The question was motivated by this question of Anton Petrunin.

By a *metric continuum* we understand a connected compact metric space.

Let $p$ be a positive real number. A metric continuum $X$ is called *$\ell_p$-almost path-connected* if for any points $x,y\in X$ and any $\varepsilon>0$ here exists a family $\big((a_n,b_n)\big)_{n\in\omega}$ of pairwise disjoint open intervals in the unit segment $[0,1]$ and a continuous map $\gamma:[0,1]\setminus\bigcup_{n\in\omega}(a_n,b_n)\to X$ such that $\gamma(0)=x$, $\gamma(1)=y$ and $\sum_{n=0}^\infty d_X(\gamma(a_n),\gamma(b_n))^p<\varepsilon$.

It is easy to see that each almost $\ell_p$-connected metric continuum is $\ell_q$-almost connected for any $q\ge p$.

By my answer to the question of Anton Petrunin, each plane continuum is almost $\ell_1$-connected. By analogy it can be shown that each continuum in $\mathbb R^3$ is $\ell_2$-connected.

**Problem.** Is there a metric continuum which is not almost $\ell_1$-path connected? not almost $\ell_p$-connected for every $p<\infty$?

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of squares of $d$ dimensional Fourier harmonics up to degree $n$.

My question is if $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$ and can be represented as combination of spherical harmonics dimension $d$, then does there exist some spherical harmonic polynomials $g_1,\ldots, g_k$ of degree $n$ such that $p=g_1^2+\cdots g_k^2$ is a sum of squares?

i edit the question after Zach Teitler's comment.

The interval $[-\pi,\pi]^d$ means we concern the trigonometric polynomials positive on frequency domains.

The optimization problems about the polynomials positive on frequency domain $[-\pi,\pi]^d$ can be implemented via SDP approach(Gram matrix Rpresentation).

Given a positive polynomial represented as combination of spherical harmonics dimension $d$, Obviously, it is sum of squares of $d$ dimensional Fourier harmonics. Furthermore, it implies the symmetry relationship between $[-\pi,\pi] \times [0,\pi]$ and $[-\pi,\pi] \times [-\pi,0]$ on 2-sphere as an example. May be there is less information on sphere than cube?

So, is it the sum of squares of spherical harmonics?

Please feel free to provide any advices. Any comments and references (in English) will also be very welcome !

Thank you very much in advance!

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $Rep_V(\mathfrak{g}) \subset Rep(\mathfrak{g})$ to be the smallest symmetric monoidal, idempotent complete, abelian subcategory with duals (so, closed under tensor products, retracts, direct sums and duals) which contains $V$.

**Question:** Does there always exist an irreducible $V$ for which $Rep_V(\mathfrak{g}) \cong Rep(\mathfrak{g})$? When it exists, is there
a unique minimal one (in terms of the order on the weights) such $V$ (up to dualizing)? If not is there a unique self-dual such representation? If it doesn't exist, what is the minimal dimensional $V$
(possibly reducible) which satisfies this condition? Is it unique in some sense?

I'm interested in the question for all $\mathfrak{g}$ of type $A,B,C$ and $D$ (the exceptionals are a luxury). I think the standard representation $V$ in the case of type $A$ generates the entire category in this sense so that the answer is positive for this case but i'm not sure about any of the other cases.

While my question topic is that of mathematical writing of papers, which is a broad subject, the particular question is specific.

I am writing a paper, in which we have a section called "Outline of Proof". (It's Section 2.)
The outline is fairly informal, and we omit some technical details, making approximations.
However, among these approximations, my co-author wants to state (and label) *important* definitions and results (lemmas, equations, etc). He then wants to, later in the paper when we are doing the corresponding part carefully and rigorously, refer back to these (say, "by equation (2.4)", or "by Lemma 2.2"). Moreover, he is very against redundancies, so does not like things being stated twice precisely (including in the outline) -- once precisely in the text and once approximately in the outline is fine.

To me, this seems insane. (Of course, I did not use such a phrase when speaking with him!) When I read a paper, I never carefully read the outline: I just read it, and try to get an overview (or 'outline') of the proof; if there are parts that I don't really understand, I don't get hung up on them, trusting that with the more rigorous explanation later I'll be able to make sense of what the authors are saying.

*However*, I'm a pretty junior author -- 2nd year of PhD -- while he is a first year postdoc. That doesn't mean that I haven't read a reasonable number of papers (and in fact my lack of experience and knowledge means that I am less able to understand poorly written papers); moreover, he has said that he feels writing papers well isn't his best attribute.

So my question is this:

(a) is it standard to read an outline of a proof carefully?

(b) is it standard (or at least not discouraged) to state precisely important, even key, results/definitions that will be referred back to in the main body of the paper when giving proofs?

Just as some extra comments... I'm not here to try to get people to tell me that I'm right and my co-author is wrong and/or being silly! I know that *sometimes* some people come to Stack Exchange for such comments (see, particular, Workplace/Interpersonal Skills SEs!). I should have perhaps made the following clear: *if everything my co-author does is standard in the field, and I'm in the wrong, I definitely want to know that and will accept it!* -- I'm here to learn :-) please have no qualms about hearing criticism! (assume that it's constructive, of course)

I'm currently studying Iwasawa theory.

1) There are many $\mathbb{Z}_p$-modules on which some Galois groups act. So I often face some facts on the group representation over local fields or p-adic integer ring. But I can't find any references yet.

Of course, there are articles on p-adic representation. But I want references that are not too deep. I want references using just easy-to-follow arguments of algebra and representation theory.

Can you suggest any references?

2) Currently, I'm reading the paper "On the Iwasawa Invariants of Totally Real Number Fields" written by Ralph Greenberg. There I cannot understand a line which I have underlined with red line.

I'm afraid that there are many counter-examples against the line.(For example we can take cyclic group of order prime to the order of the group of units.) Can you please explain the line to me?

Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$.

Assume $d \ge 3$ and that $df=0$ at some point. Is it true that $f$ is constant?

A proof for the Euclidean case, can be found in "Geometric Function Theory and Non-linear Analysis", by Iwaniec and Martin. Their proof uses the fact both $M,N$ are Euclidean.

If I am not mistaken, a conformal map (in $d\ge3$) is determined by its 2-jet at a point, so it suffices to prove all the second derivatives at the point where $df$ vanishes are zero. (But maybe this wrong; it's possible that the "2-jet determination" only holds for conformal maps whose differentials are everywhere non-zero. I am not sure.)

I only know how to differentiate the conformal equation covariantly, which at a point where $df=0$ does not provide any information.

The ordinary circle packing problem in the variant with equal radii asks for the largest radius $r_{max}$ that allows placing $n$ non-overlapping circles with radius $r_{max}$ e.g. in the unit square, in the unit circle or, on the unit sphere (Tammes' problem).

Now, I would like to solve a somehow opposite problem:

**Question:**

given a number $n\in\mathbb{N}$, what is the smallest radius $r_{min}\in\mathbb{R}^+$ that permits a non-overlapping, rigid placement of $n$ circles with radius $r_{min}$ in the unit square, or in the unit circle or, on the unit sphere?

Under a rigid configuration I understand a configuration, where every open halfplane, resp. hemisphere defined by a hyperplane through a circle's center contains at least one contact point with another circle or, with the boundary of the containing region.

Are there already algorithms and/or theoretical results available for that problem?

I need to build a function like this with the following parameters:

A - peak area

h - peak height

w - peak width in the middle

b/a - asymmetry factor

I have a function described in the file, and I'm trying to use it to get the peak with the necessary parameters, fitting the the tau and sigma, but error(the square of the sum of the error for each parameter in percent) is still to big. Does anyone know a function that can build a peak with given parameters? P.s. sorry for my "russian" english

Let $R$ be a real closed field. Recall that $x,y \in R$ are *comparable* if there are $m,n \in \mathbb Z$ such that $mx > y$ and $ny > z$. Recall that the *ladder* of $R$ is the linear order divisible ordered abelian group obtained by quotienting by comparability a certain equivalence relation.

Note that $R$ has trivial ladder iff $R$ is a subfield of $\mathbb R$. If $R$ has trivial ladder and $L$ is a linear order divisible ordered abelian group, let $R\langle\!\langle x^L\rangle \!\rangle$ be the real closure of the purely transcendental extension $R(L)$ with transcendence basis $L$, ordered as in $L$ field of Puiseux series. Then $R\langle \!\langle x^L\rangle \!\rangle$ has ladder $L$. Conversely, from any real closed field, we may extract a maximal subfield of $\mathbb R$ and a ladder. I'm wondering whether that's all there is to it, i.e. whether every real closed field is of the form $R\langle\! \langle x^L \rangle\! \rangle$. Let me state this more formally:

**Questions:** Let $R$ be a real closed field with ladder $L$, and let $R_0 \subseteq R$ be a maximal subfield with trivial ladder.

Is $R$ characterized up to isomorphism by $R_0$ and $L$?

Is $R$ characterized up to isomorphism over $R_0$ by $L$?

The relative version: for any extension $R \to S$ of real closed fields such that $R_0 = S_0$, is $S$ characterized up to isomorphism over $R$ by the induced map on ladders?

As indicated in the title question, I'm happy to assume that $R_0 = \mathbb R$ if that simplifies things.

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$

we obtain $1/2$-Hölder continuity for the square-root.

I would like to know if $x,y$ are positive Hilbert-Schmidt operators. Does it follow then that for some $C>0$

$$\left\lVert \sqrt{x}-\sqrt{y} \right\rVert_{HS} \le C \left\lVert x-y\right\rVert_{HS}^{\frac{1}{2}}.$$

Sounds natural, but on the other hand, it is less obvious to me how this should follow.

One remark however is that if it would hold for finite-rank operators, then a density argument yields the claim.

Let $\mathcal{P}_n$ be a fixed $n$-sided regular polygon with area $A:=\vert \mathcal{P}_n\vert>0$. For any $c\in (0,A)$, I would like to find the shape of the domain $D\subset \mathcal{P}_n$ such that $\vert D \vert=c$ and the perimeter $\vert\partial D \vert$ is minimal. Do you have an idea how to tackle the problem or is there any related work you know of?

Of course, for small $c$, we can choose $D$ to be a disc because it is a global solution to the isoperimetric problem. But what happens if $c$ is too large such that the corresponding disc can not stay within the polygon?

Best wishes

Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition:

We say $A$ has infinite difference length, if

**(a)** For every integer $n$ there exist a number $k=2^q$ (for some positive integer $q$) and $a_1,\cdots,a_k\in A$ such that
$$
n=a_1+\cdots+a_{\frac{k}{2}}-(a_{\frac{k}{2}+1}+\cdots+a_k).
$$

Now, denote by $k(n)$ the least $k$ obtained from (a).

**(b)** The set of all $k(n)$, where $n$ runs over all integers, is unbounded above.

For example, if $\gcd\{a,b\}=1$ then $A=\{a,b\}$ has infinite difference length, but not $A=\mathbb{Z}^+$ (it does not have the second condition (b)).

Now, my questions are:

**(1)** Does the set of all Fibonacci numbers have infinite difference length?
(see https://math.stackexchange.com/questions/1989375/representation-of-integers-by-fibonacci-numbers)

**(2)** What about the Euler numbers?

**(3)** Does anybody know some important well-known integer sequences with infinite difference lengths?

**(4)** Did anyone see something like the above property (definition) yet?

Thanks in advance

Given integers $a,\ell$ and prime $p$ we need to find the roots of the algebraic equation $x^\ell\equiv a\bmod p$. We know there are at most $\ell$ such $x$.

What is the best method to find all such $x$?

What is the complexity (is it $O(poly(\ell\log p)$?)?

John Derbyshire in his book PRIME OBSESSION says on page 366: CHAPTER 3 10.

"Here is an example of e turning up unexpectedly. Select **a random number**
between **0 and 1**. Now **select another** and add it to the first. Keep
doing this, **piling on random numbers**.

How many **random numbers**, **on average**, do you need to make the **total greater than 1**?

Answer: **2.71828….**"

**My question:**
Can you provide a proof of the above statement?
Can you indicate an experimental verification method (code and data) with the computer?

Let a "promenade" on a tree be a walk going through every edge of the tree at least once, and such that the starting point and endpoint of the walk are distinct. What we mean by isomorphic promenades should be clear (in particular, the underlying trees have to be isomorphic). What is the total number $N(l,r)$ of promenades (up to isomorphism) of length $l$ with $r$ distinct edges (letting the underlying tree vary)?

It is known that the heat kernel on n-sphere satisfies $p_t(x,y)\leq Ct^{-n/2}e^{-d(x,y)^2/5t}$ for all $t\in (0,T)$. Can something be said about how big C needs to be?

Looking for a complete regular Riemannian metric in $ \Bbb R^2 $ depending on $s$ such that $x^s+y^s=1$ is a geodesic wrt. the metric, $x,y\in(0,1), s\in \Bbb R(1, \infty). $

I recently completed reading the book "Stochastic Differential Equations" by Bernt Oksendal which is the first time ever I was exposed to the topic. Now I am interested in pursuing research ( Ph.D.) SDEs and its applications in finance and I would like some help finding some recent papers related to or useful when doing research. I have already looked at some papers on MathSciNet by the same author but I would much appreciate if anyone can suggest some journals or papers/ articles that are relevant and useful in the current times. Thank you in advance!

Can u calculate 12 dollars and hour Monday through Friday for 8 hours and 14.00 per hour on Saturday paid weekly or biweekly at 14 per hour for an 8 hour shift on Saturday and how many hours of overtime I’d need to come up with about 1500 dollars for a 1200 dollar car and then adding insurance and registration and title

The question of existence of sets $x,y$ such that

$$|x|<|y| \wedge |P(x)|=|P(y)|$$

is known to be independent of $\text{ZFC}$!

But are there known examples of sets fulfilling the above condition that necessitates violation of choice?