I am looking for

a full subcategory of $\mathrm{Grp}$ or $\mathrm{Mod(A)}$ (where $\mathrm{A}$ is a commutative ring) having the amalgamation property but not general pushouts.

a full subcategory of $\mathrm{Grp}$ or $\mathrm{Mod(A)}$ without the amalgamation property.

(A category has the *amalgamation property* if spans can always be completed to commutative square, equivalently if spans have a compatible cocone.)

Honestly, I am not looking for very strange constructions in which one takes away precisely the pushouts you have, I am looking for very natural or at least natural categories that one encounters in real mathematical life.

It seems to me that there can be almost quaternionic structure on a manifold of two real dimensions, i.e., that generated by ($I,i\sigma_a$) where $I$ is the 2$\times$2 identity matrix and $\sigma_a$ are the Pauli matrices.

The three complex structures $J_a=i\sigma_a$ give rise to three Kaehler forms using $$ \Omega_a={(J_a)_{\mu}}^{\nu}g_{\nu\rho}dx^{\nu}\wedge dx^{\rho} $$ Why is it then true that we cannot have hyperkahler structure in two real dimensions?

Please see a chain of six circles associated with a conic. This is a generalization of Pascal theorem, Pappus theorem. I reformulate as following:

*Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a hyperbola. Let $1'$ be arbitrary point in the hyperbola. The circle $(121')$ meets the hyperbola at point $2'$. The circle $(232)$ meets the hyperbola again at $3'$, define points $4', 5', 6'$ similarly. Let circle $(121')$ meets the circle $(454')$ at $A, B$, Let circle $(232')$ meets the circle $(565')$ at $C, D$. Let circle $(343')$ meets the circle $(616') $ at $E, F$. Then six points $A, B, C, D, E, F$ lie on a circle.*

**Special case:**

*1. If $1'$ at $\infty$, six circles are six lines, so the theorem is Pascal theorem.*

*2. If the hyperbola is two lines, and $1'$ at $\infty$ then six circles are six lines, the theorem is Pappus theorem*

**My question:** Can generalization the result above to Higher Dimensions?

There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element".

Probably the best known analogy supporting that heuristic is the limit q->1 for number of elements in G(F_q) - for appropriate "m" it holds:

$$ \lim_{q->1} \frac { |G(F_q) | } { (q-1)^m} = |Weyl~Group~of~G| $$

For example: $|GL(n,F_q)|= [n]_q! (q-1)^{n}q^{n(n-1)/2} $ so divided by $(q-1)^n$ one gets $[n]_q! q^{n(n-1)/2} $ and at the limit $q->1$, one gets $n!$ which is the size of $S_n$ (Weyl group for GL(n)).

**Question** What are the other analogies supporting heuristis: Weyl groups = algebraic groups over field with one element ?

**Subquestion** once googling papers on F_1, I have seen quite an interesting analogy from representation theory point of view - it was some fact about induction from diagonal subgroups of symmetric groups $S_{d_1}\times ... \times S_{d_k} \subset S_n$ where $\sum d_i = n$ and similar fact for $GL(n,F_q)$.
But I cannot google it again and cannot remember the details :( (Tried quite a lot - I was sure it was on the first or second page of Soule's paper on F_1 - but it is not there, neither many other papers).

Knowing that total element count is okay, we may ask about counting elements with certain properties - like: m-tuples of commuting elements (MO271752), involutions, elements of order $m$, whatever ... From answer MO272059 one knows that there are certain analogies for such counting, however it seems the limits q->1 are not quite clear.

**Question 2** Is there any analogy for counting elements with some reasonable conditions ? Hope to see that count for G(F_q) (properly normalized) in the limit $q->1$ gives answer for Weyl group.

Let E be an elliptic curve over $F_p$. Suppose that its *j* invarient is not supersingular and that $j\neq 0 $ or 1728.

Then the modular polynomial $\Phi_l(j,T)$ has a zero $\tilde{\jmath} \in \mathbb{F}_{p^r}$ if and only if the kernel $C$ of the corresponding isogeny $E \mapsto E/C$ is a one-dimensional eigenspace of $\phi^r_p$ in $E[l]$, with $\phi_p$ the Frobenius endomorphism of $E$.

In the proof: Counting points on eliptic curve over finite field page no:236. Conversly, if $\Phi_l(j,\tilde{\jmath} )=0$, then there exists a cyclic subgroup $C$ of $E[l]$ such that the $j$ invariat of $E/C$ is equal to $\tilde{\jmath}\in \mathbb{F}_{p^r} $. Let $E^1$ be an elliptic curve over $\mathbb{F}_{p^r}$ with $j$ invariant equal to $\tilde{\jmath}$. Let $E/C \mapsto E^1$ be an $\bar{\mathbb{F}}_p$ isomorphism and let $f: E \mapsto E/C \mapsto E^1$ be the composite isogeny. It has kernel $C$.

I can see $f$ is defined over $\mathbb{F}_{p^r}$, so this implies existence of an isogeny $E \mapsto E^1$ over $\mathbb{F}_{p^r}$.

How this will imple $C$ is an eigenspace of $\phi^r_p$. How the statement that,"Frobenius endomorphisms over $\mathbb{F}_{p^r}$ satisfies the same characteristic equation" will help to see that $C$ is an eigenspace of $\phi^r_p$.

Please help me to understand this.

I learned that "If $G$ is a finite group acting on $S^n$, the sphere, freely, then $G$ has period cohomology".

My question is: Is there any other similar theorems relating the free action of a finite group and it's cohomology?

maybe this question is too vaguer.

Let $\mathfrak{gl}_n(x)= \mathfrak{gl}_n \otimes_\mathbb{C}\mathbb{C}(x)$ be the algebra of matrices taking values in rational functions.

**Definition:** Two matrices $A, B \in \mathfrak{gl}_n(x)$ are said to be **guage equivalent** iff there exists an invertible matrix $P \in
{GL}_n(\mathbb{C}(x))$ satisfying: $$A=PBP^{-1}+P^{-1}P'$$

The above is easily seen to be an equivaence relation on the matrices in $\mathfrak{gl}_n(x)$.

The classical Jordan decomposition for complex scalar matrices gives a canonical (modulo transposition of blocks) representative for every equivalence class of matrices for the equivalence relation of **similarity**. My question is basically "is there a similar thing for this case and if there is what is it?".

**Question:** Can one define a "canonical form" for matrices in $\mathfrak{gl}_n(x)$ modulo gauge equivalence? That is a **sensible and computable** assignment of a **canonical representative matrix** for every **gauge equivalence class** in $\mathfrak{gl}_n(x)$

If the above is some kind of extremely difficult untractable super-problem I'd appreciate an answer which explains why is this the case and additionally:

If the above is hopless can this be done at least for $\mathfrak{gl}_n \otimes_\mathbb{C}\mathbb{C}((x))$? With $P$ above lying in $GL_n(\mathbb{C}((x)))$ of course.

I have to estimate a gaussian-gamma mixture model with K components using EM algorithm, in more detaills i have an optical image (RGB) modeled by gaussian distribution and SAR image (grayscale) modeled by gamma distribution and each image contains K components. thus each component represent a class in two correspondence images. Thanks for providing me more formulas to implement it in matlab

I am looking for sources on Whitehead torsion in simplicial category. Specifically I am looking for a reference where the following theorem is proved (if true):

let X and Y be simplicial complexes such that there exists a sequence of collapses and expansions from X to Y in CW category (as in Cohens book) then there exists such a sequence using only simplicial collapses and expansions.

How from $L^2$ estimate we can get the following result

Let $X$ be a compact Kahler manifold of finite volume, bounded Riemannian sectional curvature and negative Ricci curvature and $x \in X$. There exists a positive integer $m$ such that the pluricanonical map defined by $\Gamma^0(X,K_X^m)=\cup_{\alpha>0}\Gamma^\alpha(X,K^m)$ (where $\Gamma^\alpha(X,K^m)$ is the space of $L^\alpha$ holomorphic sections of $K^m$)provides an embedding of neighbourhood of $x$ to its image.

Let $E$ be an elliptic curve over finite field $\mathbb{F}_q$ and $n$ be a positive integer. I am looking for an algorithm to find the generators of $E[n]$. Is there a fast algorithm in special cases such as supersingular elliptic curves or numbers like $n=p^t$?

Let $X$ be a singular projective variety. (We can assume $X$ to be seminormal if it helps.) Let $W$ be a codimension $>1$ subvariety of $X$ so that $W$ is in the singular locus of $X$. Let $\pi \colon Y\rightarrow X$ be a proper map which is isomorphic outside $W$ and on W its a $\mathbb{P}^n$ bundle for $n\neq 0$. Is it true that $Y$ is the blow up of $X$ along $W$? Can anyone give a reference of some general result related to this? What is the condition for a binational map to be sequence of blow ups?

I want to know how Google sets gaol in Google's analytic tools.

so please help,

Suppose that $\mathcal{X} \subseteq \mathbb{R}^d$ is compact.

Let there be $n$ distinct points $X = \{ x_1,...,x_n \} \subseteq \mathcal{X}$ and $k = \lfloor n^\alpha \rfloor$ where $0 < \alpha < 1$. Assume $\alpha$ and $d$ are fixed.

Define the $k$-NN radius of $x \in \mathcal{X}$ as $r_k(x) := \inf \{ r : |B(x, r) \cap X| \ge k \}$ where $B(a, r) := \{a' \in \mathcal{X} : |a - a'| \le r \}$.

Define the $k$-NN set of $x \in \mathcal{X}$ to be $N_k(x) := X \cap B(x, r_k(x))$. This can be viewed as the $k$ closest points in $X$ to $x$ (unless there are ties).

Let $M$ be the number of distinct $k$-NN sets over $\mathcal{X}$, that is, $M := |\{ N_k(x) : x \in \mathcal{X} \}|$.

Is $M$ bounded polynomially in $n$?

If not, then if we assume $X$ is sampled i.i.d. from some distribution, then do we at least have $M = O(poly(n))$ with high probability?

I've noticed that for every natural number $n\in\mathbb{N}$, there is a finite Heyting algebra with cardinality $C(n)$, where $C(n)$ is the $n$th Catalan number, $$1,1,2,5,14,42,132,\ldots$$ I'm looking for a reference, if this fact is known (to be known).

Below I will explain where the Heyting algebra structure comes from, in case it helps. When $n=0$ or $n=1$, we have $C(n)=1$, and there is a unique Heyting algebra structure on a set with one element, so suppose $n\geq 2$.

For any $m\in\mathbb{N}$, let $[m]:=\{0,1,\ldots,m\}$ and for any $0\leq a\leq b\leq m$, write $[a,b]$ for the subinterval $\{a,a+1,\ldots,b\}\subseteq[m]$. These subintervals form a poset, which we consider as a topological space with the Alexandrov topology: points are subintervals $[a,b]$ and open sets are down-closed subsets. Write $\Omega[m]$ for the poset of open sets in this space, so it has the structure of a Heyting algebra. It remains to show that the cardinality of $\Omega[m]$ is $C(m+2)$.

It is well-known that the Catalan number $C(n)$ counts the Dyck paths of length $2n$. These are paths in a triangle of dots (see below for $n=5$), starting at the southwest point, ending at the northeast point, where each edge in the path moves one unit either northward or eastward.

Position the elments of $[m]$ in the $(m+2)$-triangle, as shown here in the case $m=3$: $$ \begin{array}{ccccccccccc} \bullet&&\bullet&&\bullet&&\bullet&&\bullet&&\bullet\\ &&&&&&&3\\ \bullet&&\bullet&&\bullet&&\bullet&&\bullet\\ &&&&&2\\ \bullet&&\bullet&&\bullet&&\bullet\\ &&&1\\ \bullet&&\bullet&&\bullet\\ &0\\ \bullet&&\bullet\\ \\ \bullet\\\\ \end{array} $$ In this setup, a Dyck path $p$ of length $m+2$ can be identified with a downclosed subset, $S(p)\in\Omega[m]$. For example, the Dyck path $p_0$ shown below $$ \begin{array}{ccccccccccc} \bullet&&\bullet&&\bullet&&\bullet&-&\bullet&-&\bullet\\ &&&&&&|&3\\ \bullet&&\bullet&-&\bullet&-&\bullet&&\bullet\\ &&|&&&2\\ \bullet&&\bullet&&\bullet&&\bullet\\ &&|&1\\ \bullet&&\bullet&&\bullet\\ &0&|\\ \bullet&-&\bullet\\ |\\ \bullet\\\\ \end{array} $$ represents the set $S(p_0)=\mathord{\downarrow}[1,2]\cup\mathord{\downarrow}[3]$.

In fact, all these Heyting algebras $\Omega[m]$ fit together in a single topos, as we now explain. Consider the additive monoid of natural numbers as a category $BN$ with one object. Let $\mathbf{Int}:=\mathrm{Tw}(BN)$ be the twisted arrow category, and consider the presheaf topos $\mathrm{Psh}(\mathbf{Int})$. The subobject classifier for this topos is a functor $$\Omega'\colon\mathbf{Int}^\mathrm{op}\to\mathbf{Set}.$$ so for each object $n\in\mathbb{N}=\mathrm{Ob}(\mathbf{Int})$, we have a set $\Omega'(n)$. Moreover this set carries the structure of a Heyting algebra. Finally, $\Omega'(n)$ has a well-known description in terms of sieves, i.e. subfunctors of the representable functor $\mathbf{Int}(-,n)$. Unwrapping the definition, these are exactly the open sets of $[n]$. In other words, we have a bijection $\Omega'(n)\cong\Omega[n]$.

How can one tell whether a point $P$ not on a closed rectifiable curve $C$ is inside or outside $C$?

If $C$ is piecewise smooth one can throw a ray $R$ from $P$ in a random direction and count the number of intersections between $C$ and $R$. Odd - $P$ is in; even - $P$ is out. If the a line segment of $C$ lies on $R$ one can either collapse it into a single point.

Now, if $C$ is merely rectifiable then it's not clear that the number of intersections between $C$ and $R$ is going to be finite. Thus the questions:

Is it possible for a rectifiable curve $C$ and a point $P$ be in such position that for *a* ray $R$ the number of intersections between $C$ and $R$ will be infinite (apart of line segments of $C$ lying on $R$)?

Is it possible for the above travesty to happen for *every* ray $R$ from a fixed point $P$?

Suppose there are $N$ unit-mass particles whose initial positions
are uniformly distributed in a unit-radius disk.
Each particle is assigned a randomly oriented initial velocity vector $v_i$ of length $1$
(green below).
*Added**:* Robert Israel's incisive comment suggests that I should also stipulate
that $\sum_i v_i = 0$.
Then the particles act upon one another via inverse-square gravity.

Dots show initial positions inside unit disk.
Green vectors: initial velocity; red vectors: final velocity.

** Q**. What is the probability that $k \ge 1$ of the $N$ particles remain within
a disk of some radius $R \ge 1$ forever?

In the illustration above, $N=8$ and $R=3$. (But I do not trust my crude simulations.)

I am wondering if the answer is: **zero**, independent of $k$ and $R$ and the
gravitational constant?
Perhaps the answer differs for points in $\mathbb{R}^2$ vs. points in $\mathbb{R}^3$ (or $\mathbb{R}^d$, $d > 3$)?

In general, it is well known that, on the real line, say on $[0,1]$, if a function $f$ is of (pointwise) bounded variation, meaning that $$ \sum_{i=1}^n |f(x_i)-f(x_{i-1})| <+\infty $$ for every partition ${x_i}_{0}^n$ of $[0,1]$, then $f$ can be written as the difference of two monotone functions, hence it is differentiable a.e. w.r.t. the Lebesgue measure.

I am wondering if the same is true for $BV$ functions in $\mathbb R^d$ for $d \ge 2$.

Of course, the right definition of $BV$ in $d$-dimensional domains passes through the theory of distributions: $f \colon \Omega \subset \mathbb R^d \to \mathbb R$ is in $BV(\Omega)$ if it is an $L^1$ function whose distirbutional gradient $Df$ can be represented by a finite Radon measure (see here).

**Question 1.** Let $f \colon \Omega \subset \mathbb R^d \to \mathbb R$ be in $BV(\Omega)$. Is it true that $f$ is differentiable a.e. with
respect to the Lebesgue measure?

What I know is that they are approximately differentiable a.e. (in view of Calderon-Zygmund Theorem) so an approximate differential exists a.e. but I am not aware of any link between the approximate differentiability and the pointwise a.e. one.

Addendum.In view of Mizar's answer, it seems that the answer to **Q1** is negative, as it has been exhibited a $BV$ function which does not have even a continuous a.e. representative (in $L^1$).

While checking the details of the answer I received, I would like to ask another version of question above (do not know if still meaningful or not).

**Question 2.** Let $f \colon \Omega \subset \mathbb R^d \to \mathbb R$ be in $BV(\Omega)$. Assume further $f \in C^0(\Omega)$ i.e. it is continuous. Is it true that $f$ is differentiable a.e. with
respect to the Lebesgue measure?

*Notes.*

- See also this for related topic (in particular, I suspect that $BV^s([0,1])$ functions should be a.e. differentiable, but I have not a formal proof of this fact, nor a counterexample)

Let $f(n,q)=\prod_{j=1}^na(q^j)$ for a polynomial $a(q)$ and let $d(n)=\det(f(i+j,q))_{i,j=0}^n$ be its Hankel determinant.

Computer experiments suggest that $$\lim_{q\to1}\frac{d(n)}{(q-1)^\binom{n+1}{2}}=(a(1)a'(1))^\binom{n+1}{2}\prod_{j=0}^nj!.$$ Has anyone an idea how to prove this?

**Remark**: For $a(q)=a+qb$ it is easy to verify that
$$d(n)=((q-1)b)^\binom{n+1}{2}q^\frac{n(n+1)(2n+1)}{6}{\prod_{j=1}^n[j]_{q}!(a+q^jb)^{n+1-j}},$$
if $[n]_{q}=\frac{1-q^n}{1-q}$ and $[n]_q!=[1]_q[2]_q\dots[n]_q.$

Therefore the conjecture is true for linear polynomials and also for $a(q)=q^m.$

Given a universe $U = \{e_1 , . . . , e_n\}$ of
elements, and given a collection $S = \{S_1 , . . . , S_m \}$ of subsets of $U$, each of size $\le k$, the subcollection $S' \subseteq S$ is a *unique coverage* of $V \subseteq U$ if each $e \in V$ is uniquely covered, i.e., appears in exactly one set of $S'$. For simplicity, we assume that $\cup {S_i}=U$.

**Question 1:** Give a lowerbound on the maximum size of $V$, as a function $f(n, k)$.

**Question 2:** Does it help if $S$ is a Sperner family?

Notice that here we are not interested in computational and algorithmic aspects. I think I can come up with a $n/k^4$ lowerbound (and constructive).

**Background**

The problem of maximizing the unique cover for $k \ge 3$ is NP-hard. The approximation algorithms are studied in 1. Approximation algorithms for Generalizations are studied in [2].

**References**

1 V. Guruswami and L. Trevisan, The complexity of making unique choices: Approximating 1-in-k SAT, 2005.

[2] ERIK D. DEMAINE , URIEL FEIGE , et al, Combination can be hard: Approximability of the unique coverage problem