Math Overflow Recent Questions


How to plot time series data in MatLab if the time array is not uniform?

Thu, 02/21/2019 - 09:47

I try to plot array $X$, $Y$ with given time array $T$. It is easy to plot these with uniform 'pause', but it does not use time array $T$.

My problem is to create a plot movie with $X$, $Y$ and $T$.

For example if $X=$$\{1,2,3,4,...\}$ , $Y=$$\{1,2,2,3,...\}$ and $T=$$\{1.1,2.2,2.5,3,...\}$ , that means the particle is at $(1,1)$ at $1.1$ second, at $(2,2)$ at $2.2$ second, at $(3,2)$ at $2.5$ second etc.

I need a moving plot of points at coordinates from array $X$ and array $Y$ which moves according to time from array $T$.

Solving or bounding the real part of the integral $\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} dt$

Thu, 02/21/2019 - 09:23

I would be interested in finding a closed form or, at least, bounding (in terms of $m$ as it becomes larger) the real part of the following itnegral:

$$f(m,a):=\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} dt$$

where $m \in \mathbb{N}, m>>0$ and $a \in \mathbb{C}, \mathfrak{Re}( a )>>0$.

For this purpose, I have tried three different methods.

Attempt 1: Firstly, I just tried to express the integral as a sum of integrals and made a change of variables, so that

$$f(m,a)= i \sum_{n=0}^{m-1} \int_{2 \pi n}^{2 \pi (n+1)} \frac{e^{-it}}{it-a} dt$$

Then, I compared those integrals whith the following:

$$\int_{\gamma} \frac{1}{\log{z}-a} dz $$

Where the path of integration is the unit circle clockwise. Each integral from sum of integrals correspond to different branches of the complex logarithm. To analyse this last one, I used a keyhole contour, taking the branch cut of the logarithm at the positive real axis. I obtained, for example, for the first summand (corresponding to the first branch of the logarithm),

$$\int_{0}^{2 \pi} \frac{e^{-it}}{it-a} dt = iR\int_0^{2 \pi} \frac{e^{it}}{2 \pi i - \log{R} -i t -a}dt - \int_1^R \frac{2 i \pi}{(a+\log{t})(a+\log{t}-2 i \pi)}dt$$

For $R \in \mathbb{R}, R > |a|$. My problem is that this family of integrals seems even more difficult to bound (even letting $R \to \infty$), so I have not gained anything.

Attempt 2: We could multiply and divide by $e^a$, so that we get:

$$f(m,a)=e^{-a} \int_0^{2 \pi i m} \frac{e^{-(t-a)}}{t-a} dt$$

From here, we could relate it to the Exponential Integral. However, the bounds that can be obtained this way seem to be not accurate at all for big $m$ and $\mathfrak{Re} a$, and it is difficult to extract the real part from them.

Attempt 3: We could separe the real and imaginary part of the original integral:

$$f(m,a)= -i \int_0^{2 \pi m} \frac{(\cos{t} + i \sin{t})(i\left (t + \mathfrak{Im}(a) \right) + \mathfrak{Re}(a))}{\mathfrak{Re}(a)^2+\left (\mathfrak{Im}(a) + t \right) ^2} dt$$

$$\mathfrak{Re}(f(m,a))= \int_0^{2 \pi m} \frac{\left (\mathfrak{Im}(a) + t \right) \cos{t} + \mathfrak{Re}(a) \sin{t}}{\mathfrak{Re}(a)^2+\left (\mathfrak{Im}(a) + t \right) ^2} dt$$

However, I do not know how to solve or bound this last integral neither.

Attempt 4: Using the following notation:

$$\mathscr{F}\left\{f(x)\right\} = F(s) = \int_{-\infty}^{\infty} {f(x)e^{-2\pi i sx} }dx$$ $$ \mathrm{sinc}(x) = \dfrac{\sin(\pi x)}{\pi x}$$ $$ \Pi(x) = \begin{cases} 1 \quad |x|<\frac{1}{2} \\ 0 \quad |x|>\frac{1}{2}\\ \end{cases}$$ $$ H(x) = \begin{cases} 0 \quad x < 0 \\ 1 \quad x > 0\\ \end{cases}$$

We can express our function as:

$$\int_{-\infty}^{\infty}\dfrac{1}{x - \dfrac{a}{2\pi i m}}\Pi\left(x-\frac{1}{2}\right)e^{-2\pi i mx}dx$$

Let's use the substitution $z_0 = \dfrac{a}{2\pi i m}$. Then, by making use of

$$\mathscr{F}\left\{\dfrac{1}{x-z_0}\right\} = -i\pi e^{-2\pi i s z_0}\left[\mathrm{sgn}(s)-\mathrm{sgn}\left(\Im\left[z_0\right]\right)\right]$$

We have:

$$\begin{align*} f(m,a) &= \mathscr{F}\left\{\dfrac{1}{x - z_0}\cdot\Pi\left(x-\frac{1}{2}\right)\right\} \\ &= \mathscr{F}\left\{\dfrac{1}{x - \dfrac{a}{2\pi i m}}\right\} * \mathscr{F}\left\{\Pi\left(x-\frac{1}{2}\right)\right\}\\ \\ &= -i\pi e^{-2\pi i m z_0}\left[\mathrm{sgn}(m)-\mathrm{sgn}\left(\mathfrak{Im}\left[z_0\right]\right)\right] * e^{-i\pi m}\mathrm{sinc}(m) \end{align*}$$

We can develop this a little bit more, but afer writing down the convolution as an integral we would end up with expressions that are pretty similar to the integral we are trying to solve, so that it would be useless.

This post has been cross-posted from MSE, where it received some upvotes but no definitive answer. I am curious about this problem, since the integral is seemingly inoffensive at first sight.

Any help will be welcomed.

Thank you.

Defining the value of a distribution at a point

Thu, 02/21/2019 - 08:59

Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is a regular distribution given by a continuous function $f\in C(U)$, then for every $\phi\in C^\infty_c(\mathbb R^n)$ with $\int_{\mathbb R^n}\phi(x)d x=1$ we can define a Dirac sequence $\{\phi^p_j\}_{j\in \mathbb N}\subset D(\mathbb R^n)$ by $\phi^p_j(x):=j^n\phi(j(x-p))$ which fulfills $$ \omega(\phi^p_j)\to f(p)\quad \text{ as }j\to \infty. $$ This shows that we can recover the value $\omega(p)\equiv f(p)$ of the distribution $\omega$ at the point $p$ via a limit of such Dirac sequences.

Now, suppose that for some $\omega\in D'(\mathbb R^n)$ and $p\in \mathbb R^n$ we just know that $ \lim_{j\to \infty}\omega(\phi^p_j) $ exists for every $\phi\in C^\infty_c(\mathbb R^n)$ with $\int_{\mathbb R^n}\phi(x)d x=1$ and is independent of $\phi$. In view of the above it then seems reasonable to define $\omega(p):=\lim_{j\to \infty}\omega(\phi^p_j)$ and to say that $\omega$ has a well-defined value at the point $p$.

Q: Is this definition useful in any sense? I have the feeling that it might be fundamentally flawed. In that case, I'd find it interesting to know what's the greatest generality in which one can make sense of "the value of a distribution at a point".

Least amount of comparisons for finding median of 5 numbers

Thu, 02/21/2019 - 08:44

I'm doing an assignment in which I have to find the number of comparisons to find medians among a group of numbers and also list them

  1. 3 numbers

I was able to do this by comparing A1 with A2 then if A1 was smaller then we compare A1 and A3. If A1 was bigger, we get A1 as the median. Similarly, did this for all the elements and got it in 3 comparisons ( A1 and A2 ) ( A2 and A3 ) ( A1 and A3 ). I did it in a form of tree where the left branch was if the left element was smaller or right if right was smaller. The root was the comparison eg - (A1:A2)...

  1. 5 numbers - I know the minimum combinations and comparisons needed are 6 but I am having trouble to finish this...

Locally nilpotent derivations on rings with zero divisors

Thu, 02/21/2019 - 08:36

Almost all books that I have found deal with derivation on several types of rings (or algebras) (for instance, commutative, noncommutative, domains, non-domains etc).

However, each paper about locally nilpotent derivations (that I know) suppose the ring is a domain.

Question: what happens with rings containing zero divisors and the study of locally nilpotent derivations? Does exist any phenomenon on them?

I appreciate any reference.

how to filter measurement noise out of a set of data

Thu, 02/21/2019 - 07:54

I need to find the best way to filter white noise from a set of data $(x_0,y_0),...,(x_n,y_n) /n>1000$


$y:physical$ $quantity$

The noise is comming from the sensor taking the mesurements. We know that the noise follows the standard normal probability law (gaussien distribution with and esperence equal to zero $m=0$), but we have no information about the value of the variance ($\sigma$ unknown).

So to deal with this problem, I taught about doing linear regression, but i dont know what the model to use.

In fact, the mesurements correspond to the thermograph of an electric equipement whether he is operational or dormant (not operating but confronted to the heat of its environement)

note that the mesurement are not taken with a constant sampling period. for exemple the value $x_3-x_2$ differs from $x_4-x_3$

So my question is: How to deal with the white noise ?

ps : Im sorry for my bad english

Periodicity of oscillators in Langton's Ant and powers of $2$

Thu, 02/21/2019 - 07:31

This question based on previous one by me. As Christopher Purcell noticed in his comment, there exist conjecture (which has a lot of counterexamples) that if you take a pair of ants $(n,n+1)$ apart (of the same colour, facing the same direction) then you will get an oscillating pattern.

Obviously, that if we place ants too far and areas where they oscillate do not intersects, each ant build a highway. If we work with ants which look in vertical direction and denote their coordinates as $(x_1,y_1)$ and $(x_2,y_2)$, $a=|x_1-x_2|$, $b=|y_1-y_2|$, $n,m$ - nonnegative integer so ants:

  • meet in one cell (and destroy each other) for some $a=2n, b=2m$

  • create an oscillator for some $a=2n, b=2m+1$ and $a=2n+1, b=2m$

  • build highways for any $a=2n+1, b=2m+1$ (and for some $a=2n, b=2m$, $a=2n, b=2m+1$ and $a=2n+1, b=2m$)

If we denote $p_1$ as periodicity of any oscillator with $a=2n+1, b=2m$ (and $p_2$ for $a=2n, b=2m+1$), $k$ - integer so:

  • $\gcd(p_1,2^k)=4$ for any $k>1$

  • $\gcd(p_2,2^k)\geqslant8$ for any $k>2$

Is there any explanation of this phenomenon?

Symplectic Chern class of holomorphic symplectic manifold

Thu, 02/21/2019 - 07:21

I've posted this question already on math.stackexchage. Unfortunately, it did not receive any answers even though there was a bounty on it. So maybe somebody of you could help me. I apologize if this does not fit the scope of this site.

Let $M$ a complex surface and $\omega\in H^0(\Omega_M^2,M)$ a non degenerate holomomorphic form.
I've read somewhere (without proof), that then the first chern class of the symplecitc manifold $(M,Re~ \omega)$ vanishes.

Why is this true? As far as I know, the Chern class of $(M, Re~ \omega)$ is the chern class of any complex vector bundle with almost complex structure compatible with $Re ~\omega$. My first guess was that the original complex structure is compatible with $Re ~\omega$. But this is not true, as then $Re ~\omega$ would be of type $(1,1)$ (with repect to the original almost complex structure).

Continuity of harmonic functions

Thu, 02/21/2019 - 07:10

I have a question about harmonic functions with respect to symmetric Markov processes.

Let $E$ be a locally compact separable metric space and $\mu$ a positive Radon measure on $E$.

Let $X=(X_t,P_x)$ be a $\mu$-symmetric Hunt process on $E$.

Def. A Borel measurable function $h$ on $E$ is called $X$-harmonic function if $$t \mapsto h(X_{t \wedge \tau_U}) \text{ is a uniformly integrable $P_x$-martingale }$$ for any relatively compact open subset $U \subset E$ and any $x \in U$. Here, $\tau_{U}=\inf\{t>0 \mid X_t \notin U\}$.

The above definition is essentially the same as the definition of harmonic functions adopted in this paper Ch.

I am interested in the following question:

Under what conditions on $X$ will $X$-harmonic functions be continuous on $E$?

If $X$ is strong Feller, the above question is true?

I would like to know if there are checkable conditions for the above question.

Is every accessible category well-powered?

Thu, 02/21/2019 - 06:51

Every locally presentable category is well-powered: since it is a full reflective subcategory of a presheaf topos, its subobject lattices are subsets of those of the latter.

Every accessible category with pushouts (hence also every locally presentable category) is well-copowered: this is shown in Theorem 2.49 of Locally presentable and accessible categories by Adámek and Rosický, and in Proposition 6.1.3 of Accessible categories by Makkai and Paré. The question of whether all accessible categories are well-copowered seems to depend on set theory (it follows from Vopenka's principle by Corollary 6.8 of Adámek and Rosický, and implies the existence of arbitrarily large measurable cardinals by Example A.19).

Are all accessible categories well-powered? I have been unable to find a mention of this one way or the other in either of these standard references.

Can you calculate the weight of shares when you only got the total amount of shares? [on hold]

Thu, 02/21/2019 - 05:50

I have a math problem and sadly not enough mathematical vocabulary to find a solution for it. Maybe you can help me.

We need to split value into shares, but the weight of these shares differ depending on how many people participated.

In an excel sheet I could of course split each job into it's own column, but I wanted to ask if there is a single formula way to calculate this on one go.

For example

In Run One 5 People participated and in run Two 3 people. Both runs are worth 10,000 points so that means the people who did both get 10000/5 + 10000/3 and the people who only did run 1 get 10000/5.

But when I have thousands of jobs I'd like to just set the amount of jobs they participated in, each run always has the same total value only the share weight shifts depending on how much people where present.

So how would the formula look for the above example?

Person 1 - 2 Person 2 - 2 Person 3 - 2 Person 4 - 1 Person 5 - 1

I tried taking the total amount of shares in this case 8 and divide the total money (20,000) by it and then times the shares, but this is obviously wrong.

Can someone point me in the right direction? I don't know where to turn to.

Which groups can be reconstructed from a single invariant subspace?

Thu, 02/21/2019 - 05:15

Let $G\subseteq\mathrm{Perm}(\Bbb R^n)$ be a matrix group consisting of permutation matrices acting on $\Bbb R^n$. Let $U\subseteq\Bbb R^n$ be an irreducible invariant subspace w.r.t. $G$. Now, define the reconstructed group

$$G_U:=\{T\in\mathrm{Perm}(\Bbb R^n) \mid TU=U\},$$

of permutation matrices that leave $U$ invariant. Obviously $G\subseteq G_U$.

Question: For which groups $G\subseteq\mathrm{Perm}(\Bbb R^n)$ there is some irreducible invariant subspace $U\subseteq\Bbb R^n$ that already determines $G$, i.e. $G=G_U?$

Some notes

  • This problem is related to reconstructing such a group from all its invariant subspaces, and this is related to finding GRRs (graphical regular representations). E.g. a cyclic group cannot be reconstructed from any invariant subspace, as a suitable dihedral group has the exact same invariant subspaces, but contains more permutation matrices.

  • So assuming that $G$ can be reconstructed from knowing all invariant subspaces $U_1,...,U_k$, i.e. $$G=\{T\in\mathrm{Perm}(\Bbb R^n)\mid TU_i=U_i,i=1,...,k\},$$ when can we find a single subspace $U_i$ that suffices for reconstruction? Actually, I do not know a counterexample where we cannot find such a subspace.

  • I am especially interested in the case wher $G$ acts regularly (hence transitively) on the canonical base vectors of $\Bbb R^n$.

Graph pattern matching

Thu, 02/21/2019 - 04:53

Given a weighted, oriented, connected graph with $10^7$ vertices and $10^{10}$ edges I need to implement the algorithm for searching various patterns on this graph for less than polynomial time.

Graph node degree can vary from $1$ to $2*10^5$, nodes and edges also have categorical attributes, that can be indexed and used during the pattern search.

Patterns are subgraphs with 5-100 nodes, some examples are shown on the pictures below, multiple edges from one node and looping is possible. The algorithm can be greedy and find only part of all pattern matchings.

I'm looking for a greedy library or method that performs multiple parallel walks on the graph to find the required pattern. It looks like "hub" nodes with a high degree make DFS ineffective for this problem.

A discontinuous construction

Thu, 02/21/2019 - 00:43

Suppose we have an uncountable family of functions $f_r: [0, 1] \to R$ indexed by $r \in [0, 1]$ such that for each $r$, there exists a unique $x$ in $[0, 1]$ such that $f_{r}$ is positive on $x$ and $0$ elsewhere.

Define the pointwise sum function $S[a, b]: [0, 1] \to R$ as $S[a, b] (x) = \sum_{r \in [a, b]} f_r (x)$.

If $S[0, 1]$ is well defined, then so is $S[a, b]$ for any $a, b \in R$.

Suppose that $S[0, 1]$ is well defined and that for every $x \in [0, 1]$, the set $\{r \in [0, 1]: f_r (x) > 0\}$ is dense in $[0, 1]$. Is it true that for a.e. $r \in [0, 1]$, the function $S[0, r]$ is discontinuous a.e.?

Concentration of the load of the maximally loaded bin ($m$ balls $n$ bins) with nonuniform bin probabilities

Thu, 02/21/2019 - 00:33

There is a common argument used when investigating the concentration of the maximally loaded bin (say $X$ is the maximum load) when $m$ balls are thrown into $n$ bins under the uniform distribution. I give the argument for $m=n,$ showing that $X$ is approximately $\ln n/\ln \ln n$ with high probability. Using the union bound, and letting $X_i$ be the number of balls in the $i^{th}$ bin $$ \mathbb{P}(X_i=k)=\binom{n}{k}\left(\frac{1}{n}\right)^k \left(1-\frac{1}{n}\right)^{n-k}\leq \binom{n}{k} \left(\frac{1}{n}\right)^k\leq \left(\frac{ne}{k}\right)^k\left(\frac{1}{n}\right)^k= \left(\frac{e}{k}\right)^k $$ yielding $$ \mathbb{P}(X_i\geq k)\leq \sum_{j=k}^n \left(\frac{e}{j}\right)^j \leq \left(\frac{e}{k}\right)^k \left(1+\frac{e}{k}+\frac{e^2}{k^2}+\cdots\right). $$ Now let $k^{\ast}=\lceil e \ln n/\ln\ln n\rceil,$ giving $$ \mathbb{P}(X_i\geq k)\leq \left(\frac{e}{k^{\ast}}\right)^{k^{\ast}} \left[\frac{1}{1-e/k^{\ast}}\right]\leq n^{-2}, $$ and using the union bound, since there are $n$ bins $$ \mathbb{P}\left(\bigcup_{i=1}^n X_i\geq k\right)\leq \frac{1}{n},\quad (1) $$ giving the concentration. What if we now have $p=(p_1,\ldots,p_n)$ with $p_i$ the probability of each ball falling into bin $i$, in an independent manner.

As far as I can tell (sort the bins so $p_1\geq p_2\geq \cdots\geq p_1>0$) as long as the maximum probability obeys $p_1\leq \frac{\ln n}{n}$, a version of this argument works.

What about distributions with larger $p_1$? What can we say? Say we allow the quantity on the RHS of (1) to be $\frac{1}{\sqrt{n}}$, for example.

I am most interested in $m=n,$ or slightly larger $m$ say $m=n (\log n)^a.$

I suppose for $p_1$ large enough wrt the other probabilities its load will highly likely be the maximum. So a kind of convex combination argument is needed...

Remark This related question here was unanswered

Does there exist a discrete gauge theory as a TQFT detecting the figure-8 knot?

Wed, 02/20/2019 - 18:07

My question: Does there exist a discrete gauge theory as TQFT detecting the figure-8 knot?

By detecting, I mean that computing the path integral (partition function with insertions of the knot/link configuration from line operators of TQFTs), and there is a nontrivial expectation value $$ Z=<\text{knot/link configuration}> \neq \# \exp{(i \theta)}. $$ A nontrivial expectation value means that the $\theta$ is not 0 mod 2 \pi. The number $\#$ depends on the global topology of the spacetime which the link/knot is evaluated at, such as a 3-sphere.

If the answer is negative, is there a way to prove such a discrete gauge TQFT detecting the figure-8 knot make NON-sense?

Background info: One of the best-studied complement space of 3-sphere is that of the figure eight knot, or Listing’s knot, say this space is called M8 (in Thurston book). The analysis is facilitated by the simplicity of the combinatorial structure of M8: it is the union of two tetrahedra, with a corresponding combinatorial decomposition of its universal cover being the tessellation of hyperbolic 3-space by regular ideal hyperbolic tetrahedra.

I know various literature using the discrete gauge theory (such as Dijkgraaf-Witten gauge theory and its generalization) as TQFT to detect their interesting corresponding link invariants. This approach from discrete gauge theory/TQFT (such as a $\mathbb{Z}_N$ gauge theory as level-$N$ $BF$with an action $S=\int_{M^3} BdA$ theory over 3-manifold $M^3$) makes up a nice list, a correspondence can be following:

from this article: Braiding statistics and link invariants of bosonic/fermionic topological quantum matter in 2+1 and 3+1 dimensions

Annals of Physics Volume 384, September 2017, Pages 254-287

Back to My question: the above list of TQFTs as discrete gauge theory do not detect the figure-8 knot. Are there possible other discrete gauge theories as TQFT or more general TQFT can detect the figure-8 knot?

For example, we know the 3d SU(2)$_k$ Chern-Simons gauge theory can detect the figure-8 knot. But the 3d SU(2)$_k$ Chern-Simons gauge theory is a 3d TQFT but NOT a 3d discrete gauge theory.

What category of toposes is monadic over the 2-category of groupoids?

Wed, 02/20/2019 - 15:37

Excuse my lack of understanding of monadicity, but I have been looking at toposes and monads. I see Lambek showed that the category of Toposes are monadic over the category of categories. I see the 2-category of groupoids as possibly another candidate for which there may be a category of toposes that are monadic over it. Is this the case?

A question has been asked about the 2-morphisms in my category of toposes that I think is monadic over the 2-category of groupoids. From my limited understanding, the 2-category of toposes will have functors as morphisms. The 2-morphisms would thus be natural transformations. I have limited understanding of functors between toposes, knowing only one special kind which is the geometric morphism. This means I have even less insight as to what the two morphisms would be. Perhaps natural transformations of geometric morphisms? I have to leave this question open in the sense that I am looking for any 2-category of toposes that is monadic over the 2-cat of groupoids. Does this mean there are many 2-cats of toposes that will be monadic?

Connectedness of moduli space

Wed, 02/20/2019 - 14:23

Let $X$ be a smooth projective variety. Given fixed Chern classes with positive rank $r$, let $M$ be the moduli space of semistable sheaves with such Chern classes. In general we only know $M$ is a projective scheme. (if $\mathrm{dim}(X)$ is $1,2$ a lot more can be said)

1st Question: Is there an example where $M$ is smooth but not connected?

2nd Question: I am trying to show the following scenario cannot happen: $M$ is smooth of dimension $d\geq 1$(Edit: first version did not contain $\geq 1$, see Jason Starr's comments below for an interesting example for $d=0$) but disconnected and one of the connected components consists merely of stable vector bundles. I feel we might be able to use the fact that the component only contains vector bundles to contradicts the projectivity but not sure how.

Thanks in advance!

$H$-space structure on coloured algebras

Wed, 02/20/2019 - 12:35

If $\mathcal{O}$ is a (classical) topological operad with unit $1\in \mathcal{O}(1)$, $\mathcal{O}(0)=\{0\}$ and multiplications $(m;a_1,\dotsc,a_r)\mapsto m(a_1,\dotsc,a_r)$. Let $X$ be an algebra over $\mathcal{O}$. Then each choice of $m\in \mathcal{O}(2)$ gives us a binary product $$X\times X\to X, (x,x')\mapsto x\cdot x' := m\cdot (x,x').$$ It is well-known that the topological structure of $\mathcal{O}$ characterises the product:

  • If $1,m(0,1),m(1,0)\in \mathcal{O}(1)$ lie in the same component, then $e:=0\cdot ()\in X$ is an $H$-unit.
  • If $m(m,1),m(1,m)\in \mathcal{O}(3)$ lie in the same component, then the product is $H$-associative.

I thought about an analogous construction for coloured operads $\mathcal{O}\binom{n}{k_1,\dotsc,k_r}$ and algebras $(X_n)$ over it. Each choice $m^n_{k_1,k_2}\in \mathcal{O}\binom{n}{k_1,k_2}$ gives a binary product $$X_{k_1}\times X_{k_2}\to X_n, (x,x')\mapsto x\cdot_n x' := m^n_{k_1,k_2}\cdot (x,x').$$ A system $(m^n_{k_1,k_2})$ may be called multiplication system. We see:

  • If all $\mathcal{O}\binom{n}{n}$ are connected, then $e_k:=0_k\cdot ()\in X_k$ is an $H$-unit in the sense that $$(-\cdot_n e_k)|_{n}\simeq (e_k\cdot_n -)|_{n}\simeq \mathrm{id}_n$$
  • If all $\mathcal{O}\binom{n}{k_1,k_2,k_3}$ are connected, then the restricted product is $H$-associative.

My hope was that if we fix such a multiplication system, we get an interesting $H$-something. As a coloured operad is the natural “horizontal categorification” of an operad, my first idea was that we obtain an $H$-category with colours as objects, and the above graded product as compositions and the $e_k$ as identities, but apparently, this is not the kind of structure we have.

(If you prefer algebraic categories, apply singular homology $H_*(-;R)$ to the above system and look at the Pontryagin structure. I first thought that the result is an $R$-algebroid, but again, the structure looks different.)

Does someone see what this is? A “monoid” where we can choose which product we want to use dependind on where we want to land?

Is there any physical or computational justification for non-constructive axioms such as AC or excluded middle?

Tue, 02/19/2019 - 19:38

I became interested in mathematics after studying physics because I wanted to better understand the mathematical foundations of various physical theories I had studied such as quantum mechanics, relativity, etc.

But after years of post-graduate study in mathematics, I've become a bit disillusioned with the way mathematics is done, and I often feel that much of it has become merely a logical game with objects that have no meaning outside of the confines of the game; e.g., existence of enough injectives/projectives, existence of bases for any vector space, etc.

I am not really an applied person---I love abstract theories such as category theory---but I would like to feel that the abstractions have some meaning outside of the logical game. To me it seems that using non-constructive proof techniques divorces the theory from reality.

Essentially, my question boils down to the following:

  1. If I adopt a constructive foundation (like say an intuitionistic type theory), is there any (apparently) insurmountable difficulty in modelling the physical universe and the abstract processes that emerge from it (quantum mechanics, relativity, chess, economies, etc.).

  2. Is there any physical/computational justification for assuming excluded middle or some sort of choice axiom?