Denote the elementary symmetric functions in $n$ variables by $e_k(x_1, x_2,\dots, x_n)$. In the special case $x_j=j$, simply write $e_k(n)$ for $e_k(1, 2, \dots, n)$. Next, define the sequence

$$a_{+}(n)=\sum_{k\geq0}(-1)^ke_{2k}(n).$$

I am interested in the following:

**Question.** If $\lambda_n=\frac{3-(-1)^{\lfloor n/2\rfloor}}2$, then is it true that We have
$a_{+}(n)\equiv\lambda_{n+1}\mod3?$

Is there an example of a $n$ dimensional manifold $M$ and a natural number $k<n$ with a Lie subalgebra $L$ of $\chi^{\infty}(M)$ with the following property:

For every $x\in M$ the space $\{V_x \in T_x M\mid V\in L\} $ is a $k$ dimensional vector space $D_x$ but the distribution $D$ consisting of all $D_x,\;x\in M$ is not an integrable distribution.

In the other word we search for a non integrable distribution $D$ of a manifold and a Lie algebra $L$ of vector fields such that $L$ is $D$-ample where $D$- ample means that the evaluation $L_x$ of $L$ at every point $x$ is equal to $D_x$.

Let $p=p(x,y),q=q(x,y) \in \mathbb{C}[x,y]$ be a Jacobian pair, namely, $p_xq_y-p_yq_x \in \mathbb{C}^*$. Denote $a:= \deg(p)$ and $b:= \deg(q)$, where $\deg()$ denotes the total degree ($(1,1)$-degree).

There are several nice results about when a Jacobian pair is an automorphic pair (= $f: (x,y) \mapsto (p,q)$ is an automorphism of $\mathbb{C}[x,y]$) involving $a$ and $b$.

For example, each of the following conditions implies that $f$ is an automorphism:

**(1)** $\gcd(a,b)=1$; Magnus.

**(2)** $\gcd(a,b) \leq 2$; Nakai-Baba

**(3)** $\gcd(a,b)=P$, $P$ is a prime number; Appelgate-Onishi-Nagata.

**(4)** $a=PQ$, $\{P,Q\}$ are prime numbers; Nowicki.

Is there any progress about the following two cases:

**(3)'** $\gcd(a,b)=PQ$, $\{P,Q\}$ are prime numbers.

**(4)'** $a=PQR$, $\{P,Q,R\}$ are prime numbers.

**Remarks:** **(I)** I can prove that $(4)'$ implies $(3)'$ (as well as that $(4)$ implies $(3)$).

**(II)** Probably it is possible to replace $\mathbb{C}$ by any algebraically closed field of characteristic zero or even by any field of characteristic zero, but I do not mind to work over $\mathbb{C}$.

Any hints and comments are welcome!

Here is the condition, which arose in contemplating polytopes associated to matroid quotients:

Let $M$ and $N$ be matroids on $E$. If $X \subseteq Y \subseteq E$ such that $X$ is indepedent in $N$ and $Y$ is spanning in $M$, then $X$ is independent in $M$ and $Y$ is spanning in $N$.

If $M \to N$ is a matroid quotient, then this condition does hold.

*Proof:* First, recall that $M \to N$ is a matroid quotient if and only if the following exchange condition holds:
if $B_N$ is a basis of $N$, $B_M$ is a basis of $M$, and $i \in B_N - B_M$, then there exists $j \in B_M - B_N$ such that $B_N-i+j$ and $B_M-j+i$ are both bases. This follows since the fundamental cocircuit of $(B_N,i)$ and fundamental circuit of $(B_M,i)$ must intersect in more than one element.

If $X\subseteq Y \subseteq E$ such that $X$ is independent in $N$ and $Y$ is spanning in $M$, then there is a basis $B_N \supseteq X$ of $N$ and a basis $B_M \subseteq Y$ of $M$, respectively. To show $Y$ is spanning in $N$: Let $B \supset X$ be a basis of $N$ such that $|B-Y|$ is minimized. Then if $x \in B - Y$, $x$ is not in $B_M$, and the above exchange condition gives a $y \in B_M - B$ such that $B - x + y$ is a basis of $N$, which contradicts our construction of $B$. Hence, $B \subseteq Y$ and so $Y$ is spanning in $N$. The argument for $X$ is dual.

The proof of this condition only uses one half of the quotient basis exchange condition at a time. However, for a single matroid, the strong and weak basis exchange conditions are equivalent, which gives me hope (although the proof of this equivalence does not seem to generalize to this problem).

One problem that has bugged me for some time (though I only seriously thought about it for a month several years ago) is to give a physical proof of the L^2 boundedness of Bourgain maximal function for averages along the squares.

To be precise, define the averages: for a function $f: \mathbb{Z} \to \mathbb{C}$, let $$ A_Nf(x) := N^{-1} \sum_{n=1}^N f(x-n) $$ and the maximal function $$ Mf(x) := \sup_{N \in \mathbb{N}} |A_Nf(x)| $$ for $x \in \mathbb{Z}$.

**Question**: Without resorting to the circle method, can one prove that $M: \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$?

In particular, I suspect that there a physical proof analogous to the proofs for $L^2$ boundedness of Stein's spherical maximal function theorem as sketched out by Laba here: https://ilaba.wordpress.com/2009/05/23/bourgains-circular-maximal-theorem-an-exposition/ or as in Schlag's work: https://math.uchicago.edu/~schlag/papers/dukecircles.pdf In particular, I am happy with a physical proof of restricted weak-type $L^2$ boundedness.

My suspicion is that Bourgain's proof, which uses the circle method, suggests that we should consider understand what happens on arithmetic progressions, decompose the characteristic function of a set into how arithmetic progressions and combine them in some way. However I could not see how to successfully deploy this strategy...

Suppose that $C \in \sigma(\{A_{n} \times B_{n} : n=1,2,...\})$ and $C_{x} = \{y \in Y: (x,y) \in C\}$. Proof that if $x_{1}$ and $x_{2}$ belong to the same $A_{n}$ then $C_{x_{1}} = C_{x_{2}}$

(a) By graphing the function f(x) = (cos 2x − cos x)/x2 and zooming in toward the point where the graph crosses the y-axis, estimate the value of lim x → 0 f(x).

(b) Check your answer in part (a) by evaluating
f(x)
for values of x that approach 0. (Round your answers to six decimal places.)
f(0.1) =

f(0.01) =

f(0.001) =

f(0.0001) =

f(−0.1) =

f(−0.01) =

f(−0.001) =

f(−0.0001) =

lim
x→0
f(x) =

I got a function

$y = f(x)$ where $y \in \mathbb{R}$ and $x \in P \subset \mathbb{R}^3$ and $P$ is a convex polytope.

I want to find the minimum and maximum of $y$ provided that $x \in P$.

I can show that $f$ is monotonic over the set $P$, i.e. the elements of the gradient do not change their signs.

My question: For the case that $P$ is an interval, I can evaluate $f$ at the vertices of this interval to find the minimum and maximum. Is the same true if $P$ is a convex polytope?

What method can be used to solve following second order system of ODE? Second order system of ODE

Let $V$ be a vector bundle on the unit disc $\Delta$, $W \subset V$ be a sub-bundle and $H \subset V$ a sub-lattice of constant rank, in the sense that $H$ is a local system on $\Delta$, contained in $V$ and for all $t \in \Delta$, the fiber $H_t$ is a lattice. Suppose that for all $t \not= 0$, $H_t \cap W_t=0$ and $H_0 \cap W_0 \cong \mathbb{Z}$. Then, what is the singularity of the quotient $(V/W)/(H/H \cap W)$? Is it a complex manifold or an orbifold or worse? Any reference/hint will be most welcome.

There have been much research related to webgraphs and social graphs. They can be thought of a kind of random graphs, but the point is that they are different from the well-known Erdős–Rényi model.

**Question:** consider citatations graphs in some field of research
are there any known mathematical models for them ? Emperical observations on mathematical properties of such graphs (similar to mentioned below) ?

**Definition:** Citation graphs are graphs with vertices given by publications and edges by citations.

Here are main properties of webgraphs, I wonder about something similar for citation graphs.

1) Diameter is quite small (~6) or Law of Six degrees of separation

2) "Sparsity" - adjacency matrix is sparse matrix,

3) Power law of distribution of degrees of vertices: C/d^(2.1) - number of vertices with "d" edges

And there is suggestion for the model how webgraphs are growing:
Barabási–Albert model, with the key idea of **Preferential attachment**:

Preferential attachment means that the more connected a node is, the more likely it is to receive new links. Nodes with higher degree have stronger ability to grab links added to the network. Intuitively, the preferential attachment can be understood if we think in terms of social networks connecting people. Here a link from A to B means that person A "knows" or "is acquainted with" person B. Heavily linked nodes represent well-known people with lots of relations.

For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian):

finite groups $\leftrightarrow$ finite groups

discrete groups $\leftrightarrow$ compact groups

discrete torsion groups $\leftrightarrow$ profinite groups

discrete groups where each element is annhilated by some power of $p$ $\leftrightarrow$ pro $p$-groups

etc.

So I was wondering if we have a similar description of the Pontryagin dual of the category of abelian locally profinite groups, i.e. locally compact totally disconnected groups. Since locally profinite groups include discrete groups and profinite groups, the dual category will need to include discrete torsion groups and compact groups. Is there more we can say?

Consider configurations consisting of 4 distinct circles on the sphere. Two configurations are equivalent if they can be mapped onto each other by a homeomorphism of the sphere. How to enumerate/classify such configurations?

Equivalent problem: classify the arrangements of 4 hyperbolic planes in the hyperbolic space, up to homeomorphisms of the space.

Before voting to close this question as trivial, you may look at the classification of generic configurations which we obtained by brute force:

Each region bounded by more than 3 sides is labeled by the number of its boundary sides. This is used to show that all configurations are non-equivalent.

Questions: Is this new? Is there a scientific method to obtain this? Is there any structure on these 35 configurations?

There is a large research area about hyperplane arrangements in a Euclidean space. How about hyperbolic space? There is also a large body of research on hyperbolic tetrahedra. But it is always assumed that the tetrahedron is compact (or has only vertices at infinity).

We encountered this question in our studies of the Heun and Painlevé VI equations with real coefficients. (See Appendix II). Projective monodromy groups associated to these equations are generated by 4 reflections in circles.

For theories with well known proof-theoretic-ordinals, (what) is there a correspondence between their proof-theoretic-ordinal and (ordinal indexed families of?) fast growing functions provable total in a given theory ?

Recently I came across a paper which I really need to read through and which uses language of Morava $E$-theory. Since I'm not comfortable with this cohomology theory, I've been looking for quite a while some source to read (at least the basics) but wasn't able to find out something concrete. Does anyone know where can someone learn about Morava $E$-theory? I highly doubt that some self-contained textbook exists that covers this particular material, therefore any instructive paper or preprint is what am looking for most likely!

P.S. The same question had been asked yesterday on MSE, where I got no answer or comment (I deleted today). I didn't know if it is or not suitable for MSE from the beginning, therefore I ask here!

Let $R$ be a ring (not necessarily commutative, but with a unit). Recall that an $R$-module $M$ is of type $FP_n$ if $M$ has a partial projective resolution of length $n$ whose terms are all finitely generated (thus $FP_0$ means finitely generated and $FP_1$ means finitely presented). Consider an extension of modules

$$0 \longrightarrow M' \longrightarrow M \longrightarrow M'' \longrightarrow 0.$$

Assume that $M$ is of type $FP_n$ and that $M''$ is of type $FP_m$. What can we say about $M'$? I suppose that I am also interested in the other possibilities (where finiteness properties of $M'$ and $M$, or of $M'$ and $M''$, are given), but the one I indicated above is the most relevant one for what I am doing. Any references for these kinds of finiteness conditions in homological algebra are also welcome (the only one I know is Brown's book on group cohomology, and of course it focuses on examples coming from group cohomology).

There is a famous result of Banyaga stating that if two closed symplectic manifolds $(M_1, \omega_1)$ and $(M_2, \omega_2)$ have isomorphic groups of Hamiltonian diffeomorphisms $\mathrm{Ham}(M_1, \omega_1)\simeq \mathrm{Ham}(M_2, \omega_2)$ then there exists a diffeomorphism $f:M_1\rightarrow M_2$ such that $f^*\omega_2=\lambda \omega_1$ for some $\lambda\in\mathbb{R}^{*}$. In other words, a symplectic structure on a closed manifold is determined (up to rescaling) by the group of Hamiltonian diffeomorphisms.

The result of Banyaga tells us that, in principle, it should be possible to understand the topology of $M$ only from $\mathrm{Ham}(M, \omega)$. However, as far as I understand, it's pretty hard to actually give an algorithm for doing that.

So my question is: are there any explicit procedures for recovery of algebro-topological invariants of $M$ from $\mathrm{Ham}(M, \omega)$? I am particularly interested in recovery of the fundamental group $\pi_1(M)$.

A weaker question would be: is there any explicit condition on $\mathrm{Ham}(M, \omega)$ which guarantees that $M$ is simply-connected?

I am aware of one comparatively weak result in this direction. Banyaga has constructed a map (called flux map) $f:\pi_1(\mathrm{Symp}_0(M, \omega))\rightarrow H^1(M, \mathbb{R})$. He has also shown that there is an isomorphism $$ \mathrm{Symp}_0(M, \omega)/\mathrm{Ham}(M, \omega) \simeq H^1(M, \mathbb{R})/\mathrm{ker}\:f. $$ Therefore, if we know both $\mathrm{Symp}^0(M, \omega)$ and $\mathrm{Ham}(M, \omega)$ we can at least recover a quotient of $H^1(M, \mathbb{R})$ (so we have a lower bound for its rank, for example).

P.S.: A clear exposition of some results on the group of Hamiltonian diffeomorphisms can be found in Polterovich's book "The geometry of the group of symplectic diffeomorphisms" (chapter 14 is particularly relevant to the question).

Consider a PDF $\pi(x)$ for $x\in[0,1]$, and the following functional $$ F(\pi) = \mathbb{E}_\pi |x-y| $$ It is minimized by any point mass, so to avoid such degeneracy I'd like to lower-bound the entropy of $\pi$: $$ \mathbb{E}_\pi \left[ \ln \pi \right] \geq C $$

Is there an analytical solution to $\min_\pi F(\pi)$ subject to such an entropy constraint? In case of multiple solutions, I'd like the one(s) closest to some given $\pi_0$ (in the $L_p$ sense for a convenient $p$).

In the absence of analytical solutions, numerical methods would be useful. The entropy constraint can be addressed by a Lagrange multiplier, but perhaps there is some elegant way to deal with its non-linearity.

Now, the functional I'd really like to minimize over $L_1$ is $$ F_\alpha(\pi) = \mathbb{E_\pi} \left[ |x-y| \cdot (\pi^\alpha (x) + \pi(y)^\alpha) \right] $$ for $1 \leq \alpha \leq 2$. Perhaps, start with $\alpha=1$.

The entropy constraint isn't critical, but I will start with some $\pi_{init}$ and would like to prevent unnecessary entropy loss if possible.

Quantum computing is a very active and rapidly expanding field of research. Many companies and research institutes are spending a lot on this futuristic and potentially game-changing technology. Some even built toy models for a quantum computer in the lab. For instance, see IBM's 50-qubit quantum computer.

However, some scientists are not that optimistic when it comes to the predicted potential advantages of quantum computers in comparison with the classical ones. They believe there are *theoretical obstacles* and *fundamental limitations* that significantly reduce the efficiency of quantum computing.

One mathematical argument against quantum computing (and the only one that I am aware of) is based on the Gil Kalai's idea concerning the sensitivity of the quantum computation process to noise, which he believes may essentially affect the computational efficiency of quantum computers.

**Question.** I look for some references on similar theoretical (rather than practical) mathematical arguments against quantum computing — if there are any. Papers and lectures on potential theoretical flaws of quantum computing as a concept are welcome.

**Remark.** The theoretical arguments against quantum computing may remind the so-called *Goedelian arguments* against the artificial intelligence, particularly the famous Lucas-Penrose's idea based on the Goedel's incompleteness theorems. Maybe there could be some connections (and common flaws) between these two subjects, particularly when one considers the recent innovations in QAI such as the Quantum Artificial Intelligence Lab.

In this question on math.stackexchange.com I have made two conjectures the first of which I have proved. The second has not been settled. I post it here to seek a proof.

Given a quadratic surd $\sqrt d$ where $d$ is a natural number and not a perfect square. $(c_i)_{i=1}^\infty$ is the sequence of convergents of the continued fraction of $x$. Let $r_i:=\frac{c_{i+1}-\sqrt d}{c_i-\sqrt d},\,\forall i\in\mathbf N$. Let $n$ be the period of the continued fraction. It has been shown $\exists \,l_r:=\lim_\limits{i\rightarrow\infty}r_{in+r},\, \forall r\in\{0,1,\cdots,n-1\}$. Is the following statement, suggested by a numerical experiment, true?

For $n\ge 3$, there exists at least two distinct $l_r$'s. For $n\le 2$, $l_0=l_1$.