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Wednesday, February 28th, 2024
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2:46 pm
A reliable reference for the statement every $k$-tree is uniquely $(k + 1)$-colorable

I see that every $k$-tree is uniquely $(k + 1)$-colorable in Uniquely_colorable_graph. Wikipedia does not cite any references, even though I know that its proof is not difficult by using mathematical induction.

Are there any suitable books to refer to?

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2:17 pm
On intersection of null geodesics

Let us consider a globally hyperbolic Lorentzian manifold $(M,g)$ with empty cut locus. Suppose that $p$ is a point in $M$ and consider $C^-(p)$ to be the past null cone in $M$ emanating from the point $p$. Next, suppose that $q_1,q_2 \in C^-(p)$ and denote by $\gamma_{q_1}$ and $\gamma_{q_2}$ the null geodesics that emanate from $q_1$ and $q_2$ respectively in the direction of the null vector that is normal to the null cone there.

Is it true that $\gamma_{q_1}$ and $\gamma_{q_2}$ can never intersect in $M\setminus C^-(p)$?

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1:51 pm
Unusual boundary value problem

In F.V. Atkinson's book, Discrete and continuous boundary value problems, (1964), there is the following BVP described in chapter 2.11: \begin{equation} \begin{cases} y'=-\frac{i}{\lambda}y,\quad &-1\le x\le 0; \\ y'={i}{\lambda}y,\quad &0< x\le 1;\\ y(-1)=y(1)\neq 0. \end{cases} \end{equation}

Eigenvalues of this BVP are determined from $\lambda-\lambda^{-1}=2\pi n$, $n\in\mathbb{Z}$ and the corresponding eigenfunctions satisfy the orthogonality relation $$ \int_0^1\overline{y(x,\lambda_r)}y(x,\lambda_s)\,dx+(\lambda_r\lambda_s)^{-1}\int_{-1}^0\overline{y(x,\lambda_r)}y(x,\lambda_s)\,dx=h_r\delta_{rs}. $$

Has anybody encountered such BVP before? Atkinson does not give any additional information or references.

Q: Where one can find more information concerning this type of BVPs?

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1:51 pm
Embedding $\mathrm{SL}_n(3)$ into $\mathrm{SL}_n(p)$

$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime. It is easy to show that $\SL_2(3)$ can be embedded into $\SL_2(p)$.

Now, let $n$ be an integer larger than $2$.

Question: In which circumstances, $\SL_n(3)$ can be embedded into $\SL_n(p)$?

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1:17 pm
Reference request for a subfamily of regular graphs

[Repost of same question math stack exchange which got no answers]

I'm looking for literature on the following family of graphs:

Call a regular graph $G=(V,E)$ (of regularity degree $d$) nice if there exists a coloring $C:V\to \{ 1,\dots,d \}$ such that each vertex $v$ has precisely one neighbour of each color.

In particular, I'm wondering

Qustion. Is the graph on $V=S_n$ with $x\sim y$ iff $y=x\tau$ for some transposition $\tau$ nice?

Here is what I know so far about nice graphs:

By double counting $\{(x,y):x\sim y, C(y)=i\}$ we see that in a nice graph we have $d\mid n = |V|$ and each color appears on $n/d$ vertices. However this property is not sufficient, as a $6$ cycle is $2$ regular but not nice.

An example of a nice graph: $V=\{0,1\}^8$, $x\sim y$ iff they differ in precisely one bit (Hamming distance 1). The coloring of $(x_0,\dots,x_7)$ is $(x_0\cdot 0)\oplus (x_1\cdot 1) \oplus \dots \oplus (x_7\cdot 7)$, with $\oplus$ denoting xor. (Of course $8$ can be replaced with any power of two).

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1:17 pm
Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process

Consider the modified Ornstein–Uhlenbeck process $$\mathop{dx_t}=\theta(y_t-x_t)\mathop{dt}+\sigma\mathop{dW_t}$$ for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the sufficiently smooth function $\phi:\mathbb{R}\rightarrow\mathbb{R}$ such that $\phi(x):=\lim_{t\rightarrow\infty}\mathbb{E}\left[y_t\mid x_t=x\right]$ and $y_t$ is deterministically dependent on $x_t$ somehow (i.e. $\mathop{dy_t}=f(x_t,y_t)\mathop{dt}$). The limit ensures that we are referring to the stationary conditional expectation only depending on the value of $x_t$, rather than one which also varies over time. Implicitly it is assumed that both $x_t$ and $y_t$ are stationary, mean-square differentiable random processes.

By Itô's lemma, $$\mathop{d\phi}=\left(\theta(y_t-x_t)\phi'(x_t)+\frac{\sigma^2}{2}\phi''(x_t)\right)\mathop{dt}+\sigma\phi'(x_t)\mathop{dW_t}.$$

Is the claim that $$\lim_{t\rightarrow\infty}\mathbb{E}\left[\left.\frac{dy}{dt}\right|x_t=x\right]=\theta(\phi(x)-x)\phi'(x)+\frac{\sigma^2}{2}\phi''(x)$$ correct and, if not, is it possible to express the above limit in terms of $\phi$ and its derivatives?

Edit: Unless I've made an error, the claim in my question implies that \begin{align} \mathbb{E}\left[\left.\frac{dy}{dt}\right|x_t=x\right]&=\mathbb{E}\left[\left.\frac{d}{dt}\mathbb{E}\left[\phi_t\right]\right|x_t=x\right]\\ &=\mathbb{E}\left[\left.\frac{d}{dt}\mathbb{E}\left[\mathbb{E}\left[y_t\mid x_t\right]\right]\right|x_t=x\right]\\ &=\mathbb{E}\left[\left.\frac{d}{dt}\mathbb{E}\left[y_t\right]\right|x_t=x\right] \end{align} in the stationary limit. Is this true?

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12:45 pm
Multidimensional power series with coefficients equal to an order of stabilizer of a set of powers

I have encountered a necessity to work with a series of the following form.

There are $N$ variables $x_1,\ldots x_N$. It is convenient to introduce monomial symmetric polynomials $m_{\lambda}$. They are defined for any power $r$ and a partition $\lambda$ of $r$ into $N$ parts, some of which can be 0, as sums of monomials with powers being elements of $\lambda$. For example, for $r=3$ and $N=3$ we have $m_{(3)}=x_1^3+x_2^3+x_3^3$, $m_{(2,1)}=x_1^2x_2+x_1^2x_3+x_2^2x_1+x_2^2x_3+x_3^2x_1+x_3^2x_2$, $m_{(1,1,1)}=x_1x_2x_3$. The power series is

$S(x_1,\ldots x_N)=\sum\limits_{r=0}^{\infty}\sum\limits_{\lambda_r}\frac{m_{\lambda_r}(x_1,\ldots x_N)}{L_{r,\lambda}}$.

Here the summation over $\lambda_r$ goes over all possible partitions of $r$ into $N$ parts with possible 0's, and $L_{r,\lambda}$ is a number of terms in $m_{\lambda_r}(x_1,\ldots x_N)$. In the example above, for $N=3$ and $r=3$ we will have, for $L_{3,(3)}=3$, $L_{3,(2,1)}=6$, $L_{3,(1,1,1)}=1$.

An alternative form of this series is

$S(x_1,\ldots x_N)=\frac{1}{N!}\sum\limits_{r=0}^{\infty}\sum\limits_{\lambda_r}N_{r,\lambda}m_{\lambda_r}(x_1,\ldots x_N)$

Here $N_{r,\lambda}$ is an order of the stabilizer of the parts of $\lambda$ under the action of the symmetric group $S_N$ acting on the variables $x_1,\ldots x_N$. Again, in the example above, for $N=3$ and $r=3$, $N_{3,(3)}=2$, $N_{3,(2,1)}=1$, $N_{3,(1,1,1)}=6$.

How can one compute a sum of such a power series? Or, maybe, a sum of some similar series with a factor of $1/L_{r,\lambda}$?

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12:15 pm
On the linearized evolution equations in general relativity

The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one recasts the Einstein equations as a first-order evolutionary problem with constraints. The main idea is the following: Let $\Sigma$ be a 3-dimensional (smooth, connected) manifold and fix one-parameter families of functions $\{N_{t}\}_{t}\subset C^{\infty}(\Sigma)$ and vector fields $\{X_{t}\}_{t}\subset\Gamma^{\infty}(T^{\ast}\Sigma)$ on $\Sigma$, smoothly depending on $t$ in a suitable sense. Then, we consider the equations \begin{align*} \text{Evolution}: \begin{cases} \partial_{t}h=&-2Nk+\mathcal{L}_{X}h\\ \partial_{t}k=&N\bigg[\mathrm{Ric}(h)+\mathrm{tr}_{h}(k)k-2(k\times k)\bigg]-\mathrm{Hess}_{h}(N)+\mathcal{L}_{X}k \end{cases} \end{align*} \begin{align*} \text{Constraints}: \begin{cases} \Vert k\Vert^{2}_{h}-\mathrm{tr}_{h}(k)^{2}-\mathrm{Scal}(h)&=0\\ \nabla\mathrm{tr}(k)-\mathrm{div}(k)&=0. \end{cases} \end{align*} where $\Vert k\Vert_{h}:=k^{ij}k_{ij}$, $(k\times k)_{ij}:={k_{i}}^{k}k_{kj}$ and $\mathrm{Hess}_{h}(N):=\nabla_{i}^{h}\nabla_{j}^{h}N$. These are equations for a $t$-dependent Riemannian metric $h_{t}$ on $\Sigma$ and a symmetric 2-tensor field $k_{t}$. Then, if $(-\varepsilon,\varepsilon)\ni t\mapsto (h_{t},k_{t})$ is a (local) solution to this system, the Lorentzian metric $g$ on $M_{\varepsilon}:=(-\varepsilon,\varepsilon)\times\Sigma$ given by $$g=-(N^{2}+X^{i}X_{i})\mathrm{d} t^{2}+2X_{i}d t\otimes \mathrm{d}x^{i}+h_{ij}\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}$$

is a (local in time) solution to the Einstein equations $\mathrm{Ric}(g)-\frac{1}{2}\mathrm{Scal}(g)g=0$, where $k_{t}$ is the second fundamental form of $\Sigma$. There has been a vast literature on the mathematical side, especially by Fischer, Marsden and Moncrief in the 1970s and 1980s on this approach (see e.g. [1] and [2] below for reviews). The constraint equations can be seen as a hyperbolic system and hence provide constraints on the choice of initial data. The solvability of these equations for a given $(N,X)$ is reviewed extensively in [3].

Now, lets say I am interested in the problem of linearized gravity in this setting. As a background, I want to consider a globally-hyperbolic manifold, i.e. a manifold globally of the form $M=\mathbb{R}\times\Sigma$ with $g=-\beta^{2}d t^{2}+h_{t}$ where $\Sigma$ is a Cauchy-hypersurface, $\beta$ a positive function and $h_{t}$ a one-parameter family of Riemannian metrics on $\Sigma$. Then, the Einstein equations for $g$ are equivalent to the ADM equations with $N:=\beta$ and $X:=0$. Now, lets say I linearize these equations around $(h,k)$ where $k:=\frac{1}{2\beta}\partial_{t}h$. Then, I get linear equations on the perturbations $(\overline{h},\overline{k})$.

Question: What should I do with the lapse and shift in the linearized setting? Now, my background manifold by definition has $N:=\beta$ and $X:=0$ globally. But if I just linearize these equations around $(h,k)$, then I only obtain perturbations $\overline{g}$ with the property $\overline{g}_{0i}=0$ (which somehow seems to be related to a gauge choice for the perturbation), which seems somehow restricted compared to the usual approach of linearized gravity (i.e. taking the 4D Einstein equations). On the other hand, if I take the full ADM equations above with non-trivial $N,X$ and linearize $(h,k,N,X)$ with background $(N,X)=(\beta,0)$, then I don't really know how I should handle the perturbations $(\overline{N},\overline{X})$, since they are in some sense just external parameters, which are freely specifiable.

Let me mention that the linearized constraint equations are discussed frequently in the literature, especially in relation to stability problems in mathematical relativity. The linearized evolution equations, however, are discussed much less, at least as far as I am aware of. An exception is a short discussion by Fischer-Marsden contained in Section 4 of the review [2] below.


Literature:

  • [1]: Fischer, Marsden: The initial value problem and the dynamical formulation of general relativity. In S. W. Hawking and W. Israel, editors, General relativity: an Einstein centenary survey, pages 138–211. Cambridge University Press 1979.
  • [2]: Fischer, Marsden: Topics in the Dynamics of General Relativity. In J. Ehlers, editors, Isolated gravitating systems in general relativity, pages 322–395. North Holland Publishing Company 1979.
  • [3]: Choquet-Bruhat. General Relativity and the Einstein Equations. Oxford University Press 2009.
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12:15 pm
Limit distribution of the self-normalized sum of Cauchy random variables

This is something that has come up in my research. I originally posted this question on CrossValidated but realized it might be better suited for this site. I have deleted the question there (in case it annoys someone).

Let $X_1, X_2, \dots, $ be a sequence of iid Cauchy$(x, 1)$ random variables for some $x \in \mathbb{R}$. Define, $$ S_n(x) = \frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n |X_i|},\quad n \in \mathbb{N}. $$ What, if it exists, is the limit of $S_n(x)$ as $n \to \infty$? I don't think an almost sure limit exists. Monte Carlo simulations suggest there should be a non-trivial limit. Perhaps, $S_n$ converges in distribution or in probability.

A paper I found in an answer to another question studies the above limit for stable distributions. It conveniently ignores the Cauchy case. They cite Feller's second volume but I haven't been able to find a result there. I wonder if any references consider the above limit for Cauchy random variables.

Another excellent answer I found concerning the median of the sum of half-Cauchy random variables. I think it suggests that the half-Cauchy might somehow belong to the domain of attraction of an asymmetric $\alpha=1$ stable law. If this is so, it might help understand the limit of $S_n$. But I am not sure.

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11:46 am
Is this a valid way to prove that for all nonzero algebraic X, e raised to the X is transcendental? [closed]

  1. Let multiplication signify conjunction

  2. Let addition signify disjunction

  3. Let N signify negation

  4. Let A signify X is nonzero

  5. Let NA signify that X is zero

  6. Let B signify that X is algebraic

  7. Let NB signify that X is transcendental

  8. Let C signify that e raised to the X is algebraic

  9. Let NC signify that e raised to the X is transcendental

  10. ∃X(ABC)→∀X∃X(AB→C)

  11. ∀X∃X(AB→C)=∀X∃X(B→NA+C)

  12. ∀X∃X(B→NA+C)=∀X∃X(BNC→NA)

  13. (12) is false, because it translates to for all X, if X is algebraic and e raised to the X is transcendental, then one of the Xs is zero. Since (12) is false, the antecedent in (10) is false, therefore ∀X(AB→NC) is true. ∀X(AB→NC) translates to, for all X, if X is algebraic and nonzero, then e raised to the X is transcendental.

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11:46 am
Regularity requirements for Sard's Theorem

The most common formulation of Sard's Theorem is that for $f\in C^{n-m+1}(\mathbb R^n, \mathbb R^m)$ with $n\ge m$, the set $f(C_f)$ has Lebesgue measure 0, where $C_f=\{x, df(x)=0\}$.

Question. Is it possible to weaken the assumption (and keep the conclusion) by assuming only $f\in C^{n-m}(\mathbb R^n, \mathbb R^m)$, $f$ is $n-m+1$ times differentiable with a locally bounded $n-m+1$ derivative?

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10:30 am
Sum of reciprocals of primes dividing Mersenne numbers

Let $p$ be a prime. Then it has been proven by Erdős that $$\sum_{p\mid 2^{n}-1}\frac1{p} \ll \log \log \log n.$$ Let $c<1$ be a positive constant. Then can we prove that $$\sum_{p\mid 2^{n}-1}\frac1{p^{c}}\ll \frac{\log ^{1-c} n}{\log \log n}?$$

It seems to me that arguments of Erdős should apply to this case and one should be able to obtain this bound. Thanks in advance for any assistance.

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10:06 am
Actual zeros of tropical Laurent polynomial

I consider the tropical semi-ring $(\mathbb{R},\oplus,\odot)=(\mathbb{R},\max,+)$. I know that the tropiclisation of any (Laurent) polynomial $p\in\mathbb{R}[x_1^{\pm1},...,x_n^{\pm1}]$ given a valuation $\nu$ simply is given by

$$\mathrm{Trop}(p)(X_1,...,X_n)=\bigoplus_{(i_1,...,i_n)} \nu(c_{i_1,...,i_n})\odot X_{i_1}^{\pm 1}\odot...\odot X_{i_n}^{\pm 1}$$

where $c_{i_1,...,i_n}$ are the coefficients of $p$. Now the zeros of $p$ interpreted as a hypersurface are translated via tropicalization to the non-differentiable "regions" of $\mathrm{Trop}(p)$, which give for example in the $n=2$ case rise to tropical curves via a possible construction using Newton polygons. This can, as far as I understood and so from examples, be extended to Laurent polynomials without any bigger issues.

Now, I am wondering, what the inverse would look like. So Assume that you have a tropical Laurent polynomial $L$ given by

$$L(X_1,...,X_n)=\bigoplus_{(i_1,...,i_n)} d_{i_1,...,i_n}\odot X_{i_1}^{\pm 1}\odot...\odot X_{i_n}^{\pm 1}$$

and some valuation $\mu$. Is there an interpretation of the solutions to $L(X_1,...,X_n)=0$, where $0$ is the neutral element with respect to $\odot$? If I take as an example $\mu$ on $\mathbb{C}$ as $\mu(z)=-\log(|z|)$, then the "un-tropicalization" looks like we are investigating the poles of $q=\mathrm{Trop}^{-1}(L)$.

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10:06 am
Some new questions on Rademacher complexity

For $A\subset R^n$,$A=(a_1,a_2,\dots, a_n)$, $\sigma_i$ are Rademacher random variable.

  1. Is $|\mathbb{E}_\sigma \inf_{a\in A}\sum_{i=1}^n\sigma_ia_i| \le |\mathbb{E}_\sigma \sup_{a\in A}\sum_{i=1}^n\sigma_ia_i| $

  2. Is there any result on the absolute value version of Rademacher complexity, i.e. $S(A) = \mathbb{E}_\sigma \sup_{a\in A}|\sum_{i=1}^n\sigma_ia_i|$

the "result" means, e.g. contract inequality, upper bound in some cases

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10:06 am
Is such a Banach space $X$ isometrically isomorphic to a Hilbert space or not?

Let $X$ be a real or complex Banach space. $X$ satisfies:There exists real or complex series $\{a_k\}_{k=1}^n,\{b_k\}_{k=1}^n$ (which satisfies that:$\begin{cases} a_k,b_k\in \mathbb{R}\ \forall k=1,\dots n,\ \sum_{k=1}^n a_k b_k =\frac{1}{2},& X\text{ is a real banach space}\\ a_k,b_k\in \mathbb{C}\ \forall k=1,\dots n,\ \sum_{k=1}^n a_k \overline{b_k} =1,\sum_{k=1}^n a_k b_k =0,& X\text{ is a complex banach space} \end{cases} $) such that for any continuous linear functional $f$ on $X$, there exists $y_f\in X$ such that $\forall x\in X,f(x)=\sum_{k=1}^{n}a _k(\|x+b_{k}y_{f}\|^2-\|x\|^2-\|b_{k}y_{f}\|^2)$, then is X isometric isomorphic to a Hilbert space?

I ask this question to generalize a conclusion I found(a converse of Riesz representation theorem):

Proposition. For banach space $X$,if $X$ satisfies that $\forall f \in X^*,\exists y_f \ \text{s.t.} \ \forall x\in X,f(x)=[x,y_f]$,where $\begin{equation} [x,y_f]:= \begin{cases} \frac{\|x+y_f\|^2-\|x-y_f\|^2}{4}& X\text{ is a real banach space}\\\\ \frac{\|x+y_f\|^2-\|x-y_f\|^2+i\|x+iy_f\|^2-i\|x-iy_f\|^2}{4}& X\text{ is a complex banach space} \end{cases} \end{equation}$(which is polarization identity in inner product space), then $X$ is isometric isomorphic to a Hilbert space.

For those interested in the specific proof process,please click on this Chinese link.

My main idea in proving above proposition is:

  1. Show that subset $M:=\{y_f:\forall f\in X^*\}$ is a subspace of $X$.
  2. Prove that $M$ is an inner product space and is isometric isomorphic to $X^*$.
  3. At this point, $X^*$ is a Hilbert space, so $X^*$ is reflexive, and $X$ is a banach space and thus reflexive, i.e., $X$ is isometrically isomorphic to the Hilbert space $X^{**}$.

But for the question I posed, I can't even prove that $M$ is a subspace of $X$.

                                                       --2024.02.07

The Cauchy-Schwarz inequality-like inequality seems to be very important in proving Proposition, so maybe start with that:

To what extent does the "Cauchy-Schwarz inequality" hold for a normed vector space not inner product space?

                                                       --2024.02.18

Consider that $f(x)=\frac{f(x)-f(-x)}{2}$, We can take $f(x)=\frac{1}{2}\sum_{k=1}^{n}a_{k}(\|x+b_{k}y_{f}\|^2-\|x-b_{k}y_{f}\|^2)$ (maybe more convenience?).

                                                       --2024.02.28
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9:30 am
Dimension and cardinality of fibers over the real numbers

If $X \subseteq \mathbb{C}^n$ and $Y \subseteq \mathbb{C}^m$ are irreducible affine varieties, and $f : X \to Y$ is a dominant polynomial map, then we know that (every irreducible component of) the fiber has dimension $\dim(X) - \dim(Y)$ over a Zariski-open subset of points in $Y$. Moreover, if $\dim(X) = \dim(Y)$, then the general fibers also have the same cardinality, which is the degree of $\mathbb{C}(X)$ over $\mathbb{C}(Y)$ with the inclusion induced by $f$.

The situation is more complicated over $\mathbb{R}$. Again, let $X \subseteq \mathbb{R}^n$ and $Y \subseteq \mathbb{R}^m$ be irreducible affine varieties. (I am not very familiar with scheme theory, and by an affine variety I still mean the vanishing locus of some polynomials.) Let $f : X \to Y$ be a dominant polynomial map. Hardt's theorem (Theorem 4.1 here) implies that $Y$ can be partitioned to finitely many nice (semialgebraic) parts such that the fibers over each part are "semialgebraically homeomorphic". I imagine (although I am not completely sure) that this implies that the fibers over each of these parts have the same dimension, and the same cardinality if zero-dimensional.

My question is: can we say more?

  1. Is there a full-dimensional semialgebraic subset of $Y$ over which the fibers have dimension $\dim(X) - \dim(Y)$?
  2. If $\dim(X) = \dim(Y)$, is there a full-dimensional semialgebraic subset of $Y$ over which the fibers have the same cardinality as the degree of $\mathbb{R}(X)$ over $\mathbb{R}(Y)$ (again with the inclusion induced by $f$)?
  3. If 1. and 2. are true, are they also true for semialgebraic sets? I.e., when $X$ is semialgebraic and $Y = f(X)$. In the case of 2. one would be comparing the cardinality of the fiber to the degree of $\mathbb{R}(\overline{X})$ over $\mathbb{R}(\overline{Y})$, that is, we take Zariski closures first.

I expect that 1. is true for both affine varieties and semialgebraic sets, but I have no intuition about 2.

I would also appreciate partial answers!

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8:18 am
Auslander-Reiten sequences where irreducible morphisms are all epi/mono

Let's work in the setting of modules over an Artin algebra $A$, or a finite-dimensional $k$-algebra $A$, or if you like, modules over a connected quiver $Q$ without oriented cycles.

Let $M$ be such a module, and suppose $M$ is indecomposable and non-injective. Then $M$ fits into an A-R sequence, $$0 \rightarrow M \xrightarrow{(f_i)} \bigoplus_i E_i \rightarrow N \rightarrow 0,$$ where $N = \tau^- M$, and the $E_i$ are the indecomposable summands of the middle term. We write $f_i \colon M \rightarrow E_i$ for the maps to each summand. These $f_i$ are irreducible morphisms, hence mono or epi.

Is there a nice condition under which all the $f_i$ are epi? This happens, e.g., for some modules for zigzag quivers of type $A_n$. But I can't find any sort of general treatment. I'm new to the whole quiver and Auslander-Reiten theory, so perhaps I can't find the right words to search for.

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8:18 am
A mean field SDE with hitting time

Let $b\in \mathbb R$ and $\sigma>0$ be given. For a fixed probability distribution $\mu_0$ on $\mathbb R$ s.t.

$$\int_{(0,\infty)}\mu_0(dx)=1,$$

consider the mean field SDE :

$$dX_t = \mathbf{1}_{\{X_t>0\}} \left[ bdt + \frac{\sigma}{1 + m_t {\bf 1}_{\{b>0\}}} dW_t \right],\quad \mbox{for all } t\ge 0,~~~~~~~~~~~~~~~~(\ast)$$

where $X_0\sim \mu_0$ is independent of the Brownian motion $(W_t)_{t\ge 0}$ and

$$m_t:=\int_{(0,\infty)}\mu_t(dx),\quad \mbox{for all } t\ge 0.$$

How can we show the existence and uniqueness of the (weak) solution to $(\ast)$?

Any answers, remarks or references are highly appreciated!

REMARK :

The case for $b\le 0$ is trivial. Indeed, $(\ast)$ reduces to $dX_t = \mathbf{1}_{\{X_t>0\}} \big[ bdt + \sigma dW_t \big]$ and the solution is given as $X_t=Y_{t\wedge \tau}$, where $Y_t:=X_0+bt+\sigma W_t$ and $\tau:=\inf\{t\ge 0: Y_t\le 0\}$. For the case $b>0$, $(\ast)$ turns to be

$$dX_t = \mathbf{1}_{\{X_t>0\}} \left[ bdt + \frac{\sigma}{1 + m_t} dW_t \right].$$

I do not find any literature on the existence of its solution.

REMARK 2 :

A heuristic argument is as follows : Let $\ell:\mathbb R_+\to [0,1]$ be some "nice" function. Consider the process

$$Y^{\ell}_t: = X_0+ bt+\int_0^t\frac{\sigma}{1+\ell(s)}dW_s.$$

Then we have $(Y^{\ell}_{t\wedge \tau^{\ell}})_{t\ge 0}$ is a solution to $(\ast)$ if $\mathbb P[\tau^{\ell}>t]=m_t$ for all $t\ge 0$, where $\tau^{\ell}:=\inf\{t\ge 0: Y^{\ell}_t\le 0\}$. Thus it remains to calculate the probability $\mathbb P[\tau^{\ell}>t]$ in terms of $\ell$. Is there any reference for this computation?

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7:47 am
Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic K Theory intersection

I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic topology/homotopy theory, and algebraic K theory. I would like to stress that this is a broad interest, and I am not exactly sure what to look at. One program in this intersection which has recently caught my eye is the homotopy theory of schemes and in particular motivic homotopy theory. Another program that has interested me is derived algebraic geometry.

I have looked at the following posts for guidance but still feel a little overwhelmed with the breadth of these areas of research.

About my background:

Of the relevant topics, I feel the most comfortable with algebraic geometry. I have some familiarity with classical varieties, schemes, and sheaf cohomology (via Hartshorne and large portions of Vakil’s notes). I am secondly most comfortable with algebraic topology (at the level of Hatcher’s book) and homotopy theory (May’s Concise). I am also roughly familiar with some topological K theory at the level of Milnor-Stasheff.

On the other hand, my understanding of stable homotopy theory and higher algebra/homotopical algebra is quite weak. I am also not familiar with the modern weaponry in algebraic geometry such as stacks and étale cohomology.

Question(s). I would like some suggestions as for:

  1. What directions to move toward given my interests

  2. Roadmaps and references for those directions, and in particular derived algebraic geometry and motivic homotopy theory given my background

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7:15 am
Convergence of slice in an equivalent renorming

Let us consider $\ell_2$ space with $\Vert \cdot \Vert_2$ norm. Let us define a new norm equivalent to $\Vert \cdot \Vert_2$ norm as follows: $$ \Vert x \Vert_0 = \max \{ \Vert x \Vert_2, \sqrt{2} \Vert x \Vert_\infty \}, $$ where $\Vert \cdot \Vert_\infty$ is the supremum norm. Let us denote $X=(\ell_2, \Vert \cdot \Vert_0)$. Then $X$ is a reflexive, non-rotund Banach space that does not have the Kadec-Klee property. Now, I want to check whether "for every $f \in S_{X^*}$ and for every $(x_n) \subseteq B_X$ satisfying $f(x_n) \to 1$, $d(x_n, S(X, f, 0)) \to 0$ or not". Here $$S(X, f, 0)= \{ x \in S_X : f(x) \geq 1 \}. $$ Can anyone help me with this? I have tried to find an example that does not follow the above criteria, but I am confused whether I am correct or not. So, I have taken $x=\frac{e_1+e_2}{\sqrt{2}}$, $x_n = \frac{e_1+e_n}{\sqrt{2}}$ and $f=(\sqrt{2}, 0, \cdots)$. Then $f(x_n) =1$ and $x \in S(X, f, 0)$; but $\Vert x_n - x \Vert_0 =1$. Could anyone please clarify whether this example is correct or not? Thank you.

One additional question I want to add here: If I slightly change the new norm to $$ \Vert x \Vert_1=\max \{ \frac{1}{\sqrt{2}} \Vert x \Vert_2, \Vert x \Vert_\infty \}.$$ Does $(\ell_2, \Vert \cdot \Vert_1)$ satisfies the above property?

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