I am currently going through Pollard's article on Lattice Sieving and have a few confusions. Firstly,how to figure that C and D in the two -dimensional array so that every (c,d) pair corresponds to a lattice element (a,b) because if I consider c \in [-C,C] and d \in [1,D] where C is to be chosen greater than D ; there are many combinations of type cV_1 + dV_2 which go outside the lattice . Here, V_1 and V_2 are the reduced basis of the lattice.

*Pollard, J.M.*, The lattice sieve, Lenstra, A. K. (ed.) et al., The development of the number field sieve. Berlin: Springer-Verlag. Lect. Notes Math. 1554, 43-49 (1993). ZBL0806.11066.Secondly, this 2-D array can't cover entire L(q),so aren't we missing a lot of smooth pairs?

Let B be a positive diagonal matrix, and C a PSD symmetric matrix with eigenvalues $\Lambda$, with $\lambda_i = {0,1}$. I'm trying to find a reference for the following bounds or similar

$B^{1\over 2}DB^{1\over 2} \preceq \Lambda B$

$DB \preceq \Lambda B$.

I've checked this answer but they bound the eigenvalues according to the weak majorization order. Also, this answer shows a bound $\lambda_i(AB)\leq \lambda_i(A)\lambda_i(B)$, where $\lambda_i(A)$ denotes the ith eigenvalue of A, but it is only for positive definite matrices.

Suppose that $\pi:(V_1,F_1)\to V_2$ is a linear surjective map, where $V_1$ and $V_2$ are vector spaces and $F_1$ is a Minkowski norm on $V_1$. Let $\Sigma_1$ be the indicatrix on $V_1$, that is the unitary sphere with respect to $F_1$. Define $\Sigma_2:=\pi(\Sigma_1)$.

If $\Sigma_2$ is an indicatrix with respect to some Mnikowski norm on $V_2$?

The theory of such surfaces goes back to the book by Alexandrov and Zalgaller (1967 English translation) and from a more analytic viewpoint, work by Reshetnyak where everything is translated into Radon measures. It is proved in AZ that any CAT(0) surface can be approximated by a polyhedral surface also with finite total curvature. Can this be done while respecting the CAT(0) condition?

A Zero Range Process is the Feller process with generator

$Af(\eta)=\sum_{x,y} p(y-x)f(\eta(x)) (f(\eta^{x,y}-f(\eta))$

where $\eta\in \mathbb{N}_0^S$ denotes a configuration of particles at sites $S$, $p=(p(x,y)_{x,y})$ is doubly-stochastic translation invariant and $g$ some nondecreasing function. We assume first $S$ is finite. The question arises what kind of conditions have to be imposed to guarantee existence of a Feller process with the given generator. I often see $g^*:=\sup_k |g(k+1)-g(k)|<\infty$, but I fail to see how this condition could be helpful in the case of finite site space $S$ (maybe necessary for the hydrodynamic limit?), as in that case I assume you would simply apply the state space $\{ \eta: \sum_x \eta_x =K\}$ for some $K$ and then end up with a bounded operator. A state space of the form $\{ \eta: \sum_x \eta_x <\infty\}$ just decomposes into irreducibility classes of the above form.

I have a question. I will be thank you if you give me some hints. This is the question:

**Let $R$ be a ring and for any simple R-module T, we have End(E(T)) is isomorphic to End(T) which is a division ring. Does we conclude that every simple R-module is injective**

*(in fact I proved that in this case, every simple R-module is injective (i.e., R is right V-ring) if and only if for any simple R-module T, Hom_{R}(E(T),T) is nonzero).*

Note that E(T) means the **injective envelope** of T,
End(T) means **the endomorphism ring of T**.

Thank you very much.

Best regards

Suppose $E=[0,1]$, and every $0\le x \le 1$ represents a coin flipper that flips an identical coin. Then $\{0,1\}^E$ is the collection of all results. Given that the probability of "Head" is p, how can I define a probability measure on $\{0,1\}^E$?

I have been searching for the answer and found this post: Can we put a Probability Measure on every $\sigma$-Algebra?. However, the OP didn't mention how a "coin-flipping measure with infinite exponent" can be written out. I've searched around and can't seem to figure out.

Any help is much appreciated! Thanks!

Hi i was reading a paper "Propagating Distributions on a Hypergraph by Dual Information Regularization by Koji Tsuda", and one section stood out to me.

hypergraphs have more flexibility in describing prior knowledge, because known clusters can be directly encoded as hyperedges. It might be possible to convert a hypergraph to a conventional graph, for example, by converting a hyperedge to a complete subgraph.

My question is how exactly can one encode a small cluster into a hyperedge and how can one go in reverse. I've tried googling it but I've come up short.

Fix $n$ a (small) integer.

Let $N$ be a (big) integer. Consider $N$ random points in the $n$-dimensional unit cube $[0, 1]^n$. The $N$ points are independently uniformly distributed.

Define $V(N)$ to be the expectation of the number of extreme points of the convex hull of the $N$ points.

Question: when $N$ grows to infinity, how fast does $V$ grow?

For $n = 1$, one has $V = 2$;

For $n = 2$, my intuition suggests $V \simeq N^{1/2}$, but I'm not quite sure;

Similarly, for general $n$, my intuition suggests $V \simeq N^{(n - 1)/n}$.

Are there known results on this problem or similar problems?

Consider a discrete time Markov chain on a countable state space which is irreducible, aperiodic, and has a given invariant distribution $\pi$. Then the chain is necessarily positive recurrent and ergodic (so the invariant distribution is unique and is the limiting distribution regardless of the initial distribution). Further assume that the chain is actually *geometrically* ergodic, so that a Lyapunov-Foster function exists, but we do not know what it is.

Can we use our prior knowledge of what the invariant distribution is in order to construct a Lyapunov-Foster function for the chain? It seems that some nicely decreasing function of $\pi$ should suffice. If it helps you may assume the chain also satisfies detailed balance with respect to $\pi$.

This may seem like a weird question; ordinarily you would construct a Lyapunov function to *check* stability, not when you already know it. My actual problem is to study the stability of a chain that is a perturbation of this "nice" chain, and for that I need a Lyapunov-Foster function for the unperturbed chain.

Let $G$ be a semisimple Lie group with the set of positive simple roots $\prod$. Let $\prod^{-}$ be the set of negative simple roots and let $\mathfrak{M}$ be the semigroup freely generated by $\prod$ and $\prod^{-}$. Then the **braid semigroup** is the quotient of the semigroup $\mathfrak{M}$ by some **braid relations** which related to the Cartan matrix $C$.

In the paper 'Cluster $\mathcal{X}$ -varieties, amalgamation and Poisson-Lie groups' https://arxiv.org/pdf/math/0508408.pdf written by V. V. Fock and A. B. Goncharov. On page 16, in order to explain the fact a mutation of a cluster seed $J(D)$ is not always a cluster seed corresponding to another word of the semigroup.I am at a loss for the following example.

**Example** Let $\prod=\{\gamma,\Delta,\eta\}$ be the root system of type $A_3$ with $C_{\eta\gamma}=0$. Then $\mu_{\binom{\gamma}{1}}J(\gamma \Delta \eta\gamma\Delta\gamma)=J(\Delta\gamma\Delta\eta\Delta\gamma)$, but $\mu_{\binom{\Delta}{1}}J(\gamma \Delta \eta\gamma\Delta\gamma)$ is a seed which does not correspond to any word.

Since $C_{\eta\gamma}=0$, then $\eta\gamma=\gamma\eta$ by the braid relation, which implies that $\gamma \Delta \eta\gamma\Delta\gamma=\gamma \Delta \gamma\eta\Delta\gamma$. Thus $\mu_{\binom{\gamma}{1}}J(\gamma \Delta \gamma\eta\Delta\gamma)=J(\Delta\gamma\Delta\eta\Delta\gamma)$. But I don't know how to mutate $\mu_{\binom{\Delta}{1}}J(\gamma \Delta \eta\gamma\Delta\gamma)$? Who can give me some mutation rule? Any help will be appreciated.

$\Omega\subset \mathbb{R}^{2}$ bounded with lipschitz boundary. Consider the fnction $f\in W^{2,\infty}(\Omega)$.

I am expecting that $f\in C^{1}(\overline\Omega)$. Indeed : Clearly $f\in W^{1,\infty}(\Omega)$ then $f$ is lipschitz.

and also $\nabla f \in W^{1,\infty}(\Omega)$ so : $\nabla f $ is also lipschitz .

this imply that $f\in C^{1}(\Omega) $.

Is that true ?

Consider a diagonal action of $\mathbb Z_n$ on $\mathbb C^2$ generated by $(z_1,z_2)\to (\mu^pz_1,\mu^qz_2)$, with $\mu^n=1$.

**Question.** Is it always possible to find a *smooth* blow up $X\to \mathbb C^2$ such that the $\mathbb Z_n$-action lifts to $X$ and such that $X/\mathbb Z_n$ is *smooth* as well?

The same question can be asked for action of any finite group $G$ on any smooth variety (but I am especially interested in the above example).

For a totally real (resp. imaginary) biquadratic number field $K$ with quadratic subfields $K_1$, $K_2$ and $K_3$, is there an explicit method to determine Hasse unit index $(U_K:U_{K_1}U_{K_2}U_{K_3})$?

Let $F$ be the set of bijective Borel-measurable functions $f \colon [0,1] \to [0,1]$ that preserve the Lebesgue measure.

Is it the case that for every non-Lebesgue-measurable set $A \subset [0,1]$, there exists a countable family $\{f_n\}_{n \in \mathbb{N}} \subset F$ such that $\ \bigcup_{n \in \mathbb{N}} f_n(A)\,$ contains a Lebesgue-measurable set of positive measure?

If so, then there is no *natural* way to extend the Lebesgue measure to include more null sets.

(Conversely, if not, then it seems reasonable to regard all the counterexemplary sets as "kind-of-null sets".)

Q: Why does not a zeta zero counting function, on the critical line, $N_0(T)=1+\frac{\vartheta(T)}{\pi}+\frac{\arg(\zeta(\frac12+i T))}{\pi}$ behave exactly in a neighborhood that all violations of Gram's law are occurred?

$N_0(T)=1-\frac{T\log(2\pi)}{2\pi}-\frac{1}{2\pi i}\left(\log\Gamma(\frac12-i T)-\log(\zeta(\frac12+i T))+\log\left(\frac{\zeta(\frac12+i T)}{(2\pi)^{i t}\Gamma(\frac12-i T)}\right)\right)$,

$N_0'(T)=0$ and

$\int{N_0(T)}{dT}=T N_0(T)$,

where $N_0(t)$ is a differentiable and integrable form on the symbolic computation.

I do not understand the difference between elementary and non-elementary mathematics. For example, the Fourier methods, statistics and differential equations with real and complex variables are commonly considered as elementary mathematics. However, some branches in math, like Banach algebra or analytic number theory, are considered to be non-elementary. It looks strange for me. I do not see any difference between elementary and non-elementary mathematics since all branches of mathematics are based on the real and complex variables. Moreover, I could not find any literature that could clearly differentiate elementary and non-elementary mathematics. Any suggestions?

[EDIT: To maintain a FRIENDLY environment in this forum please remain within the topic, be constructive and avoid sarcastic and inappropriate comments.]

Let $n$ be a given even positive integer. We have the following integral \begin{eqnarray} &&\int_0^1\cdots\int_0^1\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots dy_n\\ &=&\int_0^1\cdots\int_0^1\left(\int_0^1\prod\limits_{j=1}^n(x-y_j)dx\right)^ndy_1\cdots dy_n\\ &>&0. \end{eqnarray} Let's consider a similar integral where $n$ is also an even positive integer: $$A_n=\int_0^1\cdots\int_0^1\int_0^{2\pi}\cdots\int_0^{2\pi}\prod\limits_{i=1}^n\prod\limits_{j=1}^n\left(x_i-y_j+i\cos(\alpha_i-\beta_j)\right)dx_1\cdots dx_ndy_1\cdots dy_nd\alpha_1\cdots d\alpha_nd\beta_1\cdots d\beta_n.$$ It is easy to see that $A_n$ is a real number for every $n$. My question is whether $A_n$ is positive or not.

dg: is for differential graded

Suppose that $F: C\rightarrow D$ is a dg-functor between small dg-categories such that:

F: Objects of $C$ $\rightarrow$ Objects of $D$ is injective.

$Hom_{C}(a,b)\rightarrow Hom_{D}(F(a),F(b))$ induces an isomorphism in homology for any $a, b \in C$

Let $\hat{C}$ be the category of dg-$C$-modules. Is it true that the induced dg-functor $$\hat{F}:\hat{C}\rightarrow \hat{D}$$ is faithfull in the sense that

for any $x,y\in \hat{C} $, $Hom_{\hat{C}}(a,b)\rightarrow Hom_{\hat{D}}(\hat{F}(a),\hat{F}(b))$ induces an isomorphism in homology.

**Edit:** I'm also interested in the particular case when $F:C\rightarrow D$ is an embedding of dg-categories e.g. when $C$ is a full dg-subcategory of $D$.

Assume we have two smooth projective rational varieties $X_i$ over $\mathbb{C}$. Let $D_i\hookrightarrow X_i$ be an irreducible divisor, which has positive Kodaira dimension, $\kappa(D_i)>-\infty$.

Given $x_i\in \operatorname{Br}(k(X_i))$ a nontrivial Brauer class in the Brauer group of the function field, which ramifies only along $D_i$ in $X_i$, that is $x_i$ comes from a nontrivial element $\xi_i\in \operatorname{Br}(X_i\setminus D_i)$.

Let $f_1: Y\rightarrow X_1$ and $f_2:Y\rightarrow X_2$ be two blow ups along smooth irreducible centers such that the birational map $\phi = f_1^{-1}\circ f_2: X_1 \dashrightarrow X_2$ has the property $\phi^{*}(x_2)=x_1$.

$\textbf{Question 1:}$ Does this imply that $\phi$ restricts to a birational map $D_1\dashrightarrow D_2$?

The condition $\phi^{*}(x_2)=x_1$ implies $f_1^{*}(x_1)=f_2^{*}(x_2)$ on $Y$. This shows that $\operatorname{Ram}_Y(f_1^{*}(x_1))=\operatorname{Ram}_Y(f_2^{*}(x_2))$, here $\operatorname{Ram}_Y(-)$ denotes the ramification divisor of a Brauer class on $Y$. But $\operatorname{Ram}_Y(f_1^{*}(x_1))\subseteq\{\tilde{D}_1,E_{f_{1}}\}$, where $\tilde{D}_1$ is the strict transform of $D_1$ under $f_1$ and $E_{f_{1}}$ is the exceptional divisor, which is a projective bundle over the center. Similarly $\operatorname{Ram}(f_2^{*}(x_2))\subseteq\{\tilde{D}_2,E_{f_{2}}\}$. We note that $f_i^{*}(x_i)$ must ramify along $\tilde{D}_i$ but not necessarily along the exceptional divisor. Since projective bundles have Kodaira dimension $-\infty$, we see that we must have $\tilde{D}_1=\tilde{D}_2$, thus $D_1$ and $D_2$ are birational. Is this reasoning correct?

$\textbf{Question 2:}$ If we have an arbitrary birational map $\varphi: X_1\dashrightarrow X_2$ does the result also hold? What is the behaviour of the exceptional divisors in this case? Does every execeptional divisor (or one of its strict transforms) get contracted by a blow down, due to the condition $\operatorname{Ram}_{X_1}(x_1)=\operatorname{Ram}_{X_1}(\varphi^{*}(x_2))$?

In this case we cannot resolve $\varphi$ by two blow ups, but rather by a sequence of blow ups and blow downs. But in this case the blow downs cannot contract the $D_i$ or any of their strict transforms, due to the Kodaira dimension condition. So my guess is, the argument above (with some extra care) also works in this case?