 |
| Fig. 1 from the linked paper |
The authors of the paper in question wanted to make fibers from starch, which is nicely biocompatible for medical applications. So, starting from wheat flour and formic acid, they worked out viable parameters and were able to electrospin fibers of wheat starch (including some gluten - sorry, for those of you with gluten intolerances) into nanofibers 300-400 nm in diameter. The underlying material is amorphous (so, no appreciable starch crystallization). The authors had fun with this and called the result "nanopasta", but it may actually be useful for certain applications.
together with an L-labeling 
with one vertex and with one edge for each element of L. An L-labeling of a graph 
the same as a map of graphs
that is, the category of graphs equipped with a map to 
is a ‘discrete fibration’, and this is why we can pull back L-labelings along maps of graphs.
the fiber of 
as a set.
which in turn correspond to maps from 
This makes the fiber contravariantly functorial. That is, any map of graphs
along any map of graphs 
to get an L-labeling of 
to 
to 
is the same as a map of graphs 
and each map of graphs 
there exists a unique map of L-labeled graphs 
such that 
As a further consequence, 
is the result of pulling back 
where:
is a ‘discrete opfibration’, and this lets us ‘push forward’ L-labelings along maps of graphs.
the functor 
there exists a unique map of L-graphs 
Again, this is easy to see from the definitions.
we must have
to 
defined as above. We say this map lets us push forward any L-labeling of 
to get an L-labeling of 
is the same as a map of graphs 
is not representable, while the contravariant functor 
was. The reason is that the procedures this time use the fact that our set of labels is a commutative monoid. It’s possible I’m missing a trick or two.
directly affects 
because the first edge is labeled 
but 
because the second edge is labeled 
To figure out whether 
and 
using ‘and’. We get 
so 
means that one quantity directly affects another in a positive way:
means one quantity directly affects another in a negative way, while an edge labeled with 
means no direct effect.
multiplication is defined in the obvious way, and addition is also obvious except when we add 
What’s the sum of a positive number and a negative number? It could be positive, negative or zero! So we make addition multivalued, write it as 
and say
and start adding, subtracting and multiplying to our heart’s content, what do we get? We get a bunch of subsets of 
and 
equals this:
or indeterminate is short for the set 
In this language we say the sum of 
This nicely captures the idea that when you add up two quantities, and all you know is that one is positive and the other is negative, the sign of the result is indeterminate.
we can extend its addition and multiplication to the collection 
of nonempty subsets of 
as follows. To multiply two nonempty subsets 
we define
of two elements of a hyperring is already a subset of that hyperring.
as its additive unit and 
as its multiplicative unit.
be the smallest subrig of 
where 
This subrig may be ‘good enough for causal loop diagrams’.
in this special case. It’s already used for polarities in causal loop diagrams. I’m just trying to understand it better—for example, I’ve never heard anyone say it’s a rig, or take advantage of the fact that it’s a rig. We need some mathematicians to get into this game, make the math nice and shiny, and make it more powerful!
is the von Mangoldt function, 
was some suitable approximant to that function, 
was a nilsequence, and ![{[x,x+H]}](https://s0.wp.com/latex.php?latex=%7B%5Bx%2Cx%2BH%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
was a reasonably short interval in the sense that 
for some 
and some small 
. In that paper, we were able to obtain non-trivial estimates for 
as small as 
, and for some other functions such as divisor functions 
for small values of 
, we could lower 
, 
, 
of 
, that is to say the study of primes in short intervals, until recently the best value of 
, although this was very recently improved to 
and the supremum is over all nilsequences of a certain Lipschitz constant on a fixed nilmanifold 
. This generalizes local Fourier uniformity estimates of the form 
(where, as per the Möbius pseudorandomness conjecture, the approximant 
should be taken to be zero, at least in the absence of a Siegel zero). This is because if one could get estimates of this form for any 
that grows sufficiently slowly in 
(in particular 
), this would imply the (logarithmically averaged) Chowla conjecture, as I 
, that is to say prime number theorems in almost all short intervals, until very recently the best value of 
, recently lowered to 
by Guth and Maynard (and can be lowered all the way to zero on the 
in the error terms; for divisor functions, one can even get power savings in this regime. 
, recovering 
, and divisor correlation estimates on arithmetic progressions for almost all 
.
, and some suitable functions 
supported on medium scales. So the bad case is when for most 
that depends on 
on 
(and I am being extremely vague as to what the relation “
” means here). By a higher order (and quantitatively stronger) version of Walsh’s contagion analysis (which is ultimately to do with separation properties of Farey sequences), we can show that this implies that these polynomials 
(which exert influence over intervals ![{[x', x'+Ha]}](https://s0.wp.com/latex.php?latex=%7B%5Bx%27%2C+x%27%2BHa%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
with some new polynomials 
and various 
, which are related to many of the previous polynomials by a relationship that looks very roughly like 
and some primes 
with 
for all 
, then one should have 
for some 
, where I am being vague here about what 
for some 
that is not too large (polynomial size in 
functions to conclude.
people, each of which has a probability 
of voting for one candidate, and 
for another… and one can then try to analyze the problem from a dispassionate mathematical perspective. This type of mindset can be illuminating in many contexts. Real world events have real consequences, however, and in light of an event as consequential as the last election, a math lecture on contour integration or the central limit theorem may seem meaningless. 
was equal to 
. Some older crowdsourced computational projects, such as the 
equipped with a binary operation 
. Initially, no additional axioms on this operation 
are imposed, and as such magmas by themselves are somewhat boring objects. Of course, with additional axioms, such as the identity axiom or the associative axiom, one can get more familiar mathematical objects such as groups, semigroups, or monoids. Here we will be interested in (constant-free) equational axioms, which are axioms of equality involving expressions built from the operation 
are indeterminate variables in the magma 
would not be considered an equational axiom here, as it involves a constant 
(the identity element), which we will not consider here.
. Specializing to 
, we conclude 
is idempotent: 
: 
with the addition operation 
. More generally, the diagram asserts the following non-implications, which (together with the indicated implications) completely describes the poset of implications between the eleven axioms:
of equations that use the magma operation 
is the number of ways one can form an expression out of 
applications of a binary operation 
placeholder variables); and, given a string of 
placeholder variables, the 
is the number of ways (up to relabeling) to assign names to each of these variables, where some of the placeholders are allowed to be assigned the same name. As a consequence, ignoring symmetry, the number of equations that involve at most four operations is 
equations come in pairs by the symmetry of equality, so the total size of 
comparisons, which looks rather daunting; but of course due to the axioms of a partial order, one could presumably identify the poset by a much smaller number of comparisons. I am thinking that it should be possible to crowdsource the exploration of this poset in the form of submissions to a central repository (such as the 
comparisons, to avoid redundant effort. Most submissions could be handled automatically, with relatively little human moderation required; and the status of the poset could be updated after each such submission.
laws are implied by either of these two laws. The poset 
, which in principle cuts down the total work by a factor of about two. Specific examples of magmas, such as the natural numbers with the addition operation, obey some set of equations in 
in 
possible implications between the these 4694 laws.
implications to resolve, 
have been proven to be true, 
have been proven to be false, and only 
remain open, although even within this set, there are 
implications that we conjecture to be false and for which we are likely to be able to formally disprove soon. For reasons of compilation efficiency, we do not record the proof of every single one of these assertions in Lean; we only prove a smaller set of 
implications in Lean, which then imply the broader set of implications through transitivity (for instance, using the fact that if Equation 
and Equation 
, then Equation 
, which I have nicknamed the “
). And here is a 
; I’ll leave this as a challenge (a four-line proof in Lean is possible).
in both commutative and non-commutative rings; free magmas associated to “
, then [x^m, y^m] is central with an m^2 in the upper right-hand corner; then you can multiply by another 
or so copies of [x,y] to get the desired element in about 
to travel 
in that direction, but when 
per unit of distance goes to infinity! 
graded by non-negative integers, with finitely generated free pieces in each degree.
satisfying the vertex algebra axioms over the integers.
from the complexification to the usual monster vertex algebra. This is the “integral form” property.
, which are known from monstrous moonshine.![\mathbb{Z}[1/2]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D%5B1%2F2%5D&bg=ffffff&fg=444444&s=0&c=20201002)
, and they gave some tantalizing hints for refining this to an integral form. In particular, they pointed out that we can make a self-dual integral form from self-dual forms over ![\mathbb{Z}[1/3]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D%5B1%2F3%5D&bg=ffffff&fg=444444&s=0&c=20201002)
, if they are isomorphic over ![\mathbb{Z}[1/6]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D%5B1%2F6%5D&bg=ffffff&fg=444444&s=0&c=20201002)
. In algebraic geometry language, this is “descent by a Zariski cover”.![\mathbb{Z}[1/3, \zeta_3]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D%5B1%2F3%2C+%5Czeta_3%5D&bg=ffffff&fg=444444&s=0&c=20201002)
(now we know this is possible), and Borcherds and Ryba suggested a strategy for refining this to ![\mathbb{Z}[1/pq, \zeta_{pq}]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D%5B1%2Fpq%2C+%5Czeta_%7Bpq%7D%5D&bg=ffffff&fg=444444&s=0&c=20201002)
without doing any explicit work.
, together with an 
-module 
and an 
-module 
.
-module 
such that 
and 
?
, since both sides would be 
with different choices of parentheses. However, this is not sufficient, unless the diagram 
and quasicoherent sheaves on the sources of the arrows. We would like to have a quasicoherent sheaf on 
that pulls back to the sheaves we had. Clearly, if the scheme maps are not jointly surjective, then the sheaf on 
, and define a 2-element étale cover given by arbitrary surjective étale maps to each component: 
. The gluing data gives no information, since the intersection is empty, so we can’t in general descend a sheaf along the surjective étale maps.
copies of a message. Into how few bits (units of information) can I compress the 
together with the knowledge that quantum gates are unitary, so the Hermitian conjugate of each gate doubles as its inverse. But what if you didn’t know quantum mechanics, or even linear algebra? Could this problem be solved through logical reasoning alone? This is where I think that the visuals really help; players should be able to infer several key points from geometry alone:
, guaranteed to be either the 
or 
eigenstate, and tasks the player to determine which state they have been given. Those familiar with quantum computing will recognize this as a relatively simple problem in state tomography. There are many viable strategies that could be employed to solve this task (and I am not even sure of the optimal one myself). However, by circumventing the need for a mathematical calculation, the Qubit Factory allows players to easily and quickly explore different approaches. Hopefully this could allow players to find effective strategies through trial-and-error, gaining some understanding of state tomography (and why it is challenging) in the process.
