Where do these characters come from?

Let $G$ be a finite group. Fix a prime $p$. For a given $p$-Brauer character $\varphi$, define a class function on $G$: $$\widehat{\varphi}(x)=\varphi(x_{p'})$$ where $x_{p'}$ is the $p'$-part of $x$. Using Brauer's induction theorem, one can prove that this is a generalized character of $G$.

We have defined a group homomorphism: $$\mathbb{Z}[\text{IBr}(G)]\rightarrow\mathbb{Z}[\text{Irr}(G)]$$ which is known as the Brauer lift.

We may also define: $$\widetilde{\varphi}(x)=\begin{cases}|G|_p\varphi(x)&x\in G_{p'}\\0&\text{otherwise}\end{cases}$$ where $G_{p'}$ is the set of $p$-regular elements. This is also a generalized character. (I read about these things in Navarro's book "Characters and Blocks of Finite Groups".)

In Lusztig's paper "The Discrete Series of $GL_n$ over a finite field", a construction of the Brauer lift on the level of virtual modules is given.

Setup:

Let $k$ be a finite field and $\mathcal{O}$ be a complete DVR whose residue field is $k$.

For a $kG$-module $M$, let $\varphi_M$ denote its Brauer character (taking values in $\mathcal{O})$.

For a $\mathcal{O}G$-module $M$, let $\chi_M$ denote its character.

Lusztig's construction:

Roughly speaking (and assuming I haven't misunderstood) for a given $kG$-module $V$, Lusztig constructs a bounded chain complex $B(V)_\bullet$ of $\mathcal{O}G$-modules such that: $$\sum_i(-1)^i\chi_{B(V)_i}=\widehat{\varphi_M}$$ where $\chi_M$ denotes the character afforded by $M$.

Unfortunately for me, the paper appears to rely in an essential way on algebraic geometry. I hope to be able to read it eventually, but for now I have the following questions:

Question

1. Does this construction define a functor ${}_{kG}\mathbf{mod}\rightarrow\mathcal{D}^b({}_{\mathcal{O}G}\mathbf{mod})$?

2. Is there a more elementary & direct construction of the same bounded chain complex, that preferably does not involve algebraic geometry?

3. Is there a construction that works even when $k$ isn't finite?

4. Is there a construction of the second map $\varphi\mapsto\widetilde{\varphi}$ on the level of modules, along similar lines?

(Comment on this)

Are there any snarks with the following property?

I am working on a problem in graph theory and stumbled upon the following question about snarks: A snark is a 3-connected cubic graph where the edges have no 3-coloring. (often other conditions are also required but those aren't important to me) Are there any snarks $G$ such that there is a cycle $C$ with no chords and such that $V(G)-V(C)$ is an independent set?

Graphs with this property are interesting to me because you can transform such a graph to a hamiltonian graph $G'$ where every vertex not in $C$ is replaced by a triangle. It is known that this graph is 3-colorable (https://doi.org/10.1016/0012-365X(92)90588-7).

(For my purposes it would be fine to solve this for $\frac{|V(C)|}{3}\equiv i \mod 2$, for either $i=0$ or $1$)

(Comment on this)

On a numerical constant $B\approx 0.486$ arising from stratified sums of prime count residuals

I have been investigating the error term $E(x) = \pi(x) - \mathrm{li}(x)$ using a stratified sampling approach. Specifically, let $S(s)$ be the sum:

$$S(s) = \sum_{n=1}^{\infty} \frac{E(e^{n s})}{e^{n s}}$$

Numerical data for $s \in [1.5, 4.0]$ suggests a Gaussian-like decay of the form:

$$S(s) \approx \exp(-B \cdot s^2), \quad \text{where } B \approx 0.486$$

This constant $B$ also appears to be related to the series $\sum_{n=2}^{\infty} \frac{1}{\pi(e^n)}$ . My analysis, supported by Rosser-Schoenfeld bounds, suggests that this behavior might be linked to the spectral distribution of the Riemann zeta zeros, possibly through a regularized sum involving $1/\gamma^2$ .

Has this constant $B \approx 0.486 $ or this specific stratified sum of residuals $S(s)$ been identified in the literature concerning the statistical properties of $\pi(x)$ or the GUE (Gaussian Unitary Ensemble) hypothesis for zeta zeros? Any pointers to similar structures in analytic number theory would be greatly appreciated.

(Comment on this)

Classification of Principal $G$ bundles and vector bundles in smooth sense

Suppose $G$ is a Topological group then classification theorem of Principal $G$ bundles says that

there is a Principal $G$ bundle $EG\rightarrow BG$ such that any principal $G$ bundle over a decent topological space $X$ has to be pullback of a continuous map $f:X\rightarrow BG$.

Can we replace Topological group by Lie group and Topolgical space by Smooth manifold. Do we get all Principal $G$ bundles over smooth manifold in this case? Is $BG$ a smooth manifold??

In his book Fiber bundles, Dale Husemoller does not say anything (I could not see anything) about smooth version of that classification result. Now I have a doubt if that Milnor constriction $BG$ for a Lie group $G$ gives a smooth manifold or is this classification only for topological Principal $G$ bundles.

In similar way, when doing classification of vector bundles we construct what is called Grassmannian $G_n$ for each $n$ and a topological vector bundle $E_n\rightarrow G_n$ and say that for a decent topological space $X$, any rank $n$ vector bundle (in Topological sense, not smooth sense) over $X$ should be pullback of a continuous map $X\rightarrow G_n$. Here also we are classifying only topological vector bundles, not smooth vector bundles, right? I was thinking $G_n$ is a manifold and it classifies all smooth vector bundles but then realise I am thinking wrong.

Is the classification only restricted to Topological set up?

Is there similar classification in smooth set up? Like classifying smooth Principal $G$ bundles and classifying smooth vector bundles?

(Comment on this)

What are the applications of the Atiyah-Bott Yang Mills paper?

I recently finished a seminar going through Atiyah and Bott's paper ''The Yang-Mills Equations over Riemann surfaces''. The ideas going into the proof were surprising and very beautiful to me.

However, beyond its proof's beauty, I'm having trouble seeing the use of what I've just read. For instance, as I understand it the main result of the paper is an inductive formula for the cohomology of the space $\mathcal{C}(n,k)$ (the holomorphic vector bundles of rank $n$ and Chern class $k$ over Riemann surface $M$). This makes what the $\mathcal{C}$ look like a little clearer to me, but I've heard that if $g(M)\ne 0,1$ no very explicit of the $\mathcal{C}$ are known, so the only application I can think of (helping obtain an explicit description of the $\mathcal{C}$) seems not to have worked yet.

That naive train of thought lead me to ask:

What subsequent mathematics has heavily used the results of the Atiyah-Bott paper? Or, more petulantly, what's the point of the result?

(I know that there was a lot of activity on the Yang-Mills ideas which appear in the proof by Donaldson etc., but I'm asking about more direct applications as opposed to something like that.)

(Comment on this)

Lower bounds for the probability mass function of the Poisson Binomial distribution

Let $Y_1, \dots, Y_n$ be independent Bernoulli random variables with parameters $p_1, \dots, p_n \in [0, 1]$. Let $S_n = \sum_{i=1}^n Y_i$ denote their sum, which follows a Poisson Binomial distribution.

I am interested in lower bounds for the individual probabilities $\mathbb{P}(S_n = k)$ for $k \in \{0, \dots, n\}$.

Most literature focuses on concentration inequalities (tail bounds like Chernoff or Hoeffding) or Berry-Esseen type normal approximations. I am specifically looking for lower bounds on the probability mass function (PMF) itself rather than the cumulative distribution function (CDF).

Any references to papers or theorems dealing with these specific lower estimates would be greatly appreciated.

(Comment on this)

A strong exponential structure conjecture for $a+b=c$ with constant 1/4

Let $a, b, c$ be pairwise coprime positive integers satisfying $a+b=c$ with $c > 2$. Let $h(n)$ and $H(n)$ denote the minimum and maximum exponents in the prime factorization of $n$, respectively. (Convention: $h(1)=H(1)=+\infty$, so $1/h(1)=1/H(1)=0$).

In a previous question Is new $n$-conjecture as follows correct?, I proposed that $\min(h(a), h(b), h(c)) \le 3$. Recently, based on a comprehensive computational search over the entire "abc hits" database (for $c < 2^{63}$) and other extremal families (including Fermat-Catalan solutions), I have observed a much stronger and quantitative pattern.

I propose the following Exponential Structure Conjecture:

1. The Conjecture:

For all coprime triples $a+b=c$, the following inequality holds:

$$\sum_{x \in \{a,b,c\}} \frac{1}{h(x)} + \frac{1}{4} \sum_{x \in \{a,b,c\}} \frac{1}{H(x)} \ge 1$$

2. Empirical Evidence:Sharpness:

• The constant $1/4$ appears to be the sharp lower bound.

• Uniqueness: The equality holds uniquely in the checked dataset for the triple:$$1 + 23^3 = 2^3 \cdot 3^2 \cdot 13^2 \quad (1 + 12167 = 12168)$$Here, $\sum \frac{1}{h} = \frac{5}{6}$ and $\sum \frac{1}{H} = \frac{2}{3}$, yielding $\frac{5}{6} + \frac{1}{4}(\frac{2}{3}) = 1$.

• Consistency: This inequality strictly implies my previous conjecture. If $h(x) \ge 4$ for all $x$, the LHS is at most $0.9375 < 1$, which is a contradiction.

3. Mathematical Context & Reformulation

To place this conjecture in the context of modern Diophantine geometry, the inequality can be interpreted in terms of truncated counting functions.

3.1. Reformulation in Diophantine–Geometric Language

Let $(a,b,c)$ be pairwise coprime integers with $a+b=c$. For an integer $n$, write $$ v_p(n) \text{ for the $p$-adic valuation}, \qquad h(n) := \min_{p \mid n} v_p(n), \qquad H(n) := \max_{p \mid n} v_p(n). $$ The inequality $$ \sum_{x \in \{a,b,c\}} \frac{1}{h(x)} + \frac{1}{4} \sum_{x \in \{a,b,c\}} \frac{1}{H(x)} \ge 1 $$ can be interpreted in terms of truncated counting functions in the sense of Vojta.

Indeed, note that $$ \frac{1}{h(n)} \asymp \frac{1}{\log n} \sum_{p \mid n} \min\{1, v_p(n)\} \log p = \frac{1}{\log n} \, N^{(1)}(n), $$ where $N^{(1)}(n)$ denotes the level--$1$ truncated counting function. Similarly, the contribution involving $H(n)$ can be viewed as a higher-order truncation, controlling the depth of ramification rather than merely its support.

Thus, the above inequality may be heuristically rewritten as a quantitative constraint of the form $$ N^{(1)}(a) + N^{(1)}(b) + N^{(1)}(c) \;+\; \frac{1}{4} \, N^{(\infty)}(abc) \;\gtrsim\; h_{height}(a,b,c), $$ where $h_{height}$ denotes a suitable logarithmic height on $\mathbb{P}^1 \setminus \{0,1,\infty\}$, and $N^{(\infty)}$ reflects untruncated (or weakly truncated) counting at highly ramified primes. In this sense, the conjecture may be viewed as a "mixed truncation" version of the $abc$--phenomenon.

3.2. Comparison with Vojta's Conjecture

Vojta's Main Conjecture for $\mathbb{P}^1 \setminus \{0,1,\infty\}$ predicts, for every $\epsilon>0$, an inequality of the form $$ N^{(1)}(a) + N^{(1)}(b) + N^{(1)}(c) \;\ge\; (1-\epsilon)\, h_{height}(a,b,c) - O_\epsilon(1), $$ outside a proper exceptional set. The conjecture proposed here differs in two notable ways:

1. It incorporates explicit information on higher ramification through the quantities $H(a),H(b),H(c)$.
2. It replaces the indeterminate $\epsilon$ by a fixed numerical constant $\tfrac{1}{4}$, yielding a concrete quantitative boundary.

In particular, assuming stability under pullback along finite covers (e.g., via Belyi maps), the conjecture predicts finiteness of solutions in precisely the regime where Vojta's conjecture forbids Zariski-dense sets of integral points. From this perspective, the conjecture may be viewed as an explicit, arithmetic refinement of Vojta's inequality.

4. Questions:

Q1: Is this specific inequality ($k=1/4$) known in the literature regarding exponential Diophantine equations?

Q2: Is $k=1/4$ the true global lower bound, or is it possible that for sufficiently large $c$, there exist "worse" triples that would require a strictly higher coefficient (e.g., $k > 1/4$) to satisfy the inequality?

Q3: Are there any heuristic arguments suggesting that the boundary case $1+23^3$ is unique?

(Comment on this)

Is new $n$-conjecture as follows correct?

Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$

Define the functions $h(P)$ by $h(1)=+\infty$ and $h(P)=\min(a_1, a_2,\ldots,a_k).$

Is new the $n$-conjecture, formulated as follows, correct?

Conjecture: if ${P_1,P_2,...,P_n}$ are positive integer and pairwise coprime, then,

$$\min\{h(P_1), h(P_2),...,h(P_n), h(P_1+P_2+...+P_n)\} \leq n+1.$$

I proposed the case $n=2$ two years ago here (Is the conjecture A+B=C following correct?). Now I reformulate that question as follows:

Let ${P_1,P_2}$ are coprime, then: $$\min\{h(P_1), h(P_2), h(P_1+P_2)\} \leq 3$$

UPDATE: Adjust $h(1)=+\infty$ to be in sync with the general conjecture

(Comment on this)

A simple group with 3 generators

All finite simple groups are generated by two elements, though apparently the proof is tens of thousands of pages long. Here’s an obvious follow-up:

What’s an example of a finitely presentable simple group with 3 generators (and not 2)?

(Comment on this)

Why does an isolated singularity imply that the diagonal intersects the Newton boundary?

I am not completely sure whether this question is appropriate for mathoverflow, but I would greatly appreciate any clarification or references.

I am currently reading the paper

“Minimal Characteristic Exponent of the Gauss–Manin Connection of Isolated Singular Point and Newton Polyhedron” by Fritz Ehlers and Kam-Chan Lo.

Let me recall the definition to fix notation. Let $f(x) = \sum_{k\in \mathbb{N}^n} a_k x^k$ be a power series. We define $\Gamma_+(f) := \operatorname{Conv}\Bigl( \bigcup_{k\in\mathbb{N}^n,\ a_k\neq 0} (k+\mathbb{R}_+^n) \Bigr)$, and denote by $\Gamma(f)$ the Newton polyhedron (Newton boundary) of $f$, i.e. the union of the compact boundary faces of $\Gamma_+(f)$.

On page 434, the authors state the following theorem:

If $f$ is non-degenerate, then the minimal exponent is $t_0^{-1}$, where $(t_0,\ldots,t_0)$ is the intersection point of the diagonal line $t \mapsto (t,\ldots,t)$ with $\Gamma(f)$.

Immediately after this, they remark:

Note that since the singularity of f is isolated, this intersection is non-empty.

My question is about this last statement.

I do not clearly understand why the assumption that the singularity of f is isolated implies that the intersection of the diagonal line $\{(t,\ldots,t)\}$ with the Newton boundary $\Gamma(f)$ is non-empty.

In particular: • Since $\Gamma(f)$ is defined as the union of compact boundary faces of $\Gamma_+(f)$, how can one show that the diagonal actually meets one of these compact faces? • How does the compactness of the Newton boundary enter the argument? • At which point is the assumption that the singularity of f is isolated used in an essential way?

Any explanation, geometric intuition, or reference would be very helpful.

Additional question. Suppose that the above conclusion does not hold in general. Are there additional assumptions under which one can guarantee that the diagonal line $ \{(t,\ldots,t) \mid t > 0\} \cap \Gamma(f) \neq \varnothing. $

For instance: Does the statement become true if one assumes that

f is (weighted) homogeneous? Or if one strengthens the non-degeneracy condition? I am aware that the intersection is guaranteed when $f$ is convenient (i.e. $Γ(f)$ meets every coordinate axis), but it is not clear to me whether convenience is essential here, or whether the isolated singularity assumption alone (possibly together with non-degeneracy) already forces this property. Any clarification on the precise role of these additional assumptions would be very helpful.

(Comment on this)

Gram-Schmidt orthogonalization of $[i^{j-1}]$, and non-trivial zero elements inside

Consider the sequence of matrices $[i^{j-1}]_{(i,j)\in n\times n}$ Gram-Schmidt orthogonalized, the first ones are as follows, up to a normalization constant for each column. apologize if that's too many examples.

Note that besides trivial zeros (in the middle row of odd-ranked ones), like in the rank 7 and 13 ones, there are other zeroes not on the middle row (marked $\color{red}{\text{red}}$).

My first question is, are these matrices already named? (asked AI, no result, it suggested me like "discrete Legendre polynomial basis" or "particular Vandermonde matrix"). Then, how does non trivial zeros occur in larger ones? Do they occur infinitely, and does any zero occur in even-ranked ones? Are there closely related problems?

$$\left[\begin{matrix}1\end{matrix}\right]$$

$$\left[\begin{array}{r}1 & -1\\1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -1 & 1\\1 & 0 & -2\\1 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -3 & 1 & -1\\1 & -1 & -1 & 3\\1 & 1 & -1 & -3\\1 & 3 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -2 & 2 & -1 & 1\\1 & -1 & -1 & 2 & -4\\1 & 0 & -2 & 0 & 6\\1 & 1 & -1 & -2 & -4\\1 & 2 & 2 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -5 & 5 & -5 & 1 & -1\\1 & -3 & -1 & 7 & -3 & 5\\1 & -1 & -4 & 4 & 2 & -10\\1 & 1 & -4 & -4 & 2 & 10\\1 & 3 & -1 & -7 & -3 & -5\\1 & 5 & 5 & 5 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -3 & 5 & -1 & 3 & -1 & 1\\1 & -2 & \color{red}{0} & 1 & -7 & 4 & -6\\1 & -1 & -3 & 1 & 1 & -5 & 15\\1 & 0 & -4 & 0 & 6 & 0 & -20\\1 & 1 & -3 & -1 & 1 & 5 & 15\\1 & 2 & \color{red}{0} & -1 & -7 & -4 & -6\\1 & 3 & 5 & 1 & 3 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -7 & 7 & -7 & 7 & -7 & 1 & -1\\1 & -5 & 1 & 5 & -13 & 23 & -5 & 7\\1 & -3 & -3 & 7 & -3 & -17 & 9 & -21\\1 & -1 & -5 & 3 & 9 & -15 & -5 & 35\\1 & 1 & -5 & -3 & 9 & 15 & -5 & -35\\1 & 3 & -3 & -7 & -3 & 17 & 9 & 21\\1 & 5 & 1 & -5 & -13 & -23 & -5 & -7\\1 & 7 & 7 & 7 & 7 & 7 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -4 & 28 & -14 & 14 & -4 & 4 & -1 & 1\\1 & -3 & 7 & 7 & -21 & 11 & -17 & 6 & -8\\1 & -2 & -8 & 13 & -11 & -4 & 22 & -14 & 28\\1 & -1 & -17 & 9 & 9 & -9 & 1 & 14 & -56\\1 & 0 & -20 & 0 & 18 & 0 & -20 & 0 & 70\\1 & 1 & -17 & -9 & 9 & 9 & 1 & -14 & -56\\1 & 2 & -8 & -13 & -11 & 4 & 22 & 14 & 28\\1 & 3 & 7 & -7 & -21 & -11 & -17 & -6 & -8\\1 & 4 & 28 & 14 & 14 & 4 & 4 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -9 & 6 & -42 & 18 & -6 & 3 & -9 & 1 & -1\\1 & -7 & 2 & 14 & -22 & 14 & -11 & 47 & -7 & 9\\1 & -5 & -1 & 35 & -17 & -1 & 10 & -86 & 20 & -36\\1 & -3 & -3 & 31 & 3 & -11 & 6 & 42 & -28 & 84\\1 & -1 & -4 & 12 & 18 & -6 & -8 & 56 & 14 & -126\\1 & 1 & -4 & -12 & 18 & 6 & -8 & -56 & 14 & 126\\1 & 3 & -3 & -31 & 3 & 11 & 6 & -42 & -28 & -84\\1 & 5 & -1 & -35 & -17 & 1 & 10 & 86 & 20 & 36\\1 & 7 & 2 & -14 & -22 & -14 & -11 & -47 & -7 & -9\\1 & 9 & 6 & 42 & 18 & 6 & 3 & 9 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -5 & 15 & -30 & 6 & -3 & 15 & -5 & 5 & -1 & 1\\1 & -4 & 6 & 6 & -6 & 6 & -48 & 23 & -31 & 8 & -10\\1 & -3 & -1 & 22 & -6 & 1 & 29 & -33 & 73 & -27 & 45\\1 & -2 & -6 & 23 & -1 & -4 & 36 & 2 & -68 & 48 & -120\\1 & -1 & -9 & 14 & 4 & -4 & -12 & 28 & -14 & -42 & 210\\1 & 0 & -10 & 0 & 6 & 0 & -40 & 0 & 70 & 0 & -252\\1 & 1 & -9 & -14 & 4 & 4 & -12 & -28 & -14 & 42 & 210\\1 & 2 & -6 & -23 & -1 & 4 & 36 & -2 & -68 & -48 & -120\\1 & 3 & -1 & -22 & -6 & -1 & 29 & 33 & 73 & 27 & 45\\1 & 4 & 6 & -6 & -6 & -6 & -48 & -23 & -31 & -8 & -10\\1 & 5 & 15 & 30 & 6 & 3 & 15 & 5 & 5 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -11 & 55 & -33 & 33 & -33 & 11 & -55 & 11 & -11 & 1 & -1\\1 & -9 & 25 & 3 & -27 & 57 & -31 & 225 & -61 & 79 & -9 & 11\\1 & -7 & 1 & 21 & -33 & 21 & 11 & -251 & 119 & -227 & 35 & -55\\1 & -5 & -17 & 25 & -13 & -29 & 25 & -83 & -65 & 303 & -75 & 165\\1 & -3 & -29 & 19 & 12 & -44 & 4 & 204 & -74 & -102 & 90 & -330\\1 & -1 & -35 & 7 & 28 & -20 & -20 & 140 & 70 & -210 & -42 & 462\\1 & 1 & -35 & -7 & 28 & 20 & -20 & -140 & 70 & 210 & -42 & -462\\1 & 3 & -29 & -19 & 12 & 44 & 4 & -204 & -74 & 102 & 90 & 330\\1 & 5 & -17 & -25 & -13 & 29 & 25 & 83 & -65 & -303 & -75 & -165\\1 & 7 & 1 & -21 & -33 & -21 & 11 & 251 & 119 & 227 & 35 & 55\\1 & 9 & 25 & -3 & -27 & -57 & -31 & -225 & -61 & -79 & -9 & -11\\1 & 11 & 55 & 33 & 33 & 33 & 11 & 55 & 11 & 11 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -6 & 22 & -11 & 99 & -22 & 22 & -33 & 11 & -2 & 6 & -1 & 1\\1 & -5 & 11 & \color{red}{0} & -66 & 33 & -55 & 121 & -55 & 13 & -49 & 10 & -12\\1 & -4 & 2 & 6 & -96 & 18 & 8 & -103 & 89 & -32 & 166 & -44 & 66\\1 & -3 & -5 & 8 & -54 & -11 & 43 & -75 & -19 & 31 & -285 & 110 & -220\\1 & -2 & -10 & 7 & 11 & -26 & 22 & 65 & -71 & 6 & 210 & -165 & 495\\1 & -1 & -13 & 4 & 64 & -20 & -20 & 100 & 10 & -30 & 78 & 132 & -792\\1 & 0 & -14 & 0 & 84 & 0 & -40 & 0 & 70 & 0 & -252 & 0 & 924\\1 & 1 & -13 & -4 & 64 & 20 & -20 & -100 & 10 & 30 & 78 & -132 & -792\\1 & 2 & -10 & -7 & 11 & 26 & 22 & -65 & -71 & -6 & 210 & 165 & 495\\1 & 3 & -5 & -8 & -54 & 11 & 43 & 75 & -19 & -31 & -285 & -110 & -220\\1 & 4 & 2 & -6 & -96 & -18 & 8 & 103 & 89 & 32 & 166 & 44 & 66\\1 & 5 & 11 & \color{red}{0} & -66 & -33 & -55 & -121 & -55 & -13 & -49 & -10 & -12\\1 & 6 & 22 & 11 & 99 & 22 & 22 & 33 & 11 & 2 & 6 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -13 & 13 & -143 & 143 & -143 & 143 & -143 & 13 & -13 & 13 & -13 & 1 & -1\\1 & -11 & 7 & -11 & -77 & 187 & -319 & 473 & -59 & 77 & -97 & 119 & -11 & 13\\1 & -9 & 2 & 66 & -132 & 132 & -11 & -297 & 79 & -163 & 288 & -464 & 54 & -78\\1 & -7 & -2 & 98 & -92 & -28 & 227 & -353 & 7 & 107 & -392 & 968 & -154 & 286\\1 & -5 & -5 & 95 & -13 & -139 & 185 & 95 & -65 & 89 & 125 & -1045 & 275 & -715\\1 & -3 & -7 & 67 & 63 & -145 & -25 & 375 & -25 & -105 & 279 & 231 & -297 & 1287\\1 & -1 & -8 & 24 & 108 & -60 & -200 & 200 & 50 & -90 & -216 & 792 & 132 & -1716\\1 & 1 & -8 & -24 & 108 & 60 & -200 & -200 & 50 & 90 & -216 & -792 & 132 & 1716\\1 & 3 & -7 & -67 & 63 & 145 & -25 & -375 & -25 & 105 & 279 & -231 & -297 & -1287\\1 & 5 & -5 & -95 & -13 & 139 & 185 & -95 & -65 & -89 & 125 & 1045 & 275 & 715\\1 & 7 & -2 & -98 & -92 & 28 & 227 & 353 & 7 & -107 & -392 & -968 & -154 & -286\\1 & 9 & 2 & -66 & -132 & -132 & -11 & 297 & 79 & 163 & 288 & 464 & 54 & 78\\1 & 11 & 7 & 11 & -77 & -187 & -319 & -473 & -59 & -77 & -97 & -119 & -11 & -13\\1 & 13 & 13 & 143 & 143 & 143 & 143 & 143 & 13 & 13 & 13 & 13 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -7 & 91 & -91 & 1001 & -1001 & 143 & -13 & 91 & -91 & 91 & -7 & 7 & -1 & 1\\1 & -6 & 52 & -13 & -429 & 1144 & -286 & 39 & -377 & 494 & -624 & 59 & -71 & 12 & -14\\1 & -5 & 19 & 35 & -869 & 979 & -55 & -17 & 415 & -901 & 1631 & -205 & 313 & -65 & 91\\1 & -4 & -8 & 58 & -704 & 44 & 176 & -31 & 157 & 344 & -1724 & 356 & -766 & 208 & -364\\1 & -3 & -29 & 61 & -249 & -751 & 197 & -3 & -311 & 659 & -159 & -253 & 1067 & -429 & 1001\\1 & -2 & -44 & 49 & 251 & -1000 & 50 & 25 & -275 & -250 & 1568 & -121 & -649 & 572 & -2002\\1 & -1 & -53 & 27 & 621 & -675 & -125 & 25 & 125 & -675 & -27 & 297 & -363 & -429 & 3003\\1 & 0 & -56 & 0 & 756 & 0 & -200 & 0 & 350 & 0 & -1512 & 0 & 924 & 0 & -3432\\1 & 1 & -53 & -27 & 621 & 675 & -125 & -25 & 125 & 675 & -27 & -297 & -363 & 429 & 3003\\1 & 2 & -44 & -49 & 251 & 1000 & 50 & -25 & -275 & 250 & 1568 & 121 & -649 & -572 & -2002\\1 & 3 & -29 & -61 & -249 & 751 & 197 & 3 & -311 & -659 & -159 & 253 & 1067 & 429 & 1001\\1 & 4 & -8 & -58 & -704 & -44 & 176 & 31 & 157 & -344 & -1724 & -356 & -766 & -208 & -364\\1 & 5 & 19 & -35 & -869 & -979 & -55 & 17 & 415 & 901 & 1631 & 205 & 313 & 65 & 91\\1 & 6 & 52 & 13 & -429 & -1144 & -286 & -39 & -377 & -494 & -624 & -59 & -71 & -12 & -14\\1 & 7 & 91 & 91 & 1001 & 1001 & 143 & 13 & 91 & 91 & 91 & 7 & 7 & 1 & 1\end{array}\right]$$

$$\left[\begin{array}{r}1 & -15 & 35 & -455 & 273 & -143 & 65 & -195 & 65 & -91 & 21 & -35 & 5 & -15 & 1 & -1\\1 & -13 & 21 & -91 & -91 & 143 & -117 & 533 & -247 & 455 & -133 & 273 & -47 & 167 & -13 & 15\\1 & -11 & 9 & 143 & -221 & 143 & -39 & -143 & 221 & -715 & 307 & -849 & 187 & -821 & 77 & -105\\1 & -9 & -1 & 267 & -201 & 33 & 59 & -423 & 149 & 95 & -243 & 1219 & -393 & 2301 & -273 & 455\\1 & -7 & -9 & 301 & -101 & -77 & 87 & -157 & -133 & 575 & -133 & -453 & 413 & -3887 & 637 & -1365\\1 & -5 & -15 & 265 & 23 & -131 & 45 & 235 & -205 & 53 & 229 & -825 & -55 & 3575 & -1001 & 3003\\1 & -3 & -19 & 179 & 129 & -115 & -25 & 375 & -25 & -505 & 141 & 649 & -341 & -429 & 1001 & -5005\\1 & -1 & -21 & 63 & 189 & -45 & -75 & 175 & 175 & -315 & -189 & 693 & 231 & -3003 & -429 & 6435\\1 & 1 & -21 & -63 & 189 & 45 & -75 & -175 & 175 & 315 & -189 & -693 & 231 & 3003 & -429 & -6435\\1 & 3 & -19 & -179 & 129 & 115 & -25 & -375 & -25 & 505 & 141 & -649 & -341 & 429 & 1001 & 5005\\1 & 5 & -15 & -265 & 23 & 131 & 45 & -235 & -205 & -53 & 229 & 825 & -55 & -3575 & -1001 & -3003\\1 & 7 & -9 & -301 & -101 & 77 & 87 & 157 & -133 & -575 & -133 & 453 & 413 & 3887 & 637 & 1365\\1 & 9 & -1 & -267 & -201 & -33 & 59 & 423 & 149 & -95 & -243 & -1219 & -393 & -2301 & -273 & -455\\1 & 11 & 9 & -143 & -221 & -143 & -39 & 143 & 221 & 715 & 307 & 849 & 187 & 821 & 77 & 105\\1 & 13 & 21 & 91 & -91 & -143 & -117 & -533 & -247 & -455 & -133 & -273 & -47 & -167 & -13 & -15\\1 & 15 & 35 & 455 & 273 & 143 & 65 & 195 & 65 & 91 & 21 & 35 & 5 & 15 & 1 & 1\end{array}\right]$$

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I had the same question on zhihu. Also python (sympy) script generating them.

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A question on weak sequential completeness

For $(a_{n})_{n=1}^{\infty}\in c_{00}$, we define a norm by $$\|(a_{n})_{n=1}^{\infty}\|:=\sum_{n=1}^{\infty}|a_{n}-a_{n+1}|.$$ Let $X$ be the completion of $(c_{00},\|\cdot\|)$.

Let $Y$ be the linear space of scalars defined by $$Y:=\{(b_{n})_{n=1}^{\infty}\subset \mathbb{K}:\sup_{n}|\sum_{j=1}^{n}b_{j}|<\infty\}$$ endowed with the norm $$\|(b_{n})_{n=1}^{\infty}\|_{Y}:=\sup_{n}|\sum_{j=1}^{n}b_{j}|.$$ A standard argument shows that $X^{*}$ is isometrically isomorphic to $Y$ via the mapping $x^{*}\mapsto \langle x^{*},e_{n}\rangle$.

Question. Is $X$ weakly sequentially complete ?

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Stability of Pontryagin difference

In my scientific work a lot depends on the so called Pontryagin difference which is sort of the opposite operation for the famous Minkowski sum and can be defined as follows$\colon$

Definition

For two subsets $X, Y\subset\mathbb{R}^n$, we denote $\{s\in\mathbb{R}^n\vert s+X\subset Y\}$ by $Y\ominus X$. It is called the Pontryagin difference.

end of the definition

We will say that $Y$ is $X$ decomposable if $Y=X+(Y\ominus X)$.

We may wonder does this property hold when we shift the facets of $Y$ a little bit using the set of parameters $t_i$. Namely

Definition

Let $Y$ be given by the system of linear inequalities $\{\alpha_i.x\leq b_i\}_{i=1}^{F}$ where $F$ is the number of facets of $Y$. We will denote $\{\alpha_i.x\leq tb_i(1+s_i)\}_{i=1}^{F}$ by $X(t,s)$ for any $s=(s_1,\ldots,s_i,\ldots, s_F)$. We will say that $Y$ is stably $X$ decomposable if there exists $\epsilon>0$ such that $Y(1,s)=X+(Y(1,s)\ominus X)$ for all $s_i$ from the range $0<s_i<\epsilon$.

end of the definition

In my preprint, section 5, I have shown the following fact which I believe is correct since I didn't find any mistakes in it yet, but I cannot know for sure because it is very technical, though it relies on a property of $\ominus$ which is not very difficult (see theorem 5.6 on the page 18).

Proposition

Let $X, Y\subset\mathbb{R}^n$ and $Y$ is $X$ decomposable, then $tY$ is stably $X$ decomposable for any $t>1$.

end of proposition

Intuition standing behind this is that if you can color a polytope with another polytope from the inside without rotating them, then you can still do it after slightly shifting the facets of the first polytope even though its geometry might change. For example the top vertex of a pyramid can split if you move its side wall but you can still color it with a much smaller version of the initial pyramid. It is not a proof of the proposition of course just a picture to keep in mind.

Just wanted to ask is anybody familiar with questions like this or studied them in the past. Because I would really like to look through similar results in literature and/or maybe even find a mistake in my proof if it's wrong.

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Distribution of the change in Hamming distance between two sequences

Suppose I have two strings $s_1$ and $s_2$ of equal length $L$ with an alphabet size of $k \geq 2$. Suppose further that these two strings initially have a Hamming distance equal to $d_0 = H(s_1,s_2)$.

I then proceed through each character in one of the strings (say $s_2$), changing with probability $\mu$ the character to another distinct value in the alphabet, uniformly from the $k-1$ choices remaining. The number of edits is therefore distributed $\text{Binomial}(\mu,L)$.

Denote the new string $s'_2$. Can I say anything about the distribution of $d_0 - H(s_1,s'_2)$?

I can see that the likelihood of a positive change versus a negative change must be dependent on $d_0$, as clearly edits to two strings with a large number of matches is more likely to induce an increase in Hamming distance rather than a decrease.

I can also see that it might not be possible to give a complete answer for the case where the alphabet size $k>2$, since a flip of an unmatched character may or may not result in a match given we don't know the exact contents of the string $s_1$ it is being compared to. In this instance, we can assume that the string $s_1$ and $s_2$ were initially sampled uniformly from the space of possible strings, which I think should make it tractable.

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Up to projectivities, which configurations of four lines in $\mathbb{P}^3$ can one distinguish?

Background

I am interested in the projective classification of reduced curves of degree four in $\mathbb{P}^3(\mathbb{R})$ (and more generally of degree $n+1$ in $\mathbb{P}^n(\mathbb{R})$). More precisely, I am looking at the case where the curve is a union of four distinct lines. I need this classification because I want to make sure that I consider all possible cases in a problem in interpolation theory.

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For instance, there are two types of configurations of three lines in $\mathbb{P}^2$. Either three lines meet in a single point, or three lines meet in three distinct points. More generally, according to this integer sequence, there are 3 configurations of four lines in $\mathbb{P}^2$, 5 configurations of five lines in $\mathbb{P}^2$, and 18 configurations of 6 lines in $\mathbb{P}^2$. These configurations are shown in this figure (except for the configurations in which all lines are concurrent).

I believe there are six configurations of three lines in $\mathbb{P}^3$: Two configurations for which the three lines lie in a plane, three configurations for which precisely two of the three lines lie in a plane, and one configuration where none of the lines intersect.

My (related) questions are now as follows:

1. How many configurations are there of four lines in $\mathbb{P}^3$ (and more generally of $n+1$ lines in $\mathbb{P}^n$)?
2. Is there a convenient way to enumerate these?

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Unramified action of elliptic curve with supersingular reduction

Let $E$ be an elliptic curve over a $p$-adic local field $K$. Suppose $E$ has good supersingular reduction. According to Serre's famous paper on open image theorem or this paper, there are two cases of the action of the inertia $I_K$ on $E[p]$ depending on the Newton polygon, which I think I understood.

How to describe the unramified part, equivalent the Frobenius element, of the Galois action? Not all cases are included in the papers. In Serre's paper, he only described the case when the ramification index is $1$. It seems messier when $e>1$.

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A direct construction of the K(1)-local Atiyah-Bott-Shapiro orientation of KO

Throughout, I fix a prime $p$ (odd, if necessary), and I let $\mathrm{KO}_p$ be $p$-adically completed real $K$-theory.

From Section 7 of Ando-Hopkins-Rezk, we know that the space of $E_\infty$-string orientations on $\mathrm{KO}_p$ (which, specifically, identifies with the space of $E_\infty$-ring maps $\mathrm{map}_{E_\infty}(\mathrm{MString}, \mathrm{KO}_p)$) is non-empty, and upon choosing a point identifies with $\mathrm{map}(\mathrm{KO}_p, \mathrm{KO}_p)$. In this case, there is a canonical choice of point (see the proof of Corollary 7.12) coming from the fact that the $K(1)$-localisation of $\mathit{bstring}$ is $\mathrm{KO}_p$ itself.

The precise recipe from Section 7 of AHR, as far as I understand it, for getting a self-map of $\mathrm{KO}_p$ given an $E_\infty$-string orientation of $\mathrm{KO}_p$ is as follows: An $E_\infty$-string orientation of $\mathrm{KO}_p$ is a map $\alpha : gl_1(\mathbb{S}) / \mathrm{string} \to gl_1(\mathrm{KO}_p)$ of spectra such that the composition $$gl_1(\mathbb{S}) \to gl_1(\mathbb{S}) / \mathrm{string} \to gl_1(\mathrm{KO}_p)$$ is the functor $gl_1$ applied to the unit map $\mathbb{S}\to \mathrm{KO}_p$, where $\mathrm{string} := \Sigma^{-1}(\tau_{\ge 8}\mathrm{ko})$ and where $\Sigma \mathrm{string} = \tau_{\ge 8}\mathrm{ko} \to \Sigma gl_1(\mathbb{S})$ is the j-homomorphism; I denote by $gl_1(\mathbb{S})/\mathrm{string}$ the cofibre of this latter map. As a consequence of the Adams conjecture (see diagrams (7.7) and (7.8) of AHR), one gets a diagram

\begin{array}{ccccc} gl_1(\mathbb S) & \xrightarrow{\ } & gl_1(\mathbb{S})/\mathrm{string} & \xrightarrow{\ } & \Sigma \mathrm{string} \\ \big\downarrow\!{} && \big\downarrow\!{\small{\Phi B_c^{-1}}} && \big\downarrow \\ L_{K(1)}\mathbb S & \xrightarrow{} & \mathrm{KO}_p & \xrightarrow{\,\psi^c-1\,} & \mathrm{KO}_p \end{array} where the rows are fibre sequences, the vertical arrows present the bottom row as $K(1)$-localisations of the top, and the $\psi^c$ denotes the Adams operations by the topological generator $c\in \mathbb{Z}_p^\times/\{\pm 1\}$. I am using $\Phi B_c^{-1}$ for the middle vertical arrow in accordance with the notation of AHR. With this, the self-map of $\mathrm{KO}_p$ associated to the map $\alpha$ is then the composition $$\mathrm{KO}_p \xrightarrow{\Phi B_c} L_{K(1)} (gl_1(\mathbb{S}) / \mathrm{string}) \xrightarrow{L_{K(1)} \alpha} L_{K(1)} gl_1\mathrm{KO}_p \xrightarrow{\log_1} \mathrm{KO}_p$$ where $\log_1$ is Rezk's log for $\mathrm{KO}_p$.

Viewing the ABS orientation as an element $g_{ABS}\in \mathrm{map}(\mathrm{KO}_p, \mathrm{KO}_p)$, we even know what it does on homotopy groups: namely (by Corollary 9.10 of AHR), $g_{ABS}$ acts on $\pi_{2k}$ (for $k\ge 4$) by mutliplication by $-(1-p^{k-1})(1-c^k)\frac{B_{k}}{2k}$ where $B_{k}$ are the Bernoulli numbers and $c$ is a topological generator of $\mathbb{Z}_p^\times/\{\pm 1\}$.

My main question is the following:

Does anyone know of an "algebraic", perhaps purely $K(1)$-local, direct construction of the Atiyah-Bott-Shapiro orientation as a map $\mathrm{KO}_p \to \mathrm{KO}_p$? I.e., rather than starting with $\alpha$, can we construct the composition given above directly? By algebraic here, I mean something that doesn't resort to passing to the underlying space of the spectra involved.

The reason I believe such a construction should exist (apart from that it would be nice) is that such a construction already seems to be implicitly present (though somewhat obscured) in Lectures 26-29 lecture notes by Hopkins (transcribed by Mathew); indeed, Hopkins name-drops ABS at the end of Lecture 26.

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Splitting of normal bundle sequence for the blowup of a projective space

Let $Z$ be a smooth closed subvariety of $\mathbb{P}^n$. Let $X\rightarrow \mathbb{P}^n$ be the blowup of $\mathbb{P}^n$ along $Z$, and let $E = \mathbb{P}(\mathcal{N}_{Z/\mathbb{P}^n})$ be the exceptional. Now let $Y \subseteq E$ be a smooth closed subvariety. I'm considering the splitting of the normal bundle sequence $$0 \rightarrow N_{Y/E} \rightarrow N_{Y/X} \rightarrow N_{E/X} \rightarrow 0. $$ There is a paper of Aluffi and Faber https://arxiv.org/abs/alg-geom/9206001 which on page 8 has a geometric argument that seems to imply the splitting as follows: let $y\in Y\subseteq E$ and let $v$ be a vector normal to $E$. Let $\tilde{v}$ be a lift of $v$ to a tangent vector. Then $f_*(\tilde{v})$ is a tangent vector to $f(y)$ in $\mathbb{P}^n$ which is normal to $Z$. Now extend $f_*(\tilde{v})$ to a tangent line $\ell\subseteq \mathbb{P}^n$ (here working in a projective space, where tangent directions can be extended canonically to lines, is crucial) and denote by $\tilde{\ell}$ the strict transform. Then $y\in E = \mathbb{P}(N_{Z/\mathbb{P}^n})$ can amazingly be recovered as the intersection $\tilde{\ell} \cap E_{f(y)}$. But more so, $\tilde{\ell}$ gives a tangent vector (up to scaling) to $y$ which which is normal to $E$. So we define the image of $(y, v)$ in $N_{Y/X}$ as the unique normal vector parallel to $\tilde{\ell}$ and whose image is $v$. It seems reasonably clear to me that this construction does not depend on our choice of $\tilde{v}$, since the image of a tangent vector to $E$ should be tangent to $Z$.

So now the question I have is (firstly whether my interpretation of the Aluffi-Faber argument is even correct!) whether there is a "direct" way to prove this (more specifically using homological algebra)? Indeed, this seems like a statement about computing extension classes of vector bundles. And indeed, given some assumptions (such as when $Y$ was a projective subbundle and $H^1(Z, N_{Z/X}) = 0$ if I recall correctly) then I believe that the relevant $\mathrm{Ext}$-group was zero. But I couldn't get anywhere in general, and I had no idea how to use the fact that the original variety was $\mathbb{P}^n$, which seemed crucial to the argument of Aluffi and Faber.

Any help would be appreciated.

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How to find/guess a polynomial sequence?

My question is motivated by the recent question and more recent appearance of its author Bruce Westbury. Most of you know that the best way to find a sequence of integers is looking for it on The On-Line Encyclopedia of Integer Sequences. (My personal success of using this powerful database is however less than 10%.) Another strategy (especially, when you suspect that your sequence is holonomic) would be trying a guessing package in a computer algebra system, like $\operatorname{gfun}$ in Maple.

What can we do if we need to identify a sequence of polynomials? (for simplicity with integer coefficients)

One recipe (which was used by my colleague for solving the problem in this question) is again to use the OEIS, since the latter contains many 2D examples as well (like the whole Pascal triangle of binomial coefficients and several subcollections from it). The chances are miserable (as Bruce's sequence shows). Even having some additional information (like knowing that the polynomials are $q$-analogues of a known integer sequence), there seems to be no general machinery or database to assist in identification. Are there algorithms (better implemented) for polynomials analogous to $\operatorname{gfun}$?

Thanks!

P.S. The tag "soft-question" here means that the question is indeed soft but also on software.

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Fundamental group admitting specialization map in the non-proper setting

We know that given a henselian/complete discrete valuation ring $A$ and a proper morphism $f:X \to \mathrm{Spec}(A)$, there is a well-defined specialization morphism from the etale fundamental group of the generic fiber to that of the special fiber. If I understand correctly, such a specialization map does not exist if $f$ is non-proper. I was wondering if there is any other notion of fundamental group (eg. Tannakian fundamental group/pro-etale fundamental group, etc.) for which there is a well-defined specialization morphism in the case when $f$ is a smooth, quasi-projective, surjective morphism of finite type to $\mathrm{Spec}(A)$, but not necessarily proper?

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