Are cosum-preserving PF bijections matching excedances to occupancy descents necessarily nonlocal?

Fix $n\ge 1$. Let $PF(n)$ be the set of parking functions of length $n$. For $\pi=(\pi_1,\dots,\pi_n)\in PF(n)$ define $$ \mathrm{cosum}(\pi)=\binom{n+1}{2}-\sum_{i=1}^n \pi_i,\qquad \mathrm{exced}(\pi)=|\{i\in[n]:\pi_i>i\}|. $$ Apply the standard parking procedure to $\pi$ and let $\mathrm{oc}(\pi)\in S_n$ be the permutation of $[n]$ whose inverse sends a spot to the index of the car that parks there. Write $\mathrm{des}(\sigma)$ for the descent number.

Call a family of maps $\Phi=\{\Phi_n:PF(n)\to PF(n)\}_{n\ge1}$ radius-$k$ local if for every $n$, every index $i$, and all $\pi,\pi'\in PF(n)$, $$ (\pi_{i-k},\dots,\pi_{i+k})=(\pi'_{i-k},\dots,\pi'_{i+k}) \ \Rightarrow\ (\Phi_n(\pi))_i=(\Phi_n(\pi'))_i, $$ where out-of-range entries are ignored at the boundary. In other words, the $i$th output coordinate depends only on a fixed-size window of the input centered at $i$.

For every fixed $k$, does there exist $N(k)$ such that no radius-$k$ local family $\Phi$ can satisfy simultaneously, for all $n\ge N(k)$ and all $\pi\in PF(n)$, $$ \mathrm{cosum}(\Phi_n(\pi))=\mathrm{cosum}(\pi) \quad\text{and}\quad \mathrm{exced}(\pi)=\mathrm{des}(\mathrm{oc}(\Phi_n(\pi)))\ $$?

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Variant of van der Corput's dth-derivative estimate

I ask this question as a non-expert in analytic number theory who needs to use a number theory result. Lemma 2.2 of this paper by Chan, Kumchev and Wierdl reads as follows:

Lemma 2: Let $k \ge 2$ be an integer and put $K = 2^k$. Suppose that $a \le b \le a + N$ and that $f : [a,b] \to \mathbb{R}$ has continuous $k$th derivative that satisfies the inequality $0 < \lambda \le |f^{(k)}(x)| \le h\lambda \text{ for all } x \in [a,b].$ Then $$ \sum_{a \le n \le b} e(f(n)) \;\ll\; hN\bigl( \lambda^{1/(K-2)} + N^{-2/K} + (N^k \lambda)^{-2/K} \bigr). $$ As a proof they just say ‘it is a variant of Satz 4 in this paper of van der Corput’; Satz 4 can be translated to English as follows:

Satz 4: Let $a < b$, where $a$ and $b$ are integers, and let $k \ge 2$ be an integer. Put $\varkappa = 2^{k}$. Suppose that $f(t)$ is a real function on the interval $a \le t \le b$, $k$ times differentiable (possibly only one-sidedly at the endpoints), and that $f^{(k)}(t)$ is either always $\ge r$ or always $\le -r$, where $r$ is a positive constant independent of $t$.

Set $$ R = \frac{1}{b - a}\,\bigl| f^{(k-1)}(b) - f^{(k-1)}(a) \bigr|. $$ Then the following inequality holds: $$ \Biggl| \sum_{x = a}^{b} e^{2\pi i f(x)} \Biggr| < 21\,(b - a)\, \Biggl( \Bigl(\frac{r}{R^{2}}\Bigr)^{-1/(\varkappa - 2)} + \bigl(r(b - a)^{k}\bigr)^{-2/\varkappa} + \Bigl(\frac{r(b - a)}{R}\Bigr)^{-2/\varkappa} \Biggr). $$

At first I thought maybe Lemma 2.2 could follow from Satz 4 by an adequate change of variables, but if so I do not see how. I have several questions:

• If Lemma 2.2 follows from Satz 4, how?
• If not, is there a reference for Lemma 2.2, or how is it obtained as a `variant' of Satz 4?
• The $\ll$ in Lemma 2.2, what does it mean exactly without asymptotic notation? Does it basically mean the following? Perhaps changing the $10^{10}$ by a bigger/smaller constant, but that constant being independent of all the variables involved, as in Satz 4.

Let $k \ge 2$ be an integer and put $K = 2^k$. Suppose that $a \le b \le a + N$ and that $f : [a,b] \to \mathbb{R}$ has continuous $k$th derivative that satisfies the inequality $0 < \lambda \le |f^{(k)}(x)| \le h\lambda \text{ for all } x \in [a,b].$ Then $$ \Biggl|\,\sum_{a \le n \le b} e(f(n))\,\Biggr| \;\leq10^{10}\; hN\bigl( \lambda^{1/(K-2)} + N^{-2/K} + (N^k \lambda)^{-2/K} \bigr). $$

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Hölder continuity of Radon transform of smooth function

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by $$ R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)\,dx, $$ for every $(w,b) \in \mathbb R^{n+1}$. Here, $\delta$ is the Dirac delta distribution.

Question. In terms of smoothness of $f$, what is a sufficient condition to ensure that (1) $\|R[f]\|_\infty < \infty$, and (2) there exist constants $\alpha,C \in (0,\infty)$ such that for every $w \in \mathbb R^d$, the function $b \mapsto R[f](w,b)$ is $(\alpha,C)$-Hölder continuous, i.e., $$ \big|R[f](w,b')-R[f](w,b)\big| \le C|b'-b|^\alpha, \tag{+} $$ for all $w \in \mathbb R^d$ and $b,b' \in \mathbb R$ ?

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Related: Smoothness of Radon transform

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Convexity principle in several complex variables

I would like a reference for an analogue of the Phragmen-Lindelof in several complex variables. Specifically, if f is analytic in a region $A \subset \mathbb C^n$ and, by Bochner's theorem it extends to the convex closure of $A$, I am interested in bounds on the convex closure of $A$ in terms of bounds on $A$.

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Unprovable statements and generic properties

I should start with the following disclaimer that I know virtually no logic, sorry forgive me if my questions are ill-posed. I appreciate that all of this is probably completely obvious to a logician, but I've gone round asking all these to a lot of mathematicians around me and have so far failed to get any sort of satisfactory answer/reference on the topic.

I'm going to start with a pair of formal questions, and then explain where these come from.

• Could it be that the statement "$\pi$ is a disjunctive number" is undecidable?
• Could it be that the statement "The circle diffeomorphism $x \mapsto x + \sqrt{2} + 0.1 \sin(2\pi x)$ has irrational rotation number" is undecidable?

(One more disclaimer, I don't know the formal definition of the concept of undecidability. I take it that a statement is undecidable if assuming that it is true or false in a coherent logical system doesn't create any incoherence).

Undecidability and unprovable statements Like many people I suppose, I've always regarded the question of undecidability as a fairly esoteric one which I shouldn't concern myself with, as it is unlikely I ever come across an undecidable statement. This is mostly because the only example of undecidable I knew about was the continuum hypothesis, and until I'd come across a concrete, reasonable set whose cardinality couldn't be established it would always feel like an empty statement.

Now it's come to my attention that there could be statements which are true but that one couldn't prove. For instance, you could imagine two somewhat explicit functions $f,g : \mathbb{N} \longrightarrow \mathbb{N}$ such that the statement $\forall n, f(n) > g(n)$ is undecidable. That's where my understanding of logic becomes too shaky, but I can't imagine a statement of this form not being, for any reasonable definition of the concepts, either true or false. For all $n$, I can check whether it is true. So if it were to be false, I could exhibit a particular $n$ for which $f(n) \leq g(n)$.

If the statement is undecidable, it's just that it can't be proved.

Now (sorry if it's getting too philosophical), most statements that we try to prove we have reasons to think that they are true; a proof is not just a formal logical reasoning, it's also taming some sort of phenomenon which we thought was behind the statement.

There could well be some (simple) statements, which turn out to be true for no particular phenomenological reason. Such statements, I would actually be surprised if they could be proven to be true, as in my semi-religious belief a proof is a phenomenological justification for why something is true. It's bumping into such a statement (the second one above) that I started wondering about these kind of things.

The rotation number as a potential source of undecidability For those who do not know much dynamical systems, to a circle homeomorphism $T : S^1 \longrightarrow S^1$, one can associate a number $\rho(T) \in S^1 = \mathbb{R}/\mathbb{Z}$ which has some dynamical significance. It's a well-known theorem that in parameter families like $T_{\alpha, \epsilon} := x \mapsto x + \alpha + \epsilon \sin(2 \pi x)$, the probability that a parameter $(\alpha, \epsilon)$ corresponds to a rational/irrational rotation number has positive probability.

The definition of the rotation number involves taking a limit of infinitely many iterations of the map $T$, and somehow is highly transcendental with respect to the parameters $(\alpha, \epsilon)$. I want to speculate that the precise value of $(\alpha, \epsilon)$ is somewhat uncorrelated to the actual dynamical behaviour of $T_{\alpha, \epsilon}$. If there were a proof that $x \mapsto x + \sqrt{2} + 0.1 \sin(2\pi x)$ has irrational or rational rotation number, it would suggest that arithmetic properties of $\alpha$ and $\epsilon$ are reflected in the highly transcendental process of taking a limit of many iterations of $T_{\alpha, \epsilon}$. I would find this hard to believe, and therefore I would incline to believe that such a statement cannot be proven!

$\pi$ is disjunctive A number is disjunctive if its decimal expansion contains every finite sequence. It is relatively easy to prove that almost every real number is disjunctive. But now if one takes a number which is not defined in terms of its decimal expansion and that its decimal expansion cannot be made somewhat explicit using the (a) way it is defined, I would expect that

• it is disjunctive (because almost every number is)
• it is impossible to prove if the way the number is defined is "transcendental with respect to the decimal expansion".

Are such statements undecidable? The more I do maths, the more I come across conjectures about particular objects, the heuristical reasons why they are believed to be true seem to be true only generically for a wider class of objects. Has anyone considered the fact that such statements could be very good candidates for being undecidable statements?

A general question to conclude would then be : do logicians have interesting things to say about problems with this general structure? (a statement $S$ about a particular object $A$, known to hold true only generically for a wider class of objects $\mathcal{C}$, the definition of $A$ being "transcendental" with respect to the properties used in the proof that generic objects of $\mathcal{C}$ satisfy $S$).

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Weighted sum of standard Brownian bridges

Let $\{B_j\}_{j=1}^k$ be a sequence of Brownian bridges.

Let us consider $$X(t)=\sum_{j=1}^m w_j(t)B_j(t),$$ where $w_j$ are positive weight functions.

Then what can we say about (distribution or may be mean and variance) $X(t)$??

Clearly, $X(t)$ may not be a Brownian bridge as any standardized version of $X(t)$ has has its covariance function of the form $$\mathrm{cov}(X(t),X(t'))=\frac{(\min(t,t')-tt')\sum_{j=1}^m w_j(t)w_j(t')}{\sqrt{\sum_{j=1}^m w^2_j(t)\sum_{j=1}^m w^2_j(t')}}$$ which cannot be written as only as $\min(t,t')-tt'$. This tells us that we cannot standardise $X(t)$ to have a Brownian bridge.

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On the impossibility of infinite growth in Collatz Hypothesis [closed]

https://zenodo.org/records/17465117 Is this proof on the impossibility of infinite growth correct?

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Second Stiefel Whitney class of quotients of odd spheres

I don't know much of algebraic topology so the following question could be very silly. Let $G$ a finite subgroup of $U(n)$ that acts linearly (the action induced by the action of $U(n)$ on $\mathbb{C}^{n}$) and freely on the unit sphere $S^{2n-1}\subset \mathbb{C}^{n}$. Let $X$ be the quotient manifold $$X:=S^{2n-1}/G$$
I want to compute $w_{2}(X)\in H^{2}(X,\mathbb{Z}_{2})$ i.e. the second Stiefel-Whitney class of the tangent bundle of $X$. More precisely i'd like to know if it is $0$ or not and under which hypotheses on $G$ it vanishes. Is this a standard problem? Is there some reference for this type of calculation or something similar?

Thank you in advance!

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Defining Lebesgue non measurable sets with countable information

Is there a formula $\phi$ in the language of set theory such that $$ \text{ZFC proves } \exists x \in \mathbb{R}:\text{ the set }A_x​:=\{y\in\mathbb{R}:\phi(x,y)\} \text{ is not Lebesgue measurable?} $$ In other words, is there a process that takes a countable amount of information as input and, given the right input, outputs a non Lebesgue measurable set of real numbers?

Of course if one strengthens ZFC with, say V=L then this is possible. In ZFC + (V=L) one can construct non measurable sets without needing to make any arbitrary choices.

I am asking this question also in connection with descriptive set theory. We know that it is (probably) consistent with ZFC that all projective sets are measurable, so one would have to go beyond the projective hierarchy to find such a $\phi$. How far beyond this hierarchy do we at least have to go in order to have any chance at this?

If we did the opposite, namely adding the negation of this as an axiom schema to ZFC, so $$ \text{ZFC} + \forall x\in \mathbb{R}:\{y\in\mathbb{R} : \phi(x,y)\}\text{ is Lebesgue measurable}, $$ with the restriction that $x$ and $y$ are the only free variables of $\phi$, is there any chance that this formal system might be consistent?

My original question is equivalent to this system being inconsistent (since any inconsistency proof can only use finitely many formulas $\phi_i$, $i=1,\ldots,n$ and the resulting definable relations on $\mathbb{R}$ can be mapped to relations from $]i,i+1[$ to $\mathbb{R}$ and then rolled into one by taking union). Intuitively this axiom scheme says that any set of reals definable from countable information is Lebesgue measurable. In particular this formal system proves that all projective sets are measurable, but it obviously proves much more.

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Optimal bound on the growth of Riemann zeta along the critical line

According to the Lindelöf hypothesis, $|\zeta(\frac{1}{2}+it)|=O(t^\epsilon)$ for all $\epsilon>0$ as $t\rightarrow\infty$. I am wondering if there is any expectation for the optimal asymptotic upper bound beyond Lindelöf. Indeed, square root cancellation in the approximate functional equation suggests $|\zeta(\frac{1}{2}+it)|=O((\log t)^{1/2})$. Is this the expected optimal bound?

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Extension of eigenfunction of Laplacian with Robin boundary condition

Suppose $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with $C^1$ boundary, and consider the following (third) boundary value problem equation $$ \begin{cases} -\Delta u=\lambda u & \text{ on }\Omega, \\ hu+\dfrac{\partial u}{\partial\nu}=0 &\text{on }\partial\Omega, \end{cases} $$ Let $u$ be a eigenfunction corresponding to the principle eigenvalue $\lambda_0$: then $u\in H^1(\Omega)$. Now suppose $\Omega\subset\Omega_1$: can I get a function $\tilde{u}\in H^1(\Omega_1)$ such that $\tilde{u}|_{\Omega}=u$ ?

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Characteristic function of a domain to have higher order variation

For any $k\in\mathbb{N}$, the space $BV^k(\mathbb{R}^n)$ is defined as the set of all functions $f\in L^1(\mathbb{R}^n)$ such that there exists a Radon masure $D^\alpha f$ and for any $\phi\in C^\infty_c(\mathbb{R}^d)$, $$\int_{\mathbb{R}^d} f D^\alpha \phi = (-1)^{|\alpha|} \int_{\mathbb{R}^n}\phi d D^\alpha f$$ where $\alpha$ is a multi-index satisfying $|\alpha|\le k$. In particular when $k=1$, $BV^1$ is the space of all functions with bounded variation.

Suppose that $\Omega$ is a bounded open subset of $\mathbb{R}^n$, $n\ge 2$. What smoothness conditions on the boundary of $\Omega$ would ensure that $\chi_\Omega\in BV^2(\mathbb{R}^n)$?

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Successive Riemann integrability of products of successively Riemann integrable functions

In teaching multivariable Riemann integration, I was trying to develop the theory of successive Riemann integrals (so all start with the one-dimensional case familiar to the students) as far as reasonably possible within the confine of the course, before introducing Jordan sets and Jordan contents. The idea is to use successive Riemann integration to develop a "good" theory of volumes, with one advantage of not having to do a careful analysis in order to well-define the volume of a paved set (finite union of rectangles with sides parallel to the coordinate axis). A side question arises in the process that seems more subtle than I had anticipated, which I would like to share here.

First, allow me to fix the terminology and notation. We fix a dimension $ n \in \mathbb{N}^+ $ and an order with respect to which we shall perform the successive integrals, encoded by a permutation $\sigma \in S_n$, where $S_n$ is of course the symmetric group on $\{1,\ldots,n\}$. Let $f: \mathbb{R}^n \to \mathbb{R}$ be a bounded compactly supported function (here by support, I mean the smallest closed set outside which the function vanishes). Let $L_i$ (resp. $U_i$) denote the lower (resp. upper) Riemann integral with respect to the $i$-th variable. The lower (resp. upper) successive Riemann integral of $f$ with respect to $\sigma$ is defined to be the number $L_\sigma(f):=(L_{\sigma(1)} \circ \cdots L_{\sigma(n)})(f)$ (resp. $U_\sigma:= (U_{\sigma(1)} \circ \cdots U_{\sigma(n)})(f)$), i.e. we first integrate out the $\sigma(n)$-th variable, then the $\sigma(n-1)$-th etc. Naturally, we call a function $f: \mathbb{R}^n \to \mathbb{R}$ successively Riemann integrable with respect to $\sigma$, if $f$ is bounded, compactly supported and $U_\sigma(f) = L_\sigma(f)$, in which case, we call the number $I_\sigma(f):= U_\sigma(f)=L_\sigma(f)$ the successive Riemann integral of $f$ with respect to $\sigma$. We call a subset $A \subset \mathbb{R}^n$ successively contented with respect to $\sigma$, if the characteristic function $\chi_A$ of $A$ is successively Riemann integrable with respect to $\sigma$, and call the number $I_\sigma(\chi_A)$ the successive content/volume of $A$ with respect to $\sigma$.

Notation. We use $\mathcal{RS}_\sigma(\mathbb{R}^n)$ to denote the space of all real-valued functions on $\mathbb{R}^n$ that are successively Riemann integrable with respect to $\sigma$.

Let $L$ (resp. $U$) denote the one-dimensional lower/upper Riemann integrals of bounded compactly supported real-valued functions on $\mathbb{R}$. It is clear that $L(f + g) \ge L(f) + L(g)$ and $U(f + g) \le U(f) + U(g)$. Using these inequalities, it is easy to see that $\mathcal{RS}_\sigma(\mathbb{R}^n)$ is a vector space under the usual obvious addition and scalar multiplication, and $I_\sigma: \mathcal{RS}_\sigma(\mathbb{R}^n) \to \mathbb{R}$ is linear. From this, many basic properties that one expects of a "good" theory of volumes for subsets of $\mathbb{R}^n$ can already be established. In particular, we can already establish a "good" theory of volumes for paved sets in $\mathbb{R}^n$ without much tedious manipulation of the consistency of volumes of a paved set using different decompositions into rectangles. This being said, here comes the question.

Question 1. If $A, B \subset \mathbb{R}^n$ are successively contented, does it follow that $A \cap B$ remains so?

Or to put more generally for the integrals, I would like to know the answer to the following

Question 2. Does $f, g \in \mathcal{RS}_\sigma(\mathbb{R}^n)$ imply $fg \in \mathcal{RS}_\sigma(\mathbb{R}^n)$ ($fg$ denotes of course the pointwise product)?

To start things up, here's a rather brief sketch of how to tackle the $n = 1$ case. Using Riemann/Darboux sums, it is rather easy to establish $$ U(fg) - L(fg) \le M_f(Ug - Lg) + M_g(Uf - Ug), $$ from which it follows that question 2 in the above has an affirmative answer. The problem for general $n$ is that if we want to successively upper/lower integrate the above inequality to obtain a similar result for general $n$ and $\sigma$, we will find that the inequalities are always in the wrong sense. On the other hand, from the theorem of Fubini as well as the well-known characterization of Riemann integrable functions (not successive Riemann integrable as above, for which I don't expect such a neat characterization) as the bounded compactly supported ones with negligible sets of discontinuity, we know that even if there is a counter-example to the statement in question 2, it will not be very easy to describe.

With the above said, I would also like to present a variant of the above questions, which is still of great interest for the purpose of developing a good theory of $n$-dimensional volumes from successive Riemann integrals.

More notation. Let $\mathcal{RS}(\mathbb{R}^n)$ be the subspace of $\bigcap_{\sigma \in S_n}\mathcal{RS}_\sigma(\mathbb{R}^n)$ consisting of functions whose successive Riemann integrals with respect to any $\sigma \in S_n$ are the same.

Question 3. Is $\mathcal{RS}(\mathbb{R}^n)$ stable under taking pointwise product of functions? And if it turns out to be easier, what about the special case of characteristic functions?

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Any bound for the Betti number of a variety with rational singularities?

Let $X$ be a complex variety with rational singularities and $f: Y \to X$ be a resolution. It is well-known that $f^*:H^2(X,\mathbb{C}) \to H^2(Y,\mathbb{C})$ is injective by Leray spectral sequence. Hence $b_2(X) \le b_2(Y)$.

I am curious if there is a bound for $b_3(X)$ by the topology of $Y$, possibly with some extra assumptions.

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How to prove the convergence of the maximum point random variable of random concave function sequence？

I am really wondering how to prove this lemma from the book 'Counting processes and survival analysis'. No need for the first and second point, just the third point, why does the maximum random variable sequence $X_n$ converge to the single real value point $x$ in probability？Deeply appreciate your help.

Note: To clarify, I think the so called ‘random concave function’ means that $F_n=F_n(\omega,x)$，for each $\omega\in\Omega$，$F_n(\omega,\cdot)$ is a concave function, while for each $x\in E$，$F_n(\cdot,x)$ is a random variable.

I also want to post a little bit of my thoughts. For an open set $E$, every concave function is continuous on $E$. If $F_n$ are uniformly converging to $f$，I assume it would not be that hard to prove the unique maximum $X_n$ converges to $x$ in probability. So the second point is important here. But how can I find a compact set which is big enough to make sure that $$ P(X_n\in A)=1,\quad n=1,2,\cdots? $$ Besides that, another difficult point is how to deal with the convergence 'in probability'. We do not necessarily have the independence of $F_1,F_2,\cdots,F_n,\cdots$, so we do not necessarily have the independence of $X_1,X_2,\cdots,X_n,\cdots$. So the usual Borel-Cantelli lemma does not work here. I am really confused to further develop in this path. By the way, the author said and I quote as 'The proof, a simple $\epsilon-\delta$ argument, is left to the reader'. I just felt......

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Where lies the development prospect of knot theory？

I have been focusing on Khovanov homology, and I also want to explore other related fields. Of course, publishing academic papers is also a top priority for me.

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Frobenius action on component group of Néron model

Let $p$ be an odd prime. Suppose an elliptic curve $E/\mathbb{Q}$ has reduction of type $I_n$ at $p$. Let $\mathcal{E}$ be its Néron model, and assume that the component group $\tilde{\mathcal{E}}/\tilde{\mathcal{E}_0}(\overline{\mathbb{F}}_p)$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$. Now, when the reduction at $p$ is non-split, the group $\tilde{\mathcal{E}}/\tilde{\mathcal{E}_0}(\mathbb{F}_p)$ has order $1$. Why does this happen?

In Silverman, this is explained by choosing an extension $K/\mathbb{Q}_p$ over which the reduction becomes split multiplicative, and then applying the theory of $p$-adic uniformization. However, it seems that if one could show that Frobenius acts as multiplication by $-1$ on the component group, the result would follow immediately.

Could you please explain a proof in this direction, using the action of Frobenius?

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Simplicial Complexes arising from Young diagrams

My coauthors and I are finishing a paper in combinatorial and topological algebra, in which we define a class of simplicial complexes arising from Young diagramin the following way.

Let $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_r)$ be a partition with $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_r \geq 1$, and let $|\lambda| = \lambda_1 + \cdots + \lambda_r$. We identify $\lambda$ with its Young diagram: a top-justified array of $r$ rows of boxes of lengths $\lambda_1, \ldots, \lambda_r$, with weakly decreasing row sizes.
Fix $t \in \mathbf{N}$. For each $1 \leq j \leq r$, define
$$ \mathcal{S}_j = \{(r - j)t + i : 1 \leq i \leq \lambda_j\}. $$
Fill the boxes in the $j$-th row of the Young diagram of $\lambda$ from left to right with the elements of $\mathcal{S}_j$ in increasing order. Then, we define a $(\lambda_1 - 1)$–dimensional pure simplicial complex as follows: $$ \Big\langle \{ a_1, a_2, \ldots, a_{\lambda_1} \} \ \Big| \ a_1 < a_2 < \cdots < a_{\lambda_1}, \ a_j \text{ lies in the $j$-th column of } \lambda \text{ for all } 1 \leq j \leq \lambda_1 \Big\rangle. $$

For example, for $\lambda = (5,4,2)$ and $t=3$, the filled diagram is:

\begin{array}{lllll} \boxed{7} & \boxed{8} & \boxed{9} & \boxed{10} & \boxed{11}\\[4pt] \boxed{4} & \boxed{5} & \boxed{6} & \boxed{7} & \\[4pt] \boxed{1} & \boxed{2} & & & \end{array}

In this case, the facets of the related simplicial complex are: $$ \{1,2,6,7,11\},\ \{1,2,6,10,11\},\ \{1,2,9,10,11\},\ \{1,5,6,7,11\},\ \{1,5,6,10,11\},\ \{1,5,9,10,11\},\ \{1,8,9,10,11\},\ \{4,5,6,7,11\},\ \{4,5,6,10,11\},\ \{4,5,9,10,11\},\ \{4,8,9,10,11\},\ \{7,8,9,10,11\}. $$ Intuitively, one can think of this as taking an ascending chain of entries from $1$ to $11$ in the diagram.

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We are aware that simplicial complexes associated with Young diagrams or tableaux have appeared in the literature (for instance, in work by Knutson, Miller, and Yong).

Anyway, we are relatively new to this topic and not deeply familiar with the existing literature.

Question. We wonder whether this class of simplicial complexes is related to something already known, whether it has been defined under a different name, or if it has connections with other well-known complexes that would be worth mentioning.

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Induced analytic ring structure $f_!=f_*$

I understand that the six functor formalism on analytic rings takes the class of proper maps to be the maps with induced analytic ring structure. When we have such a map $f$, we know that $f^*$ has a colimit preserving right adjoint $f_*$ that satisfies base change and the projection formula. I have not been able to find a direct proof of the fact that $f_*=f_!$ in this context (there's a proof of this in Mann's thesis Prop A.5.10, but I have no idea how to extract this statement from that proof). My question here is not specific to the context of analytic rings: I'm just asking why an $f_*$ satisfying the properties above should also be a left adjoint of $f^*$. Thanks for the help!

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Existence of full-weight codeword in a linear q-ary code

I'm new to coding theory but would like to ask the following question:

Let $C$ be a linear $q$-ary code over $\mathbb{F}_q$ of length $n$ and dimension $k$ where $\mathbb{F}_q$ is a finite field with $q$ elements. Assume $C$ is fully supported in each position, that is for each position, every element in $\mathbb{F}_q$ appears in some codeword of $C$ at that position.

Question: under what conditions with respect to $(n,k,q)$ is a full-weight codeword (that is with nonzero entries in every position) in $C$ guaranteed to exist or not exist?

There seems to be answers to the question in several situations:

1. If $q = 2$ the question seems to be well understood.
2. There is at least partial answer for the so-called general MDS-codes. For instance see Theorem 3 in the following article: https://arxiv.org/pdf/1910.05634.pdf
3. Any result on the so-called full-weight spectrum codes or maximum weight spectrum codes certainly addresses this question. For instance see another article of Alderson: https://toc.ui.ac.ir/article_23512_5d71778dc799071faf41dab6222a7662.pdf

But I wonder if there are more general results directly targeting the above question. In a way this question stems from another recent post of mine. Instead of updating that post, allow me to reformulate the question in a more coding theoretic language hoping that the experts in this field can share some insights. Existence of fully supported element in a finite-dimensional vector space over $\mathbb{F}_p$ (and in finite abelian groups)

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