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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

\displaystyle A(f) \leq B(f) \ \ \ \ \ (1)

for all {f} in some space {X} (in many cases {X} will be a function space, and {f} a function in that space), where {A(f)} and {B(f)} are some functionals of {f} (that is to say, real-valued functions of {f}). For instance, {B(f)} might be some function space norm of {f} (e.g. an {L^p} norm), and {A(f)} might be some function space norm of some transform of {f}. In addition, we assume we have some group {G} of symmetries {T: X \rightarrow X} acting on the underlying space. For instance, if {X} is a space of functions on some spatial domain, the group might consist of translations (e.g. {Tf(x) = f(x-h)} for some shift {h}), or perhaps dilations with some normalisation (e.g. {Tf(x) = \frac{1}{\lambda^\alpha} f(\frac{x}{\lambda})} for some dilation factor {\lambda > 0} and some normalisation exponent {\alpha \in {\bf R}}, which can be thought of as the dimensionality of length one is assigning to {f}). If we have

\displaystyle A(Tf) = A(f)

for all symmetries {T \in G} and all {f \in X}, we say that {A} is invariant with respect to the symmetries in {G}; otherwise, it is not.

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

\displaystyle A(f) \leq \inf_{T \in G} B(Tf).

In particular, it is often the case that there is a way to send {T} off to infinity in such a way that the functional {B(Tf)} has a limit {B_\infty(f)}, in which case we obtain the amplification

\displaystyle A(f) \leq B_\infty(f) \ \ \ \ \ (2)

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

\displaystyle \sup_{T \in G} A(Tf) \leq B(f)

and in particular (if {A(Tf)} has a limit {A_\infty(f)} as {T \rightarrow \infty})

\displaystyle A_\infty(f) \leq B(f). \ \ \ \ \ (3)

If neither {A(f)} nor {B(f)} has a {G}-symmetry, one can still use the {G}-symmetry by replacing {f} by {Tf} and taking a limit to conclude that

\displaystyle A_\infty(f) \leq B_\infty(f),

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 {G} instead of taking a limit as {T \rightarrow \infty}, 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 {A_\infty} and {B_\infty} exist, the limiting functionals {A_\infty(f)} and {B_\infty(f)} are often simpler in form, or more tractable analytically, than their non-limiting counterparts {A(f)} and {B(f)} (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 {A(f)} by some auxiliary quantity {C(f)}, and then bounding {C(f)} by {B(f)}, thus obtaining (1) by chaining together two inequalities

\displaystyle A(f) \leq C(f) \leq B(f). \ \ \ \ \ (4)

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 {C} that is not {G}-invariant. Assuming for sake of argument that (4) were actually true, we could amplify the first half {A(f) \leq C(f)} of this inequality to conclude that

\displaystyle A(f) \leq \inf_{T \in G} C(Tf)

and also amplify the second half {C(f) \leq B(f)} of the inequality to conclude that

\displaystyle \sup_{T \in G} C(Tf) \leq B(f)

and hence (4) amplifies to

\displaystyle A(f) \leq \inf_{T \in G} C(Tf) \leq \sup_{T \in G} C(Tf) \leq B(f). \ \ \ \ \ (5)

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

\displaystyle \frac{\sup_{T \in G} C(Tf)}{\inf_{T \in G} C(Tf)} \leq \frac{B(f)}{A(f)}. \ \ \ \ \ (6)

Informally, (6) asserts that in order for the strategy (4) used to prove (1) to work, the extent to which {C} fails to be {G}-invariant cannot exceed the amount of “room” present in (1). In particular, when dealing with those “extremal” {f} for which the left and right-hand sides of (1) are comparable to each other, one can only have a bounded amount of non-{G}-invariance in the functional {C}. If {C} fails so badly to be {G}-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 {C} is doomed to failure – even if one has already produced some clever proof of one of the two inequalities {A(f) \leq C(f)} or {C(f) \leq B(f)} needed to make this strategy work. And even if it did work, one could amplify (4) to a simpler inequality

\displaystyle A(f) \leq C_\infty(f) \leq B(f) \ \ \ \ \ (7)

(assuming that the appropriate limit {C_\infty(f) = \lim_{T \rightarrow \infty} C(Tf)} 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 {T}, and take a limit as {T \rightarrow \infty} 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

\displaystyle xy \leq x^2 + y^2 \ \ \ \ \ (8)

for all {x,y>0}. Both sides of this inequality are invariant with respect to interchanging {x} with {y}, 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

\displaystyle 0 \leq (x - y/2)^2 = x^2 - xy + y^2/4

to conclude that

\displaystyle xy \leq x^2 + y^2/4

and then use the obvious inequality {x^2 + y^2/4 \leq x^2+y^2} to conclude the proof. Here, the intermediate quantity {x^2 + y^2/4} is not invariant with respect to interchange of {x} and {y}, but the failure is fairly mild (changing {x} and {y} only modifies the quantity {x^2 + y^2/4} by a multiplicative factor of {4} at most), and disappears completely in the most extremal case {x=y}, 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

\displaystyle xy \leq \frac{x^2 + y^2}{2}

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

Similarly, consider the task of proving the triangle inequality

\displaystyle |z+w| \leq |z| + |w| \ \ \ \ \ (9)

for complex numbers {z, w}. One could try to leverage the triangle inequality {|x+y| \leq |x| + |y|} for real numbers by using the crude estimate

\displaystyle |z+w| \leq |\hbox{Re}(z+w)| + |\hbox{Im}(z+w)|

and then use the real triangle inequality to obtain

\displaystyle |\hbox{Re}(z+w)| \leq |\hbox{Re}(z)| + |\hbox{Re}(w)|

and

\displaystyle |\hbox{Im}(z+w)| \leq |\hbox{Im}(z)| + |\hbox{Im}(w)|

and then finally use the inequalities

\displaystyle |\hbox{Re}(z)|, |\hbox{Im}(z)| \leq |z| \ \ \ \ \ (10)

and

\displaystyle |\hbox{Re}(w)|, |\hbox{Im}(w)| \leq |w| \ \ \ \ \ (11)

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

\displaystyle |z+w| \leq 2(|z| + |w|).

One can “blame” this loss on the fact that while the original inequality (9) was invariant with respect to phase rotation {(z,w) \mapsto (e^{i\theta} z, e^{i\theta} w)}, 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 {|z|^2 = |\hbox{Re}(z)|^2 + |\hbox{Im}(z)|^2}; this reduces the loss from {2} to {\sqrt{2}} 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

\displaystyle (\sum_{n=1}^\infty |a_n+b_n|^2)^{1/2} \leq (\sum_{n=1}^\infty |a_n|^2)^{1/2} + (\sum_{n=1}^\infty |b_n|^2)^{1/2} \ \ \ \ \ (12)

for (say) real square-summable sequences {(a_n)_{n=1}^\infty}, {(b_n)_{n=1}^\infty}, and was tasked to conclude the corresponding inequality

\displaystyle (\sum_{n \in {\bf Z}} |a_n+b_n|^2)^{1/2} \leq (\sum_{n \in {\bf Z}} |a_n|^2)^{1/2} + (\sum_{n \in {\bf Z}} |b_n|^2)^{1/2} \ \ \ \ \ (13)

for doubly infinite square-summable sequences {(a_n)_{n \in {\bf Z}}, (b_n)_{n \in {\bf Z}}}. The quickest way to do this is of course to exploit a bijection between the natural numbers {1,2,\dots} 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 {2} or {\sqrt{2}}. 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 {(a_n)_{n \in {\bf Z}}}, {(b_n)_{n \in {\bf Z}}} by some shift {h} to arrive at {(a_{n-h})_{n \in {\bf Z}}, (b_{n-h})_{n \in {\bf Z}}}, 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 {n=0}, whereas the inequality (13) treats all values of {n} 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

\displaystyle (\sum_{n=-N}^\infty |a_n+b_n|^2)^{1/2} \leq (\sum_{n=-N}^\infty |a_n|^2)^{1/2} + (\sum_{n=-N}^\infty |b_n|^2)^{1/2}

for any {N}, and on sending {N \rightarrow \infty} 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 {f} “without loss of generality” in a “normal form”, “convenient coordinate system”, or a “good gauge”. Abstractly: suppose that there is some subset {Y} of {X} with the property that every {f \in X} can be expressed in the form {f = Tg} for some {T \in G} and {g \in Y} (that is to say, {X = GY}). Then, if one wishes to prove an inequality (1) for all {f \in X}, and one knows that both sides {A(f), B(f)} of this inequality are {G}-invariant, then it suffices to check (1) just for those {f} in {Y}, as this together with the {G}-invariance will imply the same inequality (1) for all {f} in {GY=X}. By restricting to those {f} in {Y}, one has given up (or spent) the {G}-invariance, as the set {Y} will in typical not be preserved by the group action {G}. 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 {f \in Y}, for if they did not, then the arguments would work in the more general setting {f \in X}, 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 {f \in Y} that will be of maximal utility in calculations (for instance by setting as many annoying factors and terms in one’s analysis to be {0} or {1} 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 {(z,w) \mapsto (e^{i\theta} z, e^{i\theta} w)}. 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 {z} and {w}. It turns out that the most efficient way to spend the symmetry is to achieve the normalisation of {z+w} being a nonnegative real; this is of course possible since any complex number {z+w} can be turned into a nonnegative real by multiplying by an appropriate phase {e^{i\theta}}. Once {z+w} is a nonnegative real, the imaginary part disappears and we have

\displaystyle |z+w| = \hbox{Re}(z+w) = \hbox{Re}(z) + \hbox{Re}(w),

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, {z} to be a non-negative real, then one is no closer to establishing (9) than before one had spent the symmetry.)

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 {A = (a_{ij})_{1 \leq i,j \leq N}} and {B = (b_{ij})_{1 \leq i,j \leq N}} are two {N \times N} Hermitian matrices that are positive semi-definite, then their Hadamard product

\displaystyle  A \circ B := (a_{ij} b_{ij})_{1 \leq i,j \leq N}

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 {A = {\bf u} {\bf u}^*} and {B = {\bf v} {\bf v}^*} are rank one positive semi-definite matrices, since in this case {A \circ B = ({\bf u} \circ {\bf v}) ({\bf u} \circ {\bf v})^*} 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 {P(z) = c_0 + c_1 z + \dots + c_d z^d} with non-negative coefficients {c_n \geq 0} is entrywise positivity preserving on the space {{\mathbb P}_N({\bf C})} of {N \times N} positive semi-definite Hermitian matrices, in the sense that for any matrix {A = (a_{ij})_{1 \leq i,j \leq N}} in {{\mathbb P}_N({\bf C})}, the entrywise application

\displaystyle  P[A] := (P(a_{ij}))_{1 \leq i,j \leq N}

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

\displaystyle  P[A] = c_0 A^{\circ 0} + c_1 A^{\circ 1} + \dots + c_d A^{\circ d},

where {A^{\circ i}} is the Hadamard product of {i} copies of {A}, and the claim now follows from the Schur product theorem and the fact that {{\mathbb P}_N({\bf C})} is a convex cone.

A slight variant of this argument, already observed by Pólya and Szegö in 1925, shows that if {I} is any subset of {{\bf C}} and

\displaystyle  f(z) = \sum_{n=0}^\infty c_n z^n \ \ \ \ \ (1)

is a power series with non-negative coefficients {c_n \geq 0} that is absolutely and uniformly convergent on {I}, then {f} will be entrywise positivity preserving on the set {{\mathbb P}_N(I)} of positive definite matrices with entries in {I}. (In the case that {I} is of the form {I = [0,\rho]}, such functions are precisely the absolutely monotonic functions on {I}.)

In the work of Schoenberg and of Rudin, we have a converse: if {f: (-1,1) \rightarrow {\bf C}} is a function that is entrywise positivity preserving on {{\mathbb P}_N((-1,1))} for all {N}, then it must be of the form (1) with {c_n \geq 0}. Variants of this result, with {(-1,1)} replaced by other domains, appear in the work of Horn, Vasudeva, and Guillot-Khare-Rajaratnam.

This gives a satisfactory classification of functions {f} that are entrywise positivity preservers in all dimensions {N} simultaneously. However, the question remains as to what happens if one fixes the dimension {N}, in which case one may have a larger class of entrywise positivity preservers. For instance, in the trivial case {N=1}, a function would be entrywise positivity preserving on {{\mathbb P}_1((0,\rho))} if and only if {f} is non-negative on {I}. For higher {N}, there is a necessary condition of Horn (refined slightly by Guillot-Khare-Rajaratnam) which asserts (at least in the case of smooth {f}) that all derivatives of {f} at zero up to {(N-1)^{th}} order must be non-negative in order for {f} to be entrywise positivity preserving on {{\mathbb P}_N((0,\rho))} for some {0 < \rho < \infty}. In particular, if {f} is of the form (1), then {c_0,\dots,c_{N-1}} must be non-negative. In fact, a stronger assertion can be made, namely that the first {N} 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 {N} positive terms. If {f} is of the form (1) is entrywise positivity preserving on the larger set {{\mathbb P}_N((0,+\infty))}, one can furthermore show that any negative term in (1) must also be followed (though not necessarily immediately) by at least {N} 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 {n}, let {\epsilon_n \in \{-1,0,+1\}} be a sign.

  • Suppose that any negative sign {\epsilon_M = -1} is preceded by at least {N} positive signs (thus there exists {0 \leq n_0 < \dots < n_{N-1}< M} with {\epsilon_{n_0} = \dots = \epsilon_{n_{N-1}} = +1}). Then, for any {0 < \rho < \infty}, there exists a convergent power series (1) on {(0,\rho)}, with each {c_n} having the sign of {\epsilon_n}, which is entrywise positivity preserving on {{\bf P}_N((0,\rho))}.
  • Suppose in addition that any negative sign {\epsilon_M = -1} is followed by at least {N} positive signs (thus there exists {M < n_N < \dots < n_{2N-1}} with {\epsilon_{n_N} = \dots = \epsilon_{n_{2N-1}} = +1}). Then there exists a convergent power series (1) on {(0,+\infty)}, with each {c_n} having the sign of {\epsilon_n}, which is entrywise positivity preserving on {{\bf P}_N((0,+\infty))}.

One can ask the same question with {(0,\rho)} or {(0,+\infty)} replaced by other domains such as {(-\rho,\rho)}, or the complex disk {D(0,\rho)}, 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 {\mathrm{det} P[ {\bf u} {\bf u}^* ]} of polynomials {P} applied entrywise to rank one matrices {uu^*}; the positivity of these determinants can be used (together with a continuity argument) to establish the positive definiteness of {P[uu^*]} for various ranges of {P} and {u}. Using the Cauchy-Binet formula, one can rewrite such determinants as linear combinations of squares of magnitudes of generalised Vandermonde determinants

\displaystyle  \mathrm{det}( u_i^{n_j} )_{1 \leq i,j \leq N},

where {{\bf u} = (u_1,\dots,u_N)} and the signs of the coefficients in the linear combination are determined by the signs of the coefficients of {P}. 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

\displaystyle  \mathrm{det}( u_i^{n_j} )_{1 \leq i,j \leq N} = \pm V({\bf u}) s_\lambda({\bf u})

of the generalised Vandermonde determinant into the ordinary Vandermonde determinant

\displaystyle  V({\bf u}) = \prod_{1 \leq i < j \leq N} (u_j - u_i)

and a Schur polynomial {s_\lambda} applied to {{\bf u}}, where the weight {\lambda} of the Schur polynomial is determined by {n_1,\dots,n_N} 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 {(0,\rho)} or {(0,+\infty)}, the entries of {{\bf u}} 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 {P[A]} to be positive definite when {A} is a rank one positive definite matrix {A = {\bf u} {\bf u}^*}.

If we allow the exponents {n_1,\dots,n_N} 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 {A}. 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 {A} as a linear combination of a rank one matrix {{\bf u} {\bf u}^*} and another positive semi-definite matrix {B} that vanishes on the last row and column (and is thus effectively a positive definite {N-1 \times N-1} matrix). Using the fundamental theorem of calculus to continuously deform the rank one matrix {{\bf u} {\bf u}^*} to {A} in the direction {B}, one can then obtain positivity results for {P[A]} from positivity results for {P[{\bf u} {\bf u}^*]} combined with an induction hypothesis on {N}.

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

\displaystyle  \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_0(n+h_0) g_1(n+h_1)}{n} = 0

whenever {1 \leq \omega_m \leq x_m} were sequences going to infinity, {h_0,h_1} were distinct integers, and {g_0,g_1: {\bf N} \rightarrow {\bf C}} were {1}-bounded multiplicative functions which were non-pretentious in the sense that

\displaystyle  \liminf_{X \rightarrow \infty} \inf_{|t_j| \leq X} \sum_{p \leq X} \frac{1-\mathrm{Re}( g_j(p) \overline{\chi_j}(p) p^{it_j})}{p} = \infty \ \ \ \ \ (1)

for all Dirichlet characters {\chi_j} and for {j=0,1}. Thus, for instance, one had the logarithmically averaged two-point Chowla conjecture

\displaystyle  \sum_{n \leq x} \frac{\lambda(n) \lambda(n+h)}{n} = o(\log x)

for fixed any non-zero {h}, where {\lambda} 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

\displaystyle  \sum_{n \leq x} \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = o(\log x) \ \ \ \ \ (2)

for all odd {k} and all integers {h_1,\dots,h_k} (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 {n}).

For the more general Elliott conjecture, we can show that

\displaystyle  \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_1(n+h_1) \dots g_k(n+h_k)}{n} = 0

for any {k}, any integers {h_1,\dots,h_k} and any bounded multiplicative functions {g_1,\dots,g_k}, unless the product {g_1 \dots g_k} weakly pretends to be a Dirichlet character {\chi} in the sense that

\displaystyle  \sum_{p \leq X} \frac{1 - \hbox{Re}( g_1 \dots g_k(p) \overline{\chi}(p)}{p} = o(\log\log X).

This can be seen to imply (2) as a special case. Even when {g_1,\dots,g_k} does pretend to be a Dirichlet character {\chi}, we can still say something: if the limits

\displaystyle  f(a) := \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_1(n+ah_1) \dots g_k(n+ah_k)}{n}

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

Among other things, this allows us to show that all {16} possible length four sign patterns {(\lambda(n+1),\dots,\lambda(n+4)) \in \{-1,+1\}^4} of the Liouville function occur with positive density, and all {65} possible length four sign patterns {(\mu(n+1),\dots,\mu(n+4)) \in \{-1,0,+1\}^4 \backslash \{-1,+1\}^4} 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

\displaystyle  f(a) := \lim_{X \rightarrow \infty} \frac{1}{\log X} \sum_{n \leq X} \frac{\lambda(n) \lambda(n+a) \dots \lambda(n+(k-1)a)}{n}, \ \ \ \ \ (3)

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 {f}. The first proceeds by the entropy decrement method mentioned earlier, which roughly speaking works as follows. Firstly, we pick a prime {p} and observe that {\lambda(pn)=-\lambda(n)} for any {n}, which allows us to rewrite (3) as

\displaystyle  (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X}

\displaystyle  \sum_{n \leq X} \frac{\lambda(pn) \lambda(pn+ap) \dots \lambda(pn+(k-1)ap)}{n}.

Making the change of variables {n' = pn}, we obtain

\displaystyle  (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X}

\displaystyle \sum_{n' \leq pX} \frac{\lambda(n') \lambda(n'+ap) \dots \lambda(n'+(k-1)ap)}{n'} p 1_{p|n'}.

The difference between {n' \leq pX} and {n' \leq X} is negligible in the limit (here is where we crucially rely on the log-averaging), hence

\displaystyle  (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X} \sum_{n \leq X} \frac{\lambda(n) \lambda(n+ap) \dots \lambda(n+(k-1)ap)}{n} p 1_{p|n}

and thus by (3) we have

\displaystyle  (-1)^k f(a) = f(ap) + \lim_{X \rightarrow \infty} \frac{1}{\log X}

\displaystyle \sum_{n \leq X} \frac{\lambda(n) \lambda(n+ap) \dots \lambda(n+(k-1)ap)}{n} (p 1_{p|n}-1).

The entropy decrement argument can be used to show that the latter limit is small for most {p} (roughly speaking, this is because the factors {p 1_{p|n}-1} behave like independent random variables as {p} 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 {\lambda} factors). We thus obtain the approximate isotopy property

\displaystyle  (-1)^k f(a) \approx f(ap) \ \ \ \ \ (4)

for most {a} and {p}.

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

\displaystyle  f(a) = \int_X g(x) g(T^a x) \dots g(T^{(k-1)a} x)\ d\mu(x)

for some probability space {(X,\mu)} equipped with a measure-preserving invertible map {T: X \rightarrow X}. Using results of Bergelson-Host-Kra, Leibman, and Le, this allows us to obtain a decomposition of the form

\displaystyle  f(a) = f_1(a) + f_2(a) \ \ \ \ \ (5)

where {f_1} is a nilsequence, and {f_2} 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 {X}, 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 {f_1} so that one still has good control when restricting to primes, or constant multiples of primes.

Ignoring the small error {f_2(a)}, we can now combine (5) to conclude that

\displaystyle  f(a) \approx (-1)^k f_1(ap).

Using the equidistribution theory of nilsequences (as developed in this previous paper of Ben Green and myself), one can break up {f_1} further into a periodic piece {f_0} and an “irrational” or “minor arc” piece {f_3}. The contribution of the minor arc piece {f_3} can be shown to mostly cancel itself out after dilating by primes {p} and averaging, thanks to Vinogradov-type bilinear sum estimates (transferred to the primes). So we end up with

\displaystyle  f(a) \approx (-1)^k f_0(ap),

which already shows (heuristically, at least) the claim that {f} can be approximated by periodic functions {f_0} which are isotopic in the sense that

\displaystyle  f_0(a) \approx (-1)^k f_0(ap).

But if {k} is odd, one can use Dirichlet’s theorem on primes in arithmetic progressions to restrict to primes {p} that are {1} modulo the period of {f_0}, and conclude now that {f_0} 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 {p} using the “{W}-trick”, as well as known facts about the Gowers uniformity of the von Mangoldt function, one can obtain an approximation of the form

\displaystyle  (-1)^k f(a) \approx {\bf E}_{b: (b,W)=1} f(ab)

where {b} ranges over a large range of integers coprime to some primorial {W = \prod_{p \leq w} p}. On the other hand, by iterating (4) we have

\displaystyle  f(a) \approx f(apq)

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

\displaystyle  f(a) \approx {\bf E}_{b: (b,W)=1} f(ab).

For {k} odd, one can combine the two approximations to conclude that {f(a)=0}. (This argument is not given in the current paper, but we plan to detail it in a subsequent one.)

The complete homogeneous symmetric polynomial {h_d(x_1,\dots,x_n)} of {n} variables {x_1,\dots,x_n} and degree {d} can be defined as

\displaystyle h_d(x_1,\dots,x_n) := \sum_{1 \leq i_1 \leq \dots \leq i_d \leq n} x_{i_1} \dots x_{i_d},

thus for instance

\displaystyle h_0(x_1,\dots,x_n) = 0,

\displaystyle h_1(x_1,\dots,x_n) = x_1 + \dots + x_n,

and

\displaystyle h_2(x_1,\dots,x_n) = x_1^2 + \dots + x_n^2 + \sum_{1 \leq i < j \leq n} x_i x_j.

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

\displaystyle \sum_{d=0}^\infty h_d(x_1,\dots,x_n) t^d = \frac{1}{(1-t x_1) \dots (1-tx_n)}. \ \ \ \ \ (1)

We will think of the variables {x_1,\dots,x_n} as taking values in the real numbers. When one does so, one might observe that the degree two polynomial {h_2} is a positive definite quadratic form, since it has the sum of squares representation

\displaystyle h_2(x_1,\dots,x_n) = \frac{1}{2} \sum_{i=1}^n x_i^2 + \frac{1}{2} (x_1+\dots+x_n)^2.

In particular, {h_2(x_1,\dots,x_n) > 0} unless {x_1=\dots=x_n=0}. This can be compared against the superficially similar quadratic form

\displaystyle x_1^2 + \dots + x_n^2 + \sum_{1 \leq i < j \leq n} \epsilon_{ij} x_i x_j

where {\epsilon_{ij} = \pm 1} are independent randomly chosen signs. The Wigner semicircle law says that for large {n}, the eigenvalues of this form will be mostly distributed in the interval {[-\sqrt{n}, \sqrt{n}]} using the semicircle distribution, so in particular the form is quite far from being positive definite despite the presence of the first {n} positive terms. Thus the positive definiteness is coming from the finer algebraic structure of {h_2}, and not just from the magnitudes of its coefficients.

One could ask whether the same positivity holds for other degrees {d} than two. For odd degrees, the answer is clearly no, since {h_d(-x_1,\dots,-x_n) = -h_d(x_1,\dots,x_n)} in that case. But one could hope for instance that

\displaystyle h_4(x_1,\dots,x_n) = \sum_{1 \leq i \leq j \leq k \leq l \leq n} x_i x_j x_k x_l

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 {d}. In fact, a modification of his argument gives a little bit more:

Theorem 1 Let {n \geq 1}, let {d \geq 0} be even, and let {x_1,\dots,x_n} be reals.

  • (i) (Positive definiteness) One has {h_d(x_1,\dots,x_n) \geq 0}, with strict inequality unless {x_1=\dots=x_n=0}.
  • (ii) (Schur convexity) One has {h_d(x_1,\dots,x_n) \geq h_d(y_1,\dots,y_n)} whenever {(x_1,\dots,x_n)} majorises {(y_1,\dots,y_n)}, with equality if and only if {(y_1,\dots,y_n)} is a permutation of {(x_1,\dots,x_n)}.
  • (iii) (Schur-Ostrowski criterion for Schur convexity) For any {1 \leq i < j \leq n}, one has {(x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j}) h_d(x_1,\dots,x_n) \geq 0}, with strict inequality unless {x_i=x_j}.

Proof: We induct on {d} (allowing {n} to be arbitrary). The claim is trivially true for {d=0}, and easily verified for {d=2}, so suppose that {d \geq 4} and the claims (i), (ii), (iii) have already been proven for {d-2} (and for arbitrary {n}).

If we apply the differential operator {(x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j})} to {\frac{1}{(1-t x_1) \dots (1-tx_n)}} using the product rule, one obtains after a brief calculation

\displaystyle \frac{(x_i-x_j)^2 t^2}{(1-t x_1) \dots (1-tx_n) (1-t x_i) (1-t x_j)}.

Using (1) and extracting the {t^d} coefficient, we obtain the identity

\displaystyle (x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j}) h_d(x_1,\dots,x_n)

\displaystyle = (x_i-x_j)^2 h_{d-2}( x_1,\dots,x_n,x_i,x_j). \ \ \ \ \ (2)

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 {h_d} by an arbitrary symmetric, continuously differentiable function. To establish this criterion, we induct on {n} (this argument can be made independently of the existing induction on {d}). If {(y_1,\dots,y_n)} is majorised by {(x_1,\dots,x_n)}, it lies in the permutahedron of {(x_1,\dots,x_n)}. If {(y_1,\dots,y_n)} lies on a face of this permutahedron, then after permuting both the {x_i} and {y_j} we may assume that {(y_1,\dots,y_m)} is majorised by {(x_1,\dots,x_m)}, and {(y_{m+1},\dots,y_n)} is majorised by {(x_{m+1},\dots,x_n)} for some {1 \leq m < n}, and the claim then follows from two applications of the induction hypothesis. If instead {(y_1,\dots,y_n)} lies in the interior of the permutahedron, one can follow it to the boundary by using one of the vector fields {(x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j})}, and the claim follows from the boundary case.

Finally, to obtain (i), we observe that {(x_1,\dots,x_n)} majorises {(x,\dots,x)}, where {x} is the arithmetic mean of {x_1,\dots,x_n}. But {h_d(x,\dots,x)} is clearly a positive multiple of {x^d}, and the claim now follows from (ii). \Box

If the variables {x_1,\dots,x_n} 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.

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 {\partial_{tt} u = -\nabla V(u)}, here we try to embed dynamical systems into the incompressible Euler equations

\displaystyle  \partial_t u + \nabla_u u = - \mathrm{grad}_g p \ \ \ \ \ (1)

\displaystyle  \mathrm{div}_g u = 0

on a Riemannian manifold {(M,g)}. (One is particularly interested in the case of flat manifolds {M}, particularly {{\bf R}^3} or {({\bf R}/{\bf Z})^3}, 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 {M} (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 {u}. It is thus natural to compare the Euler equations with quadratic ODE of the form

\displaystyle  \partial_t y = B(y,y) \ \ \ \ \ (2)

where {y: {\bf R} \rightarrow {\bf R}^n} is the unknown solution, and {B: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}^n} 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 {(M,g)}, which means that there is an injective linear map {U: {\bf R}^n \rightarrow \Gamma(TM)} from {{\bf R}^n} to smooth vector fields on {M}, as well as a bilinear map {P: {\bf R}^n \times {\bf R}^n \rightarrow C^\infty(M)} to smooth scalar fields on {M}, such that the map {y \mapsto (U(y), P(y,y))} takes solutions to (2) to solutions to (1), or equivalently that

\displaystyle  U(B(y,y)) + \nabla_{U(y)} U(y) = - \mathrm{grad}_g P(y,y)

\displaystyle  \mathrm{div}_g U(y) = 0

for all {y \in {\bf R}^n}.

For simplicity let us restrict {M} 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 {B} in (2) has to obey a cancellation condition

\displaystyle  \langle B(y,y), y \rangle = 0 \ \ \ \ \ (3)

for some positive definite inner product {\langle, \rangle: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}} on {{\bf R}^n}. The main result of the paper is the converse to this statement: if {B} 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 {(M,g)}; the catch is that this manifold will depend on the form {B} and on the dimension {n} (in fact in the construction I have, {M} is given explicitly as {SO(n) \times ({\bf R}/{\bf Z})^{n+1}}, with a funny metric on it that depends on {B}).

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 {g} by partially reformulating the Euler equations (1) in terms of the covelocity {g \cdot u} (which is a {1}-form) instead of the velocity {u}. Using the freedom to modify the dimension of the underlying manifold {M}, one can also decouple the metric {g} 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.

I’ve just uploaded to the arXiv my paper “On the universality of potential well dynamics“, submitted to Dynamics of PDE. This is a spinoff from my previous paper on blowup of nonlinear wave equations, inspired by some conversations with Sungjin Oh. Here we focus mainly on the zero-dimensional case of such equations, namely the potential well equation

\displaystyle  \partial_{tt} u = - (\nabla F)(u) \ \ \ \ \ (1)

for a particle {u: {\bf R} \rightarrow {\bf R}^m} trapped in a potential well with potential {F: {\bf R}^m \rightarrow {\bf R}}, with {F(z) \rightarrow +\infty} as {z \rightarrow \infty}. This ODE always admits global solutions from arbitrary initial positions {u(0)} and initial velocities {\partial_t u(0)}, thanks to conservation of the Hamiltonian {\frac{1}{2} |\partial_t u|^2 + F(u)}. As this Hamiltonian is coercive (in that its level sets are compact), solutions to this equation are always almost periodic. On the other hand, as can already be seen using the harmonic oscillator {\partial_{tt} u = - k^2 u} (and direct sums of this system), this equation can generate periodic solutions, as well as quasiperiodic solutions.

All quasiperiodic motions are almost periodic. However, there are many examples of dynamical systems that admit solutions that are almost periodic but not quasiperiodic. So one can pose the question: are the dynamics of potential wells universal in the sense that they can capture all almost periodic solutions?

A precise question can be phrased as follows. Let {M} be a compact manifold, and let {X} be a smooth vector field on {M}; to avoid degeneracies, let us take {X} to be non-singular in the sense that it is everywhere non-vanishing. Then the trajectories of the first-order ODE

\displaystyle  \partial_t u = X(u) \ \ \ \ \ (2)

for {u: {\bf R} \rightarrow M} are always global and almost periodic. Can we then find a (coercive) potential {F: {\bf R}^m \rightarrow {\bf R}} for some {m}, as well as a smooth embedding {\phi: M \rightarrow {\bf R}^m}, such that every solution {u} to (2) pushes forward under {\phi} to a solution to (1)? (Actually, for technical reasons it is preferable to map into the phase space {{\bf R}^m \times {\bf R}^m}, rather than position space {{\bf R}^m}, but let us ignore this detail for this discussion.)

It turns out that the answer is no; there is a very specific obstruction. Given a pair {(M,X)} as above, define a strongly adapted {1}-form to be a {1}-form {\phi} on {M} such that {\phi(X)} is pointwise positive, and the Lie derivative {{\mathcal L}_X \phi} is an exact {1}-form. We then have

Theorem 1 A smooth compact non-singular dynamics {(M,X)} can be embedded smoothly in a potential well system if and only if it admits a strongly adapted {1}-form.

For the “only if” direction, the key point is that potential wells (viewed as a Hamiltonian flow on the phase space {{\bf R}^m \times {\bf R}^m}) admit a strongly adapted {1}-form, namely the canonical {1}-form {p dq}, whose Lie derivative is the derivative {dL} of the Lagrangian {L := \frac{1}{2} |\partial_t u|^2 - F(u)} and is thus exact. The converse “if” direction is mainly a consequence of the Nash embedding theorem, and follows the arguments used in my previous paper.

Interestingly, the same obstruction also works for potential wells in a more general Riemannian manifold than {{\bf R}^m}, or for nonlinear wave equations with a potential; combining the two, the obstruction is also present for wave maps with a potential.

It is then natural to ask whether this obstruction is non-trivial, in the sense that there are at least some examples of dynamics {(M,X)} that do not support strongly adapted {1}-forms (and hence cannot be modeled smoothly by the dynamics of a potential well, nonlinear wave equation, or wave maps). I posed this question on MathOverflow, and Robert Bryant provided a very nice construction, showing that the vector field {(\sin(2\pi x), \cos(2\pi x))} on the {2}-torus {({\bf R}/{\bf Z})^2} had no strongly adapted {1}-forms, and hence the dynamics of this vector field cannot be smoothly reproduced by a potential well, nonlinear wave equation, or wave map:

On the other hand, the suspension of any diffeomorphism does support a strongly adapted {1}-form (the derivative {dt} of the time coordinate), and using this and the previous theorem I was able to embed a universal Turing machine into a potential well. In particular, there are flows for an explicitly describable potential well whose trajectories have behavior that is undecidable using the usual ZFC axioms of set theory! So potential well dynamics are “effectively” universal, despite the presence of the aforementioned obstruction.

In my previous work on blowup for Navier-Stokes like equations, I speculated that if one could somehow replicate a universal Turing machine within the Euler equations, one could use this machine to create a “von Neumann machine” that replicated smaller versions of itself, which on iteration would lead to a finite time blowup. Now that such a mechanism is present in nonlinear wave equations, it is tempting to try to make this scheme work in that setting. Of course, in my previous paper I had already demonstrated finite time blowup, at least in a three-dimensional setting, but that was a relatively simple discretely self-similar blowup in which no computation occurred. This more complicated blowup scheme would be significantly more effort to set up, but would be proof-of-concept that the same scheme would in principle be possible for the Navier-Stokes equations, assuming somehow that one can embed a universal Turing machine into the Euler equations. (But I’m still hopelessly stuck on how to accomplish this latter task…)

Kaisa Matomaki, Maksym Radziwill, and I have uploaded to the arXiv our paper “Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges“, submitted to Proceedings of the London Mathematical Society. This paper is concerned with the estimation of correlations such as

\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+h) \ \ \ \ \ (1)

for medium-sized {h} and large {X}, where {\Lambda} is the von Mangoldt function; we also consider variants of this sum in which one of the von Mangoldt functions is replaced with a (higher order) divisor function, but for sake of discussion let us focus just on the sum (1). Understanding this sum is very closely related to the problem of finding pairs of primes that differ by {h}; for instance, if one could establish a lower bound

\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+2) \gg X

then this would easily imply the twin prime conjecture.

The (first) Hardy-Littlewood conjecture asserts an asymptotic

\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+h) = {\mathfrak S}(h) X + o(X) \ \ \ \ \ (2)

as {X \rightarrow \infty} for any fixed positive {h}, where the singular series {{\mathfrak S}(h)} is an arithmetic factor arising from the irregularity of distribution of {\Lambda} at small moduli, defined explicitly by

\displaystyle {\mathfrak S}(h) := 2 \Pi_2 \prod_{p|h; p>2} \frac{p-2}{p-1}

when {h} is even, and {{\mathfrak S}(h)=0} when {h} is odd, where

\displaystyle \Pi_2 := \prod_{p>2} (1-\frac{1}{(p-1)^2}) = 0.66016\dots

is (half of) the twin prime constant. See for instance this previous blog post for a a heuristic explanation of this conjecture. From the previous discussion we see that (2) for {h=2} would imply the twin prime conjecture. Sieve theoretic methods are only able to provide an upper bound of the form { \sum_{n \leq X} \Lambda(n) \Lambda(n+h) \ll {\mathfrak S}(h) X}.

Needless to say, apart from the trivial case of odd {h}, there are no values of {h} for which the Hardy-Littlewood conjecture is known. However there are some results that say that this conjecture holds “on the average”: in particular, if {H} is a quantity depending on {X} that is somewhat large, there are results that show that (2) holds for most (i.e. for {1-o(1)}) of the {h} betwen {0} and {H}. Ideally one would like to get {H} as small as possible, in particular one can view the full Hardy-Littlewood conjecture as the endpoint case when {H} is bounded.

The first results in this direction were by van der Corput and by Lavrik, who established such a result with {H = X} (with a subsequent refinement by Balog); Wolke lowered {H} to {X^{5/8+\varepsilon}}, and Mikawa lowered {H} further to {X^{1/3+\varepsilon}}. The main result of this paper is a further lowering of {H} to {X^{8/33+\varepsilon}}. In fact (as in the preceding works) we get a better error term than {o(X)}, namely an error of the shape {O_A( X \log^{-A} X)} for any {A}.

Our arguments initially proceed along standard lines. One can use the Hardy-Littlewood circle method to express the correlation in (2) as an integral involving exponential sums {S(\alpha) := \sum_{n \leq X} \Lambda(n) e(\alpha n)}. The contribution of “major arc” {\alpha} is known by a standard computation to recover the main term {{\mathfrak S}(h) X} plus acceptable errors, so it is a matter of controlling the “minor arcs”. After averaging in {h} and using the Plancherel identity, one is basically faced with establishing a bound of the form

\displaystyle \int_{\beta-1/H}^{\beta+1/H} |S(\alpha)|^2\ d\alpha \ll_A X \log^{-A} X

for any “minor arc” {\beta}. If {\beta} is somewhat close to a low height rational {a/q} (specifically, if it is within {X^{-1/6-\varepsilon}} of such a rational with {q = O(\log^{O(1)} X)}), then this type of estimate is roughly of comparable strength (by another application of Plancherel) to the best available prime number theorem in short intervals on the average, namely that the prime number theorem holds for most intervals of the form {[x, x + x^{1/6+\varepsilon}]}, and we can handle this case using standard mean value theorems for Dirichlet series. So we can restrict attention to the “strongly minor arc” case where {\beta} is far from such rationals.

The next step (following some ideas we found in a paper of Zhan) is to rewrite this estimate not in terms of the exponential sums {S(\alpha) := \sum_{n \leq X} \Lambda(n) e(\alpha n)}, but rather in terms of the Dirichlet polynomial {F(s) := \sum_{n \sim X} \frac{\Lambda(n)}{n^s}}. After a certain amount of computation (including some oscillatory integral estimates arising from stationary phase), one is eventually reduced to the task of establishing an estimate of the form

\displaystyle \int_{t \sim \lambda X} (\sum_{t-\lambda H}^{t+\lambda H} |F(\frac{1}{2}+it')|\ dt')^2\ dt \ll_A \lambda^2 H^2 X \log^{-A} X

for any {X^{-1/6-\varepsilon} \ll \lambda \ll \log^{-B} X} (with {B} sufficiently large depending on {A}).

The next step, which is again standard, is the use of the Heath-Brown identity (as discussed for instance in this previous blog post) to split up {\Lambda} into a number of components that have a Dirichlet convolution structure. Because the exponent {8/33} we are shooting for is less than {1/4}, we end up with five types of components that arise, which we call “Type {d_1}“, “Type {d_2}“, “Type {d_3}“, “Type {d_4}“, and “Type II”. The “Type II” sums are Dirichlet convolutions involving a factor supported on a range {[X^\varepsilon, X^{-\varepsilon} H]} and is quite easy to deal with; the “Type {d_j}” terms are Dirichlet convolutions that resemble (non-degenerate portions of) the {j^{th}} divisor function, formed from convolving together {j} portions of {1}. The “Type {d_1}” and “Type {d_2}” terms can be estimated satisfactorily by standard moment estimates for Dirichlet polynomials; this already recovers the result of Mikawa (and our argument is in fact slightly more elementary in that no Kloosterman sum estimates are required). It is the treatment of the “Type {d_3}” and “Type {d_4}” sums that require some new analysis, with the Type {d_3} terms turning to be the most delicate. After using an existing moment estimate of Jutila for Dirichlet L-functions, matters reduce to obtaining a family of estimates, a typical one of which (relating to the more difficult Type {d_3} sums) is of the form

\displaystyle \int_{t - H}^{t+H} |M( \frac{1}{2} + it')|^2\ dt' \ll X^{\varepsilon^2} H \ \ \ \ \ (3)

for “typical” ordinates {t} of size {X}, where {M} is the Dirichlet polynomial {M(s) := \sum_{n \sim X^{1/3}} \frac{1}{n^s}} (a fragment of the Riemann zeta function). The precise definition of “typical” is a little technical (because of the complicated nature of Jutila’s estimate) and will not be detailed here. Such a claim would follow easily from the Lindelof hypothesis (which would imply that {M(1/2 + it) \ll X^{o(1)}}) but of course we would like to have an unconditional result.

At this point, having exhausted all the Dirichlet polynomial estimates that are usefully available, we return to “physical space”. Using some further Fourier-analytic and oscillatory integral computations, we can estimate the left-hand side of (3) by an expression that is roughly of the shape

\displaystyle \frac{H}{X^{1/3}} \sum_{\ell \sim X^{1/3}/H} |\sum_{m \sim X^{1/3}} e( \frac{t}{2\pi} \log \frac{m+\ell}{m-\ell} )|.

The phase {\frac{t}{2\pi} \log \frac{m+\ell}{m-\ell}} can be Taylor expanded as the sum of {\frac{t_j \ell}{\pi m}} and a lower order term {\frac{t_j \ell^3}{3\pi m^3}}, plus negligible errors. If we could discard the lower order term then we would get quite a good bound using the exponential sum estimates of Robert and Sargos, which control averages of exponential sums with purely monomial phases, with the averaging allowing us to exploit the hypothesis that {t} is “typical”. Figuring out how to get rid of this lower order term caused some inefficiency in our arguments; the best we could do (after much experimentation) was to use Fourier analysis to shorten the sums, estimate a one-parameter average exponential sum with a binomial phase by a two-parameter average with a monomial phase, and then use the van der Corput {B} process followed by the estimates of Robert and Sargos. This rather complicated procedure works up to {H = X^{8/33+\varepsilon}} it may be possible that some alternate way to proceed here could improve the exponent somewhat.

In a sequel to this paper, we will use a somewhat different method to reduce {H} to a much smaller value of {\log^{O(1)} X}, but only if we replace the correlations {\sum_{n \leq X} \Lambda(n) \Lambda(n+h)} by either {\sum_{n \leq X} \Lambda(n) d_k(n+h)} or {\sum_{n \leq X} d_k(n) d_l(n+h)}, and also we now only save a {o(1)} in the error term rather than {O_A(\log^{-A} X)}.

In July I will be spending a week at Park City, being one of the mini-course lecturers in the Graduate Summer School component of the Park City Summer Session on random matrices.  I have chosen to give some lectures on least singular values of random matrices, the circular law, and the Lindeberg exchange method in random matrix theory; this is a slightly different set of topics than I had initially advertised (which was instead about the Lindeberg exchange method and the local relaxation flow method), but after consulting with the other mini-course lecturers I felt that this would be a more complementary set of topics.  I have uploaded an draft of my lecture notes (some portion of which is derived from my monograph on the subject); as always, comments and corrections are welcome.

<I>[Update, June 23: notes revised and reformatted to PCMI format. -T.]</I>

 

 

Suppose {F: X \rightarrow Y} is a continuous (but nonlinear) map from one normed vector space {X} to another {Y}. The continuity means, roughly speaking, that if {x_0, x \in X} are such that {\|x-x_0\|_X} is small, then {\|F(x)-F(x_0)\|_Y} is also small (though the precise notion of “smallness” may depend on {x} or {x_0}, particularly if {F} is not known to be uniformly continuous). If {F} is known to be differentiable (in, say, the Fréchet sense), then we in fact have a linear bound of the form

\displaystyle  \|F(x)-F(x_0)\|_Y \leq C(x_0) \|x-x_0\|_X

for some {C(x_0)} depending on {x_0}, if {\|x-x_0\|_X} is small enough; one can of course make {C(x_0)} independent of {x_0} (and drop the smallness condition) if {F} is known instead to be Lipschitz continuous.

In many applications in analysis, one would like more explicit and quantitative bounds that estimate quantities like {\|F(x)-F(x_0)\|_Y} in terms of quantities like {\|x-x_0\|_X}. There are a number of ways to do this. First of all, there is of course the trivial estimate arising from the triangle inequality:

\displaystyle  \|F(x)-F(x_0)\|_Y \leq \|F(x)\|_Y + \|F(x_0)\|_Y. \ \ \ \ \ (1)

This estimate is usually not very good when {x} and {x_0} are close together. However, when {x} and {x_0} are far apart, this estimate can be more or less sharp. For instance, if the magnitude of {F} varies so much from {x_0} to {x} that {\|F(x)\|_Y} is more than (say) twice that of {\|F(x_0)\|_Y}, or vice versa, then (1) is sharp up to a multiplicative constant. Also, if {F} is oscillatory in nature, and the distance between {x} and {x_0} exceeds the “wavelength” of the oscillation of {F} at {x_0} (or at {x}), then one also typically expects (1) to be close to sharp. Conversely, if {F} does not vary much in magnitude from {x_0} to {x}, and the distance between {x} and {x_0} is less than the wavelength of any oscillation present in {F}, one expects to be able to improve upon (1).

When {F} is relatively simple in form, one can sometimes proceed simply by substituting {x = x_0 + h}. For instance, if {F: R \rightarrow R} is the squaring function {F(x) = x^2} in a commutative ring {R}, one has

\displaystyle  F(x_0+h) = (x_0+h)^2 = x_0^2 + 2x_0 h+ h^2

and thus

\displaystyle  F(x_0+h) - F(x_0) = 2x_0 h + h^2

or in terms of the original variables {x, x_0} one has

\displaystyle  F(x) - F(x_0) = 2 x_0 (x-x_0) + (x-x_0)^2.

If the ring {R} is not commutative, one has to modify this to

\displaystyle  F(x) - F(x_0) = x_0 (x-x_0) + (x-x_0) x_0 + (x-x_0)^2.

Thus, for instance, if {A, B} are {n \times n} matrices and {\| \|_{op}} denotes the operator norm, one sees from the triangle inequality and the sub-multiplicativity {\| AB\|_{op} \leq \| A \|_{op} \|B\|_{op}} of operator norm that

\displaystyle  \| A^2 - B^2 \|_{op} \leq \| A - B \|_{op} ( 2 \|B\|_{op} + \|A - B \|_{op} ). \ \ \ \ \ (2)

If {F(x)} involves {x} (or various components of {x}) in several places, one can sometimes get a good estimate by “swapping” {x} with {x_0} at each of the places in turn, using a telescoping series. For instance, if we again use the squaring function {F(x) = x^2 = x x} in a non-commutative ring, we have

\displaystyle  F(x) - F(x_0) = x x - x_0 x_0

\displaystyle  = (x x - x_0 x) + (x_0 x - x_0 x_0)

\displaystyle  = (x-x_0) x + x_0 (x-x_0)

which for instance leads to a slight improvement of (2):

\displaystyle  \| A^2 - B^2 \|_{op} \leq \| A - B \|_{op} ( \| A\|_{op} + \|B\|_{op} ).

More generally, for any natural number {n}, one has the identity

\displaystyle  x^n - x_0^n = (x-x_0) (x^{n-1} + x^{n-2} x_0 + \dots + x x_0^{n-2} + x_0^{n-1}) \ \ \ \ \ (3)

in a commutative ring, while in a non-commutative ring one must modify this to

\displaystyle  x^n - x_0^n = \sum_{i=0}^{n-1} x_0^i (x-x_0) x^{n-1-i},

and for matrices one has

\displaystyle  \| A^n - B^n \|_{op} \leq \| A-B\|_{op} ( \|A\|_{op}^{n-1} + \| A\|_{op}^{n-2} \| B\|_{op} + \dots + \|B\|_{op}^{n-1} ).

Exercise 1 If {U} and {V} are unitary {n \times n} matrices, show that the commutator {[U,V] := U V U^{-1} V^{-1}} obeys the inequality

\displaystyle  \| [U,V] - I \|_{op} \leq 2 \| U - I \|_{op} \| V - I \|_{op}.

(Hint: first control {\| UV - VU \|_{op}}.)

Now suppose (for simplicity) that {F: {\bf R}^d \rightarrow {\bf R}^{d'}} is a map between Euclidean spaces. If {F} is continuously differentiable, then one can use the fundamental theorem of calculus to write

\displaystyle  F(x) - F(x_0) = \int_0^1 \frac{d}{dt} F( \gamma(t) )\ dt

where {\gamma: [0,1] \rightarrow Y} is any continuously differentiable path from {x_0} to {x}. For instance, if one uses the straight line path {\gamma(t) := (1-t) x_0 + tx}, one has

\displaystyle  F(x) - F(x_0) = \int_0^1 ((x-x_0) \cdot \nabla F)( (1-t) x_0 + t x )\ dt.

In the one-dimensional case {d=1}, this simplifies to

\displaystyle  F(x) - F(x_0) = (x-x_0) \int_0^1 F'( (1-t) x_0 + t x )\ dt. \ \ \ \ \ (4)

Among other things, this immediately implies the factor theorem for {C^k} functions: if {F} is a {C^k({\bf R})} function for some {k \geq 1} that vanishes at some point {x_0}, then {F(x)} factors as the product of {x-x_0} and some {C^{k-1}} function {G}. Another basic consequence is that if {\nabla F} is uniformly bounded in magnitude by some constant {C}, then {F} is Lipschitz continuous with the same constant {C}.

Applying (4) to the power function {x \mapsto x^n}, we obtain the identity

\displaystyle  x^n - x_0^n = n (x-x_0) \int_0^1 ((1-t) x_0 + t x)^{n-1}\ dt \ \ \ \ \ (5)

which can be compared with (3). Indeed, for {x_0} and {x} close to {1}, one can use logarithms and Taylor expansion to arrive at the approximation {((1-t) x_0 + t x)^{n-1} \approx x_0^{(1-t) (n-1)} x^{t(n-1)}}, so (3) behaves a little like a Riemann sum approximation to (5).

Exercise 2 For each {i=1,\dots,n}, let {X^{(1)}_i} and {X^{(0)}_i} be random variables taking values in a measurable space {R_i}, and let {F: R_1 \times \dots \times R_n \rightarrow {\bf R}^m} be a bounded measurable function.

  • (i) (Lindeberg exchange identity) Show that

    \displaystyle  \mathop{\bf E} F(X^{(1)}_1,\dots,X^{(1)}_n) - \mathop{\bf E} F(X^{(0)}_1,\dots,X^{(0)}_n)

    \displaystyle = \sum_{i=1}^n \mathop{\bf E} F( X^{(1)}_1,\dots, X^{(1)}_{i-1}, X^{(1)}_i, X^{(0)}_{i+1}, \dots, X^{(0)}_n)

    \displaystyle - \mathop{\bf E} F( X^{(1)}_1,\dots, X^{(1)}_{i-1}, X^{(0)}_i, X^{(0)}_{i+1}, \dots, X^{(0)}_n).

  • (ii) (Knowles-Yin exchange identity) Show that

    \displaystyle  \mathop{\bf E} F(X^{(1)}_1,\dots,X^{(1)}_n) - \mathop{\bf E} F(X^{(0)}_1,\dots,X^{(0)}_n)

    \displaystyle = \int_0^1 \sum_{i=1}^n \mathop{\bf E} F( X^{(t)}_1,\dots, X^{(t)}_{i-1}, X^{(1)}_i, X^{(t)}_{i+1}, \dots, X^{(t)}_n)

    \displaystyle - \mathop{\bf E} F( X^{(t)}_1,\dots, X^{(t)}_{i-1}, X^{(0)}_i, X^{(t)}_{i+1}, \dots, X^{(t)}_n)\ dt,

    where {X^{(t)}_i = 1_{I_i \leq t} X^{(0)}_i + 1_{I_i > t} X^{(1)}_i} is a mixture of {X^{(0)}_i} and {X^{(1)}_i}, with {I_1,\dots,I_n} uniformly drawn from {[0,1]} independently of each other and of the {X^{(0)}_1,\dots,X^{(0)}_n, X^{(1)}_0,\dots,X^{(1)}_n}.

  • (iii) Discuss the relationship between the identities in parts (i), (ii) with the identities (3), (5).

(The identity in (i) is the starting point for the Lindeberg exchange method in probability theory, discussed for instance in this previous post. The identity in (ii) can also be used in the Lindeberg exchange method; the terms in the right-hand side are slightly more symmetric in the indices {1,\dots,n}, which can be a technical advantage in some applications; see this paper of Knowles and Yin for an instance of this.)

Exercise 3 If {F: {\bf R}^d \rightarrow {\bf R}^{d'}} is continuously {k} times differentiable, establish Taylor’s theorem with remainder

\displaystyle  F(x) = \sum_{j=0}^{k-1} \frac{1}{j!} (((x-x_0) \cdot \nabla)^j F)( x_0 )

\displaystyle + \int_0^1 \frac{(1-t)^{k-1}}{(k-1)!} (((x-x_0) \cdot \nabla)^k F)((1-t) x_0 + t x)\ dt.

If {\nabla^k F} is bounded, conclude that

\displaystyle  |F(x) - \sum_{j=0}^{k-1} \frac{1}{j!} (((x-x_0) \cdot \nabla)^j F)( x_0 )|

\displaystyle \leq \frac{|x-x_0|^k}{k!} \sup_{y \in {\bf R}^d} |\nabla^k F(y)|.

For real scalar functions {F: {\bf R}^d \rightarrow {\bf R}}, the average value of the continuous real-valued function {(x - x_0) \cdot \nabla F((1-t) x_0 + t x)} must be attained at some point {t} in the interval {[0,1]}. We thus conclude the mean-value theorem

\displaystyle  F(x) - F(x_0) = ((x - x_0) \cdot \nabla F)((1-t) x_0 + t x)

for some {t \in [0,1]} (that can depend on {x}, {x_0}, and {F}). This can for instance give a second proof of fact that continuously differentiable functions {F} with bounded derivative are Lipschitz continuous. However it is worth stressing that the mean-value theorem is only available for real scalar functions; it is false for instance for complex scalar functions. A basic counterexample is given by the function {e(x) := e^{2\pi i x}}; there is no {t \in [0,1]} for which {e(1) - e(0) = e'(t)}. On the other hand, as {e'} has magnitude {2\pi}, we still know from (4) that {e} is Lipschitz of constant {2\pi}, and when combined with (1) we obtain the basic bounds

\displaystyle  |e(x) - e(y)| \leq \min( 2, 2\pi |x-y| )

which are already very useful for many applications.

Exercise 4 Let {H_0, V} be {n \times n} matrices, and let {t} be a non-negative real.

  • (i) Establish the Duhamel formula

    \displaystyle  e^{t(H_0+V)} = e^{tH_0} + \int_0^t e^{(t-s) H_0} V e^{s (H_0+V)}\ ds

    \displaystyle  = e^{tH_0} + \int_0^t e^{(t-s) (H_0+V)} V e^{s H_0}\ ds

    where {e^A} denotes the matrix exponential of {A}. (Hint: Differentiate {e^{(t-s) H_0} e^{s (H_0+V)}} or {e^{(t-s) (H_0+V)} e^{s H_0}} in {s}.)

  • (ii) Establish the iterated Duhamel formula

    \displaystyle  e^{t(H_0+V)} = e^{tH_0} + \sum_{j=1}^k \int_{0 \leq t_1 \leq \dots \leq t_j \leq t}

    \displaystyle e^{(t-t_j) H_0} V e^{(t_j-t_{j-1}) H_0} V \dots e^{(t_2-t_1) H_0} V e^{t_1 H_0}\ dt_1 \dots dt_j

    \displaystyle  + \int_{0 \leq t_1 \leq \dots \leq t_{k+1} \leq t}

    \displaystyle  e^{(t-t_{k+1}) (H_0+V)} V e^{(t_{k+1}-t_k) H_0} V \dots e^{(t_2-t_1) H_0} V e^{t_1 H_0}\ dt_1 \dots dt_{k+1}

    for any {k \geq 0}.

  • (iii) Establish the infinitely iterated Duhamel formula

    \displaystyle  e^{t(H_0+V)} = e^{tH_0} + \sum_{j=1}^\infty \int_{0 \leq t_1 \leq \dots \leq t_j \leq t}

    \displaystyle e^{(t-t_j) H_0} V e^{(t_j-t_{j-1}) H_0} V \dots e^{(t_2-t_1) H_0} V e^{t_1 H_0}\ dt_1 \dots dt_j.

  • (iv) If {H(t)} is an {n \times n} matrix depending in a continuously differentiable fashion on {t}, establish the variation formula

    \displaystyle  \frac{d}{dt} e^{H(t)} = (F(\mathrm{ad}(H(t))) H'(t)) e^{H(t)}

    where {\mathrm{ad}(H)} is the adjoint representation {\mathrm{ad}(H)(V) = HV - VH} applied to {H}, and {F} is the function

    \displaystyle  F(z) := \int_0^1 e^{sz}\ ds

    (thus {F(z) = \frac{e^z-1}{z}} for non-zero {z}), with {F(\mathrm{ad}(H(t)))} defined using functional calculus.

We remark that further manipulation of (iv) of the above exercise using the fundamental theorem of calculus eventually leads to the Baker-Campbell-Hausdorff-Dynkin formula, as discussed in this previous blog post.

Exercise 5 Let {A, B} be positive definite {n \times n} matrices, and let {Y} be an {n \times n} matrix. Show that there is a unique solution {X} to the Sylvester equation

\displaystyle  AX + X B = Y

which is given by the formula

\displaystyle  X = \int_0^\infty e^{-tA} Y e^{-tB}\ dt.

In the above examples we had applied the fundamental theorem of calculus along linear curves {\gamma(t) = (1-t) x_0 + t x}. However, it is sometimes better to use other curves. For instance, the circular arc {\gamma(t) = \cos(\pi t/2) x_0 + \sin(\pi t/2) x} can be useful, particularly if {x_0} and {x} are “orthogonal” or “independent” in some sense; a good example of this is the proof by Maurey and Pisier of the gaussian concentration inequality, given in Theorem 8 of this previous blog post. In a similar vein, if one wishes to compare a scalar random variable {X} of mean zero and variance one with a Gaussian random variable {G} of mean zero and variance one, it can be useful to introduce the intermediate random variables {\gamma(t) := (1-t)^{1/2} X + t^{1/2} G} (where {X} and {G} are independent); note that these variables have mean zero and variance one, and after coupling them together appropriately they evolve by the Ornstein-Uhlenbeck process, which has many useful properties. For instance, one can use these ideas to establish monotonicity formulae for entropy; see e.g. this paper of Courtade for an example of this and further references. More generally, one can exploit curves {\gamma} that flow according to some geometrically natural ODE or PDE; several examples of this occur famously in Perelman’s proof of the Poincaré conjecture via Ricci flow, discussed for instance in this previous set of lecture notes.

In some cases, it is difficult to compute {F(x)-F(x_0)} or the derivative {\nabla F} directly, but one can instead proceed by implicit differentiation, or some variant thereof. Consider for instance the matrix inversion map {F(A) := A^{-1}} (defined on the open dense subset of {n \times n} matrices consisting of invertible matrices). If one wants to compute {F(B)-F(A)} for {B} close to {A}, one can write temporarily write {F(B) - F(A) = E}, thus

\displaystyle  B^{-1} - A^{-1} = E.

Multiplying both sides on the left by {B} to eliminate the {B^{-1}} term, and on the right by {A} to eliminate the {A^{-1}} term, one obtains

\displaystyle  A - B = B E A

and thus on reversing these steps we arrive at the basic identity

\displaystyle  B^{-1} - A^{-1} = B^{-1} (A - B) A^{-1}. \ \ \ \ \ (6)

For instance, if {H_0, V} are {n \times n} matrices, and we consider the resolvents

\displaystyle  R_0(z) := (H_0 - z I)^{-1}; \quad R_V(z) := (H_0 + V - zI)^{-1}

then we have the resolvent identity

\displaystyle  R_V(z) - R_0(z) = - R_V(z) V R_0(z) \ \ \ \ \ (7)

as long as {z} does not lie in the spectrum of {H_0} or {H_0+V} (for instance, if {H_0}, {V} are self-adjoint then one can take {z} to be any strictly complex number). One can iterate this identity to obtain

\displaystyle  R_V(z) = \sum_{j=0}^k (-R_0(z) V)^j R_0(z) + (-R_V(z) V) (-R_0(z) V)^k R_0(z)

for any natural number {k}; in particular, if {R_0(z) V} has operator norm less than one, one has the Neumann series

\displaystyle  R_V(z) = \sum_{j=0}^\infty (-R_0(z) V)^j R_0(z).

Similarly, if {A(t)} is a family of invertible matrices that depends in a continuously differentiable fashion on a time variable {t}, then by implicitly differentiating the identity

\displaystyle  A(t) A(t)^{-1} = I

in {t} using the product rule, we obtain

\displaystyle  (\frac{d}{dt} A(t)) A(t)^{-1} + A(t) \frac{d}{dt} A(t)^{-1} = 0

and hence

\displaystyle  \frac{d}{dt} A(t)^{-1} = - A(t)^{-1} (\frac{d}{dt} A(t)) A(t)^{-1}

(this identity may also be easily derived from (6)). One can then use the fundamental theorem of calculus to obtain variants of (6), for instance by using the curve {\gamma(t) = (1-t) A + tB} we arrive at

\displaystyle  B^{-1} - A^{-1} = \int_0^1 ((1-t)A + tB)^{-1} (A-B) ((1-t)A + tB)^{-1}\ dt

assuming that the curve stays entirely within the set of invertible matrices. While this identity may seem more complicated than (6), it is more symmetric, which conveys some advantages. For instance, using this identity it is easy to see that if {A, B} are positive definite with {A>B} in the sense of positive definite matrices (that is, {A-B} is positive definite), then {B^{-1} > A^{-1}}. (Try to prove this using (6) instead!)

Exercise 6 If {A} is an invertible {n \times n} matrix and {u, v} are {n \times 1} vectors, establish the Sherman-Morrison formula

\displaystyle  (A + t uv^T)^{-1} = A^{-1} - \frac{t}{1 + t v^T A^{-1} u} A^{-1} uv^T A^{-1}

whenever {t} is a scalar such that {1 + t v^T A^{-1} u} is non-zero. (See also this previous blog post for more discussion of these sorts of identities.)

One can use the Cauchy integral formula to extend these identities to other functions of matrices. For instance, if {F: {\bf C} \rightarrow {\bf C}} is an entire function, and {\gamma} is a counterclockwise contour that goes around the spectrum of both {H_0} and {H_0+V}, then we have

\displaystyle  F(H_0+V) = \frac{-1}{2\pi i} \int_\gamma F(z) R_V(z)\ dz

and similarly

\displaystyle  F(H_0) = \frac{-1}{2\pi i} \int_\gamma F(z) R_0(z)\ dz

and hence by (7) one has

\displaystyle  F(H_0+V) - F(H_0) = \frac{1}{2\pi i} \int_\gamma F(z) R_V(z) V F_0(z)\ dz;

similarly, if {H(t)} depends on {t} in a continuously differentiable fashion, then

\displaystyle  \frac{d}{dt} F(H(t)) = \frac{1}{2\pi i} \int_\gamma F(z) (H(t) - zI)^{-1} H'(t) (z) (H(t)-zI)^{-1}\ dz

as long as {\gamma} goes around the spectrum of {H(t)}.

Exercise 7 If {H(t)} is an {n \times n} matrix depending continuously differentiably on {t}, and {F: {\bf C} \rightarrow {\bf C}} is an entire function, establish the tracial chain rule

\displaystyle  \frac{d}{dt} \hbox{tr} F(H(t)) = \hbox{tr}(F'(H(t)) H'(t)).

In a similar vein, given that the logarithm function is the antiderivative of the reciprocal, one can express the matrix logarithm {\log A} of a positive definite matrix by the fundamental theorem of calculus identity

\displaystyle  \log A = \int_0^\infty (I + sI)^{-1} - (A + sI)^{-1}\ ds

(with the constant term {(I+tI)^{-1}} needed to prevent a logarithmic divergence in the integral). Differentiating, we see that if {A(t)} is a family of positive definite matrices depending continuously on {t}, that

\displaystyle  \frac{d}{dt} \log A(t) = \int_0^\infty (A(t) + sI)^{-1} A'(t) (A(t)+sI)^{-1}\ dt.

This can be used for instance to show that {\log} is a monotone increasing function, in the sense that {\log A> \log B} whenever {A > B > 0} in the sense of positive definite matrices. One can of course integrate this formula to obtain some formulae for the difference {\log A - \log B} of the logarithm of two positive definite matrices {A,B}.

To compare the square root {A^{1/2} - B^{1/2}} of two positive definite matrices {A,B} is trickier; there are multiple ways to proceed. One approach is to use contour integration as before (but one has to take some care to avoid branch cuts of the square root). Another to express the square root in terms of exponentials via the formula

\displaystyle  A^{1/2} = \frac{1}{\Gamma(-1/2)} \int_0^\infty (e^{-tA} - I) t^{-1/2} \frac{dt}{t}

where {\Gamma} is the gamma function; this formula can be verified by first diagonalising {A} to reduce to the scalar case and using the definition of the Gamma function. Then one has

\displaystyle  A^{1/2} - B^{1/2} = \frac{1}{\Gamma(-1/2)} \int_0^\infty (e^{-tA} - e^{-tB}) t^{-1/2} \frac{dt}{t}

and one can use some of the previous identities to control {e^{-tA} - e^{-tB}}. This is pretty messy though. A third way to proceed is via implicit differentiation. If for instance {A(t)} is a family of positive definite matrices depending continuously differentiably on {t}, we can differentiate the identity

\displaystyle  A(t)^{1/2} A(t)^{1/2} = A(t)

to obtain

\displaystyle  A(t)^{1/2} \frac{d}{dt} A(t)^{1/2} + (\frac{d}{dt} A(t)^{1/2}) A(t)^{1/2} = \frac{d}{dt} A(t).

This can for instance be solved using Exercise 5 to obtain

\displaystyle  \frac{d}{dt} A(t)^{1/2} = \int_0^\infty e^{-sA(t)^{1/2}} A'(t) e^{-sA(t)^{1/2}}\ ds

and this can in turn be integrated to obtain a formula for {A^{1/2} - B^{1/2}}. This is again a rather messy formula, but it does at least demonstrate that the square root is a monotone increasing function on positive definite matrices: {A > B > 0} implies {A^{1/2} > B^{1/2} > 0}.

Several of the above identities for matrices can be (carefully) extended to operators on Hilbert spaces provided that they are sufficiently well behaved (in particular, if they have a good functional calculus, and if various spectral hypotheses are obeyed). We will not attempt to do so here, however.

Suppose one has a bounded sequence {(a_n)_{n=1}^\infty = (a_1, a_2, \dots)} of real numbers. What kinds of limits can one form from this sequence?

Of course, we have the usual notion of limit {\lim_{n \rightarrow \infty} a_n}, which in this post I will refer to as the classical limit to distinguish from the other limits discussed in this post. The classical limit, if it exists, is the unique real number {L} such that for every {\varepsilon>0}, one has {|a_n-L| \leq \varepsilon} for all sufficiently large {n}. We say that a sequence is (classically) convergent if its classical limit exists. The classical limit obeys many useful limit laws when applied to classically convergent sequences. Firstly, it is linear: if {(a_n)_{n=1}^\infty} and {(b_n)_{n=1}^\infty} are classically convergent sequences, then {(a_n+b_n)_{n=1}^\infty} is also classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} (a_n + b_n) = (\lim_{n \rightarrow \infty} a_n) + (\lim_{n \rightarrow \infty} b_n) \ \ \ \ \ (1)

and similarly for any scalar {c}, {(ca_n)_{n=1}^\infty} is classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} (ca_n) = c \lim_{n \rightarrow \infty} a_n. \ \ \ \ \ (2)

It is also an algebra homomorphism: {(a_n b_n)_{n=1}^\infty} is also classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} (a_n b_n) = (\lim_{n \rightarrow \infty} a_n) (\lim_{n \rightarrow \infty} b_n). \ \ \ \ \ (3)

We also have shift invariance: if {(a_n)_{n=1}^\infty} is classically convergent, then so is {(a_{n+1})_{n=1}^\infty} with

\displaystyle  \lim_{n \rightarrow \infty} a_{n+1} = \lim_{n \rightarrow \infty} a_n \ \ \ \ \ (4)

and more generally in fact for any injection {\phi: {\bf N} \rightarrow {\bf N}}, {(a_{\phi(n)})_{n=1}^\infty} is classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} a_{\phi(n)} = \lim_{n \rightarrow \infty} a_n. \ \ \ \ \ (5)

The classical limit of a sequence is unchanged if one modifies any finite number of elements of the sequence. Finally, we have boundedness: for any classically convergent sequence {(a_n)_{n=1}^\infty}, one has

\displaystyle  \inf_n a_n \leq \lim_{n \rightarrow \infty} a_n \leq \sup_n a_n. \ \ \ \ \ (6)

One can in fact show without much difficulty that these laws uniquely determine the classical limit functional on convergent sequences.

One would like to extend the classical limit notion to more general bounded sequences; however, when doing so one must give up one or more of the desirable limit laws that were listed above. Consider for instance the sequence {a_n = (-1)^n}. On the one hand, one has {a_n^2 = 1} for all {n}, so if one wishes to retain the homomorphism property (3), any “limit” of this sequence {a_n} would have to necessarily square to {1}, that is to say it must equal {+1} or {-1}. On the other hand, if one wished to retain the shift invariance property (4) as well as the homogeneity property (2), any “limit” of this sequence would have to equal its own negation and thus be zero.

Nevertheless there are a number of useful generalisations and variants of the classical limit concept for non-convergent sequences that obey a significant portion of the above limit laws. For instance, we have the limit superior

\displaystyle  \limsup_{n \rightarrow \infty} a_n := \inf_N \sup_{n \geq N} a_n

and limit inferior

\displaystyle  \liminf_{n \rightarrow \infty} a_n := \sup_N \inf_{n \geq N} a_n

which are well-defined real numbers for any bounded sequence {(a_n)_{n=1}^\infty}; they agree with the classical limit when the sequence is convergent, but disagree otherwise. They enjoy the shift-invariance property (4), and the boundedness property (6), but do not in general obey the homomorphism property (3) or the linearity property (1); indeed, we only have the subadditivity property

\displaystyle  \limsup_{n \rightarrow \infty} (a_n + b_n) \leq (\limsup_{n \rightarrow \infty} a_n) + (\limsup_{n \rightarrow \infty} b_n)

for the limit superior, and the superadditivity property

\displaystyle  \liminf_{n \rightarrow \infty} (a_n + b_n) \geq (\liminf_{n \rightarrow \infty} a_n) + (\liminf_{n \rightarrow \infty} b_n)

for the limit inferior. The homogeneity property (2) is only obeyed by the limits superior and inferior for non-negative {c}; for negative {c}, one must have the limit inferior on one side of (2) and the limit superior on the other, thus for instance

\displaystyle  \limsup_{n \rightarrow \infty} (-a_n) = - \liminf_{n \rightarrow \infty} a_n.

The limit superior and limit inferior are examples of limit points of the sequence, which can for instance be defined as points that are limits of at least one subsequence of the original sequence. Indeed, the limit superior is always the largest limit point of the sequence, and the limit inferior is always the smallest limit point. However, limit points can be highly non-unique (indeed they are unique if and only if the sequence is classically convergent), and so it is difficult to sensibly interpret most of the usual limit laws in this setting, with the exception of the homogeneity property (2) and the boundedness property (6) that are easy to state for limit points.

Another notion of limit are the Césaro limits

\displaystyle  \mathrm{C}\!-\!\lim_{n \rightarrow \infty} a_n := \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N a_n;

if this limit exists, we say that the sequence is Césaro convergent. If the sequence {(a_n)_{n=1}^\infty} already has a classical limit, then it also has a Césaro limit that agrees with the classical limit; but there are additional sequences that have a Césaro limit but not a classical one. For instance, the non-classically convergent sequence {a_n= (-1)^n} discussed above is Césaro convergent, with a Césaro limit of {0}. However, there are still bounded sequences that do not have Césaro limit, such as {a_n := \sin( \log n )} (exercise!). The Césaro limit is linear, bounded, and shift invariant, but not an algebra homomorphism and also does not obey the rearrangement property (5).

Using the Hahn-Banach theorem, one can extend the classical limit functional to generalised limit functionals {\mathop{\widetilde \lim}_{n \rightarrow \infty} a_n}, defined to be bounded linear functionals from the space {\ell^\infty({\bf N})} of bounded real sequences to the real numbers {{\bf R}} that extend the classical limit functional (defined on the space {c_0({\bf N}) + {\bf R}} of convergent sequences) without any increase in the operator norm. (In some of my past writings I made the slight error of referring to these generalised limit functionals as Banach limits, though as discussed below, the latter actually refers to a subclass of generalised limit functionals.) It is not difficult to see that such generalised limit functionals will range between the limit inferior and limit superior. In fact, for any specific sequence {(a_n)_{n=1}^\infty} and any number {L} lying in the closed interval {[\liminf_{n \rightarrow \infty} a_n, \limsup_{n \rightarrow \infty} a_n]}, there exists at least one generalised limit functional {\mathop{\widetilde \lim}_{n \rightarrow \infty}} that takes the value {L} when applied to {a_n}; for instance, for any number {\theta} in {[-1,1]}, there exists a generalised limit functional that assigns that number {\theta} as the “limit” of the sequence {a_n = (-1)^n}. This claim can be seen by first designing such a limit functional on the vector space spanned by the convergent sequences and by {(a_n)_{n=1}^\infty}, and then appealing to the Hahn-Banach theorem to extend to all sequences. This observation also gives a necessary and sufficient criterion for a bounded sequence {(a_n)_{n=1}^\infty} to classically converge to a limit {L}, namely that all generalised limits of this sequence must equal {L}.

Because of the reliance on the Hahn-Banach theorem, the existence of generalised limits requires the axiom of choice (or some weakened version thereof); there are presumably models of set theory without the axiom of choice in which no generalised limits exist, but I do not know of an explicit reference for this.

Generalised limits can obey the shift-invariance property (4) or the algebra homomorphism property (2), but as the above analysis of the sequence {a_n = (-1)^n} shows, they cannot do both. Generalised limits that obey the shift-invariance property (4) are known as Banach limits; one can for instance construct them by applying the Hahn-Banach theorem to the Césaro limit functional; alternatively, if {\mathop{\widetilde \lim}} is any generalised limit, then the Césaro-type functional {(a_n)_{n=1}^\infty \mapsto \mathop{\widetilde \lim}_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N a_n} will be a Banach limit. The existence of Banach limits can be viewed as a demonstration of the amenability of the natural numbers (or integers); see this previous blog post for further discussion.

Generalised limits that obey the algebra homomorphism property (2) are known as ultrafilter limits. If one is given a generalised limit functional {p\!-\!\lim_{n \rightarrow \infty}} that obeys (2), then for any subset {A} of the natural numbers {{\bf N}}, the generalised limit {p\!-\!\lim_{n \rightarrow \infty} 1_A(n)} must equal its own square (since {1_A(n)^2 = 1_A(n)}) and is thus either {0} or {1}. If one defines {p \subset 2^{2^{\bf N}}} to be the collection of all subsets {A} of {{\bf N}} for which {p\!-\!\lim_{n \rightarrow \infty} 1_A(n) = 1}, one can verify that {p} obeys the axioms of a non-principal ultrafilter. Conversely, if {p} is a non-principal ultrafilter, one can define the associated generalised limit {p\!-\!\lim_{n \rightarrow \infty} a_n} of any bounded sequence {(a_n)_{n=1}^\infty} to be the unique real number {L} such that the sets {\{ n \in {\bf N}: |a_n - L| \leq \varepsilon \}} lie in {p} for all {\varepsilon>0}; one can check that this does indeed give a well-defined generalised limit that obeys (2). Non-principal ultrafilters can be constructed using Zorn’s lemma. In fact, they do not quite need the full strength of the axiom of choice; see the Wikipedia article on the ultrafilter lemma for examples.

We have previously noted that generalised limits of a sequence can converge to any point between the limit inferior and limit superior. The same is not true if one restricts to Banach limits or ultrafilter limits. For instance, by the arguments already given, the only possible Banach limit for the sequence {a_n = (-1)^n} is zero. Meanwhile, an ultrafilter limit must converge to a limit point of the original sequence, but conversely every limit point can be attained by at least one ultrafilter limit; we leave these assertions as an exercise to the interested reader. In particular, a bounded sequence converges classically to a limit {L} if and only if all ultrafilter limits converge to {L}.

There is no generalisation of the classical limit functional to any space that includes non-classically convergent sequences that obeys the subsequence property (5), since any non-classically-convergent sequence will have one subsequence that converges to the limit superior, and another subsequence that converges to the limit inferior, and one of these will have to violate (5) since the limit superior and limit inferior are distinct. So the above limit notions come close to the best generalisations of limit that one can use in practice.

We summarise the above discussion in the following table:

Limit Always defined Linear Shift-invariant Homomorphism Constructive
Classical No Yes Yes Yes Yes
Superior Yes No Yes No Yes
Inferior Yes No Yes No Yes
Césaro No Yes Yes No Yes
Generalised Yes Yes Depends Depends No
Banach Yes Yes Yes No No
Ultrafilter Yes Yes No Yes No

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