gelfand-Tsetlin pattern based at .

One can generalise (6) by introducing the skew Schur functions

for , whenever is a partition of length and a partition of length for some , thus the Schur polynomial is also the skew Schur polynomial with . (One could relabel the variables here to be something like instead, but this labeling seems slightly more natural, particularly in view of identities such as (8) below.)

By construction, we have the decomposition

whenever , and are partitions of lengths respectively. This gives another recursive way to understand Schur polynomials and skew Schur polynomials. For instance, one can use it to establish the generalised Jacobi-Trudi identity

with the convention that for larger than the length of ; we do this below the fold.

The Schur polynomials (and skew Schur polynomials) are “discretised” (or “quantised”) in the sense that their parameters are required to be integer-valued, and their definition similarly involves summation over a discrete set. It turns out that there are “continuous” (or “classical”) analogues of these functions, in which the parameters now take real values rather than integers, and are defined via integration rather than summation. One can view these continuous analogues as a “semiclassical limit” of their discrete counterparts, in a manner that can be made precise using the machinery of geometric quantisation, but we will not do so here.

The continuous analogues can be defined as follows. Define a real partition of length to be a tuple where are now real numbers. We can define the relation of interspersion between a length real partition and a length real partition precisely as before, by requiring that the inequalities (1) hold for all . We can then define the continuous Schur functions for recursively by defining

and

for and of length , where and the integral is with respect to -dimensional Lebesgue measure, and as before. Thus for instance

and

More generally, we can define the continuous skew Schur functions for of length , of length , and recursively by defining

and

for . Thus for instance

and

By expanding out the recursion, one obtains the analogue

of (6), and more generally one has

We will call the tuples in the first integral real Gelfand-Tsetlin patterns based at . The analogue of (8) is then

where the integral is over all real partitions of length , with Lebesgue measure.

By approximating various integrals by their Riemann sums, one can relate the continuous Schur functions to their discrete counterparts by the limiting formula

as for any length real partition and any , where

and

More generally, one has

as for any length real partition , any length real partition with , and any .

As a consequence of these limiting formulae, one expects all of the discrete identities above to have continuous counterparts. This is indeed the case; below the fold we shall prove the discrete and continuous identities in parallel. These are not new results by any means, but I was not able to locate a good place in the literature where they are explicitly written down, so I thought I would try to do so here (primarily for my own internal reference, but perhaps the calculations will be worthwhile to some others also).

Dodgson condensation from Schur complementation

The determinant of an matrix (with coefficients in an arbitrary field) obey many useful identities, starting of course with the fundamental multiplicativity for matrices . This multiplicativity can in turn be used to establish many further identities; in particular, as shown in this previous post, it implies the Schur determinant identity

whenever is an invertible matrix, is an matrix, is a matrix, and is a matrix. The matrix is known as the Schur complement of the block .

I only recently discovered that this identity in turn immediately implies what I always found to be a somewhat curious identity, namely the Dodgson condensation identity (also known as the Desnanot-Jacobi identity)

for any and matrix , where denotes the matrix formed from by removing the row and column, and similarly denotes the matrix formed from by removing the and rows and and columns. Thus for instance when we obtain

for any scalars . (Charles Dodgson, better known by his pen name Lewis Caroll, is of course also known for writing “Alice in Wonderland” and “Through the Looking Glass“.)

The derivation is not new; it is for instance noted explicitly in this paper of Brualdi and Schneider, though I do not know if this is the earliest place in the literature where it can be found. (EDIT: Apoorva Khare has pointed out to me that the original arguments of Dodgson can be interpreted as implicitly following this derivation.) I thought it is worth presenting the short derivation here, though.

Firstly, by swapping the first and rows, and similarly for the columns, it is easy to see that the Dodgson condensation identity is equivalent to the variant

Now write

where is an matrix, are column vectors, are row vectors, and are scalars. If is invertible, we may apply the Schur determinant identity repeatedly to conclude that

and the claim (2) then follows by a brief calculation (and the explicit form of the determinant). To remove the requirement that be invertible, one can use a limiting argument, noting that one can work without loss of generality in an algebraically closed field, and in such a field, the set of invertible matrices is dense in the Zariski topology. (In the case when the scalars are reals or complexes, one can just use density in the ordinary topology instead if desired.)

The same argument gives the more general determinant identity of Sylvester

whenever , is a -element subset of , and denotes the matrix formed from by removing the rows associated to and the columns associated to . (The Dodgson condensation identity is basically the case of this identity.)

A closely related proof of (2) proceeds by elementary row and column operations. Observe that if one adds some multiple of one of the first rows of to one of the last two rows of , then the left and right sides of (2) do not change. If the minor is invertible, this allows one to reduce to the case where the components of the matrix vanish. Similarly, using elementary column operations instead of row operations we may assume that vanish. All matrices involved are now block-diagonal and the identity follows from a routine computation.

The latter approach can also prove the cute identity

for any , any column vectors , and any matrix , which can for instance be found in page 7 of this text of Karlin. Observe that both sides of this identity are unchanged if one adds some multiple of any column of to one of ; for generic , this allows one to reduce to have only the first two entries allowed to be non-zero, at which point the determinants split into and determinants and we can reduce to the case (eliminating the role of ). One can now either proceed by a direct computation, or by observing that the left-hand side is quartilinear in and antisymmetric in and which forces it to be a scalar multiple of , at which point one can test the identity at a single point (e.g. and for the standard basis ) to conclude the argument. (One can also derive this identity from the Sylvester determinant identity but I think the calculations are a little messier if one goes by that route. Conversely, one can recover the Dodgson condensation identity from Karlin’s identity by setting , (for instance) and then permuting some rows and columns.)

An addendum to “amplification, arbitrage, and the tensor power trick”

In one of the earliest posts on this blog, I talked about the ability to “arbitrage” a disparity of symmetry in an inequality, and in particular to “amplify” such an inequality into a stronger one. (The principle can apply to other mathematical statements than inequalities, with the “hypothesis” and “conclusion” of that statement generally playing the role of the “right-hand side” and “left-hand side” of an inequality, but for sake of discussion I will restrict attention here to inequalities.) One can formalise this principle as follows. Many inequalities in analysis can be expressed in the form

for all in some space (in many cases will be a function space, and a function in that space), where and are some functionals of (that is to say, real-valued functions of ). For instance, might be some function space norm of (e.g. an norm), and might be some function space norm of some transform of . In addition, we assume we have some group of symmetries acting on the underlying space. For instance, if is a space of functions on some spatial domain, the group might consist of translations (e.g. for some shift ), or perhaps dilations with some normalisation (e.g. for some dilation factor and some normalisation exponent , which can be thought of as the dimensionality of length one is assigning to ). If we have

for all symmetries and all , we say that is invariant with respect to the symmetries in ; otherwise, it is not.

Suppose we know that the inequality (1) holds for all , but that there is an imbalance of symmetry: either is -invariant and is not, or vice versa. Suppose first that is -invariant and is not. Substituting by in (1) and taking infima, we can then amplify (1) to the stronger inequality

In particular, it is often the case that there is a way to send off to infinity in such a way that the functional has a limit , in which case we obtain the amplification

of (1). Note that these amplified inequalities will now be -invariant on both sides (assuming that the way in which we take limits as is itself -invariant, which it often is in practice). Similarly, if is -invariant but is not, we may instead amplify (1) to

and in particular (if has a limit as )

If neither nor has a -symmetry, one can still use the -symmetry by replacing by and taking a limit to conclude that

though now this inequality is not obviously stronger than the original inequality (1) (for instance it could well be trivial). In some cases one can also average over instead of taking a limit as , thus averaging a non-invariant inequality into an invariant one.

As discussed in the previous post, this use of amplification gives rise to a general principle about inequalities: the most efficient inequalities are those in which the left-hand side and right-hand side enjoy the same symmetries. It is certainly possible to have true inequalities that have an imbalance of symmetry, but as shown above, such inequalities can always be amplified to more efficient and more symmetric inequalities. In the case when limits such as and exist, the limiting functionals and are often simpler in form, or more tractable analytically, than their non-limiting counterparts and (this is one of the main reasons why we take limits at infinity in the first place!), and so in many applications there is really no reason to use the weaker and more complicated inequality (1), when stronger, simpler, and more symmetric inequalities such as (2), (3) are available. Among other things, this explains why many of the most useful and natural inequalities one sees in analysis are dimensionally consistent.

One often tries to prove inequalities (1) by directly chaining together simpler inequalities. For instance, one might attempt to prove (1) by by first bounding by some auxiliary quantity , and then bounding by , thus obtaining (1) by chaining together two inequalities

A variant of the above principle then asserts that when proving inequalities by such direct methods, one should, whenever possible, try to maintain the symmetries that are present in both sides of the inequality. Why? Well, suppose that we ignored this principle and tried to prove (1) by establishing (4) for some that is not -invariant. Assuming for sake of argument that (4) were actually true, we could amplify the first half of this inequality to conclude that

and also amplify the second half of the inequality to conclude that

and hence (4) amplifies to

Let’s say for sake of argument that all the quantities involved here are positive numbers (which is often the case in analysis). Then we see in particular that

Informally, (6) asserts that in order for the strategy (4) used to prove (1) to work, the extent to which fails to be -invariant cannot exceed the amount of “room” present in (1). In particular, when dealing with those “extremal” for which the left and right-hand sides of (1) are comparable to each other, one can only have a bounded amount of non--invariance in the functional . If fails so badly to be -invariant that one does not expect the left-hand side of (6) to be at all bounded in such extremal situations, then the strategy of proving (1) using the intermediate quantity is doomed to failure – even if one has already produced some clever proof of one of the two inequalities or needed to make this strategy work. And even if it did work, one could amplify (4) to a simpler inequality

(assuming that the appropriate limit existed) which would likely also be easier to prove (one can take whatever proofs one had in mind of the inequalities in (4), conjugate them by , and take a limit as to extract a proof of (7)).

Here are some simple (but somewhat contrived) examples to illustrate these points. Suppose one wishes to prove the inequality

for all . Both sides of this inequality are invariant with respect to interchanging with , so the principle suggests that when proving this inequality directly, one should only use sub-inequalities that are also invariant with respect to this interchange. However, in this particular case there is enough “room” in the inequality that it is possible (though somewhat unnatural) to violate this principle. For instance, one could decide (for whatever reason) to start with the inequality

to conclude that

and then use the obvious inequality to conclude the proof. Here, the intermediate quantity is not invariant with respect to interchange of and , but the failure is fairly mild (changing and only modifies the quantity by a multiplicative factor of at most), and disappears completely in the most extremal case , which helps explain why one could get away with using this quantity in the proof here. But it would be significantly harder (though still not impossible) to use non-symmetric intermediaries to prove the sharp version

of (8) (that is to say, the arithmetic mean-geometric mean inequality). Try it!

Similarly, consider the task of proving the triangle inequality

for complex numbers . One could try to leverage the triangle inequality for real numbers by using the crude estimate

and then use the real triangle inequality to obtain

and

and then finally use the inequalities

and

but when one puts this all together at the end of the day, one loses a factor of two:

One can “blame” this loss on the fact that while the original inequality (9) was invariant with respect to phase rotation , the intermediate expressions we tried to use when proving it were not, leading to inefficient estimates. One can try to be smarter than this by using Pythagoras’ theorem ; this reduces the loss from to but does not eliminate it completely, which is to be expected as one is still using non-invariant estimates in the proof. But one can remove the loss completely by using amplification; see the previous blog post for details (we also give a reformulation of this amplification below).

Here is a slight variant of the above example. Suppose that you had just learned in class to prove the triangle inequality

for (say) real square-summable sequences , , and was tasked to conclude the corresponding inequality

for doubly infinite square-summable sequences . The quickest way to do this is of course to exploit a bijection between the natural numbers and the integers, but let us say for sake of argument that one was unaware of such a bijection. One could then proceed instead by splitting the integers into the positive integers and the non-positive integers, and use (12) on each component separately; this is very similar to the strategy of proving (9) by splitting a complex number into real and imaginary parts, and will similarly lose a factor of or . In this case, one can “blame” this loss on the abandonment of translation invariance: both sides of the inequality (13) are invariant with respect to shifting the sequences , by some shift to arrive at , but the intermediate quantities caused by splitting the integers into two subsets are not invariant. Another way of thinking about this is that the splitting of the integers gives a privileged role to the origin , whereas the inequality (13) treats all values of equally thanks to the translation invariance, and so using such a splitting is unnatural and not likely to lead to optimal estimates. On the other hand, one can deduce (13) from (12) by sending this symmetry to infinity; indeed, after applying a shift to (12) we see that

for any , and on sending we obtain (13) (one could invoke the monotone convergence theorem here to justify the limit, though in this case it is simple enough that one can just use first principles).

Note that the principle of preserving symmetry only applies to direct approaches to proving inequalities such as (1). There is a complementary approach, discussed for instance in this previous post, which is to spend the symmetry to place the variable “without loss of generality” in a “normal form”, “convenient coordinate system”, or a “good gauge”. Abstractly: suppose that there is some subset of with the property that every can be expressed in the form for some and (that is to say, ). Then, if one wishes to prove an inequality (1) for all , and one knows that both sides of this inequality are -invariant, then it suffices to check (1) just for those in , as this together with the -invariance will imply the same inequality (1) for all in . By restricting to those in , one has given up (or spent) the -invariance, as the set will in typical not be preserved by the group action . But by the same token, by eliminating the invariance, one also eliminates the prohibition on using non-invariant proof techniques, and one is now free to use a wider range of inequalities in order to try to establish (1). Of course, such inequalities should make crucial use of the restriction , for if they did not, then the arguments would work in the more general setting , and then the previous principle would again kick in and warn us that the use of non-invariant inequalities would be inefficient. Thus one should “spend” the symmetry wisely to “buy” a restriction that will be of maximal utility in calculations (for instance by setting as many annoying factors and terms in one’s analysis to be or as possible).

As a simple example of this, let us revisit the complex triangle inequality (9). As already noted, both sides of this inequality are invariant with respect to the phase rotation symmetry . This seems to limit one to using phase-rotation-invariant techniques to establish the inequality, in particular ruling out the use of real and imaginary parts as discussed previously. However, we can instead spend the phase rotation symmetry to restrict to a special class of and . It turns out that the most efficient way to spend the symmetry is to achieve the normalisation of being a nonnegative real; this is of course possible since any complex number can be turned into a nonnegative real by multiplying by an appropriate phase . Once is a nonnegative real, the imaginary part disappears and we have

and the triangle inequality (9) is now an immediate consequence of (10), (11). (But note that if one had unwisely spent the symmetry to normalise, say, to be a non-negative real, then one is no closer to establishing (9) than before one had spent the symmetry.)

On the sign patterns of entrywise positivity preservers in fixed dimension

Apoorva Khare and I have just uploaded to the arXiv our paper “On the sign patterns of entrywise positivity preservers in fixed dimension“. This paper explores the relationship between positive definiteness of Hermitian matrices, and entrywise operations on these matrices. The starting point for this theory is the Schur product theorem, which asserts that if and are two Hermitian matrices that are positive semi-definite, then their Hadamard product

is also positive semi-definite. (One should caution that the Hadamard product is not the same as the usual matrix product.) To prove this theorem, first observe that the claim is easy when and are rank one positive semi-definite matrices, since in this case is also a rank one positive semi-definite matrix. The general case then follows by noting from the spectral theorem that a general positive semi-definite matrix can be expressed as a non-negative linear combination of rank one positive semi-definite matrices, and using the bilinearity of the Hadamard product and the fact that the set of positive semi-definite matrices form a convex cone. A modification of this argument also lets one replace “positive semi-definite” by “positive definite” in the statement of the Schur product theorem.

One corollary of the Schur product theorem is that any polynomial with non-negative coefficients is entrywise positivity preserving on the space of positive semi-definite Hermitian matrices, in the sense that for any matrix in , the entrywise application

of to is also positive semi-definite. (As before, one should caution that is not the application of to by the usual functional calculus.) Indeed, one can expand

where is the Hadamard product of copies of , and the claim now follows from the Schur product theorem and the fact that is a convex cone.

A slight variant of this argument, already observed by Pólya and Szegö in 1925, shows that if is any subset of and

is a power series with non-negative coefficients that is absolutely and uniformly convergent on , then will be entrywise positivity preserving on the set of positive definite matrices with entries in . (In the case that is of the form , such functions are precisely the absolutely monotonic functions on .)

In the work of Schoenberg and of Rudin, we have a converse: if is a function that is entrywise positivity preserving on for all , then it must be of the form (1) with . Variants of this result, with replaced by other domains, appear in the work of Horn, Vasudeva, and Guillot-Khare-Rajaratnam.

This gives a satisfactory classification of functions that are entrywise positivity preservers in all dimensions simultaneously. However, the question remains as to what happens if one fixes the dimension , in which case one may have a larger class of entrywise positivity preservers. For instance, in the trivial case , a function would be entrywise positivity preserving on if and only if is non-negative on . For higher , there is a necessary condition of Horn (refined slightly by Guillot-Khare-Rajaratnam) which asserts (at least in the case of smooth ) that all derivatives of at zero up to order must be non-negative in order for to be entrywise positivity preserving on for some . In particular, if is of the form (1), then must be non-negative. In fact, a stronger assertion can be made, namely that the first non-zero coefficients in (1) (if they exist) must be positive, or equivalently any negative term in (1) must be preceded (though not necessarily immediately) by at least positive terms. If is of the form (1) is entrywise positivity preserving on the larger set , one can furthermore show that any negative term in (1) must also be followed (though not necessarily immediately) by at least positive terms.

The main result of this paper is that these sign conditions are the only constraints for entrywise positivity preserving power series. More precisely:

Theorem 1 For each , let be a sign.

• Suppose that any negative sign is preceded by at least positive signs (thus there exists with ). Then, for any , there exists a convergent power series (1) on , with each having the sign of , which is entrywise positivity preserving on .
• Suppose in addition that any negative sign is followed by at least positive signs (thus there exists with ). Then there exists a convergent power series (1) on , with each having the sign of , which is entrywise positivity preserving on .

One can ask the same question with or replaced by other domains such as , or the complex disk , but it turns out that there are far fewer entrywise positivity preserving functions in those cases basically because of the non-trivial zeroes of Schur polynomials in these ranges; see the paper for further discussion. We also have some quantitative bounds on how negative some of the coefficients can be compared to the positive coefficients, but they are a bit technical to state here.

The heart of the proofs of these results is an analysis of the determinants of polynomials applied entrywise to rank one matrices ; the positivity of these determinants can be used (together with a continuity argument) to establish the positive definiteness of for various ranges of and . Using the Cauchy-Binet formula, one can rewrite such determinants as linear combinations of squares of magnitudes of generalised Vandermonde determinants

where and the signs of the coefficients in the linear combination are determined by the signs of the coefficients of . The task is then to find upper and lower bounds for the magnitudes of such generalised Vandermonde determinants. These determinants oscillate in sign, which makes the problem look difficult; however, an algebraic miracle intervenes, namely the factorisation

of the generalised Vandermonde determinant into the ordinary Vandermonde determinant

and a Schur polynomial applied to , where the weight of the Schur polynomial is determined by in a simple fashion. The problem then boils down to obtaining upper and lower bounds for these Schur polynomials. Because we are restricting attention to matrices taking values in or , the entries of can be taken to be non-negative. One can then take advantage of the total positivity of the Schur polynomials to compare these polynomials with a monomial, at which point one can obtain good criteria for to be positive definite when is a rank one positive definite matrix .

If we allow the exponents to be real numbers rather than integers (thus replacing polynomials or power series by Pusieux series or Hahn series), then we lose the above algebraic miracle, but we can replace it with a geometric miracle, namely the Harish-Chandra-Itzykson-Zuber identity, which I discussed in this previous blog post. This factors the above generalised Vandermonde determinant as the product of the ordinary Vandermonde determinant and an integral of a positive quantity over the orthogonal group, which one can again compare with a monomial after some fairly elementary estimates.

It remains to understand what happens for more general positive semi-definite matrices . Here we use a trick of FitzGerald and Horn to amplify the rank one case to the general case, by expressing a general positive semi-definite matrix as a linear combination of a rank one matrix and another positive semi-definite matrix that vanishes on the last row and column (and is thus effectively a positive definite matrix). Using the fundamental theorem of calculus to continuously deform the rank one matrix to in the direction , one can then obtain positivity results for from positivity results for combined with an induction hypothesis on .

The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures

Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures“, submitted to Duke Mathematical Journal. This paper builds upon my previous paper in which I introduced an “entropy decrement method” to prove the two-point (logarithmically averaged) cases of the Chowla and Elliott conjectures. A bit more specifically, I showed that

whenever were sequences going to infinity, were distinct integers, and were -bounded multiplicative functions which were non-pretentious in the sense that

for all Dirichlet characters and for . Thus, for instance, one had the logarithmically averaged two-point Chowla conjecture

for fixed any non-zero , where was the Liouville function.

One would certainly like to extend these results to higher order correlations than the two-point correlations. This looks to be difficult (though perhaps not completely impossible if one allows for logarithmic averaging): in a previous paper I showed that achieving this in the context of the Liouville function would be equivalent to resolving the logarithmically averaged Sarnak conjecture, as well as establishing logarithmically averaged local Gowers uniformity of the Liouville function. However, in this paper we are able to avoid having to resolve these difficult conjectures to obtain partial results towards the (logarithmically averaged) Chowla and Elliott conjecture. For the Chowla conjecture, we can obtain all odd order correlations, in that

for all odd and all integers (which, in the odd order case, are no longer required to be distinct). (Superficially, this looks like we have resolved “half of the cases” of the logarithmically averaged Chowla conjecture; but it seems the odd order correlations are significantly easier than the even order ones. For instance, because of the Katai-Bourgain-Sarnak-Ziegler criterion, one can basically deduce the odd order cases of (2) from the even order cases (after allowing for some dilations in the argument ).

For the more general Elliott conjecture, we can show that

for any , any integers and any bounded multiplicative functions , unless the product weakly pretends to be a Dirichlet character in the sense that

This can be seen to imply (2) as a special case. Even when does pretend to be a Dirichlet character , we can still say something: if the limits

exist for each (which can be guaranteed if we pass to a suitable subsequence), then is the uniform limit of periodic functions , each of which is –isotypic in the sense that whenever are integers with coprime to the periods of and . This does not pin down the value of any single correlation , but does put significant constraints on how these correlations may vary with .

Among other things, this allows us to show that all possible length four sign patterns of the Liouville function occur with positive density, and all possible length four sign patterns occur with the conjectured logarithmic density. (In a previous paper with Matomaki and Radziwill, we obtained comparable results for length three patterns of Liouville and length two patterns of Möbius.)

To describe the argument, let us focus for simplicity on the case of the Liouville correlations

assuming for sake of discussion that all limits exist. (In the paper, we instead use the device of generalised limits, as discussed in this previous post.) The idea is to combine together two rather different ways to control this function . The first proceeds by the entropy decrement method mentioned earlier, which roughly speaking works as follows. Firstly, we pick a prime and observe that for any , which allows us to rewrite (3) as

Making the change of variables , we obtain

The difference between and is negligible in the limit (here is where we crucially rely on the log-averaging), hence

and thus by (3) we have

The entropy decrement argument can be used to show that the latter limit is small for most (roughly speaking, this is because the factors behave like independent random variables as varies, so that concentration of measure results such as Hoeffding’s inequality can apply, after using entropy inequalities to decouple somewhat these random variables from the factors). We thus obtain the approximate isotopy property

for most and .

On the other hand, by the Furstenberg correspondence principle (as discussed in these previous posts), it is possible to express as a multiple correlation

for some probability space equipped with a measure-preserving invertible map . Using results of Bergelson-Host-Kra, Leibman, and Le, this allows us to obtain a decomposition of the form

where is a nilsequence, and goes to zero in density (even along the primes, or constant multiples of the primes). The original work of Bergelson-Host-Kra required ergodicity on , which is very definitely a hypothesis that is not available here; however, the later work of Leibman removed this hypothesis, and the work of Le refined the control on so that one still has good control when restricting to primes, or constant multiples of primes.

Ignoring the small error , we can now combine (5) to conclude that

Using the equidistribution theory of nilsequences (as developed in this previous paper of Ben Green and myself), one can break up further into a periodic piece and an “irrational” or “minor arc” piece . The contribution of the minor arc piece can be shown to mostly cancel itself out after dilating by primes and averaging, thanks to Vinogradov-type bilinear sum estimates (transferred to the primes). So we end up with

which already shows (heuristically, at least) the claim that can be approximated by periodic functions which are isotopic in the sense that

But if is odd, one can use Dirichlet’s theorem on primes in arithmetic progressions to restrict to primes that are modulo the period of , and conclude now that vanishes identically, which (heuristically, at least) gives (2).

The same sort of argument works to give the more general bounds on correlations of bounded multiplicative functions. But for the specific task of proving (2), we initially used a slightly different argument that avoids using the ergodic theory machinery of Bergelson-Host-Kra, Leibman, and Le, but replaces it instead with the Gowers uniformity norm theory used to count linear equations in primes. Basically, by averaging (4) in using the “-trick”, as well as known facts about the Gowers uniformity of the von Mangoldt function, one can obtain an approximation of the form

where ranges over a large range of integers coprime to some primorial . On the other hand, by iterating (4) we have

for most semiprimes , and by again averaging over semiprimes one can obtain an approximation of the form

For odd, one can combine the two approximations to conclude that . (This argument is not given in the current paper, but we plan to detail it in a subsequent one.)

Schur convexity and positive definiteness of the even degree complete homogeneous symmetric polynomials

The complete homogeneous symmetric polynomial of variables and degree can be defined as

thus for instance

and

One can also define all the complete homogeneous symmetric polynomials of variables simultaneously by means of the generating function

We will think of the variables as taking values in the real numbers. When one does so, one might observe that the degree two polynomial is a positive definite quadratic form, since it has the sum of squares representation

In particular, unless . This can be compared against the superficially similar quadratic form

where are independent randomly chosen signs. The Wigner semicircle law says that for large , the eigenvalues of this form will be mostly distributed in the interval using the semicircle distribution, so in particular the form is quite far from being positive definite despite the presence of the first positive terms. Thus the positive definiteness is coming from the finer algebraic structure of , and not just from the magnitudes of its coefficients.

One could ask whether the same positivity holds for other degrees than two. For odd degrees, the answer is clearly no, since in that case. But one could hope for instance that

also has a sum of squares representation that demonstrates positive definiteness. This turns out to be true, but is remarkably tedious to establish directly. Nevertheless, we have a nice result of Hunter that gives positive definiteness for all even degrees . In fact, a modification of his argument gives a little bit more:

Theorem 1 Let , let be even, and let be reals.

• (i) (Positive definiteness) One has , with strict inequality unless .
• (ii) (Schur convexity) One has whenever majorises , with equality if and only if is a permutation of .
• (iii) (Schur-Ostrowski criterion for Schur convexity) For any , one has , with strict inequality unless .

Proof: We induct on (allowing to be arbitrary). The claim is trivially true for , and easily verified for , so suppose that and the claims (i), (ii), (iii) have already been proven for (and for arbitrary ).

If we apply the differential operator to using the product rule, one obtains after a brief calculation

Using (1) and extracting the coefficient, we obtain the identity

The claim (iii) then follows from (i) and the induction hypothesis.

To obtain (ii), we use the more general statement (known as the Schur-Ostrowski criterion) that (ii) is implied from (iii) if we replace by an arbitrary symmetric, continuously differentiable function. To establish this criterion, we induct on (this argument can be made independently of the existing induction on ). If is majorised by , it lies in the permutahedron of . If lies on a face of this permutahedron, then after permuting both the and we may assume that is majorised by , and is majorised by for some , and the claim then follows from two applications of the induction hypothesis. If instead lies in the interior of the permutahedron, one can follow it to the boundary by using one of the vector fields , and the claim follows from the boundary case.

Finally, to obtain (i), we observe that majorises , where is the arithmetic mean of . But is clearly a positive multiple of , and the claim now follows from (ii).

If the variables are restricted to be nonnegative, the same argument gives Schur convexity for odd degrees also.

The proof in Hunter of positive definiteness is arranged a little differently than the one above, but still relies ultimately on the identity (2). I wonder if there is a genuinely different way to establish positive definiteness that does not go through this identity.

On the universality of the incompressible Euler equation on compact manifolds

I’ve just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds“, submitted to Discrete and Continuous Dynamical Systems. This is a variant of my recent paper on the universality of potential well dynamics, but instead of trying to embed dynamical systems into a potential well , here we try to embed dynamical systems into the incompressible Euler equations

on a Riemannian manifold . (One is particularly interested in the case of flat manifolds , particularly or , but for the main result of this paper it is essential that one is permitted to consider curved manifolds.) This system, first studied by Ebin and Marsden, is the natural generalisation of the usual incompressible Euler equations to curved space; it can be viewed as the formal geodesic flow equation on the infinite-dimensional manifold of volume-preserving diffeomorphisms on (see this previous post for a discussion of this in the flat space case).

The Euler equations can be viewed as a nonlinear equation in which the nonlinearity is a quadratic function of the velocity field . It is thus natural to compare the Euler equations with quadratic ODE of the form

where is the unknown solution, and is a bilinear map, which we may assume without loss of generality to be symmetric. One can ask whether such an ODE may be linearly embedded into the Euler equations on some Riemannian manifold , which means that there is an injective linear map from to smooth vector fields on , as well as a bilinear map to smooth scalar fields on , such that the map takes solutions to (2) to solutions to (1), or equivalently that

for all .

For simplicity let us restrict to be compact. There is an obvious necessary condition for this embeddability to occur, which comes from energy conservation law for the Euler equations; unpacking everything, this implies that the bilinear form in (2) has to obey a cancellation condition

for some positive definite inner product on . The main result of the paper is the converse to this statement: if is a symmetric bilinear form obeying a cancellation condition (3), then it is possible to embed the equations (2) into the Euler equations (1) on some Riemannian manifold ; the catch is that this manifold will depend on the form and on the dimension (in fact in the construction I have, is given explicitly as , with a funny metric on it that depends on ).

As a consequence, any finite dimensional portion of the usual “dyadic shell models” used as simplified toy models of the Euler equation, can actually be embedded into a genuine Euler equation, albeit on a high-dimensional and curved manifold. This includes portions of the self-similar “machine” I used in a previous paper to establish finite time blowup for an averaged version of the Navier-Stokes (or Euler) equations. Unfortunately, the result in this paper does not apply to infinite-dimensional ODE, so I cannot yet establish finite time blowup for the Euler equations on a (well-chosen) manifold. It does not seem so far beyond the realm of possibility, though, that this could be done in the relatively near future. In particular, the result here suggests that one could construct something resembling a universal Turing machine within an Euler flow on a manifold, which was one ingredient I would need to engineer such a finite time blowup.

The proof of the main theorem proceeds by an “elimination of variables” strategy that was used in some of my previous papers in this area, though in this particular case the Nash embedding theorem (or variants thereof) are not required. The first step is to lessen the dependence on the metric by partially reformulating the Euler equations (1) in terms of the covelocity (which is a -form) instead of the velocity . Using the freedom to modify the dimension of the underlying manifold , one can also decouple the metric from the volume form that is used to obtain the divergence-free condition. At this point the metric can be eliminated, with a certain positive definiteness condition between the velocity and covelocity taking its place. After a substantial amount of trial and error (motivated by some “two-and-a-half-dimensional” reductions of the three-dimensional Euler equations, and also by playing around with a number of variants of the classic “separation of variables” strategy), I eventually found an ansatz for the velocity and covelocity that automatically solved most of the components of the Euler equations (as well as most of the positive definiteness requirements), as long as one could find a number of scalar fields that obeyed a certain nonlinear system of transport equations, and also obeyed a positive definiteness condition. Here I was stuck for a bit because the system I ended up with was overdetermined – more equations than unknowns. After trying a number of special cases I eventually found a solution to the transport system on the sphere, except that the scalar functions sometimes degenerated and so the positive definiteness property I wanted was only obeyed with positive semi-definiteness. I tried for some time to perturb this example into a strictly positive definite solution before eventually working out that this was not possible. Finally I had the brainwave to lift the solution from the sphere to an even more symmetric space, and this quickly led to the final solution of the problem, using the special orthogonal group rather than the sphere as the underlying domain. The solution ended up being rather simple in form, but it is still somewhat miraculous to me that it exists at all; in retrospect, given the overdetermined nature of the problem, relying on a large amount of symmetry to cut down the number of equations was basically the only hope.

Maryam Mirzakhani

I am totally stunned to learn that Maryam Mirzakhani died today yesterday, aged 40, after a severe recurrence of the cancer she had been fighting for several years.  I had planned to email her some wishes for a speedy recovery after learning about the relapse yesterday; I still can’t fully believe that she didn’t make it.

My first encounter with Maryam was in 2010, when I was giving some lectures at Stanford – one on Perelman’s proof of the Poincare conjecture, and another on random matrix theory.  I remember a young woman sitting in the front who asked perceptive questions at the end of both talks; it was only afterwards that I learned that it was Mirzakhani.  (I really wish I could remember exactly what the questions were, but I vaguely recall that she managed to put a nice dynamical systems interpretation on both of the topics of my talks.)

After she won the Fields medal in 2014 (as I posted about previously on this blog), we corresponded for a while.  The Fields medal is of course one of the highest honours one can receive in mathematics, and it clearly advances one’s career enormously; but it also comes with a huge initial burst of publicity, a marked increase in the number of responsibilities to the field one is requested to take on, and a strong expectation to serve as a public role model for mathematicians.  As the first female recipient of the medal, and also the first to come from Iran, Maryam was experiencing these pressures to a far greater extent than previous medallists, while also raising a small daughter and fighting off cancer.  I gave her what advice I could on these matters (mostly that it was acceptable – and in fact necessary – to say “no” to the vast majority of requests one receives).

Given all this, it is remarkable how productive she still was mathematically in the last few years.  Perhaps her greatest recent achievement has been her “magic wand” theorem with Alex Eskin, which is basically the analogue of the famous measure classification and orbit closure theorems of Marina Ratner, in the context of moduli spaces instead of unipotent flows on homogeneous spaces.  (I discussed Ratner’s theorems in this previous post.  By an unhappy coincidence, Ratner also passed away this month, aged 78.)  Ratner’s theorems are fundamentally important to any problem to which a homogeneous dynamical system can be associated (for instance, a special case of that theorem shows up in my work with Ben Green and Tamar Ziegler on the inverse conjecture for the Gowers norms, and on linear equations in primes), as it gives a good description of the equidistribution of any orbit of that system (if it is unipotently generated); and it seems the Eskin-Mirzakhani result will play a similar role in problems associated instead to moduli spaces.  The remarkable proof of this result – which now stands at over 200 pages, after three years of revision and updating – uses almost all of the latest techniques that had been developed for homogeneous dynamics, and ingeniously adapts them to the more difficult setting of moduli spaces, in a manner that had not been dreamed of being possible only a few years earlier.

Maryam was an amazing mathematician and also a wonderful and humble human being, who was at the peak of her powers.  Today was a huge loss for Maryam’s family and friends, as well as for mathematics.

[EDIT, Jul 16: New York times obituary here.]

[EDIT, Jul 18: New Yorker memorial here.]