Consider a polynomial ring $\mathbb C[x_1,\ldots,x_n]$ that is $\mathbb Z_{\ge 0}$-graded by degree. Let $I$ and $J$ be two homogeneous ideals therein with the same Hilbert series, i.e. with their homogeneous components of degree $n$ having the same dimension for any $n\in\mathbb Z_{\ge 0}$.

In its most general form my question is whether some kind of relationship between $I$ and $J$ follows from this condition. Or, in slightly different terms, whether one may somehow describe the set of all ideals with a given Hilbert series.

For instance, the aforementioned set of ideals is a poset with respect to taking initial ideals. Are there any general results concerning this poset?

Let $(X,F)$ a one-dimensional folication over a smooth variety $X$ over $\mathbb{Z}$ . Let $(X_p,F_p)$ the modulus $p$ reduction of $(X,F)$. We assume that $(X_p,F_p)$ is a foliation in positive characteristic, for almost every prime $p$. The Ekedahl- Barron $F$-conjecture says that with it hypothesis, the leaves of $(X,F)$ are algebraic curves.

Let $L$ be a very ample line bundle on $X$. Let $P$ a non-singular point of $(X,F)$. Let $C$ a leave of $(X,F)$ that contain $P$.

In https://drive.google.com/open?id=1_SNpE8FxC8BmO0n6sin8Ali8bKhtrL73 I have shown that if there existe a colection of $F_p$-invariant curves $C_p$ with genus $g_p$ for every prime $p$ then:

$$\chi(C,L)(n)\leq \limsup_{p \text{ prime}} (g_p+h^0(X,L))n$$

Were $\chi(C,L)(n)$ is the Hilbert- Samuel polynomial. It means that if $g_p$ is bounded for every $p$ then the leaves of $C$ are algebraic curves. It can prove it conjecture. My question is:

_ Is this inequality well known?.

_ There exist a way to bound the $g_p$'s?.

_ There exist more references about the Ekedahl-Barron $F$-Conjecture?.

I am looking for a standard name (if it exists) for the following property of a Schauder basis $(e_i)_{i=1}^\infty$ in a Banach space $X$:

$$\|\sum_{i\in F}x_ie_i\|\le\|x\|$$for any $x=\sum_{i=1}^\infty x_ie_i\in X$ and any finite subset $F\subset\mathbb N$.

This condiion implies that the Schauder basis is unconditional.
Can a Schauder basis with this property called *monotone uncounditional basis*? Or the latter term usually means something else?

My goal is to generate an irreducible polynomial over $GF(2^{12})$ with degree $t$, which can get fairly big, let's say up to $t=200$ or so. I've found this very helpful paper that walks me through the Ben-Or irreducibility test. I've implemented it, and it works perfectly for, um, $t\le5$.

Part of the algorithm requires computing $x^{q^i}-x \mod f$, where $f$ is a randomly-generated degree $t$ polynomial, the order $q=2^{12}$, and $i$ gets as high as $\frac{t}{2}$. I have a decently efficient adaptation of long division to compute the modulus. Unfortunately I'm using a polynomial library (JLinAlg, for Java) where the degree is represented as a 32-bit signed integer: $2^{q^i}$ is too large for $i>2$.

One option, of course, would be to re-implement polynomials to represent the degree with an arbitrarily large number. But I wonder, since I'm working in a field with characteristic $p=2$ and my dividend is so specific, if there's a better solution that doesn't require the big numbers at all?

Key Problem : Is there any theorem about eigenvalues or positive semi-definiteness of small size matrices with small integer elements?

I have to check positive semi-definiteness of many symmetric matrices with integer elements. First I used eigenvalues, but floating point round error happens : eig(in numpy) sometimes gives small negative eigenvalues when a matrix is actually positive semi-definite.

I know Sylvester's criterion for positive semi-definiteness can avoid this problem. But I really don't want to use it since it requires computation of determinant of all principal minors, and I have to deal with really many matrices($ > 10^{20}$). I have to do anything to reduce number of calculations.

All the elements are integer and have small absolute values($\ <3 $), sizes of matrices are also small($\ < 10 \times 10 $). It seems pretty good condition, so I believe someone already researched this kind of matrices. Does anyone know useful theorems for this situation?

P.S: I'm pretty newbie in English Internet community and not native English speaker. So if you find something awkward, pardon me and let me know.

Suppose $G$ is a locally compact compactly generated group. Let $\mu$ be a probability measure that is:

Adapted to $G$, i.e. there is no proper subgroup $H$ such that $\mu(H)=1$.

Symmetric, i.e. $\mu(A)=\mu(A^{-1})$.

compactly supported, and $1_G$ is in the support.

$\mu<<\lambda$, where $\lambda$ is Haar.

Let $K$ be a compact set with $\lambda(K)>0$. Let $\mu^{*n}$ be the $n$-fold convolution $\mu*...*\mu$.

Do there exist $c>0$ and $n>0$ such that $\frac{d\mu^{*n}}{d\lambda}\geq c>0$ on $K$?

In other words, do a $\mu$ random walk, when hitting $K$, after many steps, hits it (in a sense) uniformly?

Is there any easy way of finding supremum of the quantity $\sum_{i,j=1}^n|z_i-z_j|$ where $|z_i|=1$ for $1\leq i\leq n.$ We are considering complex variables of course.

Here :

it is conjectured that the number $$(2^k-1)\cdot 10^m+2^{k-1}-1$$ where $m$ is the number of decimal digits of $2^{k-1}-1$ (See also the title), is never prime when it is of the form $7s+6$. Amazingly, primes with the other residues exists although the residue $6$ occurs twice often than $1$.

Upto $k=120\ 000$, there is no counter-example, but probably a counter-example exists because the growth rate is roughly $4^k$ and $1$ out of $9$ $k's$ lead to a number of the form $7s+6$.

How large will the smallest counterexample probably be ?

Let $(u_1, u_2, u_3, u_4)$ and $(v_1, v_2, v_3, v_4)$ be vectors in $\mathbb R_+^4$. Is the following inequality true?

\begin{align*} \left(\sum_{{[4] \choose 3}} \sqrt{u_i u_j u_k}\right)^{2/3} + \left(\sum_{{[4] \choose 3}} \sqrt{v_i v_j v_k}\right)^{2/3} \leq \left(\sum_{{[4] \choose 3}} \sqrt{(v_i+u_i) (v_j+u_j) (v_k+u_k)}\right)^{2/3} \end{align*} Here $\sum_{{[4] \choose 3}}$ refers to $\sum_{1\leq i < j < k \leq 4}$.

I ran a million Matlab simulations for random vectors and it did not yield any counterexample.

Note: Previously asked on MSE (https://math.stackexchange.com/questions/2647608/a-minkowski-like-inequality-for-symmetric-sums)

I'm studying the difference between regularization in RKHS regression and linear regression, but I have a hard time grasping the crucial difference between the two.

Given input-output pairs $(x_i,y_i)$, I want to estimate a function $f(\cdot)$ as follows \begin{equation}f(x)\approx u(x)=\sum_{i=1}^m \alpha_i K(x,x_i),\end{equation} where $K(\cdot,\cdot)$ is a kernel function. The coefficients $\alpha_m$ can either be found by solving \begin{equation} {\displaystyle \min _{\alpha\in R^{n}}{\frac {1}{n}}\|Y-K\alpha\|_{R^{n}}^{2}+\lambda \alpha^{T}K\alpha},\end{equation} where, with some abuse of notation, the $i,j$'th entry of the kernel matrix $K$ is ${\displaystyle K(x_{i},x_{j})} $. This gives \begin{equation} \alpha^*=(K+\lambda nI)^{-1}Y. \end{equation} Alternatively, we could treat the problem as a normal ridge regression/linear regression problem: \begin{equation} {\displaystyle \min _{\alpha\in R^{n}}{\frac {1}{n}}\|Y-K\alpha\|_{R^{n}}^{2}+\lambda \alpha^{T}\alpha},\end{equation} with solution \begin{equation} {\alpha^*=(K^{T}K +\lambda nI)^{-1}K^{T}Y}. \end{equation}

What would be the crucial difference between these two approaches and their solutions?

Usually the question whether the diamond principle $\diamondsuit(\kappa)$ holds for some large cardinal $\kappa$ only concerns large cardinal notions of very low consistency (among the weakly compacts). Partly since it *does* hold for all subtle cardinals, which are only barely stronger than the weakly compacts, and pretty much every large cardinal notion below a weakly compact has been shown to consistently *not* satisfy it (see Failure of diamond at large cardinals and Ben Neria ('17)).

That subtle cardinals satisfy diamond of course means that almost all large cardinals *do* satisfy it as well, but there are some strange ones lying around though, including Woodin cardinals and inaccessible Jónsson cardinals. Is anything known about diamond holding for any of these two?

Let $M$ be a (possibly simply connected) compact manifold $M$. Are there always non-zero classes in the homotopy or homology of $\mathrm{Diff}(M)$ that directly arise from the topology of $M$ itself?

As an example of the type of answers I am looking for I construct non-zero classes in the homotopy and homology of the loop space $\Omega(M)$, which come from the topology of $M$.

Let $M$ be simply connected. Then there is a smallest positive dimension $d$ where $H^d(M)$ is nonzero. Hence $\pi_d(M)\cong H_d(M)$ is non-zero by Hurewicz' Theorem. The long exact sequence in homotopy of the pathspace fibration shows that $\pi_{d-1}(\Omega M)\cong \pi_d(M)$. Applying Hurewicz' Theorem again we see that $H_{d-1}(\Omega M)\cong \pi_{d-1}(\Omega M)\cong \pi_d(M)\cong H_d(M)$. Thus the homology and homotopy have non-trivial elements that come from the topology of $M$.

Is there a known relation between the Hochschild cohomology of group algebras and cohomology of groupoids?

Clarification: It is known that 1-dimensional Hochschild cohomology of the Group algebra C[G] is isomorphic to so-called external derivations of C[G]. On the other hand, it is known that the space of derivations can be identified with so-called characters on a groupoid aG of adjoint actions of the group G.

Therefore 1-dimensional Hochschild cohomology can be identified with 1-dimensional cohomology of the Cayley complex of the groupoid aG in the case when the group G is a finite presented group. I would like to know if a similar identification is known for the Hochschild cohomologies of higher dimensions

For a commutative ring $R$ with unity, I am looking for an equivalent condition for an ideal $T$ to have the property that $T$ contains a unique maximal proper subideal, equivalently, the sum of proper subideals of $T$ is not equal to $T$.

I am reading the lecture notes INTRODUCTION TO DONALDSON–THOMAS INVARIANTS. I have a question in the end of page 1 about the proof of a map is an automorphism.

Let $m>0$ be an integer. Let $\overline{A} = Q[x_1, x_2]$ be an algebra with multiplication: \begin{align} x_1^ax_2^b \cdot x_1^c x_2^d = (-1)^{m(ad-bc)} x_1^{a+c} x_2^{b+d}. \end{align}

For any $a,b \in \mathbb{N}^2\backslash \{0\}$, define a homomorphism $T_{a,b}: \overline{A} \to \overline{A}$ by \begin{align} x_1 \mapsto x_1 \cdot (1-x_1^a x_2^b)^{-mb}, \\ x_2 \mapsto x_2 \cdot (1-x_1^a x_2^b)^{ma}. \end{align} It is said that this map is an automorphism. How to show that $T_{a,b}$ is a bijection?

It is said that the algebra $\overline{A}$ is commutative. But according to the definition of the multiplication, this algebra is not commutative. Am I correct?

Thank you very much.

Is the higher order algebraic k group has some direct analogy with the corresponding topological k group ? since the 0th,1th algebraic k group are direct algebraic version of topological k group,but for higher order, the relation seems a bit obscure, for example, is the 2nd(or others) algebraic k group the algebraic version of the 2nd topological k group?

This is a kind of a follow-up to Question on Hessian of a function (probability question). Suppose I give you a continuous function $f:\mathbb{R}^n \to \mathbb{R}.$ Is it true that there exists a ($C^2$) function $g:\mathbb{R}^n \to \mathbb{R},$ such that $\det(\mbox{Hess}(g)) = f?$ (where $\mbox{Hess}$ is the Hessian, of course)?

Let $C$ a point of $\partial \bar{M}_{g,n}$, corresponding to a curve with exactly one node $p$. Then the normal space of $\partial \bar{M}_{g,n}$ inside $\bar{M}_{g,n}$ at $C$ is given by $T_p(C_1)\otimes T_p(C_2)$.

Let $C$ be a singular stable $n$-pointed curve of genus $g$, i.e. a point of $\partial \bar{H}_{g,n}$, with $r\geq 1$ nodes. What is the normal space of $\partial \bar{H}_{g,n}$ inside $\bar{H}_{g,n}$ at $C$?

The reason I am interested in this is the following. I would like to apply the "method of test curves" to $\bar{H}_{g,n}$, and the only examples I have come from the case of $\bar{M}_{g,n}$. In this case, given a family of curves $X\to B$ entirely contained in a certain divisor $\Delta$ of $\partial\bar{M}_{g,n}$, to compute the degree of the line bundle associated to $\Delta$ I would consider the normalization $\widetilde{X}\to B$, then base change until the singular components of dimension $1$ are disjoint, and then take the self-intersection of these component and add it to the self-intersection of the isolated nodes. Can the same procedure be carried out on $\bar{H}_{g,n}$, or this method does not work anymore?

I wonder how I could find a formula that would give me the maximal number of different sets of arrays following certain constraints. I will try to explain what I would like to achieve with an example:

Imagine their is 3 tables with 3 chairs. Every chairs have a number. I would like to find the maximum chairs-changing it is possible to do when the chairs can only meet one time.

How many times can I change the chairs of place and match them with new ones until I have to match them with ones they already had been match with?

I'm sorry if my english is not perfect, I hope my question is well understood and appropriate to this particular stack exchange.

I encounter a function $$f(x,y)=-4xy \pi( cos(y) sin^2(x\pi))+sin(y)(-2y^2 x \pi +2x^3 \pi^3+(y^2+x^2 \pi^2)sin(2x\pi)) $$ defined for $x\times y\in[0,2]\times[0,\pi]$. This function seems to be nonnegative in this compact domain. However, I just can prove this result after tedious analysis on the local property of $f$ around the zero points. Since this function seems to be simple, I wonder whether there is a straightforward technique to do so .Any hints or references in this respect would be greatly appreciated!