Recent MathOverflow Questions

Can we prove that for any real valued $d\times d$ matrix $A$, $A$ can be decomposed to finite product of such matrices

$A=\prod_{i=1}^n (I+R_i)$

where $I$ is the identity matrix and $rank(R_i)=1$.

As far as I know, if there is $LDU$ or $LU$ decomposition to $A$, then we can easily figure out each $R_i$.

I do not quite understand the construction of Voevodsky motives yet. Let $k$ be a field (possibly not algebraically closed), $X$ be a connected smooth projective $k$-scheme. Does the motive of $X$ in the abelian category of pure numerical motives determine its Voevodsky motive? The coefficients are either $\mathbb{Z}$ or $\mathbb{Q}$ (the answer should address both). Please provide a reference.

If $f$ is Lipschitz, then the following holds for the Hausdorff dimension: $$\dim_H f(A) \le \dim_H A.$$

Is the same true for the box counting dimension?

Let $k$ be an algebraically closed field, $DM$ be the category of $\mathbb{Z}$-motives over $k$. Are there two smooth projective $k$-schemes that have isomorphic motives in $DM$ but have non-isomorphic etale fundamental groups? Note that for $\mathbb{Q}$-motives the question has been addressed on MO.

I have a random variable $x \in [a, b]$ with PDF $f(x)$ and an event $E$ which satisfies the following property for any $x'<b$.

$$\Pr[E|x > x'] \geq \Pr[E]$$

My question is whether or not the following inequality holds.

$$\int_{a}^{b} uf(u)\Pr[E|x=u]du \geq \Pr[E]\int_{a}^{b} uf(u)du$$

Let $f:X\rightarrow Y$ be a map of $H$-spaces such that image of homology groups $H_i(X,\mathbb{Z})$ for $i\geq 1$ under $f_*$ are finitely generated. Does this imply that the image of homotopy groups under $f_*$ are also finitely generated? Few tips that might be helpful:

This is true for $\pi_1$ since it is Abelian.

This is true rationally.

This fact is not true: If $f_*$ is zero on homologies then it is so on homotopy groups. Here is the counterexample if you like to see it.

I am trying to understand some of the recent research in number theory. There is apparently a certain lemma, called Ihara's lemma, which can be established in some contexts and is unknown in other contexts. Occasionally, one can still prove its consequences unconditionally. This acrobatics is called Ihara avoidance. What are some important papers containing arguments like this? Also, what are the papers containing Ihara avoidance-type argument that are technically easy to understand?

I feel like I will never be able to penetrate this sea of indices so if there is some easy paper which relies on that idea, I could try to understand it well and other papers then would become less scary.

Cauchy's Interlacing Theorem says that given an $n \times n$ symmetric matrix $A$, let $B$ be an $(n-1) \times (n-1)$ principal submatrix of it, then the eigenvalues of $A$ and those of $B$ interlace.

Using this property, one can obtain a lower bound on the $k$-th largest eigenvalue of a $t \times t$ principal submatrix of $A$, using the $(k+n-t)$-th largest eigenvalue of $A$. This lower bound is best possible, for example when $A$ is diagonal. But for many interesting (fixed) matrices, such bound is usually far from being optimal. For example, let $$A=\begin{bmatrix} 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 1\\ 1 & 0 & 0 & 1\\ 0 & 1 & 1 & 0 \end{bmatrix}$$

The eigenvalues of $A$ are $2, 0, 0, -2$. So if we would like to bound from below the largest eigenvalue of its $3 \times 3$ principal submatrix using Cauchy's Theorem, we only get a lower bound of $0$. However it is straightforward to check that it is always at least $\sqrt{2}$.

I am wondering if there is a more "quantitative" Interlacing Theorem, say if your matrix satisfies some additional properties (non-negative, binary, etc.), then one can obtain a better lower bound on the $k$-th largest eigenvalue of a $t \times t$ principal submatrix of $A$?

I am from Europe and I visit the "High School" there. Let me walk you through what we did already in school:

• Linear Functions
• Fraction equations (Both Algebra and Graphical Solutions)
• Power laws

These are the main algebra things. Ofc we solved equations and all that. But I am into machine learning and I know how to code the models and all that. And now I would just like to know how I should continue with my journey ?

Thanks in advance. Leon

Suppose I give SnapPy a cusped hyperbolic 3-manifold (using, say, the link editor) and specify some filling. SnapPy can then provide a presentation of the fundamental group of the filled manifold. Can it tell me what the core curve of the added solid torus is, as a word in the fundamental group?

I am currently studying crystalline cohomology and while all the talk about crystalline topoi is nice, I would like to see some explicit computations. What are some references on this subject which contain computations for smooth projective varieties over $\mathbb{F}_{p^n}$ or $\overline{\mathbb{F}_p}$ defined by explicit equations (with the action of Frobenius worked out too)? Or if you want, you can post an example of your own.

To begin, we have two constraints \begin{equation} C1=x>0\land z>0\land y>0\land x+2 y+3 z<1 \end{equation} and \begin{equation} C2=x>0\land y>0\land x+2 y+3 z<1\land x^2+x (3 z-2 y)+(y+3 z)^2<3 z. \end{equation} $C1$ ensures the nonnegative-definiteness of a class of $9 \times 9$ ("two-qutrit") density matrices ($\rho$).

$C2$ also ensures this, as well as the nonnegative-definiteness of the "partial transpose" ($\rho^{PT}$) of $\rho$.

The integration--subject to $C1$--of the value 36 over the unit cube $\{x,y,z\} \in [0,1]^3$ yields 1.

The integration--subject to $C2$--of the value 36 over the unit cube yields (the Hilbert-Schmidt positive partial transpose probability) $\frac{8 \pi}{27 \sqrt{3}} \approx 0.537422$.

Now, we are interested in similarly enforcing both $C1 \land C3$ (yielding an "entanglement probability") and $C2 \land C3$ (yielding a "bound-entanglement probability"), where, the entanglement constraint $C3$ is \begin{equation} b \left(-x \left(a^2-a (b+2)+2 b+1\right)+2 y (a (-a+b+2)+b-1)-3 z (a-b-1) (a+b-1)+(a-1)^2\right)<0, \end{equation} with its three parameters $a,b,c$ subject to \begin{equation} C4= b>0\land c>0\land 0<a<1\land a+b+c=2\land (a-1)^2=b c. \end{equation}

Now, I suspect these last two problems are too difficult to resolve in their full generality (leaving $a,b,c$ unspecified).

But, to begin, if we take \begin{equation} \{a,b,c\}=\left\{\frac{1}{4} \left(3-\sqrt{5}\right),\frac{1}{2},\frac{1}{4} \left(3+\sqrt{5}\right)\right\}, \end{equation} the integration of 36 over the unit cube subject to $C1 \land C3$ yields $\frac{5}{132} \left(5+\sqrt{5}\right) \approx 0.274093$.

Alternatively, for \begin{equation} \{a,b,c\} = \left\{\frac{1}{3},\frac{1}{3},\frac{4}{3}\right\}, \end{equation} the integration of 36 over the unit cube subject to $C1 \land C3$ yields $\frac{125}{486} \approx 0.257202$.

However, I have not been so far able to obtain the counterparts for these last two results for $C2 \land C3$. (Using numerical integration, we get the much lower values of 0.001497721920258410 and 0.003256122941383665, respectively.)

A set of values of $a,b,c$ which satisfy $C4$ are \begin{equation} \left\{\frac{2}{3} (\cos (\alpha )+1),\frac{2}{3} \left(-\frac{1}{2} \sqrt{3} \sin (\alpha )-\frac{\cos (\alpha )}{2}+1\right),\frac{2}{3} \left(\frac{1}{2} \sqrt{3} \sin (\alpha )-\frac{\cos (\alpha )}{2}+1\right)\right\}, \end{equation} for $\frac{\pi}{3} \leq \alpha \leq \frac{5 \pi}{3}$.

So, I would like to obtain results of integration of the value 36 over $[0,1]^3$ of $C1 \land C3$ and $C2 \land C3$ for either specific values of $a,b,c$, satisfying $C4$, or even without particular values being specified.

Let $S_n(R)$ denote the number of lattice points in an $n$-dimensional "sphere" with radius $R$. For clarification, I am interested in lattice points found both strictly inside the sphere, and on its surface. I want to count exactly how many such points there are. This count corresponds to the amount of sets of integers that are solutions to the equation $$a_1^2+a_2^2+a_3^2+...+a_{n-1}^2+a_n^2= R^2$$ where the $a_i$'s are required to be integers, not necessarily positive, and sets that differ just by the order of the summands are also considered distinct, i.e for the case $n=3, R=5$, the following are both counted: $3^2+(-4)^2+0^2$ and $(-4)^2+0^2+3^2$. This can also be interpreted as the sum of squares function - $r_d(k)$ denotes in how many such ways I can write $k$ as the sum of $d$ squares. This means that $$S_n(R) = \sum_{k=1}^{R^2}r_n(k)$$

I am asking this question as a general one, but I am mostly concerned about the cases of $n=2, n=3,n=4$.

In order not to be too lengthy and remain focused, I won't explain why but summing this way requires to factor each $k$ first. This is very time consuming, so I searched for better ways.

For the case of $n=2$, which is essentially just the count of lattice points in a circle of radius $R$, there is a lot of information - this is known as the Gauss Circle Problem and I have managed to find that $$S_2(R)= 1+\sum_{i=1}^\infty\biggl(\biggl \lfloor\frac{R^2}{4i+1}\biggl \rfloor-\biggl \lfloor\frac{R^2}{4i+3}\biggl \rfloor\biggl )$$For the case of $n=4$, I found: $$S_4(R)= 1+8\sum_{k=1}^{R^2}\sum_{d|k \atop {4\nmid d}}d$$ Unfortunately, for the case of a sphere ($n=3$), I have found no formula, neither for the count of lattice points inside nor on the surface of the sphere.

Although these formulas do indeed provide the count of lattice points, they are very slow in computational terms, as their running time complexity is $O(R^2)$. I was wondering if a more efficient way exists to count the number of such lattice points, perhaps $O(R)$ or $O(R^{1/2+\epsilon})$, so that I can work with radii as big as $10^9$ or even more. I suspect there is a way, but I can't find it. Not necessarily a different approach to the problem, but rather just a more clever way to reduce the order of the sum. Also, any insight about $S_3(R)$ will be appreciated.

In a computation of characters of certain representations of finite cyclic groups which appear as equivariant indices of Dirac operators (using the Atiyah-Bott fixed point formula, cf. [1, Theorem 8.35]), the following elementary problem emerged.

Let $d > 2$ be an integer and set $n = \lceil\frac{d}{2}\rceil -1$. For $k \in \{1, \dotsc, n\}$ with $\gcd(k,d)=1$, we set $$ v_k = \begin{pmatrix} \csc^2(\frac{k \pi}{d})& \csc^2(\frac{2k \pi}{d}) &\cdots&\csc^2(\frac{n k\pi}{d}) \end{pmatrix} \in \mathbb{R}^n, $$ where $\csc x = \frac{1}{\sin x}$ denotes the cosecant. If $\gcd(k,d) \neq 1$, we let $v_k = (0\ \cdots\ 1\ \cdots 0)$ be the standard basis vector with entry $1$ at the $k$-th position.

Question: Do the vectors $v_1, \dots, v_n$ generate the vector space $\mathbb{R}^n$?

As this appears somewhat intimidating at first glance, let us simplify it to the case where $d$ is an odd prime number. Then it reads as follows:

Let $p$ be an odd prime number.

Question: Is the following $\frac{p-1}{2} \times \frac{p-1}{2}$-matrix invertible? $$ \begin{pmatrix} \csc^2(\frac{\pi}{p}) & \csc^2(\frac{2\pi}{p}) &\cdots & \csc^2(\frac{(p-1)\pi}{2p}) \\ \csc^2(\frac{2\pi}{p}) & \csc^2(\frac{4\pi}{p}) &\cdots & \csc^2(\frac{2(p-1)\pi}{2p}) \\ \vdots & \vdots & \ddots & \vdots\\ \csc^2(\frac{(p-1)\pi}{2p}) & \csc^2(\frac{2(p-1)\pi}{2p}) &\cdots & \csc^2(\frac{(p-1)^2\pi}{4p}) \end{pmatrix} $$

The questions appears to be of a number-theoretic nature. Potentially relevant formulas involving the cosecant appear for instance in Cauchy's elementary solution to the classical Basel problem.

Moreover, numerical experiments suggest that

• Question 1 has an affirmative answer at least for $d \leq 200$.
• The determinant of the matrix in Question 2 is an integer times $\frac{1}{\sqrt{p}}$ if $p \equiv 1 \mod 4$, and an integer otherwise. This suggests a relation to quadratic reciprocity.

Spotting a concrete formula for the determinants from numerical computations is difficult, however, because the numbers grow very rapidly. But in [2, Lemma 3.1], the formula $$ \prod_{j=1}^{\lfloor\frac{d}{2}\rfloor} \sin(\frac{j \pi}{d}) = \frac{\sqrt{d}}{2^{\frac{d-1}{2}}} $$ is provided which is reminiscent of our numerical observations. A more complicated formula involving quadratic reciprocity is provided in [2, Lemma 3.2].

All of this makes it plausible that it should be possible to derive a concrete formula for the determinants of the relevant matrices appearing in Question 1 and 2, but it remained elusive to me so far.

[1] Atiyah, Michael F.; Bott, Raoul, A Lefschetz fixed point formula for elliptic complexes. II: Applications, Ann. Math. (2) 88, 451-491 (1968). ZBL0167.21703.

[2] Miatello, Roberto J.; Podestá, Ricardo A., Eta invariants and class numbers, Pure Appl. Math. Q. 5, No. 2, 729-753 (2009). ZBL1183.58021.

Let

• $(\Omega,\mathcal A,\operatorname P)$ be a probability space
• $(E,\mathcal E)$ be a measurable space
• $n\in\mathbb N$
• $X_1,\ldots,X_n$ be $(E,\mathcal E)$-valued random variables on $(\Omega,\mathcal A,\operatorname P)$

I'm interested in the following question: Given a partial order $\le$ on $E$, I want to construct $(E,\mathcal E)$-valued random variables $X_{1:n},\ldots,X_{n:n}$ on $(\Omega,\mathcal A,\operatorname P)$ such that $X_{k:n}$ is the $k$th smallest element among $X_1,\ldots,X_n$ (where $X_i$ is considered as smaller than $X_j$ whenever $X_i=X_j$ and $i<j$).

I would like to define them by something like $X_{1:n}:=\min\left\{X_1,\ldots,X_n\right\}$ and $$X_{i:n}:=\min\left\{X_j:X_j>X_{i-1:n}\right\}\;\;\;\text{for }j\in\left\{2,\ldots,n\right\}\tag1,$$ but this won't be well-defined if not all $X_i$ are distinct and it won't be measurable in general. Equivalently, we may ask if there is a random permutation $\pi:\Omega\times\left\{1,\ldots,n\right\}\to\left\{1,\ldots,n\right\}$ such that $X_{\pi(1)}\le\cdots\le X_{\pi(n)}$ and each $X_{\pi(k)}$ is measurable.

The examples I've got in mind include $E=\mathbb R$ with the usual order or $E=\mathbb R^d$ and the order given by the smallest distance to a fixed element $x\in\mathbb R^d$.

Which conditions on the relation between $\mathcal E$ and $\le$ do we need to impose and how do we actually need to define $X_{k:n}$ (or $\pi$)?

My definition of a site is a pair $(\mathcal{C},\DeclareMathOperator{\Cov}{Cov}\Cov (\mathcal{C}))$ where $\mathcal{C}$ is a category and $\Cov (\mathcal{C})$ is a set consisting of elements of the form $\{U_i\rightarrow U\}_{i\in I}$, and $\Cov (\mathcal{C})$ satisfies three axioms which I gonna omit since this is not the problem.

I have searched a few books about the definition of an big/small étale/fppf/... site.

For example, a small étale site of a scheme $X$, $X_{ét}$, has the underlying category whose objects are étale morphisms $U\rightarrow X$, and the covering set $\Cov (X_{ét})$ consists of all surjective families of étale morphisms $\{f_i:U_i\rightarrow U\}_{i\in I}$, i.e. $f_i$ is étale and $U=\cup_i f_i(U_i)$.

But those books don't mention that why the "covering set" is a set.

"Introduction to étale cohomology" by Gunter Tamme, Manfred Kolster did that.

"Étale Cohomology" by James Milne did not mention the definition of an arbitrary site, he just define a big/small $E$-site for a class of $E$-morphisms of schemes.

"Lecture notes on étale cohomology" by James Milne on the website see this link, he defines an arbitrary site s.t. for each object $U$, we have a distinguished set of covering families, and this is a system of coverings, didn't say the whole thing is a set.

Stacks project, uses a confusing way to construct the underlying scheme of a small/big étale/fppf/... site, so I have to ignore it.

SGA is in French which I have an incredibly slow reading speed, so ignored.

Which definition of site (arbitrarily) and small/big étale/fppf/... site should I accept?

We fix the standard inner products on $\mathbb{C}^n$ and $\mathbb{C}^n\otimes \mathbb{C}^n$.

Is there an algebra structure on $\mathbb{C}^n$ with multiplication $m$ such that the adjoint operator $m^*: \mathbb{C}^n \to \mathbb{C}^n \otimes \mathbb{C}^n$ is a coproduct (coalgebraic operation)?

Is there a bialgebra structure whose product and coproduct are adjoints of each other? Is there a Hopf algebra with the latter property, and the additional condition that the antipode map is an isometry?

Note: The adjoint operators are taken with respect to the corresponding inner products.

I am hoping the following is true. Mention of related ideas/topics are appreciated.

Suppose $F:\mathbb{C}^2 \to \mathbb{C}^2$ is a injective holomorphic mapping such that $F(0)=0$ and $dF(0) = I_2$ where $I_2$ is the $2 \times 2$ identity matrix. Let $\partial B$ denote the boundary of the unit ball centered at the origin in $\mathbb{C}^2$. Let $M = \sup_{x \in \partial B} ||F(x)||$ where $|| \cdot ||$ is the usual Euclidean norm. Then $ \{ x \in \partial B: M = ||F(x)|| \}$ is equal to $\partial B$ or a finite union of curves and points.

Suppose one is written a math/computer science paper, but more focused in the math part of it. I had a very complicated function and need it to find it's maximum, so I used Mathematica (Wolfram) to do it. How do I explain that? "Using wolfram we find the maximum of $f$ to be $1.0328...$ therefore...". It's look very sloppy.

I have a question about singularities of conformal mappings.

Let $\mathbb{H} \subset \mathbb{C} \cong \mathbb{R}^2$ be the upper half-place and let $D$ be a Jordan domain. Let $\varphi:\mathbb{H} \to D$ denote a conformal map.

I am concerned with the following quantity: \begin{align*} I(z,r)=\int_{\mathbb{H} \cap B(z,r)}\log(|z-w|^{-1})|\varphi'(w)|^2\,dm(w),\quad z \in \bar{\mathbb{H}},\ r>0. \end{align*} Here $m$ denotes the two-dimensional Lebesgue measure and $B(z,r)$ denotes the open ball centered at $z$ with radius $r>0$. Of course, $I(z,r)$ is a variant of logarithmic potential. $I(z,r)$ roughly represents a singularity of $\varphi$ around the point $z$.

In fact, the quantity $I(z,r)$ naturally appears in the context of "random time-change" in probability theory. This controls local behaviors of the reflected Brownian motion in $D$ in some sense.

I am interested when $$\text{(A)}\quad \lim_{r \to 0}I(z,r)=0 \text{ uniformly in $z$ over each compact subset of $\bar{\mathbb{H}}$}$$ (of course, when $\partial D$ is smooth, this question is not interesting).

Has such a thing been studied in the context of conformal mappings and logarithmic potential theory?