Let $R$ be a commutative unital ring and let $i$ be a non-negative integer such that $K^i_{alg}(R)$ is finitely generated abelian group. Is it possible that there does not exist weak homotopy type of finite CW complex $X$ such that $KO^i(X)\approx K^i_{alg}(R)$?

I am not an expert in calculus of variations and I am getting pretty lost in the vast literature. I've been studying the following functional $$ \int_\Omega (|\nabla f|+|\nabla g|)^2 dxdy $$ where $\Omega \subset \mathbb{R}^2$ is open and bounded and $f,g$ are real valued.

Fixing boundary values for $f$ and $g$, do you know any regularity result that can be applied to the minimizer of this functional?

If we assume $g$ is fixed, the minimizer will satisfy the Euler-Lagrange equation $$ \mathrm{div}\left(\left(1+\frac{|\nabla g|}{|\nabla f|}\right)\nabla f\right)=0$$

which is singular. Do you know any reference, where this kind of singularity has been studied?

The Azuma-Hoeffding Inequality says that if $X_1,X_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X_i - X_{i-1}\| \le 1$ say, then we should not expect the difference $\|X_N - X_0\|$ to grow *too fast*. Formally we have

$$P\left(|X_N -X_0| > \epsilon N\right) \le \exp\Big ( \frac{- \epsilon^2 N}{2 }\Big) \mbox{ for any }\epsilon > 0.$$

Note the inequality has nothing to do with how *spread out* the variables $X_i$ are. It only uses how the differences are bounded. In case all $X_i$ have small variance we'd expect stronger concentration results.

For example suppose the $A_1 , A_2, \ldots$ are drawn independently from $\mathcal N(0,\sigma^2)$ distributions cut off outside $[-1,1]$ and each $X_i = A_1 + \ldots + A_i$. The above inequality does not distinguish between the cases when $\sigma^2$ is large and small. When it is smaller we should expect a stronger concentration around the mean. In the degenerate case when $\sigma^2=0$ and all $A_i \equiv 0$ the left-hand-side will be exactly zero.

Are there any modifications of Azuma-Hoeffding that take into account the variances of the conditined variiables $ X_{i+1} | X_i, \ldots, X_1$ ? So far I have only found this paper in information theory. The theorem 2 is a version of AH that involves the variance. However that paper is quite recent, and it seems likely the problem has been considered by probabilists in the past.

Can anyone point me in the right direction?

Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$ such that $\sum\limits_{i=1}^K z_i = 0$, does there exist an eigenfunction of the Laplacian $f: \mathbb{T}^d \to \mathbb{R}$ such that $sgn(f(\theta_i)) = z_i$ for all $i$?

Here sgn is signum function.

Inspired by this question

Let $X$ be an integral quasi-compact scheme, $\pi : U \to X$ an étale morphism and $g : \eta \to X$ the generic point of $X$. Then $\Gamma(U_{\eta}, \mathscr{O}) = R(U)$, the ring of rational functions?

This relates the "étale version of sheaf of Cartier divisors". (I want to show that $g_* \mathbb{G}_m = R^* $, the sheaf of the unit group of the rational functions on the étale site on $X$. If this is true, the Cartier divisor sheaf is the cokernel of the canonical injection $ \mathbb{G}_{m,X} \to g_*\mathbb{G}_{m, \eta}$.)

Since for an étale morphism $ f : A \to B$ and a prime ideal $\mathfrak{p}$ of $B$ we have $f ^{-1}(\mathfrak{p}) = 0 \iff \dim B_\mathfrak{p} = 0 \iff \mathfrak{p}$ is minimal, it's easy in the case that $X$ and $U$ are affine. Please show this in general.

Thank you very much.

Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\pi^{-1}(s)$ is a curve in $\mathbb{C}^2$ (under the restriction of the inclusion of $\mathcal{C}$ into $\mathbb{C}^2 \times S$). Suppose further that $S$ is an integral scheme. If for all $s \in S$, the $\delta$-invariant of the corresponding curve $\mathcal{C}_s:=\pi^{-1}(s)$ is constant, can we then say that $\pi$ is flat? I think this is true if $S$ is smooth. Here, I weaken it to integrality.

Any hint/reference will be most welcome.

Let us suppose that I'm in the following situation: I have a chain complex $(C,\partial)$ and say a finite group $G$ acting over $C$ up to homotopy, meaning that for each $g \in G$ I have a self chain homotopy equivalence $g_\#:C\to C$ such that $h_\# \circ g_\#\simeq(f\circ g)_\#$, where $\simeq$ is a chain homotopy.

If $\partial=0$ the situation would collapse to the one of a group acting over $C$ (simply a module in this case) and I could consider the group cohomology of $G$ with coefficients in $C$. So my question is:

Is there a meaningful notion of cohomology of $G$ with coefficients in a chain complex?

PS: of course I can take the homology and get a proper action of $G$ on $H_*(C)$ but I find this boring somehow.

For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\mathbb{R}$, finer than the usual topology and compatible with the (additive) group structure (i.e., $+$ and $-$ are continuous), such that $\lambda_d$ is, up to some normalization, the Haar measure for $(\mathbb{R},+,\mathscr{T}_d)$?

(For $d=1$ the usual topology provides a positive answer. For $d=0$ the discrete topology does. So the question is whether we can do something in between.)

Bonus points if $\mathscr{T}_d$ can somehow be made "canonical".

Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$.

Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ a smooth and proper morphism, $X := Y\times_SS_0$.

Berthelot’s comparison Theorem in crystalline cohomology says that we have:

$$R\Gamma(\text{Cris}(X/S),\mathcal{F}) = R\Gamma(Y,\mathcal{F}_Y\otimes_{\mathcal{O}_Y}\Omega^*_{Y/S})$$

where $\Omega_{Y/S}^*$ is the usual algebraic de Rham complex of $Y/S$, $\mathcal{F}$ is a finite type crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules, $\mathcal{F}_Y$ is given by formal GAGA after restricting $\mathcal{F}$ to $Y\times_S\text{Spec}(A/I^n)$ for every $n\ge 1$ and algebrizing.

In the literature usually one then says “and then one sees that the right side is functorial in $X$”.

This sounds to me like a bit of hand waving. The right side is functorial in $Y$ and who knows if endomorphisms of $X$ over $S_0$ lift to endomorphisms of $Y$ over $S$.

Here is the question:

Does this sentence actually mean that one can **declare** that the effect of an endomorphism $f : X\to X$ over $S_0$ on the right side **is** by decree the effect on the left side, pre and post-composed with the above canonical comparison map?

For background on this comparison isomorphism see the Stacks Project or Berthelot-Ogus.

Let $K$ be a field endowed with a rank (height) one valuation, which is not discrete. Let $R$ be its valuation ring.

Let $L$ be a separable finite field extension of $K$ and fix an extension of the valuation of $K$ on $L.$ Suppose that the residue fields of $K$ and $L$, as well as the value groups of their respective valuations are the same. Let $V$ be the valuation ring of $L$ (or equivalently, the integral closure of $R$ in $L$).

Is the inclusion $R \rightarrow V$ étale? I am not sure it is finitely presented.

I am following some lecture notes on Ricci flow and reached the section where we linearize the Ricci tensor and obtain the principal symbol for the resulting operator. We have $T \in \: \Gamma(Sym^2 T^{*}M)$ smooth, fixed and positive definite and then compute the time derivative for the divergence of $G(T)$:

$\bigg(\frac{\partial}{\partial t}\delta G(T) \bigg)Z = -T \bigg((\delta G(h))^{\#},Z\bigg) + \bigg<h,\nabla T(.,.,Z) - \frac{1}{2}\nabla_{Z}T \bigg>, $

where $\frac{\partial g}{\partial t}=h$, $Z$ an arbitrary vector field and $G(T)=T-\frac{1}{2}(tr \: T)h$.

The notes then say that this implies

$\frac{\partial}{\partial t}T^{-1}\delta G(T) = -\delta G(h)\: + \:\: ...$

where the dots indicate terms which don't have derivatives of $h$. I see that you can apply $T^{-1}$ as it doesn't depend on $t$ and then lose the $Z$ as it is arbitrary, but not sure exactly what happens to the right-hand side such that we end up with $-\delta G(h)$ or where the sharp goes.

The parities of the number of partitions of $n!/6$ and $n!/2$ appear to be non-random initially, as follows — is there an explanation for this other than chance? With $p$ being the partition function and $\lfloor \rfloor$ the floor,

$$ \begin{align} & p(n!/6) \equiv \lfloor n/3\rfloor\ (\text{mod}\ 2), && \text{for}\ 3\leq n\leq 19\\ & p(n!/2) \equiv 1\ (\text{mod}\ 2), && \text{for}\ 2\leq n\leq 15 \end{align} $$

However, neither statement is true for all $n$: $p(20!/6) \equiv 1\ (\text{mod}\ 2)$ and $p(n!/2) \equiv 0\ (\text{mod}\ 2)$ for $16\leq n\leq 19$. This is as far as I have computed (using SageMath with the "flint" algorithm).

Also, is there a faster algorithm for computing the parity of the number of partitions than first computing the number of partitions with the "flint" algorithm (which uses the Hardy-Ramanujan-Rademacher formula in an especially efficient way, as I understand)?

Any way to use the special form $n!/k!$ to speed evaluation, e.g., of a recurrence relation for $p$?

**tl;dr:** Are there known convergence estimates for approximating a function with a radial basis family?

**Details:** Let $\mathcal{G}$ be a family of radial basis functions, e.g. $\mathcal{G}=\{\exp(-\tfrac1{2\sigma^2}(x-\mu)^2) : \mu\in\mathbb{R}, \sigma\ge 0\}$ and let $\mathcal{G}_{m,\alpha}$ denote the set of all linear combinations of functions from $\mathcal{G}$ with at most $m$ terms and whose nonzero coefficients are bounded away from $\alpha$ (for completeness, $\alpha=0$ is allowed):
$$
\mathcal{G}_{m,\alpha}
= \Bigg\{\sum_{k=1}^m a_k g_k : g_k\in\mathcal{G},\, |a_k|\ge\alpha\cdot1(a_k\ne 0)\Bigg\}.
$$

Let also $\mathcal{F}$ be a (sufficiently regular) family of functions. Let $f\in\mathcal{F}$ and $g^*_{m,\alpha}$ be such that $$ d(f,g^*_{m,\alpha}) = \inf_{g\in\mathcal{G}_{m,\alpha}} d(f,g), $$

where $d$ is some metric on functions. (I am including the case $\alpha=0$ for this question. The reason for considering $\alpha > 0$ is that I am interested in avoiding pathologies with weights that tend to zero in the above projection as $m$ increases.)

I am interested in bounds of the form $d(f,g^*_{m,\alpha})\le C/m^\beta$ for some constant $C$ (which may or may not depend on $f$ and $\alpha$--I am mostly focused on the dependence on $m$). My questions are as follows:

- In the first place, are there known bounds of this form for nontrivial choices of $(d,\mathcal{F},C)$? I am being deliberately vague here in order to understand what types of results along these lines are available.
- Ideally, I am interested in the case where $\mathcal{F}$ is all smooth densities on $\mathbb{R}$, $\mathcal{G}_{m,\alpha}$ is restricted to
*convex combinations*of Gaussians, $\alpha>0$, and $d$ is a standard probability metric such as Hellinger or total variation.

**Some additional background:** These types of results are quite standard in the approximation theory literature when $\mathcal{G}$ is a set of algebraic or trigonometric polynomials (e.g. Bernstein approximation). Of course, it is well-known that $d(f,g^*_m)\to0$ when $\mathcal{G}$ is the set of Gaussian densities, however, I am not aware of any quantitative convergence estimates for this family. The one paper I am aware of is Li and Barron (2000) who consider the case $\alpha=0$.

Consider a metric space $(X,d)$ and fix a base point $w$. A horofunction is a function of the form $$\beta_y(x)=d(x,y)-d(w,y).$$ The map $y\mapsto \beta_y$ is an embedding of $X$ into the space of $1$-Lipschitz map vanishing at $w$. The closure of $X$ into this space is called the horofunction compactification of $X$, which is denoted by $\overline{X}^h$ in the following. Then, $\partial^hX:=\overline{X}^h\setminus X$, the complement of $X$, is called the horofunction boundary.

Finding the homeomorphism type of the horofunction boundary is often a difficult problem. We can still say something in some examples.

The horofunction boundary for $\mathbb{R}^d$ endowed with the Euclidean distance is a sphere of dimension $d-1$. We can actually describe the topology of the compactification itself: a sequence $x_n$ converges to a point in the horofunction boundary if and only if $x_n$ goes to infinity and $\frac{x_n}{\|x_n\|}$ converges to some $\theta\in \mathbb{S}^{d-1}$.

There are several possible generalizations of this example. For instance, the horofunction compactification of a CAT(0) space coincide with the visual compactification, see Bridson and Haefliger book metric spaces of non-positive curvature for more details.

Another way of generalizing the Euclidean case is to consider a nilpotent Lie group. Let us focus on Carnot groups. Such a group is endowed with a sub-Riemannian metric, which comes from the first step of nilpotency. The path-length distance associated to this sub-Riemannian metric is called the Carnot-Caratheodory distance.

**My questions are the following**:

1) Is there a description of the topology of the horofunction compactification/boundary for the Carnot-Caratheodory distance on a Carnot group ? The answer is positive for the Heisenberg group (see the paper of Klein and Nicas The horofunction boundary of the Heisenberg group: the Carnot-Caratheodory metric.

2) At least, can we construct two distinct Carnot-Caratheodory distances on a Carnot group such that the horofunction compactifications/boundaries do not have the same homeomorphism type ?

Let $x=(z_1,\ldots,z_n)$ be real vector and $(p_1,\ldots,p_n)$ be a probability vector.

Question$\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i \le ???$

ObservationThis paper allows us to upper-bound things like $\Delta_f(z,p):=f(\sum_{i}z_i p_i) - \sum_i f(p_iz_i)$, thus providing a kind of reversed Jensen's inequality.

Indeed, it was shown that

If $f$ is concave, $a:=\min_i z_i$ and $b := \max_i z_i$, then $$ \Delta_f(z,p) \le \max_{p,q \ge 0,\;p+q=1} pf(b)+qf(a) - f(pb+qa). $$

I could probably use this with $f=\log$ to bound $\Delta_{\log} (z,p)$.

Let $f \colon \mathbb R^N \to \mathbb R$ be a smooth function. Let $\mu$ be a probability measure on $[0,1]$ and $X_1, \ldots , X_N$ be i.i.d. random variables on $\mathbb R$.

**Question.** What is the maximum value of the expectation
$$
\mathbb E[\vert f(X_1, \ldots , X_N) \vert] = \int_{\mathbb [0,1]^N} \vert f(x_1, \ldots , x_N) \vert d\mu(x_1) \ldots d\mu(x_N)
$$
among all probability measures $\mu$ on $[0,1]$?

This question arises from this post and from this recent answer of Sangchul Lee which seem however tailored for specific functions $f$ and particularly for the case $N=2$.

I am very interested in the case $N\ge 3$; the function $f$ maybe be as smooth as needed (e.g. a polynomial). I have troubles in extending the (very elegant) variational approach of Sangchul Lee's to more variables, as no "bilinear form" is available.

If I am not mistaken, if one considers only a.c. measures then the question becomes: maximize $$ \int_{[0,1]^N} |f(x_1, \ldots, x_N)| g(x_1) g(x_2) \ldots g(x_N) dx_1 dx_2 \ldots dx_N, $$ among functions $g \ge 0$ such that $\int_0^1 g(s)\, ds = 1$. Can this be handled by the general Holder's inequality?

*Disclaimer*: I have asked this also on MSE but I have not found any answer yet.

Imagine a graph where the vertices and edges model an n dimensional hypercube (a line, a square, a cube and so on). A red vertex must have a minimum distance of 3 from every other red vertex. The problem is to maximise the number of red vertices for a given n.

I have absolutely no idea where to start. Any help is appreciated.

I am a second year (Pure) Math and (Theoretical) Physics undergraduate in India. I want to do a masters in Applied/Computational Science or Math, for which I have apply after next 7 months.

I have following questions:

Does having only theoretical knowledge of subjects, make me worthless candidate for applied stuff?

Since masters in many universities is not funded, are there any programmes or scholarships that can pay me at least for my tuition fee?

I have lots of electives left for next two semesters, what are some of the courses I should take( or you’ll take if you were me)?

With having no prior experience in applied stuff, what exactly should I write on my SOP when I apply?

(Sorry for my poor english skill..)

Let $N$ be a large integer and the set $X$ be the subset of $\mathbb{Z}/N\mathbb{Z}$. For two sets $A$ and $B$, we define \begin{equation} A+B:=\{a+b : a\in A, b\in B\}. \end{equation} Is there a bound of size X that satisfies $X+X=\mathbb{Z}/N\mathbb{Z}$?

My question is about how to prove

k(x,t)=i(x-t) is symmetric also k(x,t)=i(x+t) is not symmetric