I'm having some fun playing around with string diagrams for monoidal categories, expressing familiar constructions from Riemannian geometry and linear algebra in terms of elegant string diagrams.

I've been wondering if there is a nice extension of this diagram calculus to bimonoidal categories (also known as rig categories)? This would allow us to work with direct sums (of bundles, or representations etc.) diagrammatically as well.

If $\mathcal{R}'$ is a *closed* subsurface of a hyperbolic surface $\mathcal{R}$, then according to theorem 3.18 page 84 of the book *a primer on mapping-class group* there is an inclusion homomorphism between the mapping class groups:

$$\text{Mod}(\mathcal{R}')\longrightarrow \text{Mod}(\mathcal{R})$$

I am concerned with the situation where $\mathcal{R}''$ is a general subsurface of $\mathcal{R}$. Such a surface has some handles, a number of boundaries, and a number of punctures. The presence of punctures makes the surface non-closed assuming that the boundary curves belong to the boundaries of $\mathcal{R}''$. It is possible some of the boundaries and/or punctures of $\mathcal{R}''$ are also boundaries and/or punctures of $\mathcal{R}$, i.e :

$$\partial\mathcal{R}''\cap \partial\mathcal{R}\ne \emptyset$$

Here a mapping-class **fixes the boundary but can permute the punctures**.

There are two questions:

- Is there an inclusion homomorphism between the mapping-class groups in this case, i.e. does a homomorphism $\text{Mod}(\mathcal{R}'')\longrightarrow \text{Mod}(\mathcal{R})$ exist?
- If yes, when is such a homomorphism injective? In particular, when $\text{Mod}(\mathcal{R}'')$ is a subgroup of $\text{Mod}(\mathcal{R})$?

A good reference is highly appreciated.

The case where the mapping-classes fix the punctures and all of the boundaries of $\mathcal{R}''$ belong to the interior of $\mathcal{R}$ is treated in Geometric Subgroups of Mapping Class Groups.

Please observe the following 16 two-valued logical operators on propositions p and q.

As can be seen they are naturally ordered from contradiction to tautology or backwards.

Moreover, each two-valued logical operator has a complement.

Are there formal mathematical researches that are focusing on the natural order among logical operators and the fact that each two-valued logical operator has a complement (please provide concrete examples)?

In my research, I found this identity and as I experienced, it's surely right. But I can't give a proof for it. Could someone help me? This is the identity: let $a$ and $b$ be two positive integers; then:

$\sum_{i,j \ge 0} \binom{i+j}{i}^2 \binom{(a-i)+(b-j)}{a-i}^2=\frac{1}{2} \binom{(2a+1)+(2b+1)}{2a+1}$.

Let $G = \langle V, E \rangle$ be an undirected, connected and weighted multigraph, with the weights given by a function $w: E \rightarrow N$. Consider any spanning tree $T$. Denote the edges of $T$ by $e_1, e_2, \ldots , e_{|V|-1}$. Define $P_T = \frac{\sum_{i = 1}^{|V| - 1|} w(e_i)}{|V| - 1}$, the arithmetic mean of weights of edges in $T$. Define $F_T = \sum_{i = 1}^{|V| - 1|} (w(e_i) - P_T)^2$. I want to find a spanning tree $T$ which minimizes $F_T$. Is there any known polynomial time algorithm for this problem?

**UPD:** What is the fastest algorithm that can solve it?

Posted also on: https://cs.stackexchange.com/questions/82538/spanning-tree-minimizing-f-t-sum-i-1v-1-we-i-p-t2 since further optimizations may require tricky, and programming-related ideas.

Recently I started working on a problem in Differential Geometry (where I'm not a specialist, so I apologize if this question turns out to have a trivial answer) and I had to consider the following situation.

Let $M \subset \mathbb{R}^n$ be a smooth submanifold of dimension $n-k$. By a *normal tube* of radius $\varepsilon$ centered at $M$, I mean a tubular neighborhood of $M$ given by a *disjoint* union $$\mathscr{B}(M, \, \varepsilon):=\bigsqcup_{p \in M} B(p, \, \varepsilon),$$ where $B(p, \, \varepsilon)$ is a $k$-dimensional ball of radius $\varepsilon$ centred at $p \in M$ and contained in the (embedded) normal subspace $N_pM \subset \mathbb{R}^n$.

**Q.** Under which conditions on $M$ does a normal tube $\mathscr{B}(M, \, \varepsilon)$ exist (for $\varepsilon$ sufficiently small)?

It seems to me that this is always the case when $M$ is compact, because then we can identify $\mathscr{B}(M, \, \varepsilon)$ with the open neighborhood $B_M(\varepsilon)$ of $M$ in $\mathbb{R}^n$ given by $$B_M(\varepsilon) := \{x \in \mathbb{R}^n \, | \, d(x, \, M) < \varepsilon\}.$$ On the other hand, there are also examples where $\mathscr{B}(M, \, \varepsilon)$ exists even if $M$ is not compact, for instance in the case where $M$ is a linear subspace (in this case, in fact, the normal spaces to $M$ are pairwise disjoint).

My feeling is that $\mathscr{B}(M, \, \varepsilon)$ should always exist when the curvature of $M$ is bounded in some suitable sense, but I'm not able to specify it better.

Any answer, example/counterexample and reference to the relevant literature will be greatly appreciated.

Let $X$ be a noetherian local normal scheme, we may even assume that $X$ is complete if necessary.

Consider $X\times (\mathbb{A}^2-\{0\})$, is it true that the Picard group of this scheme vanishes?

Let $X$ be a set and let $\Phi(X)$ denote the collection of filters on $X$. For $x\in X$ we denote by $P_x$ the filter $P_x=\{A\subseteq X:x\in A\}$. A *convergence space* is a pair $(X,\to)$, where $X$ is a set, and $\to$ is a subset of $\Phi(X)\times X$ with the following properties:

- $P_x \to x$ for all $x\in X$ (we write ${\cal F}\to x$ instead of $({\cal F},x)\in \to$), and
- If ${\cal F}\subseteq {\cal G}\in \Phi(X)$ and ${\cal F}\to x$, then ${\cal G}\to x$.

If $(X,\to_X)$ and $(Y,\to_Y)$ are convergence spaces, then $f: X\to Y$ is said to be *continuous* if ${\cal F}\in \Phi(X), x\in X$ and ${\cal F}\to_X x$ imply $f({\cal F}) \to_Y f(x)$, where $f({\cal F})$ is the image filter of ${\cal F}$.

The category ${\bf Conv}$ consists of convergence spaces with continous maps between them. There is a functor ${\bf Top}\to {\bf Conv}$ constructed in the following way:

Objects: $(X,\tau)$ maps to $(X,\to_\tau)$ where ${\cal F} \to_\tau x$ if and only if ${\cal F} \supseteq {\cal N}_x$, where ${\cal N}_x$ is the neighborhood filter of $x$ in $(X,\tau)$.

Maps: It is easy to see that a continous map in the topological sense is continuous in the convergence space sense, so the functor is the identity on the morphisms.

**Question.** Does this functor have an adjoint going back from ${\bf Conv}$ to ${\bf Top}$?

Note: Many topological properties can be transferred to convergence spaces. For instance, a convergence space $(X,\to)$ is *compact* if for every ultrafilter ${\cal U}\in \Phi(X)$ there is $x\in X$ such that ${\cal U}\to x$. Also, Tychonoff's theorem can be proved for compact convergence spaces. Moreover, a convergence space is said to be $T_2$ if ${\cal F}\to x$ and ${\cal F} \to y$ imply $x=y$.

**Definition:** The ordered interal $I=[m,M]\subset X$ is defined as
$$[m,M]=\{x\in X: m\leq x\leq M\}$$

**Question:** Let $I=[m,M]$ be an ordered interval in an ordered normed space $X$.
Is there any example of a convex operator $F:X\rightarrow I $ such that $m\leq F(x)\leq M$ for all $x\in X$.

*This question was the motivation behind an earlier question of mine; having thought about it some more, that question seems nontrivial and the connection is actually pretty tenuous anyways, so I've decided to ask this directly:*

For $\mathcal{K}\subseteq\omega_1$, we let $G_{CD}(\mathcal{K})$ be the game defined as follows *(which is a special case of a broad class of games defined by Montalban)*: players $1$ and $2$ alternate building objects by finite extensions (with passing allowed) - in player $1$'s case, a single linear order $L^1$, and in player $2$'s case an array of linear orders $L^2_i$ ($i\in\omega$). Player $2$ wins iff $(i)$ each $L_i^2$ is in $\mathcal{K}$, and $(ii)$ if $L^1\in\mathcal{K}$ then $L^1\cong L^2_i$ for some $i\in\omega$. Put another way, player $1$ is trying to *either* build an element of $\mathcal{K}$ which player $2$ doesn't build, *or* trick player $2$ into building something not in $\mathcal{K}$.

If $\mathcal{K}$ is unbounded, then $G_{CD}(\mathcal{K})$ can't be a win for player $2$ by $\Sigma^1_1$-bounding. Conversely, if CH holds it's easy to build an unbounded $\mathcal{K}\subseteq\omega_1$ such that $G_{CD}(\mathcal{K})$ isn't won by player $1$ either:

Fix an enumeration $\{\Sigma_\alpha:\alpha<\omega_1\}$ of all strategies for player $2$.

Let $\mathcal{K}_0=\emptyset$ and let $\mathcal{K}_\lambda=\bigcup_{\alpha<\lambda} \mathcal{K}_\alpha$ for $\lambda$ limit.

For successor stages, having constructed $\mathcal{K}_\alpha$ for $\alpha\in\omega_1$ we fix some enumeration $\Pi_\alpha$ of $\mathcal{K}_\alpha$; we then let $\mathcal{K}_{\alpha+1}=\mathcal{K}_\alpha\cup\{\omega_1^{\Sigma_\alpha\oplus\Pi_\alpha}\}$.

Now let $\mathcal{K}=\bigcup_{\alpha<\omega_1}\mathcal{K}_\alpha$. For each $\alpha\in\omega_1$, the strategy for player $1$ which simply enumerates $\mathcal{K}_\alpha$ according to $\Pi_\alpha$ beats $\Sigma_\alpha$ in $G_{CD}(\mathcal{K})$. At the same time, $\mathcal{K}$ is unbounded in $\omega_1$. So $G_{CD}(\mathcal{K})$ is undetermined.

However, in lieu of CH things seem much harder. Intuitively, there are continuum-many strategies but only $\omega_1$-many "choices" involved in creating $\mathcal{K}$. "Obviously" ZFC alone should prove that there ar many $\mathcal{K}\subseteq\omega_1$ such that $G_{CD}(\mathcal{K})$ is undetermined, but I don't see how to do that. So my question is:

Does ZFC **alone** prove that there is some $\mathcal{K}\subseteq\omega_1$ such that $G_{CD}(\mathcal{K})$ is undetermined?

I am new to optimization stuff. I need to formulate and solve this optimization problem.

$$\min \sum_{t\in\mathcal{T}}p_t$$

s.t. $$\sum_{t\in \mathcal{T}}w_t\log_2\left(1+\frac{h}{w_tn_0}p_t\right)= D$$

or

$$\sum_{t\in \mathcal{T}}w_t\ln\left(1+\frac{h}{w_tn_0}p_t\right)= S$$

Here, $p_t$ is the optimization variable.

Here, $h$, $w_t$, $n_0$ and $D$/$S$ are real and positive and great than $0$, and they are known. $\mathcal{T}$ is index set with $T$ elements, i.e., $\mathcal{T}=\{1,2,\cdots, T\}$.

Somone please help me to solve this.How can I express $p_t$.

Let $a \in L^{\infty}(\Omega)$ and suppose $a$ is continuous near $\partial \Omega$. Assume $w\in H^1(\Omega)$ solves

$\Delta w=a w$ in $\Omega$ with $w=\frac{\partial w}{\partial \nu}=0$ on $\partial \Omega$.

Is $w\equiv 0$? By Hopf's lemma $w$ has to changes sign in every neighborhood of any point $x\in \partial \Omega$. Is this enough to show $w\equiv 0$?

Let $K\subset R^2$ be a *convex body*, i.e., a compact convex set with interior points. A *plank* $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ *properly* provided that both boundary lines of $P$ intersect $K$.

**Question:** Let $K\subset R^2$ be a convex body with diameter $d$. Suppose there exists a measure $\mu$ on $K$ such that $\mu(P\cap K)$ is constant for all planks of some fixed width $0<h<d$ which properly intersect $K$. Must $K$ then be a disk?

A disk $D\subset R^2$ does indeed admit such a measure. Let $S\subset R^3$ be a sphere with $D$ as an equatorial disk, and $\mu$ be the measure on $D$ induced by the projection of $S$ (see the picture below which is from this paper of Kupavskii and Pach). Then $\mu$ will be plank invariant due to the Archimedes hatbox theorem.

The motivation for the above question is that a positive answer here would yield a positive answer to this earlier question, on the *Converse of the Archimedean property of the sphere*, as follows.

Suppose that there exits a convex surface $S\subset R^3$ with the Archimedean slab invariant area property mentioned in the earlier question. Then the projection of $S$ into any plane would yield a convex body with the plank invariant property with respect to the induced measure. Hence the projection would be a disk, if the answer to the above question were yes, which would in turn imply that $S$ is a sphere (it is well-known that the sphere is the only convex surface with round projections).

For some results in the direction of the above problem see this paper of Gardner. Another nice paper discussing some relevant background and questions is the expository paper of King in the Monthly. More references can be found in the recent paper of Kupavskii and Pach mentioned above. Also see the answer to this related question asked earlier, and another similar question which has remained unanswered.

So I'm new to Category Theory and I'm not sure if I should post my question here. Anyways.. I'm in the category of vector spaces $Vect_{\mathbb{F}}$ over a field $\mathbb{F}$. I want to find the object $X$ such that

$\hskip3in$

is a pushout, where $\{1\}$ is a set with one element (weird notation). I thougt about using coproducts and projection maps to find $X$. My question is: what would be the object $X$ and could it be generalized?

PS: Sorry for grammar mistakes, if any.

I am having trouble trying to solve this question. I'm not to sure what the order of an element is either.

Show that every element in the group (Zn × Zn,) has oder less or equal than n. (Hint: Show that the nth power of any element in the group is equal to the identity element).

I'm trying to solve a second order recurrence relations. Can someone help to solve $x_n+bnx_{n-1}+an(n-1)x_{n-2}=0$, where $b,a$ are some real numbers with $x_0=1,x_1=b$.

Thanks a lot!

The following problem got solved by me, but it's wrong. please help indicate my mistake. Thanks in advance.

I have plotted solutions $(a,b,c)$ to $a^2 + b^2 + c^2 = n$ for $12000 \leq n < 12100$, rescaled to $S^2$ and projected onto the first two coordinates. (these are read from the lower left, across and upwards. Sorry.)

While, over all $ a \leq n \leq b$ there is clear tendency towards equidistribution, for a fixed $n$, there are marked patterns (or even no solution at all). I've been struggling to find language for the types of patterns that are observing. There are solution-free regions, there are linear patterns, circular patterns and other sub-varieties.

I suspect for fixed $n$, the solutions are clustering along the intersection of two vareties, $S^2 \cap V $ and I am trying to characterize the equations of $V$.

This is my theory of why equidistribution might be so hard to prove; it's because there are in fact patterns.

Let $M$ be a simply connected topological 4-manifold with intersection form given by the E8 lattice. Does anyone know of examples of continuous self-maps of $M$ of degree 2 or 3? Or of degree any other prime for that matter?

This is a combined question, strictly speaking I am asking three questions concerning, respectively, homotopy type theory, stable homotopy theory and Yetter-Drinfeld modules. But I believe in the present context they can be viewed as a single question.

After a lecture on HoTT by Voevodsky (at TACL 2013 in Nashville) I asked him whether Homotopy Type Theory can be useful for stable homotopy theory. Without much hesitation he replied that HoTT is not really well suited to describe stable phenomena. We never talked after that, and now we will never talk at all, so I still do not know where the main difficulty lies.

Discussing this with several other people (sorry, don't exactly remember with whom) I gradually developed impression that the main difficulty relates to the fact that there seems to be no known good formalism combining constructive type theory and substructural logics, that would lead to some sort of linear type theory. Among possible models one should have triangulated monoidal categories, like stable homotopy category, and it seems not to be known what is the analog of dependent types there.

The first question then is whether what I said above makes any sense, and whether there is any work in this direction.

A situation where one has something like "linear dependent types" is in the context of Yetter-Drinfeld modules over Hopf algebras. Heuristically, the starting point is the fact that in any category $\mathscr C$ with (Cartesian) products every object $X$ has a unique comonoid structure, and the category of comodules over this comonoid is equivalent to the slice $\mathscr C/X$. One could then try to imitate this in (non-Cartesian) monoidal categories, considering comodules over comonoids as variable families over that comonoid. When this comonoid is equipped with a Hopf monoid structure, one can get a "linear" analog of Freyd-Yetter's crossed G-sets.

The second question is then, whether anybody studied features of categories of Yetter-Drinfeld modules over varying Hopf algebras which would resemble either what happens in HoTT or in stable homotopy category.

And the third question comes from the fact that while there are plenty of examples of monoids in stable homotopy categories (ring spectra) I cannot remember any considerations of **co**monoids or Hopf objects there. Does anybody know some works on this? Or even more simple question (actually fourth, sorry): in unstable homotopy theory one knows that for a, say, topological group $G$ there are certain homotopy categories of $G$-spaces and of spaces over $BG$ which are equivalent. Is there an analog of this fact in stable homotopy theory?

Or maybe Voevodsky had in mind that there seems to be no analog of univalence in the stable context? (Well that was fifth, I certainly must stop here.)

I found only one related question, Are there connections between Homotopy type theory and Grothendieck's theory of motives?, about possible connections between HoTT and motivic homotopy theory. I discovered my pessimistic comment there, remembering that answer by Voevodsky and arguing that motivic homotopy is in a sense even "more linear" than stable homotopy, so it might be even more difficult to relate these two. Anyway.