in the positive quadrant. You and your opponent take turns: the allowable moves are to go up, right, or both at once (i.e. add (0,1), add (1,0), or add (1,1).) First person to leave the quarter-circle wins. What does it look like if you color a starting point black for “first-player loss” and yellow for “first-player win”? It looks like this:
, which is a cartesian product on this category, and 
, which is a symmetric monoidal structure. There’s much more to linear logic than this (since there are other connectives), and maybe also less (since we may want our category to be a mere poset), but never mind. I want to focus on the weird business of having two kinds of ‘and’.
as usual in logic.
This other kind of ‘and’ is about resources: from one copy of a thing 
you can’t get two copies.
is “I have a sandwich”, 
is like “I have a sandwich and I have a sandwich”, while 
is like “I have two sandwiches”.
as “
implies 
”. The least upper bound of any subset of proposition is their ‘or’. Their greatest lower bound is their ‘and’. But we also have the commutative monoid operation, which we call 
This operation distributes over least upper bounds.
(not just for pairs of propositions, but arbitrary sets of them) and the 
. Indeed, my student Jade has analyzed this in detail and shown the resulting functor from the category of Petri nets to the category of commutative monoidal categories is a left adjoint:
Moreover, the resulting functor 
preserves products. As a result, every commutative monoidal category gives a commutative monoidal poset: that is, a commutative monoid object in the category of Posets.
whose elements are downsets of 
such that
gives a downset
, the 
is a large number, then the largest prime gap 
in ![{[1,x]}](https://s0.wp.com/latex.php?latex=%7B%5B1%2Cx%5D%7D&bg=ffffff&fg=000000&s=0 "{[1,x]}")
is of size 
. (
, where 
. Here we use the asymptotic notation 
for 
, 
for 
, 
for 
, and 
for 
.)
are fixed distinct integers, then the number of numbers ![{n \in [1,x]}](https://s0.wp.com/latex.php?latex=%7Bn+%5Cin+%5B1%2Cx%5D%7D&bg=ffffff&fg=000000&s=0 "{n \in [1,x]}")
with 
all prime is 
as 
, where the singular series 
is defined by the formula
with 
were asymptotically distributed according to an 
, in the sense that 
. Roughly speaking, the way this is established is by using the Hardy-Littlewood conjecture to control the mean values of 
for fixed 
, where 
ranges over the primes in 
is even and
; setting 
and taking means, one then gets upper and lower bounds for the probability that the interval 
is free of primes. The most difficult step is to control the mean values of the singular series 
ranges over 
-tuples in a fixed interval such as ![{[0, \lambda \log x]}](https://s0.wp.com/latex.php?latex=%7B%5B0%2C+%5Clambda+%5Clog+x%5D%7D&bg=ffffff&fg=000000&s=0 "{[0, \lambda \log x]}")
.
, one is then led to Cramér’s conjecture, since the right-hand side of 
when 
is significantly larger than 
, which was already known (see 
). An inspection of the argument shows that if one wished to extend 
, as well as the magnitudes of the shifts 
, were also allowed to grow with 
, and with shifts 
, with a power savings in the error term, then 
.
and shifts 
for all 
, with a power savings in the error term, then 
.
. Here we found it useful to switch to a probabilistic interpretation of this series. For technical reasons it is convenient to work with a truncated, unnormalised version
; it turns out that when studying prime tuples of size 
, the most convenient cutoff 
is the “
); by 
. One can interpret 
probabilistically as
is the randomly sifted set of integers formed by removing one residue class 
uniformly at random for each prime 
. The Hardy-Littlewood conjecture can be viewed as an assertion that the primes 
behave in some approximate statistical sense like the random sifted set 
, and one can prove the above theorem by using the Bonferroni inequalities both for the primes 
of the primes, in which each natural number 
is selected for membership in 
. This model matches the primes well in some respects; for instance, it almost surely obeys the “Riemann hypothesis”![\displaystyle | {\mathcal C} \cap [1,x] | = \int_2^x \frac{dt}{\log t} + O( x^{1/2+o(1)})](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7C+%7B%5Cmathcal+C%7D+%5Ccap+%5B1%2Cx%5D+%7C+%3D+%5Cint_2%5Ex+%5Cfrac%7Bdt%7D%7B%5Clog+t%7D+%2B+O%28+x%5E%7B1%2F2%2Bo%281%29%7D%29&bg=ffffff&fg=000000&s=0 "\displaystyle | {\mathcal C} \cap [1,x] | = \int_2^x \frac{dt}{\log t} + O( x^{1/2+o(1)})")
was almost surely 
. On the other hand, it does not obey the Hardy-Littlewood conjecture; more precisely, it obeys a simplified variant of that conjecture in which the singular series 
is absent.
to Cramér’s random model ![{[x,2x]}](https://s0.wp.com/latex.php?latex=%7B%5Bx%2C2x%5D%7D&bg=ffffff&fg=000000&s=0 "{[x,2x]}")
) all residue classes 
for 
for a certain threshold 
, and then places each surviving natural number 
in 
. One can verify that this model obeys the Hardy-Littlewood conjectures, and Granville showed that the largest gap 
in this model was almost surely 
, for instance, have shown that 
in this range. This is not necessarily a conflict, however; Granville’s analysis relies on inspecting gaps in an extremely sparse region of natural numbers that are more devoid of primes than average, and this region is not well explored by existing numerics. See 
. After some experimentation, we were able to produce a tractable random model 
for the primes which obeyed the Hardy-Littlewood conjectures with power savings, and which reproduced Granville’s gap prediction of 
for both models, though we expect the lower bound to be closer to the truth); to us, this strengthens the case for Granville’s version of Cramér’s conjecture. The model can be described as follows. We select one residue class 
, and as before we let 
be the sifted set of integers formed by deleting the residue classes 
for any fixed 
and for shifts 
. This is unfortunately a tiny bit weaker than what Theorem 
), although there is a variant of Theorem 
that involves the optimal bounds for the linear sieve (or interval sieve), in which one deletes one residue class modulo ![{[0,y]}](https://s0.wp.com/latex.php?latex=%7B%5B0%2Cy%5D%7D&bg=ffffff&fg=000000&s=0 "{[0,y]}")
for all primes 
as ![{n \in [0,y]}](https://s0.wp.com/latex.php?latex=%7Bn+%5Cin+%5B0%2Cy%5D%7D&bg=ffffff&fg=000000&s=0 "{n \in [0,y]}")
to primes 
conjecture), which I previously discussed in 
denote the positive integers (with 
the natural numbers), and let 
be the map defined by setting 
equal to 
when 
when 
be the minimal element of the Collatz orbit 
. Then we have
for all 
. 
. This was then improved 
for almost all 
, and extended later 
. In this paper we obtain the following further improvement (at the cost of weakening natural density to logarithmic density):
be any function with 
. Then we have 
for almost all 
for almost all 
for times 
of 
; the aforementioned results of Terras, Allouche, and Korec, for instance, are of this time. However, to get 
one needs something more like an “(almost) global-in-time” result, where the evolution remains under control for so long that the orbit has nearly reached the bounded state 
.
one picks at random an integer ![{[1,x^{\theta}]}](https://s0.wp.com/latex.php?latex=%7B%5B1%2Cx%5E%7B%5Ctheta%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{[1,x^{\theta}]}")
. Similarly, if one picks an integer 
at random from ![{[1,x^\theta]}](https://s0.wp.com/latex.php?latex=%7B%5B1%2Cx%5E%5Ctheta%5D%7D&bg=ffffff&fg=000000&s=0 "{[1,x^\theta]}")
, then in most cases, the orbit of ![{[1,x^{\theta^2}]}](https://s0.wp.com/latex.php?latex=%7B%5B1%2Cx%5E%7B%5Ctheta%5E2%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{[1,x^{\theta^2}]}")
. It is then tempting to concatenate the two statements and conclude that for most ![{N \in [1,x]}](https://s0.wp.com/latex.php?latex=%7BN+%5Cin+%5B1%2Cx%5D%7D&bg=ffffff&fg=000000&s=0 "{N \in [1,x]}")
reaches ![{M \in [1,x^\theta]}](https://s0.wp.com/latex.php?latex=%7BM+%5Cin+%5B1%2Cx%5E%5Ctheta%5D%7D&bg=ffffff&fg=000000&s=0 "{M \in [1,x^\theta]}")
that do not make it into 
, defined on the odd numbers 
by setting 
, where 
is the largest power of 
that divides 
at each iteration step, which makes the map better behaved when performing “
is never divisible by 
, the number of times 
that 
can be seen to have a 
with probability 
. Such a geometric random variable is twice as likely to be odd as to be even, which is what gives the above irregularity. There are similar irregularities modulo higher powers of 
will take the residue classes 
with probabilities 
will be distributed according to the law of a random variable 
on 
that we call a Syracuse random variable, and can be described explicitly as 
are iid copies of a geometric random variable of mean 
in a certain total variation sense. More precisely, in the paper we establish the estimate 
and any 
. This type of stabilisation is plausible from entropy heuristics – the tuple 
of geometric random variables that generates 
, which is significantly larger than the total entropy 
of the uniform distribution on 
before running into memory and CPU issues), but it turns out to be surprisingly delicate to make this precise.
vary over tuples in 
. Unfortunately, the process of reducing mod 
creates a lot of collisions (as must happen from the pigeonhole principle); however, by a simple “Lefschetz principle” type argument one can at least show that the reductions 
(as drawn using the geometric distribution) as long as 
is a bit smaller than 
(basically because the rational number appearing in 
with 
and 
(roughly speaking, when the argument is optimised, it shows that this random variable cannot concentrate in any subset of 
for some large absolute constant 
). To get from this to a stabilisation property 
that is not divisible by 
. The point here is that after conditioning on the 
to be fixed, the random variables 
remain independent (though the distribution of each 
depends on the value that we conditioned 
to), and so the above expression is a conditional sum of independent random variables. This lets one express the characeteristic function of 
, to be precise) of indices 
. (Actually, for technical reasons we have to also restrict to those 
for which 
, but let us ignore this detail here.) To put it another way, if we let 
denote the set of pairs 
for which ![\displaystyle \frac{\xi 3^{2j-2} (2^{-l+1} \mod 3^n)}{3^n} \in [-\varepsilon,\varepsilon] + {\bf Z},](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B%5Cxi+3%5E%7B2j-2%7D+%282%5E%7B-l%2B1%7D+%5Cmod+3%5En%29%7D%7B3%5En%7D+%5Cin+%5B-%5Cvarepsilon%2C%5Cvarepsilon%5D+%2B+%7B%5Cbf+Z%7D%2C+&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{\xi 3^{2j-2} (2^{-l+1} \mod 3^n)}{3^n} \in [-\varepsilon,\varepsilon] + {\bf Z},")
and 
? The question splits physicists into two camps. Pure quantum information (QI) theorists use uncertainty relations, whereas condensed-matter and high-energy physicists prefer out-of-time-ordered correlators. Let’s meet the camps in turn.![\Delta \hat{A} \, \Delta \hat{B} \geq \frac{1}{i \hbar} \langle [\hat{A}, \hat{B}] \rangle](https://s0.wp.com/latex.php?latex=%5CDelta+%5Chat%7BA%7D+%5C%2C+%5CDelta+%5Chat%7BB%7D+%5Cgeq+%5Cfrac%7B1%7D%7Bi+%5Chbar%7D+%5Clangle+%5B%5Chat%7BA%7D%2C+%5Chat%7BB%7D%5D+%5Crangle&bg=ffffff&fg=333333&s=0 "\Delta \hat{A} \, \Delta \hat{B} \geq \frac{1}{i \hbar} \langle [\hat{A}, \hat{B}] \rangle")
.
and measuring 
of yielding the outcome 
. Different trials will yield different 
. We define 
analogously. 
denotes Planck’s constant, a number that characterizes our universe as the electron’s mass does.![[\hat{A}, \hat{B}]](https://s0.wp.com/latex.php?latex=%5B%5Chat%7BA%7D%2C+%5Chat%7BB%7D%5D&bg=ffffff&fg=333333&s=0 "[\hat{A}, \hat{B}]")
denotes the observables’ commutator. The numbers that we use in daily life commute: 
. Quantum numbers, or operators, represent ![[\hat{A}, \hat{B}] = \hat{A} \hat{B} - \hat{B} \hat{A}](https://s0.wp.com/latex.php?latex=%5B%5Chat%7BA%7D%2C+%5Chat%7BB%7D%5D+%3D+%5Chat%7BA%7D+%5Chat%7BB%7D+-+%5Chat%7BB%7D+%5Chat%7BA%7D&bg=ffffff&fg=333333&s=0 "[\hat{A}, \hat{B}] = \hat{A} \hat{B} - \hat{B} \hat{A}")
represents how little 
, a 
quantifies the difficulty of predicting the outcome 
is defined analogously. 
is called the overlap. It quantifies your ability to predict what happens if you prepare your system with a well-defined 
represent the poke. Suppose that the system evolves chaotically for a time 
afterward, the qubits interacting. Information about the poke spreads through many-body entanglement, or scrambles.
of a few qubits far from the 
. The 
represents the time evolution.
, the operators agree, and the OTOC 
. At late times, the operators disagree loads, and the OTOC 
.
, and 
, then certain norms of the solution must also go to infinity. Here are some examples (not an exhaustive list) of such blowup criteria:
blows up at a finite time 
, then 
for an absolute constant 
.
, where 
.
, where 
is the 
for some absolute constant 
.
.
for any 
.
for any 
.
for any 
, had to become large. This result now has many proofs; for instance, many of the subsequent blowup criterion results imply the Escauriaza-Seregin-Sverak one as a special case, and there are also additional proofs by 
norm will go to infinity (along a subsequence of times).
with 
, then
and 
.
where there is a frequency 
for which one has the bound 
, where 
is a Littlewood-Paley projection to frequencies 
. (This can be compared with the upper bound of 
for the quantity on the left-hand side that follows from 
normalises the left-hand side of 
cannot get too large; in particular, we end up establishing a bound of the form
norm of 
to also show up at many other locations than 
, which eventually contradicts 
is somewhat involved (though simpler than the much more convoluted schemes I initially envisaged for this argument):
in spacetime, with 
also close to 
, 
, and 
). This can be viewed as a sort of contrapositive of a “local regularity theorem”, such as the ones established by Caffarelli, Kohn, and Nirenberg. A key point here is that the lower bound 
in the conclusion 
propagating backwards in time; by using estimates ultimately resulting from the dissipative term in the energy identity, one can extract such a sequence in which the 
increase geometrically with time, the 
are comparable (up to polynomial factors in 
, and one has 
. Using the “epochs of regularity” theory that ultimately dates back to Leray, and tweaking the 
slightly, one can also place the times 
(of length comparable to a small multiple of 
) in which the solution is quite regular (in particular, 
enjoy good 
bounds on 
).
norm of the vorticity 
near 
of this equation obey good 
bounds, allowing the machinery of Carleman estimates to come into play. Using a Carleman estimate that is used to establish unique continuation results for backwards heat equations, one can propagate this lower bound to also give lower 
bounds on the vorticity (and its first derivative) in annuli of the form 
for various radii 
, although the lower bounds decay at a gaussian rate with 
.
where (a slightly normalised form of) the entrosphy is small at time 
; using a version of the localised enstrophy estimates from ![{\{ (t,x) \in [t_n,t_0] \times {\bf R}^3: R_n \leq |x-x_n| \leq R'_n\}}](https://s0.wp.com/latex.php?latex=%7B%5C%7B+%28t%2Cx%29+%5Cin+%5Bt_n%2Ct_0%5D+%5Ctimes+%7B%5Cbf+R%7D%5E3%3A+R_n+%5Cleq+%7Cx-x_n%7C+%5Cleq+R%27_n%5C%7D%7D&bg=ffffff&fg=000000&s=0 "{\{ (t,x) \in [t_n,t_0] \times {\bf R}^3: R_n \leq |x-x_n| \leq R'_n\}}")
in which one has good estimates; in particular, one has 
, establishing a lower bound for the vorticity on the spatial annulus 
. By some basic Littlewood-Paley theory one can parlay this lower bound to a lower bound on the 
norm of the velocity 
be some domain (such as the real numbers). For any natural number 
denote the space of symmetric real-valued functions 
on 
, thus
. For instance, for any natural numbers 
, the 
. With the pointwise product operation, 
, in which case 
consists solely of the real constants.
of ![{[F^{(k)}]_{k \rightarrow p} \in L(\Omega^p)_{sym}}](https://s0.wp.com/latex.php?latex=%7B%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D+%5Cin+L%28%5COmega%5Ep%29_%7Bsym%7D%7D&bg=ffffff&fg=000000&s=0 "{[F^{(k)}]_{k \rightarrow p} \in L(\Omega^p)_{sym}}")
of ![\displaystyle [F^{(k)}]_{k \rightarrow p}(x_1,\dots,x_p) = \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq p} F^{(k)}(x_{i_1}, \dots, x_{i_k})](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D%28x_1%2C%5Cdots%2Cx_p%29+%3D+%5Csum_%7B1+%5Cleq+i_1+%3C+i_2+%3C+%5Cdots+%3C+i_k+%5Cleq+p%7D+F%5E%7B%28k%29%7D%28x_%7Bi_1%7D%2C+%5Cdots%2C+x_%7Bi_k%7D%29+&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)}]_{k \rightarrow p}(x_1,\dots,x_p) = \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq p} F^{(k)}(x_{i_1}, \dots, x_{i_k})")
ranges over all injections from 
to 
(the latter formula making it clearer that ![{[F^{(k)}]_{k \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[F^{(k)}]_{k \rightarrow p}}")
is symmetric). Thus for instance![\displaystyle [F^{(1)}(x_1)]_{1 \rightarrow p} = \sum_{i=1}^p F^{(1)}(x_i)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BF%5E%7B%281%29%7D%28x_1%29%5D_%7B1+%5Crightarrow+p%7D+%3D+%5Csum_%7Bi%3D1%7D%5Ep+F%5E%7B%281%29%7D%28x_i%29&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(1)}(x_1)]_{1 \rightarrow p} = \sum_{i=1}^p F^{(1)}(x_i)")
![\displaystyle [F^{(2)}(x_1,x_2)]_{2 \rightarrow p} = \sum_{1 \leq i < j \leq p} F^{(2)}(x_i,x_j)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BF%5E%7B%282%29%7D%28x_1%2Cx_2%29%5D_%7B2+%5Crightarrow+p%7D+%3D+%5Csum_%7B1+%5Cleq+i+%3C+j+%5Cleq+p%7D+F%5E%7B%282%29%7D%28x_i%2Cx_j%29&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(2)}(x_1,x_2)]_{2 \rightarrow p} = \sum_{1 \leq i < j \leq p} F^{(2)}(x_i,x_j)")
![\displaystyle e_k^{(p)}(x_1,\dots,x_p) = [x_1 \dots x_k]_{k \rightarrow p}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+e_k%5E%7B%28p%29%7D%28x_1%2C%5Cdots%2Cx_p%29+%3D+%5Bx_1+%5Cdots+x_k%5D_%7Bk+%5Crightarrow+p%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle e_k^{(p)}(x_1,\dots,x_p) = [x_1 \dots x_k]_{k \rightarrow p}.")
![\displaystyle [1]_{k \rightarrow p} = \binom{p}{k} = \frac{p(p-1)\dots(p-k+1)}{k!}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B1%5D_%7Bk+%5Crightarrow+p%7D+%3D+%5Cbinom%7Bp%7D%7Bk%7D+%3D+%5Cfrac%7Bp%28p-1%29%5Cdots%28p-k%2B1%29%7D%7Bk%21%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle [1]_{k \rightarrow p} = \binom{p}{k} = \frac{p(p-1)\dots(p-k+1)}{k!}.")
, and is equal to 
if 
. We also have the transitivity![\displaystyle [F^{(k)}]_{k \rightarrow p} = \frac{1}{\binom{p-k}{p-l}} [[F^{(k)}]_{k \rightarrow l}]_{l \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D+%3D+%5Cfrac%7B1%7D%7B%5Cbinom%7Bp-k%7D%7Bp-l%7D%7D+%5B%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+l%7D%5D_%7Bl+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)}]_{k \rightarrow p} = \frac{1}{\binom{p-k}{p-l}} [[F^{(k)}]_{k \rightarrow l}]_{l \rightarrow p}")
.![{[]_{k \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7Bk+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{k \rightarrow p}}")
is a linear map from 
to 
, one has ![\displaystyle [x_1]_{1 \rightarrow p} [x_1]_{1 \rightarrow p} = (\sum_{i=1}^p x_i)^2 \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Bx_1%5D_%7B1+%5Crightarrow+p%7D+%5Bx_1%5D_%7B1+%5Crightarrow+p%7D+%3D+%28%5Csum_%7Bi%3D1%7D%5Ep+x_i%29%5E2+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0 "\displaystyle [x_1]_{1 \rightarrow p} [x_1]_{1 \rightarrow p} = (\sum_{i=1}^p x_i)^2 \ \ \ \ \ (1)")
![\displaystyle = [x_1^2]_{1 \rightarrow p} + 2 [x_1 x_2]_{1 \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+%5Bx_1%5E2%5D_%7B1+%5Crightarrow+p%7D+%2B+2+%5Bx_1+x_2%5D_%7B1+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle = [x_1^2]_{1 \rightarrow p} + 2 [x_1 x_2]_{1 \rightarrow p}")
![\displaystyle \neq [x_1^2]_{1 \rightarrow p}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cneq+%5Bx_1%5E2%5D_%7B1+%5Crightarrow+p%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle \neq [x_1^2]_{1 \rightarrow p}.")
![\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow p} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow p} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BF%5E%7B%28k%29%7D%28x_1%2C%5Cdots%2Cx_k%29%5D_%7Bk+%5Crightarrow+p%7D+%5BG%5E%7B%28l%29%7D%28x_1%2C%5Cdots%2Cx_l%29%5D_%7Bl+%5Crightarrow+p%7D+%3D+%5Csum_%7Bk%2Cl+%5Cleq+m+%5Cleq+k%2Bl%7D+%5Cfrac%7B1%7D%7Bk%21+l%21%7D+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow p} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow p} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (2)")
![\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%5Csum_%7B%5Cpi%2C+%5Crho%7D+F%5E%7B%28k%29%7D%28x_%7B%5Cpi%281%29%7D%2C%5Cdots%2Cx_%7B%5Cpi%28k%29%7D%29+G%5E%7B%28l%29%7D%28x_%7B%5Crho%281%29%7D%2C%5Cdots%2Cx_%7B%5Crho%28l%29%7D%29%5D_%7Bm+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow p}")
and 
, where 
range over all injections 
, 
with 
. Combinatorially, the identity 
and 
with total image 
of cardinality 
, and furthermore there exist precisely 
triples 
of injections 
such that 
and 
.
, one has![\displaystyle [x_1 x_2]_{2 \rightarrow p} [x_1]_{1 \rightarrow p} = [\frac{1}{2! 1!}( 2 x_1^2 x_2 + 2 x_1 x_2^2 )]_{2 \rightarrow p} + [\frac{1}{2! 1!} 6 x_1 x_2 x_3]_{3 \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Bx_1+x_2%5D_%7B2+%5Crightarrow+p%7D+%5Bx_1%5D_%7B1+%5Crightarrow+p%7D+%3D+%5B%5Cfrac%7B1%7D%7B2%21+1%21%7D%28+2+x_1%5E2+x_2+%2B+2+x_1+x_2%5E2+%29%5D_%7B2+%5Crightarrow+p%7D+%2B+%5B%5Cfrac%7B1%7D%7B2%21+1%21%7D+6+x_1+x_2+x_3%5D_%7B3+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [x_1 x_2]_{2 \rightarrow p} [x_1]_{1 \rightarrow p} = [\frac{1}{2! 1!}( 2 x_1^2 x_2 + 2 x_1 x_2^2 )]_{2 \rightarrow p} + [\frac{1}{2! 1!} 6 x_1 x_2 x_3]_{3 \rightarrow p}")
![\displaystyle = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow p} + [3x_1 x_2 x_3]_{3 \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+%5Bx_1%5E2+x_2+%2B+x_1+x_2%5E2%5D_%7B2+%5Crightarrow+p%7D+%2B+%5B3x_1+x_2+x_3%5D_%7B3+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow p} + [3x_1 x_2 x_3]_{3 \rightarrow p}")
of formal sums![\displaystyle F^{(*)} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F%5E%7B%28%2A%29%7D+%3D+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D&bg=ffffff&fg=000000&s=0 "\displaystyle F^{(*)} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}")
is an element of ![{[]_{k \rightarrow *}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7Bk+%5Crightarrow+%2A%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{k \rightarrow *}}")
being formally linear, thus![\displaystyle [F^{(k)}]_{k \rightarrow *} + [G^{(k)}]_{k \rightarrow *} := [F^{(k)} + G^{(k)}]_{k \rightarrow *}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D+%2B+%5BG%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D+%3A%3D+%5BF%5E%7B%28k%29%7D+%2B+G%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)}]_{k \rightarrow *} + [G^{(k)}]_{k \rightarrow *} := [F^{(k)} + G^{(k)}]_{k \rightarrow *}")
![\displaystyle c [F^{(k)}]_{k \rightarrow *} := [cF^{(k)}]_{k \rightarrow *}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+c+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D+%3A%3D+%5BcF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D&bg=ffffff&fg=000000&s=0 "\displaystyle c [F^{(k)}]_{k \rightarrow *} := [cF^{(k)}]_{k \rightarrow *}")
and scalars 
, and with multiplication given by the analogue ![\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow *} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow *} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (3)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BF%5E%7B%28k%29%7D%28x_1%2C%5Cdots%2Cx_k%29%5D_%7Bk+%5Crightarrow+%2A%7D+%5BG%5E%7B%28l%29%7D%28x_1%2C%5Cdots%2Cx_l%29%5D_%7Bl+%5Crightarrow+%2A%7D+%3D+%5Csum_%7Bk%2Cl+%5Cleq+m+%5Cleq+k%2Bl%7D+%5Cfrac%7B1%7D%7Bk%21+l%21%7D+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow *} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow *} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (3)")
![\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow *}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%5Csum_%7B%5Cpi%2C+%5Crho%7D+F%5E%7B%28k%29%7D%28x_%7B%5Cpi%281%29%7D%2C%5Cdots%2Cx_%7B%5Cpi%28k%29%7D%29+G%5E%7B%28l%29%7D%28x_%7B%5Crho%281%29%7D%2C%5Cdots%2Cx_%7B%5Crho%28l%29%7D%29%5D_%7Bm+%5Crightarrow+%2A%7D+&bg=ffffff&fg=000000&s=0 "\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow *}")
![\displaystyle [x_1]_{1 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2]_{1 \rightarrow *} + 2 [x_1 x_2]_{2 \rightarrow *}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Bx_1%5D_%7B1+%5Crightarrow+%2A%7D+%5Bx_1%5D_%7B1+%5Crightarrow+%2A%7D+%3D+%5Bx_1%5E2%5D_%7B1+%5Crightarrow+%2A%7D+%2B+2+%5Bx_1+x_2%5D_%7B2+%5Crightarrow+%2A%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [x_1]_{1 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2]_{1 \rightarrow *} + 2 [x_1 x_2]_{2 \rightarrow *}")
![\displaystyle [x_1 x_2]_{2 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow *} + [3 x_1 x_2 x_3]_{3 \rightarrow *}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Bx_1+x_2%5D_%7B2+%5Crightarrow+%2A%7D+%5Bx_1%5D_%7B1+%5Crightarrow+%2A%7D+%3D+%5Bx_1%5E2+x_2+%2B+x_1+x_2%5E2%5D_%7B2+%5Crightarrow+%2A%7D+%2B+%5B3+x_1+x_2+x_3%5D_%7B3+%5Crightarrow+%2A%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle [x_1 x_2]_{2 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow *} + [3 x_1 x_2 x_3]_{3 \rightarrow *}.")
![{[1]_{0 \rightarrow *}}](https://s0.wp.com/latex.php?latex=%7B%5B1%5D_%7B0+%5Crightarrow+%2A%7D%7D&bg=ffffff&fg=000000&s=0 "{[1]_{0 \rightarrow *}}")
. (I do not know if this algebra has previously been studied in the literature; it is somewhat analogous to the abstract algebra of finite linear combinations of Schur polynomials, with multiplication given by a Littlewood-Richardson rule. )![{[]_{* \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7B%2A+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{* \rightarrow p}}")
from ![\displaystyle [\sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}]_{* \rightarrow p} := \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D%5D_%7B%2A+%5Crightarrow+p%7D+%3A%3D+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle [\sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}]_{* \rightarrow p} := \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}.")
![{[x_1]_{1 \rightarrow *}}](https://s0.wp.com/latex.php?latex=%7B%5Bx_1%5D_%7B1+%5Crightarrow+%2A%7D%7D&bg=ffffff&fg=000000&s=0 "{[x_1]_{1 \rightarrow *}}")
to ![{[x_1]_{1 \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5Bx_1%5D_%7B1+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[x_1]_{1 \rightarrow p}}")
and ![{[x_1 x_2]_{2 \rightarrow *}}](https://s0.wp.com/latex.php?latex=%7B%5Bx_1+x_2%5D_%7B2+%5Crightarrow+%2A%7D%7D&bg=ffffff&fg=000000&s=0 "{[x_1 x_2]_{2 \rightarrow *}}")
to ![{[x_1 x_2]_{2 \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5Bx_1+x_2%5D_%7B2+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[x_1 x_2]_{2 \rightarrow p}}")
. From ![{[]_{* \rightarrow p}: L(\Omega^*)_{sym} \rightarrow L(\Omega^p)_{sym}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7B%2A+%5Crightarrow+p%7D%3A+L%28%5COmega%5E%2A%29_%7Bsym%7D+%5Crightarrow+L%28%5COmega%5Ep%29_%7Bsym%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{* \rightarrow p}: L(\Omega^*)_{sym} \rightarrow L(\Omega^p)_{sym}}")
is an algebra homomorphism, even though the maps ![{[]_{k \rightarrow *}: L(\Omega^k)_{sym} \rightarrow L(\Omega^*)_{sym}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7Bk+%5Crightarrow+%2A%7D%3A+L%28%5COmega%5Ek%29_%7Bsym%7D+%5Crightarrow+L%28%5COmega%5E%2A%29_%7Bsym%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{k \rightarrow *}: L(\Omega^k)_{sym} \rightarrow L(\Omega^*)_{sym}}")
and ![{[]_{k \rightarrow p}: L(\Omega^k)_{sym} \rightarrow L(\Omega^p)_{sym}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7Bk+%5Crightarrow+p%7D%3A+L%28%5COmega%5Ek%29_%7Bsym%7D+%5Crightarrow+L%28%5COmega%5Ep%29_%7Bsym%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{k \rightarrow p}: L(\Omega^k)_{sym} \rightarrow L(\Omega^p)_{sym}}")
are not homomorphisms. By inspecting the 
component of 
on the space 
on every product space 
. To avoid degeneracies we will assume that the integral 
is strictly positive. Assuming suitable measurability and integrability hypotheses, a function 
can then be integrated against this product measure to produce a number
![\displaystyle \int_{{\bf R}^p} [x_1]_{1 \rightarrow p}\ d\mu^p = p (\int_{\bf R} x\ d\mu(x)) (\int_{\bf R} \mu)^{p-1} \ \ \ \ \ (4)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%7B%5Cbf+R%7D%5Ep%7D+%5Bx_1%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+%3D+p+%28%5Cint_%7B%5Cbf+R%7D+x%5C+d%5Cmu%28x%29%29+%28%5Cint_%7B%5Cbf+R%7D+%5Cmu%29%5E%7Bp-1%7D+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{{\bf R}^p} [x_1]_{1 \rightarrow p}\ d\mu^p = p (\int_{\bf R} x\ d\mu(x)) (\int_{\bf R} \mu)^{p-1} \ \ \ \ \ (4)")
![\displaystyle \int_{{\bf R}^p} [x_1 x_2]_{2 \rightarrow p}\ d\mu^p = \binom{p}{2} (\int_{{\bf R}^2} x_1 x_2\ d\mu(x_1) d\mu(x_2)) (\int_{\bf R} \mu)^{p-2}. \ \ \ \ \ (5)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%7B%5Cbf+R%7D%5Ep%7D+%5Bx_1+x_2%5D_%7B2+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+%3D+%5Cbinom%7Bp%7D%7B2%7D+%28%5Cint_%7B%7B%5Cbf+R%7D%5E2%7D+x_1+x_2%5C+d%5Cmu%28x_1%29+d%5Cmu%28x_2%29%29+%28%5Cint_%7B%5Cbf+R%7D+%5Cmu%29%5E%7Bp-2%7D.+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{{\bf R}^p} [x_1 x_2]_{2 \rightarrow p}\ d\mu^p = \binom{p}{2} (\int_{{\bf R}^2} x_1 x_2\ d\mu(x_1) d\mu(x_2)) (\int_{\bf R} \mu)^{p-2}. \ \ \ \ \ (5)")
![\displaystyle \int_{\Omega^p} [F^{(*)}]_{* \rightarrow p}\ d\mu^p = \sum_{k=0}^\infty \binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu^k) (\int_\Omega\ d\mu)^{p-k}. \ \ \ \ \ (6)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5COmega%5Ep%7D+%5BF%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+%3D+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cbinom%7Bp%7D%7Bk%7D+%28%5Cint_%7B%5COmega%5Ek%7D+F%5E%7B%28k%29%7D%5C+d%5Cmu%5Ek%29+%28%5Cint_%5COmega%5C+d%5Cmu%29%5E%7Bp-k%7D.+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [F^{(*)}]_{* \rightarrow p}\ d\mu^p = \sum_{k=0}^\infty \binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu^k) (\int_\Omega\ d\mu)^{p-k}. \ \ \ \ \ (6)")
as a polynomial 
in ![\displaystyle F^{(p)} = [F^{(*)}]_{* \rightarrow p} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F%5E%7B%28p%29%7D+%3D+%5BF%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%3D+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle F^{(p)} = [F^{(*)}]_{* \rightarrow p} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}")
and ![\displaystyle [F^{(*)}]_{* \rightarrow p} + [G^{(*)}]_{* \rightarrow p} := [F^{(*)} + G^{(*)}]_{* \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BF%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%2B+%5BG%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%3A%3D+%5BF%5E%7B%28%2A%29%7D+%2B+G%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(*)}]_{* \rightarrow p} + [G^{(*)}]_{* \rightarrow p} := [F^{(*)} + G^{(*)}]_{* \rightarrow p}")
![\displaystyle c [F^{(*)}]_{* \rightarrow p} := [c F^{(*)}]_{* \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+c+%5BF%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%3A%3D+%5Bc+F%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle c [F^{(*)}]_{* \rightarrow p} := [c F^{(*)}]_{* \rightarrow p}")
![\displaystyle [F^{(*)}]_{* \rightarrow p} [G^{(*)}]_{* \rightarrow p} := [F^{(*)} G^{(*)}]_{* \rightarrow p}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BF%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%5BG%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%3A%3D+%5BF%5E%7B%28%2A%29%7D+G%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(*)}]_{* \rightarrow p} [G^{(*)}]_{* \rightarrow p} := [F^{(*)} G^{(*)}]_{* \rightarrow p}.")
of 
no longer makes sense as a set, and the formal measure 
is a random variable; on any power 
, we let 
be the usual independent copies of 
on 
for 
. Then for any real or complex ![\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5COmega%5Ep%7D+%5BX_1%5D_%7B1+%5Crightarrow+p%7D%5E2%5C+d%5Cmu%5Ep+&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p")
![\displaystyle [X_1]_{1 \rightarrow p}^2 = [X_1^2]_{1 \rightarrow p} + 2[X_1 X_2]_{2 \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BX_1%5D_%7B1+%5Crightarrow+p%7D%5E2+%3D+%5BX_1%5E2%5D_%7B1+%5Crightarrow+p%7D+%2B+2%5BX_1+X_2%5D_%7B2+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [X_1]_{1 \rightarrow p}^2 = [X_1^2]_{1 \rightarrow p} + 2[X_1 X_2]_{2 \rightarrow p}")
to conclude that![\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = \binom{p}{1} \int_{\Omega} X^2\ d\mu + 2 \binom{p}{2} \int_{\Omega^2} X_1 X_2\ d\mu^2](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5COmega%5Ep%7D+%5BX_1%5D_%7B1+%5Crightarrow+p%7D%5E2%5C+d%5Cmu%5Ep+%3D+%5Cbinom%7Bp%7D%7B1%7D+%5Cint_%7B%5COmega%7D+X%5E2%5C+d%5Cmu+%2B+2+%5Cbinom%7Bp%7D%7B2%7D+%5Cint_%7B%5COmega%5E2%7D+X_1+X_2%5C+d%5Cmu%5E2&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = \binom{p}{1} \int_{\Omega} X^2\ d\mu + 2 \binom{p}{2} \int_{\Omega^2} X_1 X_2\ d\mu^2")
![\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = p \mathbf{Var}(X) + p^2 \mathbf{E}(X)^2. \ \ \ \ \ (7)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5COmega%5Ep%7D+%5BX_1%5D_%7B1+%5Crightarrow+p%7D%5E2%5C+d%5Cmu%5Ep+%3D+p+%5Cmathbf%7BVar%7D%28X%29+%2B+p%5E2+%5Cmathbf%7BE%7D%28X%29%5E2.+%5C+%5C+%5C+%5C+%5C+%287%29&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = p \mathbf{Var}(X) + p^2 \mathbf{E}(X)^2. \ \ \ \ \ (7)")
are jointly independent copies of 
has expectation 
and variance 
. One can thus view ![\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^4\ d\mu^p = p \mathbf{Var}(X^2) + p(3p-2) (\mathbf{E}(X^2))^2](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5COmega%5Ep%7D+%5BX_1%5D_%7B1+%5Crightarrow+p%7D%5E4%5C+d%5Cmu%5Ep+%3D+p+%5Cmathbf%7BVar%7D%28X%5E2%29+%2B+p%283p-2%29+%28%5Cmathbf%7BE%7D%28X%5E2%29%29%5E2&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^4\ d\mu^p = p \mathbf{Var}(X^2) + p(3p-2) (\mathbf{E}(X^2))^2")
. This is a shame, because otherwise one could hope to start endowing 
with some sort of commutative von Neumann algebra type structure (or the abstract probability structure discussed in 
is non-negative, then so is![\displaystyle \int_{\Omega^p} [F]_{1 \rightarrow p}\ d\mu^p = p (\int_\Omega F\ d\mu) (\int_\Omega\ d\mu)^{p-1}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5COmega%5Ep%7D+%5BF%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+%3D+p+%28%5Cint_%5COmega+F%5C+d%5Cmu%29+%28%5Cint_%5COmega%5C+d%5Cmu%29%5E%7Bp-1%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [F]_{1 \rightarrow p}\ d\mu^p = p (\int_\Omega F\ d\mu) (\int_\Omega\ d\mu)^{p-1}.")
![{[1]_{0 \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5B1%5D_%7B0+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[1]_{0 \rightarrow p}}")
against 
on 
. Let ![\displaystyle \frac{d}{dt} \int_{\Omega^p} F^{(p)}\ d\mu(t)^p = \int_{\Omega^p} F^{(p)} [a(t)]_{1 \rightarrow p}\ d\mu(t)^p \ \ \ \ \ (10)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdt%7D+%5Cint_%7B%5COmega%5Ep%7D+F%5E%7B%28p%29%7D%5C+d%5Cmu%28t%29%5Ep+%3D+%5Cint_%7B%5COmega%5Ep%7D+F%5E%7B%28p%29%7D+%5Ba%28t%29%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%29%5Ep+%5C+%5C+%5C+%5C+%5C+%2810%29&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{d}{dt} \int_{\Omega^p} F^{(p)}\ d\mu(t)^p = \int_{\Omega^p} F^{(p)} [a(t)]_{1 \rightarrow p}\ d\mu(t)^p \ \ \ \ \ (10)")
that are independent of 
to now depend on 
![\displaystyle = \int_{\Omega^p} \frac{d}{dt} F^{(p)}(t) + F^{(p)}(t) [a(t)]_{1 \rightarrow p}\ d\mu(t)^p,](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+%5Cint_%7B%5COmega%5Ep%7D+%5Cfrac%7Bd%7D%7Bdt%7D+F%5E%7B%28p%29%7D%28t%29+%2B+F%5E%7B%28p%29%7D%28t%29+%5Ba%28t%29%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%29%5Ep%2C+&bg=ffffff&fg=000000&s=0 "\displaystyle = \int_{\Omega^p} \frac{d}{dt} F^{(p)}(t) + F^{(p)}(t) [a(t)]_{1 \rightarrow p}\ d\mu(t)^p,")
![{F^{(p)} = [F^{(k)}]_{k \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7BF%5E%7B%28p%29%7D+%3D+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{F^{(p)} = [F^{(k)}]_{k \rightarrow p}}")
for some symmetric function ![\displaystyle \frac{d}{dt} [\binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu(t)^k) (\int_\Omega\ d\mu(t))^{p-k}]. \ \ \ \ \ (12)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdt%7D+%5B%5Cbinom%7Bp%7D%7Bk%7D+%28%5Cint_%7B%5COmega%5Ek%7D+F%5E%7B%28k%29%7D%5C+d%5Cmu%28t%29%5Ek%29+%28%5Cint_%5COmega%5C+d%5Cmu%28t%29%29%5E%7Bp-k%7D%5D.+%5C+%5C+%5C+%5C+%5C+%2812%29&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{d}{dt} [\binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu(t)^k) (\int_\Omega\ d\mu(t))^{p-k}]. \ \ \ \ \ (12)")
are the standard 
on 
, we can write this expression using ![\displaystyle \int_{\Omega^p} [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p}\ d\mu^p](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5COmega%5Ep%7D+%5BF%5E%7B%28k%29%7D+%28a_1+%2B+%5Cdots+%2B+a_k%29%5D_%7Bk+%5Crightarrow+p%7D+%2B+%5B+F%5E%7B%28k%29%7D+%5Cast+a+%5D_%7Bk%2B1+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p}\ d\mu^p")
is the symmetric function
![\displaystyle [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p} = [F^{(k)}]_{k \rightarrow p} [a]_{1 \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BF%5E%7B%28k%29%7D+%28a_1+%2B+%5Cdots+%2B+a_k%29%5D_%7Bk+%5Crightarrow+p%7D+%2B+%5B+F%5E%7B%28k%29%7D+%5Cast+a+%5D_%7Bk%2B1+%5Crightarrow+p%7D+%3D+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D+%5Ba%5D_%7B1+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p} = [F^{(k)}]_{k \rightarrow p} [a]_{1 \rightarrow p}")
can be interpreted as a classical integral. In this case, the identity ![\displaystyle \frac{d}{dt} d\mu(t)^p = [a(t)]_{1 \rightarrow p} d\mu(t)^p.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdt%7D+d%5Cmu%28t%29%5Ep+%3D+%5Ba%28t%29%5D_%7B1+%5Crightarrow+p%7D+d%5Cmu%28t%29%5Ep.&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{d}{dt} d\mu(t)^p = [a(t)]_{1 \rightarrow p} d\mu(t)^p.")
be a solution to the heat equation 
with initial data 
a rapidly decreasing finite non-negative Radon measure, or more explicitly
. Then for any 
, the quantity
for 
, constant for 
, and monotone non-increasing for 
.
is absolutely continuous, with Radon-Nikodym derivative a test function; this is more than enough regularity to justify the arguments below.
, let 
denote the Radon measure
can be written as a fractional dimensional integral
![\displaystyle \frac{d}{dt} Q_p(t) = -\frac{d}{2t} Q_p(t) + t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [\frac{|x-y|^2}{4t^2}]_{1 \rightarrow p} d\mu(t,x)^p\ dx \ \ \ \ \ (13)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdt%7D+Q_p%28t%29+%3D+-%5Cfrac%7Bd%7D%7B2t%7D+Q_p%28t%29+%2B+t%5E%7B-d%2F2%7D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5B%5Cfrac%7B%7Cx-y%7C%5E2%7D%7B4t%5E2%7D%5D_%7B1+%5Crightarrow+p%7D+d%5Cmu%28t%2Cx%29%5Ep%5C+dx+%5C+%5C+%5C+%5C+%5C+%2813%29&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{d}{dt} Q_p(t) = -\frac{d}{2t} Q_p(t) + t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [\frac{|x-y|^2}{4t^2}]_{1 \rightarrow p} d\mu(t,x)^p\ dx \ \ \ \ \ (13)")
of 
.
, we have
![\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}\ d\mu(t,x)^p\ dx = - \int_{({\bf R}^d)^p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x_j%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx+%3D+-+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5B%5Cfrac%7Bx_j-y_j%7D%7B2t%7D%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx.&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}\ d\mu(t,x)^p\ dx = - \int_{({\bf R}^d)^p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.")
![\displaystyle d Q_p(t) = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+d+Q_p%28t%29+%3D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+x_j+%5B%5Cfrac%7Bx_j-y_j%7D%7B2t%7D%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx&bg=ffffff&fg=000000&s=0 "\displaystyle d Q_p(t) = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx")
![\displaystyle = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+x_j+%5B%5Cfrac%7Bx_j-y_j%7D%7B2t%7D%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx&bg=ffffff&fg=000000&s=0 "\displaystyle = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx")
is any reasonable function depending only on ![\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}[F_j(y)]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x_j%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D%5BF_j%28y%29%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx+&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}[F_j(y)]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx")
![\displaystyle = - \int_{({\bf R}^d)^p} [F_j(y)]_{1 \rightarrow p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+-+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5BF_j%28y%29%5D_%7B1+%5Crightarrow+p%7D+%5B%5Cfrac%7Bx_j-y_j%7D%7B2t%7D%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx&bg=ffffff&fg=000000&s=0 "\displaystyle = - \int_{({\bf R}^d)^p} [F_j(y)]_{1 \rightarrow p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx")
![\displaystyle 0 = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [F_j(y) \frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+0+%3D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5BF_j%28y%29+%5Cfrac%7Bx_j-y_j%7D%7B2t%7D%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx.&bg=ffffff&fg=000000&s=0 "\displaystyle 0 = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [F_j(y) \frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.")
![\displaystyle \frac{d}{2t} Q_p(t) = t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} (x_j - [F_j(y)]_{1 \rightarrow p}) [\frac{(x_j-y_j)}{4t}]_{1 \rightarrow p} d\mu(t,x)^p\ dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7B2t%7D+Q_p%28t%29+%3D+t%5E%7B-d%2F2%7D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%28x_j+-+%5BF_j%28y%29%5D_%7B1+%5Crightarrow+p%7D%29+%5B%5Cfrac%7B%28x_j-y_j%29%7D%7B4t%7D%5D_%7B1+%5Crightarrow+p%7D+d%5Cmu%28t%2Cx%29%5Ep%5C+dx+&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{d}{2t} Q_p(t) = t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} (x_j - [F_j(y)]_{1 \rightarrow p}) [\frac{(x_j-y_j)}{4t}]_{1 \rightarrow p} d\mu(t,x)^p\ dx")
![\displaystyle [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p} - (x_j - [F_j(y)]_{1 \rightarrow p}) [x_j - y_j]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%28x_j-y_j%29%28x_j-y_j%29%5D_%7B1+%5Crightarrow+p%7D+-+%28x_j+-+%5BF_j%28y%29%5D_%7B1+%5Crightarrow+p%7D%29+%5Bx_j+-+y_j%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx.&bg=ffffff&fg=000000&s=0 "\displaystyle [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p} - (x_j - [F_j(y)]_{1 \rightarrow p}) [x_j - y_j]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.")
that then achieves the most cancellation turns out to be 
(this cancels the terms that are linear or quadratic in the ![{x_j - [F_j(y)]_{1 \rightarrow p} = \frac{1}{p} [x_j - y_j]_{1 \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7Bx_j+-+%5BF_j%28y%29%5D_%7B1+%5Crightarrow+p%7D+%3D+%5Cfrac%7B1%7D%7Bp%7D+%5Bx_j+-+y_j%5D_%7B1+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{x_j - [F_j(y)]_{1 \rightarrow p} = \frac{1}{p} [x_j - y_j]_{1 \rightarrow p}}")
. Repeating the calculations establishing ![\displaystyle \int_{({\bf R}^d)^p} [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p}\ d\mu^p = p \mathop{\bf E} |x-Y|^2 (\int_{{\bf R}^d}\ d\mu)^{p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5B%28x_j-y_j%29%28x_j-y_j%29%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+%3D+p+%5Cmathop%7B%5Cbf+E%7D+%7Cx-Y%7C%5E2+%28%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D%5C+d%5Cmu%29%5E%7Bp%7D&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{({\bf R}^d)^p} [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p}\ d\mu^p = p \mathop{\bf E} |x-Y|^2 (\int_{{\bf R}^d}\ d\mu)^{p}")
![\displaystyle \int_{({\bf R}^d)^p} [x_j-y_j]_{1 \rightarrow p} [x_j-y_j]_{1 \rightarrow p}\ d\mu^p](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5Bx_j-y_j%5D_%7B1+%5Crightarrow+p%7D+%5Bx_j-y_j%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{({\bf R}^d)^p} [x_j-y_j]_{1 \rightarrow p} [x_j-y_j]_{1 \rightarrow p}\ d\mu^p")
is the random variable drawn from 
. Since 
, one thus has 
, equal to zero for 
here as 
if desired, though it is not strictly necessary to do so for the proof.) 
, where 
are real or complex numbers. The functions 
to integrate here lie in the tensor product space 
, with the usual tensor product identifications and algebra operations. One can evaluate fractional dimensional integrals of such functions against “virtual product measures” 
, with 
a measure on 
, by the natural formula
to an element ![{[F_j^{(m)}]_{m \rightarrow p; j}}](https://s0.wp.com/latex.php?latex=%7B%5BF_j%5E%7B%28m%29%7D%5D_%7Bm+%5Crightarrow+p%3B+j%7D%7D&bg=ffffff&fg=000000&s=0 "{[F_j^{(m)}]_{m \rightarrow p; j}}")
of the space ![\displaystyle [F_j^{(m)}]_{m \rightarrow p; j} := 1^{\otimes j-1} \otimes [F_j^{(m)}]_{m \rightarrow p} \otimes 1^{\otimes k-j}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5BF_j%5E%7B%28m%29%7D%5D_%7Bm+%5Crightarrow+p%3B+j%7D+%3A%3D+1%5E%7B%5Cotimes+j-1%7D+%5Cotimes+%5BF_j%5E%7B%28m%29%7D%5D_%7Bm+%5Crightarrow+p%7D+%5Cotimes+1%5E%7B%5Cotimes+k-j%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle [F_j^{(m)}]_{m \rightarrow p; j} := 1^{\otimes j-1} \otimes [F_j^{(m)}]_{m \rightarrow p} \otimes 1^{\otimes k-j}.")
and 
are functions and 
are measures on 
respectively, then (assuming sufficient measurability and integrability) then the multiple fractional dimensional integral![\displaystyle \int_{\Omega_1^{p_1} \times \Omega_2^{p_2}} [F_1]_{1 \rightarrow p_1; 1} [F_2]_{1 \rightarrow p_2;2}\ d\mu_1^{p_1} d\mu^{p_2}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5COmega_1%5E%7Bp_1%7D+%5Ctimes+%5COmega_2%5E%7Bp_2%7D%7D+%5BF_1%5D_%7B1+%5Crightarrow+p_1%3B+1%7D+%5BF_2%5D_%7B1+%5Crightarrow+p_2%3B2%7D%5C+d%5Cmu_1%5E%7Bp_1%7D+d%5Cmu%5E%7Bp_2%7D&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega_1^{p_1} \times \Omega_2^{p_2}} [F_1]_{1 \rightarrow p_1; 1} [F_2]_{1 \rightarrow p_2;2}\ d\mu_1^{p_1} d\mu^{p_2}")
are natural numbers, one can view the “virtual” integrand ![{[F_1]_{1 \rightarrow p_1; 1} [F_2]_{1 \rightarrow p_2;2}}](https://s0.wp.com/latex.php?latex=%7B%5BF_1%5D_%7B1+%5Crightarrow+p_1%3B+1%7D+%5BF_2%5D_%7B1+%5Crightarrow+p_2%3B2%7D%7D&bg=ffffff&fg=000000&s=0 "{[F_1]_{1 \rightarrow p_1; 1} [F_2]_{1 \rightarrow p_2;2}}")
here as an actual function on 
, namely
varies with respect to a time parameter 
is a time-varying function in 
, a vector-valued function 
, and a matrix-valued function 
taking values in real symmetric positive semi-definite 
matrices. Let 
and 
, we define the modified measures
, and also replacing 
.)
would obey some heat-type partial differential equation, and the situation is now very analogous to Proposition 
to vary slightly in the 
do not directly obey any PDE.
, we have a matrix-valued function 
defined by![\displaystyle A_* := \sum_{j=1}^k [A_j]_{1 \rightarrow p_j; j}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+A_%2A+%3A%3D+%5Csum_%7Bj%3D1%7D%5Ek+%5BA_j%5D_%7B1+%5Crightarrow+p_j%3B+j%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle A_* := \sum_{j=1}^k [A_j]_{1 \rightarrow p_j; j}.")
is then a scalar element of the algebra 
and let 
-matrix valued function
of a 
, where 
as
case. As such, the fractional dimensional calculus (based heavily on derivative identities such as 
, in the sense that 
(where we use for instance the operator norm to measure the size of matrices, and we allow implied constants in the 
notation to depend on 
, and the 
). Then we can write 
for some bounded matrix 
, and then we can write![\displaystyle A_* = \sum_{j=1}^k [A_j^0]_{1 \rightarrow p_j;j} + \varepsilon [B_j]_{1 \rightarrow p_j;j} = \sum_{j=1}^k p_j A_j^0 + \varepsilon \sum_{j=1}^k [B_j^0]_{1 \rightarrow p_j;j}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+A_%2A+%3D+%5Csum_%7Bj%3D1%7D%5Ek+%5BA_j%5E0%5D_%7B1+%5Crightarrow+p_j%3Bj%7D+%2B+%5Cvarepsilon+%5BB_j%5D_%7B1+%5Crightarrow+p_j%3Bj%7D+%3D+%5Csum_%7Bj%3D1%7D%5Ek+p_j+A_j%5E0+%2B+%5Cvarepsilon+%5Csum_%7Bj%3D1%7D%5Ek+%5BB_j%5E0%5D_%7B1+%5Crightarrow+p_j%3Bj%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle A_* = \sum_{j=1}^k [A_j^0]_{1 \rightarrow p_j;j} + \varepsilon [B_j]_{1 \rightarrow p_j;j} = \sum_{j=1}^k p_j A_j^0 + \varepsilon \sum_{j=1}^k [B_j^0]_{1 \rightarrow p_j;j}.")
and the coefficients of the matrix 
are some polynomial combination of the coefficients of ![{[C_j^0]_{1 \rightarrow p_j;j}}](https://s0.wp.com/latex.php?latex=%7B%5BC_j%5E0%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%7D&bg=ffffff&fg=000000&s=0 "{[C_j^0]_{1 \rightarrow p_j;j}}")
, with all coefficients in this polynomial of bounded size. As a consequence, and on expanding out all the fractional dimensional integrals, one obtains a formula of the form
is strictly positive definite and 
is small enough, this quantity 
as 
![\displaystyle \mathrm{det}(A_*) \sum_{j=1}^k [\langle A_j(x-y_j),(x-y_j) \rangle]_{1 \rightarrow p_j;j}\ d\mu(t,x)^{\vec p}\ dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cmathrm%7Bdet%7D%28A_%2A%29+%5Csum_%7Bj%3D1%7D%5Ek+%5B%5Clangle+A_j%28x-y_j%29%2C%28x-y_j%29+%5Crangle%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%5C+d%5Cmu%28t%2Cx%29%5E%7B%5Cvec+p%7D%5C+dx&bg=ffffff&fg=000000&s=0 "\displaystyle \mathrm{det}(A_*) \sum_{j=1}^k [\langle A_j(x-y_j),(x-y_j) \rangle]_{1 \rightarrow p_j;j}\ d\mu(t,x)^{\vec p}\ dx")
as the coordinate for the copy of 
.
are the standard basis for ![\displaystyle d Q_{\vec p}(t) = \frac{2\pi}{t^{\frac{d}{2}+1}} \int_{{\bf R}^d} \int_{\Omega^{\vec p}}\mathrm{det}(A_*) \sum_{j=1}^k x_i [\langle A_j(x-y_j), e_i\rangle]_{1 \rightarrow p_j;j}\ d\mu(t,x)^{\vec p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+d+Q_%7B%5Cvec+p%7D%28t%29+%3D+%5Cfrac%7B2%5Cpi%7D%7Bt%5E%7B%5Cfrac%7Bd%7D%7B2%7D%2B1%7D%7D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%5COmega%5E%7B%5Cvec+p%7D%7D%5Cmathrm%7Bdet%7D%28A_%2A%29+%5Csum_%7Bj%3D1%7D%5Ek+x_i+%5B%5Clangle+A_j%28x-y_j%29%2C+e_i%5Crangle%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%5C+d%5Cmu%28t%2Cx%29%5E%7B%5Cvec+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle d Q_{\vec p}(t) = \frac{2\pi}{t^{\frac{d}{2}+1}} \int_{{\bf R}^d} \int_{\Omega^{\vec p}}\mathrm{det}(A_*) \sum_{j=1}^k x_i [\langle A_j(x-y_j), e_i\rangle]_{1 \rightarrow p_j;j}\ d\mu(t,x)^{\vec p}")
. Also, similarly to before, we suppose we have an element 
of ![\displaystyle \int_{{\bf R}^d} \int_{\Omega^{\vec p}} \sum_{j=1}^k F_i [\langle A_j(x-y_j), e_i\rangle]_{1 \rightarrow p_j;j}\ d\mu(t,x)^{\vec p} = 0](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%5COmega%5E%7B%5Cvec+p%7D%7D+%5Csum_%7Bj%3D1%7D%5Ek+F_i+%5B%5Clangle+A_j%28x-y_j%29%2C+e_i%5Crangle%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%5C+d%5Cmu%28t%2Cx%29%5E%7B%5Cvec+p%7D+%3D+0&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{{\bf R}^d} \int_{\Omega^{\vec p}} \sum_{j=1}^k F_i [\langle A_j(x-y_j), e_i\rangle]_{1 \rightarrow p_j;j}\ d\mu(t,x)^{\vec p} = 0")
,![\displaystyle \int_{{\bf R}^d} \int_{\Omega^{\vec p}} \sum_{j=1}^k \langle [A_j(x-y_j)]_{1 \rightarrow p_j;j}, F \rangle\ d\mu(t,x)^{\vec p} = 0.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%5COmega%5E%7B%5Cvec+p%7D%7D+%5Csum_%7Bj%3D1%7D%5Ek+%5Clangle+%5BA_j%28x-y_j%29%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%2C+F+%5Crangle%5C+d%5Cmu%28t%2Cx%29%5E%7B%5Cvec+p%7D+%3D+0.&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{{\bf R}^d} \int_{\Omega^{\vec p}} \sum_{j=1}^k \langle [A_j(x-y_j)]_{1 \rightarrow p_j;j}, F \rangle\ d\mu(t,x)^{\vec p} = 0.")
is the element of ![\displaystyle G := \mathrm{det}(A_*) \sum_{j=1}^k [\langle A_j(x-y_j),(x-y_j) \rangle]_{1 \rightarrow p_j;j} \ \ \ \ \ (22)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+G+%3A%3D+%5Cmathrm%7Bdet%7D%28A_%2A%29+%5Csum_%7Bj%3D1%7D%5Ek+%5B%5Clangle+A_j%28x-y_j%29%2C%28x-y_j%29+%5Crangle%5D_%7B1+%5Crightarrow+p_j%3Bj%7D+%5C+%5C+%5C+%5C+%5C+%2822%29&bg=ffffff&fg=000000&s=0 "\displaystyle G := \mathrm{det}(A_*) \sum_{j=1}^k [\langle A_j(x-y_j),(x-y_j) \rangle]_{1 \rightarrow p_j;j} \ \ \ \ \ (22)")
![\displaystyle - \langle \sum_{j=1}^k [A_j(x-y_j)]_{1 \rightarrow p_j;j}, \mathrm{det}(A_*) x - F \rangle.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%5Clangle+%5Csum_%7Bj%3D1%7D%5Ek+%5BA_j%28x-y_j%29%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%2C+%5Cmathrm%7Bdet%7D%28A_%2A%29+x+-+F+%5Crangle.&bg=ffffff&fg=000000&s=0 "\displaystyle - \langle \sum_{j=1}^k [A_j(x-y_j)]_{1 \rightarrow p_j;j}, \mathrm{det}(A_*) x - F \rangle.")
that are quadratic in ![\displaystyle \langle x, A_* F - \mathrm{det}(A_*) \sum_{j=1}^k [A_j y_j]_{1 \rightarrow p_j; j} \rangle.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Clangle+x%2C+A_%2A+F+-+%5Cmathrm%7Bdet%7D%28A_%2A%29+%5Csum_%7Bj%3D1%7D%5Ek+%5BA_j+y_j%5D_%7B1+%5Crightarrow+p_j%3B+j%7D+%5Crangle.&bg=ffffff&fg=000000&s=0 "\displaystyle \langle x, A_* F - \mathrm{det}(A_*) \sum_{j=1}^k [A_j y_j]_{1 \rightarrow p_j; j} \rangle.")
![\displaystyle F = A_*^{-1}\mathrm{det}(A_*) \sum_{j=1}^k [A_j y_j]_{1 \rightarrow p_j; j} .](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F+%3D+A_%2A%5E%7B-1%7D%5Cmathrm%7Bdet%7D%28A_%2A%29+%5Csum_%7Bj%3D1%7D%5Ek+%5BA_j+y_j%5D_%7B1+%5Crightarrow+p_j%3B+j%7D+.&bg=ffffff&fg=000000&s=0 "\displaystyle F = A_*^{-1}\mathrm{det}(A_*) \sum_{j=1}^k [A_j y_j]_{1 \rightarrow p_j; j} .")
does not necessarily exist. However, because of the weight factor 
, which is such that 
where 
is the identity matrix. We therefore set ![\displaystyle F := \mathrm{adj}(A_*) \sum_{j=1}^k [A_j y_j]_{1 \rightarrow p_j; j}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F+%3A%3D+%5Cmathrm%7Badj%7D%28A_%2A%29+%5Csum_%7Bj%3D1%7D%5Ek+%5BA_j+y_j%5D_%7B1+%5Crightarrow+p_j%3B+j%7D&bg=ffffff&fg=000000&s=0 "\displaystyle F := \mathrm{adj}(A_*) \sum_{j=1}^k [A_j y_j]_{1 \rightarrow p_j; j}")
![\displaystyle G = \mathrm{det}(A_*) \sum_{j=1}^k [\langle A_j(\overline{y}-y_j),(\overline{y}-y_j) \rangle]_{1 \rightarrow p_j;j}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+G+%3D+%5Cmathrm%7Bdet%7D%28A_%2A%29+%5Csum_%7Bj%3D1%7D%5Ek+%5B%5Clangle+A_j%28%5Coverline%7By%7D-y_j%29%2C%28%5Coverline%7By%7D-y_j%29+%5Crangle%5D_%7B1+%5Crightarrow+p_j%3Bj%7D&bg=ffffff&fg=000000&s=0 "\displaystyle G = \mathrm{det}(A_*) \sum_{j=1}^k [\langle A_j(\overline{y}-y_j),(\overline{y}-y_j) \rangle]_{1 \rightarrow p_j;j}")
![\displaystyle - \langle \sum_{j=1}^k [A_j(\overline{y}-y_j)]_{1 \rightarrow p_j;j}, \mathrm{det}(A_*) \overline{y} - F \rangle](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%5Clangle+%5Csum_%7Bj%3D1%7D%5Ek+%5BA_j%28%5Coverline%7By%7D-y_j%29%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%2C+%5Cmathrm%7Bdet%7D%28A_%2A%29+%5Coverline%7By%7D+-+F+%5Crangle&bg=ffffff&fg=000000&s=0 "\displaystyle - \langle \sum_{j=1}^k [A_j(\overline{y}-y_j)]_{1 \rightarrow p_j;j}, \mathrm{det}(A_*) \overline{y} - F \rangle")
is an arbitrary element of ![\displaystyle G = \mathrm{det}(A_*) \sum_{j=1}^k [\langle A_j(\overline{y}-y_j),(\overline{y}-y_j) \rangle]_{1 \rightarrow p_j;j}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+G+%3D+%5Cmathrm%7Bdet%7D%28A_%2A%29+%5Csum_%7Bj%3D1%7D%5Ek+%5B%5Clangle+A_j%28%5Coverline%7By%7D-y_j%29%2C%28%5Coverline%7By%7D-y_j%29+%5Crangle%5D_%7B1+%5Crightarrow+p_j%3Bj%7D+&bg=ffffff&fg=000000&s=0 "\displaystyle G = \mathrm{det}(A_*) \sum_{j=1}^k [\langle A_j(\overline{y}-y_j),(\overline{y}-y_j) \rangle]_{1 \rightarrow p_j;j}")
![\displaystyle - \langle \sum_{j=1}^k [A_j(\overline{y}-y_j)]_{1 \rightarrow p_j;j}, \mathrm{adj}(A_*) \sum_{j'=1}^k [A_j(\overline{y} - y_j)]_{1 \rightarrow p_j; j} \rangle.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%5Clangle+%5Csum_%7Bj%3D1%7D%5Ek+%5BA_j%28%5Coverline%7By%7D-y_j%29%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%2C+%5Cmathrm%7Badj%7D%28A_%2A%29+%5Csum_%7Bj%27%3D1%7D%5Ek+%5BA_j%28%5Coverline%7By%7D+-+y_j%29%5D_%7B1+%5Crightarrow+p_j%3B+j%7D+%5Crangle.&bg=ffffff&fg=000000&s=0 "\displaystyle - \langle \sum_{j=1}^k [A_j(\overline{y}-y_j)]_{1 \rightarrow p_j;j}, \mathrm{adj}(A_*) \sum_{j'=1}^k [A_j(\overline{y} - y_j)]_{1 \rightarrow p_j; j} \rangle.")
![\displaystyle G = \mathrm{det}(A_*) \sum_{j=1}^k [\|w_j\|^2]_{1 \rightarrow p_j;j}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+G+%3D+%5Cmathrm%7Bdet%7D%28A_%2A%29+%5Csum_%7Bj%3D1%7D%5Ek+%5B%5C%7Cw_j%5C%7C%5E2%5D_%7B1+%5Crightarrow+p_j%3Bj%7D+&bg=ffffff&fg=000000&s=0 "\displaystyle G = \mathrm{det}(A_*) \sum_{j=1}^k [\|w_j\|^2]_{1 \rightarrow p_j;j}")
![\displaystyle - \langle \sum_{j=1}^k [B_j w_j]_{1 \rightarrow p_j;j}, \mathrm{adj}(A_*) \sum_{j'=1}^k [B_j w_j]_{1 \rightarrow p_j; j} \rangle.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%5Clangle+%5Csum_%7Bj%3D1%7D%5Ek+%5BB_j+w_j%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%2C+%5Cmathrm%7Badj%7D%28A_%2A%29+%5Csum_%7Bj%27%3D1%7D%5Ek+%5BB_j+w_j%5D_%7B1+%5Crightarrow+p_j%3B+j%7D+%5Crangle.&bg=ffffff&fg=000000&s=0 "\displaystyle - \langle \sum_{j=1}^k [B_j w_j]_{1 \rightarrow p_j;j}, \mathrm{adj}(A_*) \sum_{j'=1}^k [B_j w_j]_{1 \rightarrow p_j; j} \rangle.")
, thus we can write 
. Inserting this into the previous expression (and expanding out 
![\displaystyle G^0 = \mathrm{det}(A^0_*) (\sum_{j=1}^k [\|w_j\|^2]_{1 \rightarrow p_j;j}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+G%5E0+%3D+%5Cmathrm%7Bdet%7D%28A%5E0_%2A%29+%28%5Csum_%7Bj%3D1%7D%5Ek+%5B%5C%7Cw_j%5C%7C%5E2%5D_%7B1+%5Crightarrow+p_j%3Bj%7D&bg=ffffff&fg=000000&s=0 "\displaystyle G^0 = \mathrm{det}(A^0_*) (\sum_{j=1}^k [\|w_j\|^2]_{1 \rightarrow p_j;j}")
![\displaystyle - \langle \sum_{j=1}^k B^0_j [w_j]_{1 \rightarrow p_j;j}, (A^0_*)^{-1} \sum_{j'=1}^k B^0_j [w_j]_{1 \rightarrow p_j; j} \rangle )](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%5Clangle+%5Csum_%7Bj%3D1%7D%5Ek+B%5E0_j+%5Bw_j%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%2C+%28A%5E0_%2A%29%5E%7B-1%7D+%5Csum_%7Bj%27%3D1%7D%5Ek+B%5E0_j+%5Bw_j%5D_%7B1+%5Crightarrow+p_j%3B+j%7D+%5Crangle+%29&bg=ffffff&fg=000000&s=0 "\displaystyle - \langle \sum_{j=1}^k B^0_j [w_j]_{1 \rightarrow p_j;j}, (A^0_*)^{-1} \sum_{j'=1}^k B^0_j [w_j]_{1 \rightarrow p_j; j} \rangle )")
is some polynomial combination of the 
(or more precisely, of the quantities ![{[D_j]_{1 \rightarrow p_j;j}}](https://s0.wp.com/latex.php?latex=%7B%5BD_j%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%7D&bg=ffffff&fg=000000&s=0 "{[D_j]_{1 \rightarrow p_j;j}}")
, ![{[w_j]_{1 \rightarrow p_j;j}}](https://s0.wp.com/latex.php?latex=%7B%5Bw_j%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%7D&bg=ffffff&fg=000000&s=0 "{[w_j]_{1 \rightarrow p_j;j}}")
, ![{[D_j w_j]_{1 \rightarrow p_j;j}}](https://s0.wp.com/latex.php?latex=%7B%5BD_j+w_j%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%7D&bg=ffffff&fg=000000&s=0 "{[D_j w_j]_{1 \rightarrow p_j;j}}")
, ![{[\|w_j\|^2]_{1 \rightarrow p_j;j}}](https://s0.wp.com/latex.php?latex=%7B%5B%5C%7Cw_j%5C%7C%5E2%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%7D&bg=ffffff&fg=000000&s=0 "{[\|w_j\|^2]_{1 \rightarrow p_j;j}}")
) that is quadratic in the 
![\displaystyle + O( \varepsilon t^{-\frac{d}{2}+2} \int_{{\bf R}^d} \int_{\Omega^{\vec p}} \sum_{j=1}^k [\| w_j \|^2]_{1 \rightarrow p_j;j}d\mu(t,x)^{\vec p}\ dx).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+O%28+%5Cvarepsilon+t%5E%7B-%5Cfrac%7Bd%7D%7B2%7D%2B2%7D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%5COmega%5E%7B%5Cvec+p%7D%7D+%5Csum_%7Bj%3D1%7D%5Ek+%5B%5C%7C+w_j+%5C%7C%5E2%5D_%7B1+%5Crightarrow+p_j%3Bj%7Dd%5Cmu%28t%2Cx%29%5E%7B%5Cvec+p%7D%5C+dx%29.&bg=ffffff&fg=000000&s=0 "\displaystyle + O( \varepsilon t^{-\frac{d}{2}+2} \int_{{\bf R}^d} \int_{\Omega^{\vec p}} \sum_{j=1}^k [\| w_j \|^2]_{1 \rightarrow p_j;j}d\mu(t,x)^{\vec p}\ dx).")
. We let
; for each 
this is just a vector in 
, leading to the identities![\displaystyle [\|w_j\|^2]_{1 \rightarrow p_j;j} = p_j \|\overline{w_j}\|^2 + 2 \langle \overline{w_j}, [w_j - \overline{w_j}]_{1 \rightarrow p_j;j}\rangle](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%5C%7Cw_j%5C%7C%5E2%5D_%7B1+%5Crightarrow+p_j%3Bj%7D+%3D+p_j+%5C%7C%5Coverline%7Bw_j%7D%5C%7C%5E2+%2B+2+%5Clangle+%5Coverline%7Bw_j%7D%2C+%5Bw_j+-+%5Coverline%7Bw_j%7D%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%5Crangle+&bg=ffffff&fg=000000&s=0 "\displaystyle [\|w_j\|^2]_{1 \rightarrow p_j;j} = p_j \|\overline{w_j}\|^2 + 2 \langle \overline{w_j}, [w_j - \overline{w_j}]_{1 \rightarrow p_j;j}\rangle")
![\displaystyle + [\| w_j - \overline{w_j} \|^2]_{1 \rightarrow p_j;j}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+%5B%5C%7C+w_j+-+%5Coverline%7Bw_j%7D+%5C%7C%5E2%5D_%7B1+%5Crightarrow+p_j%3Bj%7D&bg=ffffff&fg=000000&s=0 "\displaystyle + [\| w_j - \overline{w_j} \|^2]_{1 \rightarrow p_j;j}")
![\displaystyle \sum_{j=1}^k B^0_j [w_j]_{1 \rightarrow p_j;j} = \sum_{j=1}^k p_j B^0_j \overline{w_j} + \sum_{j=1}^k B^0_j [(w_j - \overline{w_j})]_{1 \rightarrow p_j;j}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bj%3D1%7D%5Ek+B%5E0_j+%5Bw_j%5D_%7B1+%5Crightarrow+p_j%3Bj%7D+%3D+%5Csum_%7Bj%3D1%7D%5Ek+p_j+B%5E0_j+%5Coverline%7Bw_j%7D+%2B+%5Csum_%7Bj%3D1%7D%5Ek+B%5E0_j+%5B%28w_j+-+%5Coverline%7Bw_j%7D%29%5D_%7B1+%5Crightarrow+p_j%3Bj%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle \sum_{j=1}^k B^0_j [w_j]_{1 \rightarrow p_j;j} = \sum_{j=1}^k p_j B^0_j \overline{w_j} + \sum_{j=1}^k B^0_j [(w_j - \overline{w_j})]_{1 \rightarrow p_j;j}.")
is problematic, but we can eliminate it as follows. By construction one has (supressing the dependence on ![\displaystyle \sum_{j=1}^k p_j B^0_j \overline{w_j} \int_{\Omega^p} \ d\mu^{\vec p} = \int_{\Omega^p} \sum_{j=1}^k B^0_j [w_j]_{1 \rightarrow p_j; j} \ d\mu^{\vec p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bj%3D1%7D%5Ek+p_j+B%5E0_j+%5Coverline%7Bw_j%7D+%5Cint_%7B%5COmega%5Ep%7D+%5C+d%5Cmu%5E%7B%5Cvec+p%7D+%3D+%5Cint_%7B%5COmega%5Ep%7D+%5Csum_%7Bj%3D1%7D%5Ek+B%5E0_j+%5Bw_j%5D_%7B1+%5Crightarrow+p_j%3B+j%7D+%5C+d%5Cmu%5E%7B%5Cvec+p%7D+&bg=ffffff&fg=000000&s=0 "\displaystyle \sum_{j=1}^k p_j B^0_j \overline{w_j} \int_{\Omega^p} \ d\mu^{\vec p} = \int_{\Omega^p} \sum_{j=1}^k B^0_j [w_j]_{1 \rightarrow p_j; j} \ d\mu^{\vec p}")
![\displaystyle = \int_{\Omega^p}\sum_{j=1}^k B^0_j [B_j(\overline{y} - y_j)]_{1 \rightarrow p_j; j} \ d\mu^{\vec p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+%5Cint_%7B%5COmega%5Ep%7D%5Csum_%7Bj%3D1%7D%5Ek+B%5E0_j+%5BB_j%28%5Coverline%7By%7D+-+y_j%29%5D_%7B1+%5Crightarrow+p_j%3B+j%7D+%5C+d%5Cmu%5E%7B%5Cvec+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle = \int_{\Omega^p}\sum_{j=1}^k B^0_j [B_j(\overline{y} - y_j)]_{1 \rightarrow p_j; j} \ d\mu^{\vec p}")
![\displaystyle = \int_{\Omega^p}\sum_{j=1}^k B^0_j [B_j]_{1 \rightarrow p_j;j} \overline{y} - \int_{\Omega^p}\sum_{j=1}^k B^0_j [B_j y_j]_{1 \rightarrow p_j; j} \ d\mu^{\vec p}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+%5Cint_%7B%5COmega%5Ep%7D%5Csum_%7Bj%3D1%7D%5Ek+B%5E0_j+%5BB_j%5D_%7B1+%5Crightarrow+p_j%3Bj%7D+%5Coverline%7By%7D+-+%5Cint_%7B%5COmega%5Ep%7D%5Csum_%7Bj%3D1%7D%5Ek+B%5E0_j+%5BB_j+y_j%5D_%7B1+%5Crightarrow+p_j%3B+j%7D+%5C+d%5Cmu%5E%7B%5Cvec+p%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle = \int_{\Omega^p}\sum_{j=1}^k B^0_j [B_j]_{1 \rightarrow p_j;j} \overline{y} - \int_{\Omega^p}\sum_{j=1}^k B^0_j [B_j y_j]_{1 \rightarrow p_j; j} \ d\mu^{\vec p}.")
![\displaystyle \int_{\Omega^p}\sum_{j=1}^k B^0_j [B_j]_{1 \rightarrow p_j;j}\ d\mu^{\vec p} = (\sum_{j=1}^k p B_0^j B_0^j + O(\varepsilon)) \int_{\Omega^p}\ d\mu^{\vec p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%5COmega%5Ep%7D%5Csum_%7Bj%3D1%7D%5Ek+B%5E0_j+%5BB_j%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%5C+d%5Cmu%5E%7B%5Cvec+p%7D+%3D+%28%5Csum_%7Bj%3D1%7D%5Ek+p+B_0%5Ej+B_0%5Ej+%2B+O%28%5Cvarepsilon%29%29+%5Cint_%7B%5COmega%5Ep%7D%5C+d%5Cmu%5E%7B%5Cvec+p%7D+&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p}\sum_{j=1}^k B^0_j [B_j]_{1 \rightarrow p_j;j}\ d\mu^{\vec p} = (\sum_{j=1}^k p B_0^j B_0^j + O(\varepsilon)) \int_{\Omega^p}\ d\mu^{\vec p}")
is positive definite and 
so that the expression 
vanishes. Making this choice, we then have![\displaystyle \sum_{j=1}^k [B^0_j w_j]_{1 \rightarrow p_j;j} = \sum_{j=1}^k [B^0_j (w_j - \overline{w_j})]_{1 \rightarrow p_j;j}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bj%3D1%7D%5Ek+%5BB%5E0_j+w_j%5D_%7B1+%5Crightarrow+p_j%3Bj%7D+%3D+%5Csum_%7Bj%3D1%7D%5Ek+%5BB%5E0_j+%28w_j+-+%5Coverline%7Bw_j%7D%29%5D_%7B1+%5Crightarrow+p_j%3Bj%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle \sum_{j=1}^k [B^0_j w_j]_{1 \rightarrow p_j;j} = \sum_{j=1}^k [B^0_j (w_j - \overline{w_j})]_{1 \rightarrow p_j;j}.")
![\displaystyle \langle \overline{w_j}, [w_j - \overline{w_j}]_{1 \rightarrow p_j;j} \rangle](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Clangle+%5Coverline%7Bw_j%7D%2C+%5Bw_j+-+%5Coverline%7Bw_j%7D%5D_%7B1+%5Crightarrow+p_j%3Bj%7D+%5Crangle&bg=ffffff&fg=000000&s=0 "\displaystyle \langle \overline{w_j}, [w_j - \overline{w_j}]_{1 \rightarrow p_j;j} \rangle")
![\displaystyle \langle [w_j - \overline{w_j}]_{1 \rightarrow p_j;j}, M [w_{j'} - \overline{w_{j'}}]_{1 \rightarrow p_{j'};j'}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Clangle+%5Bw_j+-+%5Coverline%7Bw_j%7D%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%2C+M+%5Bw_%7Bj%27%7D+-+%5Coverline%7Bw_%7Bj%27%7D%7D%5D_%7B1+%5Crightarrow+p_%7Bj%27%7D%3Bj%27%7D+&bg=ffffff&fg=000000&s=0 "\displaystyle \langle [w_j - \overline{w_j}]_{1 \rightarrow p_j;j}, M [w_{j'} - \overline{w_{j'}}]_{1 \rightarrow p_{j'};j'}")
and arbitrary constant matrices 
vanishes. As a consequence, we can now simplify the integral 
![\displaystyle + \langle [w_j - \overline{w_j}]_{1 \rightarrow p_j;j}, (1 - B_0^j (A^0_*)^{-1} B_0^{j}) [w_j - \overline{w_j}]_{1 \rightarrow p_j; j} \rangle\ d\mu^{\vec p}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+%5Clangle+%5Bw_j+-+%5Coverline%7Bw_j%7D%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%2C+%281+-+B_0%5Ej+%28A%5E0_%2A%29%5E%7B-1%7D+B_0%5E%7Bj%7D%29+%5Bw_j+-+%5Coverline%7Bw_j%7D%5D_%7B1+%5Crightarrow+p_j%3B+j%7D+%5Crangle%5C+d%5Cmu%5E%7B%5Cvec+p%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle + \langle [w_j - \overline{w_j}]_{1 \rightarrow p_j;j}, (1 - B_0^j (A^0_*)^{-1} B_0^{j}) [w_j - \overline{w_j}]_{1 \rightarrow p_j; j} \rangle\ d\mu^{\vec p}.")
![\displaystyle \langle [w_j - \overline{w_j}]_{1 \rightarrow p_j;j}, (1 - B_0^j (A^0_*)^{-1} B_0^{j}) [w_j - \overline{w_j}]_{1 \rightarrow p_j; j} \rangle](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Clangle+%5Bw_j+-+%5Coverline%7Bw_j%7D%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%2C+%281+-+B_0%5Ej+%28A%5E0_%2A%29%5E%7B-1%7D+B_0%5E%7Bj%7D%29+%5Bw_j+-+%5Coverline%7Bw_j%7D%5D_%7B1+%5Crightarrow+p_j%3B+j%7D+%5Crangle&bg=ffffff&fg=000000&s=0 "\displaystyle \langle [w_j - \overline{w_j}]_{1 \rightarrow p_j;j}, (1 - B_0^j (A^0_*)^{-1} B_0^{j}) [w_j - \overline{w_j}]_{1 \rightarrow p_j; j} \rangle")
![\displaystyle [\langle w_j - \overline{w_j}, (1 - B_0^j (A^0_*)^{-1} B_0^{j}) (w_j - \overline{w_j}) \rangle ]_{1 \rightarrow p_j; j}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%5Clangle+w_j+-+%5Coverline%7Bw_j%7D%2C+%281+-+B_0%5Ej+%28A%5E0_%2A%29%5E%7B-1%7D+B_0%5E%7Bj%7D%29+%28w_j+-+%5Coverline%7Bw_j%7D%29+%5Crangle+%5D_%7B1+%5Crightarrow+p_j%3B+j%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [\langle w_j - \overline{w_j}, (1 - B_0^j (A^0_*)^{-1} B_0^{j}) (w_j - \overline{w_j}) \rangle ]_{1 \rightarrow p_j; j}")
![\displaystyle 2[\langle w_j(\omega_1) - \overline{w_j}, (1 - B_0^j (A^0_*)^{-1} B_0^{j}) (w_j(\omega_2) - \overline{w_j}) \rangle ]_{2 \rightarrow p_j; j}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+2%5B%5Clangle+w_j%28%5Comega_1%29+-+%5Coverline%7Bw_j%7D%2C+%281+-+B_0%5Ej+%28A%5E0_%2A%29%5E%7B-1%7D+B_0%5E%7Bj%7D%29+%28w_j%28%5Comega_2%29+-+%5Coverline%7Bw_j%7D%29+%5Crangle+%5D_%7B2+%5Crightarrow+p_j%3B+j%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle 2[\langle w_j(\omega_1) - \overline{w_j}, (1 - B_0^j (A^0_*)^{-1} B_0^{j}) (w_j(\omega_2) - \overline{w_j}) \rangle ]_{2 \rightarrow p_j; j}.")
. Thus we have simplified ![\displaystyle + [\langle w_j - \overline{w_j}, (1 - B_0^j (A^0_*)^{-1} B_0^{j}) (w_j - \overline{w_j}) \rangle ]_{1 \rightarrow p_j; j} \ d\mu^{\vec p}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+%5B%5Clangle+w_j+-+%5Coverline%7Bw_j%7D%2C+%281+-+B_0%5Ej+%28A%5E0_%2A%29%5E%7B-1%7D+B_0%5E%7Bj%7D%29+%28w_j+-+%5Coverline%7Bw_j%7D%29+%5Crangle+%5D_%7B1+%5Crightarrow+p_j%3B+j%7D+%5C+d%5Cmu%5E%7B%5Cvec+p%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle + [\langle w_j - \overline{w_j}, (1 - B_0^j (A^0_*)^{-1} B_0^{j}) (w_j - \overline{w_j}) \rangle ]_{1 \rightarrow p_j; j} \ d\mu^{\vec p}.")
, where the ordering is in the sense of positive definite matrices. Then we have the pointwise bound
![\displaystyle \sum_{j=1}^k [\|\overline{w_j}\|^2 + \| w_j - \overline{w_j} \|^2 - O(\varepsilon) \|w_j\|^2 ]_{1 \rightarrow p_j;j}\ d\mu^{\vec p}\ dx.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bj%3D1%7D%5Ek+%5B%5C%7C%5Coverline%7Bw_j%7D%5C%7C%5E2+%2B+%5C%7C+w_j+-+%5Coverline%7Bw_j%7D+%5C%7C%5E2+-+O%28%5Cvarepsilon%29+%5C%7Cw_j%5C%7C%5E2+%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%5C+d%5Cmu%5E%7B%5Cvec+p%7D%5C+dx.&bg=ffffff&fg=000000&s=0 "\displaystyle \sum_{j=1}^k [\|\overline{w_j}\|^2 + \| w_j - \overline{w_j} \|^2 - O(\varepsilon) \|w_j\|^2 ]_{1 \rightarrow p_j;j}\ d\mu^{\vec p}\ dx.")
![{[]_{1 \rightarrow p_j;j}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7B1+%5Crightarrow+p_j%3Bj%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{1 \rightarrow p_j;j}}")
is non-negative, and we conclude the monotonicity
, let 
be positive semi-definite real symmetric 
be such that 
. Then for any positive measure spaces 
and any functions 
on 
for a sufficiently small 
) we have
Dirac masses, and each 
, then the key condition 
, and one arrives at the multilinear Kakeya inequality
are infinite tubes in 
, for a sufficiently small absolute constant Guest post by Nadezda Chernyavskaya - ETH Zurich
Bavarian Evening at the Lindau Nobel Laureate Meetings - Picture Credit: Christian Flemming
This continues to be a very very busy time, but here are a few interesting things to read:
- "Voodoo fusion" - an article from the APS Forum on Physics and Society that pretty much excoriates all of the fusion-related startup efforts, basically saying that they're about as legitimate as Theranos. Definitely an interesting read.
- https://arxiv.org/abs/1910.06389 - This is Kenneth Libbrecht's tour de force monograph on snow crystals. Talk about an example of emergence: From the modest water molecule comes, when many of them get together, the remarkable, intricate structure of snowflakes, with highly complex six-fold rotational symmetry.
- Somehow this tenure announcement just showed up in my newsfeed. It's very funny, and the titles and abstracts of his talks are in a similar vein. Perhaps I need to start writing up my physics talks this way.
- Beware of unintended consequences of ranking metrics.
- https://arxiv.org/abs/1910.05813 - insert snarky comment here about how this is a mathematical model for hiring/awards/funding/publication.